TRANSFER MATRIX APPROACH TO PROPAGATION OF ANGULAR
PLANE WAVE SPECTRA THROUGH METAMATERIAL MULTILAYER
STRUCTURES
Thesis
Submitted to
The School of Engineering of the
UNIVERSITY OF DAYTON
In Partial Fulfillment of the Requirements for
The Degree
Master of Science in Electro-Optics
By
Han Li
UNIVERSITY OF DAYTON
Dayton, Ohio
December, 2011
TRANSFER MATRIX APPROACH TO PROPAGATION OF ANGULAR
PLANE WAVE SPECTRA THROUGH METAMATERIAL MULTILAYER
STRUCTURES
Name: Li, Han
APPROVED BY:
___________________________
Partha P. Banerjee, Ph.D.
Advisor Committee Chairman
Professor
Department of Electrical Engineering
And Electro-Optics Program
___________________________
Joseph W. Haus, Ph.D.
Committee Member
Professor
Department of Electro-Optics
Program
__________________________
Andrew Sarangan, Ph.D.
Committee Member
Professor
Department of Electro-Optics Program
___________________________
John G. Weber, Ph.D.
Associate Dean
School of Engineering
__________________________
Tony E. Saliba, Ph.D.
Dean, School of Engineering
& Wilke Distinguished Professor
ii
c Copyright by
○
Han Li
All rights reserved
2011
iii
ABSTRACT
TRANSFER MATRIX APPROACH TO PROPAGATION OF ANGULAR
PLANE WAVE SPECTRA THROUGH METAMATERIAL MULTILAYER
STRUCTURES
Name: Li, Han
University of Dayton
Advisor: Partha P. Banerjee
The development of electromagnetic metamaterials for perfect lensing
and optical cloaking has given rise to novel multilayer bandgap structures using
stacks of positive and negative index materials. Gaussian beam propagation
through such structures has been analyzed using transfer matrix method (TMM)
with paraxial approximation, and unidirectional and bidirectional beam
propagation methods (BPMs).
In this thesis, TMM is used to analyze
non-paraxial propagation of transverse electric (TE) and transverse magnetic
(TM) angular plane wave spectra in 1 transverse dimension through a stack
containing layers of positive and negative index materials.
The TMM
calculations are exact, less computationally demanding than finite element
methods, and naturally incorporate bidirectional propagation.
iv
ACKNOWLEDGMENTS
I would like to specially thank Dr. Partha Banerjee, for all his help and
advice in my studies, for directing this thesis and careful modifications. His
patience, time and vast knowledge are the biggest encouragement for me.
I
would also like to thank my committee members Dr. Joseph Haus and Dr.
Andrew Sarangan for their assistance and helpful comments.
Additionally, I also would like to thank Dr. Haus for his encouragement,
Dr. Qiwen Zhan for his help with literature search, Drs. Sarangan, Bradley
Duncan, Peter Powers, John Loomis and Georges Nehmetallah for their great
classes. I would also like to express my thanks to Dr. Rola Aylo for all her
help on this research.
Finally, I would like to thank all in my family for their love, support and
encouragement.
v
TABLE OF CONTENTS
ABSTRACT ....................................................................................................... .iv
ACKNOWLEDGEMENTS ..................................................................................v
TABLE OF CONTENTS.................................................................................... vi
LIST OF FIGURES ......................................................................................... .ix
LIST OF ABBREVIATIONS AND NOTATIONS ......................................... xiv
CHAPTER I. INTRODUCTION ..........................................................................1
1.1 Background .............................................................................................1
1.2 Objective and brief introduction .............................................................2
1.3 Thesis outline ..........................................................................................3
CHAPTER II. UNIDIRECTIONAL BEAM PROPAGATION METHOD ........5
2.1 Introduction .............................................................................................5
2.2 One and two-dimensional Fourier transforms of Gaussian function ......6
2.3 UBPM in a homogeneous medium .........................................................8
2.4 UBPM in an inhomogeneous medium ..................................................14
2.5 Conclusion ............................................................................................19
CHAPTER III. PLANE WAVE PROPAGATION THROUGH AN OPTICAL
BOUNDARY ......................................................................................................21
3.1 Introduction ...........................................................................................21
3.2 Plane waves and Snell’s law .................................................................22
3.3 Reflection and transmission of TE and TM waves ...............................25
3.4 Principle of reversibility .......................................................................29
vi
3.5 Conclusion ............................................................................................30
CHAPTER IV. PLANE WAVE PROPAGATION THROUGH MULTILAYER
STRUCTURES ...................................................................................................31
4.1 Introduction ...........................................................................................31
4.2 Reflection and transmission coefficients of a thin layer .......................31
4.3 Matrix formulation of TMM for a thin film..........................................37
4.4 Extension to multilayer system .............................................................39
4.5 Conclusion ............................................................................................45
CHAPTER V. PROPAGATION OF ANGULAR PLANE WAVE SPECTRA
THROUGH MULTILAYER STRUCTURES ...................................................47
5.1 Introduction ...........................................................................................47
5.2 Comparison of TMM and FEM for TE plane wave incidence .............48
5.3 Comparison of TMM and FEM for TM plane wave incidence ............50
5.4 Propagation of angular plane wave spectrum through multilayer
structure using TMM ..........................................................................54
5.5 TE case: Propagation of a collection of plane waves with Gaussian
profile ..................................................................................................55
5.6 TM case: Propagation of a collection of plane waves with Gaussian
profile ..................................................................................................59
5.7 Conclusion ............................................................................................63
CHAPTER VI. CONCLUSION AND FUTURE WORK ................................64
BIBLIOGRAPHY ...............................................................................................67
APPENDIX A. MATLAB CODES ....................................................................69
A.1 1_D_FFT_GAUSSIAN.m..................................................................69
A.2 unidirectional_beam_propagation.m..................................................71
vii
A.3 propagation_in_layers_movie.m ........................................................73
A.4 project_BPM.m ..................................................................................75
A.5 multilayerstructureplanewave.m ........................................................77
A.6 layeryehTE.m .....................................................................................79
A.7 TEwave ..............................................................................................81
A.8 layeryehH.m .......................................................................................83
A.9 wave ...................................................................................................85
A.10 TE .....................................................................................................87
A.11 ExEz .................................................................................................89
viii
LIST OF FIGURES
Figure 2.1. The 1-D amplitude distribution of a Gaussian function.
The
horizontal axis is in microns, in all Gaussian profile plots in the Chapter, unless
other stated ............................................................................................................6
Figure 2.2. Plane wave propagating at angle w.r.t. z ......................................9
Figure 2.3. Gaussian profile before (blue) and after (red) propagation by a
distance equal to Rayleigh range ........................................................................10
Figure 2.4. Gaussian profile before (blue) and after (red) propagation by a
distance equal to twice the Rayleigh range .........................................................11
Figure 2.5. Initial 2-D Gaussian profile; x-projection.........................................12
Figure2.6. Initial 2-D Gaussian profile, contour view ........................................12
Figure 2.7. Final 2-D Gaussian profile; x-projection ..........................................13
Figure2.8. Final 2-D Gaussian profile, contour view .........................................13
Figure 2.9. Initial 2-D Gaussian profile; x-projection.........................................15
Figure2.10. Final 2-D Gaussian profile at end of 1st medium; x-projection .......15
Figure 2.11. Representative 2-D Gaussian profile within 2nd medium after a
short distance of propagation; x-projection.........................................................16
Figure2.12. Representative 2-D Gaussian profile within 2nd medium after a
larger distance of propagation; x-projection .......................................................16
Figure2.13. Representative 2-D Gaussian profile a short distance in 3rd medium;
x-projection .........................................................................................................17
ix
Figure2.14. Representative 2-D Gaussian profile in 3rd medium after longer
distance of propagation; x-projection..................................................................17
Figure 2.15. The refractive index profile of two-slab waveguide.......................18
Figure2.16. Gaussian propagation showing coupling in the two-slab waveguide
.............................................................................................................................19
Figure 3.1. Transmitted wavevector and reflected wavevector at a boundary ...23
Figure 3.2. Reflection and transmission scenario for TE case.
The electric
fields in this figure are directed out of the page for all waves ............................24
Figure 3.3. The transmission coefficient for TE case as a function of the
incident angle ......................................................................................................26
Figure 3.4. The reflection coefficient for TE case as a function of the incident
angle ....................................................................................................................27
Figure 3.5. The transmission coefficient for TM case as a function of the
incident angle ......................................................................................................28
Figure 3.6. The reflection coefficient for TM case as a function of the incident
angle ....................................................................................................................29
Figure 4.1. A thin layer of dielectric material .....................................................32
Figure 4.2. Absolute value of reflection coefficient for TE case as a function of
the incident angle for single layer structure ........................................................35
Figure 4.3. Absolute value of transmission coefficient for TE case as a function
of the incident angle for single layer structure....................................................35
Figure 4.4. Absolute value of reflection coefficient for the TM case as a
function of the incident angle for single layer structure .....................................36
Figure 4.5. Absolute value of transmission coefficient for the TM case as a
function of the incident angle for single layer structure .....................................36
x
Figure 4.6. Same as Figure 4.1, but with coefficients Ai , Bi , Ai ' , Bi ' inserted ....37
Figure 4.7. Periodic structure composed of 2 materials with refractive index n1
and n2 ................................................................................................................40
Figure 4.8. Absolute value of electric field amplitude in multilayer structure
described above. The structure includes 7 periods, each period including two
layers with n1 2.0, n 2 1.5 , each layer with a thickness d i 1.5 106 m . The
envelope of the absolute value of electric field amplitude is decaying when it is
propagating in the multilayer structures .............................................................42
Figure 4.9. Absolute value of electric field amplitude for the 7 period structure,
with each period including two layers with n1 2.0, n 2 1.5 , each layer with
a thickness d i 1.5 106 m . The envelope of the absolute value of electric field
amplitude is decaying when it is propagating in the multilayer structures .........43
Figure 4.10. The absolute value of electric field amplitude is constant when it is
propagating in the multilayer structure. The structure includes 7 periods, each
period including two layers with n1 2.0, n 2 1.5 , each layer with a
thickness d i 1.5 106 m ....................................................................................44
Figure 4.11. The real part of the electric field amplitude is oscillatory when it is
propagating in the multilayer structures.
The structure includes 7 periods,
each period including two layers with n1 2.0, n 2 1.5 , each layer with a
thickness d i 1.5 106 m ....................................................................................45
Figure 5.1. (a) The magnitude squared of the y-component of the electric field,
(b) the magnitude squared of the x-component of the magnetic field in the
incidence medium, the structure and the substrate for the two incidence angles
i 0, / 6 simulated using the TMM technique and FEM, respectively .............49
xi
Figure 5.2. Layered structure composed of materials with refractive indices
with thickness
...........................................................................................50
Figure 5.3. (a) The magnitude squared of the magnetic field, (b) the magnitude
squared of the x-component of the electric field in the incidence medium, the
structure and the substrate for the two incidence angles i 0, / 6 simulated
using the TMM technique and FEM, respectively..............................................53
Figure 5.4. A profile in the transverse spatial domain is equated to the
superposition of plane waves with different traveling directions and amplitudes
.............................................................................................................................55
Figure 5.5. Initial and transmitted Gaussian profile (using TMM and FEM) after
propagation in the structure defined in Section 5.2, for two beam waists (a)
w0 , (b)
w0 10 respectively. The MATLAB code can be found in
Appendix A under wave.m and layeryehTE.m ...................................................56
Figure 5.6. The magnitude squared of the electric field as it propagates inside
the metamaterial structure with same parameters as in Figure 5.5(a) using (a)
TMM and (b) FEM respectively. The MATLAB code can be found in
Appendix A under TE.m and layeryehTE.m ......................................................57
Figure 5.7. Initial and transmitted Gaussian profile (using TMM and FEM) after
propagation in the structure defined in Section 3.2 but with rp 2.25, rp 1,
rn 1.44 , rn 1 , and for beam waist w0 . The magnitude squared of the
electric field as it propagates inside the metamaterial structure using (b) TMM
and (c) FEM respectively
.............................................................................................................................58
Figure 5.8. (a) Initial and transmitted Gaussian profile (using TMM and FEM)
after propagation in the same structure as the previous example with beam waist
w0 , (b) Initial and transmitted Gaussian profile (using TMM and FEM) after
propagation in a structure same as in (a) but with p 2.25, p 1 and NIM
xii
layers of n 1.44 , n 1 .
The MATLAB code can be found in
Appendix A under wave.m and layeryehH.m .....................................................60
Figure 5.9. |Hy|2 as it propagates inside the metamaterial structure of the same
parameters as in Figure 5.8(a) using (a) TMM and (b) FEM respectively.
|Ex|2
the magnitude squared of the x-component of the electric field using (c) TMM
and (d) FEM respectively....................................................................................61
Figure 5.10. (a) |Hy|2 during propagation inside the metamaterial structure with
p 2.25, p 1, n 1.44 , n 1 , and for beam waist w0 , and for TM
incidence, (b) |Ex|2, (c) |Ez|2 . All simulations are done using TMM, and are in
excellent agreement with FEM simulations. The MATLAB code can be found
in Appendix A under ExEz.m and layeryehH.m ................................................62
xiii
LIST OF ABBREVIATIONS AND NOTATIONS
ABBREVIATIONS DEFINITIONS
UBPM
Unidirectional beam propagation method
DFT
Discrete Fourier transform
BBPM
Bidirectional beam propagation method
BPM
Beam propagation method
TMM
Transfer matrix method
TE
Transverse electric
TM
Transverse magnetic
PIM
Positive index material
NIM
Negative index material
FEM
Finite element method
SVEA
Slowly- varying envelope approximation
xiv
NOTATIONS
DEFINITIONS
A
Complex amplitude
A( z )
Transmitted electric amplitude
B( z )
Reflected electric amplitude
c
Velocity of light
d
Distance of one layer
DTE
Dynamical matrix (TE)
DTM
Dynamical matrix (TM)
Ei
Incident electric field
Et
Transmitted electric field
Er
Reflected electric field
E 0,i
Incident complex electric field amplitude
E 0,t
Transmitted complex electric field amplitude
E 0,r
Reflected complex electric field amplitude
E sz
Complex electric field in substrate (z-component)
Esx
Complex electric field in substrate (x-component)
E ( k x )
Fourier transform of E ( x)
E ( x)
Inverse Fourier transform of E ( k x )
k
Wavevector
kx
x-component of the wavevector
ky
y-component of the wavevector
L
Total propagation distance
xv
P
Propagation matrix
r
Position vector
r
Reflection coefficient
R
Power reflection coefficient or reflectivity
T
Transmission coefficient
T
Power transmission coefficient or transmittivity
w
Gaussian profile waist
n
Refractive index
x
variable in the x dimension
y
variable in the y dimension
z
Longitudinal distance along the propagation path
Phase of the plane wave
Permittivity
Permeability
Curl
xvi
CHAPTER I
INTRODUCTION
1.1 Background
The development of electromagnetic (EM) metamaterials for perfect
lensing [1] and optical cloaking devices [2] has given rise to the design and
fabrication of novel multilayer bandgap structures using stacks of positive and
negative index materials. Traditional beam propagation methods (BPMs) for
analyzing propagation through longitudinally inhomogeneous media are pretty
straightforward and helpful when the refractive index varies sufficiently slowly
along the propagation direction so that the accumulated reflections can be
ignored.
But in some practical applications such as photonic bandgap
structures, the longitudinal refractive index changes may be significant, and
hence there exists reflections at the interfaces between different refractive
indices, so that the numerical methods which account for the forward and
backward propagation become indispensible.
Several numerical procedures
have been investigated to solve this problem such as the finite difference time
domain (FDTD) approach [3] and finite element method (FEM) [4].
For
saving computational time and computer memory, the bidirectional beam
propagation method (BBPM) which tracks both forward and backward
traveling waves and based on the transfer matrix method (TMM) [5] has been
developed in this thesis.
1
1.2 Objective and brief introduction
In this thesis, the objective is to systematically investigate propagation of
plane waves and arbitrary (e.g., Gaussian) profiles through a multilayer
structure.
The understanding of transfer matrices for any structure is
important in the design of anti-reflection films and optical filters. The TMM,
developed for plane wave incidence, naturally incorporates interface reflections,
as well as the polarization state of the electric field.
This approach can be
used to calculate the reflected and transmitted waves for a single layer structure
and can be readily extended to multilayer structures.
In this thesis, this
approach is further extended and applied to the propagation of arbitrary profiles
of arbitrary polarizations through multilayer structures decomposing the spatial
profile into a collection or spectrum of plane waves.
Using this approach, the
electric (and magnetic) field distributions of, say, Gaussian profiles of arbitrary
polarizations are simulated and observed at any point inside and outside the
multilayer structure.
Results are compared to those obtained using FEM.
TMM is based on the fact that, according to Maxwell’s equations, there
are simple continuity conditions for the electric and magnetic fields across
boundaries from one medium to the next.
If the field is known at the
beginning of a layer, the field at the end of the layer can be derived from a
simple matrix operation. A stack of layers can then be represented as a system
matrix, which is the product of the individual layer matrices. The final step of
the method involves converting the system matrix back into reflection and
transmission coefficients [6].
In this thesis, TMM is first used simulate propagation of a single plane
wave through a stack containing layers of positive and negative index materials
2
(PIMs/NIMs) [7] for TE/TM incidence.
Thereafter, this method is extended to
analyze a collection or spectrum of plane waves, e.g., with a Gaussian angular
plane wave spectrum in 1 transverse dimension (x) propagating through a stack
of PIM/NIMs. The spatial variation of the electric field at any plane (z) during
bidirectional propagation through the stack is found from the composite angular
plane wave spectra.
The numerical results from TMM are compared with
numerical simulations using FEM techniques.
The TMM calculations are
exact, less computationally demanding, not limited by the thickness of the
structures and can be performed for arbitrary angular plane wave spectra for
both paraxial and non-paraxial propagation.
The BBPM developed in this
thesis can be readily applied to a wide variety of other cases, such as
propagation through induced reflection gratings in nonlinear medium, and in
the design of anti-reflective coatings, filters and dielectric mirrors.
1.3 Thesis outline
Chapter II in this thesis is a demonstration of the unidirectional BPM
(UBPM), which is an efficient way to simulate (scalar) beam propagation
through a homogeneous or nearly homogeneous material. Examples of
Gaussian beam diffraction and Gaussian beam propagation through a guided
structure are shown as examples of UBPM.
This method is neither suitable for
accounting for reflections (which makes it unsuitable for multilayer structures),
nor can it account for the polarization state of the beam.
Chapter III summarizes the basic principle of plane wave propagation
through a single optical boundary.
This is the background to TMM, which is
used to develop the BBPM, which makes up for the deficiencies of the UBPM.
3
Chapter IV is a continuation of the development of Chapter III, and
analyzes one plane wave incident on a thin layer which includes two optical
interfaces and one propagation distance within the layer. This simple concept
is thereafter generalized to the theory for one plane wave incident on multilayer
structures, which may include PIMs and NIMs. An example of a PIM/NIM
structure is given, along with illustrative examples.
Chapter V summarizes the principle and provides examples of Gaussian
profiles composed of many plane waves incident on and propagating through
the multilayer structure.
Both TE and TM polarization states of the Gaussian
profile are investigated. The simulation results are compared to results obtained
using FEM. It is concluded that the BBPM using TMM is faster and takes less
memory space than standard FEM techniques.
Finally, Chapter VI concludes the thesis and provides a summary of
ongoing and future work.
4
CHAPTER II
UNIDIRECTIONAL BEAM PROPAGATION METHOD
2.1 Introduction
The unidirectional beam propagation method (UBPM or simply, BPM) is
a numerical technique method to determine the optical field profile in a medium.
The principle of UBPM is to decompose an optical profile in the spatial domain
into a superposition of plane waves with various propagation directions, and
recompose the spatial profile back after propagation of the plane waves through
the dielectric structure.
This process needs the use of Fourier transforms to
convert from spatial domain to spatial frequency domain and back again.
Therefore the discrete Fourier transform (DFT) is first discussed in this Chapter
which is used as a numerical tool used through entire thesis. DFT is
implemented using the fast Fourier transform (FFT) algorithm in MATLAB,
and used to simulate the propagation of optical fields through homogeneous and
nearly homogeneous structures. The basic principle of UBPM is then
summarized, followed by some examples and simulation results.
As
mentioned in Chapter I, BBPM is better than UBPM for the practical
applications to be discussed in the next several Chapters.
5
2.2 One and two-dimensional Fourier transforms of Gaussian functions
A one-dimensional (1-D) scalar optical Gaussian envelope profile is defined as
Ee ( x ) e
x2
w2
,
(2.2-1)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Figure 2.1. The 1-D amplitude distribution of a Gaussian function. The
horizontal axis is in microns, in all Gaussian profile plots in the Chapter,
unless other stated.
and is plotted in Figure 2.1 for w 2 .
Such an optical field in the spatial
domain can be represented as a superposition of plane waves [8], since
Ee ( x )
1
2
E
e
( k x )e jk x x dk x
(2.2-2)
which implies that the phasor
E p ( x)
1
2
E ( k
e
x
)e jk x x e jk z z dk x
using the concept of the Fourier Transform where
6
(2.2-3)
E e (k x )
E ( x)e
jk x x
e
dx .
(2.2-4)
From Equation (2.2-3), it is clear that E p ( x ) comprises a collection of
plane waves traveling in directions given by kx and with complex amplitudes
E e ( k x ) .
Equations (2.2-2) and (2.2-4) comprise the 1-D Fourier transform.
The two dimensional (2-D) Fourier transform is given by
E e (k x , k y )
E ( x, y)e
j ( k x xk y y )
e
dxdy ,
(2.2-5)
and the inverse Fourier transform is
Ee ( x, y )
1
4
E (k , k )e
e
2
x
j ( kx xk y y )
y
dk x dk y ,
(2.2-6)
In Equations (2.2-5) and (2.2-6), kx and k y are the x and y components
of the propagation vector. The BPM method entails propagating the plane
waves in the spatial frequency domain, by considering the additional phase shift
of each wave traveling at an arbitrary angle w.r.t. a wave traveling along the z
direction.
The discrete Fourier transform (DFT) is used to calculate the Fourier
transform. DFT is easy to implement in the computer and is a way of
numerically approximating the continuous Fourier transform of a function.
Given a discrete function f (n ), n 0,...N 1 , where is the sampling
interval in x and f p (n) can be written as [9]:
f p (n )
f (n rN ) ,
r
the DFT of f p ( n ) is defined as
7
(2.2-7)
N 1
Fp (mK ) f p (n )e jmnK ,
(2.2-8)
n 0
where K
2
. The inverse DFT is defined as
N
N 1
f p (n) Fp (mK )e jmnK ,
(2.2-9)
m 0
The Fast Fourier transform (FFT) is an efficient numerical way to
evaluate the DFT using computer programs.
In the following Chapters, FFT
will be used very frequently for simulations.
Standard MATLAB FFT
operations will be used.
2.3 UBPM in a homogeneous medium
BPM is a computational technique which is used to solve the Helmholtz
equation under the condition of a time-harmonic wave, and under slowlyvarying envelope approximation (SVEA), for linear and nonlinear equations
[10]. There are two parts to UBPM, one to take into account wave diffraction
when it is propagating, the other to incorporate medium inhomogeneously,
either linear or nonlinearly included. If the medium is inhomogeneous, then
n is a variable and represents on the local index of refraction.
Diffraction during propagation can be mathematically modeled using the
transfer function for propagation, which incorporates the phase differences
between propagating plane waves at different angles, when monitored at a
certain (z) plane.
The physical meaning of this phase can be seen from the
following discussion.
In Figure 2.2, assume that a plane wave propagate a
distance z=L in the z direction while a second plane wave propagates with an
angle with respect to the z direction.
While the propagation distance of the
first wave is L, the propagation distance for the second wave is
8
L / cos L (1
2
2
).
Upon defining k0 as the propagation vector of the
waves, sin 1 ( k x / k0 ) , where k0 k0 .
Upon using this and the path
difference between the two waves, the transfer function follows.
In general, one can assume many plane waves after the Fourier transform
action on, say, a Gaussian beam in the spatial domain.
array form
N
X
to
( N 2)
Hence kx should be a
with the space between two samples being
X
2
where X is the size of the spatial domain of the field profile, N being the
X
samples on X , and with k0
2
.
x
L / cos
L
Figure 2.2. Plane wave propagating at angle w.r.t. z.
Because of diffraction, the Gaussian profile spreads out, and since the
total energy does not change due to absence of absorption, the amplitude
decreases and the beam width increases.
Figure 2.3 and Figure 2.4 are
examples of propagation of the Gaussian profile of Figure 2.1, and shows
9
z
diffraction.
Figure 2.3 shows the diffraction of the Gaussian profile after
propagation by a Rayleigh range zR , with the wavelength assumed to be
11.11µm.
Figure 2.4 is the computed final Gaussian profile after propagation
by 2 zR .
Note that with longer propagation distance, the “wings” of the
Gaussian develop numerical errors; this can be removed by re-sampling the
Gaussian profile.
The MATLAB code can be found in Appendix A under
1D_FFT_GAUSSIAN.m.
1
0.9
0.8
0.7
am plitude
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
x
2
4
6
8
10
Figure 2.3. Gaussian profile before (blue) and after (red) propagation by a
distance equal to Rayleigh range.
10
1
0.9
0.8
0.7
amplitude
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
x
2
4
6
8
10
Figure 2.4. Gaussian profile before (blue) and after (red) propagation by a
distance equal to twice the Rayleigh range.
Figures 2.5 and 2.6 are the x-projection and brightness plots of 2-D initial
Gaussian profiles.
Figures 2.7 and 2.8 are corresponding plots after
propagation through the Rayleigh range in air.
The MATLAB code can be
found in Appendix A under unidirectional_beam_propagation.m.
11
Figuree 2.5. Initia
al 2-D Gausssian profile; x-projecction.
Figuree 2.6. Initiall 2-D Gaussian profilee, contour view.
v
12
Figuree 2.7. Finall 2-D Gausssian profilee, x- projecttion.
Figuree 2.8. Final 2-D Gausssian profilee, contour view.
v
13
2.4 UBPM in an inhomogeneous medium
Gaussian beam propagation in a nearly homogeneous or weakly
inhomogeneous medium can also be studied by using UBPM, where n is the
variable corresponding to the local index of refraction.
It can also be used to
model propagation inside of materials with different refractive indices, as long
as reflections are neglected.
As an example, a three-layer structure with
refractive indices of 1, -1.2 and 1 is considered.
The propagation distance for
each layer is the Rayleigh range in air, then because of diffraction, the Gaussian
beam spreads out in the first layer and begins shrinking in the second layer due
to focusing in the negative index medium; while in the third layer, it spreads out
again. The total energy does not change inside of the structures since there is
no loss.
Figure 2.9 and Figure 2.10 show the 2-D Gaussian beams at the
beginning and end of the first layer.
Figure 2.11 and Figure 2.12 are
representative plots at two distances in the second layer, and show the focusing
of the Gaussian beam due to the negative index.
Finally, Figure 2.13 and
Figure 2.14 show representative plots of the Gaussian beams in the third layer,
showing defocusing. The MATLAB code can be found in Appendix A under
Propagation_in_layers_movie.m.
14
Figuree 2.9. Initiall 2-D Gaussian profilee; x- projecction.
Figuree 2.10. Finaal 2-D Gausssian profille at end off 1st medium
m; x-projecction.
15
Figure 2.11. Reprresentative 2-D Gaussian profile within 2nd medium after a
short distance
d
of propagatio
p
on; x-projecction.
Figure 2.12. Reprresentative 2-D Gaussian profile within 2nd medium after a
larger distance
d
off propagatio
on; x-projeection.
16
Figure 2.13. Representativee 2-D Gausssian profiile a short distance in
i 3rd
medium
m; x-projecction.
Figure 2.14. Rep
presentative 2-D Gau
ussian proofile in 3rdd medium after
longer distance off propagatiion; x-projeection.
17
The
T UBPM approach also
a
can be used to exxplore the performance
p
e of a
coupled
d waveguidde [11].
A is well known, ev
As
vanescent waves
w
can shift
energy from one waveguidee to anotheer in a couupled waveeguide struucture.
Here, an
a examplee for the Gaussian
G
beeam inside the coupleed waveguiide is
shown.
The refraactive index
x profile is plotted
p
in Figure
F
2.15 which is a cross
section of a two slab wavegguide.
Figure 2.16 shhows periodic beam power
p
transferr between tthe two waaveguides as
wave
a a conseq
quence of evanescent
e
couplin
ng. This sim
mulation iss also veryy useful too help testt the waveeguide
propertiies.
The MATLAB
B code cann be foun
nd in App
pendix A under
project__BPM.m.
F
Figure
2.15.. The refra
active indexx profile of two-slab waveguide.
w
18
Figure 2.16. Gaussian proopagation showing coupling
c
in the twoo-slab
wavegu
uide.
BPM
B
is a quuick and easy method of
o solving fo
or fields in integrated
i
o
optical
devicess. It is typiically used only in soolving for intensity
i
annd modes within
w
shaped (bent, tappered, term
minated) waveguide
w
structures, as opposeed to
scatterinng problem
ms [12]. However,
H
inn other praactical applications suuch as
multilay
yer structures, there is reflectioon at surfaaces and innterfaces of
o the
structurre.
This m
means the UBPM
U
can give
g
rise to significantt errors. T
That is
why a revised
r
metthod called bidirectional Beam Prropagation Method
M
(BB
BPM)
needs too be introduuced.
2.5 Con
nclusion
T basic prrinciple of UBPM
The
U
has been discuussed, with some illusttrative
examples.
A profile in the spatial
s
domaain can be described
d
as a superpoosition
19
of plane waves through its plane wave spectrum, which can be propagated
using the transfer function for propagation.
The propagated spectrum is
inverse Fourier transformed to go back to the spatial domain profile.
beam propagation in 1-D and 2-D are shown as examples.
Gaussian
Propagation in a
homogeneous medium, through a layer of a NIM, and through a coupled planar
waveguide is shown. However, UBPM cannot take into account reflections,
and cannot determine the reflected wave.
In the following Chapters, the basic technique of BBPM is introduced
based on the TMM.
20
CHAPTER III
PLANE WAVE PROPAGATION THROUGH AN OPTICAL BOUNDARY
3.1 Introduction
In Chapter II, it has been shown that BPMs for analyzing propagation
through longitudinally inhomogeneous media are pretty straightforward and
helpful when the refractive index varies sufficiently slowly along the
propagation direction so that the accumulated reflections can be ignored.
But
in some practical applications such as photonic bandgap structures, there may
exist cumulative reflections from the interfaces between different refractive
indices.
Hence, there is a need to develop a bidirectional beam propagation
method (BBPM) which accounts for the forward and backward fields, and is
not limited by the shortcomings of UBPM.
TMM, which has the added
advantage of incorporating the polarization information, is suitable; however, it
has been primarily developed for analysis of one plane wave through such
strongly inhomogeneous structures.
In this Chapter, a summary of the theory
of reflection and transmission of plane waves at a simple boundary between two
isotropic materials is introduced, which is the basis of TMM, is first presented.
Also, relevant notations for solving multilayer structures in the future Chapters
are introduced.
21
3.2 Plane waves and Snell’s law
An incident plane wave on the optical interface between two isotropic
media is split into two plane waves, with the plane wave propagating beyond
the optical interface being called transmitted wave while the other is the
reflected wave.
The incident ( i ), transmitted ( t ) and reflected ( r ) plane
waves are defined as:
E i ,t ,r Re{E ei ,t ,r exp j (t k i ,t ,r R) } ,
(3.2-1)
where E ei ,t ,r represent the complex envelopes, k i ,t ,r represent the wavevectors,
and represents the angular frequency. Boundary conditions demand that
these field amplitudes in Equation (3.2-1) at the interface z 0 satisfy the
equation:
( k i R ) z 0 ( k r R ) z 0 ( kt R ) z 0 .
(3.2-2)
Referring to Figure 3.1, where n1 and n2 represent the refractive indices of
the two media,
ki kr
c
n1 , kt
c
n2 .
(3.2-3)
From Equation (3.2-2), it follows that the tangential components of all
the wavevectors must be equal, and hence from the definitions of the incident,
reflected, transmitted angles as in Figure 3.1, Snell’s law follows:
n1 sin i n2 sin t n1 sin r .
22
(3.2-4)
n2
n1
kr
r
ki
kt
t
i
Figure 3.1. Transmitted wavevector and reflected wavevector at a
boundary.
In Figure 3.2, some additional notation is defined.
The permittivites
and permeabilities of regions 1 and 2 are taken to be 1 , 1 and 2 , 2 ,
respectively. The plane wave incident from medium 1 is assumed to have a
wavevector k1 , correspondingly k2 , k1 ' are the transmitted and reflected
wavevectors.
Here
ki , kr , kt
have been replaced with
convenience in calculations in multilayers.
k1 , k1' , k 2
for
E2 ' is the electric field from the
reflected wave from a second interface, which is the case for multilayer
structures.
In this way all participating fields can be setted up to use for TMM
analysis for multilayer structures in future Chapters.
23
1 , 1
Region 1
H 1'
E 1'
2 , 2
Region 2
E2
2
H2
H 2'
1
E1
E 2'
H1
Figure 3.2. Reflection and transmission scenario for TE case. The electric
fields in this figure are directed out of the page for all waves.
The total electric field phasor in Regions 1 and 2 can be therefore written
as:
jkR jk 'R
1
(Ee
E ' e 1 )
Ep 1 1
jk R
jk 'R
(E2e 2 E2 ' e 2 )
z 0
,
z 0
(3.2-5)
while the magnetic field phasor H p can be obtained from Maxwell’s
equations:
j
Hp
E p.
(3.2-6)
In the case of Figure 3.2 as drawn, E1 , E1' and E2 are the incident,
reflected and transmitted amplitudes, respectively. E2 ' is zero because in this
case, there is no reflected light in region 2.
24
3.3 Reflection and transmission of TE and TM waves
Generally any elliptic polarization can be split into a linear combination
of a TE wave and a TM wave.
Thus, the reflection and transmission
coefficients of the TE and TM waves need to be determined.
Figure 3.2 is the
TE case because all the electric fields are directed out of the page along the y
direction.
According to the boundary condition for electric fields, all
tangential or E y components are continuous,
E1TE E1TE ' E2TE E2TE '
Similarly, all H x components are also continuous.
(3.3-1)
By using boundary
conditions:
1
2
( E1TE E1TE ') cos 1
( E E2TE ') cos 2 ,
1
2 2TE
(3.3-2)
using equations (3.3-1) and (3.3-2),
E1TE
E2TE
DTE (1)
DTE (2)
,
E1TE '
E2TE '
(3.3-3)
where
1
DTE (i ) i
cos i
i
, i 1, 2 .
i
cos i
i
1
DTE (i ) is called “dynamical matrix” of the TE wave.
(3.3-4)
The reflection (r)
and transmission (t) coefficients for a single interface (implying E2 ' 0 ) for the
TE case can be found from Equations (3.3-1) and (3.3-2) as [13]
25
n1
cos1
n2
cos 2
E1TE
1
2
;
rTE
E1TE n1 cos n2 cos
1
2
'
1
2
(3.3-5)
n1
tTE
2 cos1
E2TE
1
,
n
n2
E1TE
1
cos 1 cos 2
1
2
where E1TE ' is the reflected amplitude, E1TE is the incident amplitude, E2TE
is transmitted amplitude, 1 is the incident angle and 2 is the refracted angle.
Figures 3.3 and 3.4 show the transmission and reflection coefficients for the TE
case, respectively, with light incident from air ( n1 1 ) to glass ( n2 1.5 ).
0.8
0.7
transmission coefficients
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle (radian)of the TE wave
1.4
1.6
Figure 3.3. The transmission coefficient for TE case as a function of the
incident angle.
26
-0.2
-0.3
refection coefficients
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
0
0.2
0.4
0.6
0.8
1
1.2
incident angle (radian)of the TE wave
1.4
1.6
Figure 3.4. The reflection coefficient for TE case as a function of the
incident angle.
In the case of TM waves, the H field is perpendicular to the plane of
incidence at z 0 . As before, from the boundary condition and Maxwell
equations, the dynamical matrix of the TM wave can be derived as [14]
cos i
DTM (i ) i
i
cosi
i , i 1, 2,
i
(3.3-6)
When H 2 ' 0 , the reflection (r) and transmission (t) coefficients for a
single optical interface for the TM case become:
n1
cos 2
n2
cos1
E1TM '
2
rTM 1
n1
n
E1TM
cos 2 2 cos1
1
2
27
(3.3-7)
E2TM
2 cos 1
tTM
n n
E1TM
（ 2 / 1 ）cos 1 cos 2
(3.3-8)
2 1
where E1TM ' is the reflected amplitude, E1TM is the incident amplitude, E2TM
is transmitted amplitude, 1 is the incident angle and 2 is the refracted
angle.
Figures 3.5 and 3.6 show the transmission and reflection coefficients for
the TM case respectively with light incident from air ( n1 1 ) into GaAs
( n2 3.6 ).
0.45
0.4
transmission coefficients
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle (radian)of the TM wave
1.4
1.6
Figure 3.5. The transmission coefficient for TM case as a function of the
incident angle.
28
1
0.8
refection coefficients
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
0.2
0.4
0.6
0.8
1
1.2
incident angle (radian)of the TM wave
1.4
1.6
Figure 3.6. The reflection coefficient for TM case as a function of the
incident angle.
3.4 Principle of reversibility
The principle of reversibility states that light follows exactly the same
path if its direction of travel is reversed. Thus far, the case where a plane wave
is incident from medium 1 to 2 has been considered, and the reflection and
transmission coefficients have been denoted as r and t, respectively. When a
plane wave is incident from medium 2 to medium 1, the reversed reflection and
transmission coefficients are:
r21 (
E2
E'
); t21 ( 1 )
E2 '
E2 '
(3.4-1)
From Equations (3.3-1) and (3.3-2), it follows that [10]:
r21 r12 ,
t12t21 r12 r21 1 ,
with the reflectance and transmission defined as, respectively:
29
(3.4-2)
(3.4-3)
2
Rij rij ; Tij 1 Rij .
(3.4-4)
From Equations (3.4-2), (3.4-3) and (3.4-4), it is easily seen that:
R12 R21 ; T12 T21 .
(3.4-5)
Therefore the Fresnel reflectance and transmittance for plane wave
incidence from medium 1 to 2 are equal to the reflection and transmission for
plane wave incidence from medium 2 to 1, respectively. This law is vital for
future studies on wave propagation in multilayer structures.
3.5 Conclusion
As an introduction to the development of BBPM based on TMM, in this
Chapter plane wave propagation on one optical interface has been discussed. It
is shown that from the boundary conditions and Maxwell’s equations, the
reflection and transmission coefficients of a simple optical boundary can be
determined.
In the next Chapter, plane wave propagation inside multilayer structures
for both TE and TM cases is investigated. Using TMM, the reflection and
transmission coefficients of a multilayer structure can be evaluated.
30
CHAPTER IV
PLANE WAVE PROPAGATION THROUGH MULTILAYER
STRUCTURES
4.1 Introduction
The method described in Chapter III can be applied to explore the
amplitude of the fields of any optical interface. However, for EM fields
propagating through multilayers, the computations of the amplitude within
these layers can become complicated due to the large amount of calculations
needed.
Therefore, as an example, in the beginning of this Chapter, the
reflection and transmission coefficients of a thin layer, which includes two
optical interfaces and one propagation distance is investigated.
Thereafter
TMM is discussed at length. MATLAB can conveniently be used for TMM
since it can manipulate matrices efficiently. Plane waves at normal incidence
are discussed first.
Also, in the multiplayer case, examples involving
PIM-NIM layers are investigated.
4.2 Reflection and transmission coefficients of a thin layer
An example of a plane wave propagating in a thin layer is shown in
Figure 4.1 below.
31
n1
n0
ns
A1
A2
A
B1
B2
B
'
3
'
3
z=d
z=0
Figure 4.1. A thin layer of dielectric material.
Here an incident TE wave is assumed to enter into a thin layer from the
left-hand side where the TE wave was assumed.
Therefore, the electric fields
can be written as:
A1e jk0 z z B1e jk0 z z ,
E py A2e jk1 z z B2e jk1 z z ,
' jksz ( z d )
,
A 3e
z0
0 zd
(4.2-1)
dz
where ki represents the propagation constants inside each layer along z.
Using Maxwell’s equations and Equation (4.2-1), the magnetic fields can be
written as:
32
H px
k0 z
jk0 z z
B1e jk0 z z ),
( A1e
k
1z ( A2e jk1 z z B2e jk1 z z ),
ksz ' jksz ( z d )
( A 3e
),
z0
0 zd
(4.2-2)
dz
Boundary conditions at the interfaces state that the E y and H x are continuous
at z 0 and z d , which yields:
A1 B1 A2 B2 ,
k ( A B ) k ( A B ),
1z
2
2
0z 1 1
jk1 z d
jk1 z d
'
B2e
A 3,
A2e
jk
d
jk
k ( A e 1 z B e 1 z d ) k A' ,
sz
2
3
1z 2
(4.2-3)
so that using Equation (4.2-3),
r01TE
r1sTE
t01TE
t1sTE
where k0 z
2
n0 cos0 ,
k1z
n0
cos 0
n0
cos 0
n1
cos1
0
0
1
n1
1
cos1
2
n0
0
1
2
0
n1
1
cos1
n1
cos 1
ns
cos s
1
1
s
ns
s
,
(4.2-4)
cos s
,
cos 0
cos 0
2
n1
n0
n1
n1
1
cos 1
,
(4.2-5)
cos 1
cos1
ns
s
cos s
,
n1 cos1 , 1 sin 1 (
and 0 s 1 [11].
33
n0
sin 0 ) ,
n1
n0 ns n1 ,
Using Equations (4.2-3), (4.2-4) and (4.2-5), the overall amplitude
reflection and transmission coefficients for the TE case can be written as [15]
rTE &TM
B1 r01 r1s e2 j
,
A1 1 r01r1s e 2 j
(4.2-6)
tTE &TM
A'3
t t e j
01 1s 2 j ,
A 1 r01r1s e
(4.2-7)
where
k1z d
2 d
n1 cos1 .
(4.2-8)
The reflection and the transmission coefficients of TM waves can also be
derived in a similar way. In this case, Equations (4.2-6)-(4.2-8) can again be
used, but with
n0
n
cos1 1 cos 0
1
r01TM 0
,
n
n1
0
cos 1 cos 0
0
1
n1
n
cos s s cos 1
1
s
,
r1sTM n
ns
1
cos s cos1
1
s
(4.1-9)
n
2 0 cos 0
0
t01TM
,
n0
n
cos1 1 cos 0
0
1
n
2 1 cos 1
1
.
t1sTM n
ns
1
cos s cos 1
1
s
(4.1-10)
34
As an example, let d 1104 m , 10 106 m and n0 ns 1,
n1 1.5 .
Figures 4.2 and 4.3 show the reflection and the transmission
coefficients for the TE case, respectively. Light is incident from air ( n0 1 )
onto glass ( n1 1.5 ) with the substrate (region s) assumed to be air ( ns 1 ).
1
0.9
0.8
reflection coefficients
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle(radian) of the TE wave
1.4
1.6
Figure 4.2. Absolute value of reflection coefficient for TE case as a function
of the incident angle for single layer structure.
1
0.9
transmission coefficients
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle(radian) of the TE wave
1.4
1.6
Figure 4.3. Absolute value of transmission coefficient for TE case as a
function of the incident angle for single layer structure.
35
Figures 4.4 and 4.5 show the reflection and transmission coefficients for
the TM wave case, respectively. Here light is incident from air ( n0 1 ) onto
GaAs ( n1 3.6 ) and with the substrate (region s) assumed to be air ( ns 1 ).
1
0.9
0.8
reflection coefficients
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle(radian) of the TM wave
1.4
1.6
Figure 4.4. Absolute value of reflection coefficient for the TM case as a
function of the incident angle for single layer structure.
1.2
transmission coefficients
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
incident angle(radian) of the TM wave
1.4
1.6
Figure 4.5. Absolute value of transmission coefficient for the TM case as a
function of the incident angle for single layer structure.
36
4.3 Matrix formulation of TMM for a thin film
From the previous examples, it is clear that forward and backward
traveling plane waves exist in a layer.
While the x-component of the
wave-vector is presented, the z-component comprises forward and backward
traveling components.
Hence, the electric field can be expressed generically
as:
E p E p ( z ) e jk x x ,
(4.3-1)
where kx is the wavenumber along the x direction.
n0
n1
ns
A0
A '1
A1 A ' s
B0
B '1
B 1 B 's
z=0
z=d
Figure 4.6. Same as Figure 4.1, but with coefficients Ai , Bi , Ai ' , Bi ' inserted.
From Equation (4.3-1), and referring to Figure 4.6,
E p ( z ) A( z ) B ( z ) Ae jkz z Be jkz z
37
(4.3-2)
where E p ( z ) is defined as the phasor part of standing wave, Ai ( z ), Ai ' ( z ) are
the forward traveling amplitudes, and Bi ( z ), Bi ' ( z ) are the backward traveling
amplitudes.
From Equation (3.3-3), obviously, A( z ), B( z ) are not continuous
at each interface and the relation can be written as:
A0
A '1
A '1
1
D0 D1 D01 .
B '1
B '1
B0
(4.3-3)
Over the distance z=0 to z=d,
A '1 e j1
B '1 0
A1
0
P1
e
B1
(4.3-4)
j1
where P is the propagation matrix with k z d .
At the interface z=d,
As '
A' s
A1
1
D
D
D
s
1
1s
'
B '
B1
Bs
s
(4.3-5)
From Chapter III, the dynamical matrices can be summarized as
1
1
i cos i cos for TE wave
i
i
i
i
Di
,
cos
cos
i
i
i for TM wave
i
i
i
where i 1, 2,3 and i is incident angel of the plane wave.
(4.3-7)
According to
Equation (4.1-3), (4.1-4) and (4.1-5), the amplitudes A0 , B0 and A' s , B ' s can be
related by
A' s
A' s
A0
1
1
D0 D1P1D1 Ds ' M '
B0
Bs
Bs
where
38
(4.3-8)
M
M 11
M 21
M 12
1
1
D1 D2 P2 D2 D3
M 22
(4.3-9)
is the transfer matrix as applied to a sample single layer system.
4.4 Extension to multilayer system
The multilayer system is governed by the same principles used in the
single thin film case.
As shown in Figure 4.7 as an example, the layer
thicknesses d l may vary with the position of the z axis, viz.,
d1 z1 z0 ;
d 2 z2 z1;
......
(4.4-1)
d n zn zn1
......
The electric field distribution inside the layers is given by
A0e jk0 z ( z z0 ) B0e jk0 z ( z z0 ) ,
E p Al e jklz ( z zl ) Bl e jklz ( z zl ) ,
' jksz ( z zn )
Bs 'e jksz ( z zn ) ,
As e
with klz nl
c
z z0
zl 1 z zl
(4.4-2)
zn z
cosl , which is the wavenumber along the z direction. A0 is
the initial wave amplitude incident on the first interface from layer “0”. As is
the transmitted wave amplitude in the substrate layer.
39
Figure 4.7. Perioodic structture compoosed of 2 materials with refraactive
index n1 and n2 .
Upon
U
extendding the conncept in thee previous Section,
S
it iss readily veerified
that
N
A' s
A0
1
1
Pl l Ds ' ,
D0 Dl PD
l 1
B s
B0
(44.4-3)
t transfer matrix cann be written as
so that the
M
M 11
M 21
N
M 12
1
1
D0 Dl Pl Dl Ds .
M 22
l 1
2
For
F
multilaayer structtures, the calculation
n of
(44.4-4)
M ij is formid
dable.
Therefo
ore, it is logical to use software
s
succh as MATL
LAB for thee calculations.
Iff light is inccident from
m the left sidde and propaagating throough a multtilayer,
it has been shown that the refflection andd transmissio
on coefficieents are givven by
40
B0
r A ,
0
A'
t s .
A0
(4.4-5)
Alternatively, using the notation in Section 4.2, the reflection and
transmission coefficients can be rewritten in terms of the transfer matrix
coefficients as
M 21
r M ,
11
1
t
.
M 11
(4.4-6)
Indeed, this can be readily verified for the single layer case, using the
definition of the matrix M.
An example of the periodic structures only
containing positive refractive index materials is given below, and where a TE
plane wave is propagating from left side to the right side, as shown in Figure
4.7.
In Figure 4.8, each period is composed of 2 layers, each layer with a
thickness d i 1.5 106 m . Assume that the incident angle 0 , the
semi-infinite incident medium has refractive index n0 1 , the semi-infinite
substrate medium has refractive index ns 1 , and that there are 7 periods, with
each period comprising two layers with n1 2.0, n 2 1.5 , and with 1 .
From the discussion above, assuming that the incident amplitude A0 is given,
M can be calculated by multilayer structure properties, and B0 can be found by
Equations (4.4-5) and (4.4-6). Using Equation (4.4-2), the variation of the
electric field E p is found and is shown in Figure 4.8. The MATLAB code for
41
generating the figure can be found in Appendix A under the file,
multilayerstructureplanewave.m and layeryehTE.m.
1.4
absolute value of electric amplitude
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
propagation distance(m)
2
2.5
-5
x 10
Figure 4.8. Absolute value of electric field amplitude in multilayer
structure described above. The structure includes 7 periods, each period
including two layers with n1 2.0, n 2 1.5 , each layer with a thickness
d i 1.5 106 m .
The envelope of the absolute value of electric field
amplitude is decaying when it is propagating in the multilayer structures.
A second example of periodic structures containing NIMs is shown in
Figure 4.9, where each period is composed of 2 layers, with each layer having a
thickness d i 1.5 106 m . Assume that the incident angle 0 , the
semi-infinite incident medium has refractive index n0 1 , the semi-infinite
substrate medium has refractive index ns 1 , and that there are 7 periods with
42
each period including two layers with n1 2.0, n 2 1.5 , and 1 .
Again, from the discussion above, if the incident amplitude A0 is given, M can
be calculated by multilayer structure properties, and B0 can be found by
Equations (4.4-5) and (4.4-6). Therefore, using Equation (4.4-2), the variation
of the absolute value of the electric field E p can be calculated and is shown in
Figure 4.9.
1.4
absolute value of electric amplitude
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
propagation distance(m)
2
2.5
-5
x 10
Figure 4.9. Absolute value of electric field amplitude for the 7 period
structure, with each period including two layers with n1 2.0, n 2 1.5 ,
each layer with a thickness d i 1.5 106 m . The envelope of the absolute
value of electric field amplitude is decaying when it is propagating in the
multilayer structures.
Finally, a special example of a periodic structure containing NIMs and
PIMs is shown. Note that if the characteristic impedances are all equal, there
43
is expected to be no reflection on any optical boundary. In Figure 4.10, each
period is composed of 2 layers, each layer with a thickness d i 1.5 106 m .
Assume that the incident angle 0 , the incident medium has refractive index
n0 1 , the substrate media has refractive index ns 1 , and that there are 7
periods, where each period includes a PIM and NIM with n1 1, n 2 1 . Note
that 1 when refractive index is a negative value and 1 when
refractive index is positive. As expected, the electric field amplitude does not
decay during propagation, as shown by the simulation results in Figure 4.10.
Interestingly, the real part of the total electric field shows spatial
oscillations with the periodicity of the structure as shown in Figure 4.11, which
implies that the total phase progressively changes through the structure.
1
absolute value of electric amplitude
1
1
1
1
1
1
1
0
0.5
1
1.5
propagation distance(m)
2
2.5
-5
x 10
Figure 4.10. The absolute value of electric field amplitude is constant when
it is propagating in the multilayer structure.
The structure includes 7
periods, each period including two layers with n1 2.0, n 2 1.5 , each
layer with a thickness di 1.5 106 m.
44
1
0.95
electric amplitude
0.9
0.85
0.8
0.75
0.7
0.65
0
0.5
1
1.5
propagation distance(m)
2
2.5
-5
x 10
Figure 4.11. The real part of the electric field amplitude is oscillatory when
it is propagating in the multilayer structures.
The structure includes 7
periods, each period including two layers with n1 2.0, n 2 1.5 , each
layer with a thickness di 1.5 106 m.
4.5 Conclusion
The focus of this Chapter is to introduce the basics of electric (and
magnetic) field propagation through multilayer structures using the TMM.
The conventional approach for a single layer is identical to the TMM approach,
which can be generalized for multilayer structures.
In general, the electric field envelope decays when it is propagating
inside the multilayer structure due to reflection. But when the impedances are
matched, the reflected wave reduces to zero.
As will be seen in Chapter V,
with variation in the incident angles, the reflection and the transmission
coefficients also vary. Therefore, different incident fields at different angles
45
lead to different spatial variations when propagating within a multilayer
structure.
This concept of TMM is used to analyze propagation of angular
plane wave spectra through normal and metamaterial multilayer structures in
the next Chapter, from which the profile of beams through such structure can be
calculated.
46
CHAPTER V
PROPAGATION OF ANGULAR PLANE WAVE SPECTRA THROUGH
MULTILAYER STRUCTURES
5.1 Introduction
In previous Chapters, the traditional paraxial beam propagation methods
such as UBPM have been introduced. UBPM is easy to implement and can be
used to analyze propagation through longitudinally inhomogeneous media
accurately when the refractive index varies sufficiently slowly along the
propagation direction, so that the accumulated reflections can be ignored.
However, it is not suitable for layered structures where the refractive index
variation can be large; furthermore, the polarization state is also usually
considered in simple BPM algorithms. In this Chapter, a methodology to
numerically analyze propagation of arbitrary beam profiles with arbitrary
polarizations through layered structures is discussed. This is achieved using
TMM.
The layered structure may comprise regions of positive and negative
refractive indices.
The refractive indices can, in general, be complex as well.
Example of propagation of a collection of TE or TM plane waves with a
Gaussian amplitude profile in 1 transverse dimension (x) through a stack of
PIM/NIM are shown. The spatial variation of the electric field at any plane (z)
during bidirectional propagation through the stack is found from the composite
angular plane wave spectra of the forward and backward traveling waves.
47
The numerical results from TMM are compared with numerical
simulations using finite element method (FEM) techniques.
5.2 Comparison of TMM and FEM for TE plane wave incidence
In Chapter IV, some examples for TE wave propagation inside multilayer
structures have been discussed. In what follows, more examples are shown,
including comparison of TMM with FEM.
As a first example, assume the structure consists of 5 periods of
alternating PIM layers of p 2.25, p 1 and NIM layers of n 2.25 ,
n 1 . Each layer has a thickness of d p d n 1.5 m . The incidence
medium and the substrate are assumed to have a refractive index of 1.
The
total thickness of the structure is then D 15m. The incident plane wave has a
wavelength of 11.11m and has an incidence angle i .
Figure 5.1 (a) shows the magnitude squared of the electric field Ey and
Figure 5.1 (b) shows the magnitude squared of the magnetic field Hx in the
incidence medium, the structure and the substrate for the two incidence angles
i 0, / 6 simulated using the TMM technique and FEM, respectively. The
MATLAB code can be found in Appendix A under TEwave.m and
layeryehTE.m.
As seen from the figures, there is excellent agreement between
the TMM simulations and the FEM results.
48
(aa)
(bb)
Figure 5.1. (a) Th
he magnitu
ude squareed of the y--componen
nt of the eleectric
field, (b
b) the maggnitude squ
uared of th
he x-compoonent of thee magneticc field
in the incidence medium, the structture and the substra
ate for thee two
inciden
nce angles i 0, / 6 simulated using
u
the TMM
T
techn
nique and FEM,
F
respecttively.
49
5.3 Com
mparison of
o TMM an
nd FEM forr TM planee wave incid
dence
The
T dynamical matrix derived
d
in Equation
E
(3
3.3-6) cannoot be used in
i the
TM casse because it
i pertains to
t the electrric field.
H
Here
the H field is asssumed
to be th
he initial fielld to propaggate in the m
multilayer sttructure [166].
Figure 5.2. Layeered struccture comp
posed of materials
m
with refraactive
indices
with th
hickness
.
Consider
C
thee TM case (p-wave)
(p
shhown in Figuure 5.2. Unnlike the preevious
case, th
he magneticc field is tanngential to the interfacce. The dyynamical annd the
propagaation matricces for the magnetic field
f
H willl be deriveed instead of
o the
The magneetic field in layer can be written ass:
electric field E.
'
'
ikl .r
Hleikl .r Hleikl .r eit
Ae
Bl eikl .r eit
l
Hy
'
'
ikl1.r
Hl1eikl1.r eit Al'1eikl1.r Bl'1eikl1.r eit
Hl 1e
Imposinng the contiinuity of Ex and Hy ,:
50
z0
(5.3-1)
z 0
H l H l H l1 H l1
(5.3-2 a,b)
Elx Elx El1, x El1, x
From Maxwell’s equations,:
j
j H y
E
H
xˆ
z
H y
zˆ
Ex xˆ Ez zˆ
x
(5.3-3)
Upon using Equations (5.3-3), (5.3-2b) and setting kl ,l 1z kl ,l 1 cos l ,l 1 ,
and k 'l ,l 1z k 'l ,l 1 cos l ,l 1 , it follows that
H l H l H l1 H l1
cos H H cos H H
l
l
l
l 1
l 1
l 1
l 1
l
(5.3-4 a,b)
Equations (5.3-4) can be written in matrix form as:
l cos l
1
H l
H l1
1
1
l cos l l 1 cos l 1 l 1 cos l 1
Hl
H l 1
1
(5.3-5)
Therefore the dynamical matrix for TM case can be defined as:
DlTM
1
l
cos l
l
, l 1, 2,... .
l cosl
l
1
(5.3-6)
Since it is more conventional to monitor electric fields, Maxwell’s
equations can now be used to obtain
51
sin H y ,
0c
Ex
cos H y ,
0c
Ez
(5.3-7)
to determine the x and z components of the electric fields in the required layer.
As an example of TM incidence, consider the same structure as in the TE
case.
Figure 5.3 (a) shows the magnitude squared of the magnetic field Hy and
Figure 5.3 (b) shows the magnitude square of the electric field Ex in the
incidence medium, the structure and the substrate for the two incidence angles
i 0, / 6 simulated using the TMM technique and FEM.
The MATLAB code
can be found in Appendix A under TEwave.m and layeryehH.m. Once again,
there excellent agreement between the TMM simulations and the FEM results.
52
(aa)
(bb)
Figure 5.3. (a) The
T
magniitude squaared of the magneticc field, (b
b) the
magnittude squareed of the x--componen
nt of the eleectric field in the incid
dence
medium
m, the stru
ucture and
d the subsstrate for the two in
ncidence aangles
i 0, / 6 simulateed using thee TMM tecchnique and
d FEM, resspectively.
53
5.4 Propagation of angular plane wave spectrum through multilayer
structure using TMM
In the above Sections, the basic concept of TMM for a single plane wave
incidence on a multilayer structure has been discussed. This can be readily
extended to study the propagation of a collection of plane waves (plane wave
spectrum) corresponding to a profiled beam.
Assume that E m (k xm ) represents
the complex amplitude of a plane wave (m) with a transverse propagation
constant k xm , which corresponds to an angle of propagation
k xm
k0
m sin 1
2
, k 0
.
(5.4-1)
As shown in Figure 5.4, a collection of such plane waves constitutes the
profile of the beam.
The angular plane wave spectral amplitudes can be
determined, for instance, by taking a spatial Fourier transform of a function
Em ( x) E m (k xm )e jkxm x kx
(5.4-2)
m
which represents the beam profile. Each plane wave of amplitude
E m ( k xm ) E m ( m ) and incident at an angle m is propagated through the
multilayer structure using TMM to determine the forward and backward
traveling complex amplitudes at any longitudinal position within and outside
the structure. The profile of the beam at any longitudinal position within the
multilayer structure as well as in the incident medium and the substrate can be
determined by summing all of the requisite complex amplitudes along with
their associated “phases” similar to the relation in Equation (5.4-2) above.
54
Spatial domain
Spatial
frequency
domain
kx
kz
E(x)
Figure 5.4. A profile in the transverse spatial domain is equated to the
superposition of plane waves with different traveling directions and
amplitudes.
5.5 TE case: Propagation of a collection of plane waves with Gaussian
profile
Using the same structure parameters and wavelength as above
propagation of a collection of plane waves with a Gaussian profile, is now
examined.
Figure 5.5 shows the initial and transmitted Gaussian profile (using
TMM and FEM) after propagation in such a structure for two beam waists (a)
w0 , (b) w0 10 , respectively.
As expected, there is no change between
the incident and transmitted profile in this alternating structure since the
refractive indices are equal in magnitude and opposite in sign. Figures 5.6 (a)
and (b) show the magnitude square of the electric field as it propagates inside
the metamaterial structure for the case when the beam waists w0 , using
TMM and FEM respectively. Notice the focusing and defocusing inside the
structure due to the negative index medium. Figure 5.7(a) shows the initial
and transmitted Gaussian profile (using TMM and FEM) after propagation in
the structure defined in Section 5.2 but with p 2.25, p 1, n 1.44 ,
55
n 1,
1 and for beam
b
waist w
0
. The magnitude squared off the electric
c field
as it prropagates innside the metamateria
m
al structure using TMM
M and FEM
M are
shown.
(aa)
(b
b)
Figure 5.5. Initial an
nd transmiitted Gausssian profilee (using TM
MM and FEM)
F
after propagation in the structu
ure defined
d in Section
n 5.2, for tw
wo beam waists
w
(a) w0 , (b) w0 100 respectivvely. The MATLAB
B code can
n be foun
nd in
Appendix A under wave.m and layeryehTE.m.
56
(a)
(b)
Figure 5.6. The magnitude
m
squared of
o the electtric field as
a it propaagates
inside the
t metamaaterial stru
ucture with
h same paraameters as in Figure 5.5(a)
5
using (a)
( TMM and (b) FE
EM respecctively. Th
he MATLA
AB code caan be
found in
i Appendiix A under TE.m and layeryehTE.m.
57
(a))
(b)
(c)
Figure 5.7. Initiall and transmitted Gau
ussian proffile (using TMM
T
and FEM)
after propagatio
n in the structure defined in
p
i Section
n 3.2 but with
p 2.225, p 1, n 1.444 , n 1 , and for beam waaist w0 . The
magnittude squarred of thee electric field as it propaga
ates insidee the
metamaterial stru
ucture usin
ng (b) TMM
M and (c) FE
EM respecctively.
58
5.6 TM case: Propagation of a collection of plane waves with Gaussian
profile
In this example, TM propagation of a collection of plane waves with a
Gaussian profile is investigated.
Figure 5.8 (a) shows the initial and
transmitted Gaussian profile (using TMM and FEM) after propagation in the
same structure as the previous example with beam waist w0 . As expected,
there should not be any change between the incident and transmitted profile in
this alternating structure.
Figure 5.8 (b) shows the initial and transmitted
Gaussian profile (using TMM and FEM) after propagation in a structure with
the same thickness as above consisting of 5 periods of alternating PIM layers of
p 2.25, p 1 and NIM layers of n 1.44 , n 1 .
The beam waist
w0 and the same wavelength as above are shown. Figures 5.9 (a) and (b)
shows the magnitude squared of the magnetic field as it propagates inside the
metamaterial structure of the same parameters as in Figure 5.8 (a) using TMM
and FEM respectively. Figures 5.9 (c) and (d) show the magnitude squared of
the x-component of the electric field as it propagates inside the metamaterial
structure of the same parameters as in Figure 5.8(a) using TMM and FEM
respectively.
Finally, we show one representative result for the TM incidence and for
w0 for p 2.25, p 1 , and n 1.44 , n 1 , obtained using TMM.
Figure 5.10 (a) shows the y-component of the magnetic field through the
structure, while the x and z component of the electric field, derived using
relations such as (5.3-6), are plotted in Figures 5.10(b) and 5.10(c),
respectively.
59
(aa)
(bb)
Figure 5.8. (a) In
nitial and transmitted
t
d Gaussian
n profile (u
using TMM
M and
FEM) after prop
pagation in
n the samee structure as the prrevious exaample
with beeam waist w0 , (b) Initial and
d transmitted Gaussiaan profile (using
TMM and
a FEM) after prop
pagation in
n a structurre same ass in (a) butt with
nd NIM layers of n 1.44 , n 1 .
p 2.225, p 1 an
The MAT
TLAB
code caan be found
d in Appendix A undeer wave.m and
a layeryeehH.m.
60
(a)
(b)
(c)
(d)
Figure 5.9. |Hy|2 as
a it propa
agates insid
de the metaamaterial structure
s
o the
of
same parameters
p
s as in Figure
F
5.8(a) using (a)
( TMM and (b) FEM
respecttively.
|Ex|2 the maagnitude sq
quared of the x-com
mponent of
o the
electricc field usingg (c) TMM
M and (d) FE
EM respecttively.
61
(a)
(b)
(c)
Figure 5.10. (a) |H
Hy|2 during propagation inside the metamaaterial stru
ucture
w0 , and
with p 2.25, p 1, n 1.44 , n 1 , and fo
or beam waist
w
for TM
M incidence, (b) |Ex|2, (c)
( |Ez|2 .
A simulattions are doone using TMM,
All
T
and aree in excelleent agreemeent with FE
EM simula
ations. The MATLAB
B code
can be found in Appendix
A
A under ExE
Ez.m and laayeryehH.m
m.
62
5.7 Conclusion
In this Chapter, it is shown that TMM can be successfully used to find the
electric and magnetic fields in a metamaterial structure consisting of PIM and
NIM alternating layers in both TE and TM cases.
Results of TMM are
compared with another numerical technique based on FEM method.
It is
concluded that while the results are in excellent agreement, TMM takes less
time and less memory to find the solution. This TMM based technique can be
used to study the propagation of a collection of TE or TM plane waves with any
profile in general, and incorporates both forward and backward traveling waves.
Finally, the TMM method can readily simulate beam propagation through
relatively “long” structures, spanning hundreds of wavelengths, which take
much longer time using FEM techniques. For instance, for 100 layers, there is
about a 30 fold benefit in computational time using TMM as compared to FEM,
and computational time benefit scales nonlinearly with number of layers.
63
CHAPTER VI
CONCLUSION AND FUTURE WORK
In this thesis, plane wave and beam propagation has been extensively
studied for application to multilayer structures comprising work, one or several
interfaces between materials with different refractive indices.
While the
traditional BPM, referred to as UBPM, can be used to model propagation of
beams through media that have a longitudinally slowly varying refractive index
profile, it is not suitable for media that have large changes of refractive index,
as occurs in layered structures. Plane wave propagation through an interface
is reviewed, and extended to the case of propagation through a slice of a
material with refractive index different from the surrounding medium (media).
While the problem is analytically tractable for a single slice, it becomes
insurmountable for multiple interfaces. TMM is a systematic way to analyze
propagation of plane waves through such structures, and takes into account
electric and magnetic fields of arbitrary polarizations. The exact nature of the
electric and magnetic fields through the layered structure, which comprises
forward and backward propagating waves, can be analyzed and visualized.
The method can be easily extended to arbitrary angles of incidence or arbitrary
transverse and longitudinal propagation vector components, for both
propagating and nonpropagating fields.
64
We have extended the TMM approach to analyze the propagation of a
collection or spectrum of plane waves incident on layered structures.
In
particular, we have analyzed cases where the layered structures can comprise
alternating layers of PIMs and NIMs, assuming one transverse dimension.
Both TE and TM cases have been considered.
For the latter case, the TMM
has been redeveloped, and the electric field distribution can be calculated using
the results for the magnetic field and employing Maxwell’s equations.
TMM results have also been compared with another numerical technique
based on FEM method using COMSOL. It is seen that while the FEM results
are very similar to the TMM case, the program run time in the latter is less and
takes less memory to find the solution. Typically, for an Intel Core I7 930
@2.8 GHz with an 8GB memory, for a 40 layered structure, an FEM solution
takes around 189sec while using TMM it takes 47sec.
For 100 layers, the
FEM technique takes 1702 sec while TMM takes 60sec, suggesting that the
execution time difference is nonlinear with respect to the stack length, leading
to the necessity of such a TMM based technique.
Furthermore, the TMM based spectral propagation method can be used to
analyze arbitrary profiles, and particularly to a case where the “stack”
comprises regions of varying index due to induced nonlinearity such as in a
photorefractive reflection grating.
This is part of continuing and future work.
Also, TMM will be extended to problems involving two transverse dimensions.
While simulation tools using TMM have been perfected during this work,
experimental work, not reported in this thesis, is being performed on the
fabrication of negative index metamaterials.
While co-sputtering has been
used to fabricate good quality thin films comprising nanomixtures of SiC and
65
Ag, work on deposition of nanoparticles through laser ablation is also being
pursued.
This has immediate applications in the area of near-field imaging.
It is also our aim to fabricate layered two- and three-dimensional structures of
SiC and Ag for use as an imaging tool for sub-wavelength objects in the
far-field.
Simulation of arbitrary profiles of electric fields through such
structures will be attempted using TMM as well, as an efficient alternative to
FEM methods.
66
BIBLIOGRAPHY
[1] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85,
3966-3969 (2000).
[2] R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, D. R. Smith, “Broadband
ground-plane cloak,” Science 323, 366-369 (2009).
[3] J. Yamauchi, H. Kanbara, H. Nakano, “Analysis of optical waveguides with
high-reflection coatings using the FD-TD method,” IEEE Photonic Technology
Lett. 10, 111-113 (1998).
[3] http://en.wikipedia.org/wiki/Transfer-matrix_method_(optics).
[4] D. Bouzakis, N. Vidakis, T. Leyendecker, G. Erkens, R. Wenke,
“Determination of the fatigue properties of multilayer PVD coating on various
substrates, based on the impact test and its FEM simulation,” Thin Solid Films
308-309, 315-322 (1997).
[5] J. Hong, W. P. Huang, T. Makino, “On the transfer matrix method for
distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868
(1992).
[6] http://en.wikipedia.org/wiki/Finite_element_method.
[7] V. G. Veselago, “The electrodynamics of substances with simultaneously
negative value of and ,” Sov. Phys. Usp. 10, 509-514 (1968).
[8] P. P. Banerjee, “Contemporary Optical Image Processing,” Sec 1.1, 2001.
[9] P. P. Banerjee, “Contemporary Optical Image Processing,” Sec 1.2, 2001.
[10] http://en. wikipedia.org/wiki/Beam_propagation_ method.
67
[11] C. R. Pollock, “Fundamentals of Optoelectronics,” Sec 9.7, 1995.
[12] http://en.wikipedia.org/wiki/Beam_propagation_method.
[13] C. R. Pollock, “Fundamentals of Optoelectronics,” Sec 1.9, 1995.
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68
APPENDIX A
MATLAB CODES
A.1 1_D_FFT_GAUSSIAN.m
clear all;
clc;
close all;
x1=[-10:20./64.:-10+63.*20/64];
z=3.5;
w0=3;
nref=1;
%%
u0=exp(-((x1.^2)./w0^2));
%energy=h^2*sum(sum(abs(u0.^2)))
z0=fft(u0);
%figure,contour(abs(u0))
%title('initial gaussian beam profiles')
figure(1),plot(x1,abs(u0))
title('initial1-D Gaussian profile in the time domain')
ylabel('amplitude')
xlabel('x')
grid on;
v=(exp(i*x1.*x1*.5*z));
w=fftshift(v);
69
zp=z0.*w;
yp=ifft(zp);
figure(2),plot(x1,abs(yp))
grid on
title('final1-D Gaussian profile in the time domain')
ylabel('amplitude')
xla
70
A.2 unidirectional_beam_propagation.m
clear all;
clc;
close all;
lambda=10^-3; % wavelength
w0=3;
N=500; % point in X
L=20;
%Length of X
k0=2*pi/lambda;
Zr=(w0^2*k0)/2; %rayleigh range
z=1*Zr; %propagation distance
f=Zr;
M=10; %number of point
delz=z/M;
umax(1)=1;
h = L/N ; %dx
n =[-N/2:1:N/2-1]';
x1=n*h;
% Indices
% Grid points
x2=x1';
x_min=min(x1);y_min=min(x2);
x_max=max(x1);y_max=max(x2);
z_min=0;
z_max=1.5;
a=pi/h;
e1=[-a:2*a/N:a-2*a/N]';
e2=e1';
[ee1,ee2] = meshgrid(e1);
nref=1;
%%
71
u0=umax(1)*exp(-((x1.^2)./w0^2))*exp((-x2.^2)./w0^2);
energy=h^2*sum(sum(abs(u0.^2)))
%figure,contour(abs(u0))
%title('initial gaussian beam profiles')
figure,mesh(x1,x2,abs(u0)),view(90,0)
Num=-(ee1.^2+ee2.^2);
Dem=i*2*k0*nref;
P=Num./Dem;
zprop(1)=0;…
72
A.3 propagation_in_layers_movie.m
%%% Propagation of Gaussian Beam .
clear all;
clc;
close all;
lambda=10^-3; % wavelength
w0=3;
N=300; % point in X
L=20;
%Length of X
k0=2*pi/lambda;
Zr=(w0^2*k0)/2; %rayleigh range
z=(1*Zr); % the width of each layer .. propagat distance 10*z
M=3; %number of point
delz=z/M;
umax(1)=1;
h = L/N ; %dx
a=pi/h;
n = (-N/2:1:N/2-1)';
x1 = n*h;
% Indices
% Grid points
x2=x1';
x_min=min(x1);y_min=min(x2);
x_max=max(x1);y_max=max(x2);
z_min=0;
z_max=1;
e1=[-a:2*a/N:a-2*a/N]';
e2=e1';
[ee1,ee2] = meshgrid(e1);
aviobj=avifile('prop.avi','compression','cinepak');
%%
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u0=umax(1)*exp(-((x1.^2)./w0^2))*exp((-x2.^2)./w0^2);
energy=h^2*sum(sum(abs(u0.^2)))
% figure,contour(x1,x2,abs(u0))
% title('Initial Gaussian beam profile')
mesh(x1,x2,abs(u0)),view(90,0)
axis([x_min x_max y_min y_max z_min z_max]…
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A.4 project_BPM.m
clear all;
close all;
clc;
cm=1e-2;
mm=1e-3;
um=1e-6;
nm=1e-9;
ns=1.499;
i=sqrt(-1);
nf=1.5;
nave=(ns+nf)/2;
cladwidth=200*um;
wgwidth=10*um;
sig=5*um;
dz=4*um;
atten=1500;
aper=40;
loopnum=250;
maxiterations=4000;
wgsep=14*um;
for j=1:512
coupledindex(j)=ns+(nf-ns)*(abs(abs(j-256)-wgsep*256/cladwidth)<
wgwidth*256/cladwidth);
end
n=coupledindex;
aper=round(512*aper/100);
iterations=0;
lambda=1*um;
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k0=2*pi/lambda;
od=atten*[ones(1,256-fix(aper/2)),zeros(1,aper),ones(1,256-fix((
aper+1)/2))];
a=cladwidth/2/pi;
k=[0:255 -256:-1]/a;
x=cladwidth*(-0.5+(0:511)/512);
[xx,zz]=meshgrid(x,[1:1:maxiterations]);
phase1=exp(i*dz*(k.^2)./(nave*k0+sqrt(max(0,nave^2*k0^2-k.^2))).
..
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A.5 multilayerstructureplanewave.m
clear all;
close all;
clc;
ni=1;
ns=2.25;
n=[ni 2.0 repmat([1.5 2.0],1,9) 1.5 ns];
l=[0.75*10^-6 repmat([1.5*10^-6],1,18) 0.75*10^-6];
pol=1;
mun=[1 1 repmat([1 1],1,9) 1 1]
lambda=11.11*10^-6;
thetai=0;
thi=asin(ni*sin(thetai)./n);
[A,B,T,ref]=layeryeh(n,ni,ns,[10 l 10],thetai,lambda,mun);
x=linspace(13.9*10^-8,20.99*10^-6,10000);
ii=1
Sum=[0 cumsum(l)];
for s=1:1:length(x)
check=0;
while check==0
if x(s)<Sum(ii+1)
kx=n(ii+1)*2*pi*cos(thi(ii+1))/lambda;
E(s)=A(ii+1)*exp(-1i*kx*(x(s)-Sum(ii)))+B(ii+1)*exp(1i*kx*(x(s)Sum(ii)));
check=1;
else ii=ii+1;
end
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end
end
figure
plot(x,real(E).^2 )
xlabel('propagation distance(m)');
ylabel('electric amplitude');
grid on;……………………………………….
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A.6 layeryehTE.m
function [A,B,T,R]=layeryeh(n,ni,ns,l,thetai,lambda,mun)
%s polarization
thi=asin(ni*sin(thetai)./n);
xx=length(n);
for r=1:1:length(lambda)
M=[1,0;0,1];
D1=[ 1 1; (n(1)/mun(1))*cos(thi(1))
-(n(1)/mun(1))*cos(thi(1))];
for m=2:1:length(n)-1,
D2=[ 1 1; (n(m)/mun(m))*cos(thi(m))
-(n(m)/mun(m))*cos(thi(m))];
P2=[exp(i*n(m)*2*pi*cos(thi(m))*l(m)/lambda(r)) 0 ; 0
exp(-i*l(m)*n(m)*2*pi*cos(thi(m))/lambda(r))];
M2=(D2)*P2*inv(D2);
M=M*M2;
end
D3=[ 1 1; (n(m+1)/mun(m+1))*cos(thi(m+1))
-(n(m+1)/mun(m+1))*cos(thi(m+1))]; % D3 is equal to Ds
M=inv(D1)*M*(D3);
T=abs(1/M(1,1)^2);% change
ref=M(2,1)/M(1,1);
R=abs(ref)^2;
Bs=0; B(xx)=Bs;
%coeff
A0=1; A(1)=A0;
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B0=ref*A0; B(1)=B0;
Minv=inv(M);
As=Minv(1,1)*A0+Minv(1,2)*B0; A(xx)=As;
t=1/M(1,1);
As=t*A0;A(xx)=As;
m=1
D11=[ 1 1; (n(m)/mun(m))*cos(thi(m)) …………………….
80
A.7 TEwave
clear all;
clc;
ni=1;
ns=1;
E0=1
lin=3*10^-6;%m
ls=3*10^-6;%m
n=[ni repmat([1.5 -1.5],1,5) ns];
l=repmat([1.5*10^-6],1,10);
pol=1;
mun=[1 repmat([1 -1],1,5) 1];
lambda=11.11*10^-6;
thetai=0;
thi=asin(ni*sin(thetai)./n);
[A,B,T,ref,A0,B0,As]=layeryeh(n,ni,ns,[10 l
10],thetai,lambda,mun);
x=linspace(13.9*10^-11,sum(l)-10E-9,1000);
ii=1
Sum=[0 cumsum(l)];
x_in=linspace(-lin,0,100);
u=1:1:length(x_in)
kin1=n(1)*2*pi*cos(thetai)/lambda;
Ein1=A0.*exp(-1i*kin1.*x_in)+B0.*exp(1i*kin1.*x_in); %incident
layer
%middle
for s=1:1:length(x)
check=0;
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while check==0
if x(s)<Sum(ii+1)
kx=n(ii+1)*2*pi*cos(thi(ii+1))/lambda;
E(s)=A(ii+1)*exp(-1i*kx*(x(s)-Sum(ii)))+B(ii+1)*exp(1i*kx*(x(s)Sum(ii)));
check=1;
else ii=ii+1;
end…
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A.8 layeryehH.m
function
[A,B,T,R,ref,t,A0,B0,As]=layeryehH(n,ni,ns,l,thetai,lambda,Ein,p
ol,mun)
%s polarization
thi=asin(ni*sin(thetai)./n);
xx=length(n);
for r=1:1:length(lambda)
M=[1,0;0,1];
if pol==1
D1=[ 1 1; (mun(1)/n(1))*cos(thi(1))
-(mun(1)/n(1))*cos(thi(1))]; %% TE
elseif pol==0
D1=[ cos(thi(1)) cos(thi(1)); (n(1)/mun(1))
-(n(1)/mun(1))]; %% TM
end
%D1=[ 1 1; n(1)*cos(thi(1)) -n(1)*cos(thi(1))];
for m=2:1:length(n)-1,
if pol==1
D2=[ 1 1; (mun(m)/n(m))*cos(thi(m))
-(mun(m)/n(m))*cos(thi(m))]; %% TE
elseif pol==0
D2=[ cos(thi(m)) cos(thi(m)); (n(m)/mun(m))
-(n(m)/mun(m))]; %% TM
end
%D2=[ 1 1; n(m)*cos(thi(m)) -n(m)*cos(thi(m))];
P2=[exp(i*n(m)*2*pi*cos(thi(m))*l(m)/lambda(r)) 0 ; 0
exp(-i*l(m)*n(m)*2*pi*cos(thi(m))/lambda(r))];
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M2=(D2)*P2*inv(D2);
M=M*M2;
end
if pol==1
D3=[ 1 1; (mun(m+1)/n(m+1))*cos(thi(m+1))
-(mun(m+1)/n(m+1))*cos(thi(m+1))];
%% TE
elseif pol==0
D3=[ cos(thi(m+1)) cos(thi(m+1)); (n(m+1)/mun(m))
-(n(m+1)/mun(m+1))]; %% TM
End…
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A.9 wave
clear all; close all; clc;
lin=5.5*10^-6;
ls=1.5*10^-6;
ni=1;
ns=1;
lambda=11.11*10^-6;
k0 = 2*pi/lambda;
w=15*10^(-6);
zr=pi*w^2/lambda;
n=[ni -2.4 repmat([1.5 -2.4],1,5) ns];
mun=[1 -1 repmat([1 -1],1,5) 1];
l=[0.75*10^-6 repmat([1.5*10^-6],1,9) 0.75*10^-6];
L=repmat([1.5*10^-6],1,10);
N=2^10+1; %point in X
%zr=pi*w^2/lambda;
%L=[2*zr,2*zr]; %length of material
L1=0.001; %Length of X
h = L1/N ; %dx
n1 =[-N/2:1:N/2-1];
% Indices
x1=n1*h;
% y=@(x) exp(-1/w^2.*x.^2);
y1=exp(-1/w^2.*x1.^2);
plot(x1,y1);
%figure(1)
%plot(x,y1)
energy1=h*sum(abs(y1.^2));
% axis([-0.5 0.5 0 2])
a=pi/h;
kx1=[-a:2*a/N:a-2*a/N];
f= fftshift(fft(y1));
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pol=1; %pol=1 TE %pol=0 TM
%% field inside layers and movie do not have to run
for q=1:1:length(kx1)
k0=2*pi/lambda;
thetai=asin(kx1(q)./k0);
thi=asin(ni*sin(thetai)./n);
%ni*sin(v0)=Nm*sin(vm)
[A,B,T(q),R(q),ref(q),t(q)]=layeryeh(n,ni,ns,[10 l …
86
A.10 TE
clear all; close all; clc;
lin=5.5*10^-6;
ls=1.5*10^-6;
ni=1;
ns=1;
lambda=11.11*10^-6;
k0 = 2*pi/lambda;
w=15*10^(-6);
zr=pi*w^2/lambda;
n=[ni -2.4 repmat([1.5 -2.4],1,5) ns];
mun=[1 -1 repmat([1 -1],1,5) 1];
l=[0.75*10^-6 repmat([1.5*10^-6],1,9) 0.75*10^-6];
L=repmat([1.5*10^-6],1,10);
N=2^10+1; %point in X
%zr=pi*w^2/lambda;
%L=[2*zr,2*zr]; %length of material
L1=0.001; %Length of X
h = L1/N ; %dx
n1 =[-N/2:1:N/2-1];
% Indices
x1=n1*h;
% y=@(x) exp(-1/w^2.*x.^2);
y1=exp(-1/w^2.*x1.^2);
plot(x1,y1);
%figure(1)
%plot(x,y1)
energy1=h*sum(abs(y1.^2));
% axis([-0.5 0.5 0 2])
a=pi/h;
kx1=[-a:2*a/N:a-2*a/N];
f= fftshift(fft(y1));
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pol=1; %pol=1 TE %pol=0 TM
%% field inside layers and movie do not have to run
for q=1:1:length(kx1)
k0=2*pi/lambda;
thetai=asin(kx1(q)./k0);
thi=asin(ni*sin(thetai)./n);
%ni*sin(v0)=Nm*sin(vm)
[A,B,T(q),R(q),ref(q),t(q)]=layeryeh(n,ni,ns,[10 l …
88
A.11 ExEz
clear all; close all; clc;
ni=1;
ns=1;
lin=4*10^-6;
ls=4*10^-6;
lambda=11.11*10^-6;
k0 = 2*pi/lambda;
w=lambda;
%beam waist
n=[ni repmat([1.5 -1.499],1,5) ns];
l=[repmat([1.5*10^-6],1,10)];
mun=[1 repmat([1 -1],1,5) 1];
mun0=4*pi*10^(-7);
eps0=8.85*10^(-12);
epss=[1 repmat([2.25 -1.499^2],1,5) 1];
c=3*10^(8);
zr=pi*w^2/lambda;
%L=repmat([1.5*10^-6],1,10);
N=2^10+1; %point in X
%zr=pi*w^2/lambda;
%L=[2*zr,2*zr]; %length of material
L1=0.001; %Length of X
h = L1/N ; %dx
n1 =[-N/2:1:N/2-1];
% Indices
x1=n1*h;
% y=@(x) exp(-1/w^2.*x.^2);
y1=exp(-1/w^2.*x1.^2);
plot(x1,y1);
%figure(1)
%plot(x,y1)
energy1=h*sum(abs(y1.^2));
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% axis([-0.5 0.5 0 2])
a=pi/h;
kx1=[-a:2*a/N:a-2*a/N];
f= fftshift(fft(y1));
pol=1; %pol=1 TE %pol=0 TM
%% field inside layers and movie do not have to run
for q=1:1:length(kx1)
k0=2*pi/lambda…
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