Fibre Bragg grating sensors have emerged as a simple, inexpensive, accurate, sensitive and reliable platform, a viable alternative to the traditional bulkier optical sensor platforms.
In this work we present an extensive theoretical analysis of the tilted fibre Bragg grating sensor (TFBG) with a particular focus on its polarization-dependent properties.
We have developed a highly efficient computer model capable of providing the full characterization of the TFBG device in less then $3$~minutes for a given state of incident light polarization. As a result, the polarization-dependent spectral response, the field distribution at the sensor surface as well as the fine structure of particular resonances have become accessible for theoretical analysis.
As a part of this computer model we have developed a blazingly fast full-vector complex mode solver, capable of handling cylindrical waveguides of an arbitrary complex refractive index profile.
Along with the theoretical study we have investigated optical properties of the TFBG sensor with application to polarisation-resolved sensing.
We proposed a new method of the TFBG data analysis based on tracking the grating transmission spectra along its principle axes, which were extracted from the Jones matrix.
In this work we also propose a new method of enhancing the TFBG sensor refractometric sensitivity limits, based on resonant coupling between the TFBG structure resonances and the local resonances of nanoparticles deposited on the sensor surface. The $3.5$-fold increase in the TFBG sensor sensitivity was observed experimentally.
TILTED FIBRE BRAGG GRATING
SENSORS WITH RESONANT
NANO-SCALE COATINGS
Simulation of optical properties
Aliaksandr Bialiayeu, PhD
Carleton University
To my parents
CONTENTS IN BRIEF
1 A full-vector complex mode solver for circularly symmetric optical
waveguides
1
2 Modeling of tilted Bragg grating (TFBG) structures
39
3 Experimental polarization-based optical sensing with application
to TFBG sensors
89
4 Optical properties of materials
107
5 Optical properties of nanoparticles
125
6 The optimal parameters for a nanoparticle-based coating
137
7 Modification of the sensor surface with various types of nano-
scale coatings
157
8 Conclusion
175
v
CONTENTS
List of Figures
xi
List of Tables
xxi
Foreword
xxiii
Acknowledgments
xxv
Acronyms
xxvii
List of Symbols
xxix
Introduction
xxxi
Organization of the Book
xxxvii
1
A full-vector complex mode solver for circularly symmetric
optical waveguides
1
1.1
Introduction
1
1.2
The solutions for a cylindrical waveguide
4
1.2.1
Weakly guided approximation
5
1.2.2
The exact solution for cylindrical waveguides
6
1.2.3
TE and TM modes in slab waveguides
8
vii
viii
CONTENTS
1.3
The numerical method
10
1.3.1
The scalar modes
10
1.3.2
The vectorial modes
15
1.4
Discussion
19
1.5
The orthogonality of the basis functions
29
1.5.1
The orthogonality relation for the scalar modes
29
1.5.2
The orthogonality relation for the vectorial modes
31
1.5.3
Numerical verification
34
1.6
Conclusion
36
2
Modeling of tilted Bragg grating (TFBG) structures
39
2.1
Derivation
39
2.2
The matrix elements
44
2.3
Coupled-mode theory,
the two modes approximation
56
2.4
Coupling between the core mode and many cladding modes in
the TFBG
61
2.5
Polarization-dependent coupling
75
2.6
The electric field distribution at the fibre boundary
82
3
Experimental polarization-based optical sensing with application
to TFBG sensors
89
3.1
The Optical Setup
91
3.2
The data processing technique
94
3.2.1
Measurements along principal axes of an optical system
94
3.2.2
Measurements along geometrical axes of an optical
system
98
3.2.3
Extracting transmission spectra measured along
geometrical axes from the Jones matrix or the Stokes
vector data
98
3.3
Polarization-based detection of small refractive index changes
with TFBG sensors
102
3.4
Conclusion
104
4
Optical properties of materials
107
4.1
The methods of measurement of optical constants
107
4.1.1
The methods based on single parameter measurements
108
CONTENTS
ix
4.1.2
The methods based on simultaneous measurement of
both parameters
109
4.2
Optical constants of metals
109
4.2.1
Free electron approximation, the Drude-Sommerfeld
model
110
4.2.2
The near infrared band
111
4.2.3
The visible band. The interband absorption
112
4.2.4
Dispersion curves
115
4.3
Optical properties of mixtures and rough surfaces
116
4.3.1
Local field effects and effective medium theory
116
4.3.2
The exact numerical method accounting for the
interaction between film elements.
118
4.4
Discrete dipole approximation (DDA) or coupled dipole
approximation (CDA) method
121
4.5
Conclusion
124
5
Optical properties of nanoparticles
125
5.1
Review
125
5.1.1
Characterization of light scattering by particles
125
5.2
Simulation of light scattering by small particles
129
5.2.1
Analytical solution to the problem of electromagnetic
wave scattering on spherical particles
130
5.2.2
The quasi-static approximation
133
5.2.3
The ellipsoidal shape particles
134
5.3
Conclusion
135
6
The optimal parameters for a nanoparticle-based coating
137
6.1
The idea behind sensitivity enhancement
137
6.2
Simulation of optical properties of metal nanoparticles
138
6.2.1
The quasi-static approximation
138
6.2.2
The exact solution
142
6.2.3
The shape effect. Ellipsoidal nanoparticles.
147
6.3
The optimal parameters choice for coatings based on spherical
nanoparticles
149
6.4
Conclusion
155
7
Modification of the sensor surface with various types of nano-
scale coatings
157
x
CONTENTS
7.1
Deposition and synthesis techniques
161
7.1.1
Synthesis of silver nanorods and nanowires
161
7.1.2
Synthesis of gold nanoparticles
161
7.1.3
Deposition of nanoparticles
162
7.1.4
Electroless metal coating
162
7.2
The optimal metal coating for Surface Plasmon Resonance
(SPR) excitation
164
7.3
Sensitivity enhancement with a nanoparticle based coating
166
7.3.1
Coating with silver nanowires
166
7.3.2
Discussion
172
7.3.3
Conclusion
173
8
Conclusion
175
A
MatLab Code. The full vectorial complex mode solver.
179
B
Mathematica Code for Mie scattering
191
References
195
LIST OF FIGURES
I.1
Schematic representation of the TFBG sensor coated with
sensitivity enhancing layer of nanoparticles.
xxxii
I.2
A typical spectrum of a 10 degree TFBG sensor.
xxxiii
I.3
Evolution of the TFBG spectral response during the silver
nanoparticle deposition followed by the continues film formation.
xxxiv
1.1
The refractive index profile of SMF-28 fibre immersed in water.
The eigenvalues and eigenfunctions are plotted for m = 0 and
m = 1
13
1.2
The fibre cross-section and corresponding refractive index profile
with modes bounded inside the fibre.
14
1.3
The radial eigenfunctions R
m
k
() (shown in blue color) and the
potential well function U
m
() (shown in red color) for m = 40.
15
1.4
One of the basis function (mode)
3
10
(, ) , at n = 10
(N
ef f
= 1.4181) and m = 3.
15
1.5
The structure of the sparse matrix [ ^
M ] for N = 10 and m = 0.
17
xi
xii
LIST OF FIGURES
1.6
The dispersion curves of a symmetric glass slide immersed in
water. The waveguide and the medium refractive indices are
n
W G
= 1.45 and n
M ed.
= 1.33, respectively.
19
1.7
The dispersion curves of the scalar (
1.14
) modes, here
n
W G
= 1.45 and n
M ed.
= 1.33.
21
1.8
The dispersion curves of the vector modes (
1.13
). The
waveguide and medium refractive indices are n
W G
= 1.45 and
n
M ed.
= 1.33, respectively.
21
1.9
The degeneracy of modes: the scalar (1, 1) mode and two vectorial
(0, 1) and (2, 1) modes. The waveguide and medium refractive
indices are n
W G
= 1.45 and n
M ed.
= 1.33, respectively. The
potential barrier is depicted with green color
22
1.10
The degenerate modes: the scalar (2, 2) mode and two vectorial
(1, 4) and (3, 3) modes. Here the waveguide with n
2
= 1.45 is
immersed in water n
M ed.
= 1.33.
24
1.11
The split in dispersion curves between m = 1 and m = 3 modes.
The waveguide and the medium refractive indices are n
W G
= 3
and n
M ed
= 1.33, respectively.
25
1.12
The dispersion curves, obtained by solving the exact vectorial
equation, for various refractive index ratios n = n
W G
- n
M ed.
at m = 2. The waveguide is immersed in water n
M ed.
= 1.33.
25
1.13
The split in dispersion curves between m = 0 and m = 2 mode
families, and split between the modes inside the same family at
m = 0. The waveguide and the medium refractive indices are
n
W G
= 3 and n
M ed.
= 1.33, respectively.
26
1.14
The split between the eigenvalues of pure radially polarized E
and pure angular polarized E
modes of the m = 0 family. The
potential barrier is depicted with green color, the core refractive
index n
2
= 3 is surrounded by the cladding with n
1
= 1.33.
27
1.15
The dispersion curves, obtained by solving the exact vectorial
equation, for various refractive index ratios n = n
W G
- n
M ed.
at m = 0. The waveguide is immersed in water n = 1.33.
28
1.16
The overlap matrix C
mm
jk
is computed for SMF-28 fibre immersed
in water. The matrix is an identity for modes belonging to
the same family of mode (C
mm
jk
= I for m = 0 or m = 10).
However, if a different families of modes (m = 0 and n = 10)
are considered, the overlap matrix C
mn
jk
is dense.
35
LIST OF FIGURES
xiii
1.17
The overlap matrix for the vectorial case is computed for SMF-28
fibre immersed in water. The overlap matrix C
mm
jk
, for the modes
belonging to the same family m, is an identity matrix for m = 0
or close to the identity for small azimuthal numbers. For different
families of modes, the overlap matrix is dense.
36
2.1
The schematic representation of the problem geometry.
40
2.2
Illustration of a tilted fibre grating
44
2.3
Illustration of the refractive index perturbation along the z-axis
44
2.4
a) A schematic representation of a tilted Bragg grating inside the
fibre, b) projection onto the zx and c) xy planes, d) superposition
of normal gratings taken at various angles.
45
2.5
The weighted function J
m
() caused by the grating tilt,
calculated for = 0
o
and = 10
o
grating tilt angles. The grating
period is assumed to be
G
= 0.6 m, thus for = 0
o
we get
= K
g
sin(0) = 0, and for = 10
o
we get = 1.81 m
-1
.
The number m = m
1
- m
2
is the difference between azimuthal
order of the first m
1
and the second m
2
mode families.
48
2.6
The 2
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
49
2.7
The 2
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
50
2.8
The 4
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
51
2.9
The 4
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
52
2.10
The 10
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
53
2.11
The 10
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
54
2.12
The 10
o
degree grating assisted coupling coefficients between the
core and cladding modes, n = 10
-4
.
55
2.13
The momentum diagram of contra-directional coupling (here
1
corresponds to the core mode and
2
to the cladding mode).
57
2.14
The transfer of power between the incident core mode
A
1
(z) and back-scatters cladding mode A
2
(z) in the case of
contra-directional coupling. Here
kL
= 2.4
59
xiv
LIST OF FIGURES
2.15
The power loss of the core mode as a function of the phase
mismatch, for various coupling parameters C = L.
60
2.16
The power loss of the core mode as a function of the
phase mismatch, for the fibre waveguide coated with
a thin film (h = 100 nm) made of lossy material (blue
curves) a) N
ef f
= 1.3661 - j6.1165 × 10
-4
, and b)
N
ef f
= 1.3669 - j2.2791 × 10
-5
. The red curves, used
as a reference, corresponds to the non-lossy material (
N
ef f
= 1.3661 ).
60
2.17
The phase matching condition in TFBG
61
2.18
The schematic ilustration of energy transfer from the core mode
into the cladding modes in the TFBG. The various potential
barriers correspond to different azimuthal numbers m.
62
2.19
The resonances of different families of modes, computed in
accordance with (
2.57
). The peaks arising due to the coupling
between the core mode and m = 0 family of modes, neglecting
the coupling to higher family of modes. Here the grating length
L = 10mm and the coupling constant C = 2
· 10
-4
.
63
2.20
Transforming the boundary value problem into the initial
value problem. a) solving IVP by assuming A
core
(L) = 1
and A
clad
(L) = 0, b) renormalized the solution by setting
A
core
(0) = 1.
64
2.21
Coupling between the core mode and two cladding modes.
The grating length L = 10 mm and the coupling constant are
C
1
= 2 · 10
4
and C
2
= 1 · 10
4
between the core and the two
cladding modes. Here A
1
(z) is the amplitude of the forward
propagating core mode (the red line), A
2
(z) and A
3
(z) are
amplitude of the backward propagating cladding modes (the blue
and the green lines).
65
2.22
The non-uniform grid with a finer mesh in the vicinity of resonances.
66
2.23
The dispersion of modes. The resonances of m = 0 family
of modes are computed at
1
= 1.5 m and
2
= 1.6 m
operational wavelengths.
67
2.24
The experimentally measured spectra of the 2
o
degree 1 cm long
grating.
68
2.25
The theoretically computed transmission spectra of 2
o
degree
grating. (L = 1 cm, n = 10
-4
)
68
2.26
The fine structure of the particular resonances and corresponding
coupling coefficients of 2
o
degree grating.
69
LIST OF FIGURES
xv
2.27
The experimentally measured spectra of the 4
o
degree 1 cm long
grating.
70
2.28
The theoretically computed transmission spectra of 4
o
degree
grating. (L = 1 cm, n = 10
-4
)
70
2.29
The fine structure of the particular resonances and corresponding
coupling coefficients of 4
o
degree grating.
71
2.30
The fine structure of the particular resonances and corresponding
coupling coefficients of 4
o
degree grating.
71
2.31
The experimentally measured spectra of the 10
o
degree 1 cm
long grating.
72
2.32
The theoretically computed transmission spectra of 10
o
degree
grating. (L = 1 cm, n = 10
-4
)
72
2.33
The fine structure of the particular resonances and corresponding
coupling coefficients of 10
o
degree grating.
73
2.34
The fine structure of the particular resonances and corresponding
coupling coefficients of 10
o
degree grating.
73
2.35
The fine structure of the particular resonances and corresponding
coupling coefficients of 10
o
degree grating.
74
2.36
The schematic representation of the TFBG grating and the
incident linearly polarized core mode (the E
component is
shown). The core mode is rotated about the optical device axis
by some angle . The grating is tilted by angle about the x axis.
75
2.37
The weighted function
mn
(, ) in the fibre core, for the
4
o
degree tilted grating.
76
2.38
The coupling coefficients C
k
( ) for the 4
o
degree TFBG
computed for = 0
o
, 45
o
, 90
o
polarization angles of the incident
light.
77
2.39
The polarization dependence of coupling coefficients
corresponding to two resonances of the 4
o
degree TFBG.
78
2.40
The polarization dependence of coupling coefficients
corresponding to two resonances of the 10
o
degree TFBG.
79
2.41
The coupling coefficients between the core and cladding modes
of azimuthal order m = 0, 1, 2, 3, 4, 5 computed at various angles
of linearly polarized light incident at the 4
o
degree TFBG.
80
2.42
The coupling coefficients of the 4
o
degree TFBG computed at
various angles of linearly polarized light.
81
xvi
LIST OF FIGURES
2.43
The transmission spectra of the 4
o
degree TFBG, computed at
various angles of linearly polarized light.
81
2.44
The field components E
and E
computed at the fibre boundary
for modes with m = 0, 1, 2 azimuthal symmetry.
83
2.45
The field components E
and E
at the fibre boundary for modes
with m = 3, 4, 5 azimuthal symmetry.
84
2.46
The field components E
and E
computed at 1 nm distance
away from the fibre boundary.
85
2.47
The intensity of electric field components E
and E
at the fibre
boundary for 4
o
degree TFBG, computed at various states of
linearly polarized light P = 0
o
, 45
o
, 90
o
.
86
2.48
The structure of the particular resonances of 4
o
degree TFBG.
87
2.49
The electric field at the fibre surface, corresponding to particular
resonances. The linear polorizes light incident at P = 45
o
and
P = 90
o
angles at the 4
o
TFBG.
88
3.1
A typical TFBG transmission spectrum for linearly polarized
light, and series of spectra obtained by rotating a linear polarizer
about the optical axis are shown as the density plot.
90
3.2
The optical setup based on SI720 spectrophotometer and PR2000
polarization controller.
91
3.3
The Stokes vector representation.
92
3.4
The principle of operation of an optical vector analyzer .
93
3.5
The Experimental Setup.
94
3.6
TFBG with physical ^
x, ^
y axes and principal system axes u
1
and
u
2
measured at some optical frequency .
95
3.7
Eigenvalues
1
and
2
and the angle between the geometrical
axes of the system ^
x, ^
y and the coordinate system defined by the
principal axes u
1
, u
2
, before a) and after b) the eigenvalues were
reordered. (The data were obtained by means of the OVA 5000
Luna Technologies.)
96
3.8
Rotation of the principal axes of TFBG device as a function of
optical wavelength. (The data was obtained by means of the OVA
5000 Luna Technologies.)
97
3.9
Transmission loss along the TFBG system principal axes and its
geometrical axes for perfectly align coordinate systems = 0
o
and rotated by 45
o
degrees = 45
o
. The intersession loss I
x,y
and eigenvalues
1,2
are given in linear scale.
99
LIST OF FIGURES
xvii
3.10
Two eigenvalue spectra (individual transmission along the
principal axes) and corresponding polarization-dependent loss
(PDL) parameter. The peak in PDL spectrum is denoted by "1"
and zeros by "2". Here the more common dB scale is used.
104
3.11
Position of the resonance a) in the transmission spectra I
x
and I
y
of light polarized along ^
x and ^
y geometrical axes here aligned
with the principal axes, and b) in PDL spectra (the maximum
and zero values of PDL are detected), as a function of refractive
index change. The continuous lines represent the least square
approximation to the measured data, and SD is the standard
deviation from the linear approximation
105
4.1
Optical constants n and k of Au, Ag, Cu and Al as a function of
photon energy.
115
4.2
SEM images of gold film surface morphology, taken at different
stages of chemical deposition [
1
].
118
4.3
Simulation of light scattering by a rough silver film deposited on
the glass substrate.
119
4.4
Electric field of S (a) and P (b) polarized light interacting with an
array of metallic spheres [
2
].
120
6.1
The shift in absorption peak (red) and the corresponding shift in
real refractive index of a medium consisting from nanoparticles.
138
6.2
The schematic representation of connection between high Q
resonances of TFBG and low Q resonances of nanoparticles.
139
6.3
Simulated absorption efficiency of 30 nm spherical nanoparticle
made of Au, Ag, Cu and Al metals, as a function of photon
energy (the graphs for Ag and Al were divided and multiplied by
10, respectively).
140
6.4
Simulated absorption and scattering efficiency of 30 nm gold
spherical NP, immersed in media with various refractive indexes.
140
6.5
The absorption, scattering and extinction efficiencies as functions
of the size parameter x, for a particle with the relative refractive
index m = 5 + j0.4.
143
6.6
The absorption efficiency of 30 nm spherical nanoparticle made
of various metals ( Au, Ag, Cu and Al ) as a function of photon
energy. The result were obtained with use of the exact Mie's
theory and with the quasi-static approximation theory.
143
6.7
Absorption and scattering efficiency of 30 nm gold spherical
nanoparticle, immersed in media with various refractive indexes.
144
xviii
LIST OF FIGURES
6.8
The size dependence of the scattering and absorption efficiencies
of gold NP illuminated by electromagnetic wave at = 560nm.
The particle is assumed to be in the medium with refractive index
n = 1.
145
6.9
The absorption efficiency of gold, silver, aluminum and copper
particles as a function of particle size and incident photon energy
(plotted in logarithmic scale).
146
6.10
SEM image of the fibre surface coated with gold nanorod particles
147
6.11
Comparison of the optical absorption and extinction efficiencies
of prolate ellipsoids made from various metals with the principal
axes b = 30, a = 100 nm.
148
6.12
Comparison of the absorption efficiency for prolate silver
nanoparticles with different aspect ratios between the principal
axes =
b
a
148
6.13
The asymmetry of the coating made of elongated nanoparticles.
a) The transversely polarized electric field E
encounters
dielectric permittivity and b) the tangentially polarized electric
field E
sees the
dielectric permittivity.
149
6.14
A schematic representation of a TFBG sensor coated with
spherical nanoparticles.
150
6.15
The absorption efficiency as a function of the particle size
parameter x and the real part of the relative complex refractive
index
[m] at various fixed values of [m].
151
6.16
The absorption efficiency as a function of the particle size
parameter x and the real part of the relative complex refractive
index
[m] at various fixed values of [m].
152
6.17
The absorption efficiency as a function of the particle size
parameter x and the imaginary part of the relative complex
refractive index
[m] at various fixed values of [m].
153
6.18
Refractive index of titanium dioxide, T iO
2
.
154
6.19
The optimal size of a particle made of Titanium Dioxide.
155
7.1
The SEM mages of the fibre surface coated with gold (top) and
copper (bottom) films of different morphologies and thickness [
1
].
158
7.2
The AFM (a) and SEM (b) images of the fibre surface coated
with silver nanowires [
3
].
159
7.3
The SEM images of gold nanorod based coating
159
LIST OF FIGURES
xix
7.4
The SEM images of T iO
2
spheres deposited on the fibre surface
160
7.5
The AFM and SEM images of silver nanocubes (80nm)
160
7.6
(a) The envelope of PDL spectra, taken continuously along the
course of gold film deposition, and cross sections centered at the
point of the deepest notch: wavelength = 1542 nm (b) and time
= 7 min (c)
165
7.7
The SEM (b) images of the fibre surface coated with silver
nanowires [
3
].
166
7.8
UV-vis-NIR absorption spectrum of synthesized nanowire
coating (deposited on a flat glass substrate). The insets illustrate
the relative polarization of the Plasmon oscillations that give rise
to the absorption.
167
7.9
The TFBG spectrum evolution before (a) and after (b) deposition
of nanowires, for several values of the refractive index of the
solution. The concentration of the Ethylene Glycol in water goes
from 0%, 25%, 50%, 75%, 100% from the top to the bottom of
the figure. The corresponding total refractive index change is
n = 3.81 × 10
-2
.
168
7.10
Singular values (a,b) and polarization-dependent loss parameter
(c,d) (linear scale) changes due to a small refractive index change
of n = 3.77
× 10
-4
, before (a,c) and after (b,d) deposition,
corresponding to a single resonance taken at = 1555.7 nm .
169
7.11
(a) Wavelength shift of several individual resonances of the
TFBG sensor due to the surrounding refractive index change n.
(b) The sensetivity before (open marks) and after (closed marks)
deposition [
3
].
171
7.12
Vectorial E field structure for two modes with almost identical
propagation constants (hence resonance wavelengths), but
different polarization states ((a)â^TMlike mode, (b)â^TElike
mode). Silver nanowires are shown schematically on top.
172
LIST OF TABLES
4.1
Optical masses and the relaxation times for copper, silver and
gold. [
4
]
112
4.2
Values of the Lorentz-Drude Model Parameters [
5
].
114
xxi
FOREWORD
xxiii
xxiv
FOREWORD
Fibre Bragg grating sensors have emerged as a simple, inexpensive, accurate,
sensitive and reliable platform, a viable alternative to the traditional bulkier optical
sensor platforms. In this work we present an extensive theoretical analysis of the
tilted fibre Bragg grating sensor (TFBG) with a particular focus on its polarization-
dependent properties.
We have developed a highly efficient computer model capable of providing the
full characterization of the TFBG device in less then 3 minutes for a given state of
incident light polarization. As a result, the polarization-dependent spectral response,
the field distribution at the sensor surface as well as the fine structure of particular
resonances have become accessible for theoretical analysis. As a part of this com-
puter model we have developed a blazingly fast full-vector complex mode solver,
capable of handling cylindrical waveguides of an arbitrary complex refractive index
profile.
Along with the theoretical study we have investigated optical properties of the
TFBG sensor with application to polarisation-resolved sensing. We proposed a new
method of the TFBG data analysis based on tracking the grating transmission spectra
along its principle axes, which were extracted from the Jones matrix.
In this work we also propose a new method of enhancing the TFBG sensor refrac-
tometric sensitivity limits, based on resonant coupling between the TFBG structure
resonances and the local resonances of nanoparticles deposited on the sensor surface.
The 3.5-fold increase in the TFBG sensor sensitivity was observed experimentally.
ACKNOWLEDGMENTS
xxv
xxvi
I would like to express my sincere gratitude to Dr. Jacques Albert for his support,
patience, help, encouragement and guidance. He has shown me how to approach my
work as a scientist and has reminded me of the high standards and quality of research.
Working with Dr. Jacques Albert had a profound impact on me as a scientist and as
a person. I wish to extend my most sincere thanks Dr. Anatoli Ianoul, I always knew
I could count on his help and support.
During my Ph.D program I was fortunate to build some true friendships. In par-
ticular, I am very grateful to Ksenia Yadav, Yanina Shevchenko, Nur Ahamad and
Alec Millar who shared my everyday life. I would also like to acknowledge my
colleagues. In particular, Alexander Andreyuk, Albane Laronche, Zahirul Alam and
Nina Mamaeva. I am especially grateful to Lingyun Xiong for his willingness to
help.
I am sincerely thankful to my parents who encouraged me throughout the years,
and I deeply appreciate their support.
Finally, I acknowledge the generous financial assistance from Carleton University
and Institut national doptique (INO). My past six years at Carleton University have
been a particularly enjoyable part of my life.
ACRONYMS
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
xxvii
xxviii
ACRONYMS
AFM
Atomic Force Microscope
APTMS
(3-aminopropyl)trimethoxysilane
CVD
Chemical Vapor Deposition
EM
Electromagnetic field
FBG
Fiber Bragg Grating
IR
Infrared Radiation
NIR
Near Infrared Radiation
LPG
Long Period Grating
RIU
Refractive Index Unit
ODE
Ordinary Differential Equation
OVA
Optical Vector Analyzer
SEM
Scanning Electron Microscope
SMF-28
Single mode fiber (Corning SMF-28)
SNR
signal to noise ratio
SPR
Surface Plasmon Resonance
LSPR
Local Surface Plasmon Resonance
TFBG
Tilted Fiber Bragg Grating
TE
Transverse Electric
TM
Transverse Magnetic
UV
Ultraviolet Radiation
SYMBOLS
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
xxix
xxx
LIST OF SYMBOLS
E, H
Electric and Magnetic fields
,
Dielectric and magnetic permittivity
c
Velocity of light in vacuum
Angular Frequency
Propagation constant
n
Refractive index
N
ef f
Effective refractive index
k
Wave propagating constant
k
0
Free space wave propagating constant
k
i
Incident wave vectors
k
s
Scattered wave vectors
k
B
Grating wave vectors
· The Divergence operator
× The Curl operator
t
Partial derivative with respect to the variable t
Dielectric perturbation
Phase mismatch
mn
Dirac delta
C
mn
Matrix coefficients of the coupling strength
m
Mode azimuthal symmetry number
Wavelength
I
Intensity
R
Reflectivity
T
Transmissivity
j
, u
j
Principal values and corresponding principal vectors
J
m
Bessel functions of the first kind
J
l
(r)
Spherical Bessel functions,
N
l
(r)
Spherical Neumann functions,
H
(1)
l
(r)
Spherical Hankel functions of the first kind
H
(2)
l
(r)
Spherical Hankel functions of the second kind
INTRODUCTION
xxxi
xxxii
INTRODUCTION
In the presented work we study the tilted fibre Bragg grating (TFBG) sensor and
present new methods of improving its sensitivity. The TFBG sensor has proven to be
a simple, inexpensive, accurate, sensitive, and reliable platform [
6
], with a variety of
applications, including chemical and biological sensing [
7
,
8
].
The TFBG sensor is based on a standard telecommunication fibre with a tilted
grating inscribed inside the core. The telecommunication fibre is an optical cylin-
drical waveguide consisting of a high refractive index core surrounded by a cladding
layer with a lower refractive index. The light propagating in the fibre core is confined
by the total internal reflection phenomenon.
In our research the 1-cm-long TFBGs were inscribed in the hydrogen-loaded core
of a standard telecom single mode optical fibre (Corning SMF-28) using the phase
mask technique and intense ultraviolet light (pulsed KrF excimer laser at 193 nm or
248 nm). The TFBG sensor coated with small spherical particles is shown schemat-
ically in Figure
I.1
.
Figure I.1
Schematic representation of the TFBG sensor coated with sensitivity enhancing
layer of nanoparticles.
The inscribed grating planes are slightly tilted relative to the fibre cross-section,
which allows to couple the forward-propagating light from the fibre's core to the
backward-propagating cladding modes [
9
,
10
]. More than a thousand propagating
cladding modes are usually excited. Each cladding mode can be viewed as a propa-
gating electromagnetic wave, with a unique wave vector and field distribution, thus
interacting differently with the outside medium.
The coupling between the core and the cladding modes, mediated by the grating,
produces discrete narrow attenuation bands in the transmitted spectrum (the "reso-
nances") and since the cladding modes are guided by the fibre-surrounding medium
interface, these resonances are extremely sensitive to the refractive index of the sur-
rounding medium [
11
]. The energy coupled to a particular cladding mode as well as
the corresponding propagation constants can be precisely measured by acquiring the
transmission spectrum of the grating. The spectrum is shown in Figure
I.2
, consisting
INTRODUCTION
xxxiii
of a set of narrow resonances sensitive to the refractive index change of environment.
Figure I.2
A typical spectrum of a
10 degree TFBG sensor.
The TFBG sensor has a unique set of advantages, including the possibility to
isolate temperature effects and develop compact and robust refractometers with a
wide operating range [
11
,
12
]. In addition, the simultaneous probing of the external
medium at various wavelengths with modes having unique polarization properties
and incidence angles is possible.
The sensitivity of the TFBG sensor can be further improved if modes of the sen-
sor can be coupled to an external resonant system, such as a nano-scale coating, with
properties sensitive to the surrounding medium. The cladding modes excited by the
TFBG structure have a non-zero evanescent field at the cladding boundary, therefore
the light can tunnel outside the fibre into a coating layer. If the phase matching condi-
tion is met the resonant coupling between the cladding modes and resonances of the
coating layer can occur. When some modes (among the large number of cladding
modes accessible) are phase matched to the surface plasmon resonance (SPR) of
the outer surface of the metal coating, the light energy incident from these cladding
modes is efficiently coupled to the surface plasmon-polariton oscillations [
7
]. The
energy coupled to the SPR causes a drastic change in the TFBG transmission spec-
trum at wavelengths corresponding to the resonant modes. The principle of opera-
tion of the TFBG SPR sensor is analogous to the well-known Kretschmann-Raether
setup [
13
,
14
], except that the cladding modes use wavelength and angle scanning to
probe for the SPR phase matching condition (because each cladding mode couples
at a different wavelength and "strikes" the cladding boundary at a different incident
angle).
Finally, a further increase of the selectivity of the modes coupling to SPR is pro-
vided by controlling the light polarization. This discovery led the way to a strong
increase in the accuracy with which we can follow SPR shifts associated with small
refractive index changes of the outer medium surrounding the fibre by using the
xxxiv
INTRODUCTION
polarization-dependent loss (PDL) of the TFBG transmission [
15
]. In particular,
it was found that narrowband resonances (100 pm spectral bandwidth (BW)) with
refractometric sensitivities (S) of 350 nm/RIU (refractive index units) showed up
in the PDL spectrum. These resonances have some of the highest figures of merit
(S/BW = 3500 RIU
-1
) for SPR sensors of any kind [
16
].
Several resonances in the transmission spectrum are shown in Figure
I.3
. The res-
onances were continuously observed during the process of nano-scale metal coating
deposition. The complicated structure of the resonances, with strong dependence on
the film thickness, is proportional to the time of the deposition is clearly seen.
Figure I.3
Evolution of the TFBG spectral response during the silver nanoparticle deposition
followed by the continues film formation.
Recently it was discovered that the sensitivity of TFBG sensors can be improved
by coating its surface with randomly oriented silver nanowires. Such a coating would
allow for a large number of cladding modes to interact with the deposited nanowires,
exciting localized surface plasmon resonances (LSPR) when the appropriate phase
matching condition is met. The sensitivity of the TFBG resonances to the external
refractive index change was increased by a factor of 3.5 even though the surface
coverage of the nanowires was less than 14%.
In the presented work we investigate methods of improving the TFBG sensor
sensitivity by coating its surface with various types of nano-scale films, including
metal films and nanoparticle coatings. We also discuss the polarization properties of
the sensor and present the developed highly sensitive measurement technique.
However, the main objective of our work was to simulate the behaviour of the
TFBG sensor. The sensor is an interesting object with more than a thousand inter-
acting modes and a complex polarization-dependent response, as can be seen from
ures
I.2
and
I.3
. The simplicity of the spectral response hides many non-trivial phys-
ical effects that we might wish to decode. Our goal here was to understand the TFBG
sensor behaviour, interpret the polarization-dependent spectral response and create a
computer model which would allow us to predict the outcome of experiments. More-
INTRODUCTION
xxxv
over, the process of improving the sensor sensitivity required the rather tedious ex-
perimental work of scanning the parameter space by probing particles with different
geometrical shapes, such as spheres, cubes, cages and wires, varying the nanopar-
ticle's size and the deposition densities. A theoretical guidance which would allow
the prediction of an optimal set of parameters, such as the film material, thickness,
composition and morphology is highly desirable.
ORGANIZATION OF THE BOOK
xxxvii
xxxviii
BOOKORG
This book is organized in the following manner. Chapter 1 provides an introduc-
tion to the problem and motivation.
In Chapter 2 we explain how we developed a full-vector complex mode solver
which we applied to circularly symmetric optical waveguides in order to compute
scalar and vectorial modes. The exact and approximate solutions were compared
and the orthogonality of the modes was verified. An interesting analogy between
TE and TM modes in slab waveguides and modes in cylindrical structures was es-
tablished. The analysis of a weakly-guided approximation and the exact solution
allowed us to build a connection between the scalar and vectorial equations. The
split in eigenvalues was discussed and analyzed thoroughly. To our knowledge, this
particular analysis has not been reported previously.
Although a number of commercial mode solver software packages is available,
the software interfaces usually impose certain limitations on the process automation.
For example, to simulate a TFBG spectral response more than a thousand interacting
modes should be considered. These modes should be computed, often an various
frequencies, stored and be available for further processing. If the optimal parame-
ters are searched, a number of separate computations must be done sequentially. In
addition, in order to study a particular property of the sensor only the modes with
this particular property are required, thus these modes can be targeted discriminately
without computation of the remaining modes. Thus, by targeting only specific modes
the simulation speed can be increased dramatically.
Chapter 3 is based upon solutions obtained in Chapter 2. The modes obtained for
the non perturbed case were used as basis functions to solve the problem of tilted
Bragg grating structure. The polarization-dependent effects were investigated. The
results of the simulations were compared with the experimental measurements. The
emphasis was set on polarization dependency of the coupling coefficients, which
were computed for all possible angles of incident core mode polarization, to our
knowledge this result has not been previously reported.
Chapter 4 provides a complete experimental characterization of the TFBG struc-
ture. The experimental measurement techniques with application to polarization-
based sensing were developed.
In Chapter 5 we start our discussion on the possible enhancement of TFBG sensor
sensitivity by coating its surface with nano-scale films. We review optical properties
of various materials, metals in particular.
Chapter 6 provides a detailed overview of nanoparticle optical properties, as well
as several simulation methods, in particular the Mie theory.
In Chapter 7 we present our ideas on enhancing the sensitivity of TFBG sensors,
and conduct a search for parameters that would provide the optimal nanoparticle-
based coating.
Chapter 8 presents experimental results from various nano-scale film coatings and
their associated sensitivity enhancements.
Lastly, Chapter 9 is a summary of the work findings.
CHAPTER 1
A FULL-VECTOR COMPLEX MODE
SOLVER FOR CIRCULARLY
SYMMETRIC OPTICAL WAVEGUIDES
1.1
Introduction
In this chapter we present a simple yet efficient and exact approach for fast and accu-
rate modeling of waveguides with cylindrical or elliptical symmetry, with arbitrary
real and imaginary refractive index profiles.
The analytical solution for the simplest case of a cylindrical dielectric waveg-
uide, as well as modes classification and field plots were presented by Snitzer in
1961 [
17
]. Shortly after, direct numerical integration techniques were presented, in
a noticeable paper published by Dil and Blok [
18
] on a numerical solution of four
coupled Maxwell's equation in radial parabolic dielectric waveguides.
It is well known that in the case of waveguide with a small refractive index con-
trast the weakly guided approximation can be used, allowing the system of Maxwell's
to be reduced to the Helmholtz equation, which can more easily be solved [
19
,
20
].
Here we consider the general problem of full vectorial mode solutions of waveguides
with high refractive index contrasts and a non-zero imaginary part of the refractive
index, i.e. we consider an arbitrary complex permittivity profiles.
Several methods have been proposed earlier. Usually the waveguide profile is sub-
divided into a number of piecewise homogeneous concentric layers, and the system
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
1
2
A FULL-VECTOR COMPLEX MODE SOLVER
of Maxwell's equations are solved in each layer. The solutions are next connected
with the help of boundary conditions for the tangential electric and magnetic fields
at the layer boundaries.
In cylindrical coordinates the solutions for each layer can be represented in terms
of Bessel and modified Bessel functions, and then connected through a 4x4 transfer
matrix, which incorporates the boundary conditions. This is the so-called transfer
matrix method proposed by Yeh and Lindgren [
21
]. The modes propagation constant
are obtained by finding the roots of a polynomial, over Bessel functions, with a
degree proportional to the number of layers. This method becomes increasingly
prohibitive with regards to computational resources as well as to numerical stability
with the increase in number of layers. The transfer matrix method becomes even
more complicated if the permittivity posses an imaginary part or the modes are leaky.
In such a case the roots have to be searched on the complex plane.
Although its limitations, the transfer matrix method can be successfully applied to
waveguide structures consisting only of a few uniform layers [
22
,
23
,
24
] or to struc-
tures consisting of infinitely many alternating uniform layers, the so-called Bragg
fibres [
25
]. The computational complexity of the matrix method can be reduced
by replacing Bessel functions with their asymptotic expressions, or if the periodic
cylindrical layers are approximated as planar Bragg stacks [
26
].
Another possibility is to use the pseudospectral method, also called the spectral
collocation method, where a solution is searched in terms of sine functions [
27
],
Chebyshev-Lagrange [
28
,
29
], Laguerre-Gauss [
30
,
31
], Hermite-Gauss [
32
] inter-
polating polynomials, or in terms of some other suitable basis functions. The pseu-
dospectral method can be implemented either for the entire domain or separate basis
functions can be chosen for each uniform layer. In each case the basis functions are
chosen with regards to boundary conditions. If the problem is solved for the entire
domain the approximation of mode fields might require hundreds or even thousands
of functions, and the convergence is generally a problem [
33
]. The multidomain
method is in a way similar to the transfer matrix method. The dielectric interface
conditions should be fulfilled, thus the functions between different layers are again
connected through the transfer matrix at the boundaries [
29
]. In both cases the system
of algebraic equations results in numerical eigenvalue problem, with eigenvectors
consisting of the expansion coefficients for a particular mode. A simpler but more
resource hungry method was proposed in [
34
,
35
] where instead of the global func-
tions for each homogeneous region a finite different method was applied, and then
the transverse electric or magnetic fields were matched at the radial index disconti-
nuities. The comparison between the transfer matrix method and the pseudospectral
method was conducted in [
33
,
29
] and definitely favours the pseudospectral method.
Yet another possibility is to use a standard finite difference (FDM) or finite el-
ement (FEM) method for the entire 2D waveguide profile, with the Yee's cell [
36
]
adapted for the cylindrical symmetry [
37
,
38
,
39
]. A number of commercial simula-
tion software based on the finite difference and finite element methods are available.
This approach does not require understand of engineering or physical aspects of the
problem. A given problem is viewed as a "blackbox" accepting the input data e.g.
geometrical shape, an operational wavelength, etc. and returning the required numer-
INTRODUCTION
3
ical results. However, the use of commercial software imposes certain limitations,
for example it might be challenging to automate the process of obtaining a series of
solutions for a varying parameter. It should also be noted that the FDM and FEM
method are the slowest and the most resource hungry, especially if used for 2D or
3D problems.
Here we present a simple yet efficient and fast numerical method. First the system
of Maxwell's equations was reduced to only 2 coupled ordinary differential equations
for the electric field. The variation of the dielectric permittivity was incorporated in
the equation such that equations become suitable for numerical integration through
the entire domain, in contrary to the piecewise homogeneous methods described
above. Next the equations were transformed into a system of algebraic equations
with the help of a finite difference method. Finally the problem was reduced to find-
ing eigenvalues and eigenvectors of a five-diagonal matrix. The eigenvalue problem
was effectively solved with the standard iterative method commonly applied to large
sparse linear systems. The proposed method requires less grid points if the refractive
index discontinuity is approximated with a smooth function at an interval of about
1 nm at the position of discontinuity.
4
A FULL-VECTOR COMPLEX MODE SOLVER
1.2
The solutions for a cylindrical waveguide
We start with Maxwell's equations written in the MKS system of units, for a source
free region [
40
]:
× H(r, t) =
o
(r)
c
t
E(r, t),
(1.1a)
× E(r, t) = -
o
c
t
H(r, t),
(1.1b)
· H(r, t) = 0,
(1.2a)
· (r)E(r, t) = 0.
(1.2b)
Assuming that fields E(r, t) and H(r, t) are varying harmonically with time, i.e.
proportional to the oscillating term e
it
, we can take the time derivative and next
apply the curl operator to both equations (
1.1a
) and (
1.1b
). Next plugging
× H
term into second equation (
1.1b
) and
× E term into first equation (
1.1a
) we split
Maxwell's equations into two subsystems, separating E and H fields:
× × E(r) =
(r)k
2
o
E(r),
· (r)E(r)
= 0,
(1.3)
and
×
1
(r) ×
H(r)
= k
2
o
H(r),
· H(r) = 0,
(1.4)
where k
o
=
c
=
2
, and =
1
o
o
.
Each system of equations (
1.3
) and (
1.4
) contains the full description of electro-
magnetic field. We can either use system (
1.3
) to find E field and next express field
H in terms of
× E by using (
1.1b
), or alternatively, the field H can be obtained
from equation (
1.4
) and field E can be found in terms of
× H with help of (
1.1a
).
Here we prefer to use system (
1.3
) over (
1.4
) to avoid complications caused by
the
×
1
(r)
× H(r) term.
Now considering the vector identity [
41
]:
× × E = (E) -
2
E,
(1.5)
and expansion of the Gauss equation (
1.2b
):
E = E + E · = 0,
(1.6)
we obtain the following identity:
× × E = - E ·
-
2
E,
(1.7)
THE SOLUTIONS FOR A CYLINDRICAL WAVEGUIDE
5
Thus system (
1.3
) can be written as:
2
E + k
2
o
E =
- E ·
(1.8)
The equation (
1.8
) is all that is needed to describe electromagnetic waves in inhomo-
geneous nonmagnetic media. Once the electric field E is found the corresponding
magnetic field can be expressed in terms of
× E as follows from (
1.1b
) equation.
1.2.1
Weakly guided approximation
Let us assume that the rate at which function (r) is changing with the change of r
coordinate is insignificant, i.e.
(r) is smaller than (r). Then the right hand side
of equation (
1.8
) can be neglected [
42
]:
2
E + (r)k
2
o
E = 0
(1.9)
This is the so-called weakly guided approximation. The equation (
1.9
) provides full
description of electromagnetic waves in the case of small
(r).
Let us write equation (
1.9
) in cylindrical coordinates:
2
-
1
2
+ k
2
o
-
2
2
0
2
2
2
-
1
2
+ k
2
o
0
0
0
2
+ k
2
o
E
E
E
z
= 0.
(1.10)
Here
2
is the Laplacian for a scalar function in cylindrical coordinates:
2
=
2
+
1
+
1
2
2
+
2
z
,
(1.11)
and
2
A is the vector Laplacian in cylindrical coordinates:
2
A =
1
(
) +
1
2
2
+
2
z
A
^
+ A
^
+ A
z
^z ,
(1.12)
The equation (
1.10
) can be obtained from (
1.12
) by taking into account that unit vec-
tors themselves are functions of coordinates, and noting that in cylindrical coordi-
nates there are two nonzero derivatives of the unit vectors:
^
= and
^
=
-,
the remaining derivatives are zero.
We note that equation (
1.10
) decouples into two independent equations,
the vectorial equation:
2
-
1
2
+ k
2
o
-
2
2
2
2
2
-
1
2
+ k
2
o
E
E
= 0,
(1.13)
6
A FULL-VECTOR COMPLEX MODE SOLVER
and the scalar equation:
2
E
z
+ k
2
o
E
z
= 0.
(1.14)
Here we are interested in circularly symmetric optical waveguides, assuming that
the waveguide structure is uniform along the z coordinate, and searching for a peri-
odic solution in , i.e. u(, , z)
u()e
jz
e
jm
, we can can replace
z
with j
and
with jm. Thus the equations (
1.13
) and (
1.14
) can be written in form of the
vector eigenvalue problem:
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
-
j
2m
2
j
2m
2
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
E
E
=
2
E
E
,
(1.15)
and the scalar eigenvalue problem:
d
2
+
1
d
-
m
2
2
+ k
2
o
E
z
=
2
E
z
.
(1.16)
Here =
2
are unknown eigenvalues and is the propagation constant.
The equation (
1.16
) is well known and is often considered for the weakly guided
approximation case while the equation (
1.15
) is rarely used, see for example [
43
,
44
,
45
].
Once any of these two equations are solved, the remaining components of the
electric field E can be found from Gauss's law (
1.2b
), and next the magnetic field H
can expressed in terms of electric field with use of equation (
1.2b
).
1.2.2
The exact solution for cylindrical waveguides
The exact solution can be obtained if term
is no longer neglected. The electro-
magnetic field in such a case is described by equation (
1.8
):
2
E + k
2
o
E =
- E ·
.
(1.17)
Let us assume that dielectric permittivity varies only along the the radial direction
= (), then
E ·
= E
()
()
+ E
1
()
()
+ E
z
z
()
()
=
= E
()
()
=
= E
· (ln ()) ,
(1.18)
and
E ·
= E
(ln ) =
(ln ) E
+ (ln )
E
(ln )
1
E
(ln )
z
E
.
(1.19)
THE SOLUTIONS FOR A CYLINDRICAL WAVEGUIDE
7
Now equation (
1.17
) can be rewritten in cylindrical coordinates:
2
-
1
2
+ k
2
o
+ (ln ) + (ln )
-
2
2
0
2
2
+ (ln )
1
2
-
1
2
+ k
2
o
0
(ln )
z
0
2
+ k
2
o
E
E
E
z
= 0.
(1.20)
The first two equations can again be separated, as they do not depend on the E
z
component.
Replacing
z
with j and
with jm as previously, we obtain a slightly different
from of equation (
1.15
). The vectorial eigenvalue problem (
1.21
), now depends on
the rate of change of dielectric permittivity ():
d
2
+
1
d
+ (ln ) d
-
m
2
+1
2
+ k
2
o
+ (ln )
-
j
2m
2
j
2m
2
+
jm
(ln )
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
E
E
=
2
E
E
(1.21)
Here we note again that terms
() were previously ignored in the weakly guided
approximation (
1.15
). In the following section we compare the results following
from the weakly guided approximation (
1.15
) and from the exact formulation (
1.21
).
Finally, once the fields E
and E
are known from equation (
1.21
), the remain-
ing E
z
component can be found from the Gauss's equation (
1.2b
) written in cylin-
drical coordinates:
E
z
= -
1
j
1
d
d
( E
) +
jm
E
,
(1.22)
and the magnetic field H can now be expressed in terms of electric field E = (E
, E
, E
z
)
with help of equation (
1.2b
):
H = j
c
o
× E.
(1.23)
8
A FULL-VECTOR COMPLEX MODE SOLVER
1.2.3
TE and TM modes in slab waveguides
Here we show that our approach is valid, by deriving the well known equation for
slab waveguides. Again, let us start with equation (
1.8
):
2
E + k
2
o
E = - E ·
.
(1.24)
Assuming that the slab waveguide is uniform along the
y and z axis, the dielectric permittivity
can only be function of the
x coordinate
= (x), thus equation (
1.24
) can be written in
Cartesian coordinate system as follows:
2
+ k
2
o
0
0
0
2
+ k
2
o
0
0
0
2
+ k
2
o
E
x
E
y
E
z
= -
(ln ) E
x
+ (ln )
x
E
x
(ln )
y
E
x
(ln )
z
E
x
.
(1.25)
Considering the waveguide symmetries, uniformity along the y and z axis, and as-
suming that the wave is propagating along the z axis, we can can replace
z
with j
and
y
with 0. The scalar Laplacian
2
can now be written in the following form:
2
=
2
x
+
2
y
+
2
z
=
=
d
2
x
-
2
,
(1.26)
and equation (
1.25
) can be rewritten in the the form of vectorial eigenvalue problem:
d
2
x
+ (ln ) d
x
+ k
2
o
+ (ln )
0
0
0
d
2
x
+ k
2
o
0
(ln )
0
d
2
x
+ k
2
o
E
x
E
y
E
z
=
2
E
x
E
y
E
z
. (1.27)
The first 2 equations are not coupled and can be written separately:
d
2
x
+ (ln ) d
x
+ (ln ) + k
2
o
E
x
=
2
E
x
(1.28)
d
2
x
+ k
2
o
E
y
=
2
E
y
(1.29)
The equation (
1.29
) is well known equation for the TE modes in slab waveguide [
46
,
47
].
The equation (
1.28
) can be recognized as equation for TM modes if we rewrite it
in a slightly different form:
d
x
1
d
x
+ k
2
o
( E
x
) =
2
( E
x
),
(1.30)
where we have considered that
d
x
1
d
x
( u) =
1
( u)
= u +
u
= u + (ln ) u + (ln ) u
(1.31)
THE SOLUTIONS FOR A CYLINDRICAL WAVEGUIDE
9
We note that in the case of slab waveguide the system of equations (
1.27
) for the
transverse field components E
x
and E
y
decouples into separate equations, indepen-
dently describing TE and TM modes, unlike in the case of cylindrical waveguide
where the system of equations (
1.21
) is coupled.
We also observe that in the case of weakly guided approximation the equations
for TM and TE mode (
1.30
,
1.29
) become identical, as
d
x
1
d
x
u(x)
d
2
x
u(x),
(1.32)
thus the degeneracy occurs, i.e. the two different set of eigenfunctions E
x
(x) and
E
y
(x) corresponding to the same eigenvalues.
10
A FULL-VECTOR COMPLEX MODE SOLVER
1.3
The numerical method
In this section we describe the numerical procedure we implemented to obtain a
vectorial mode solution and propagating constants of a dielectric waveguide with
circular symmetry. The waveguide can be of an arbitrary refractive index profile, i.e.
it can consist of any number of radially stratified layers made of various materials
including absorbing material such as metals with the dominant imaginary part in
the refractive index. In the following sections these modes, in other words eigen-
functions, are used as a basis for a more general problem of circularly symmetric
dielectric waveguide with a small perturbation along the z axis.
1.3.1
The scalar modes
Let us start our analysis with the weakly guided approximation and the scalar Helmholtz
equation (
1.16
):
d
2
+
1
d
-
m
2
2
+ k
2
o
E
z
() =
2
E
z
().
(1.33)
Rewriting the above equation in a slightly different notation we get:
1
d
(d
) + U
m
() R
m
k
() = (
m
k
)
2
R
m
k
(),
(1.34)
where
U
m
() = k
2
o
n
2
o
() -
m
2
2
.
(1.35)
The eigenfunctions are denoted as R
m
k
() and eigenvalues as
m
k
, known as propa-
gation constants in waveguide theory. It should be noted that for each index m a set
of different solutions will be obtained, keeping this in mind, let us omit the index m
from the notation for simplicity.
First, let us build a matrix representation of equation (
1.34
). This can be achieved
by introducing an uniform grid (
1
,
2
, ...,
N
) and representing the unknown func-
tion R() as a vector with components R
j
= R(
j
). Next, applying the finite differ-
ence method we can rewrite equation (
1.33
) in the following form:
^
L
R() =
2
R(),
^
L
= d
2
+
1
d
-
m
2
2
+ k
2
o
,
(1.36)
with the central difference approximation:
a
j
R
j
-1
+ b
j
R
j
+ c
j
R
j
+1
=
2
R
j
,
(1.37)
THE NUMERICAL METHOD
11
where
a
j
=
1
h
2
-
1
j
1
2h
,
b
j
= -
2
h
2
-
m
2
2
j
+ k
2
o
n
2
j
,
c
j
=
1
h
2
+
1
j
1
2h
.
(1.38)
Here n
j
= n(
j
) is the refractive index profile of a given structure. The refractive
index can be a complex number to take into account absorbing materials.
Thus equation (
1.36
) can be written in the matrix form:
[L]R =
2
R,
(1.39)
where [L] is the sparse 3-diagonal matrix with b
j
on the main diagonal, a
j
on the
lower sub-diagonal and c
j
on the upper sub-diagonal.
Considering that equation (
1.37
) is the second order differential equation, the two
proper boundary conditions should be imposed:
1. The condition at infinity
.
This condition can be imposed by introducing an extra point R
N
+1
, outside the
computational domain. Here we are interested in guided modes, thus we can
assume that there is no field at infinity and simply impose the zero boundary
condition. By choosing a sufficiently large computational window we ensure
that the field value is negligible at the computational boundary.
R
N
+1
= 0.
(1.40)
Plugging this condition into (
1.37
) we have
a
N
R
N
-1
+ b
N
R
N
+ c
N
· 0 =
2
R
N
.
(1.41)
We note that this condition is already imposed by the initial equation (
1.37
),
thus there is no need to modify this equation.
2. The condition at = 0.
(a) Assuming that m = 1,
±2, ±3, ... we should require
R( = 0) = 0.
(1.42)
By introducing an imaginary point R
0
outside the computational domain on
the left, we can write:
a
1
R
0
+ b
1
R
1
+ c
1
R
2
=
2
R
1
,
(1.43)
12
A FULL-VECTOR COMPLEX MODE SOLVER
considering (
1.42
) we get:
a
1
· 0 + b
1
R
1
+ c
1
R
2
=
2
R
1
,
(1.44)
Again, we see that this condition is already imposed be the default form of
the equation (
1.37
).
(b) Assuming that m = 0 we get a maximum in the field distribution at the
center of the waveguide, and thus should require a zero derivative of the
field:
dR()
d
|
=
1
= 0,
(1.45)
using the central difference the above relation can be written as:
R
0
= R
2
= 0.
(1.46)
Plugging the above condition into (
1.43
) we get:
b
1
R
1
+ (c
1
+ a
1
)R
2
=
2
R
1
,
(1.47)
Therefore the matrix [L] has to be modified:
L
12
= L
12
+ a
1
(1.48)
This is the only modification of (
1.37
) which is required.
Finally considering modification (
1.48
) the problem (
1.37
) can be stated in the
matrix form:
[ ^
L]R =
2
R.
(1.49)
Here we note that the zero boundary condition can be imposed in both cases if the
problem is solved on the internal
[-R, R] instead of [0, R].
The problem (
1.49
) is a well known numerical eigenvalue problem, which can
be solved with variety of numerical techniques. Because we are dealing with a 3-
diagonal sparse matrix, very efficient iterative methods for large sparse linear sys-
tems can be implemented. We used MATLAB software, which allowed us to con-
sider up to one million elements. Even for a very large matrix of 10
6
×10
6
elements,
the required eigenvalues and corresponding eigenvectors, in the range of interest,
were found in less than 1 minute. The matrix does not actually includes all of the
10
12
numbers, but rather 3
· 10
6
numbers written in terms of three separate vectors
corresponding to the three matrix diagonals. The computational algorithm is based
on the iterative routine converging from a random guess [
48
] towards the exact eigen-
vectors and corresponding eigenvalues in the given range. If, instead of a sparse
matrix, a dense matrix is used, the computation might take a prohibitive amount of
time for the particular problem presented in this work (for example, it would take a
couple of days to compute eigenvalues of a 5000
× 5000 matrix in Matlab with the
default computational routine).
THE NUMERICAL METHOD
13
After the numerical eigenvalue problem is solved, a set of eigenvalues
k
and cor-
responding eigenvectors R
k
are accessible. It is convenient to use a potential barrier
analogy with quantum mechanics. The potential energy function is defined by equa-
tion (
1.35
). Basically, the square of refractive index profile n
2
() plays a role similar
to the role of a potential energy barrier in quantum mechanics. The eigenvalue can
be though of as "energy states", and should be expressed in the same units as the
potential barrier, or refractive index. The eigenfunctions R
k
() represent the electric
field profile, that is E
z
component if the scalar equation
1.33
is considered, or E
and E
components for the vectorial case, as will be discussed later.
Here we introduce the so-called effective refractive indices i.e. N
ef f,k
= k
0
k
,
and plot the normalized eigenfunctions R
k
() centering them at the corresponding
N
ef f,k
with analogy to quantum mechanics. The result is shown in Figure
1.1
.
Figure 1.1
The refractive index profile of SMF-28 fibre immersed in water. The eigenvalues
and eigenfunctions are plotted for
m = 0 and m = 1
As can be seen from Figure
1.1
that in the case when N
ef f,k
becomes smaller than
the refractive index of the surrounding medium, the field energy is leaking outside the
fibre boundaries, and hence the energy is no longer confined inside the fibre as shown
in Figure
1.2
. Such modes are called the leaky modes. The proposed method should
14
A FULL-VECTOR COMPLEX MODE SOLVER
be modified for the proper treatment of leaky modes. The zero boundary condition
should be replaced with a numerical absorbing boundary condition, ensuring that
the incident travelling waves are absorbed and no energy is reflected back into the
waveguide, and hence the formation of standing waves is prohibited. Such absorbing
boundary layer is called the perfectly matched layer (PML). An efficient numerical
technique for implementing the absorbing boundary condition in the FDTD method
was developed by Mur in 1981 [
49
].
Figure 1.2
The fibre cross-section and corresponding refractive index profile with modes
bounded inside the fibre.
We can note the further similarities with quantum mechanics. The number m
in the equation (
1.34
) plays a role similar to the role of orbital quantum number in
quantum mechanics. The larger the angular momentum of a particle, the closer its
wavefunction is to the periphery, which is also consistent with classical mechanics.
Unless the critical value is exceeded, the particle stays confined by central potential,
although some fraction of its energy leaks outside the potential barrier boundaries.
The described effect can be observed in Figure
1.3
. The potential function U
m
() is
defined by the equation (
1.35
) and is shown in red color. We can note in Figure
1.3
how significant is the role of angular momentum potential U
m
() =
m
2
2
.
Finally, the complete basis functions
m
k
(, ) can be constructed by considering
the angular dependence as well:
m
k
(, ) = R
m
k
()e
jm
.
(1.50)
A particular solution can be constructed by choosing an orbital number m, thus
defining the potential function U
m
k
(), and picking a particular radial eigenfunc-
tion R
m
k
() for the chosen mode family m. An example of such particular solution
is shown in Figure
1.4
.
THE NUMERICAL METHOD
15
Figure 1.3
The radial eigenfunctions
R
m
k
() (shown in blue color) and the potential well
function
U
m
() (shown in red color) for m = 40.
Figure 1.4
One of the basis function (mode)
3
10
(, ) , at n = 10 (N
ef f
= 1.4181) and
m = 3.
1.3.2
The vectorial modes
Let us first consider the vectorial equation (
1.15
) for the weakly guided approxima-
tion:
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
-
j
2m
2
j
2m
2
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
E
E
=
2
E
E
.
(1.51)
Again, we can implement the finite difference method to replace the system of two
second order coupled ordinary differential equations (
1.51
) with a system of alge-
16
A FULL-VECTOR COMPLEX MODE SOLVER
braic equations:
[ ^
M ]E
T
=
2
E
T
.
(1.52)
Before constructing the finite difference implementation we note that matrix in (
1.51
)
contains complex numbers. Although the complex numbers does not impose any
limitation on the numerical method, the computational speed can be increased if the
matrix is real. The memory requirement is also reduced by a factor of two for the
real numbers. Multiplying the first equation in (
1.51
) by imaginary unit j and pulling
it under the E
component we get:
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
2m
2
2m
2
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
jE
E
=
2
jE
E
.
(1.53)
The operator matrix now is converted to a pure real form, unless the material permit-
tivity is complex. The same equation was previously obtained elsewhere [
45
,
43
].
The finite difference matrix [ ^
M ] can be constructed in the following way:
[M]E
T
=
2
E
T
,
(1.54)
where
[ ^
M ]
=
^
L
a
a
^
L
,
a
i
=
2m
2
i
,
^
L
= [D]
2
+
1
[D] -
m
2
+ 1
2
+ k
2
o
,
(1.55)
here [D]
2
and [D] are the second order and the first order finite difference matrices,
respectively, assembled with the help of the central difference approximation in a
fashion similar to (
1.37
) and (
1.38
). We note that operator ^
L becomes identical to
the scalar operator (
1.36
) if m
2
+ 1 is replaced by m
2
.
The structure of matrix [ ^
M ] is shown in Figure
1.5
, where the number of nodes
was limited to N = 10 for the convenience of representation, however for a practical
application at least several thousands of elements should be considered.
In can clearly be seen that the matrix is sparse with only five main diagonals. The
right most and left most diagonals represent term
2m
2
which is responsible for the
coupling between E
and E
field components. In the case when m = 0 the matrix
becomes a three diagonal, with uncoupled E
and E
components.
In the similar way we can approach the exact equation (
1.21
):
d
2
+
1
d
+ (ln ) d
-
m
2
+1
2
+ k
2
o
+ (ln )
2m
2
2m
2
+
m
(ln )
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
jE
E
=
2
jE
E
,
(1.56)
THE NUMERICAL METHOD
17
Figure 1.5
The structure of the sparse matrix
[ ^
M ] for N = 10 and m = 0.
here we pulled the imaginary unit j out of the matrix.
The system of algebraic equation takes the following form:
^
L
2
a
b
^
L
E
T
=
2
E
T
,
(1.57)
where a and ^
L are defined previously in (
1.55
), and
b
i
= a
i
+
m
(ln ) ,
^
L
2
= ^
L + (ln ) [D] + (ln ) .
(1.58)
The matrix for the exact case has the same structure as shown in Figure
1.5
.
Alternatively, the exact system of equations (
1.21
) can be rewritten in the Sturm-
Liouville form:
d
d
1
d
d
+ -
m
2
2
+ k
2
o
2m
2
1
2m
2
+
m
(ln )
d
d
1
d
d
+ -
m
2
2
+ k
2
o
j E
E
=
2
j E
E
,
(1.59)
where we assumed
1
r
(ru)
= u +
1
r
u
-
1
r
2
u,
1
r
(r u)
= u +
1
r
u
+ (ln ) u -
1
r
2
u + (ln ) u.
(1.60)
We discuss this subject in more detail in the following section. Next, the numerical
finite difference method explicitly designed for Sturm-Liouville problems can be
18
A FULL-VECTOR COMPLEX MODE SOLVER
applied. The main idea of this approach is to derive a central difference equation for
the
d
d
p()
d
d
operator.
In the next section we compare results obtained by numerically solving the scalar
(
1.14
) and vector (
1.13
) equations for the weakly guided approximation with the
exact solution obtained from (
1.21
).
DISCUSSION
19
1.4
Discussion
In this section we verify our results and discuss properties of the modes. First let us
start by verifying the developed numerical method. The analytical solution can be
easily obtained for slab waveguides, as shown for example in [
46
,
50
].
The solution for TE and TM modes guided in a thin symmetric glass slide im-
mersed in water as well as the dispersion curves were presented in [
46
].
Figure 1.6
The dispersion curves of a symmetric glass slide immersed in water.
The waveguide and the medium refractive indices are
n
W G
= 1.45 and n
M ed.
= 1.33,
respectively.
Defining the normalized frequency V and normalized propagation constant b as:
V
= Rk
o
n
2
1
- n
2
2
R
,
b
=
N
2
ef f
- n
2
2
n
2
1
- n
2
2
N
2
ef f
,
(1.61)
and solving equations (
1.28
,
1.29
) numerically using the proposed method we plot
the dispersion curves for TE and TM modes, as shown in Figure
1.6
. The identical
graph was obtained in [
46
]. Here R is the waveguide width (or diameter), n
1
and n
2
are the refractive indices of the waveguide and the surrounding medium, respectively.
In the similar manner we can verify the results for cylindrical waveguides, see for
example results obtained elsewhere [
17
,
18
,
44
,
45
,
46
,
50
].
It was shown in the previous section that the Maxwell equations can be decoupled
into the scalar (
1.14
) and vector (
1.13
) equations if the weakly guided approximation
applies. Solving these equations independently we obtain dispersion curves as shown
in Figure
1.7
and Figure
1.8
.
20
A FULL-VECTOR COMPLEX MODE SOLVER
The eigenvalues for the scalar and vectorial problems are identical. Indeed, this
result can be expected as both equations were obtained from the same initial system
of Maxwell's equations, and are in a way simply a different representation of the
same physical system.
We note that solutions to the vectorial equation are degenerate, except for the
modes with the zero angular part, i.e. m = 0. The equations for the E
and E
field
components become uncoupled at m = 0, thus E
component can take arbitrary val-
ues independent on the value of E
component. In other words the modes belonging
to the m = 0 family can have an arbitrary polarization, with no restriction imposed
by the cylindrical symmetry of the problem. In such cases modes can be represented
either by a superposition of two orthogonal linearly polarized waves or as a super-
position of left and right circularly polarized waves. In the literature these modes
are known as linearly polarized (LP) modes [
17
,
43
], although the term circularly
polarized (CP) modes would also be appropriate [
43
].
The mode profile at m = 0 is shown in Figure
1.9
. Indeed, it can be noted that
two modes
E
0
and
0
E
have an identical propagation constant, thus can be su-
perposed to represent an arbitrary polarization, including the circularly polarization.
Hence the widely used term the linearly polarized modes are in a way misleading. A
more detailed discussion on this subject can be found in [
43
].
DISCUSSION
21
Figure 1.7
The dispersion curves of the scalar (
1.14
) modes, here
n
W G
= 1.45 and
n
M ed.
= 1.33.
Figure 1.8
The dispersion curves of the vector modes (
1.13
). The waveguide and medium
refractive indices are
n
W G
= 1.45 and n
M ed.
= 1.33, respectively.
22
A FULL-VECTOR COMPLEX MODE SOLVER
Figure 1.9
The degeneracy of modes: the scalar
(1, 1) mode and two vectorial (0, 1)
and
(2, 1) modes. The waveguide and medium refractive indices are n
W G
= 1.45 and
n
M ed.
= 1.33, respectively. The potential barrier is depicted with green color
DISCUSSION
23
In addition we note that the first mode in a cylindrical waveguide, or the core
mode in a single mode fibre, is obtained by assuming that m = 0 in the scalar case
or m = 1 in the vectors case, as can be seen from Figures
1.7
and
1.8
. In the vectorial
case the second mode is observed at m = 0 , with which the so-called LP modes are
associated. Therefore the LP modes can propagate only in the cladding of a single
mode fibre.
In the case when the weakly guided approximation can be applied, each mode can
be obtained by solving either the scalar equation (
1.14
) or vector equation (
1.13
). The
vectorial modes are mostly degenerate. It is interesting to compare radial profiles of
such identical modes. For example, in Figure
1.9
the radial profile of (1, 1) scalar
mode and (0, 1), (2, 1) vectorial modes are shown. All three modes have an identical
propagation constant. We note that the LP modes (0, 1), with the radial and angular
components decoupled, coincide with the (2, 1) mode, with coupled E
and E
components. It is interesting to note that all the modes have the identical radial
profile, even though the scalar mode was obtained by solving the scalar equation at
m = 0 for the E
z
component, where the vectorial modes were solved for E
and E
at m = 1 and m = 2. Each mode
E
E
is normed to unity, hence the difference in
amplitudes between (0, 1) and (2, 1) modes. The (0, 1) mode has only single nonzero
component:
E
0
or
0
E
, whereas in the mode (2, 1) the both components E
and E
are nonzero.
Let us consider the vectorial modes with m = 0. The field components E
and
E
are always coupled in this instance, thus one can not be chosen independently of
another. In the case of weakly guided approximation the modes are mostly degener-
ate. For example, the radial profile of (2, 2) scalar mode and (1, 4), (3, 3) vectorial
modes are shown in Figure
1.10
. Again, we note the similarity in the radial profile
of these modes.
If an exact solution is considered the degeneracy is removed. The modes with a
different m number start to behave differently, diverging apart from each other with
the increase in refractive index difference between the waveguide and the medium,
as shown in Figure
1.11
. The modes (1, 2) and (3, 1), (1, 4) and (3, 3), (1, 6) and
(3, 5) no longer coincide, as in the case of weakly guided approximation. In the
same graph the single modes (1, 1), (1, 3) and (1, 5), without a pair mode, are also
plotted. In general, each mode obtained for a small refractive index difference, splits
into two modes, each with a distinct dispersion curve.
From Figure
1.11
we note that not only the dispersions curves of different mode
families m diverge apart from each other, but even the modes belonging to the same
family start to behave differently, for example the (1, 3), (1, 5) and (1, 7) modes.
This effect can be studied in more detail by plotting the dispersions curves corre-
sponding to different refractive index differences n = n
W G
- n
M ed
between the
waveguide and the medium, as shown in Figure
1.12
for the m = 2 family.
We note that different modes inside the same mode family m, behave differently.
For example, the modes (2, 2) and (2, 4) have barely changed in location on the dis-
24
A FULL-VECTOR COMPLEX MODE SOLVER
Figure 1.10
The degenerate modes: the scalar
(2, 2) mode and two vectorial (1, 4) and
(3, 3) modes. Here the waveguide with n
2
= 1.45 is immersed in water n
M ed.
= 1.33.
persion plane, whereas modes (2, 1), (2, 4) and (2, 5) are highly divergent, especially
at the cutoff region. We discuss this effect for the m = 0 case in more detail later.
We conclude our discussion here by pointing that if m = 0, and the refractive
index difference between the waveguide and the medium is significant, so that the
weakly guided approximation can no longer be applied, the majority of modes splits
into two separate modes, thus the degeneracy is removed. Once this property is
understood we can move on to a more complicated case of m = 0.
If the vectorial equation is solved for the case of m = 0 the degeneracy observed
for a small refractive index difference is removed as was discussed above. But now
not only the modes with different m numbers are distinctly seen as separated (for
example, as was shown in Figure
1.11
), but additionally the modes inside the same
family m = 0 are split into two subfamilies. Thus a single mode is split into three
distinct modes, as shown in Figure
1.13
.
The split between modes belonging to different families m = 0 and m = 2 was
discussed above. Now, we focus our attention to the extra split within the same
family of modes at m = 0. To understand this split we shall go back to the vectorial
DISCUSSION
25
Figure 1.11
The split in dispersion curves between
m = 1 and m = 3 modes.
The waveguide and the medium refractive indices are
n
W G
= 3 and n
M ed
= 1.33,
respectively.
Figure 1.12
The dispersion curves, obtained by solving the exact vectorial equation, for
various refractive index ratios
n = n
W G
- n
M ed.
at
m = 2. The waveguide is immersed
in water
n
M ed.
= 1.33.
equation (
1.21
):
d
2
+
1
d
+ (ln ) d
-
m
2
+1
2
+ k
2
o
+ (ln )
-
j
2m
2
j
2m
2
+
jm
(ln )
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
E
E
=
2
E
E
,
(1.62)
26
A FULL-VECTOR COMPLEX MODE SOLVER
Figure 1.13
The split in dispersion curves between
m = 0 and m = 2 mode families, and
split between the modes inside the same family at
m = 0. The waveguide and the medium
refractive indices are
n
W G
= 3 and n
M ed.
= 1.33, respectively.
which splits into two uncoupled separated differential equations at m = 0:
d
2
+
1
d
+ (ln ) d
+ k
2
o
+ (ln )
E
=
2
E
,
(1.63)
and
d
2
+
1
d
+ k
2
o
E
=
2
E
.
(1.64)
In the weakly guided approximation case the term (ln ) can be neglected, thus equa-
tion (
1.63
) becomes identical to equation (
1.64
), hence the same propagation cases
are obtained from both equations, and hence the corresponding modes are degener-
ate.
However, if the refractive index difference between the waveguide and the medium
is significant, the term (ln ) can no longer be neglected, and thus different propa-
gation constants would result from the solution to equations (
1.63
) and (
1.64
). The
initial degeneracy would be removed.
The result for two modes at m = 0 is shown in Figure
1.14
for a significant re-
fractive index difference. The modes
E
0
and
0
E
are obtained by solving
equation (
1.63
) and (
1.64
), respectively. The split between the corresponding prop-
agation constants is clearly seen. Thus the modes no longer coincide and hence can
no longer be used to represent an arbitrary polarization, such as linear polarization,
therefore the typically used term "LP modes" does not completely reflect the physi-
cal properties of such modes.
DISCUSSION
27
Figure 1.14
The split between the eigenvalues of pure radially polarized
E
and pure angular
polarized
E
modes of the
m = 0 family. The potential barrier is depicted with green color,
the core refractive index
n
2
= 3 is surrounded by the cladding with n
1
= 1.33.
The vector E of the
E
0
mode is orientated radially, i.e. is aligned transver-
sally with respect to the interface along the ^
vector, whereas the vector E of the
mode
0
E
is aligned tangentially to the interface, along the ^
vector. The remain-
ing E
z
component gives a small contribution to both radial and tangential compo-
nents.
The equations (
1.63
) and (
1.64
) can be rewritten in a more elegant Sturm-Liouville
form:
d
d
1
d
d
+ k
2
o
v =
2
v,
(1.65a)
d
d
1
d
d
+ k
2
o
u =
2
u.
(1.65b)
Here v = E
and u = j E
. The equations were derived with the help of (
1.60
).
We note the similarity between equations (
1.65b
,
1.65a
) and the eigenvalue equa-
tions for the TE
1.29
and TM
1.30
modes in the case of slab waveguides:
d
2
dx
2
+ k
2
o
v =
2
v
(1.66a)
d
dx
1 d
dx
+ k
2
o
u =
2
u,
(1.66b)
Here v = E
y
and u = E
x
. Because of this similarity the solutions obtained
form (
1.65a
) and (
1.65b
) equations sometimes called TE and TM modes, respec-
28
A FULL-VECTOR COMPLEX MODE SOLVER
tively. The role of E
x
and E
y
components in a slab waveguide is played by E
and
E
components in the case of a cylindrical waveguide.
Both equations (
1.66b
) for the TM modes in the case of a slab waveguide and the
equation (
1.65b
) for the TMlike modes in the cylindrical waveguide case contain
terms
d
dx
1 d
dx
and
d
d
1
d
d
, respectively, are sensitive to the rate of change
of permittivity. Therefore, we conclude that the TM and TMlike modes should be
particular sensitive to changes in refractive index.
Figure 1.15
The dispersion curves, obtained by solving the exact vectorial equation, for
various refractive index ratios
n = n
W G
- n
M ed.
at
m = 0. The waveguide is immersed
in water
n = 1.33.
The dispersion curves for various index contrasts are shown in Figure
1.15
. In-
deed, it can be clearly seen that the TMlike modes for the E
component are seen
shifting significantly with the increase in index contrast, whereas the TElike modes
for the E
component are almost unmovable. The operator
d
d
1
d
d
is sensi-
tive to changes in the refractive index profile, affecting the E
solution. On the other
hand, the rate of change in the refractive index is not incorporated in the E
solution,
hence if the corresponding propagation constant is plotted along the b =
N
2
ef f
-n
2
2
n
2
1
-n
2
2
axis the corresponding dispersion curve should stay intact.
THE ORTHOGONALITY OF THE BASIS FUNCTIONS
29
1.5
The orthogonality of the basis functions
In the next Chapter we are going to look for a solution to the more general problem
of a waveguide with a small perturbation along the z axis. The modes obtained for
the unperturbed case might be used as a basis function in terms of which the general
solution can be expressed. The calculations can be significantly simplified if the
basis functions are orthogonal. In this section we derive the orthogonality relation in
terms of a weighted product, ensuring the orthogonality of modes.
To our knowledge the orthogonality relation for the vectorial modes of the form
E =
E
E
in cylindrical waveguides with an arbitrary radial refractive index pro-
file is not available in literature.
1.5.1
The orthogonality relation for the scalar modes
Let us start by checking the orthogonality property of scalar modes. The equa-
tion (
1.16
)
d
2
d
2
+
1
d
d
-
m
2
2
+ k
2
o
E
z
=
2
E
z
(1.67)
can be rewritten in the self-adjoint or Sturm-Liouville form [
51
,
52
]:
d
d
p()
d
d
+ q() + w() e() = 0,
(1.68)
that is
d
d
d
d
+
k
2
o
-
m
2
-
2
e() = 0,
(1.69)
where
p()
=
q()
=
k
2
o
-
m
2
w()
=
(1.70)
It can be proven that the eigenfunctions e
k
() obtained by solving equation (
1.68
),
written in the Sturm-Liouville form, are orthogonal with respect to the weighted
function w(), i.e.:
< e
k
|w|e
j
>=
b
a
w()e
k
()e
j
()d =
kj
,
(1.71)
where e
k
() and
k
are the eigenfunctions and eigenvalues of (
1.69
), respectively.
Here we provide a short proof of the fact that the scalar modes obtained from
equation (
1.68
) are indeed orthogonal, and later we will build our original prove for
30
A FULL-VECTOR COMPLEX MODE SOLVER
the vectorial case upon it. Detailed discussions of the scalar case can be found in [
51
]
and [
52
], although a more general proof for the vectorial case was not provided.
We start with the eigenvalue problem written in the Sturm-Liouville form (
1.68
):
Le
k
() =
k
w()e
k
().
(1.72)
where
L =
d
d
p
d
d
+ q.
(1.73)
Considering
Lu
= wu,
Lv
= wv,
(1.74)
We can construct the following expression:
vLu
- uLv = ( - )wuv.
(1.75)
Inserting the operator L and using the Lagrange's Identity [
51
,
52
] we obtain
d
(pud
v
- pvd
u) = (
- )wuv.
(1.76)
After integration over the interval
[a, b]
b
a
d
(pud
v
- pvd
u) =
b
a
( - )w()u()v()d,
(1.77)
we come to the Green's Identity [
51
,
52
]:
p() u()
dv()
d
- v()
du()
d
b
a
= ( - )
b
a
w()u()v()d.
(1.78)
The left hand side vanishes if we consider that the field decays outside the waveguide
boundary and is approximately zero at the numerical boundaries.
u(a)
= v(a) = 0,
u(b)
= v(b) = 0.
(1.79)
Finally we obtain
0 = ( - )
b
a
w()u()v()d.
(1.80)
For non-degenerate modes the above relation can be true only if
b
a
w()u()v()d = 0 , for = .
(1.81)
Thus we have proven the orthogonality relation for the scalar case (
1.69
):
< E
z,
|E
z,
>=
b
a
E
z,
()E
z,
()d = 0 , for = .
(1.82)
In the next section, in a similar manner, we prove the orthogonality relation for the
vector case.
THE ORTHOGONALITY OF THE BASIS FUNCTIONS
31
1.5.2
The orthogonality relation for the vectorial modes
Let us assume that we have a system of equations:
^
L
1
a
b
^
L
2
u
1
u
2
=
w
11
0
0
w
22
u
1
u
2
,
(1.83)
where operators ^
L
1
and ^
L
2
are given in the Sturm-Liouville form:
^
L
1
=
d
d
p
1
()
d
d
+ q
1
(),
^
L
2
=
d
d
p
2
()
d
d
+ q
2
().
(1.84)
Following the derivation in the previous section let us start with two modes:
[M]u = [W ]u,
[M]v = [W ]v.
(1.85)
Our goal here is to find the orthogonality relation between the two modes u and v.
First, let us construct the scalar product:
v
[M]u = v
[W ]u,
(1.86)
here v
= (v
T
)
. Rewriting the above relation in coordinate form we obtain:
(v
1
, v
2
)
^
L
1
g
h
^
L
2
u
1
u
2
= (v
1
, v
2
)
w
11
0
0
w
11
u
1
u
2
,
(1.87)
or
v
1
^
L
1
u
1
+ gv
1
u
2
+ hv
2
u
1
+ v
2
^
L
2
u
2
= v
[W ]u.
(1.88)
Now the expression
v
[M]u - u
[M]v = v
[W ]u - u
[W ]v
(1.89)
can be rewritten in the form
(v
1
u
2
- u
1
v
2
)(g - h) + (v
1
^
L
1
u
1
+ v
2
^
L
2
u
2
) - (u
1
^
L
1
v
1
+ u
2
^
L
2
v
2
) =
= ( - ) (w
11
u
1
v
1
+ w
22
u
2
v
2
) , (1.90)
or rearranging the terms we get:
(v
1
^
L
1
u
1
- u
1
^
L
1
v
1
) + (v
2
^
L
2
u
2
- u
2
^
L
2
v
2
) =
= ( - )(w
11
u
1
v
1
+ w
22
u
2
v
2
) + (v
1
u
2
- u
1
v
2
)(h - g).
(1.91)
32
A FULL-VECTOR COMPLEX MODE SOLVER
Finally, applying the Lagrange's Identity we obtain:
p
1
() v
1
du
1
d
- u
1
dv
1
d
b
a
+ p
2
() v
2
du
2
d
- u
2
dv
2
d
b
a
=
= ( - )
b
a
(w
11
()u
1
()v
1
() + w
22
()u
2
()v
2
())d +
+
b
a
((v
1
()u
2
() - u
1
()v
2
())(h() - g()))d,
(1.92)
Assuming as previously that at the computational boundaries all the field components
vanish, we get the following expression
( - )
b
a
(w
11
()u
1
()v
1
() + w
22
()u
2
()v
2
())d =
=
b
a
((v
1
()u
2
() - u
1
()v
2
())(h() - g()))d.
(1.93)
For the two different modes, i.e. = , the expression
< u
|W |v >=
b
a
(w
11
()u
1
()v
1
() + w
22
()u
2
()v
2
())d = 0
(1.94)
is non zero, thus we can not consider modes u and v to be orthogonal in the gen-
eral case. However, as we will see later, for a small azimuthal number m we can
proximately consider modes to be orthogonal:
< u
|W |v > 0 , for = .
(1.95)
Now, let us apply the orthogonality expression (
1.94
) to a particular case of modes
propagating in cylindrical waveguide of an arbitrary profile (
1.21
):
d
2
+
1
d
+ (ln ) d
-
m
2
+1
2
+ k
2
o
+ (ln ) -
2
-
j
2m
2
j
2m
2
+
jm
(ln )
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
-
2
E
E
= 0
(1.96)
First, let us rewrite equation (
1.96
) in the Sturm-Liouville form (
1.83
). Consider-
ing that
1
r
(ru)
= u +
1
r
u
-
1
r
2
u,
1
r
(r u)
= u +
1
r
u
+ (ln ) u -
1
r
2
u + (ln ) u
(1.97)
we can rewrite the matrix (
5.24
) equation in the following form:
d
d
1
d
d
+
1
-
m
2
2
+ k
2
o
-
2
-
1
j
2m
2
1
j
2m
2
+
jm
(ln )
d
d
1
d
d
+
1
-
m
2
2
+ k
2
o
-
2
E
E
= 0
(1.98)
THE ORTHOGONALITY OF THE BASIS FUNCTIONS
33
Now we can pull and under the E
and E
field components, respectively:
^
L
1
g
h
^
L
2
E
E
=
2
1
0
0
1
E
E
,
(1.99)
here
g()
= -
1
2jm
2
h()
=
1
2jm
2
+ (ln )
jm
(1.100)
^
L
1
=
d
d
1
d
d
+
1
-
m
2
2
+ k
2
o
^
L
2
=
d
d
1
d
d
+
1
-
m
2
2
+ k
2
o
(1.101)
Thus, the eigenvalue equation for cylindrical waveguide (
1.96
) can be written in
the form of equation (
1.83
). Considering that
h()
- g() =
1
2jm
2
+ (ln )
jm
+
1
2jm
2
=
=
jm
2
+ 1
2
+ (ln )
1
= 0 , for m = 0,
(1.102)
a standard orthogonality relation follows form (
1.93
):
< u
|W |v >=
b
a
(w
11
()u
1
()v
1
() + w
22
()u
2
()v
2
())d = 0 , for = .
(1.103)
The equation (
1.103
) approximately holds for small azimuthal numbers m, as we will
see in the next section. The above expression can be rewritten in terms of vectorial
modes E =
E
E
in cylindrical waveguides as follows:
< u
|W |v >=
b
a
(w
11
()u
1
()v
1
() + w
22
()u
2
()v
2
())d =
=
b
a
1
( E
,
)
( E
,
) +
1
(E
,
)
(E
,
) d
(1.104)
Hence, for the same mode family with a relatively small azimuthal number m:
< E
|E
>
b
a
E
,
E
,
+ E
,
E
,
d = 0 , for = .
(1.105)
The above equation is used extensively in the following sections. We note that
the vectorial orthogonality relation (
1.105
) resembles the scalar orthogonality re-
lation (
1.82
). The same weighted function is used in both cases, although the extra
weighted function () is also used to weight the E
component.
34
A FULL-VECTOR COMPLEX MODE SOLVER
1.5.3
Numerical verification
Solving the eigenvalue problem (
1.16
) or (
1.21
) numerically we obtain a set of ba-
sis eigenvectors
{e
k
()}. We can verify the solution by checking whether the ob-
tained set of eigenvectors satisfies the orthogonality relations (
1.82
) for the scalar
case (
1.16
), or (
1.105
) for the vector case (
1.21
).
We note that for a different m number we get a different Sturm-Liouville problem,
with a different set of eigenvalues
{
m
k
} and eigenvectors {e
k
()
m
}. For example,
for the scalar case (
1.16
) we have:
d
(d
) + k
2
o
n
2
o
() -
m
2
-
m
k
R
m
k
() = 0,
(1.106)
here index m denotes the family of modes obtained for a given angular momentum
defined by the m number.
According to the equation (
1.82
) the orthogonality relation is defined as
< R
m
k
|R
m
j
>=
0
R
n
k
()R
m
j
()d,
(1.107)
here we considered the proper boundary conditions on [a, b] = [0,
].
Thus equation (
1.106
) allows us to construct a complete set of basis functions, or-
thogonal in the sense of (
1.107
). As we noted previously, the orthogonality property
will substantially simplify the analysis of a more general problem in the following
chapter.
Now let us construct the overlap matrix
|R
m
j
>< R
n
k
| = C
mn
jk
representing the
overlap integral between j-th and k-th modes, taken from the m-th and n-th families.
|R
m
j
>< R
m
k
| =
0
R
m
k
()R
m
j
()d = C
mm
jk
= 0
if
j = k.
(1.108)
For convenience, let us norm each basis function to unity:
0
|R
m
k
()|
2
d = 1.
(1.109)
We can view the normalization procedure as a construction of a new basis:
R
m
k
() =
R
m
k
()
0
|R
m
k
)
|
2
d
.
(1.110)
The resulting overlap matrix C
mn
jk
is shown in Figure
1.16
. As can be seen, all
the modes belonging to the same family of modes, with the identical m number, are
mutually orthogonal, regardless of the refractive index profile n(). However, if the
overlap integral is computed between modes of different families, for instance R
0
k
()
and R
10
j
(), the result is non zero in most of the cases. The overlap matrix is dense.
Indeed, each family of modes is a solution to a different Sturm-Liouville problem,
uniquely defined by a different potential barrier (
1.35
).
THE ORTHOGONALITY OF THE BASIS FUNCTIONS
35
Figure 1.16
The overlap matrix
C
mm
jk
is computed for SMF-28 fibre immersed in water. The
matrix is an identity for modes belonging to the same family of mode (
C
mm
jk
= I for m = 0
or
m = 10). However, if a different families of modes (m = 0 and n = 10) are considered,
the overlap matrix
C
mn
jk
is dense.
A similar analysis can be conducted for the vectorial case. The orthogonality
relation for a small azimuthal number m can be written in accordance with (
1.105
)
as follows:
C
mm
jk
= |R
m
j
>< R
m
k
| =
b
a
R
m
,j
R
m
,k
+ R
m
,j
R
m
,k
d
0 if
j = k,
(1.111)
The radial function can be normed for convenience:
R
m
j
() =
R
m
j
()
b
a
R
m
,j
R
m
,j
+ R
m
,j
R
m
,j
d
.
(1.112)
The overlap matrix C
mn
jk
is shown in Figure
1.17
.
We note that for the vectorial case the number of modes is approximately doubled
in comparison with the scalar case due to the mode splitting, as was discussed in
Section
1.4
.
As can be seen from Figure
1.17
, the orthogonality relation holds almost exactly
for small azimuthal numbers m. In the next section we will see that we need modes
only from a few first families with a small m number. In addition, we will show that
due to the phase matching condition, the overlap within a family of modes can be
neglected, except for the m = 1 family.
We conclude that radial components of the basis functions are not mutually or-
thogonal for different families of modes, i.e. families with different azimuthal num-
bers, in both scalar and vectorial cases. However, within a particular family m the
radial components can be considered to be mutually orthogonal. In the following
section we will see that although radial components belonging to different families
36
A FULL-VECTOR COMPLEX MODE SOLVER
Figure 1.17
The overlap matrix for the vectorial case is computed for SMF-28 fibre immersed
in water. The overlap matrix
C
mm
jk
, for the modes belonging to the same family
m, is an identity
matrix for
m = 0 or close to the identity for small azimuthal numbers. For different families
of modes, the overlap matrix is dense.
of modes may not necessarily be mutually orthogonal, the complete basis functions
are always orthogonal due to the orthogonality of angular components.
1.6
Conclusion
We conclude this section by stating that the scalar model yields the same result as the
vector model if the refractive index contrast is small and hence the weakly guided
approximation can be applied. However, if the index contrast is high each scalar
mode splits, in general, into two separate modes.
In the vectorial case the first mode, with the highest effective refractive index,
occurs at m = 1 and is single. The remaining modes come in pairs (for exam-
ple (m = 0, m = 2),(m = 1, m = 3), etc.) and are observed as doublets in the spec-
CONCLUSION
37
trum, which is to say have almost identical dispersion curves in the case of a small
refractive index contrast. The modes at m = 0 have an additional split due to the
independence of radial E
and angular E
components, thus triplets should be ob-
served in the spectrum.
It is convenient to distinguish modes with the dominant E
and E
components
and call them TM and TElike modes, respectively. In the case of m = 0 we have the
full analogy with slab waveguides: the TMlike (
1.65b
) and TElike (
1.65a
) modes
in the cylindrical case are described with similar equations as used for TM (
1.30
) and
TE (
1.29
) modes in slab waveguides. The behaviour of TE and TMlike modes in
cylindrical case resembles the behaviour of TE and TM modes in slab waveguides,
which is to say TElike group of modes are relatively unmoved to changes in the
refractive index of external medium. Hence, approximately 50% of modes can be
obtained correctly with the scalar model even in the case of high refractive index
contrast.
It is interesting to note that the so-called LP modes occur at m = 0, where E
and E
components are decoupled. Thus, one of the components can be chosen
independently of the other. Hence, when the refractive index contrast between the
core and cladding is small and two modes coincide, the LP (linearly polarized) or
CP (circularly polarized ) modes can be contracted by superposing the E
and E
field components. However, this procedure is no longer correct if the refractive index
contrast is high. The modes
0
E
and
E
0
become separated and can no longer
be used as a basis for construction of LP or CP modes. The name LP modes can be in
a way misleading if the weakly guided approximation is not valid. In the general case
at m = 0 we have a set modes, coming in pairs, with a slightly different propagation
constants and distinct alignment of the vector E, either radial or tangential. Thus the
energy can be coupled separately into
0
E
or
E
0
modes.
The split between the two LP modes can be viewed as a manifestation of light's
intrinsic degree of freedom. The observed split of degenerate states resembles the
Zeeman effect where the electron energy levels are spit due to the electrons intrinsic
degree of freedom.
We also note that the presented dispersion curves showed that the modes posi-
tioned close to the cutoff region are the most sensitive to the refractive index changes
in the surrounding medium. The largest split between the TElike and TMlike
modes is also observed at some small distance away from the cutoff region, thus we
can particularly target those modes close to the cutoff region to achieve the largest
sensor sensitivity.
CHAPTER 2
MODELING OF TILTED BRAGG
GRATING (TFBG) STRUCTURES
The goal of this chapter is to obtain the exact solution to the problem of light prop-
agating through a cylindrical waveguide with a tilted periodic structure inscribed
along its longitudinal axis.
2.1
Derivation
The problem geometry is shown in Figure
2.1
. First, let us consider the scalar
Helmholtz equation (
1.14
):
(
2
+ k
2
)u(r) = f(r),
(2.1)
here f (r) is the excitation term. Assuming that the excitation is a harmonic function
f (r, t)
f(r)e
jt
, the solution for a linear system should also have a harmonic
function u(r, t)
u(r)e
jt
.
In a cylindrical coordinate system the equation (
2.1
) can be written in the follow-
ing form:
1
(
) +
1
2
2
+
2
z
+ k
2
o
n
2
(, , z) u(, , z) = f(, , z),
(2.2)
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
39
40
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.1
The schematic representation of the problem geometry.
or
Lu(, , z) = f (, , z),
(2.3)
We should note that , and z coordinates are coupled through the refractive index
n = n(, , z) function, dependent on , and z coordinates. Thus we have a
coupled partial differential equation. For a non-tilted grating the coupling would
only occur between z and coordinates, but for a tilted grating all three coordinates
( z, and ) are coupled.
If the coupling is relatively small we can use perturbation theory. First we find
solutions for the case of unperturbed decoupled equation, and then, using these solu-
tions as a basis we solve the coupled problem in terms of this basis.
It is in our interest to keep the coupled term as small as possible. Considering that
the perturbation along the z axis is significantly smaller than the perturbation along
the radial direction, the following decomposition of the refractive index profile can
be written:
n
2
(, , z) n
2
o
() + (, , z),
(2.4)
Now the coupling term (, , z) is as small as possible.
The next step is to find the basis functions e
k
(r) in terms of which we can repre-
sent the solution of the coupled problem. There are several possible ways the basis
function e
k
(r) can be chosen, however it is convenient to use the natural basis of
the radial part L
of the L operator (
2.3
). Such basis functions should satisfy the
orthogonality condition (
1.16
) with respect to the weighting functions. As well, they
should form the complete set of functions, spanning the solution space.
Before we continue let us split the operator L, defined in (
2.2
), (
2.3
), into the
radial and longitudinal parts. With this in mind, we search for a solution in the form
of the series:
u(, , z) =
m
c
m
(, z)e
jm
.
(2.5)
From now on we will use the upper indices to referencing the functions dependent
on the angular coordinate.
DERIVATION
41
Ensuring that the function u(, , z) at = 0 has the same value as at = 2 ,
we conclude that m =
±1, 2, 3, ... for [0, 2].
Plugging the series expansion (
2.5
) into the Helmholtz equation (
2.2
) and consid-
ering (
2.4
) we get:
m
1
(
) -
m
2
2
+ k
2
o
n
2
o
() +
2
z
+ k
2
o
(, , z) c
m
(, z)e
jm
= f(, , z),
(2.6)
or
m
L
m
+
2
z
+ k
2
o
(, , z) c
m
(, z)e
jm
= f(, , z).
(2.7)
Here the operator acting on is denoted as L
m
, and it is the radial part of the L
operator:
L
m
=
1
d
(d
) -
m
2
2
+ k
2
o
n
2
o
().
(2.8)
Finally the basis functions can be found by solving the eigenvalue problem:
L
m
e
m
k
() =
m
k
e
m
k
().
(2.9)
Here e
m
k
() are the eigenfunctions, spanning the solution space, which we are going
to use as the basis function to solve the coupled problem (
2.2
). The eigenvalues are
defined as
m
k
.
Let us assume that the basis functions e
m
k
() were successfully found, and we can
continue solving the initial coupled equation (
2.2
). At this point the coupling appears
through c
m
(, z) and (, , z) terms in equations (
2.5
) and (
2.6
). The coordinates
z and can be decoupled by expanding c
m
(, z) term into a series over the basis
functions e
m
k
(), similarly to what we did in (
2.5
):
c
m
(, z) =
k
C
m
k
(z)e
m
k
().
(2.10)
Upon inserting the series (
2.10
) into (
2.7
) we get:
k
m
L
m
+
2
z
+ k
2
o
(, , z) C
m
k
(z)e
m
k
()e
jm
= f(, , z).
(2.11)
The main advantage of our basis now becomes evident. Let us consider the eigen-
value equation (
2.9
) and replace L
m
e
m
k
() with
m
k
e
m
k
() :
k
m
m
k
+ d
2
z
+ k
2
o
(, , z) C
m
k
(z)e
m
k
()e
jm
= f(, , z).
(2.12)
We are left with the system of coupled ordinary differential equations of only one
variable z.
Let us simplify the following derivation by introducing variable:
m
k
(, ) = e
m
k
()e
jm
,
(2.13)
42
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
thus the equation (
2.12
) becomes:
k
m
m
k
+ d
2
z
+ k
2
o
(, , z) C
m
k
(z)
m
k
(, ) = f(, , z).
(2.14)
The upper and lower indices are used to refer to a particular mode family and a
particular mode in the given family, respectively.
Now let us consider the orthogonality properties of the
m
k
(, ) functions:
2
0
0
n
i
m
k
dd
=
2
0
0
(e
n
i
()e
jn
)(e
m
k
()e
-jm
)dd
=
0
e
n
i
()e
m
k
()d
2
0
e
jn
e
-jm
d
=
nm
ik
ik
2
mn
,
(2.15)
where the norming factor
nm
ik
is defined as:
nm
ik
=< e
n
i
|e
m
k
>=
0
e
n
i
()e
m
k
()d.
(2.16)
Here we have considered the orthogonality relation of harmonic functions and the
basis functions (
1.107
).
The above expression (
2.15
) is non zero if and only if i = k and m = n si-
multaneously, otherwise it is zero, thus we have proved that
m
k
(, ) functions are
orthogonal. Using this property let us take the scalar product of both sides of equa-
tion (
2.14
):
k,m
(
m
k
+ d
2
z
)
n
i
m
k
dd C
m
k
(z) +
k,m
n
i
m
k
k
2
o
dd C
m
k
(z) =
=
n
i
f (, , z)dd,
(2.17)
here
n
i
(, ) = e
n
i
()e
-jn
is the complex conjugate of
n
i
(, ) function.
Finally, considering the orthogonality relation we get the system of differential
equations:
(
n
i
+ d
2
z
)C
n
i
(z) +
k
m
[M
nm
ik
(z)] C
m
k
(z) = F
n
i
(z),
(2.18)
here
M
nm
ik
(z) =
1
2
1
n
i
2
0
0
k
2
o
(, , z)e
n
i
()e
m
k
()e
-jn
e
jm
dd,
F
n
i
(z) =
1
2
1
n
i
2
0
0
f (, , z)e
n
i
()e
-jn
dd,
n
i
=
0
e
n
i
()e
n
i
()d.
(2.19)
DERIVATION
43
Once the system (
2.18
) is solved for C
n
i
(z) functions, the final solution can be con-
structed with help of (
2.5
) and (
2.10
):
u(, , z) =
m
k
C
m
k
(z)e
m
k
()e
jm
.
(2.20)
The system of equations (
2.18
,
2.19
) allow us to model the process of light prop-
agating through a cylindrical waveguide, with a tilted grating inscribed along its
longitudinal axis.
Let us remark here that equations in system (
2.18
) are enumerated by an unique
pair of indices. In analogy with quantum mechanics we can refer to the upper index
as the orbital quantum number, pointing to a particular mode family with a unique
angular momentum; and refer to the lower index as the main quantum number, de-
termining a particular mode in the given family of modes.
In the most general case of vectorial modes (
1.21
) the eigenvalue equation is
defined as follows:
d
2
+
1
d
+ (ln ) d
-
m
2
+1
2
+ k
2
o
+ (ln )
-
j
2m
2
j
2m
2
+
jm
(ln )
d
2
+
1
d
-
m
2
+1
2
+ k
2
o
E
E
=
2
E
E
(2.21)
or
[L
m
]e
m
k
() =
m
k
e
m
k
().
(2.22)
Here = () is considered to be independent of z and variables. The dependence
is included as a small perturbation in the (, , z) function on the next step as it
is described above. The following derivation is identical except that now we shall
consider the vectorial basis functions e
m
k
() =
u
m
k
v
m
k
instead of the scalar basis
function e
m
k
(). As a result, the form of equations (
2.18
) is preserved, but the ma-
trix coefficients are computed differently. The scalar product has to be modified in
accordance with (
1.105
):
nm
ik
=< e
n
i
|e
m
k
>=
0
( ()u
n
i
()u
n
i
() + v
n
i
()v
n
i
()) d.
(2.23)
The remaining steps are analogous to the scalar case.
In this section we showed that the initial problem, defined by the partial differen-
tial equation (
1.14
) or (
1.21
), with coupling along all the three coordinates z, and ,
can be successfully reduced to a system of ordinary differential equations, coupled
only along the z axis. The coupling was introduced through the matrix elements
[M
nm
ik
(z)], varying along the z axis and defined by the grating profile (, , z).
The basis functions e
n
i
() are defined by the equation (
1.106
) and depend on the
waveguide radial profile. We will determine the matrix elements and basis functions
in the following sections.
44
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
2.2
The matrix elements
In this section we compute matrix elements [M
nm
ik
(z)], defined in equation (
2.18
),
for a tilted fibre Bragg grating. Next, energy transfer from the fundamental core
mode into a different set of cladding modes can be determined with the help of the
coupled-mode theory (CMT).
The approach based on the coupled-mode theory (CMT) with application to the
tilted fibre grating was initially introduced by Erdogan and Sipe [
53
], following by a
series of publications presenting numerical results [
54
,
9
]. Alternatively, the method
based on antenna theory and equivalence theorem [
55
], called the volume current
method (VCM), can be implemented as described in [
56
,
57
,
58
], where the scattered
field is represented by radiation of an array of elementary current dipole located
at the index perturbation. Thus, instead of approaching the problem from a pure
mathematical point of view, the physical analogy between a perturbation in refractive
index and an antenna can be established. The comparison between CMT and VCM
methods was presented in [
59
].
The tilted fibre Bragg grating is shown schematically in Figure
2.2
.
Figure 2.2
Illustration of a tilted fibre grating
The grating is specially modulated by a sine function written with fringe planes
that are blazed with respect to the optical axis. We will see later that the tilt of the
grating is the key element for selective and polarization-dependent light coupling.
Figure 2.3
Illustration of the refractive index perturbation along the
z-axis
Assuming a uniform and periodic grating along the z-axis, the refractive index
perturbation in the core can be represented in the following form:
n(, , z) = n
o
() + n(, , z),
(2.24)
THE MATRIX ELEMENTS
45
as shown in Figure
2.3
.
n
2
= n
2
o
+ n
o
n + (n)
2
n
2
o
+ n
o
n
(2.25)
according to (
2.24
)
n
2
= n
2
o
+
(2.26)
thus
(, , z) = n
o
()n(, , z) =
= n
o
()() cos(K
g
z )
(2.27)
Let us assume that the z -axis is the grating axis, tilted at the angle
g
with respect
to the fibre axis z. The fringe planes of the grating are written parallel to the y-axis,
as shown in Figure
2.4
.
The coordinate system Oxyz can be introduced in such a way that the z -axis
transforms into the z-axis by the coordinate system rotation above the y-axis.
Figure 2.4
a) A schematic representation of a tilted Bragg grating inside the fibre, b)
projection onto the
zx and c) xy planes, d) superposition of normal gratings taken at various
angles.
Considering the rotation about the y-axis
z
= x sin(
g
) - z cos(
g
)
(2.28)
46
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
the equation (
2.27
) can be rewritten in the following form:
(z, , ) = n
o
()() cos (K
g
x sin(
g
) - K
g
z cos(
g
)) =
= n
o
()() cos(-K
z
z + K
t
()) =
=
n
o
2
e
-jK
z
z
e
jK
t
()
+ c.c.
(2.29)
Here we considered that x = cos() as shown in Figure
2.4
, and
K
g
=
2
g
,
K
z
= K
g
cos(
g
),
K
t
() = K
x
cos() = K
g
sin(
g
) sin().
(2.30)
It should be noted that K
t
() depends on the angle , as shown in Figure
2.4
d),
and reaches the maximum value along the x-axis:
K
t
max
= K
t
( = ±
2
) = K
g
sin(
g
),
(2.31)
and the minimum value along the y-axis:
K
t
min
= K
t
( = 0, ) = 0,
(2.32)
hence, the grating period for the transverse grating depends on the angle
()
2
K
g
sin(
g
)
,
.
Therefore, mathematically such a grating may be described by superposition of
infinitely many individual gratings written perpendicular to the fibre axis.
Finally, considering (
2.28
) we can rewrite the matrix elements (
2.18
) in the fol-
lowing form:
M
nm
ik
(z) =
1
2
1
n
i
2
0
0
k
2
o
(, , z)e
n
i
()e
m
k
()e
j
(m-n)
dd =
=
1
4
k
2
o
n
i
e
-jK
z
z
0
e
n
i
()e
m
k
()n
o
()()
2
0
e
jK
t
()
e
j
(m-n)
d d + c.c.
=
1
4
k
2
o
n
i
e
-jK
z
z
0
e
n
i
()e
m
k
()n
o
()()
mn
()d + c.c.
(2.33)
Here
mn
() is the function dependent only on (a cylindrically symmetric func-
tion):
mn
() =
2
0
e
jK
t
()
e
j
(m-n)
d =
=
2
0
e
jK
g
sin(
g
) sin()
e
j
(m-n)
d =
= 2(-1)
k
J
k
(),
(2.34)
THE MATRIX ELEMENTS
47
here J
k
() - are Bessel function of first kind, k = m - n, and = K
g
sin(
g
).
Hence, in the case of a tilted grating the orthogonality between the modes is bro-
ken due to the n
o
()J
k
() term, which arises via the perturbation caused by the
tilted grating.
We are mainly interested in the energy transfer from the core modes into the
cladding modes, considering the weighted function J
k
() and the fact that the core
mode is mainly confined inside the core, and the grating perturbation also exists
inside the core, we conclude that the coupling is possible only to a limited number
of higher order azimuthal mode families, with azimuthal number m < 12, as shown
in Figure
2.5
.
If the grating in non-tilted, the equation (
2.34
) is reduced to:
mn
() =
2
0
e
j
(m-n)
d = 2
mn
,
(2.35)
thus, the modes belonging to different families are mutually orthogonal, and hence,
the energy transfer between such modes is impossible.
We conclude that the expression for coupling coefficients (
2.18
) is reduced to ex-
pression describing orthogonality between radial components of modes, except that
we have an extra weighted function n
o
()
mn
(). The extra weighted function not
only breaks orthogonality between modes within a particular family, but also be-
tween modes belonging to different families of modes with different azimuthal num-
bers m. We note that the radial components of modes belonging to different families
were not orthogonal initially, as shown in Figure
1.16
, but the complete modes with
the angular part included, in the case of a non-tilted grating, were orthogonal due
to (
2.35
).
According to orthogonality relation (
1.105
), for the vectorial case the term
e
n
i
()e
m
k
() in equation (
2.33
) should be replaced with:
e
n
i
()e
m
k
() = ()u
n
i
()u
m
k
() + v
n
i
()v
m
k
(),
(2.36)
here e
m
k
() =
u
m
k
v
m
k
are the eigenvectors of non-perturbed problem (
1.21
).
Whereas in the scalar case, in accordance with (
1.111
), we have:
e
n
i
()e
m
k
() = e
n
i
()e
m
k
(),
(2.37)
with eigenvectors e
m
k
() for the scalar eigenvalue problem (
1.16
).
Finally, considering the complex conjugate part c.c. , and assuming that all the
coupling coefficients are known, we can rewrite equation (
2.33
) in the simple form:
M
nm
ik
(z) = cos(K
z
z)[ ^
M
nm
ik
],
(2.38)
here the number ^
M
nm
ik
defines overlap (or coupling) between two modes: (n, i)
and (m, k), as it follows from equation (
2.33
). All such coupling coefficients are
assembled into the matrix [ ^
M
nm
ik
].
48
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.5
The weighted function
J
m
() caused by the grating tilt, calculated for = 0
o
and
= 10
o
grating tilt angles. The grating period is assumed to be
G
= 0.6 m, thus for
= 0
o
we get
= K
g
sin(0) = 0, and for = 10
o
we get
= 1.81 m
-1
. The number
m = m
1
- m
2
is the difference between azimuthal order of the first
m
1
and the second
m
2
mode families.
Let us compute coupling coefficients C
m
k
between the core and cladding modes
(here m is the mode family with azimuthal number m, and k is the mode order in the
family). The results are presented in Figures
2.6
,
2.7
for 2
o
degree, Figures
2.8
,
2.9
for 4
o
degree and in Figures
2.10
,
2.11
,
2.12
for 10
o
degree gratings. We note that it
is sufficient to consider only 4, 5 and 9 families of modes for 2
o
, 4
o
and 10
o
degree
tilted gratings, respectively.
THE MATRIX ELEMENTS
49
In the following section we apply the coupled mode theory to compute the spectral
response of the gratings. The computed coupling coefficients C
m
k
are going to used
as input parameters.
Figure 2.6
The
2
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
50
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.7
The
2
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
THE MATRIX ELEMENTS
51
Figure 2.8
The
4
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
52
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.9
The
4
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
THE MATRIX ELEMENTS
53
Figure 2.10
The
10
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
54
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.11
The
10
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
THE MATRIX ELEMENTS
55
Figure 2.12
The
10
o
degree grating assisted coupling coefficients between the core and
cladding modes,
n = 10
-4
.
56
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
2.3
Coupled-mode theory,
the two modes approximation
The coupled mode theory was employed in electromagnetics in the early 1950's,
and was initially applied to microwave travelling wave devices [
60
,
61
,
62
]. In the
early 1970's the coupled mode theory had been successfully applied to the modeling
of various guided wave optical systems. The noticeable papers were published by
Kogelnik [
63
,
64
], Snyder [
65
], Yariv [
66
], Marcuse [
67
] and others. The application
to parallel waveguides can be found in [
68
,
69
,
70
].
In this section we continue our consideration of the cylindrical waveguide with
a periodic grating inscribed in its core. The grating allows the transfer of energy,
initially confined in the fibres core, outside the core into the cladding modes. It is also
known that in our particular case the contra-directional phase matching condition
occurs, as we will explain it later.
Let us start with equation (
2.18
). Assuming that there is no external excitation we
get:
(
n
i
+ d
2
z
)C
n
i
(z) +
k
m
[M
nm
ik
(z)] C
m
k
(z) = 0.
(2.39)
For convenience we enumerate modes and coefficients with single index, e.g. each
pair of indices (i, n) we will be enumerated with a single index , thus we can
rewrite (
2.18
) as
(
+ d
2
z
)C
(z) = -
[M
(z)] C
(z).
(2.40)
If the perturbation can be neglected, i.e. the right hand side of (
2.40
) can be can-
celed [M
(z)] = 0, and hence the solutions would be C
(z) = A
e
i
z
, where
A
are some constants. If the perturbation is switched on, the constants should be re-
placed with a slowly varying functions A
= A
(z) along the z coordinate [
63
,
66
].
C
(z) = A
(z)e
i
z
=
= A
(z)e
i
z
,
(2.41)
here we denoted
:=
, called the propagation constants.
The final solution to (
2.20
) can be written in the following series form:
u(, , z) =
m
k
A
m
k
(z)e
m
k
()e
jm
e
i
z
.
(2.42)
Inserting (
2.41
) into (
2.40
), and considering that
d
2
z
C
(z) = d
2
z
A
(z)e
j
z
= e
j
z
[-
+ j2
d
z
+ d
2
z
]A
(z),
(2.43)
we obtain
e
j
z
[j2
d
z
+ d
2
z
]A
(z) = -
[M
(z)] A
(z)e
j
z
.
(2.44)
COUPLED-MODE THEORY,THE TWO MODES APPROXIMATION
57
The matrix coefficients M
(z) are defined in accordance with (
2.38
):
[M
(z)] = [M
] cos(K
z
z)
(2.45)
The equation (
2.44
) can be rewritten in the following form:
[j2
d
z
+ d
2
z
]A
(z) = -
[M
] A
(z) cos(K
z
z)e
j
(
-
)z
.
(2.46)
Now let us consider the oscillating terms:
cos(K
z
z)e
j
(
-
)z
=
1
2
e
jK
z
z
+ e
-jK
z
z
e
j
(
-
)
=
1
2
e
j
(
-
+K
z
)
+ e
j
(
-
-K
z
)
(2.47)
The matrix terms oscillate periodically and rapidly unless the phase is close to
zero, in which case the power coupled between modes accumulates coherently and
gives rise to a significant power exchange between the modes. Thus we can neglect
all the matrix terms except a few terms that have a small or zero phase = 0 under
the exponential functions [
71
,
66
,
50
].
Considering (
2.47
) and assuming that the energy couples from the core mode
with the propagating constant
, to one of the cladding modes with
propagating
constant, there are two possible phase matching conditions:
= K
z
+
-
= 0,
= K
z
+
-
= 0,
(2.48)
called co-directional and contra-directional coupling, respectively.
From the experiments described in the following sections we know that in our case
only the contra-directional coupling occurs, thus we limit ourselves to this particular
case. Figure
2.13
shows the phase matching condition, or momentum diagram, of
contra-directional coupling.
Figure 2.13
The momentum diagram of contra-directional coupling (here
1
corresponds
to the core mode and
2
to the cladding mode).
Now let us make a few extra assumptions. First we assume that instead of con-
sidering coupling between all the modes at once, we can consider only two modes,
the core mode and one of the cladding modes. This assumption is useful to un-
derstand the basic properties of energy transfer from the core mode to a particu-
lar cladding mode. Once the energy is transfered to a particular cladding mode it
58
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
can not be recoupled to an other cladding modes due to the significant phase mis-
match. We also assume that the amplitudes A
(z) in (
2.41
) are "slowly" varying,
i.e. d
2
z
A
(z)
d
z
A
(z), this is the so-called slowly varying amplitude approxi-
mation [
63
,
66
]. The resulting equations (
2.46
) will takes the following form:
j2
1
d
z
A
1
(z) = -e
-jz
A
2
(z),
j2
2
d
z
A
2
(z) = -
e
jz
A
1
(z).
(2.49)
Here:
=
-K
z
+
1
-
2
0 is the phase mismatch. We should keep in mind that if the
propagation constant
1
is positive then, due to the contra-directional coupling the
2
constant should be taken with the minus sign.
= [M
12
] = [M
21
]
=<
1
||
2
> is the coupling constant, and can be computed
in accordance with equation (
2.18
).
We note, by looking at the equation (
2.47
), that the off-diagonal matrix elements
have phases approaching zero in the case when the phase matching condition is sat-
isfied, thus indeed we have condition that allows for coherent power accumulation,
which leads to the power redistribution between the modes.
Let us solve equations (
2.49
) assuming that the phase matching condition is satis-
fied = 0, and the proper boundary conditions are given. Considering that the core
mode with the amplitude A
1
(z) to be incident at z = 0 on the perturbation region
z
[0, L] we can set A
1
(0) = 1. The cladding mode A
2
(z) is "generated" by the
perturbation, hence A
2
(z) should be set to zero at the end of the perturbation region
A
2
(L) = 0. The solution of (
2.49
) (for the case of = 0) is given by equations (
2.50
)
and mode power of the incident and scattered waves are shown in Figure
2.14
.
A
1
(z) = cosh
(z
- L)
2
1
2
cosh
-1
L
2
1
2
,
A
2
(z) = i
1
2
sinh
(z
- L)
2
1
2
sinh
-1
L
2
1
2
.
(2.50)
Now let us consider the amplitudes A
1
and A
2
at the perturbation boundary. Set-
ting as previously A
1
(0) = 1 and A
2
(L) = 0 we get for the general case of non zero
phase detuning ( = 0 ) the following solution:
A
1
(L) =
e
-iL
2
cosh(L
2
) - i sinh(L
2
)
,
A
2
(0) =
-
ik
2
sinh(L
2
)
cosh(L
2
) - i sinh(L
2
)
.
(2.51)
Here
=
2
1
2
-
2
,
= -K
z
+
1
-
2
,
(2.52)
COUPLED-MODE THEORY,THE TWO MODES APPROXIMATION
59
Figure 2.14
The transfer of power between the incident core mode
A
1
(z) and back-scatters
cladding mode
A
2
(z) in the case of contra-directional coupling. Here
kL
= 2.4
here L is the perturbation length,
1
and
2
are the propagation constants of the core
and cladding modes, respectively, as shown in Figure
2.13
.
We have a particular interest in the energy loss due to the coupling from the core to
the cladding modes, as this energy loss can be observed experimentally by measuring
transmission spectra of the optical system:
(x)
= |
A
1
(L, x)|
2
|A
1
(0, x)|
2
=
=
2
(x)
2
(x) cosh
2
(C
(x)
2
) + x
2
sinh
2
(C
(x)
2
)
,
(2.53)
here C is the coupling parameter defined as C := L, x :=
is the phase detuning
parameter, := /
1
2
and (x) :=
=
1 - x
2
.
The power loss (x) function of the core mode due to the coupling to the cladding
mode is shown in Figure
2.15
for various coupling parameters C = L and the phase
mismatch x =
.
In the case of coupling to the lossy mode we note that the peak becomes broader
and the base line is shifted, as shown in Figure
2.16
.
60
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.15
The power loss of the core mode as a function of the phase mismatch, for
various coupling parameters
C = L.
Figure 2.16
The power loss of the core mode as a function of the phase mismatch, for the
fibre waveguide coated with a thin film (
h = 100 nm) made of lossy material (blue curves)
a)
N
ef f
= 1.3661 - j6.1165 × 10
-4
, and b)
N
ef f
= 1.3669 - j2.2791 × 10
-5
. The red
curves, used as a reference, corresponds to the non-lossy material (
N
ef f
= 1.3661 ).
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
61
2.4
Coupling between the core mode and many cladding modes in the
TFBG
In this section we apply the coupled mode theory to our problem of interest: the
tilted fibre Bragg grating (TFBG) inscribed inside the core of a standard telecommu-
nication fibre SMF-28.
The phase matching condition in such a case is schematically shown in Fig-
ure
2.17
, where the core mode is coupled to a number of cladding modes.
Figure 2.17
The phase matching condition in TFBG
In the case of TFBG the energy confined inside the fibres core is coupled from
the core mode into a multiplicity of cladding modes, as shown schematically in Fig-
ure
2.18
. The grating-assisted coupling, for the problem of interest, is only possible
for the core and cladding modes propagating in opposite directions. It should also
be noted that for the given geometry and operational wavelength the energy cou-
pling between different cladding modes is prohibited due to the significant phase
mismatch.
If a spectrum of TFBG grating was obtained from an experiment, the grating
wavenumber K
G
can be expressed in terms of the measured value of the Bragg
resonance position
B
, by assuming that the forward propagating wave in the core is
coupled to the backward propagating wave in the core:
K
G
= 2
B
= 2N
B
2
B
.
(2.54)
The phase detuning condition condition (
2.48
) takes the following form:
k
j
() = K
G
- (
B
() +
j
()) =
= 2 2
N
B
B
-
N
B
+ N
j
()
,
(2.55)
62
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.18
The schematic ilustration of energy transfer from the core mode into the
cladding modes in the TFBG. The various potential barriers correspond to different azimuthal
numbers
m.
with
B
() = N
B
k
o
= N
B
2
,
j
() = N
j
()k
o
= N
j
()
2
.
(2.56)
Here
B
and
j
are propagation wave numbers of the core mode, with the effective
refractive index N
B
, and the j-th cladding mode with the corresponding effective re-
fractive index N
j
, computed at the operational wavelength . We have also assumed
that N
() constant approximately does not depend on the , which is true if the
radius of the core is significantly smaller then the radius of the cladding.
The Bragg Condition is obtained by assuming that there is no phase detuning
between the modes, i.e. k
j
0. Hence the position of the j-th resonance is defined
by the following equation:
res
j
=
1
2
1 +
N
j
()
N
B
B
(2.57)
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
63
Assuming the operational wavelength and considering (
2.55
) we can determine
the phase detuning for the j-th resonance in the proximity to
j
:
j
(
j
- ) = 2 2
N
B
B
-
N
B
+ N
j
()
j
- 2 2
N
B
B
-
N
B
+ N
j
()
=
= 2(N
B
+ N
j
()) -
1
j
+
1
2(N
B
+ N
j
())
-
j
2
.
(2.58)
Figure 2.19
The resonances of different families of modes, computed in accordance
with (
2.57
). The peaks arising due to the coupling between the core mode and
m = 0
family of modes, neglecting the coupling to higher family of modes. Here the grating length
L = 10mm and the coupling constant C = 2 · 10
-4
.
Using the two mode approximation and considering the dephasing expression (
2.58
)
the resonances of a particular family of modes, let us say for m = 0, can be plotted
along the wavelength of operation , as shown in Figure
2.19
. For a single family of
modes the resonance peaks have almost no overlap (Figure
2.19
), thus a resonance
can be approximately computed with the two mode approximation independent of
the other resonances. In other words, the energy coupling between the core mode
and a particular cladding mode can be computed without considering other cladding
resonances of the same family of modes.
However, in the case of TFBG, many resonances, corresponding to various fam-
ilies of modes, are present in close proximity to each other. In Figure
2.19
, reso-
nances of the first 11 families of modes are plotted along with the peaks computed
for m = 0 family. Although the two mode analysis is useful to study basic properties
of the problem, it can not be applied to our problem of interest due to the overlap
between the resonances.
Going back to the initial coupled modes equation (
2.46
) and, as previously, ne-
glecting the rapidly oscillating terms, not contributing to the change of amplitudes,
64
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
we obtain the following system of coupled equations:
d
z
A
core
(z) =
N
k
=1
i
C
k
2
core
e
-i
k
z
A
clad
k
(z),
d
z
A
clad
k
(z) = -i
C
k
2
clad
k
e
+i
k
z
A
core
(z).
(2.59)
Here C
k
=<
core
||
clad
k
> is the coupling constant between the core and the
k-th cladding mode, as was discussed in the previous section; N is the number of
coupled modes. The values of
k
(),
clad
k
(), () and C
k
() have to be computed
at a particular operational wavelength .
The analytical solution, similar to (
2.50
), can no longer be applied to the system
of equations (
2.59
). Instead this system of coupled nonlinear differential equations
can be solved numerically.
In the case of co-directional coupling the initial value problem (IVP) is considered
A
core
(0) = 1 and A
clad
k
(0) = 0, hence the system of equations can be directly inte-
grated. Unfortunately, in our case of contra-directional coupling the boundary value
problem (BVP) A
core
(0) = 1 and A
clad
k
(L) = 0 has to be considered (as the bound-
ary conditions are known at the opposite sides of the interval), and this significantly
complicates the numerical routine. Likely the problem (
2.59
) is linear with respect to
the z variable, hence we can start propagation from the opposite end z = L assuming
that A
clad
k
(L) = 0 and setting amplitude of the core mode at z = L to some arbitrary
constant C to be determined later: A
core
(L) = C. Next the system of equations can
be integrated as an initial value problem. Once the solution A
core
(0) at z = 0 is
known we can renormalise the solution to ensure that A
core
(0) = 1. The procedure
is shown in Figure
2.20
.
Figure 2.20
Transforming the boundary value problem into the initial value problem. a)
solving IVP by assuming
A
core
(L) = 1 and A
clad
(L) = 0, b) renormalized the solution by
setting
A
core
(0) = 1.
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
65
The initial value problem is next solved with the help of the standard fourth or-
der Runge-Kutta method. The result for the core mode coupled to the two closely
positioned cladding mode resonances is shown in Figure
2.21
.
Figure 2.21
Coupling between the core mode and two cladding modes. The grating length
L = 10 mm and the coupling constant are C
1
= 2 · 10
4
and
C
2
= 1 · 10
4
between the core
and the two cladding modes. Here
A
1
(z) is the amplitude of the forward propagating core
mode (the red line),
A
2
(z) and A
3
(z) are amplitude of the backward propagating cladding
modes (the blue and the green lines).
In Figure
2.21
a) the difference between the two mode approximation, when the
coupling to cladding modes is considered independently (the green and the blue
curves), and the exact approach (the red curve) is clearly seen. It is also interest-
ing to notice that the energy from the cladding mode Figure
2.21
c) (the blue curve)
can be recoupled back to the core mode (the red curve).
Finally, the coupling problem for the TFBG structure of our interest can be solved
as follows:
1. We introduce a grid vector
j
spanning the operation range of the sensor, i.e.
j
[
min
,
max
], where at each point
j
the system of coupled equations (
2.59
)
has to be solved. We note that a typical peak width is about 0.2 nm, where the
operational range is about 100 nm. Thus, at least 1000 points
j
have to be
66
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
considered to observe the resonances. Considering the necessity of solving the
system of coupled differential equations (
2.59
) at each point, the described ap-
proach can be extremely time consuming. We can overcome this difficulty by
introducing a nonuniform grid. The positions of resonances is known (
2.57
):
res
j
=
1
2
1 +
N
j
()
N
B
B
,
(2.60)
hence we can introduce a finer mesh in the vicinity of resonances, as shown in
Figure
2.22
.
Figure 2.22
The non-uniform grid with a finer mesh in the vicinity of resonances.
2. Next the detuning parameter
j
for each resonance can be computed in accor-
dance with (
2.58
):
j
() 2(N
B
+ N
j
())
-
j
2
.
(2.61)
3. Unfortunately the dispersion of modes can not be neglected, as can be seen from
Figure
2.23
. Hence, we have to find modes and coupling constants at each point
j
of the grid, or at least we can split the interval [
min
,
max
] into a set of
smaller subintervals and compute resonances and coupling coefficients for each
of them, considering them to be constant within the subintervals.
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
67
Figure 2.23
The dispersion of modes. The resonances of
m = 0 family of modes are
computed at
1
= 1.5 m and
2
= 1.6 m operational wavelengths.
4. Finally the system of coupled nonlinear differential equations (
2.59
) can be
solved independently at each point
j
of the grid:
d
z
A
core
(z,
j
) =
N
k
=1
i
C
k
(
j
)
2
core
(
j
)
e
-i
k
(
j
)z
A
clad
k
(z,
j
),
d
z
A
clad
k
(z,
j
) = -i
C
k
(
j
)
2
clad
k
(
j
)
e
+i
k
(
j
)z
A
core
(z,
j
).
(2.62)
The number of coupled modes N can be limited to 15 modes by choosing only
the modes with the smallest phase mismatch
k
(
j
). The remaining modes can
be neglected due to a weak coupling. For each
j
the choice of N coupled
modes should be reconsidered.
The theoretically computed transmission spectra along with the experimentally
measured spectra are shown in Figures
2.24
and
2.25
for the 2
o
degree grating, in
Figures
2.27
and
2.28
for the 4
o
degree grating and in Figures
2.31
and
2.32
for the
10
o
degree grating. A more detailed information can be obtained by zooming in to
particular resonances, which are shown in Figures
2.26
for the 2
o
degree grating, in
Figures
2.29
,
2.30
for the 4
o
degree grating and in Figures
2.33
,
2.34
,
2.35
for the
10
o
degree grating. The coupling coefficients are plotted on the same figures.
We note that the resonances have a similar structure along the whole operational
range of the sensor. We also note the an interesting effect of peaks alternation. There
are only two group of peaks: the peaks consisting of modes with odd azimuthal sym-
metry m = 1, 3, 5..., and modes with even azimuthal symmetry m = 0, 2, 6... . In
the spectral area where peaks are separate, these two groups of peaks are alternating.
This is an important effect explaining some of the experimental data, to be discussed
later.
68
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.24
The experimentally measured spectra of the
2
o
degree
1 cm long grating.
Figure 2.25
The theoretically computed transmission spectra of
2
o
degree grating. (
L =
1 cm, n = 10
-4
)
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
69
Figure 2.26
The fine structure of the particular resonances and corresponding coupling
coefficients of
2
o
degree grating.
70
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.27
The experimentally measured spectra of the
4
o
degree
1 cm long grating.
Figure 2.28
The theoretically computed transmission spectra of
4
o
degree grating. (
L =
1 cm, n = 10
-4
)
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
71
Figure 2.29
The fine structure of the particular resonances and corresponding coupling
coefficients of
4
o
degree grating.
Figure 2.30
The fine structure of the particular resonances and corresponding coupling
coefficients of
4
o
degree grating.
72
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.31
The experimentally measured spectra of the
10
o
degree
1 cm long grating.
Figure 2.32
The theoretically computed transmission spectra of
10
o
degree grating. (
L =
1 cm, n = 10
-4
)
As can be seen from the presented figures the developed theoretical model pro-
vides results in good agreement with the experimental measurements. However, the
small difference is present in the area of the so-called ghost modes [
11
] for 4
o
degree
tilted grating. This area is highly populated with resonances, thus our approximation
where only 15 coupled modes are considered might not be sufficient. It also should
be noted that we presume an ideal grating. In reality a physical grating might be
COUPLING BETWEEN THE CORE MODE AND MANY CLADDING MODES IN THE TFBG
73
Figure 2.33
The fine structure of the particular resonances and corresponding coupling
coefficients of
10
o
degree grating.
Figure 2.34
The fine structure of the particular resonances and corresponding coupling
coefficients of
10
o
degree grating.
subjected to various unaccounted-for effects, such as a non-uniform and one sided
illumination by the UV beam in the process of grating inscription. Nevertheless,
from the point of view of practical application, we are interested in the spectral area
with distinct resonances. This region is simulated in good agreement with the mea-
surements.
74
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.35
The fine structure of the particular resonances and corresponding coupling
coefficients of
10
o
degree grating.
POLARIZATION-DEPENDENT COUPLING
75
2.5
Polarization-dependent coupling
In this section we discuss polarization-dependent properties of the TFBG. Consider-
ing that the core mode has m = 1 azimuthal symmetry it can be orientated differently
in the plane transverse to the propagation axis. The tilt of the grating planes breaks
the cylindrical symmetry of the fibre, and defines the reference frame x
- y in which
it is convenient to analyze the system. Considering the mutual orientation of the grat-
ing planes and the incident core mode, we can expect that the coupling coefficients
might depend on the relative orientation between the polarization (or orientation) of
the incident core mode and relative orientation of the grating planes, as shown in
Figure
2.36
.
Figure 2.36
The schematic representation of the TFBG grating and the incident linearly
polarized core mode (the
E
component is shown). The core mode is rotated about the optical
device axis by some angle
. The grating is tilted by angle about the x axis.
The coupling coefficients C
nm
ik
in accordance with (
2.33
) are proportional to
C
nm
ik
( )
0
R
n
i
()R
m
k
()n
o
()()
mn
(, )d + c.c.,
(2.63)
here is the angle at which linearly polarized light is incident on the TFBG structure
and
mn
(, ) is the weighted function, shown in Figure
2.37
:
mn
(, ) =
2
0
e
jK
g
sin(
g
) sin(+)
e
j
(m-n)
d
(2.64)
As it shown in Figure
2.37
the weighted function
mn
(, ) corresponding to
coupling between the modes of the same family (in our case the core mode has the
azimuthal number m = 1 is coupled to the m = 1 cladding modes) is not affected by
76
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.37
The weighted function
mn
(, ) in the fibre core, for the 4
o
degree tilted
grating.
the incident angle of linearly polarized light, whereas coupling to different families
of modes, with the m
- n = 1, 2, 3, reveals a significant angle dependence.
Hence, considering the weighted function
mn
(, ), the polarization-dependent
coupling coefficients can be computed. For instance for the 4
o
degree tilted grating
the coupling coefficients are shown in Figure
2.38
for = 0
o
, 45
o
, 90
o
polarization
angles of the incident light.
Now let us study the polarization dependence of particular resonances in more de-
tail. As we mentioned in the previous section there are only two alternating groups
of resonances with either odd or even azimuthal symmetry, thus it is sufficient to an-
alyze two closely positioned resonances. The results for 4
o
and 10
o
degrees gratings
are shown in Figure
2.39
and Figure
2.40
, respectively.
It is clearly seen that the resonances have a complicated inner structure. Sev-
eral coupling coefficients of various polarization-dependent behaviors are bounded
together by the system of coupled mode equations. We note that by changing the
polarization of incident light the dominant coupling coefficient can be changed, thus
POLARIZATION-DEPENDENT COUPLING
77
Figure 2.38
The coupling coefficients
C
k
() for the 4
o
degree TFBG computed for
= 0
o
, 45
o
, 90
o
polarization angles of the incident light.
the energy can be predominately coupled to a particular mode with a specific field
distribution at the sensor surface.
78
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.39
The polarization dependence of coupling coefficients corresponding to two
resonances of the
4
o
degree TFBG.
POLARIZATION-DEPENDENT COUPLING
79
Figure 2.40
The polarization dependence of coupling coefficients corresponding to two
resonances of the
10
o
degree TFBG.
80
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Let us now assemble the density plot by computing the TFBG spectra at each
angle of linear polarized light (the angle of rotation about the optical axis). First
we compute the coupling coefficients for cladding modes of m = 0, 1, 2, 3, 4, 5 az-
imuthal symmetry (as was previously shown the six first families of cladding modes
are sufficient to accurately compute the spectrum of 4
o
degree TFBG).
Figure 2.41
The coupling coefficients between the core and cladding modes of azimuthal
order
m = 0, 1, 2, 3, 4, 5 computed at various angles of linearly polarized light incident at the
4
o
degree TFBG.
The coupling coefficients of various m azimuthal numbers plotted on the same
figure are shown in Figure
2.42
.
Applying the developed in the previous chapter method we compute the corre-
sponding transmission spectra for each angle of linearly polarized light incident at
the TFBG structure. The result of the computation is shown in Figure
2.43
, the
experimental measured spectra at various polarization angles are shown Figure
3.1
.
POLARIZATION-DEPENDENT COUPLING
81
Figure 2.42
The coupling coefficients of the
4
o
degree TFBG computed at various angles
of linearly polarized light.
Figure 2.43
The transmission spectra of the
4
o
degree TFBG, computed at various angles
of linearly polarized light.
82
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
2.6
The electric field distribution at the fibre boundary
In the following Chapters we discuss the possible ways of improving the TFBG sen-
sor sensitivity by depositing nanoparticles on the sensor surface. The light particle
interaction is dependent, among other parameters, on the incident electric field ori-
entation. In this section we discuss the electric field orientation at the fibre surface.
The developed vectorial mode solver allows us to determine E
and E
field com-
ponents for each of the modes. In Figure
2.44
and Figure
2.45
the field components
E
and E
are plotted along the effective refractive index of the modes with different
azimuthal symmetries m = 0, 1, 2, 3, 4, 5. We note that by default the field of each
mode is normalized to unity.
As can be seen from Figure
2.44
and Figure
2.45
the modes with a high effec-
tive refractive index are almost completely bounded inside the waveguide, with al-
most zero field at the fibre surface, whereas fields with low effective refractive index
modes leak outside the waveguide. This effect can be seen in more detail if the field
is computed at some distance from the fibre. In Figure
2.46
the field components
E
and E
are computed at the distance of 1 nm away from the fibre boundary. We
note that these modes with low effective refractive index have long exponential tails
penetrating deep into the surrounding medium, these are the so-called evanescent
waves. The long exponential tail of the modes is the reason for high sensitivity of
the sensor in the region close to the cutoff.
The Figures
2.44
,
2.45
show components of the electric field normalized to unity.
However, the field should depend on the energy coupled to a particular mode, as each
mode is excited with different strength. The energy (or light) coupled to a particular
k-th mode is proportional to the corresponding coupling coefficient C
k
. Thus the
resulting field at the sensor surface can be obtained by multiplying the electric field
components E
and E
of the k-th mode by the C
k
coupling coefficient. The result
is shown in Figure
2.47
for various states of linearly polarized light incident on the
4
o
degree TFBG structure.
THE ELECTRIC FIELD DISTRIBUTION AT THE FIBRE BOUNDARY
83
Figure 2.44
The field components
E
and
E
computed at the fibre boundary for modes
with
m = 0, 1, 2 azimuthal symmetry.
84
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.45
The field components
E
and
E
at the fibre boundary for modes
with
m = 3, 4, 5 azimuthal symmetry.
THE ELECTRIC FIELD DISTRIBUTION AT THE FIBRE BOUNDARY
85
Figure 2.46
The field components
E
and
E
computed at
1 nm distance away from the
fibre boundary.
86
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.47
The intensity of electric field components
E
and
E
at the fibre boundary for
4
o
degree TFBG, computed at various states of linearly polarized light
P = 0
o
, 45
o
, 90
o
.
THE ELECTRIC FIELD DISTRIBUTION AT THE FIBRE BOUNDARY
87
Let us zoom in on a few particular resonances. As mentioned previously each
resonance consists of modes of either odd or even azimuthal symmetry, as shown in
Figure
2.48
.
Figure 2.48
The structure of the particular resonances of
4
o
degree TFBG.
Multiplying the electric field components E
and E
at the fibre surface by the
corresponding coupling coefficients we get the value of electric field present at the
fibre surface, shown in Figure
2.49
.
We conclude by stating that the electric field at the fibre surface is predominantly
radially polarized (with electric field component normal to the fibre surface) in the
wide spectral region, as can be inferred from Figure
2.47
. However, the composi-
tion of resonances is complex (Figure
2.49
) and consists of many modes with radial
and tangential dominant polarization. Moreover, the composition depends on polar-
ization of the incident light, as shown in Figure
2.47
. Thus, by changing the core
mode polarization, the coupling coefficient can be changed, and hence the energy
couples to a particular mode with a specific field distribution at the sensor surface
can be changed as well. The spectral response of the TFBG sensor is polarization-
dependent, as shown in Figure
2.43
. The electric field at the sensor surface is po-
larized, with orientation of E dependent on the core mode polarization. Hence the
spectral response of the TFBG sensor can be further enhanced by coating its surface
with a layer, the optical properties of which are polarization-dependent.
In the following Chapter
3
we discuss experimental measurement techniques tak-
ing advantage of the polarization-dependent response of the TFBG sensor. The ob-
served polarization-dependence of the TFBG spectrum and electric field at the sur-
face of the sensor will lead us to the sensitivity enhancement technique proposed in
the following chapters.
88
MODELING OF TILTED BRAGG GRATING (TFBG) STRUCTURES
Figure 2.49
The electric field at the fibre surface, corresponding to particular resonances.
The linear polorizes light incident at
P = 45
o
and
P = 90
o
angles at the
4
o
TFBG.
CHAPTER 3
EXPERIMENTAL
POLARIZATION-BASED OPTICAL
SENSING WITH APPLICATION TO TFBG
SENSORS
In this Chapter we proposing a polarization-based sensing method developed for
TFBG sensors.
Polarization-based sensing is crucial for various types of optical sensors, in partic-
ular for stress analysis, plasmon-mediated sensing, and sensing of anisotropic media
or other forms of perturbations [
72
,
73
,
74
,
75
]. In the particular case of waveguide-
type sensors (including optical fibres), it is possible to investigate the devices optical
properties with polarized light but in general this requires very careful alignment and
control of the input polarization.
As we described in the previous sections, TFBG sensors reveal strong polarization-
dependent properties [
76
,
3
] due to the tilt of the grating planes which breaks the
cylindrical symmetry of the fibre and strongly impacts the magnitude of the cou-
pling coefficients between the incident core mode and the cladding modes excited by
the grating [
9
], as is schematically shown in Figure
2.36
.
The grating tilt causes asymmetry in coupling between the cladding modes and
the core mode, with respect to the grating tilt. In particular, for a grating tilted by an
angle in the reference y
- z plane and a linearly polarized input core mode, whose
polarization is rotated by an arbitrary angle with the reference x axis, the coupling
to individual cladding modes will depend strongly on both and .
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
89
90
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
Figure 3.1
A typical TFBG transmission spectrum for linearly polarized light, and series of
spectra obtained by rotating a linear polarizer about the optical axis are shown as the density
plot.
The asymmetry in sensing results from the different nature of interaction between
the cladding TM-like and TE-like modes, with the predominant radial and tangential
orientation of the electric field at the sensor surface, from one side, and the environ-
ment under test from another side. The asymmetry can be significantly enhanced
if the sensor surface is coated with a thin metallic film[
1
] or metal nano rods [
3
],
interacting differently with the radial and tangential oriented electric fields.
The polarization effects in optical fibres have been traditionally quantified with
a polarization-dependent loss (PDL) parameter, provided by optical vector analyzer
(OVA) instruments. Alternatively, the polarization effects can be studied with regards
to linearly polarized light aligned with a specific axis of the device under test [
1
].
A typical series of spectra measured at various angles of linearly polarized light is
shown if Figure
3.1
.
The effect of the input mode polarization can be observed by measuring optical
transmission spectra with a polarizer inserted between the light source and the grat-
THE OPTICAL SETUP
91
ing [
1
]. A typical series of spectra measured at various angles of linearly polarized
incident light is shown in Figure
3.1
for a 1 cm long TFBG immersed in water.
In the case of TFBG sensor modes with a different orientation of electric field at
the sensor surface can be excited by rotating linear polarized light about the sensor
axis [
9
]. To rotate the linearly polarized light an external polarizer is usually used,
sometimes in combination with OVA instrument.
In the presented work we compare the PDL based approach, which is based on
measurements of transmission spectra along the orthogonal principal axes, with the
measurements based on extracting linear states of polarization along a predefined
axis either from the Jones matrix or the Stokes vector.
3.1
The Optical Setup
The measuring optical system was designed to acquire transmission spectra of tilted
fibre Bragg sensor at various polarization states.
Two alternative approaches were used. The first straightforward approach as
shown in Figure
3.2
, is based on introduction of a linear polarizer in the optical
path [
1
], thus allowing the collection of spectra at various states of linear polar-
ization. The second approach was based on measuring the Stokes parameters or
alternatively the Jones matrix elements, and will be discussed later.
Figure 3.2
The optical setup based on SI720 spectrophotometer and PR2000 polarization
controller.
The experimental setup is shown schematically in Figure
3.2
consists of a SI720
spectrophotometer (Micron Optics) and a polarization controller PR2000 (JDS Uniphase).
The SI720 spectrophotometer is based on the fast sweeping tunable laser and inte-
grated photo detector. The polarization controller was set to continuously scan all
linearly polarized light states in 80 seconds.
The spectra over the full operational range (from 1520 to 1570 nm) were taken
continuously at a rate of one spectrum each 0.2 seconds, thus the spectra were ac-
quired with the resolution of less than 1
o
degree of polarization angle. Such high
resolution allowed us to collect a huge amount of data and, as a result, to precisely
describe the spectral and polarization response of the TFBG-SPR sensor.
An example of acquired spectra is shown in Figure
3.1
. The spectra are rep-
resented in the form of a density plot. The spectra are stored in the matrix, with
each row corresponding to a different angle of linearly polarized light. Two orthog-
92
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
onal states of polarization I
x
and I
y
were extracted at the data processing stage,
and correspond to the measurements along axes which the maximum and minimum
transmission spectra.
Considering that light transmitted through a device under test is experiencing the
insertion loss dependent on the state of incident light polarization, as a function of
optical frequency, two alternative approaches to optical measurements are possible.
The optical system can be fully characterized either by the Stokes parameters or the
Jones matrix.
Figure 3.3
The Stokes vector representation.
The Stokes parameters and Jones matrix were acquired with help of a JDS Uniphase
SWS-OMNI-2 system and from an optical vector analyzer (OVA) 5000 from Luna
Technologies, respectively. Both approaches are based on the same operational prin-
ciple, except that the optical vector analyzer from Luna Technologies is also capable
of measuring the phase delay in an optical device under test.
An optical vector analyzer usually consists of an optical source, a polarization
controller capable of producing one of four known polarization states and a power
meter measuring an insertion loss. Only four measurements to determine Stokes
parameters S
0
, S
1
, S
2
, S
3
are required. The Stokes parameters can be visualized with
help of a Poincar´e sphere sphere as shown in Figure
3.3
. First, an optical signal is
polarized to produce one of four known polarization states a, b, c and d, described by
the Stokes vectors S
0,a
, S
0,b
, S
0,c
and S
0,d
, and then transmitted through the device
under test. The corresponding transmitted powers T
0,a
, T
0,b
, T
0,c
and T
0,d
of the
output signal are measured with the power meter [
77
,
78
].
The transmissivity of the optical component under test can be represented by a
4 × 4 Mueller matrix [M]:
S
out
= [M]S
in
,
(3.1)
where S
in
= (S
0
, S
1
, S
2
, S
3
) and S
out
= (T
0
, T
1
, T
2
, T
3
) are the input and the out-
put states of polarization, respectively, represented by Stokes vectors. T
0
is the in-
THE OPTICAL SETUP
93
tensity of transmitted light, measured with the power meter. Hence, for each state of
polarization a, b, c, d the four transmitted intensities T
0,a
, T
0,b
, T
0,c
, T
0,d
can be mea-
sured, and four equations can be written by multiplying the first row of the Mueller
matrix by the input Stokes vector:
T
o,a
= m
00
S
0,a
+ m
01
S
1,a
+ m
02
s
2,a
+ m
0k
S
3,a
T
o,b
=
m
00
S
0,b
+ m
01
S
1,b
+ m
02
s
2,b
+ m
0k
S
3,b
T
o,c
=
m
00
S
0,c
+ m
01
S
1,c
+ m
02
s
2,c
+ m
0k
S
3,c
T
o,d
= m
00
S
0,d
+ m
01
S
1,d
+ m
02
s
2,d
+ m
0k
S
3,d
(3.2)
Solving the above system of equations the four matrix elements m
00
, m
01
, m
02
and m
03
can be found, in terms of which other parameters, such as polarization-
dependent loss (PDL), can be expressed [
77
].
Unfortunately the described technique does not allow us to gather information
about a phase delay in an optical system. The complete information about an optical
device can only be obtained if the instrument used can measure phase as a function of
frequency and polarization, thus providing the complete response of a system. One of
the possible realizations of such a device is schematically shown in Figure
3.4
[
78
].
Figure 3.4
The principle of operation of an optical vector analyzer .
The principle of operation is based on the so-called swept-homodyne interfer-
ometry [
78
] performed for various states of polarization and fully characterizes an
optical system by measuring the attenuation and phase delay as functions of opti-
cal frequency. As shown in Figure
3.4
the device consists of an unpolarized light
source, supplying unpolarized coherent light whose optical frequency is swept con-
tinuously as a function of time, which is first passed through the interferometer then
through a three-way polarization splitter coupled to individual detectors. The detec-
tion polarization splitter is designed to measure the light intensities P
1
, P
2
, P
3
by
94
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
projecting the light polarization state onto the three linearly independent preselected
Stokes axes of polarization S
1
, S
2
, S
3
, shown in Figure
3.3
, as a function of optical
frequency. The fourth measured P
4
is a total polarization-independent power. The
optical delay is measured with a Michelson interferometer, where a device under test
is coupled in one arm of the interferometer and the other arm of the interferometer
is used for the reference. As a result a 2
× 2 Jones matrix [J] is computed at each
optical frequency:
E
out
= [J]E
in
.
(3.3)
The four elements of the Jones matrix are in general complex numbers, encoding the
attenuation and phase delay of an optical system.
3.2
The data processing technique
The optical setup based on OVA implementation is shown in Figure
3.5
. In spite of
the apparent simplicity, the data analysis is significantly more complicated than in
the case where the state of polarization is rotated mechanically.
Figure 3.5
The Experimental Setup.
In this section we describe the data processing technique which allowed us to
extract polarization-based parameters, characterizing the sensor response, from the
data provided by an optical vector analyzer.
3.2.1
Measurements along principal axes of an optical system
In this section we review polarization-dependent loss (PDL) technique and provide
an approach to study the system transmission spectrum along its principal axes.
A linear optical system can be represented in terms of the 2
× 2 Jones matrix [J]
connecting an incident and transmuted electric field vectors [
79
]:
E
out
= [J]E
in
(3.4)
The transmission spectrum of a device under test usually is characterized with
two major parameters. The polarization independent parameter called the insertion
loss I() and the polarization-dependent loss (PDL) parameter, both measured as a
THE DATA PROCESSING TECHNIQUE
95
function of optical frequency . To extract these parameters from the Jones matrix,
a hermitian matrix [H] has to be constructed first.
[H] = [J]
[J]
(3.5)
Next, performing eigen decomposition of [H] matrix
[H] = [U][][U]
T
(3.6)
we find diagonal matrix [] containing eigenvalues
1
and
2
of [H] matrix.
Alternative singular value decomposition (SVD) of the Jones matrix can be com-
puted [
80
]:
[J] = [U][][V ]
T
(3.7)
with the diagonal matrix [] containing two singular values
1
and
2
such that
1
=
2
1
and
2
=
2
2
, due to the fact that [H] = [J ]
[J].
The matrix [U ] = (u
1
, u
2
) contains two orthogonal eigenvectors (due to the prop-
erties of SVD decomposition), corresponding to
1
and
2
eigenvalues, align with
principal axes of the system. The principal axes u
1
, u
2
of TFBG sensor are showed
schematically in Figure
3.6
.
Figure 3.6
TFBG with physical
^x, ^y axes and principal system axes u
1
and
u
2
measured at
some optical frequency
.
Having the singular values of the Jones matrix [J ] or eigenvalues of [H] = [J ]
[J]
matrix the transmission loss can be computed as follows [
80
].
I
out
=
E
out
|E
out
= E
in
|[J]
[J]|E
in
=
=
E
in
||E
in
=
1
+
2
2
I
in
(3.8)
Which can be re-formulated in the more frequently used dB scale as:
I
out
I
in
= 10 log
10
1
+
2
2
.
(3.9)
96
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
Similarly, the polarization-dependent loss, which is the magnitude of the difference
between the maximum and minimum device transmission over all possible input
polarization actually corresponds to the difference in the loss measured along the
system principal axes u
1
and u
2
. It is defined as follows:
P DL = 10
| log
10
1
2
|
(3.10)
Alternatively we can introduce a parameter called degree of polarization [
3
]:
P =
1
-
2
1
+
2
(3.11)
The eigenvalues
k
=
2
k
can be interpreted as an observable transmission loss
for the case when incident to an optical system electric vector E is align with the
system principal axes, i.e. E
u
k
. The PDL parameter accounts for the difference
in the loss measured along the system principal axes u
1
and u
2
.
Figure 3.7
Eigenvalues
1
and
2
and the angle
between the geometrical axes of the
system
^x, ^y and the coordinate system defined by the principal axes u
1
,
u
2
, before a) and after
b) the eigenvalues were reordered. (The data were obtained by means of the OVA 5000 Luna
Technologies.)
In addition to the eigenvalues of the system transmission matrix (plotted as a func-
tion of wavelength in Figure
3.7
), we can also find the angle of rotation () of the
principal axes about the device optical axis (z) and plot it as a function of wavelength
(bottom frame of Figure
3.7
). From the mathematical point of view, both eigenvalues
are roots of a quadric polynomial and can be ordered arbitrary, usually in descending
order: this is why
1
is always larger than
2
in Figure
3.7
a, hence the eigenstates are
interchanged when they cross. Because of this effect, the corresponding eigenvectors
THE DATA PROCESSING TECHNIQUE
97
are interchanged as well, therefore the angle experiences /2 shifts at the crossing
points, as shown on the bottom panel of Figure
3.7
a. Therefore, a strategy for restor-
ing the individual transmission spectra along the principal axes is to interchange the
eigenvalues and principal axes every time the = /2 jumps are detected.
The result of such reordering is shown Figure
3.7
b. Now the transmission spec-
trum corresponds to linearly polarized light with its electric field vector aligned with
each principal axis of the system and the rotation angle jumps are eliminated.
In general, the principal axes of an optical system are not necessarily fixed with
respect to a reference frame but may depend on the wavelength, as shown in Fig-
ure
3.8
for a TFBG sensor. Indeed, by changing the optical frequency it is expected
that an optical system would operate differently. The global behavior of the principal
axes, along the whole operational range of the TFBG sensor, is shown in Figure
3.8
b,
and the corresponding insertion loss in Figure
3.8
a. To remove the noise only the
points at which the difference between eigenvalues is noticeable (
|
2
-
1
| > 0.05)
are plotted. The noise arises from the fact that at the points of eigenvalues crossing
the transmission matrix becomes degenerate, has the eigenvectors are not well de-
fined, and oscillate rapidly with respect to the geometrical axis, as can be seen from
Figure
3.8
b at the points of crossing.
Figure 3.8
Rotation of the principal axes of TFBG device as a function of optical wavelength.
(The data was obtained by means of the OVA 5000 Luna Technologies.)
98
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
Figure
3.8
b shows that the system possesses significant polarization asymmetry,
or birefringence, in the wavelength range
[1545 - 1575]. Although the principal
axes are globally stable, near 75
o
degrees relative to the reference frame of the LUNA
interrogation system, locally they experience small oscillations, of about 8
o
degrees
about the optical axis. These oscillations are related to the resonances observed in
the spectrum (Figure
3.8
a), and reflect a wavelength dependent birefringence.
We also note the observed alternation between the peaks (Figure
3.8
a), which can
be explained by the fact that the alternating peaks have different azimuthal symme-
tries and polarizations [
9
].
3.2.2
Measurements along geometrical axes of an optical system
Instead of choosing principal axes as a reference coordinate system, geometrical
axes can be chosen alternatively. In the case of TFBG it is convenient to choose
orthogonal axes such that one of the axis is normal to the fibre axis and lay on the
plane parallel to the grating blades, as shown in Figure
2.36
. Usually the spectra
along geometrical axes are obtained by introducing an external linear polarizer, such
that state E
in
can be align along the required axis. In Figure
3.1
such orthogonal
states along ^
x and ^
y axes are shown on the density plot and denoted as I
x
and I
y
,
respectively.
The difference in transmission loss along the principal axis of the TFBG sensor
and its geometrical axes is shown in Figure
3.9
. It can be clearly seen that in the case
when geometrical axes are align with the principal axes of an optical system both
approaches yield almost identical result. The small difference comes from the afore-
mentioned wavelength dependent oscillation of the principal axes (which cannot be
compensated for in the direct measurement), and to a possible slight change in polar-
ization state between polarizer and the grating (since a non-polarization maintaining
fibre is used). The following section will demonstrate that in spite of this small
inaccuracy, the parameters extracted from the Jones matrix data provide excellent
spectral sensitivity results for refractometric sensing.
Nevertheless, although the singular value decomposition, or egen decomposition,
provide an elegant way to introduce coordinate axes along which the system can
be studied, it might be desirable to fix the reference axis permanently to an optical
system geometry, for example in the cases of TFBG sensor, when it is known that
the system behaves physically different along particular axes.
3.2.3
Extracting transmission spectra measured along geometrical axes
from the Jones matrix or the Stokes vector data
In this section we describe an approach allowing to extract transmission loss spectra
along the given geometrical axes from the Jones matrix or the Stokes vector without
introduction of external polarizer. Thus a single measurement by OVA can provide a
complete information about transmission loss along all possible geometrical axes.
THE DATA PROCESSING TECHNIQUE
99
Figure 3.9
Transmission loss along the TFBG system principal axes and its geometrical
axes for perfectly align coordinate systems
= 0
o
and rotated by
45
o
degrees
= 45
o
. The
intersession loss
I
x,y
and eigenvalues
1,2
are given in linear scale.
3.2.3.1 The Jones matrix analysis
Transmission spectrum of linear polarized
light, with the electric field vector E align along a given geometrical axis can be
extracted directly from the Jones matrix, measured for example by means of OVA
5000 from Luna Technologies.
The instrument was designed to provide a full characterization of the polarization-
dependent transmission properties of optical fibre devices as a function of wave-
length in the form of the Jones matrix elements. The spectral accuracy of the OVA is
+/ - 1.5pm and its insertion loss accuracy is +/ - 0.1 dB, over a wavelength range
from 1525 to 1610 nm.
The OVA 5000 Luna is capable of capturing four complex transfer functions (am-
plitude and phase) of a fibre as a function of wavelength and polarization. The in-
cident and transmitted light can be represented in terms of the electric field vector,
written as a column vector: E =
E
x
E
y
, known as a Jones vector [
79
], where the
field components E
x
, E
y
are projections of the electric field vector E on the x, y
100
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
axes. The optical TFBG sensor then can be represented as a four port device with
two input and output states of polarization. These four transfer functions can be as-
sembled into a Jones matrix [
79
], giving the complete characterization of a device
under test.
E
out
= [J]E
in
(3.12)
[J()] =
a()
b()
c()
d()
(3.13)
Here, J () is the Jones matrix consisting from the four complex transfer functions
a(), b(), c(), d() of the optical frequency .
The Jones matrix of an optical device under test connected to a linear polarizer is
give by:
[J
dev
+pol
] = [J
pol
][J
dev
].
(3.14)
Here [J
dev
] is the measured by an optical vector analyzer Jones matrix of a bare
device, and [J
pol
] is the Jones matrix of linear polarizer known from theory.
The Jones matrix of a linear horizontal polarizer [J
dev
()], rotated by angle with
respect to the optical device axes, is given by:
[J
pol
()] = [R()]
1 0
0 0
[R(-)] =
=
cos
2
()
cos() sin()
sin() cos()
sin
2
()
,
(3.15)
where [R()] is the rotation matrix:
[R()] =
cos() - sin()
sin()
cos()
,
(3.16)
rotating the coordinates system so that the horizontal polarizing element has the sim-
plest representation and then rotating the system back down into the original system
of coordinates.
Thus, on optical device with a known Jones matrix [J
dev
], connected to a linear
polarizer rotated by angle, is described by the Jones matrix:
[J
dev
+pol
()] =
cos
2
()
cos() sin()
sin() cos()
sin
2
()
[J
dev
].
(3.17)
Finally the transmission spectrum I(, ), for a given angle of the linear polar-
izer, can be computed as
I(, ) = 10 log
10
1
(, ) +
2
(, )
2
,
(3.18)
where
1
(, ) and
2
(, ) are the eigenvalues of H(, ) = [J
dev
+pol
(, )]
[J
dev
+pol
(, )]
matrix.
THE DATA PROCESSING TECHNIQUE
101
3.2.3.2 The Stokes vector analysis
In a similar way we can use the Stokes vec-
tor. The Stokes vector can be measured with a polarizer controller or a special in-
strument, such as JDS Uniphase SWS-OMNI-2 system.
A beam of light can be completely described by the four parameters [
81
,
82
],
represented in the form of the Stokes vector:
S =
S
0
S
1
S
2
S
3
=
I(0
o
) + I(90
o
)
I(0
o
) - I(90
o
)
I(45
o
) - I(135
o
)
I
RHS
- I
LHS
,
(3.19)
here I() is the intensity of light polarized in the direction defined by the angle in
the plane perpendicular to the direction of light propagation, and I
RHS
, I
LHC
are
the intensities of right- and left-handed polarized light, respectively.
Since the Stokes parameters are dependent upon the choice of axes, they can be
transformed into a different coordinate system with a rotation matrix [R]. Consid-
ering that a second coordinate system is obtained by rotating the original coordinate
system about the direction of light propagation on the angle , we can write [
83
]
S
() = [R()]S,
(3.20)
or
S
0
S
1
S
2
S
3
=
1
0
0
0
0
cos(2)
sin(2) 0
0 - sin(2) cos(2) 0
0
0
0
1
S
0
S
1
S
2
S
3
=
=
S
0
S
1
cos(2) + S
2
sin(2)
-S
1
sin(2) + S
2
cos(2)
S
3
,
(3.21)
here S
0
, S
1
, S
2
and S
3
are the Stokes parameters.
Now considering that
I(0
o
) + I(90
o
) = S
o
,
I(0
o
) - I(90
o
) = S
1
cos(2) + S
2
sin(2),
(3.22)
we conclude that the transmission loss spectra along the two orthogonal axis, rotated
by the angle with respect to the original system of measurements, are given by the
102
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
following expressions:
I
x
(, ) =
1
2
(S
o
() + S
1
() cos(2) + S
2
() sin(2)),
I
y
(, ) =
1
2
(S
o
() - S
1
() cos(2) - S
2
() sin(2)).
(3.23)
Hence, if the Stokes vector is known in one coordinate system, the transmission
spectrum of linearly polarized light with the electric field E align along a given
geometrical axis, defined by the angle , can be recovered.
We should also note that the Jones matrix and Mueller matrix representations are
connected, and can be expressed as [
84
]:
M = U (J
J)U
-1
,
(3.24)
where J
J is the direct product of Jones matrices, and matrix U is given by
U =
1 0
0
1
1 0
0
-1
0 1
1
0
0 i -i
0
(3.25)
Although the two approaches are related the phase information is available only in
the case when the Jones matrix is known. The phase information allows to determine
additional phase related parameters, such as Group Delay (GD).
3.3
Polarization-based detection of small refractive index changes with
TFBG sensors
In this section we investigate which polarization-based measurement techniques pro-
vide the best signal-to-noise ratio when TFBG sensors are used to detect small refrac-
tive index changes, and use the special case of a TFBG coated with gold nanorods
(as described in [
3
]). This choice is made because the polarization dependence of
waveguide-type sensors is much enhanced when metal coatings are used. As indi-
cated earlier, the interaction of guided waves with metal interfaces depends strongly
on whether the electric fields of the waves are tangential or normal to the metal
boundary. It was further mentioned that when the core-guided input light of a TFBG
is linearly polarized along the principal axes, the electric fields of high order cladding
modes are either tangential or radial (hence normal) to the cladding boundary. To be
precise, y-polarized input light (corresponding to light polarized parallel to the plane
of incidence on the tilted grating fringes, as shown in Figure
2.36
, i.e. P-polarized)
couples to radially polarized cladding modes while x-polarized light (perpendicular
to the tilt plane, or S-polarized) couples to azimuthally polarized cladding modes
POLARIZATION-BASED DETECTION OF SMALL REFRACTIVE INDEX CHANGES WITH TFBG SENSORS
103
(tangential to the boundary). Further polarization effects arise with non-uniform
metal coatings, since they have boundaries that are both tangential and radially ori-
ented relative to the cylindrical geometry of the fibre [
85
]. It is therefore desirable
to carry out two transmission measurements along the principal axis to observe di-
rectly such polarization effects on the strengths and positions of the cladding mode
resonances [
86
]. On the other hand, it has been shown here that a Jones matrix
measurement can provide this information as well, in addition to other parameters
of interest, such as PDL. While the PDL spectrum "hides" the physical effects re-
sponsible for difference in transmission due to the different polarization states, it has
been shown in the past to yield excellent limits of detection for surface plasmon res-
onance based TFBG sensors [
15
]. We now proceed to compare the signal noise for
refractive index measurements by the various polarization-dependent data extraction
techniques.
As shown in Figure
3.10
for a TFBG coated with a sparse layer of gold nanorods
and immersed in water, the PDL parameter provides the absolute value of the differ-
ence between resonances observed in the transmission spectra measured along the
principal axes. The relative position of the peaks, their amplitude, and their width
become convoluted in the PDL parameter, which provides instead a maximum lo-
cated somewhat in between the individual resonance maxima and a zero on either
side corresponding to the wavelengths where the spectra cross each other.
When the refractive index of the medium surrounding such TFBG changes the
waveguiding characteristics of the cladding are modified and the resonances ob-
served in the transmission spectrum change accordingly. Therefore, to detect changes
in refractive index we can either follow the amplitudes and positions of individual
resonances [
86
] or of the PDL features. Here, the sensor was immersed in water and
the refractive index was incrementally increased in steps of n = 1.517
× 10
-4
by
adding 10 l of ethylene glycol ( C
2
H
4
(OH)
2
) to 5 ml to the water. The impact of
each increase in refractive index on all parameters of interest is shown in Figure
3.11
for a typical slice of the spectrum (it was noted in [
3
] that there was little difference
in wavelength shifts across the TFBG spectrum for this device).
The refractive index change was chosen to have a relatively small value of ( n =
1.517 ×10
-4
) to test the sensor detection limits and a linear fitting was used because
the sensor response for such small changes is expected to be linear. The standard
deviation of the errors from the linear fit was calculated in the usual manner by:
SD =
1
N
N
i
=1
(x
i
- )
2
,
(3.26)
here is the expected value of x, and x = y
appr
- y
data
is the difference between
the measured data y
data
and its linear approximation y
appr
.
The results of the fits show that the most accurate detection of the refractive index
change is achieved with the zeros in the PDL spectrum, as this measurement provides
the smallest standard deviation of SD = 1.40 pm. The worst result were obtained
with the PDL peak (SD = 6.44 pm), while the detection of individual polarized res-
onances provides an intermediate value of the standard deviation (essentially equal
104
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
Figure 3.10
Two eigenvalue spectra (individual transmission along the principal axes) and
corresponding polarization-dependent loss (PDL) parameter. The peak in PDL spectrum is
denoted by "1" and zeros by "2". Here the more common dB scale is used.
to 3 pm). It is not surprising that zeros of PDL should provide the most accurate
results as they consist essentially of a differential measurement between two spec-
tra with well-defined crossing points that occur on the sides of the individual reso-
nances, where the spectral slope is highest. On the other hand, at the expense of an
increase in noise by a factor of approximately 2, other effects such as differences in
the change in the resonance amplitude for the two polarizations (which can be linked
to differential loss or scattering) can be studied when the principal axis spectra are
used.
3.4
Conclusion
In this Chapter we investigated the use of Jones matrix and Stokes vector based
techniques and polarization analysis to extract information from optical fibre sensors
in non-polarization maintaining fibres. In particular we showed how to calculate
transmission spectra with electric fields E aligned with the system principal axes
or along any other system axes. We also proposed the method of extraction spectra
of light polarization along a given geometrical axis either from the Jones Matrix
CONCLUSION
105
Figure 3.11
Position of the resonance a) in the transmission spectra
I
x
and
I
y
of light
polarized along
^x and ^y geometrical axes here aligned with the principal axes, and b) in PDL
spectra (the maximum and zero values of PDL are detected), as a function of refractive index
change. The continuous lines represent the least square approximation to the measured data,
and
SD is the standard deviation from the linear approximation
or the Stokes vector data. In the case of a TFBG inscribed in a non-polarization
maintaining fibre for instance, the principal axes of the system are determined by the
direction of the tilt of the grating planes (and its perpendicular). The transmission
spectra for light polarized in the tilt plane (P-polarized) or out of the tilt plane (S-
polarized) can thus be extracted without having to separately align a linear polarizer
upstream from the grating and hoping that the polarization remains linear between
the polarizer and the TFBG. Polarization-resolved spectra can also be obtained at a
faster rate, for applications in chemical deposition process monitoring for instance,
without the need to line up or rotate a polarizer between each measurement.
The transmission spectra of P- and S-polarized light were compared with the
TFBG spectral response along the principal axes. A small oscillation of about 8
degrees in the orientation of the principal axes as a function of wavelength was ob-
served.
106
EXPERIMENTAL POLARIZATION-BASED SENSING WITH TFBG
Furthermore, it was determined that the best TFBG sensor detection limits are
achieved when zeros in the PDL spectrum are followed, mainly because of the sharp-
ness of crossing points occurring on the steep sides of the individual resonances, but
at the expense of the direct observation of the sensor response to polarized light
(which are also available from the Jones matrix data). In the latter case, while the
standard deviation of the spectral sensitivity is doubled relative to PDL measure-
ments, additional information becomes available regarding the influence of the mea-
sured medium on cladding mode loss, allowing further uses from the TFBG data
sets [
87
].
CHAPTER 4
OPTICAL PROPERTIES OF MATERIALS
In the presented work we investigate methods of increasing TFBG sensor sensitivity
by coating its surface with various types of coatings, such as metal films and films
consisting of nanoparticles.
Before running numerical simulations, the optical properties of materials should
be defined. In this chapter we will focus our attention on optical properties of metals
since it is known the that sensitivity of an optical sensor can be increased by exci-
tation of surface plasmon resonances (SPR) observed in metal films and nanoparti-
cles of a proper geometrical dimension [
88
,
89
,
90
,
91
]. We also consider various
surface morphologies and review methods that allow us to account for the dipole-
dipole radiative interaction between elements of a film or between closely deposited
nanoparticles.
4.1
The methods of measurement of optical constants
It is not a trivial task to choose the correct optical parameters for thin metal films
or nanoparticles. Various sources of information do not always provide consistent
results. The theoretically computed permittivity of metals is not always consistent
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
107
108
OPTICAL PROPERTIES OF MATERIALS
with the experimental measurements, and experimental results sometimes are limited
by a particular method.
In this section we provide a brief review of various methods available for measur-
ing of optical constants and discuss limitations imposed by a particular method, so
that we can choose a reliable data source for further computation. Next we will
consider models for mixed media needed to describe rough films resulting from
nanoparticle-based coatings.
The optical constants n and k of a medium at frequency can be determined
either by (1) measuring n and k independently at a given or (2) one of the parame-
ters (n or k) should be measured over the entire frequency range, then the remaining
parameter can be deduced from the measured data.
Several alternative methods were proposed as well, including the Drude ellipsom-
etry method introduced in 1889 [
92
] in which the relative amplitude and phase shift
are measured simultaneously, however, the method is significantly affected by the
sample surface quality [
93
].
4.1.1
The methods based on single parameter measurements
Historically, this approach was based on Bode's electric network theory introduced
in 1945, where it had been shown that the attenuation and phase at the output of
an electric network are not independent of one another. In Robinson's paper from
1952 [
94
] the absorption spectrum was derived from a normal incidence reflection
spectrum. The phase change of a reflected wave was deduced from the curve of the
reflection attenuation as a function of frequency, and the optical constants n and k
were then determined with the aid of a Smith chart.
A modern treatment of the problem can be found in [
95
,
96
,
97
]. The usual
optical procedure is to measure the reflectance at normal incidence over as wide
range as possible, and then use, for example, the free-electron extrapolation for the
reflectance outside the range. Once the reflectance is approximated over the range
from zero to infinity the Kramers-Kronig (KK) integration might be implemented:
1
() =
2
P
0
·
2
()
2
-
2
d,
2
() = -
2
P
0
1
()
2
-
2
d.
(4.1)
Here the integration is done in the sense of the Cauchy principal value, defined as the
P
and () is a complex analytical function, vanishing faster than
1
||
as
|| ,
1
() := Re[()] and
2
() := Im[()].
Thus the optical constants n and k can be connected through the Kramers-Kronig
relations:
() = ~
n()
= n() + ik() =
=
1
() + i
2
()
(4.2)
OPTICAL CONSTANTS OF METALS
109
However, the results of such calculations are often quite sensitive to the shape of
the extrapolated spectrum, therefore the extrapolation of the reflectance at both ends
of the spectrum is crucial. While a reasonable low-energy extrapolation is possible
by simply assuming that R = 100%, additional experimental information is needed
to make a reasonable high-energy extrapolation. The measurements have to be ex-
tended further into the vacuum ultraviolet region in order to determine the n and k
in the visible band. The necessity of such measurements is the major disadvantage
of this method [
97
,
98
].
4.1.2
The methods based on simultaneous measurement of both pa-
rameters
Simultaneous measurement of both parameters is usually difficult in practice, es-
pecially in the region of strong absorption. To overcome this problem several new
techniques were proposed and successfully implemented in [
98
,
4
]. Instead of a sin-
gle normal incidence reflectivity measurement, the parameters for oblique incidence
at several angles and various polarizations were measured. The right choice of the in-
cident angle and polarization state can significantly improve the results in the region
where normal incidence measurements are inaccurate [
98
]. The proposed method is
also less sensitive to the surface roughness, unlike the ellipsometry method.
First the reflectance R and transmittance T of a thin film deposited on a trans-
parent substrate were calculated for the normal and oblique incidence, for s and
p polarization, and different film thicknesses. The calculation involves solution of
Maxwell's equation boundary-value problem from which the reflectance R and trans-
mittance T can be expressed in terms of the material constants n and k.
Next the obtained function R(k, n) and T (k, n) were inverted to obtain n and k
expressed in terms of the measured R and T . Since the optical constants n and k
are independent of the boundary conditions, the n and k values were expected to
depend only on the incident light wavelength (which excites the electronic transi-
tion), but not on polarization and angle of incidence (if material is isotropic) or on
the film thickness. Therefore at each wavelength, in the range of interest, reflectance
R and transmittance T at specified incident angles and polarizations were measured.
Then the film thickness was measured by means that were independent of the R
and T measurements and compared with the result deduced directly from R and T
measurements [
98
]. The process of inversion requires solution of several nonlinear
equations, and a number of iterations till the convergence is achieved [
98
].
4.2
Optical constants of metals
In our calculation we are going to use optical constants obtained in [
4
] by the method
described in the previous section [
98
], where the optical constants n and k were
obtained for copper, silver, and gold from reflection and transmission measurements.
The films were created by vacuum-evaporation with film-thickness in the range of
110
OPTICAL PROPERTIES OF MATERIALS
185 - 500 °
A, and the measurements were conducted in the spectral range of 0.5
-
6.5 eV.
It was observed that the results were independent of film thickness only above
a certain critical thickness of about 250 °
A, and were unchanged after vacuum an-
nealing or aging in air. The optical properties of evaporated thin films have been
found to be the same as for bulk materials for the thickness of the films greater than
about 300 °
A [
4
].
4.2.1
Free electron approximation, the Drude-Sommerfeld model
The obtained experimental data can be approximated with the free-electron model.
Assuming that the motion of electrons is confined to a region much smaller than the
wavelength we can implement the Drude-Sommerfeld model [
99
,
92
]. The model
does not include a restoring force, assuming free electrons, thus an equation of mo-
tion of a single electron can be written in the following form:
m
o
· d
2
t
x(t) + m
o
· d
t
x(t) = F
ext
(t),
(4.3)
where m
o
is the effective optical mass of electron, =
1
is the macroscopic damp-
ing constant due to the dispersion of the electrons caused by the crystalline structure,
is the relaxation time and F
ext
(t) is the external force applied to the electron.
Assuming harmonic excitation: F
ext
() = -eE() and taking the Laplace trans-
form of equation (
4.3
) we have:
x() =
e
m
o
1
2
+ i
E().
(4.4)
The macroscopic polarization can be written in the following form:
P ()
= N · p() = -Ne · x() = -
N e
2
m
o
1
2
+ i
E() =
=
o
()E() =
=
o
( () - 1) E(),
(4.5)
where: is the electric susceptibility, N is the number of conducting electrons per
unit volume (density),
0
is the electric permittivity of free space and is the relax-
ation time.
Excluding E() from equation (
4.5
), the expression of () can be obtained in
terms of effective optical mass and macroscopic damping of a free electron. We
denote it as
f
().
f
() = 1 -
2
p
2
+ i
,
(4.6)
here
p
=
N e
2
o
m
o
is the plasma frequency.
OPTICAL CONSTANTS OF METALS
111
The complex dielectric constant
^ =
1
+ i
2
(4.7)
and the complex index of refraction, (measured in experiments)
^n = n + ik
(4.8)
are connected by the relation ^ = ^
n
2
, so that
1
= n
2
- k
2
,
2
= 2nk.
(4.9)
Considering (
4.6
) it is possible to separate ^
f
into its real and imaginary parts:
f
1
= 1 -
2
p
2
1 +
2
2
,
f
2
=
2
p
(1 +
2
2
)
.
(4.10)
We note that free-electron approximation to the dielectric function is determined
by the relaxation time and the the plasma frequency
p
defined by the electron
optical mass m
o
.
4.2.2
The near infrared band
For metals at near-infrared frequencies we can assume
1
. Considering equa-
tions (
4.6
) and (
4.10
) we can write:
f
1
1 -
2
p
2
= 1 -
2
2
p
,
f
2
2
p
3
=
3
2
p
,
(4.11)
where
1
2
p
=
N e
2
m
o
c
2
,
= 2c.
(4.12)
The values of n() and k() can be measured experimentally, hence the values
of
1
and
2
can be calculated using (
4.9
). Assuming that the experimental results at
near-infrared bands can be satisfactorily approximated with the free-electron model
we set
=
f
.
(4.13)
112
OPTICAL PROPERTIES OF MATERIALS
Then the optical mass m
o
can be determined from the experimental results for
1
from the slope of a plot of
- vs
2
. Next, using the expression for
2
and plotting
2
/ vs
2
we can determine from the slope of the graph in the infrared band. Such
derivation was conducted in [
4
] where the effective optical mass and relaxation time
were obtained. It was shown that the effective optical mass and relaxation time can
be considered to be constant for the whole near-infrared band.
The results for copper, silver, and gold are shown in Table
4.1
[
4
]. The results for
m
o
are relatively accurate since in the infrared k
n and can be measured precisely.
The error in the measurements is larger due to the error in the n measurements in
the infrared band [
4
].
Table 4.1
Optical masses and the relaxation times for copper, silver and gold. [
4
]
Metal
m
0
(electron masses)
× 10
-15
(sec)
Copper
1.49 ± 0.06
6.9 ± 0.7
Silver
0.96 ± 0.04
31 ± 12
Gold
0.99 ± 0.04
9.3 ± 0.9
We can conclude that the properties of metals at the near-infrared band can be
approximated with the free-electron model, defined by the parameters m
o
and ,
which can be obtained from the experimental measurements of the optical constants
n and k.
4.2.3
The visible band. The interband absorption
In the visible and near-ultraviolet regions Drude's free-electron theory fails and the
interband absorption should be taken into account. The absorption in the visible and
ultraviolet band is significantly influenced by the transitions from the completely oc-
cupied d bands to an empty state above the Fermi level in the conduction band. More-
over, these transitions depend on the nanoparticle size. The interband absorption
peak becomes sharp and its peak position shifts towards the lower energy side [
100
].
We are going to review the size-induces changes in the d band in the following
chapter, here we simply state the significance of the interband absorption. The in-
terband contribution to the imaginary part of the dielectric constant can be obtained
by subtracting the free-electron contribution value
f
2
from the experimentally deter-
mined value of
2
. The comparison between the experimental data and theoretical
prediction, based on the classical free-electron Drude theory, was conducted in [
4
].
The interband contribution was also compared with the theoretical prediction
based on the band-structure model, where the joint density of states and the transition-
probability matrix elements throughout the Brillouin zone were taken into account [
101
,
102
,
103
].
OPTICAL CONSTANTS OF METALS
113
The discussion of optical constants of Ag and Cu in terms of free-electron ef-
fects, interband transitions and collective oscillations can be found in Ehrenreich
and Philipp [
104
].
We conclude that the free-electron expression of ^
f
is useful only for photon
energies below a threshold energy and can be applied for the near-infrared band
only. Above this threshold energy the form of the
2
curve depends on the specific
material band structure, and also exhibits size-dependent properties [
100
].
Assuming that interband contribution in known either from experiments or theory
we can write
() =
intra
() +
f
()
(4.14)
where () is the dielectric function of a metal,
f
() is the free-electron contribu-
tion and
intra
() is the interband contribution.
Although the interband contribution can be satisfactory calculated only in the
framework of quantum mechanics, or obtained from the experimental measurements,
the classical approximation is still possible. The electrons in the d band are not free
as valence band electrons, but rather bounded by the lattice ions. Correcting the
classical free electron model (
4.3
), by introducing a linear restoring force we come
to the Lorentz-Drude oscillator model, based on the damped harmonic oscillator
approximation:
m
d
· d
2
t
x(t) + m
d
· d
t
x(t) + m
d
2
d
x(t) = F
ext
(t),
(4.15)
thus the single induced dipole moment is:
p() = ex() =
-
e
2
m
1
-
2
d
+
2
+ i
E().
(4.16)
The macroscopic polarization P () depends on the model describing dipole-dipole
interaction. The discussion on this subject can be found in [
105
].
In practice it is not always possible to approximate interband absorption with one
specific type of electron (which has an effective mass m
d
and resonates at some
eigenfrequency
d
). Usually several different types of electrons with different effec-
tive masses bounded by different restoring forces should be taken into consideration.
Thus for each resonance in a given absorption spectrum the restoring forces and
the fraction of each type of electrons can be chosen for accurate approximation. Such
investigation was conducted in [
5
] were the six-oscillator Lorentz-Drude model was
used to fit optical functions of eleven widely-used metals in optoelectronic. The
parameters were chosen for the best fit of experimental data, obtained by the method
described in the previous section [
4
].
In accordance with Lorentz-Drude oscillator model, a complex dielectric function
() can be expressed [
106
,
5
]:
() =
f
() +
b
(),
(4.17)
where the free-electron or Drude model is represented by the first term:
f
() = 1 -
f
o
2
p
2
- i
0
,
(4.18)
114
OPTICAL PROPERTIES OF MATERIALS
and the interband part of the dielectric function is represented by the bounded Lorentz
electron model, similar to the model used for insulators:
b
() = -
N
j
=1
f
j
2
p
2
-
2
j
- i
j
.
(4.19)
Here
p
is the plasma frequency, N is the number of oscillators with resonant fre-
quency
j
, and oscillator strength f
j
, and
j
=
1
j
is the macroscopic damping
constant of the jth oscillator (
0
corresponds to the plasma damping constant ).
The obtained results can be summarized in Table
4.2
[
5
] (where
j
is given in
electron volts, and
j
in sec
-1
).
Table 4.2
Values of the Lorentz-Drude Model Parameters [
5
].
Ag
Au
Cu
Al
Be
Cr
Ni
Pd
Pt
Ti
W
p
9.01
9.03
10.83 14.98 18.51 10.75 15.92 9.72
9.59
7.29
13.22
1
0.816 0.415 0.291 0.162 0.100 0.121 0.174 0.336 0.780 0.777 1.004
2
4.481 0.830 2.957 1.544 1.032 0.543 0.582 0.501 1.314 1.545 1.917
3
8.185 2.969 5.300 1.808 3.183 1.970 1.597 1.659 3.141 2.509 3.580
4
9.083 4.304 11.18 3.473 4.604 8.775 6.089 5.715 9.249 19.43 7.498
5
20.29 13.32 0
0
0
0
0
0
0
0
0
o
0.048 0.053 0.030 0.047 0.035 0.047 0.048 0.008 0.080 0.082 0.064
1
3.886 0.241 0.378 0.333 1.664 3.175 4.511 2.950 0.517 2.276 0.530
2
0.452 0.345 1.056 0.312 3.395 1.305 1.334 0.555 1.838 2.518 1.281
3
0.065 0.870 3.213 1.351 4.454 2.676 2.178 4.621 3.668 1.663 3.332
4
0.916 2.494 4.305 3.382 1.802 1.335 6.292 3.236 8.517 1.762 5.836
5
2.419 2.214 0
0
0
0
0
0
0
0
0
f
o
0.845 0.760 0.575 0.523 0.084 0.168 0.096 0.330 0.333 0.148 0.206
f
1
0.065 0.024 0.061 0.227 0.031 0.151 0.100 0.649 0.191 0.899 0.054
f
2
0.124 0.010 0.104 0.050 0.140 0.150 0.135 0.121 0.659 0.393 0.166
f
3
0.011 0.071 0.723 0.166 0.530 1.149 0.106 0.638 0.547 0.187 0.706
f
4
0.840 0.601 0.638 0.030 0.130 0.825 0.729 0.453 3.576 0.001 2.590
f
5
5.646 4.384 0
0
0
0
0
0
0
0
0
OPTICAL CONSTANTS OF METALS
115
4.2.4
Dispersion curves
In this section, as an example, we plot the real and imaginary parts of the refractive
index ^
n() = n() + ik() of Au, Ag, Al and Cu metals. The optical properties
of metals were described phenomenologically using the LorentzDrude model, in
which parameters of six oscillators were fitted for the best consistency with experi-
ments [
5
]. The results are shown in Figure
4.1
.
Figure 4.1
Optical constants
n and k of Au, Ag, Cu and Al as a function of photon energy.
It can be seen from Figure
4.1
that the simple free electron Drude model can be
applied only below the visible band for Ag and Au, where is in the case of Al and
Cu the interband absorption resonances are present even in the IR band.
Once the dispersion curves of various metals are known, we can proceed with
simulation of various metal-based coatings.
116
OPTICAL PROPERTIES OF MATERIALS
4.3
Optical properties of mixtures and rough surfaces
In this section we review a simple yet efficient approach allowing us to describe a
nonuniform metal coating, rough surfaces and surfaces coated with nanoparticles.
4.3.1
Local field effects and effective medium theory
Probably the simplest way to calculate dielectric permittivity of a given material
with inclusions of another material is to apply the effective medium theory. The
effective medium theory describes macroscopic properties of a medium based on the
properties and the relative fractions of its components.
In 1909 Lorentz [
107
] had shown that the dielectric properties of a substance can
be related to the polarizabilities with the Clausius-Mossotti (or Lorentz-Lorenz)
relation [
108
,
109
,
40
]:
=
3
N
- 1
+ 2
,
(4.20)
where N is the number of dipole particles per unit volume and
is the dielectric
permittivity. The derivation goes as follows [
110
,
40
]: an external field E
ext
induces
local molecular dipole moments p
j
at each jth site, proportional to the local field
E
local
, with the proportionality constant - called the atomic polarizability:
p = E
local
.
(4.21)
In the general case is a complex number, meaning that the polarization may be
shifted in phase with the external electric field, dependent on the external field fre-
quency. The polarization can be a tensor for a non isotropic material. The local field
E
local
is not equivalent to the external field E
ext
and should be corrected 1) by taking
into account the field from all other dipoles (excluding the one under consideration)
and 2) in some cases by including the dipole radiation, if the external field oscillates
at a sufficiently high frequency.
The first effect is considered by subtracting a microscopic sphere from the con-
tinuous medium and finding the electric field inside the formed imaginary cavity.
The sphere diameter has to be chosen much smaller than the wavelength, hence the
average field may be assumed to be spatially homogeneous. The resulting field is the
local field E
local
(which corrects the E
ext
field) and is directly connected to the in-
duced local dipole moment p. The macroscopic polarization P is simply a sum of all
the local microscopic dipole moments. Then macroscopic observables, such as the
dielectric function of a medium, can be easily deduced from the known connection
between an external field E
ext
and the correspondent macroscopic polarization P .
The second effect (the dipole radiated field) can be neglected in many cases, and
the result can be viewed as a zero-frequency limit [
40
]. If the effect of the dipole
radiative interaction is significant it can be accounted with the discrete dipole ap-
proximation method.
In the presented work we are interested in the calculation of the dielectric permit-
tivity of particles made of the same material (the inclusions) which are embedded in
OPTICAL PROPERTIES OF MIXTURES AND ROUGH SURFACES
117
another material (the host). The derivation of the general optical mixing formula can
be found in [
105
,
40
,
111
] and written in the following form:
-
h
h
+ ( -
h
)L
=
j
p
j
j
-
h
h
+ ( -
h
)L
,
(4.22)
where
- is the effective dielectric function.
j
- the dielectric functions of the constituents.
h
- is the dielectric function of the host material.
L - is the depolarisation factor, defined by the morphology. For spherical inclusions
L = 1/3.
p
j
= V
j
/V - is the filling factor of the constituents, also called the volume fraction.
V - is the full volume occupied by mixtures,
V
j
- is the volume fraction occupied by jth constitute.
By making several assumptions the general form of equation (
4.22
) can be re-
duced to a set of particular cases, best suited for certain types of media, such as
aerosols, porous media and metamaterials.
4.3.1.1 Maxwell Garnett (MG) approach
If one of the constituents is regarded as
the host material, and the others as inclusions, the equation (
4.22
) can be rewritten
as MaxwellGarnett (MG) equation [
112
,
113
]:
-
m
m
+ ( -
m
)L
=
j
=m
p
j
j
-
m
m
+ ( -
m
)L
,
(4.23)
where
m
is the dielectric functions of the host material.
It should be noted that the result depends on whether the first material is embedded
into the second or the second material is considered to be embedded in the first
material.
4.3.1.2 Lorentz-Lorenz (LL) approach
The LorentzLorenz approach is based
on the assumption that the host material is a vacuum (i.e.
h
= 1) [
108
,
107
]:
- 1
1 + ( - 1)L
=
j
=m
p
j
j
- 1
1 + ( - 1)L
.
(4.24)
4.3.1.3 Effective medium approximation (EMA) or Bruggeman approach
As-
suming that the effective dielectric function acts as a host medium for inclusions, i.e.
=
h
, the resulting equation (
4.22
) can be rewritten with the left hand side equated
to zero [
114
]:
0 =
j
p
j
j
-
+ ( - )L
.
(4.25)
118
OPTICAL PROPERTIES OF MATERIALS
4.3.1.4 Discussion
One of the approaches is most effective depending on the par-
ticular type of medium [
105
]. The MG theory should be applied when constituents
clearly may be subdivided into the inclusions and the matrix material. The EMA
theory works best in the presence of molecular mixtures, where a clear subdivision
into inclusions and the host material is not possible. The LL approach is best suited
for porous materials. The effective medium theory can be viewed as the first order
approximation, allowing us to obtain the effective permittivity of the host material
with the inclusions.
One of the objectives of the presented work is to obtain optical constants for
various metallic films deposited on the fibre surface with either the chemical or CVD
technique. Two examples of the film morphology are shown in Figure
4.2
.
Figure 4.2
SEM images of gold film surface morphology, taken at different stages of
chemical deposition [
1
].
The effective optical constant can be obtained in fairly straightforward way using
the Bruggeman effective medium theory (EMT). An example of EMT application to
a thin film nucleation and growth on a silica wafer can be found, for example, in [
93
].
4.3.2
The exact numerical method accounting for the interaction be-
tween film elements.
The full numerical simulation of the optical properties of a film can be used as an
alternative to the simple mixtures models described in the previous section.
The absorption, and thus the imaginary part of the refractive index, can be ob-
tained if the incident and scattered intensity are known. Next, the real part of the re-
fractive index can be found by implementing the Kramers-Kronig integration method,
as was described in Section
4.1.1
. The idea here was to use the Finite-Difference
Time Domain (FDTD) method to compute the film absorption properties by illumi-
nating it with a short light pulse. The absorbed power can be found by taking the
difference between the incident and scattered intensities. In order to separate the
incident and reflected pulses in time domain, either a large simulation area in the
space domain has to be chosen or the pulse should also be sufficiently short in the
time domain. However, the pulse should be sufficiently long so that its spectral im-
OPTICAL PROPERTIES OF MIXTURES AND ROUGH SURFACES
119
age is sufficiently narrow, allowing the required spectral resolution to be achieved.
Therefore, several contradictory conditions have to be met.
We chose "FDTD-method Maxwell solver" from Lumerical Solutions for our
simulations. The software package was designed specifically for electromagnetic
field simulations and is used extensively. Examples of its applications to the simula-
tion of nanopoarticles can be found, for example, in [
115
,
116
].
In our research we simulated the rough metal film shown in Figure
4.2
. The film
was approximated by a metallic film substrate in which spherical NP inclusions were
immersed, as shown in Figure
4.3
. The metal film and immersed nanoparticles were
chosen to have an identical refractive index. Considering the enormous amount of
Figure 4.3
Simulation of light scattering by a rough silver film deposited on the glass
substrate.
time required for FDTD simulation, only a small area with randomly assign rough-
ness was simulated. The sample was assumed to be infinite. This condition was
satisfied by surrounding the sample with four planes, transverse to its surface, which
enforced the periodic boundary condition. The perfect matching layers were placed
above and beyond the sample, in order to cancel reflection from the computational
domain boundaries, as shown in Figure
4.3
. The pulse was set to incident the film
surface from within the glass substrate, which is the case for fibre optical sensors.
The incident and scattered fields were measured with twelve power sensors (2D rect-
angular plates) detecting the incident field. The first six sensors were arranged in
a 3D cube configuration including the simulation region but excluding the emitting
antenna. The last six sensors were arranged into the outer cubic shell confining all
120
OPTICAL PROPERTIES OF MATERIALS
the simulated objects including the antenna. The inner and the outer shells were
set to detect only ingoing (antenna) and outgoing (scattering) radiation energies, re-
spectively. The absorbed energy was computed by taking the difference between the
incident and scattered energies. Alternatively, the system geometry can be chosen
such that the incident and the scattered pulses can be separated in time domain. Re-
peating the measurements for a number of frequencies, in the bandwidth of interest,
the absorption spectrum was obtained.
Although the proper 3D film simulation takes considerable amount of time, more
than 24 h, some useful conclusions can be drawn from a simple 2D simulation. The
result of light interaction with an array of spherical nanospheres is shown in Fig-
ure
4.4
. The obtained simulation results helped to explain the observed anomalies in
transmission spectra of a metal coated TFBG sensor [
2
], which was my contribution
to that work.
Figure 4.4
Electric field of S (a) and P (b) polarized light interacting with an array of metallic
spheres [
2
].
It should be further noted that the 2D simulation took less than 1 h, wheres a full
3D simulation requires more than 24 h. Furthermore, if the goal is to establish polar-
ization or angular dependency, the parametric space should be scanned (by changing
values of the incident angle and polarization state) which makes this approach pro-
hibitively time expensive. In such a case it would be reasonable to choose a different
software package capable of running on a supercomputer, or to use a different sim-
ulation technique, for example the discrete dipole approximation, described in the
following section.
DISCRETE DIPOLE APPROXIMATION (DDA) OR COUPLED DIPOLE APPROXIMATION (CDA) METHOD
121
4.4
Discrete dipole approximation (DDA) or coupled dipole approxima-
tion (CDA) method
The discrete dipole approximation (DDA) is an extremely flexible and powerful nu-
merical method used for computing the scattering and absorption of electromagnetic
waves by a target of arbitrary geometry, whose dimensions are comparable or larger
than the incident wavelength.
The basic idea of the DDA was introduced in 1964 by DeVoe with application to
the optical properties of molecular aggregates [
117
,
118
]. At first the retardation ef-
fects were not included, and treatment was limited to aggregates that were small com-
pared with the wavelength. The retardation effects were added in 1973 by Purcell and
Pennypacker [
119
]. The accuracy of the DDA method had been confirmed by com-
paring it against the analytical results and experimental measurements [
120
,
121
].
The DDA method was popularized by Draine and Flatau. In 1993 they distributed
discrete dipole approximation open source code DDSCAT [
122
,
121
], which is now
freely available. DDSCAT is an open-source Fortran-90 software package applying
the discrete dipole approximation (DDA) to calculate the scattering and absorption of
electromagnetic waves by targets with arbitrary geometries and a complex refractive
index. The target may be an isolated object as well as 1-d or 2-d periodic arrays of
target unit cells. The method can be used to study absorption, scattering, as well
as electric fields. The description of the latest version DDSCAT 7.2 can be found
in [
123
]. At the present day a few other DDA implementations are available. The
highly optimized computational framework OpenDDA was written in C language
by J. M. Donald at University of Ireland [
124
]. Another implementation of DDA,
called ADDA has been developed through over a period of ten years at University of
Amsterdam by M. A. Yurkin and A. G. Hoekstra. Their software package is capable
of running on multiple processors and clusters of computers under Linux Operating
system [
125
].
The DDA method implies an approximation of a continuum target (scatterer) by
an ensemble of polarizable points in a cubic lattice. The polarizable points acquire
dipole moments in response to the local electric field. Each of the dipoles is driven
by an incident electric field and by contributions from the other dipoles, hence the
method is also known as the coupled dipole approximation (CDA) method. These
interacting dipoles give rise to a system of linear equations, from which a self-
consistent solution for the oscillating dipole moments is found, and next the absorp-
tion and scattering cross sections are computed. If DDA solutions are obtained for
two linearly independent states of polarization of the incident wave, then the com-
plete amplitude scattering matrix for a given scatterer can be determined as well. An
extensive review of the DDA method and its applications was given in [
122
,
126
].
The theory behind the DDA method was discussed in [
127
,
128
,
125
,
129
,
130
] here
we only show the idea behind the derivation.
Let us assume an array of N polarizable point dipoles located at r
j
with each
one characterized by a polarizability
j
. The system is excited by a monochromatic
incident plane wave:
122
OPTICAL PROPERTIES OF MATERIALS
E
loc
(r, t) = E
o
e
i
(kr-t)
,
(4.26)
inducing a dipole moment in each polarizable point:
P
j
= a
j
E
j,loc
,
(4.27)
where index j denotes position r
j
of the jth element and P
j
is the dipole moment of
the jth element.
Each dipole of the system is subjected to an electric field that can be split in two
contributions: the incident radiation field E
in
, plus the field radiated by all the other
induced dipoles E
dp
. The sum of both fields is the so-called local field at each dipole:
E
j,loc
= E
j,in
+ E
j,dp
.
(4.28)
Now excluding E
j,loc
from the consideration by substituting (
4.27
) into (
4.28
) we
obtain
1
a
j
P
j
- E
j,dp
= E
o
e
ikr
j
.
(4.29)
Now let's consider the field radiated by all the other dipoles E
j,dp
. First we use
the well-known expression for radiation of a dipole located at position r
k
evaluated
at point r
j
= r
k
[
40
]:
E
jk
=
e
ikR
jk
R
jk
k
2
n
jk
× (n
jk
× P
k
) +
1
R
2
jk
-
ik
R
jk
(3n
jk
· (n
jk
· P
k
) - P
k
)
(4.30)
where R
jk
= r
j
- r
k
, and n
jk
is a unit vector pointing from r
k
point to r
j
point.
Taking a close look at equation (
4.30
) we note that at the given j and k this equa-
tion simply expresses components of the E
jk
vector in terms of linear combination
of the P
j
vector components. Thus we can rewrite this equation in a matrix form:
E
jk
= [A
jk
]P
k
,
(4.31)
where [A
jk
] is a 3 × 3 matrix.
Now summing all the fields from all k = j dipoles we obtain:
E
j,dp
=
j
=k
[A
jk
]P
k
.
(4.32)
Basically we obtained a representation of E
j,dp
vector in terms of all the 3
· (N - 1)
components of all the P
j
vectors where j = k.
Finally, (
4.29
) can be rewritten as:
1
a
j
P
j
-
j
=k
[A
jk
]P
k
= E
o
e
ikr
j
.
(4.33)
DISCRETE DIPOLE APPROXIMATION (DDA) OR COUPLED DIPOLE APPROXIMATION (CDA) METHOD
123
This is the final equation that has to be solved, consisting of N coupled vectorial
equations from which N vectors P
j
can be found, thus providing a solution to the
scattering problem. It is more convenient to formulate the problem as a set of scalar
equations, therefore we can simply rewrite the above equations as:
[ ^
A] ^
P = ^
E
in
.
(4.34)
For N dipole elements the [ ^
A] is a 3N
× 3N symmetric matrix, ^
P = (P
1
, ..., P
N
)
and ^
E
in
= (E
o
e
ikr
1
, ..., E
o
e
ikr
N
) are 3N vectors.
Once the polarizations P
j
are found, the extinction, absorption and scattering
cross section can be expressed as follows [
131
]:
C
ext
=
4k
|E
in
|
2
N
j
Im[E
j,in
· P
j
],
(4.35)
the extinction cross section is computed from the forward-scattering amplitude using
the optical theorem [
131
];
C
abs
=
4k
|E
in
|
2
Im[E
j,in
· E
i
n],
(4.36)
the absorption cross section is obtain by summing rate of energy dissipation by each
of the dipoles;
C
sca
=
4k
|E
in
|
2
n
j
P
j
- n · (n · P
j
) e
-ik(nr
j
)
2
d,
(4.37)
where n() is a unit vector in direction of scattering and d is the solid angle ele-
ment.
124
OPTICAL PROPERTIES OF MATERIALS
4.5
Conclusion
In this chapter we have reviewed the optical properties of materials, briefly discussed
experimental methods which are used to determine the optical constants n and k. We
have chosen parameters for eleven metals commonly used in optics and plotted their
dispersion curves, which are extensively used in the following simulations.
The morphology of various films can be modeled with the simplest mixtures
model. We are going to use the MaxwellGarnett model to compute the effective
refractive index of rough films, and the LorentzLorenz model to determine the re-
fractive indices of liquid mixtures.
The properties of nanoparticles are discussed in the next Chapter, where we will
obtain absorption coefficients and deduce the effective optical constants of nanopar-
ticles. The obtained effective permittivity of nanoparticles can be considered as the
permittivity of inclusions in the MaxwellGarnett model. Thus it should be possible
to characterize metallic films of various roughnesses as well as films consisting of
nanoparticles.
A more accurate result can be obtained if the radiative dipole-dipole interactions
between the film elements are considered. Unfortunately the FDTD method is pro-
hibitively time expensive, although we did reveal some useful results.
Alternatively we might consider the DDA method. For the sake of simplicity,
properties of a particular particle can be found with other methods, such as the Mie
theory discussed in the following Chapter, and then used to define the starting points
of the DDA method. Therefore all the elements of nanoparticle-based coating be-
come intrinsically interconnected, thus allowing us to account for a collective effect.
However, for sparse coatings the analysis can be solely based on optical properties
of one single nanoparticle.
CHAPTER 5
OPTICAL PROPERTIES OF
NANOPARTICLES
In this chapter we discuss optical properties of metal nanoparticles. Next we present
results of our simulations. The results of simulation would allow us to deduce pa-
rameters for an optimal nanoparticle based coating enchaining the TFBG sensor sen-
sitivity.
5.1
Review
5.1.1
Characterization of light scattering by particles
We start by introducing parameters which are usually used for particles characteri-
zation. Let us consider a single particle illuminated by a plane wave with intensity
I
inc
, than the total power scattered by this particle is W
sca
, defined as:
W
sca
= C
sca
I
inc
,
(5.1)
here C
sca
is the scattering cross section, with dimensions of area.
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
125
126
OPTICAL PROPERTIES OF NANOPARTICLES
Particles can also absorb electromagnetic radiation. The rate of absorption W
abs
should also be proportional to the incident intensity:
W
abs
= C
abs
I
inc
,
(5.2)
where C
abs
is the absorption cross section.
The sum of the scattering and absorption sections is called the extinction cross
section:
C
ext
= C
sca
+ C
abs
.
(5.3)
In practice, the cross sections are usually normalized by the particles area pro-
jected onto a plane perpendicular to the incident beam, and defined as Q
ext
, Q
sca
,
Q
abs
.
5.1.1.1 polarization-dependent properties
Particles can be thought as miniature
polarizers and retarders. Assuming that the incident field is polarized, the polariza-
tion of the scattered field would depend on the nanoparticle shape, size and the di-
rection of scattering. Even unpolarized light can become partially polarized upon
particular direction of scattering.
The polarization properties of a particle can be described, for example, with Jones
matrix. Assuming that the incident wave, with the wavenumber k, propagates along
the z axis, the general relation between incident and scattered fields can be written
in the following from [
132
]:
E
sca
=
e
ik
(r-z)
-ikr
[S]E
inc
.
(5.4)
For the known direction of incident light and a particular chosen direction of the
scattered light, the scattering plane can be introduced, spanned by these two direc-
tions. Now fields can be decomposed into two orthogonal components, one parallel,
the other perpendicular to the scattering plane:
E
,sca
E
,sca
=
e
ik
(r-z)
-ikr
S
11
S
12
S
21
S
22
E
,inc
E
,inc
,
(5.5)
The elements S
jk
of the scattering matrix [S] (or Jones matrix) are complex-valued
functions of the scattering direction defined, for example, by and in polar coor-
dinate system.
The scattering Mueller matrix can also be introduced, connecting the Stokes pa-
rameters, which can be measured directly [
132
]:
I
sca
Q
sca
U
sca
V
sca
=
1
k
2
r
2
S
11
S
12
S
13
S
14
S
21
S
22
S
23
S
24
S
31
S
32
S
33
S
34
S
41
S
42
S
43
S
44
I
inc
Q
inc
U
inc
V
inc
,
(5.6)
REVIEW
127
here
I =
k
2
E E
+ E
E
,
Q =
k
2
E E
- E
E
,
(5.7)
U =
k
2
E E
+ E
E
,
V =
k
2
E E
- E
E
.
(5.8)
It can be noted that all the parameters I, Q, U, V are real numbers and can be mea-
sured by optical means with quadric detector, without the necessity to measure the
phase of the signal, unlike in the case of characterization by the Jones matrix. This
subject was discussed in more detail in Chapter
3
.
An example calculation can be found in reference [
133
], where polarization prop-
erties of light scattered by four different homogeneous particles: a prolate spheroid,
an oblate spheroid, a finite cylinder and a bisphere with touching components were
considered. The exact analytical solution exists for such scattering problems, which
made it possible to verify T-matrix and DDA methods techniques [
133
]. The parti-
cle was illuminated by a plane harmonic wave propagating along the positive z-axis.
The scattering matrix [S] = S
ij
was computed as a function of the scattering angle.
It was shown for all scattering angles the elements S
13
, S
14
, S
23
, S
24
, S
31
, S
32
, S
41
,
S
42
vanish and S
11
= S
22
, S
12
= S
21
, S
34
= -S
43
, S
33
= S
44
. Therefore only
the remaining elements S
11
, S
21
, S
33
, S
43
were considered. A strong polarization
dependence was revealed, especially in the case of prolate spheroid. Another exam-
ple, revealing a strong polarization dependence of optical absorption can be found
in [
134
], where a prolate spheroid with an aspect ratio of 1.6 and a major axis of 8
nm embedded in silica was illuminated with linear polarized plane wave.
The extensive study of polarization properties of light scattered by spherical par-
ticles can be found in the original Mie's paper [
135
] as well as in [
136
,
137
], where it
was shown that for a particle smaller than 100nm the scattered light behaves accord-
ing to the Rayleigh scattering law and has the maximum polarization at scattering
angle = 90
o
. Even if particles are illuminated by unpolarized light, the scattered
light becomes partly polarized. The scattering patterns of a spherical particles were
also considered in [
136
], where computation method was based on numerical eval-
uation of Mie's coefficients. It was shown that the maximum of polarization moves
to = 120
o
with the increase in the size of a particle. For particles with sizes larger
than 100nm the degree of polarization diminishes rapidly. Light scattered sideways
is always linearly polarized, regardless of the particle size [
136
,
137
].
We conclude, that even spherically symmetric particles have nontrivial polariza-
tion properties which depend on the scattering angle, particle size and material prop-
erties from which the particle is made. Even a small variation in material absorption
is causing significant change in the polarization properties of scattered light.
128
OPTICAL PROPERTIES OF NANOPARTICLES
5.1.1.2 Size dependent effects
The absorption and scattering phenomena can be
viewed as a size-dependent process. The first extensive review of this subject was
given in Mie's original paper [
135
], where absorption and scattering spectra were
calculated for gold nanoparticles with various sizes.
The classical Mie's approach can be applied with no limitations to nanoparticles
larger than 20 nm [
138
]. For smaller particle the material properties would differ
significantly from measured bulk material properties. The imaginary part of the di-
electric function, which is responsible for the absorption, increases by a factor of 10
compared to the bulk value for particles of 2.5 nm whereas for 20 nm particles the
difference is only of a factor 1.5
- 2, depending on wavelength [
138
].
For a particle of a few nanometers in size the mean free path of conduction elec-
trons is limited by the particle boundary, which interferes with the electron relax-
ation time and leads to the increase in absorption at a narrow wavelength range.
This effect can be taken into account by introducing an extra damping term into the
Drude-Sommerfeld free-electron model.
The experimental investigation performed by Soonnichsen [
138
] confirmed that
the quantum treatment is not necessary for metallic particles larger than 20nm. It was
also shown that the collective oscillation dephasing is adequately described by single
electron dephasing, following from the classical solution. The surface scattering
effect and chemical damping by the outside medium can be negligible as well. Thus
the bulk material properties can be successfully applied to simulation of particles
with sizes larger then 20 nm.
5.1.1.3 Shape and substrate dependent effects
As was mentioned previously,
the exact analytic solutions to the scattering problems is known only for a few simple
geometries. Numerical methods, such as discrete dipole approximation, have to be
applied in the general case.
The size dependent scattering and absorption properties of non-spherical particles
were studied, for example, in [
139
]. Discrete dipole approximation (DDA) method
was chosen for simulations of absorption and scattering spectra of gold spheres,
cubes and rods, ranging in size from 10 to 100 nm. The problem of light scatter-
ing by silver particles with various geometries (spheres, cubes, truncated cubes) was
also reviewed in [
134
]. First the polarizability was calculated using the Clausius
Mossotti relation with dielectric function of bulk silver. It was observed that the
optical response below 325 nm is independent of a particle shape and is defined by
intra-band transitions in silver.
The optical properties of particles are also strongly dependent on the substrate
properties. For example, if a nanoparticle is deposited on a substrate, the substrate
can be considered as an array of dipoles interacting with the nanoparticle [
140
]. Al-
ternatively, the model can be simplified by replacing the substrate with an induced
image of the nanoparticle, thus dipolar interaction between the particle and its im-
age can be considered instead [
134
]. However such dipolar interaction leads to the
strongly inhomogeneous field, especially when the particle is close to the substrate.
In such a case multipolar interactions have to be considered [
141
].
SIMULATION OF LIGHT SCATTERING BY SMALL PARTICLES
129
5.2
Simulation of light scattering by small particles
The problem of light scattering by a spherical particle made of an arbitrary material,
including absorbing materials such as metals, can be solved exactly by analytical
methods. The exact solutions are also known for spheroids, or infinite cylinders.
The solution is usually obtained by expanding the incident, scattered, and internal
fields in a series of vector spherical harmonics. The expansion coefficients are found
from the boundary condition ensuring continuity of tangential components of the
electric and magnetic fields across the surface of a particle. Observable quantities
are expressed in terms of the coefficients.
An interesting overview of the early work was given in [
142
,
137
,
143
], an ex-
tensive Mie's theory review can be found in [
136
,
143
]. As early as in 1863 Al-
fred Clebsch (18331872) published a general solution of the elastic wave equation
in terms of the vector wave functions [
144
] and Ludvig Lorenz (18291891) com-
pletely solved the problem in terms of the ether theory. Lord Rayleigh (John Strutt)
(18421904) introduced his theory of elastic light scattering by small dielectric par-
ticles in 1871 [
145
,
146
]. The particles were considered to be much smaller than the
wavelength of the light so that the general electromagnetic problem can be reduced
to electrostatic problem of an isotropic, homogeneous, dielectric sphere in a uniform
field.
The interaction of electromagnetic waves with a perfectly conducting sphere was
reviewed by Thomson [
147
]. The solution was generalized by Hasenorl [
148
] who
introduced a non-zero conductance for metals. His paper actually already gives the
full solution contained in Mie's later publication [
137
]. Ehrenhaft [
149
] gave a rigor-
ous treatment of the scattering of light by small absorbing spheres, which is an even
more elegant than Mie's work [
137
]. A similar treatise was given by Debye [
150
],
who provided an alternative approach by utilizing two scalar potential functions.
Finally in 1908 Gustav Mie presented his solution to the problem in recognizably
modern notation, and presented a comprehensive comparison of experimental re-
sults with theoretical findings, providing many numerical examples [
135
]. Although
in his calculation Mie was using only three terms in the expansion series it was suffi-
cient to completely explain all the optical phenomena observed until then. All Mie's
conclusions published in his paper from 1908 remain valid. Today the problem of
light scattering by a homogeneous isotropic sphere is firmly attached to the name of
Gustav Mie, although Mie was rather the last one who solved this problem.
Mie had to limit his results to particles smaller than 200 nm due to the limits
imposed by calculation. The calculations were performed by hand, and only three
first terms in the infinite series were considered. He also limits his finding to spheri-
cal particles, although he knew that ellipsoidal particles can be treated in the similar
way [
137
]. Today, with availability of computers, these limitations can be easily
removed.
The exact solutions to Maxwell's equations are known only for special geometries
such as spheres, spheroids, or infinite cylinders, so approximate methods are required
for the general case. An excellent review of various methods can be found in [
151
,
130
OPTICAL PROPERTIES OF NANOPARTICLES
152
,
153
,
154
,
129
,
127
]. Mie's theory is usually used as reference, to validate other
methods.
5.2.1
Analytical solution to the problem of electromagnetic wave scat-
tering on spherical particles
In this section we briefly outline the steps taken to solve the problem of electromag-
netic wave scattering on small spherical particles. An excellent review of the subject
can be found in [
126
,
127
,
128
,
132
].
We start with formulation of the problem. Let us assume that an incident electro-
magnetic field E
inc
, H
inc
interacts with a particle that occupies a bounded region.
The total fields E
tot
, H
tot
in the surrounding medium are equal to the sum of the
incident and the scattered fields:
E
tot
= E
inc
+ E
sca
H
tot
= H
inc
+ H
sca
.
(5.9)
We are interested in a free space solution outside the particle boundaries, therefore
we assume that there are no sources in space. Hence, Maxwell's equations can be
reduced to the vectorial Helmholtz equations, as was shown in Chapter
1
:
(
2
+ k
2
(r))E(r) = 0,
(
2
+ k
2
(r))H(r) = 0,
(5.10)
where
k
2
(r) = (r)(r)
2
c
2
= n
2
(r)k
2
o
.
(5.11)
Now let us consider the boundary condition: (i) the tangential components of
the total fields have to be continuous across the particle surface; (ii) the radiation
condition, requiring that the tangential components of the electric and magnetic fields
have to approach zero at rate 1/r as the distance r from the origin approaches infinity.
The solution to the vector Helmholtz equations (
5.10
) can be searched in terms of
scalar basis functions, obtained as a solution to the scalar Helmholtz equation:
(
2
+ k
2
(r))(r) = 0.
(5.12)
In spherical coordinates equation (
5.12
) becomes:
1
r
2
r
2
(r) +
1
r
2
sin()
sin()
+
1
r
2
sin
2
()
2
2
+ k
2
= 0.
(5.13)
Separating the variables with ansatz
(r, , ) = R(r)()(),
(5.14)
SIMULATION OF LIGHT SCATTERING BY SMALL PARTICLES
131
the three ordinary differential equations are obtained:
0 =
d
2
dr
2
+ k
2
-
l(l + 1)
r
2
(rR),
0 =
1
sin()
d
d
sin()
d
d
+ l(l + 1) -
m
2
sin
2
()
,
0 =
d
2
d
2
+ m
2
.
(5.15)
The solution to the radial equation in (
5.15
) can be expressed in terms of special
functions u
(k)
l
:
R(r) = u
(k)
l
,
(5.16)
where
u
(1)
l
= J
l
(r) are spherical Bessel functions,
u
(2)
l
= N
l
(r) are spherical Neumann functions,
u
(3)
l
= H
(1)
l
(r) = J
l
(r) + iN
l
(r) are spherical Hankel functions of the first kind
and
u
(4)
l
= H
(2)
l
(r) = J
l
(r) - iN
l
(r) are spherical Hankel functions of the second kind.
The solutions to the polar equation in (
5.15
) are the associated Legendre func-
tions:
() = P
m
l
(),
(5.17)
where m = 0,
±1, ... ± l.
Finally the solutions to the azimuthal equation are simply harmonic functions:
() = e
im
.
(5.18)
Thus the solution to the scalar Helmholtz equation in spherical coordinates can
be written in the following form:
(k)
l,m
(r, , ) = u
(k)
l
P
m
l
()e
im
.
(5.19)
Using the above scalar basis function we can create vectorial basis functions:
L
(k)
l,m
(a) = · (
(k)
l,m
· a),
M
(k)
l,m
(a) = × (
(k)
l,m
· a),
N
(k)
l,m
(a) =
1
k
× M
(k)
l,m
.
(5.20)
Here a is some constant vector or the position vector a = r, in such a case func-
tions (
5.20
) satisfy the vector Helmholtz equation in spherical coordinates (
5.10
) [
127
,
132
].
However, we assumed that Maxwell's equations are divergence-free , i.e. the
particle is immersed in a charge free medium, but the functions L
(k)
l,m
(a) are not
132
OPTICAL PROPERTIES OF NANOPARTICLES
divergence-free and therefore should be excluded from consideration. Hence, the
incident, scattered and internal electric fields can now be represented in terms of
spherical vector wave functions:
E
inc
(k
o
r)
=
L
l
=0
l
m
=-l
M
l,m
M
(1)
l,m
(k
o
r) +
N
l,m
N
(1)
l,m
(k
o
r) ,
E
sca
(k
o
r)
=
L
l
=0
l
m
=-l
M
l,m
M
(3)
l,m
(k
o
r) +
N
l,m
N
(3)
l,m
(k
o
r) ,
E
int
(kr) =
L
l
=0
l
m
=-l
c
M
l,m
M
(1)
l,m
(kr) + c
N
l,m
N
(1)
l,m
(kr) .
(5.21)
Here E
int
defines the field inside the particle and k is the wavenumber inside the
particle.
To satisfy the boundary condition at the origin a non-singular at the the origin
spherical vector wave basis functions of the first kind have to be used. The scattered
field has to satisfy the radiation condition, therefore it can be expanded in spherical
vector wave functions of the third kind.
Considering the continuity of the tangential field components across the surface
of the particle, the boundary condition can be set:
^n
+
× (E
inc
(k
o
r) + E
sca
(k
o
r)
- E
int
(kr)) = 0,
^n
+
× (H
inc
(k
o
r) + H
sca
(k
o
r)
- H
int
(kr)) = 0.
(5.22)
Here ^
n
+
denotes the outward pointing vector, normal to the boundary surface of the
particle.
The boundary conditions yield a system of linear equations, from which the un-
known expansion coefficients
M
l,m
,
N
l,m
and c
M
l,m
, c
N
l,m
can be expressed in terms
of the known expansion coefficients
M
l,m
,
N
l,m
, characterizing the known incident
field.
For spherical particles this system of linear equations can be inverted analytically,
and leads to the well-known Mie's solution [
135
,
150
].
Observable quantities such as scattering C
sca
, absorption C
abs
and extinction
C
ext
= C
sca
+ C
abs
cross sections can be expressed as follows [
132
]:
C
ext
=
2
k
2
l
=1
(2l + 1)Re(a
l
+ b
l
),
C
sca
=
2
k
2
l
=1
(2l + 1)(|a
l
|
2
+ |b
l
|
2
).
(5.23)
SIMULATION OF LIGHT SCATTERING BY SMALL PARTICLES
133
Here
a
l
=
n
l
(nx)
l
(x) -
l
(x)
l
(nx)
n
l
(nx)
l
(x) -
l
(x)
l
(nx)
,
b
l
=
l
(nx)
l
(x) - n
l
(x)
l
(nx)
l
(nx)
l
(x) - n
l
(x)
l
(nx)
,
(5.24)
with size parameter x = ka, where a is the radius of the sphere and k is the
wavenumber of the incident light in the surrounding medium,
l
and
l
are the
Riccati-Bessel functions, and n =
is particle refractive index.
Alternatively the Mie solution can be derived with help of Debye potentials [
110
].
The described procedure can be applied in any other coordinate system in which
the scalar Helmholtz equation becomes separable. Hence various particles with non-
spherical symmetry can be considered as well [
155
,
156
]. However, the basis func-
tions might no longer be orthogonal and a resulting system of linear equations can
be only inverted numerically.
5.2.2
The quasi-static approximation
In this section we review the quasi-static approximation, which can significantly
simply analysis of non spherical particles in the following section.
For a particle with size significantly smaller than the wavelength of the incident
light, the electric field of the light can be viewed as an uniform and static. Thus the
interaction between a small particle and light can be described in terms of electrostat-
ics rather than electrodynamics. The electron cloud of the particle can be viewed as
displaced by the electric field, hence a restoring force arises from Coulomb attraction
between electrons and nuclei, resulting in electron cloud oscillation.
If a homogeneous, isotropic sphere is placed in medium with a different permit-
tivity, subjected to a uniform static electric field, a charge will be induced on the
surface of the sphere. The electric fields inside and outside the sphere (of radius
R) can be described by Laplace's equation in terms of scalar potentials
1
(r, ) and
2
(r, ) written in spherical coordinates [
132
,
40
]:
2
1
(r < R, ) = 0,
2
2
(r > R, ) = 0,
(5.25)
At the boundary between sphere and medium the potentials must satisfy continu-
ity condition:
1
(R, ) =
2
(R, ),
1
1
r
1
(R, ) =
1
2
r
2
(R, ).
(5.26)
Thus we have the partial differential equation and two boundary conditions. In
addition, by requiring the field at infinity to be equal to the external homogeneous
field E
o
, the solution can be written in the following form:
134
OPTICAL PROPERTIES OF NANOPARTICLES
1
(r, ) = -
3
2
1
+ 2
2
E
o
r cos(),
2
(r, ) = -E
o
r cos() + R
3
E
o
1
-
2
1
+ 2
2
cos()
r
2
.
(5.27)
Now superposing the above two potentials and comparing the result with the po-
tential of an ideal dipole:
(r, ) =
1
4
2
p
· r
r
3
=
1
4
2
p cos
r
2
,
(5.28)
The solution (
5.27
) can rewritten in the following form:
p =
2
E
o
,
= 4R
3 1
-
2
1
+ 2
2
,
(5.29)
where (R,
1
,
2
) is the polarizability, which depends on the particle size and mate-
rial permittivity,
1
is the material permittivity, from which the particle is made and
2
is the medium permittivity.
The above result states that a sphere in an electrostatic field is equivalent to an
ideal dipole, with momentum p =
2
E
o
. Therefore the problem of electromagnetic
wave scattering by a sphere can be reduced to a well studied problem of dipole
radiation. The cross section for absorption and scattering of field by an ideal dipole
exposed to a plane wave source is given by the following relations [
132
,
40
]:
C
abs
= k (),
C
sca
=
k
4
6 |
|
2
,
(5.30)
here k = n
2
c
is the wavenumber of incident wave in the medium outside the par-
ticle. However it is more convenient to use in practice unitless parameters called
efficiency:
Q
abs
=
C
abs
R
2
,
Q
sca
=
C
sca
R
2
,
(5.31)
5.2.3
The ellipsoidal shape particles
The analytical solution for ellipsoidal shape particle can be found in a similar way as
in the case of spherical particle by introducing ellipsoidal coordinates and exploiting
particle symmetries to solve Laplace's equation. The derivation can be found for
example in [
132
,
40
]. Here we review the quasi-static approximation instead.
CONCLUSION
135
As before we solve the problem of electric charge distribution on a particle of a
given shape subjected to a uniform static electric field. The result can be written in
the similar form as in the previous section. For a cigar-shaped ellipsoid with two
equal length principle axes, defined by b and a > b (such spheroid is called prolate)
we have:
p =
2
E
o
,
x
=
4
3
ab
2
1
-
2
2
+ L
x
(
1
-
2
)
,
(5.32)
The material permittivities for the particle and the surrounding medium are defined
as
1
and
2
, respectively. The parameter L
x
is called depolarization factor and
depends on the relative orientation of the long axis a of prolate spheroid and the
polarization direction of incident light, which is incident transversely to the plane
on which particles are deposited. For incident light with the polarization direction
parallel to the long axis of the prolate ellipsoid x = a the depolarization factor is
defined as:
L
a
=
1 - e
2
e
2
1
2e
ln
1 + e
1 - e
- 1 ,
(5.33)
and for the polarization direction perpendicular to the long axis x = b the depolar-
ization factor L
b
can be defined in terms of L
a
[
132
]:
L
b
=
1
2
(1 - L
a
) ,
(5.34)
here parameter e =
1 -
2
is the eccentricity, with the aspect ratio =
b
a
.
The cross sections C
abs
, C
sca
, C
ext
defined in the previous section by equation
(
5.30
) can be applied without modification, but the efficiencies in (
5.31
) should be
modified by replacing the particle radius with the product of its principle axes:
Q
j
=
C
j
(ab
2
)
2
3
.
(5.35)
5.3
Conclusion
At this point we know how to define optical properties of metals, how to simulate
properties of separate nanoparticles, either exactly or with help of the quasi-static ap-
proximation, and how to model films with different morphology. In the next Chapter
we apply this knowledge to design the optimal coating for TFBG based refractive
index sensors.
CHAPTER 6
THE OPTIMAL PARAMETERS FOR A
NANOPARTICLE-BASED COATING
In this chapter we study optical properties of metal nanoparticles embedded in a ho-
mogeneous ambient medium, with particular attention to the absorption properties
of nanoparticles. The absorption properties allow us to compute the effective refrac-
tive index of a medium by applying the Kramers-Kronig relation and connecting the
imaginary and real part of the refractive index. Next we discuss the choice of an
optimal nanoparticle-based coating.
6.1
The idea behind sensitivity enhancement
Although the optical properties of bulk materials are known, the optical properties
of a particle can not be directly deduced. The particle size and its shape introduce
additional resonances and hence additional absorption. The absorption peaks are
not only influenced by the shape and size of the particle but also by the external
mediums refractive index. Hence, if the refractive index of the external medium is
changed the particles resonances will change their locations as well. Thus a coating
layer can be created with optical properties defined by the refractive index of an
external medium. In such a material not only the imaginary part of the refractive
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
137
138
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Figure 6.1
The shift in absorption peak (red) and the corresponding shift in real refractive
index of a medium consisting from nanoparticles.
index becomes sensitive to the environmental changes, but also the real part through
the Kramers-Kronig connection, as shown in Figure
6.1
.
We are mainly interested in the real part of a materials effective refractive index, as
it defines the positions of resonances, which can be observed in the TFBG spectrum,
and hence the change in external medium can be easily detected.
The connection between high quality factor resonances of TFBG structure and
low quality factor resonance of nanoparticles is schematically shown in Figure
6.2
.
6.2
Simulation of optical properties of metal nanoparticles
6.2.1
The quasi-static approximation
Let us first simulate optical properties of small spherical nanoparticles made of vari-
ous metals.
Following the results from Chapter
4
we obtain complete characterization of op-
tical properties of 11 metals in the wide range of frequencies (from = 0 up to
10 - 20 eV ). The metals were described phenomenologically by LorentzDrude
model, in which parameters of six oscillators were fitted for the best consistency
with experiments [
5
].
We start with the quasi-static approximation, which is valid for the problem of
light scattering on spherical particles with the particle size significantly smaller then
the wavelength. The wavelength of interest is = 1.5 m, hence the the approxi-
mation should yield the correct result for particles about 100 nm in diameter.
Considering the LorentzDrude model from Chapter
4
and using equations (
5.29
)-
(
5.31
) the absorption and scattering efficiencies can be plotted for various metals as
shown in Figure
6.3
.
SIMULATION OF OPTICAL PROPERTIES OF METAL NANOPARTICLES
139
Figure 6.2
The schematic representation of connection between high Q resonances of TFBG
and low Q resonances of nanoparticles.
The next Figure
6.4
shows the absorption and scattering efficiencies of gold nanopar-
ticle immersed in solutions with various refractive indexes.
From the above results we can draw the following conclusions:
1. From equations (
5.29
) and (
5.31
) we note that the absorption Q
abs
() and scat-
tering Q
sca
() curves are independent of the particle size, hence the position of
plasmon resonance is determined only by the dielectric functions of the particle
1
and the surrounding media
2
. The position of the resonance is given by the
pole of the polarizability in equation (
5.29
), i.e. when
1
+ 2
2
is approaches
to zero. This pole is commonly referred as a polarization mode or plasmon
resonance.
2. The position of the absorption resonance is different for different metals. As
can be seen from Figure
6.3
for Au, Ag and Cu metals the absorption resonance
is located in the visible band, wheres for Al it is located at IR band, not far away
from the interband absorption peak. We can also conclude that the Al coating
might look promising with regards to sensing application in IR band.
3. Although we used a first order approximation, the quasi-static approximation,
valid only for a particle with size significantly smaller than the wavelength, we
came to the correct qualitative as results reported in the literature. The obtained
140
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Figure 6.3
Simulated absorption efficiency of 30 nm spherical nanoparticle made of Au,
Ag, Cu and Al metals, as a function of photon energy (the graphs for Ag and Al were divided
and multiplied by 10, respectively).
Figure 6.4
Simulated absorption and scattering efficiency of 30 nm gold spherical NP,
immersed in media with various refractive indexes.
absorption spectra shown in Figure
6.4
are in a good agreement with experi-
mental measurements, predicting the red colors of light transmitted through a
suspension of fine gold particles. This effect happens due to the strong interband
absorption of the shorter wavelength by the gold.
4. With the increase of a particle size the electronic cloud displacement can ex-
hibit high-multipolar charge oscillations, thus several poles in the polarizability
function can appear and significantly affect the absorption spectrum. The quasi-
SIMULATION OF OPTICAL PROPERTIES OF METAL NANOPARTICLES
141
static approximation can no longer be applied, hence the exact Mie's theory has
to be used, as shown in the next section.
142
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
6.2.2
The exact solution
As was discussed in Section
5.2.1
the problem of light scattering by a spherical
particle can be solved exactly. The absorption, scattering and extinction efficiencies
can be written in the form of infinite series [
132
]:
Q
ext
(x) =
2
(Rk)
2
N
l
=1
(2l + 1) (a
l
(x) + b
l
(x)),
Q
sca
(x) =
2
(Rk)
2
N
l
=1
(2l + 1)(|a
l
(x)|
2
+ |b
l
(x)|
2
),
Q
abs
(x) = C
ext
(x) - C
sca
(x),
(6.1)
with a
l
(x) and b
l
(x) given by:
a
l
(x) =
m
l
(mx)d
x
l
(x) -
l
(x)d
x
l
(mx)
m
l
(mx)d
x
l
(x) -
l
(x)d
x
l
(mx)
,
b
l
(x) =
l
(mx)d
x
l
(x) - m
l
(x)d
x
l
(mx)
l
(mx)d
x
l
(x) - m
l
(x)d
x
l
(mx)
.
(6.2)
Here x = k
o
R is the size parameter, dependent on the radius R of the particle, the
wavenumber k
o
of the incident light ( k
o
=
c
) and the relative complex refractive
index of the sphere m = n
particle
/n
medium
. The wavenumber k corresponds to the
wave in the surrounding medium.
The Riccati-Bessel functions
l
(x) and
l
(x) are defined as follows:
l
(x) =
x
2
J
l
(x),
l
(x) =
x
2
H
1
l
(x),
(6.3)
where H
1
l
(x) = J
l
(x) + iY
l
(x) is the Hankel function of the first kind, and J
l
(x)
and Y
l
(x) are Bessel functions of the first and the second kind.
The number N in equation (
6.1
) is the number of terms in Mie series. In order
to apply these equation in practice the series must be limited to a finite number of
terms N
max
. Analysis of the convergence behavior of these series reveals that the
sufficient number of terms, dependent only on the size parameter x, is determined by
the following equation [
157
]:
N
max
= x + 4x
1
3
+ 1
(6.4)
The relation (
6.4
) applies to all refractive indices since the series (
6.1
) convergence
is determined entirely by Bessel functions of x alone, as can be seen from (
6.2
). Mie
himself considered only three terms as all calculations were done by hand. At the
SIMULATION OF OPTICAL PROPERTIES OF METAL NANOPARTICLES
143
present moment it's possible to compute hundreds of terms in several minutes on a
typical PC.
The terms in the Mie series were computed with the help of Mathematica software
from Wolfram Research. The code is presented in Appendix
B
, and results for the
absorption, scattering and extinction efficiencies are shown in Figure
6.5
.
Figure 6.5
The absorption, scattering and extinction efficiencies as functions of the size
parameter
x, for a particle with the relative refractive index m = 5 + j0.4.
Next, considering optical properties of metals we can compare absorption effi-
ciency for 30 nm spherical nanoparticle made of various metals, as shown in Fig-
ure
6.6
. The figure also shows the results based on the quasi-static approximation.
Figure 6.6
The absorption efficiency of
30 nm spherical nanoparticle made of various metals
( Au, Ag, Cu and Al ) as a function of photon energy. The result were obtained with use of the
exact Mie's theory and with the quasi-static approximation theory.
144
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
It can be noted that the proximate and the exact results are in good mutual qualita-
tive correspondence, although quantitative differences are also noticeable, especially
for nanoparticles made from silver. This confirms the well known fact, which states
that the optical properties of small metal particles are predominately defined by the
bulk material properties from which the nanoparticles are made.
Next we will consider the influence of the refractive index of the surrounding
medium on the absorption and scattering spectra of gold nanoparticles. The results
are shown in Figure
6.7
. Comparing these results with the quasi-static approximation
Figure 6.7
Absorption and scattering efficiency of
30 nm gold spherical nanoparticle,
immersed in media with various refractive indexes.
Figure
6.4
we can conclude that the exact Mie solution predicts a significantly differ-
ent response, with a smaller correlation between the change in the external refractive
index and the absorption and scattering spectra.
The most striking difference is observed when the absorption and scattering effi-
ciencies are plotted against the particle radius, as shown in Figure
6.8
. In the case
of the exact solution a number of local resonances is observed, where in the case
of the quasi-static approximation only a single resonance is predicted. The single
resonance is defined by the approximation equation (
5.30
) where Q
abs
R and
Q
abs
R
4
, in accordance with the Rayleigh scattering theory [
145
,
146
].
Our goal here is to find the optimal set of system parameters, such that the lo-
cal plasmon resonances of nanoparticles can be effectively coupled to the cladding
modes excited in the TFBG sensor. We have a particular interest in the absorption
efficiency Q
abs
of nanoparticles as it affects the effective refractive index of the sur-
rounding medium detected by the sensor. With this in mind we can plot Q
abs
as a
function of various parameters, and thus study the parametric space.
First, let us study the dependence on the particle radius and the material optical
properties from which particle is made. Figure
6.9
shows the absorption efficiency
Q
abs
as a function of particle radius and incident photon energy. The distinct location
of the plasmon resonances is intrinsically connected with the material from which the
SIMULATION OF OPTICAL PROPERTIES OF METAL NANOPARTICLES
145
Figure 6.8
The size dependence of the scattering and absorption efficiencies of gold NP
illuminated by electromagnetic wave at
= 560nm. The particle is assumed to be in the
medium with refractive index
n = 1.
particle is made. It is also should be noted that as the particle size increases higher
modes of plasmon oscillation can occur, such as quadrupole modes and higher order
modes. However such higher order modes are not always excited, for example if the
particle is illuminated by a plane wave, therefore the energy absorption might not be
observed in the spectrum. Such modes are often referred as dark plasmon modes.
We can summarize our findings as follows:
1. Both theories: the approximation quasi-static theory and the exact Mie theory
are predict similar qualitative results, stressing the importance of the metal di-
electric function from which the particle is made. However the predictions vary
significantly for large particles.
2. From Figure
6.9
it can be seen that there is an optimal particle size for each
given wavelength, at which the absorption peak has the greatest value. This
result was rather unexpected, but a similar conclusion can be drawn from Fig-
ure
6.8
, obtained in several other papers as well.
3. The material choice is extremely important. If we seek to excite plasmons in
the visible band, a nanoparticle made of gold is a good option. However, as is
seen from Figure
6.9
, copper can also be a good choice for a relatively large
nanoparticle. If the objective is to obtain a resonance in the IR band aluminum
might be a reasonable choice, although the resonance is extremity weak.
146
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Figure 6.9
The absorption efficiency of gold, silver, aluminum and copper particles as a
function of particle size and incident photon energy (plotted in logarithmic scale).
SIMULATION OF OPTICAL PROPERTIES OF METAL NANOPARTICLES
147
6.2.3
The shape effect. Ellipsoidal nanoparticles.
In the previous section we have established that the quasi-static approximation theory
can provide satisfactory results in the case of small particles. This conclusion was
drawn by comparing the exact scattering theory with the quasi-static approximation.
Therefore we can use the quasi-static approximation to investigate the influence of a
particle geometrical shape on its optical properties.
We have a particular interest in randomly orientated silver nanorod particles,
shown in Figure
6.10
, as some interesting experimental result were reported [
3
].
Figure 6.10
SEM image of the fibre surface coated with gold nanorod particles
Such nanorod particles can be treated as prolate spheroids, with a significant ratio
between its principle axes a and b. Random orientation can be considered simply by
taking the average polarizability [
158
]:
=
1
3
a
+
2
3
b
,
(6.5)
here
a
and
b
are polarizabilities along a and b principle axes, respectively. The
cross sections for absorption and scattering processes can be found with help of
equations
5.30
and
5.31
, as was discussed previously.
The simulation result is shown in Figure
6.11
. It can be observed that silver and
gold particles have distinct resonances in the visible band and look promising for the
further investigation.
We next show that by changing the aspect ratios between principal axes of an el-
lipsoid particle we can tune the absorption resonance into the operational range of
TFBG sensor. The figure
6.11
shows the absorption efficiency of randomly orien-
tated prolate silver particles with various aspect ratios.
The properties of the TFBG sensor coated with elongated nanoparticles might be
simulated in the usual manner, except that we have to account for the film anisotropy,
148
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Figure 6.11
Comparison of the optical absorption and extinction efficiencies of prolate
ellipsoids made from various metals with the principal axes
b = 30, a = 100 nm.
Figure 6.12
Comparison of the absorption efficiency for prolate silver nanoparticles with
different aspect ratios between the principal axes
=
b
a
particularly for the difference in optical properties of the film seen by the tangential
and transverse polarized electric field. The difference arises from the fact that the
position of the absorption resonances of an elongated nanoparticle is a polarization-
dependent function, hence the real part of the effective dielectric permittivity func-
tion is also polarization-dependent. It is convenient to introduce effective dielectric
permittivities along the tangential and transverse axes associated with the film planes,
as shown in Figure
6.13
.
THE OPTIMAL PARAMETERS CHOICE FOR COATINGS BASED ON SPHERICAL NANOPARTICLES
149
Figure 6.13
The asymmetry of the coating made of elongated nanoparticles.
a) The
transversely polarized electric field
E
encounters
dielectric permittivity and b) the
tangentially polarized electric field
E
sees the
dielectric permittivity.
The basis functions of a cylindrical waveguide coated with elongated nanoparti-
cles can be found from the modified equation (
1.21
):
d
2
+
1
d
+ (ln
) d
-
m
2
+1
2
+
k
2
o
+ (ln
) -
2
-
j
2m
2
j
2m
2
+
jm
(ln
)
d
2
+
1
d
-
m
2
+1
2
+
k
2
o
-
2
E
E
= 0,
(6.6)
here
and
are the dielectric permittivities as seen by the transversely and tan-
gentially polarized electric fields E
and E
, respectively.
Our main interest is to increase the TFBG sensor sensitivity operating the in the
IR band. We can expect that by coating the sensor with silver or gold prolate parti-
cles, with appropriate principal axes ratio of about
7..10, the sharp longitudinal
resonance of nanoparticles can be tuned to the operational range of cladding modes
resonances, thus making the TFBG sensor more sensitive to perturbations in the sur-
rounding media. The sensor can even be coated with silver nanorods of various
lengths, we still should expect that a fraction of these particle would have an ap-
propriate aspect ratio and thus would contribute to an increase of the TFBG sensor
sensitivity. We also should note that in the case of prolate particles with high aspect
ratio the quasi-static approximation might not provide a satisfactory prediction, as
was also the case with large spherical nanoparticles.
6.3
The optimal parameters choice for coatings based on spherical
nanoparticles
Our goal is to design a nanoparticle coating with a sharp absorption resonance, lo-
cated in the operational range of the TFBG sensor with location sensitive to the
refractive index of surrounding media. From the previews section we know that sil-
150
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
ver or gold prolate nanoparticles might be an excellent choice, as the position of the
plasmon resonance can be tuned at the exact location by choosing particles with the
proper aspect ratio between its axes (
7..10).
In this section we discuss choice of parameters for coatings based on spherical
nanoparticles. The sensor coated with spherical nanoparticles is schematically shown
in Figure
6.14
.
Figure 6.14
A schematic representation of a TFBG sensor coated with spherical
nanoparticles.
Let us plot absorption efficiencies for various particle sizes and materials. As a
first step we will fix the the imaginary part of the particle permittivity, and com-
pute Q
abs
as a function or real part of the relative complex refractive index
[m] =
=
[n
particle
]
n
medium
and the particle size parameter x =
2
R, as shown in Figures
6.15
and
6.16
.
Our goal here is to choose an optimal material and particle size. There are two
main parameters to be considered: first we need a sharp prominent resonance, so
that the real part of the effective refractive index can be significantly altered, and
the second is the resonance steepness. We have a particular interest in the steepness
of resonances. The steeper is the resonance the more its location is affected by the
refractive index of the external medium, hence the higher is the sensor sensitivity.
We can conclude that it is desirable to have particles made of material with a
small imaginary part of the dielectric permittivity [m]
0.05 and choose a particle
size parameter x such that
[m] 2 - 3.5. We note that the smaller is the [m]
parameter, the steeper is the resonance, unfortunately for small
[m] we can no
longer obtain a prominent resonance. Once the real part of dielectric permittivity
was chosen, we can study the effect of material absorption in more detail, as shown
in Figure
6.17
. We note that for
[m] < 0.05 the resonance amplitude starts to
decrease, hence a material with small but noticeable loss is required.
THE OPTIMAL PARAMETERS CHOICE FOR COATINGS BASED ON SPHERICAL NANOPARTICLES
151
Figure 6.15
The absorption efficiency as a function of the particle size parameter
x and the
real part of the relative complex refractive index
[m] at various fixed values of [m].
152
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Figure 6.16
The absorption efficiency as a function of the particle size parameter
x and the
real part of the relative complex refractive index
[m] at various fixed values of [m].
THE OPTIMAL PARAMETERS CHOICE FOR COATINGS BASED ON SPHERICAL NANOPARTICLES
153
Figure 6.17
The absorption efficiency as a function of the particle size parameter
x and the
imaginary part of the relative complex refractive index
[m] at various fixed values of [m].
154
THE OPTIMAL PARAMETERS FOR A NANOPARTICLE-BASED COATING
Considering the large value of the required relative refractive index m our choice
is limited to only a few materials with high refractive indices, such us Zirconium
Silicate and Titanium Dioxide.
The refractive index of Titanium Dioxide is defined by equation (
6.7
) and is shown
as a function of frequency in Figure
6.18
.
n = 5.913 +
0.2441
2
- 0.0803
(6.7)
Figure 6.18
Refractive index of titanium dioxide,
T iO
2
.
Considering the operational wavelength = 1.55m and the refractive index of
the external medium (n
medium
= 1.318 for water) we find that the relative refrac-
tive index of T iO
2
particle immersed in water is m
1.9. The horizontal line cor-
responding to m
1.9, as shown in Figure
6.19
, crosses several Mie resonances,
hence a proper particle size can be chosen at one of such crossing points. Let us
choose the particle size parameter x = 2.1 , corresponding to the second resonance.
Then the particle radius can be found from the relation x = k
o
n
m
R and for the above
mentioned parameter is R = 393 nm.
We note that we only barely met the required conditions. With the available re-
fractive index for particles we reached the very beginning of the resonance peak, as
shown in Figure
6.19
. A material with a higher refractive index in the IR band is
highly desirable, unfortunately our choice is limited.
The properties of the TFBG sensor coated with spherical nanoparticles can be
modeled with equation (
6.6
) as previously. In the case of sparsely deposited spheri-
cal particles the dipole-dipole radiative interaction between the particles can be ne-
glected. Thus the effective dielectric permittivity function can be assumed to be an
isotropic function
=
. Hence, once the particles absorption is known the re-
fractive index can be computed with help of the Kramers-Kronig relation (
4.1
) and
the MaxwellGarnett model, used to compute the effective refractive index of the
particles embedded in the surrounding medium.
CONCLUSION
155
Figure 6.19
The optimal size of a particle made of Titanium Dioxide.
6.4
Conclusion
In this chapter we presented results of our simulations of spherical and ellipsoidal
nanoparticles made of various materials. We presented the method for increasing
TFBG sensor sensitivity based on resonant coupling between the modes of TFBG
sensor and the modes of particles deposited on the fibre surface. The particle are
chosen to have a sharp resonance with its location sensitive to the external refractive
index.
It was shown that the metallic nanorod-like particles made of silver or gold look
promising for the sensitivity-enhancing coating. The resonance of nanorod particles
can be tuned exactly to the operational range of the sensor by choosing particles with
7..10 aspect ratio between its principal axes.
Alternatively, dielectric spherical particles with high refractive index can be cho-
sen. We found that particles made of Titanium Dioxide with the diameter D
800 nm
look promising for the sensitivity enhancing coating.
We should also note that the particle-based coatings should be sufficiently sparse,
so that the interaction between particles, and hence the broadening of the resonance
peak can be neglected. However, the sensitivity enhancement effect is proportional
to the number of particles deposited on the fiber the surface. The effective refractive
index of nanoparticle-based coatings can be computed with the MaxwellGarnett
model (see Section
4.3
), if the particles are assumed to be mutual independent and
non-interacting.
In the next chapter we present our experimental findings.
CHAPTER 7
MODIFICATION OF THE SENSOR
SURFACE WITH VARIOUS TYPES OF
NANO-SCALE COATINGS
In this chapter we present our experimental results on enhancing the TFBG sensor
sensitivity by coating its surface with various types of nano-scale coatings. It is
already known that the TFBG sensor performance can be improved by coating its
surface with a nano-scale metal film [
7
,
15
], thus coupling the TFBG resonances to
a surface plasmon resonance (SPR) excited in the metal film. This technique was
thoroughly investigated in [
7
,
15
]. In the presented work we are mainly interested
on a possible sensitivity improvement with the introduction of nanoparticle-based
coatings.
As was shown in Chapter
6
, we propose using an external resonant system, such
as small nanoparticles coupled to the TFBG resonances, to boost the sensor perfor-
mance. It is well known that the position of a particle resonance is strongly affected
by the surrounding medium, as shown in Figures
6.19
and
6.7
, thus a thin coating
layer created out of such particles, with optical properties strongly dependent on the
refractive index of the external environment, can be engineered to boost the TFBG
sensor sensitivity. For example, such effect was observed in [
3
] where the sensor
was coated with silver nanowires. A 3.5-fold increase in the TFBG sensor sensi-
tivity was reported. In this context we have considered various types of particles.
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
157
158
MODIFICATION OF THE SENSOR SURFACE
However, spherical and ellipsoidal particles are particularly interesting as the exact
analytical solution exists for such cases.
During the course of our research we investigated several coatings, such as metal-
lic films of various morphologies and thicknesses, as shown in Figures
7.1
and coat-
ings with nanoparticles of various shapes and materials. In particular we have tested
coatings with nanocubes of various sizes (40
- 80 nm) synthesized from silver and
gold metals, as shown in Figure
7.5
, coatings with elongated gold and silver nanopar-
ticles, as shown in Figures
7.2
and
7.3
, and coatings with nanospheres, shown in Fig-
ure
7.4
. The particles were deposited at various densities creating films of different
morphologies. The surface images were obtained by means of a scanning electronic
microscope (SEM) and an atomic force microscope (AFM).
Figure 7.1
The SEM mages of the fibre surface coated with gold (top) and copper (bottom)
films of different morphologies and thickness [
1
].
159
Figure 7.2
The AFM (a) and SEM (b) images of the fibre surface coated with silver
nanowires [
3
].
Figure 7.3
The SEM images of gold nanorod based coating
160
MODIFICATION OF THE SENSOR SURFACE
Figure 7.4
The SEM images of
T iO
2
spheres deposited on the fibre surface
Figure 7.5
The AFM and SEM images of silver nanocubes (80nm)
DEPOSITION AND SYNTHESIS TECHNIQUES
161
7.1
Deposition and synthesis techniques
In this section we briefly describe chemical methods of particles synthesis and depo-
sition techniques that were used in the presented work.
7.1.1
Synthesis of silver nanorods and nanowires
Silver nanowires were chemically synthesized as described in [
159
]. The procedure
results in highly crystalline nanowires with smooth surfaces. All the reagents used
for synthesis were obtained from Sigma-Aldrich. All glassware was cleaned using
aqua regia, rinsed in 18.2 M deionized water, and placed in an oven to dry prior to
experimentation.
A 50 mL round bottom flask containing 24.0 mL of anhydrous 99.8% ethylene
glycol (EG) and a clean stir bar were placed in an oil bath set to 150
o
C and allowed
to sit for 1 hour. Using a micropipette, 400 L of 3 mM sodium sulfide dissolved
in EG was added to the flask. Ten minutes later 6 mL of EG containing 0.12 g of
dissolved poly(vinylpyrolidone) (PVP) with a molecular weight of 55000 amu was
injected using a glass syringe immediately followed by the injection of 0.5 mL of
6 mM HCl. After an additional 5 minutes, 2.0 mL of 282 mM 99% + silver nitrate
dissolved in EG was injected slowly using a glass syringe. Upon addition of the
silver nitrate the solution immediately turned black and slowly became a transparent
yellow, then changed to an ochre color as some plating in the flask occurred. The
reaction was allowed to continue until the solution became white.
The reaction was monitored by periodically taking small samples out of the re-
action flask using a pasteur pipette and dispersing it in a quartz cuvette filled with
95% ethanol for UV-visible spectroscopy. The reaction was quenched by placing
the flask in an ice bath after the solution and had become fully white and turbid in
appearance. Purification of the nanowires was achieved by adding 20 mL of ethanol
to the solution and centrifuging it at 13800 g for 20 minutes to remove the excess
PVP, EG, and any reaction by-products. The supernatant was then discarded and the
rods re-dispersed in ethanol by sonication. This process was then repeated several
times at 400 g to separate out the heavier wires from the solution.
7.1.2
Synthesis of gold nanoparticles
The gold nanoparticles were synthesized by the reaction of citrate reduction of chloroau-
ric Acid (H[AuCl
4
]) in water, as was proposed by Turkevich [
160
,
161
,
162
]. The
reducing agent sodium citrate (N a
3
C
6
H
5
O
7
) reduces gold ions to neutral gold
atoms, when the solution becomes supersaturated gold nanoparticles start to form.
Citrate ions act as both a reducing agent and a capping agent. The repulsion of
the negatively charged citrate ions prevents the nanoparticles from forming aggre-
gates [
162
]. As the result of this process spherical gold nanoparticles in the range
from 10 to 20 nm in diameter suspended in water were created.
162
MODIFICATION OF THE SENSOR SURFACE
7.1.3
Deposition of nanoparticles
At the first step, the fibre surface was cleaned and prepared for chemical deposition.
The plastic jacket around the fibre was removed by soaking it into dichloromethane
(CH
2
Cl
2
) and further cleaned to remove organic residue through the following mul-
tistep approach. The uncoated area of the fibre was rinsed with methanol and sub-
sequently treated with freshly prepared piranha solution (mixture of H
2
O : N H
3
:
H
2
O
2
= 5 : 1 : 1) at 70
o
C for 10 minutes to remove the remaining organic residues,
and rinsed in 18.2 M deionized water.
The cleaned fibre was then submersed in a 1% (3-aminopropyl)trime-thoxysilane
(APTMS, Aldrich, 97%, 281778) in methanol for 20 minutes in order to form a uni-
form a positively charged monolayer on the fibre surface. The APTMS molecules as-
sembled on the glass by covalent bonding were exposed to hydroxyl sites (Si
-OH)
on the glass, thus forming a cross-linked self-assembled monolayer (SAM). The
APTMS treatment replaces the hydroxyl groups (OH) adsorbed on the glass (fibre)
substrate (SiO
2
) with APTMS molecules forming a siloxane bond between the Si on
one end of the APTMS molecules and an oxygen atom on the SiO
2
surface [
163
]. As
a consequence, the amino group attached on the other end of the APTMS molecule is
oriented away from the substrate. These amino groups on the APTMS molecules im-
mobilise gold particles onto the substrate because of the affinity of the amino group
to the gold [
163
].
Finally, the APTMS modified fibre was rinsed with methanol and deionized water
followed by drying with a flow of N
2
gas. After drying, the modified fibre was
submerged into a freshly prepared colloidal solution of gold nanoparticles and left
for 24 hours. The results of nanoparticles deposition are shown in Figures
7.2
,
7.3
and
7.5
,.
7.1.4
Electroless metal coating
The fact that a cylindrically symmetric fibre is used makes the deposition of an uni-
form nanometer-scale film quite challenging. Previously in our group the time cali-
brated two-step sputtering process was used in order to deposit gold films [
7
,
164
],
but we faced some reproducibility issues that could only come from coating errors.
In an effort to improve this situation we had investigated electroless plating [
165
]
because of its potential for highly uniform metal coatings on non-planar surfaces, its
low cost and simplicity of implementation (with a potential for batch production),
and its relatively low deposition rate, which facilitates the control of the process
duration. The electroless gold and silver coatings can be deposited reliably and ac-
curately for the fabrication of near infrared TFBG-SPR sensors, but most importantly
the TFBG itself can be used to monitor the deposition process and to stop it at the
optimum film thickness, for example for SPR excitation of metal coated fibres in
water. While coating conventional FBGs with copper and nickel had been described
earlier for the purpose of making the FBG response athermal, the exact thickness and
the uniformity of the films was not critical for those applications [
165
]. Further stud-
ies of thermal stress between such metal coatings and optical fibres were performed
DEPOSITION AND SYNTHESIS TECHNIQUES
163
in [
166
], where it was shown that the metal coating of FBG sensors can be used not
only as a protective layer but also as a temperature sensitivity booster.
The method we were using to coat the TFBG sensor relies on the attachment of
gold nanoparticles on a suitably prepared bare fibre surface, followed by the electro-
less plating process, based on the reduction of metallic ions from a solution to a solid
surface without applying an electrical potential [
167
,
168
].
To initiate the reaction the fibre surface was precoated with gold nanoparticles,
with the method described in the previous section. The electroless plating is based
on the process of negative charge production on the surface of gold nanoparticles
through the oxidation reaction assisted by a reducing agent. The gold nanoparticles
immobilized on the glass surface constitute excellent sites for the reduction of gold
or silver ions by the hydroxyl amine solution. Thus, the nanoparticles attached to the
fibre surface play a catalytic role during the process of electroless plating.
The modified with gold nanoparticles fibre was next dipped into the plating bath.
For the silver coating the plating solution consisted of silver nitrate (AgN O
3
, 0.01 M ),
ammonium nitrate (N H
4
N O
3
, 8.96M), acetic acid (CH
3
COOH, 2.24M), and hy-
drazine hydrate (H
2
N H
2
.H
2
O, 0.4 M ) at a volumetric ratio of 1 : 1 : 1 : 2,
respectively. For the gold plating the solution consisted of 0.01% chloroauric acid
(HAuCl
4
.3H
2
O) and the reducing agent hydroxylamine hydrochloride (N H
2
OHHCl,
0.4 M ) in a 1 : 1 volumetric ratio.
The electroless plating occurs on the surfaces of already immobilized gold nanopar-
ticles, eventually merging into a continuous thin metallic film, as shown in Fig-
ure
7.1
. The film thickness can be controlled in real-time with high precision [
1
].
The film morphology can also be controlled by varying the deposition time, the so-
lution composition, or the type and size of nanoparticles.
As a separate check we further calibrated the monitoring process by performing
Atomic Force Measurements (AFM) on sample films to correlate the TFBG response
with the physical thickness of the films. The AFM measurements provided evidence
that the electroless-deposited gold layer retained some significant granularity. Dur-
ing the deposition, we monitored the wavelengths and amplitudes of several reso-
nances in the transmission spectrum.
164
MODIFICATION OF THE SENSOR SURFACE
7.2
The optimal metal coating for Surface Plasmon Resonance (SPR)
excitation
The sensitivity of TFBG sensor can be enhanced by coating its surface with a thin
metal film, of a proper thickness, thus allowing the energy coupling between the
backward-propagating cladding modes and the collective electron oscillation on the
metal surface, the so-called surface plasmon resonances (SPR) [
13
]. The sensitivity
enhancement of the TFBG-SPR sensor was previously reported in [
7
,
15
]. Here we
focus on optimization of the coating process.
In order to optimize the SPR sensitivity and resolution, the metal coating thick-
ness and uniformity must be controlled very accurately around the fibre circumfer-
ence, for thicknesses of the order of a few tens of nm. For coatings that are too thin,
the SPR resonances broaden and weaken. On the other hand, when the thickness is
too large, light cannot tunnel across the metal to excite the plasmon polariton on the
outer surface.
Here we present a novel method for the optimal metal coating of TFBG sensors
by electroless plating, thus creating a Surface Plasmon Resonance (SPR) finely tuned
into the operational range of the TFBG sensor [
1
].
The electroless plating of gold or silver metals from solution is a simple and
efficient way to deposit a conformal coating of the required thickness and uniformity
for fibre SPR applications, and furthermore, the deposition process can be controlled
in real-time by means of a TFBG sensor and stopped at the optimum film thickness
for SPR excitation. Therefore, a precise control of the plating process (temperature,
chemical concentrations) is not necessary and reproducible coatings with thicknesses
on the order of 50 nm can be obtained regardless of the speed of the plating process.
A quantitative measure of the quality of the SPR response of the TFBG in the
plating solution is provided by tracking the polarization-dependent loss (PDL) in real
time and used as a criterion to end the plating. When the metal coating has a thick-
ness for which a plasmon wave can be excited by cladding modes, the PDL spectrum
acquires a deep notch that reveals the SPR location [
15
]. Using the polarization-
based optical sensing method, presented in Chapter
3
, we can identify the precise
moment of the plating process at which the SPR signature is observed. When the
spectral notch is maximized, the thickness is optimal for SPR operation. This is best
observed in the density plot of Figure
7.6
, where the envelopes of the PDL spectra
are represented as a colour scale in the horizontal direction at different moments of
the plating process, the vertical direction.
Indeed, it can be clearly seen that after only approximately 7 minutes of gold
film deposition a deep notch in the PDL envelope occurs (darkest blue colour in
Figure
7.6
(a)). The individual PDL spectrum corresponding to this plating time is
shown as Figure
7.6
(b), while Figure
7.6
(c) extracts the time evolution of the PDL
value at the wavelength where the SPR is observed. The optimum point is clearly
seen in the latter figure, hence the deposition process can be interrupted as soon as a
local PDL envelope minimum is detected, regardless of the plating rate.
THE OPTIMAL METAL COATING FOR SURFACE PLASMON RESONANCE (SPR) EXCITATION
165
Figure 7.6
(a) The envelope of PDL spectra, taken continuously along the course of gold
film deposition, and cross sections centered at the point of the deepest notch: wavelength
= 1542 nm (b) and time = 7 min (c)
We should note that the SPR signature appears suddenly, hence the monitoring
process is necessary. Figure
7.6
(c) further shows that the PDL envelope evolution
slows down gradually for longer plating times. These effects are due to both the
self-termination of the plating process and to the fact that as the metal layer be-
comes thicker it eventually shields the light from the cladding modes from the out-
side medium so that further thickness growth is not detected.
We conclude that using TFBG's, we have presented a new method to monitor the
growth of the plated film in situ and in real time. Most importantly however, we
showed that by monitoring a simple parameter in the polarization-dependent Loss of
the TFBG during plating, the optimum film thickness for SPR operation can be found
with great accuracy, regardless of the plating rate. It is also likely that other metals
could be plated and monitored in similar fashion. The method is inherently limited
in thickness since the evanescent field of the cladding modes must tunnel across to
the outer surface of the metal film in order to detect changes in thickness. When the
thickness exceeds the penetration depth of the light, the thickness growth appears to
saturate. Obviously for the application of fibre SPR, the thickness needed must be
smaller than the penetration depth, so this does not present an actual limitation in
this case.
166
MODIFICATION OF THE SENSOR SURFACE
7.3
Sensitivity enhancement with a nanoparticle based coating
As we discussed in the previous section the sensitivity of a TFBG sensor can be in-
creased by coating its surface with a metal film of the appropriate thickness, thus
creating conditions for SPR excitation. However, such improvement occur only
over a narrow spectral range for which the phase matching condition between fi-
bres cladding modes and the SPR wavevector is satisfied. In this section we present
a new technique for increasing the sensor sensitivity. Instead of a resonance in con-
tinuous metallic film we proposing to use resonances of nanoparticles deposited on
the sensor surface [
3
].
We showed in [
3
] that by sparsely coating the TFBG sensor with randomly ori-
ented silver nanowires, a large number of fibre modes with different wavevectors
and electric field distribution can resonate with nanowires, exciting localized sur-
face plasmon resonances (LSPR). LSPR are collective oscillations of conducting
electrons in the nanoparticles giving rise to strongly enhanced and highly localized
electromagnetic fields, which can be utilized in various sensing devices [
7
].
7.3.1
Coating with silver nanowires
Figure 7.7
The SEM (b) images of the fibre surface coated with silver nanowires [
3
].
The sensor was coated with chemically synthesized silver nanowires
100 nm in
diameter and several micrometres in length, as shown in Figure
7.2
. A UV-vis-NIR
absorption spectrum was collected on a similar sample deposited on a flat glass slide.
The spectrum shown in Figure
7.8
contains two main features: a peak at
380 -
400 nm corresponding to the excitation of transverse plasmon resonances along the
short axis and a broad peak in the NIR region corresponding to the excitation of
longitudinal plasmon resonances along the long axis.
SENSITIVITY ENHANCEMENT WITH A NANOPARTICLE BASED COATING
167
Figure 7.8
UV-vis-NIR absorption spectrum of synthesized nanowire coating (deposited on
a flat glass substrate). The insets illustrate the relative polarization of the Plasmon oscillations
that give rise to the absorption.
The spectral region of interest for TFBG (
[1525, 1590] nm) is highlighted in
grey on the figure. It is seen that a strong extinction signal for silver nanowires exists
at these frequencies. However, since the nanowires were deposited on the glass sur-
faces (fibre or flat substrate) using the self-assembly approach [
169
] (made possible
by their partially negative charge [
170
]), their orientation was not controlled (Fig-
ure
7.2
). Therefore the excitation of the longitudinal plasmons in the 1550
-1600 nm
band by polarized light sources can only occur for those wires that are parallel to the
incident light polarization (i.e. about half on average). It appears impossible to excite
the short (radial) plasmons of these nanowires at the same wavelengths.
At the first step of the characterization process, the TFBG sensor was tested to
determine its refractometric operational range before and after the nanowire coating
deposition. Mixtures of ethylene glycol (EG) and water were used with different
volume ratios, varying from pure water to pure EG, allowing to cover a range of
n = 3.81 × 10
-2
for the refractive index of the surrounding medium. A custom-
made Teflon cell was used to hold the fibre still and straight while being submerged
in varying solutions of ethylene glycol (EG) and water while transmission measure-
ments were carried out with an optical vector analyser (OVA). Figure
7.9
shows the
average insertion loss of the coated and uncoated sensor for various intermediate
values of the surrounding refractive index.
In those spectra, each downward peak corresponds to loss of light in the core, due
to coupling to cladding modes. A discontinuity in the amplitudes of the resonances is
observed on all spectra and its spectral position shifts towards longer wavelengths as
the surrounding index increases. This discontinuity corresponds to the cut-off condi-
tion where the cladding modes become leaky because their effective index becomes
larger than the surrounding index [
11
]. The behavior of both groups of spectra is
similar, apart from the fact that the amplitudes of the resonances are always smaller
168
MODIFICATION OF THE SENSOR SURFACE
Figure 7.9
The TFBG spectrum evolution before (a) and after (b) deposition of nanowires,
for several values of the refractive index of the solution. The concentration of the Ethylene
Glycol in water goes from
0%, 25%, 50%, 75%, 100% from the top to the bottom of the
figure. The corresponding total refractive index change is
n = 3.81 × 10
-2
.
for the coated grating. This is consistent with the assumption that plasmons can be
excited in the nanowires in the 1525
- 1590 nm range, thereby inducing additional
loss in the cladding modes, which results in a decrease of the coupling strength (and
of the resonance amplitude). The observation of the shift in the cut-off wavelength
of the average insertion loss provides a relatively rough (but absolute) estimate of the
refractive index of the solution.
For finer measurements (of small index change increments), observations on a sin-
gle resonance provides better accuracy. The refractometric sensitivity measurements
were performed on the TFBGs before and after the deposition of silver nanowires by
monitoring the response following small changes of the relative EG concentration.
The initial measurements were taken using a well mixed 1 : 1 mixture of EG and
H2O prepared in a large 200 mL beaker and transferred to the 5 ml Teflon cell. Sub-
sequent measurements were taken with solutions obtained by adding 10 l of water
to the cell and continuous stirring. The mechanical stirring was stopped prior to each
measurement to eliminate potentially detrimental fluid motions and acoustic distur-
SENSITIVITY ENHANCEMENT WITH A NANOPARTICLE BASED COATING
169
bances around the fibre that could result in false readings. After each dilution, the
refractive index of the solution was determined from Lorentz-Lorenz mixing equa-
tion
4.24
[
171
]:
n
2
- 1
n
2
+ 2
= p
1
n
2
1
- 1
n
2
1
+ 2
+ p
2
n
2
2
- 1
n
2
2
+ 2
.
(7.1)
Here, n represents the refractive index of the mixture, n
1
and n
2
are refractive indices
of pure components (water and EG, 1.318 and 1.394, respectively at wavelengths
near 1550 nm [
172
]), and p
1
= V 1/V , p
2
= V 2/V are the volume fractions of the
components.
Figure
7.10
(a and b) depicts the response of an individual resonance to small in-
dex variations, before and after the sensor surface was modified with silver nanowires,
and for the two singular values of the transfer function.
Figure 7.10
Singular values (a,b) and polarization-dependent loss parameter (c,d) (linear
scale) changes due to a small refractive index change of
n = 3.77 × 10
-4
, before (a,c) and
after (b,d) deposition, corresponding to a single resonance taken at
= 1555.7 nm .
We note that the most obvious effect resulting from the nanowire coating is the
increase in the splitting between the two singular values, from
30 pm to 100 pm
(comparing the central wavelengths of
1
and
2
for n
1
or n
2
). This increase should
be associated with an increase in the polarization dependence of the boundary condi-
tion for the cladding modes, as expected for metal particles. A similar effect had been
observed previously by our group for gratings covered by a sparse layer of spherical
170
MODIFICATION OF THE SENSOR SURFACE
copper nanoparticles deposited by a pulsed chemical vapor depositio (CVD) tech-
nique [
173
].
It was also observed that the singular value spectra shifted to shorter wavelengths
with decreasing refractive index. However, this shift is somewhat difficult to quan-
tify precisely because the resonances are rather broad and flat bottomed. Instead of
isolating one of the singular value to quantify the shifts (and thereby throwing away
half the data, corresponding to the other singular value) we will follow zeros in PDL
spectra, corresponding to resonances of interest, as we discussed in Chapter
3
. We
are basically measure differential spectrum between polarization states, which pro-
vides extremely sharp peaks (Figure
7.10
c and
7.10
d) that can be tracked down with
much greater accuracy than the insertion loss or singular value resonance position.
Conducting subsequent measurements for fine variations in the refractive index
obtained by the step-wise addition of 10 l quantities of water to the solution we
determine the refractometric sensitivity for several peaks. Each addition of water
caused a small decrement of n
7.5 × 10
-5
in the refractive index of the sur-
rounding medium. Those measurements can be carried out on any of the resonances.
The results of these detailed measurements are shown in Figure
7.11
, where a set of
5 different individual resonances were measured in order to determine if the spectral
position of a resonance had an impact on its sensitivity, shown in Figure
7.11
b.
Comparing the slopes in Figure
7.11
b for the resonances of the coated and un-
coated TFBG, we obtain an increase in sensitivity from 53 nm RIU
-1
for the
bare uncoated TFBG sensor to 185 nm RIU
-1
for the sensor coated with silver
nanowires (RIU stands for "Refractive index unit"). Although the reported sensitiv-
ity for metal coated SPR TFBG sensors is reported to have a higher value, of about
555 nm RIU
-1
[
174
], the operational range of SPR sensors is limited to a narrow
range of few nanometers where the resonant phenomena of SPR excitation is ob-
served. The proposed sensor with a coating of nanowires has an advantage over a
SPR sensor as the increase in sensitivity is observed over the whole operational range
of the sensor, as shown in Figure
7.11
a.
SENSITIVITY ENHANCEMENT WITH A NANOPARTICLE BASED COATING
171
Figure 7.11
(a) Wavelength shift
of several individual resonances of the TFBG sensor
due to the surrounding refractive index change
n. (b) The sensetivity before (open marks)
and after (closed marks) deposition [
3
].
172
MODIFICATION OF THE SENSOR SURFACE
7.3.2
Discussion
The observed enhancement in sensitivity was investigated by taking a close look at
the electromagnetic field interaction with the silver nanowires. The device is capable
of sensing the external medium by means of the evanescent field of cladding modes,
leaking outside the fibre cladding, as shown in Figure
2.46
. We note that the evanes-
cent field decays slowly enough to extend into the thin layer of silver nanowires,
thus conditions for coupling between the evanescent field incident from the fibre and
the nanoparticles on its surface are created. Perturbations of the evanescent field
modify how the grating couples the light from the core to each cladding mode and
hence the resonances observed in the transmission spectrum. Knowledge of the fi-
bre geometry and refractive indices, as well as of the grating period, tilt angle and
modulation amplitude, allow us to uniquely assign specific cladding modes to the
resonances observed experimentally. The electric field distribution corresponding to
two closely positioned TMlike and TElike modes of the same azimuthal family at
= 1548.1 nm and = 1552.1 nm are shown in Figure
7.12
.
Figure 7.12
Vectorial
E field structure for two modes with almost identical propagation
constants (hence resonance wavelengths), but different polarization states ((a)â^TMlike mode,
(b)â^TElike mode). Silver nanowires are shown schematically on top.
As discussed in Section
6.2.3
, the coatings with cylindrical particles give rise to
very prominent and sharp longitudinal resonances, which shift toward the infrared
band with an increase of particle elongation. The transversely polarized electric
field excites a series of blue-shifted transverse resonances in the case of elongated
nanoparticles. The particles should be chosen in such a way that the resonances are
positioned far away from the operational range of the sensor, and hence the dielectric
permittivity
is a relatively constant function in the operational range. However,
the longitudinal resonances of the particles, excited by the tangentially polarized
electric field E
, are positioned exactly in the range where the sensor operates, so
that the energy coupling can occur between the TElike modes of the sensor and local
resonances of the particles, as shown in Figure
7.12
. The imaginary part of the
SENSITIVITY ENHANCEMENT WITH A NANOPARTICLE BASED COATING
173
dielectric permittivity function has a resonance in the operational range. The position
of the resonance is defined, among other parameter, by the external refractive index
of the surrounding medium. Therefore the differential sensitivity with the TElike
and TMlike modes is possible, due to the fact that the modes are affected differently
by changes in the external refractive index.
The TFBG couples core-guided light to two different families of cladding modes
according to the polarization of the input light relative to the orientation of the tilt
plane, as shown in Figure
2.47
. To be precise, when the input core-guided light is
polarized linearly in the plane of the tilt (corresponding to P-polarized light), the
cladding modes that are excited by the TFBG belong to the TMlike family of
modes and have their electric field polarized predominantly in the radial direction
at the cladding boundary. On the other hand, the S-polarized light (again relative to
the plane of tilt), and the grating couples the core light to TElike cladding modes
whose electric field at the cladding boundary is mostly azimuthal (i.e. tangential).
The calculated wavelength spacing and refractometric properties of the two modes in
each pair reveal that they can be associated with the observed singular values of the
insertion loss resonances measured above. Therefore, we can explain the observed
splitting of the singular value pair by noting how the electric field of the correspond-
ing mode probes the nanowires: since the TMlike modes have electric fields that
are predominantly radial at the cladding boundary while TElike modes are mostly
tangential, TMlike modes cannot excite the "long axis" plasmons of the nanowires
while the TElike modes can.
We also note that these results have been obtained in a relatively stable temper-
ature environment, even though no active temperature control was used. In was re-
ported [
11
] that the underlying TFBG platform can be made temperature-independent
over several tens of degrees by referencing all wavelengths to the Bragg resonance,
bounded the fibre's core and completely insensitive to changes outside the fibre. With
this referencing scheme the only impact of temperature on the positions of the reso-
nances arises from the change in the refractive index of the nanoparticles and of the
medium that surrounds them. The temperature effect was studied with application
to TFBGs coated with uniform gold films and it was found to be insignificant (about
10 pm/
C [
175
]) compared to the shifts observed here.
7.3.3
Conclusion
The sensitivity of a TFBG refractometer can be increased at least 3.5-fold by the
addition of a sparse coating of silver nanowires. The coating is created by simple
liquid phase self-assembly process that does not require special deposition tools,
unlike other methods used to fabricate plasmon assisted devices (e-beam lithography,
chemical vapour deposition [
7
,
176
,
86
] or real time monitoring of the deposition
process [
1
] to achieve very specific layer thicknesses).
The high sensitivity of the sensor is explained by the fact that the resonances of a
1(cm) long grating have Q values in excess of 15000 at near infrared wavelengths,
due to full widths of 100(pm). As a result, the positions of these resonances can be
followed accurately, even for shifts of the order of several picometres (reproducibility
174
MODIFICATION OF THE SENSOR SURFACE
better than 3 pm was recently demonstrated experimentally with a similar measuring
system [
15
]).
Associated with our 185 nm RIU
-1
sensitivity, a 3(pm) resonance position ac-
curacy yields a minimum detection level of 1.6
× 10
5
RIU.
The second advantage is based on the fact that different resonances have similar
sensitivity and any of them can be used for refractive index sensing, as opposed
to other kinds of SPR sensor, where only one or a few resonances satisfying the
phase matching condition have high sensitivity [
15
]. The high Q value of these
resonances is a key factor in taking advantage of plasmonic effects because of the
inherent trade-off between strong plasmon excitation (and enhanced sensitivity) and
increased loss, which tends to broaden resonances: the TFBG configuration allows
the device designer a great amount of control to adjust the amount of overlap between
the evanescent field and the metal particles and also in the relative orientation of the
light polarization and particle geometry.
CHAPTER 8
CONCLUSION
The primary focus of this book has been set on theoretical analysis of the TFBG
sensor and proposing methods of enhancing its sensitivity by surface modification.
The TFBG sensor is not a trivial object. The sensor has a complex polarization-
dependent response dependent on more than a thousand interacting modes. To ad-
dress the challenge of the theoretical analysis, we developed a highly efficient and
fast numerical solver, capable of computing a TFBG spectra in approximately 3 min-
utes for a given state of incident light polarization.
Theoretical analysis
The solver we developed consists of two major parts: the
mode solver and the numerical integrator, which solve a system of coupled differen-
tial equations.
The mode solver is the key element, as it is required to compute more than a thou-
sand modes at various wavelengths. We developed a simple yet efficient and fast
full-vectorial complex mode solver, capable of handling waveguides of an arbitrary
complex refractive index profile. First, the system of Maxwell's equations was re-
duced to only two coupled ordinary differential equations for the electric field. Next,
the equations were transformed into a system of algebraic equations with the help of
a finite difference method. Finally the problem was reduced to the problem of find-
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
175
176
CONCLUSION
ing eigenvalues and eigenvectors of a five-diagonal sparse matrix. The eigenvalue
problem was effectively solved with the standard iterative method.
At the next step the modes obtained for the non-perturbed case were used as
the basis functions to solve the problem of tilted grating inscribed along the optical
axis. The problem of small perturbations along the optical axis was handled with
the coupled mode theory. As a result, the problem was reduced to a set of coupled
differential equations, representing the energy transfer between the modes. Due to
the grating tilt, the energy transfer depends on the core mode polarization. The
polarization-dependent effects were thoroughly analysed in this work. Interesting
insights into the properties of the electric field at the TFBG sensor boundary were
gained.
The results of the simulation were shown to be in a good accordance with the
experimental measurements.
Polarization-based experimental measurements
Along with the theoretical study,
we conducted an extensive experimental study of the optical properties of the TFBG
sensor. We based our study on the Jones matrix and Stokes vectors data made avail-
able by means of Optical Vector Analyser (OVA).
We proposed two alternative approaches towards the TFBG sensor data analysis.
The first method was based on tracking the grating transmission of two orthogonal
states of linear polarized light that were extracted from the measured Jones matrix or
Stokes vectors of the TFBG transmission spectra. The second method was based on
the measurements taken along the system principle axes and polarization-dependent
loss (PDL) parameter, also calculated from measured data. It was shown that the
frequent crossing of the Jones matrix eigenvalues as a function of wavelength leads
to a non-physical interchange of the calculated principal axes. A method to remove
this unwanted mathematical artefact and to restore the order of the system eigenval-
ues and the corresponding principal axes was provided. A comparison of the two
approaches revealed that the PDL method provides a smaller standard deviation and
therefore a lower limit of detection in refractometric sensing.
Furthermore, the polarization analysis of the measured spectra allows for the iden-
tification of the principal states of polarization of the sensor system and consequently
for the calculation of the transmission spectrum for any incident polarization state.
The stability of the orientation of the system principal axes was also investigated as
a function of wavelength. A small oscillation in the orientation of the principal axes
as a function of wavelength was observed.
Sensitivity enhancement
In the presented work we investigate the sensor response
to various types of nano-scale film coatings. We have proposed the method of reso-
nant coupling between the high quality factor resonances of the TFBG structure and
the local resonances of nanoparticles deposited on the sensor surface. The problem
was investigated theoretically for spherical and elliptical particles made of various
materials. The optimal parameters for particles were determined. A 3.5-fold in-
crease in the TFBG sensor sensitivity was observed experimentally by coating the
sensor surface with silver nanowires.
177
This doctoral project resulted in the following publications:
1. A. Bialiayeu, C. Caucheteur, N. Ahamad, A. Ianoul, and J. Albert, "Self-optimized
metal coatings for fiber plasmonics by electroless deposition," Optics express 19,
18742-18753 (2011).
In this paper we presented a novel method to prepare optimized metal coatings for
infrared Surface Plasmon Resonance sensors by electroless plating. We show that
Tilted Fiber Bragg grating sensors can be used to monitor in real-time the growth
of gold nanofilms up to 70 nm in thickness and to stop the deposition of the gold
at a thickness that maximizes the SPR (near 55 nm for sensors operating in the
near infrared at wavelengths around 1550 nm). The deposited films are highly
uniform around the fiber circumference and in spite of some nanoscale roughness
(RMS surface roughness of 5.17 nm) the underlying gratings show high quality
SPR responses in water.
2. A. Bialiayeu, A. Bottomley, D. Prezgot, A. Ianoul, and J. Albert, "Plasmon-
enhanced refractometry using silver nanowire coatings on tilted fibre Bragg grat-
ings," Nanotechnology 23, 444012 (2012).
In this paper we presented a novel technique for increasing the sensitivity of tilted
fibre Bragg grating (TFBG) based refractometers. The TFBG sensor was coated
with chemically synthesized silver nanowires
100 nm in diameter and several
micrometres in length. A 3.5-fold increase in sensor sensitivity was obtained
relative to the uncoated TFBG sensor. This increase was associated with the
excitation of surface plasmons by orthogonally polarized fibre cladding modes
at wavelengths near 1.5 m. Refractometric information was extracted from the
sensor via the strong polarization-dependence of the grating resonances using a
Jones matrix analysis of the transmission spectrum of the fibre.
3. W. Zhou, D. J. Mandia, M. B. Griffiths, A. Bialiayeu, Y. Zhang, P. G. Gordon, S.
T. Barry, and J. Albert, "Polarization-dependent properties of the cladding modes
of a single mode fiber covered with gold nanoparticles," Optics express 21, 245-
255 (2013).
In this paper the properties of the high order cladding modes of a standard optical
fiber were measured in real-time during the deposition of gold nanoparticle layers
by chemical vapor deposition. A correlation between the transmission spectra of
the 10
o
TFBG and the optical properties of gold particle coatings was established.
My contribution to this work allowed to explain the observed effects by the nu-
merical FDTD modelling of the gold particle coating layer, as described in Sec-
tion
4.3.2
.
4. A. Bialiayeu, A. Ianoul, and J. Albert, "Engineering a resonant nanocoating for
an optical refractive index sensor," in "Electronic, photonic, plasmonic, phononic
and magnetic properties of nanomaterials," , vol. 1590 (AIP Publishing, 2014),
vol. 1590, pp. 68-70.
178
CONCLUSION
In this work we proposed an idea to boost the performance of refractive index
sensor based on the tilted fiber Bragg grating structure by resonant coupling of
small spherical nanoparticles to the TFBG resonances. The optimal choice of
nanoparticle parameters was discussed.
5. A. Bialiayeu, J. Albert, A. Ianoul, A. Bottomley, and D. Prezgot, "Silver nanowire
coated tilted fibre Bragg gratings," in "Bragg Gratings, Photo-sensitivity, and
Poling in Glass Waveguides," (Optical Society of America, 2012), pp. BW2E1.
In this work we reported a 3.5-fold increase in sensitivity of TFBG based refrac-
tometers by coating the sensor surface with silver nanowires. A strong polariza-
tion dependence of the grating resonances was observed and analyzed.
APPENDIX A
MATLAB CODE. THE FULL VECTORIAL
COMPLEX MODE SOLVER.
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
179
180
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
The following code had been developed to obtain the exact full vectorial solution
to the problem of cylindrical and slab waveguides of an arbitrary refractive index
profile, including lossy waveguides with a non-zero imaginary part of the refractive
index.
The main routine.
1
% r in [-R, R]
2
3
clear all
4
clc
5
clf
6
7
%============== Input =====================================
8
lam = 1.6;
% wavelength um
9
ko = 2*pi/lam;
% free space wave vector
10
11
% layer vectors
12
%RL = [4.1, 62.5, 70];
13
%nL = [1.4492, 1.444, 1.3556];
14
%nL = [1.4492+ 0.001, 1.444 + 0.001, 1.33];
15
16
R = 1.5;
17
% n1 = 1.4492; n2 = 1.3556;
% refractive index
18
n1 = 3; n2 = 1.3556;
% refractive index
19
RL = [R, R + 3];
20
nL = [n1, n2];
21
22
m=1;
23
Neig = 40;
% number of eigenvalues to compute
24
N = 10000;
25
26
%============== Potential =================================
27
28
h = RL(end)/(N-0.5);
29
r = linspace(0.5*h, RL(end), N);
30
r = [- fliplr(r), r].';
31
r = r - 0.1*h;
32
33
ni = f_construct_ni2(RL, nL, N);
34
ni = [fliplr(ni), ni].';
35
36
e = ni.^2;
37
%============== Solve ===============================
38
39
%[Ls, U] = f_FD_Construct_Matrix_Cyl_Exact_Vect(r, e, ko, m);
40
[Ls, U] = f_FD_Construct_Matrix_Cyl_WGA_Vect(r, e, ko, m);
41
%[Ls, U] = f_FD_Construct_Matrix_Cyl_Exact_Vect(r, e, ko, m);
42
[V, Eig] = f_FD_eig(Ls, Neig, U);
43
44
%============== Results ===================================
45
Neff = sqrt(Eig./ko^2);
46
U = sqrt(U/ko^2);
47
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
181
48
f_Plot_vec(m, r, U, Neff, V, 3, 1, 3.2)
49
%f_Plot_Scal_Single(12, 3, -7, 7, 2.5, 3.1, m, r, sqrt(U)./ko, ...
sqrt(Eig)./ko, V, h)
The matrix assembly for the weakly guided approximation of the scalar prob-
lem.
1
function
[Ls, u] = f_FD_Construct_Matrix_Cyl_WGA_Scalar(r, e,
...
ko, m)
2
% My working code
3
% r - grid vector
4
% ki = k_o n
5
% m - if zero we get symmetric function: d_r u(r) = 0
6
7
8
h = r(3) - r(2);
% the step size
9
N = length(r);
% number of elements r
10
11
12
%============= Construct d_x matrix ============
13
B = zeros(N,3);
14
B(:,1)= -1;
15
B(:,3)= 1;
16
D = 1/(2*h)*spdiags(B,[-1,0,1],N,N);
17
18
%============= Construct d_x^2 matrix ==========
19
B = ones(N,3);
20
B(:,2)= -2;
21
D2 = 1/(h^2)*spdiags(B,[-1,0,1],N,N);
22
23
24
%============= Construct matrix ================
25
% (d_r^2 + 1/r d_r + [k^2 - m^2/r^2])u(r) = b^2 u(r)
26
27
u = e*ko^2 - m^2./r.^2;
28
U = spdiags(u,0,N,N);
29
30
% sparse matrices
31
Ri = spdiags(1./r,0,N,N);
32
33
Ls = D2 + Ri*D + U;
The matrix assembly for the weakly guided approximation of the vectorial
problem.
1
function
[Ls, u] = f_FD_Construct_Matrix_Cyl_WGA_Vect(r, e,
...
ko, m)
2
3
h = r(3) - r(2);
% the step size
4
N = length(r);
% number of elements r
5
182
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
6
%============= Construct d_x matrix ============
7
B = zeros(N,3);
8
B(:,1)= -1;
9
B(:,3)= 1;
10
D = 1/(2*h)*spdiags(B,[-1,0,1],N,N);
11
12
%============= Construct d_x^2 matrix ==========
13
B = ones(N,3);
14
B(:,2)= -2;
15
D2 = 1/(h^2)*spdiags(B,[-1,0,1],N,N);
16
17
%============ Construct diagonal matrices ======
18
% vectors
19
u = e*ko^2 - m^2./r.^2;
20
21
u1 = -(1+m^2)./r.^2 + e*ko^2;
22
u2 = (m*2)./r.^2;
23
24
% sparse matrices
25
Ri = spdiags(1./r,0,N,N);
26
U1 = spdiags(u1,0,N,N);
27
U2 = spdiags(u2,0,N,N);
28
29
L = D2 + Ri*D;
30
Ls = [L+U1, U2; U2, L+U1];
The matrix assembly for the exact solution of waveguides with cylindrical
symmetry.
1
function
[Ls, u] = f_FD_Construct_Matrix_Cyl_Exact_Vect(r, e,
...
ko, m)
2
% puled r under E_r, Need it as the dispersion curves are more ...
realistic !!!
3
4
h = r(3) - r(2);
% the step size
5
N = length(r);
% number of elements r
6
7
%============ Construct d_x matrix =============
8
B = zeros(N,3);
9
B(:,1)= -1;
10
B(:,3)= 1;
11
D = 1/(2*h)*spdiags(B,[-1,0,1],N,N);
12
13
%============ Construct d_x^2 matrix ===========
14
B = ones(N,3);
15
B(:,2)= -2;
16
D2 = 1/(h^2)*spdiags(B,[-1,0,1],N,N);
17
18
%============ The operator =====================
19
ri = 1./r;
20
ri2 = ri.^2;
21
22
u = e*ko^2 - m^2*ri2;
23
U = spdiags(u,0,N,N);
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
183
24
Ri = spdiags(ri,0,N,N);
25
26
%============== TE-like modes ==================
27
L2 = D2 - Ri*D + U;
28
29
%============== TM-like modes ==================
30
ei = 1./e;
31
de = D*e;
32
dlne = de.*ei;
33
p = ri + dlne;
34
P = spdiags(p,0,N,N);
35
L1 = D2 - P*D + U;
36
37
%================ off diagonal terms ===========
38
a = 2*m*ri2;
39
u1 = e.*a;
40
u2 = ei.*(a + m*dlne.*ri);
41
U1 = spdiags(u1,0,N,N);
42
U2 = spdiags(u2,0,N,N);
43
44
%================ Final assembly ===============
45
Ls = [L1, U1; U2, L2];
The matrix assembly for the exact solution of slab waveguides, TE and TM
modes.
1
function
[Ls, u] = f_FD_Construct_Matrix_SlabTE(r, e, ko)
2
% TE modes, BC missing for m = 0
3
4
% r - grid vector
5
% ki = k_o n
6
% m - if zero we get symmetric function: d_r u(r) = 0
7
8
N = length(r)
% number of elements r
9
h = (r(end) - r(1))/N;
% the step size
10
11
%============ Construct d_x matrix =============
12
B = zeros(N,3);
13
B(:,1)= -1;
14
B(:,3)= 1;
15
D = 1/(2*h)*spdiags(B,[-1,0,1],N,N);
16
17
%============ Construct d_x^2 matrix ===========
18
B = ones(N,3);
19
B(:,2)= -2;
20
D2 = 1/(h^2)*spdiags(B,[-1,0,1],N,N);
21
22
%============ Construct dioganal matrices ======
23
% vectors
24
u = e*ko^2;
% Potential barier
25
% sparse matrices
26
U = spdiags(u,0,N,N);
27
28
%=========== The Operator Matrix ===============
184
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
29
% (d_r^2 + U)u(r) = b^2 u(r)
30
% U = e*ko^2
31
32
Ls = D2 + U;
33
%Ls = D*D + U;
% Fail
1
function
[Ls, u] = f_FD_Construct_Matrix_SlabTM(r, k2)
2
%TM modes, BC missing for m = 0
3
4
% r - grid vector
5
% ki = k_o n
6
% m - if zero we get symmetric function: d_r u(r) = 0
7
8
N = length(r)
% number of elements r
9
h = (r(end) - r(1))/N;
% the step size
10
11
%============== Construct d_x matrix ===================
12
B = zeros(N,3);
13
B(:,1)= -1;
14
B(:,3)= 1;
15
D = 1/(2*h)*spdiags(B,[-1,0,1],N,N);
16
17
%============= Construct d_x^2 matrix ==================
18
B = ones(N,3);
19
B(:,2)= -2;
20
D2 = 1/(h^2)*spdiags(B,[-1,0,1],N,N);
21
22
%============= The Operator Matrix =====================
23
24
u = k2;
% Potential barier
25
e = k2.';
26
27
U = spdiags(e,0,N,N);
28
29
de = D*e;
30
d2e = D2*e;
31
32
dlne = de./e;
33
Dlne = spdiags(dlne,0,N,N);
34
35
d2lne = d2e./e - dlne.^2;
36
D2lne = spdiags(d2lne,0,N,N);
37
38
Ls = D2 + Dlne*D + D2lne + U;
The matrix diagonalization routine.
1
function
[V, Eig] = f_FD_eig(Ls, Neig, U)
2
% find eigenvalues and sort them
3
4
% Ls - sparse operator matrix
5
% U - potential barrier, to cut unbounded eigenvalues
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
185
6
% Neig - number of eigenvalues to find.
7
8
Umax = max(U);
% EXTREAMLY IMPORTENT: find them close to ...
the potential at the centre.
9
Umin = U(end);
10
11
% Find eigenvalues
12
tic;
13
[V,Eig] = eigs(Ls, Neig, Umax);
% [L] u = lam u
14
toc;
15
Eig = diag(Eig);
% creates vector ...
from L's
diagonal elements
16
17
%Eig
18
19
% need only with positive real part
20
b = real(Eig)>0;
21
Eig = Eig(b);
22
V = V(:,b);
23
24
% need only bounded eigenvalues
25
b = ((real(Eig)>Umin) & (real(Eig) < Umax))
26
Eig = Eig(b)
27
V = V(:,b);
28
29
% Sort Eiganvalues and eigenvectors
30
[t, ind] = sort(Eig,
'descend'
);
31
Eig = Eig(ind);
% sorted eigenvalues
32
V = V(:,ind);
% correspondent ...
eigenvectors to sorted lambdas
33
34
% Remove Degenerecy
35
% v = real(Eig);
36
% d = v(1:end-1) - v(2:end);
37
% b = abs(d) > 0.0001;
38
% Eig = Eig(b);
Construct the waveguide profile.
1
function
ni = f_construct_ni(RL, nL, N)
2
% Construct step index potential barrier
3
h = RL(end)/N;
4
5
ni = nL(1)*ones(1,N);
6
for
k=1:length(nL)-1
7
round(RL(k)/h)
8
ni(round(RL(k)/h):end) = nL(k+1);
9
end
1
function
ni = f_construct_ni2(RL, nL, N)
2
% ramp instead of steps potential barrier
3
186
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
4
h = RL(end)/N;
5
d=7;
% number of steps for jump
6
7
ni = nL(1)*ones(1,N);
8
for
k=1:length(nL)-1
9
p = round(RL(k)/h);
10
n1 = nL(k);
11
n2 = nL(k+1);
12
ni(p:end) = n2;
% next layer
13
14
% slow ramp, nit a jump.
15
ni(p-d:p+d-1) = linspace(n1, n2, d*2);
16
end
Plot eigenfunctions(modes) and eigenvalues (effective refractive indexes) for
scalar and vectorial cases.
1
function
f_Plot_vec(m, r, U, Eig, Veig, scale, ymin, ymax)
2
% plot potential well, eigenvalues and eigenfunctions
3
4
5
N = length(r);
6
h = r(3)-r(2);
7
8
clf
9
%plot(r, ki2, 'k')
10
h1 = plot(r, U,
'g'
,
'LineWidth'
,1)
11
12
hline(real(Eig))
13
hold on
14
15
Neig = length(Eig)
16
for
k=1:Neig
17
18
y = abs(Veig(1:N,k)).^2;
% E_r component
19
y = Eig(k) + y/sqrt(h)*scale;
20
plot(r, real(y),
'r-'
,
'LineWidth'
, 0.5)
21
hold on
22
23
y = abs(Veig(N+1:end,k)).^2;
% E_f component
24
y = Eig(k) + y/sqrt(h)*scale;
25
plot(r, real(y),
'b'
,
'LineWidth'
, 0.5)
26
hold on
27
28
end
29
30
31
%================== Labling ====================
32
hOx = ylabel(
'n, N_{eff}'
);
33
hOy = xlabel(
'\rho,
\mu m'
);
34
35
s = sprintf(
'm = %u'
,m)
36
hTitle = title(s);
37
set( [hOx, hOy, hTitle],
'FontWeight'
,
'normal'
,
'FontSize'
, 12);
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
187
38
39
dy = 0.04;
40
Vy = ymin:dy:ymax-dy;
% Y ticks
41
42
%axis([r(1),r(end),ymin,ymax]);
43
axis([-3.5,3.5,ymin,ymax]);
44
45
Vy = ymin:dy:ymax-dy;
% Y ticks
46
47
%==================== Axis ======================
48
set(gca,
...
49
'Box'
,
'off'
,
...
50
'FontSize'
, 8
,
...
51
'FontName'
,
'Arial'
,
...
52
'TickDir'
,
'out'
,
...
53
'TickLength'
, [.02 .02] ,
...
54
'XMinorTick'
,
'on'
,
...
55
'YMinorTick'
,
'on'
,
...
56
'YGrid'
,
'off'
,
...
57
'XGrid'
,
'off'
,
...
58
'YTick'
, Vy,
...
59
'XScale'
,
'linear'
,
...
60
'LineWidth'
, 1
);
61
62
%=================== Print =====================
63
set(gcf,
'PaperPositionMode'
,
'manual'
);
64
set(gcf,
'PaperUnits'
,
'inches'
);
65
set(gcf,
'PaperPosition'
, [0 0 4 3]);
% defines the ...
aspekt ratio
66
print(
'-dpng'
,
'-r300'
,
'test.png'
)
% saves file in ...
cur. dir
67
set(gcf,
'Color'
,
'w'
);
Orthogonality verification.
1
function
f_Plot_Orthogonality_Check(V,r)
2
% plot overlap matrix
3
4
dim = size(V)
% number of points, and ...
eigenfunctions
5
6
rsqr = sqrt(r)
7
8
for
k=1:dim(2)
9
v = V(:,k);
10
v = v.*rsqr';
% r - is weighted function
11
v = v/sqrt(sum(v.*v));
% normed, already weighted ...
functions
12
V(:,k) = v;
13
end
14
15
M = V'*V;
16
figure
17
clf, imagesc(M), colorbar
% orthogonality check
188
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
18
norm = sqrt(sum(sum(M)))
Dispersion curves.
1
clear all
2
clc
3
4
%for m = 2:6
5
m = 2
6
7
N = 15000;
% number of elements in the grid
8
Neig = 15;
% number of eigenvalues
9
Np = 30;
% number of points along x (40 is OK)
10
11
%n1 = 1.4492; n2 = 1.3556;
% Glass in Si in water, ...
refractive index
12
n1=3; n2 = 1.3556;
% Si in water
13
14
15
Vmin = 0.1;
16
Vmax = 10;
17
18
lam = 1.6;
% wavelength um
19
ko = 2*pi/lam;
% free space wave vector
20
21
V = linspace(Vmin, Vmax, Np);
22
x = V/sqrt(n1^2 - n2^2);
23
24
M = zeros(Neig+1, Np);
25
M(
end
,:) = V;
% save position in the same matrix
26
27
for
i=1:Np
28
i
29
30
R = x(i)/ko
31
RL = [R, R + 10];
32
nL = [n1, n2];
33
%============== Potential ==================
34
35
h = RL(end)/(N-0.5);
36
r = linspace(0.5*h, RL(end), N);
37
r = [- fliplr(r), r].';
38
r = r - 0.1*h;
39
40
ni = f_construct_ni2(RL, nL, N);
41
ni = [fliplr(ni), ni].';
42
43
e = ni.^2;
44
45
46
%============== Solve ======================
47
%[Ls, U] = f_FD_Construct_Matrix_Cyl_WGA_Scalar(r, e, ko, m);
48
%[Ls, U] = f_FD_Construct_Matrix_Cyl_WGA_Vect(r, e, ko, m);
49
[Ls, U] = f_FD_Construct_Matrix_Cyl_Exact_Vect(r, e, ko, m);
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
189
50
51
%[Ls, U] = f_FD_Construct_Matrix_SlabTM(r, e, ko);
52
%[Ls, U] = f_FD_Construct_Matrix_SlabTE(r, e, ko);
53
54
[V, Eig] = f_FD_eig(Ls, Neig, U);
55
56
neff2 = Eig/ko^2
57
58
b = (neff2 - n2^2)/(n1^2 - n2^2);
59
M(1:length(neff2),i) = b;
60
61
end
62
63
save([
'GL_E_'
, int2str(m)],
'M'
);
64
%save(['SiEx_', int2str(m)], 'M');
65
%save(['TE_', int2str(m)], 'M');
66
%save(['TM_', int2str(m)], 'M');
1
clear all
2
clc
3
clf
4
5
s =
'GL_S_1'
6
7
C =
'r-'
8
N = 16;
9
n=1;
10
for
k=1:N
11
f_dispersion_Plot(s,C,n)
12
n = n+1;
13
end
14
15
%=================== Set Plot ==================
16
axis([0,10,0,1])
17
18
hOy = ylabel(
'Normalized propagation constant b'
);
19
hOx = xlabel(
'Normalized frequency V'
);
20
%hL = legend('Exact', '', 'WGA','location', 'NorthWest')
21
%legend('boxoff')
22
23
%s = sprintf('Dispersion curves')
24
hTitle = title(
'Dispersion curves'
);
25
set( [hOx, hOy, hTitle],
'FontWeight'
,
'normal'
,
'FontSize'
, 10);
26
27
set(gca,
...
28
'Box'
,
'off'
,
...
29
'FontSize'
, 8
,
...
30
'FontName'
,
'Arial'
,
...
31
'TickDir'
,
'out'
,
...
32
'TickLength'
, [.02 .02] ,
...
33
'XMinorTick'
,
'on'
,
...
34
'YMinorTick'
,
'on'
,
...
35
'YGrid'
,
'off'
,
...
36
'XGrid'
,
'off'
,
...
37
'YTick'
, 0:0.2:1,
...
190
MATLAB CODE. THE FULL VECTORIAL COMPLEX MODE SOLVER.
38
'LineWidth'
, 1
);
39
40
set(gcf,
'PaperPositionMode'
,
'manual'
);
41
set(gcf,
'PaperUnits'
,
'inches'
);
42
set(gcf,
'PaperPosition'
, [0 0 5 3]);
% defines the ...
aspekt ratio
43
print(
'-dpng'
,
'-r300'
,
'test.png'
)
% saves file in ...
cur. dir
44
set(gcf,
'Color'
,
'w'
);
1
function
f_dispersion_Plot(path, col, N)
2
% path
the path to data
3
% col
color
4
5
load(path);
6
7
V = M(
end
,:);
8
y = real(M(N,:));
9
10
plot(V, y, col)
11
hold on
APPENDIX B
MATHEMATICA CODE FOR MIE SCAT-
TERING
Tilted fibre Bragg grating sensors with resonant nano-scale coatings.
By Aliaksandr Bialiayeu Copyright © 2015
191
192
MATHEMATICA CODE FOR MIE SCATTERING
The following code was used to calculate absorption, scattering and extinction
coefficients of a sphere particle submerged into solvent.
MATHEMATICA CODE FOR MIE SCATTERING
193
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