Transfer Matrix Approach to Propagation of Angular

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

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. Joseph W. Haus, Ph.D. Advisor Committee Chairman Committee Member Professor Professor Department of Electrical Engineering Department of Electro-Optics And Electro-Optics Program Program

______Andrew Sarangan, Ph.D. Committee Member Professor Department of Electro-Optics Program

______John G. Weber, Ph.D. Tony E. Saliba, Ph.D. Associate Dean Dean, School of Engineering School of Engineering & Wilke Distinguished Professor

○c Copyright by

Han Li

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.

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.

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

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

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

'' Figure 4.6. Same as Figure 4.1, but with coefficients Aiii,,,BAB i 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

6 layers with n122.0, n 1.5 , each layer with a thickness dmi 1.5 10 . 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 n12 2.0, n 1.5 , each layer with

6 a thickness dmi 1.5 10 . 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 n12 2.0, n 1.5 , each layer with a

6 thickness dmi 1.5 10 ...... 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 n12 2.0, n 1.5 , each layer with a

6 thickness dmi 1.5 10 ...... 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

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 pp 2.25,  1and 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

2 Figure 5.9. |Hy| as it propagates inside the metamaterial structure of the same 2 parameters as in Figure 5.8(a) using (a) TMM and (b) FEM respectively. |Ex| the magnitude squared of the x-component of the electric field using (c) TMM and (d) FEM respectively ...... 61

2 Figure 5.10. (a) |Hy| during propagation inside the metamaterial structure with

 pp2.25, 1,  n 1.44 ,1n   , and for beam waist w0   , and for TM

2 2 incidence, (b) |Ex| , (c) |Ez| . 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

 E0,t Transmitted complex electric field amplitude

 E0,r Reflected complex electric field amplitude

Esz Complex electric field in substrate (z-component)

Esx Complex electric field in substrate (x-component)

E()kx Fourier transform of E()x

E()x Inverse Fourier transform of E()kx  k Wavevector

kx x-component of the wavevector

ky y-component of the wavevector

L Total propagation distance

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.

1.2 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

(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.

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.

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.

2.2 One and two-dimensional Fourier transforms of Gaussian functions

A one-dimensional (1-D) scalar optical Gaussian envelope profile is defined as

x2  w2 Ee ()xe , (2.2-1)

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

1  Ex() Eke ( )jkx x dk (2.2-2) eexx 2  which implies that the phasor

1  Ex() Eke ( ) jkx x e jkz z dk (2.2-3) p  ex x 2  using the concept of the Fourier Transform where

 Ek () Exedx ()jkx x . (2.2-4) ex e 

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

Ekex(). Equations (2.2-2) and (2.2-4) comprise the 1-D Fourier transform.

The two dimensional (2-D) Fourier transform is given by

 Ekk (, ) Exye (,)jkxky()xy dxdy, (2.2-5) exy e   and the inverse Fourier transform is

 1 jkxky() Exy(, ) Ekke ( , ) xy dkdk, (2.2-6) eexyxy2  4 

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 fn(), nN 0,... 1 , where  is the sampling

interval in x and f p ()n can be written as [9]:

 f p ()n  f (nrN ), (2.2-7) r

the DFT of f p ()n is defined as

N1 jmnK FmKpp() f () n e , (2.2-8) n0

2 where K  . The inverse DFT is defined as N

N 1  jmnK fnpp() FmKe ( ) , (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 slowly- varying 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

 2  LL/cos  (1 ). Upon defining k as the propagation vector of the 2 0

1  waves,   sin (kkx /0 ) , where kk00 . 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. Hence kx should be a

 N  (2)N  array form  to with the space between two samples being X X

2 where X is the size of the spatial domain of the field profile, N being the X

2 samples on X , and with k  . 0 

L /cos

 L z 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

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.

0.9

0.8

0.7

0.6

0.5 amplitude 0.4

0.3

0.2

0.1

0 -10 -8 -6 -4 -2 0 2 4 6 8 10 x

Figure 2.3. Gaussian profile before (blue) and after (red) propagation by a distance equal to Rayleigh range.

0.9

0.8

0.7

0.6

0.5 amplitude 0.4

0.3

0.2

0.1

0 -10 -8 -6 -4 -2 0 2 4 6 8 10 x

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.

Figure 2.5. Initial 2-D Gaussian profile; x-projection.

Figure 2.6. Initial 2-D Gaussian profile, contour view.

Figure 2.7. Final 2-D Gaussian profile, x- projection.

Figure 2.8. Final 2-D Gaussian profile, contour view.

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.

Figure 2.9. Initial 2-D Gaussian profile; x- projection.

Figure 2.10. Final 2-D Gaussian profile at end of 1st medium; x-projection.

Figure 2.11. Representative 2-D Gaussian profile within 2nd medium after a short distance of propagation; x-projection.

Figure 2.12. Representative 2-D Gaussian profile within 2nd medium after a larger distance of propagation; x-projection.

Figure 2.13. Representative 2-D Gaussian profile a short distance in 3rd medium; x-projection.

Figure 2.14. Representative 2-D Gaussian profile in 3rd medium after longer distance of propagation; x-projection.

The UBPM approach also can be used to explore the performance of a coupled waveguide [11]. As is well known, evanescent waves can shift energy from one waveguide to another in a coupled waveguide structure.

Here, an example for the Gaussian beam inside the coupled waveguide is shown. The refractive index profile is plotted in Figure 2.15 which is a cross section of a two slab waveguide. Figure 2.16 shows periodic beam power transfer between the two waveguides as a consequence of evanescent wave coupling. This simulation is also very useful to help test the waveguide properties. The MATLAB code can be found in Appendix A under project_BPM.m.

Figure 2.15. The refractive index profile of two-slab waveguide.

Figure 2.16. Gaussian propagation showing coupling in the two-slab waveguide.

BPM is a quick and easy method of solving for fields in integrated optical devices. It is typically used only in solving for intensity and modes within shaped (bent, tapered, terminated) waveguide structures, as opposed to scattering problems [12]. However, in other practical applications such as multilayer structures, there is reflection at surfaces and interfaces of the structure. This means the UBPM can give rise to significant errors. That is why a revised method called bidirectional Beam Propagation Method (BBPM) needs to be introduced.

2.5 Conclusion

The basic principle of UBPM has been discussed, with some illustrative examples. A profile in the spatial domain can be described as a superposition

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. Gaussian beam propagation in 1-D and 2-D are shown as examples. 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.

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.

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:

    EEjtkRRe{ exp ( ) }, (3.2-1) itr,, eitr ,,  itr,,    where E ei,, t r represent the complex envelopes, k itr,, 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:    ()kRiz000 () kR rz () kR tz. (3.2-2)

Referring to Figure 3.1, where n1 and n2 represent the refractive indices of the two media,

  kk nk,  n. (3.2-3) ircc12 t

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:

nnn121sinitr sin sin . (3.2-4)

n 1 n2   kr k t

 r  t

 i  k 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 kkkirt,, have been replaced with kkk11,, 2 for

 convenience in calculations in multilayers. 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.

 '  ,   11,  H 1 22 Region 2 Region 1 ' E 1  E2

  2 H 2

' H 2 1

 ' E1 E 2  H 1

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:

    jk11 R  jk'  R   (')Ee11 E e  z  0 Ep        , (3.2-5) jk22 R  jk'  R z  0 (')Ee22 E e  

 while the magnetic field phasor H p can be obtained from Maxwell’s equations:

 j HEpp. (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.

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,

EE '' EE (3.3-1) 11TE TE 2 TE 2 TE

Similarly, all H x components are also continuous. By using boundary conditions:

12 (EE11TE TE ')cos 1 ( EE 2 TE 2 TE ')cos 2 , (3.3-2) 12 using equations (3.3-1) and (3.3-2),

EE12TE  TE DDTE(1) TE (2) , (3.3-3) EE12TE''  TE where

11  Di() , i 1,2 . (3.3-4) TE iicos cos ii ii

DiTE ()is called “dynamical matrix” of the TE wave. 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]

nn 12cos  cos E ' 12 r 1TE 12; TE E nn 1TE 12cos  cos 12 12 (3.3-5) n 2cos1  E  1 t 2TE 1 , TE nn E1TE 12 cos12 cos 12

' 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

0.6

0.5

0.4

0.3 transmission coefficients 0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle (radian)of the TE wave

Figure 3.3. The transmission coefficient for TE case as a function of the incident angle.

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7 refection coefficients refection

-0.8

-0.9

-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle (radian)of the TE wave

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ii cos  Di() , i 1,2, (3.3-6) TM ii  ii  When H 2 '0 , the reflection (r) and transmission (t) coefficients for a single optical interface for the TM case become:

nn 12cos  cos E ' 21 1TM r 12 (3.3-7) TM nn E1TM 12 cos21 cos 12

E 2cos 21TM t (3.3-8) TM nn E1TM 21 （）/coscos12  21

' 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

0.35

0.3

0.25

0.2

0.15 transmission coefficients

0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle (radian)of the TM wave

Figure 3.5. The transmission coefficient for TM case as a function of the incident angle.

0.8

0.6

0.4

0.2

0 refection coefficients

-0.2

-0.4

-0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle (radian)of the TM wave

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:

E21E ' rt21();() 21 (3.4-1) E22''E

From Equations (3.3-1) and (3.3-2), it follows that [10]:

rr21  12 , (3.4-2)

tt12 21 rr 12 21 1, (3.4-3) with the reflectance and transmission defined as, respectively:

2 RrTij ij;1 ij R ij . (3.4-4)

From Equations (3.4-2), (3.4-3) and (3.4-4), it is easily seen that:

R12 RT 21; 12 T 21 . (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.

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.

n n 0 1 ns

' A 1 A A 2 3

' B 1 B 2 B 3

z=0 z=d

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:

 jk00zz z jk z Ae11 Be, z 0   jk11zz z jk z EAeBezdpy   22, 0 (4.2-1)

 ' jksz () z d Ae3 , d z

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:

 k0z  jk00zz z jk z  (),Ae11 Be z 0   k1z  jk11zz z jk z HAeBezdpx   (22 ), 0 (4.2-2)   ksz ' jksz () z d  (Ae3 ), d z 

Boundary conditions at the interfaces state that the E y and H x are continuous at z  0 and zd , which yields:

AB11 AB 2 2,  kAB01zz()(), 1 kAB 12  2 (4.2-3)   jk11zz d jk d ' Ae223 Be A , kAe(), jk11zz d Be jk d kA'  12zsz 2 3 so that using Equation (4.2-3),

 nn 01cos  cos  01 01 r01TE  ,  nn01 cos01 cos  01  (4.2-4)  nn1 s cos1  coss  1 s r1sTE  , nn1 s  cos1  coss  1 s

 n 2cos0   0 0 t01TE  ,  nn01 cos01 cos  01  (4.2-5)  n1 2cos1  1 t1sTE  , nn1 s cos1 coss  1 s

2 2 1 n0 where kn000z  cos , kn111z  cos , 10 sin ( sin ) , nnn01s ,   n1

and 01s [11].

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]

2 j Brre1011 s rTE& TM  2 j , (4.2-6) Arre10111 s

'  j Atte3011s tTE& TM  2 j , (4.2-7) Arre1 01 1s where

2d  kd ncos . (4.2-8) 111z 

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

 nn 01cos  cos  10 01 r01TM  ,  nn01 cos10 cos  01  (4.1-9)  nn1 s coss  cos1  1 s r1sTM  , nn1 s  coss  cos1  1 s

 n 2cos0   0 0 t01TM  ,  nn01 cos10 cos  01  (4.1-10)  n1 2cos1  1 t1sTM  . nn1 s coss cos1  1 s

4 6 As an example, let dm110 ,  10 10 m and nn0 s 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).

0.9

0.8

0.7

0.6

0.5

0.4

reflection coefficients 0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle(radian) of the TE wave

Figure 4.2. Absolute value of reflection coefficient for TE case as a function of the incident angle for single layer structure.

0.9

0.8

0.7

0.6

0.5

0.4

transmission coefficients 0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle(radian) of the TE wave

Figure 4.3. Absolute value of transmission coefficient for TE case as a function of the incident angle for single layer structure.

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).

0.9

0.8

0.7

0.6

0.5

0.4

reflection coefficients 0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle(radian) of the TM wave

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

0.8

0.6

0.4 transmission coefficients

0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 incident angle(radian) of the TM wave

Figure 4.5. Absolute value of transmission coefficient for the TM case as a function of the incident angle for single layer structure.

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:

 jkx x EEzepp () , (4.3-1)

where kx is the wavenumber along the x direction.

n0 n 1 n s

' ' A0 A 1 A1 A s

B ' B0 B '1 1 B s

z=0 z=d

'' Figure 4.6. Same as Figure 4.1, but with coefficients Aiii,,,BAB i inserted.

From Equation (4.3-1), and referring to Figure 4.6,

 jkzz jkzz Ep () z A () z B () z Ae  Be (4.3-2)

' where Ezp () is defined as the phasor part of standing wave, Aii(),zAz () are

' the forward traveling amplitudes, and Bii(),zBz ()are the backward traveling amplitudes. From Equation (3.3-3), obviously, A(),()zBz are not continuous at each interface and the relation can be written as:

A0 1 A''11 A DD01 D 01 . (4.3-3) B011BB'' 

Over the distance z=0 to z=d,

j1 A'11e 0 A P1 (4.3-4)  j1  B '110 e B

where P is the propagation matrix with   kdz . At the interface z=d,

'' A1 1 As A s DD11ss D  (4.3-5) B ''  1 Bs B s

From Chapter III, the dynamical matrices can be summarized as

11  for TE wave iicos cos ii ii  Di   , (4.3-7) cosii cos    for TM wave ii  ii

wherei  1,2,3 and i is incident angel of the plane wave. According to

'' Equation (4.1-3), (4.1-4) and (4.1-5), the amplitudes A00,B and A s , B s can be related by

'' A0 11AAs s DDPDD0111s  M  (4.3-8) B ''  0 B s B s where

MM11 12 11 M DDPDD12223 (4.3-9) MM21 22 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 dl may vary with the position of the z axis, viz.,

dzz110  ;

dzz221; ...... (4.4-1)

dzznnn1 ......

The electric field distribution inside the layers is given by

jk00zz() z z jk 00 () z  z Ae00 Be, z z 0  jklz() z z l jk lz () z  z l EAeBep  ll, zzz ll1  (4.4-2)

 ''jkzzsz() n jkzz sz ()  n Aess Be, z n z

 with kn cos , which is the wavenumber along the z direction. A is lz lc l 0

the initial wave amplitude incident on the first interface from layer “0”. As is the transmitted wave amplitude in the substrate layer.

Figure 4.7. Periodic structure composed of 2 materials with refractive index n and n 1 2 .

Upon extending the concept in the previous Section, it is readily verified that

N ' A0 1  1   A s   D0 Dl Pl DDls  , (4.4-3) B    '  0  l1   B s  so that the transfer matrix can be written as

N  M11 M12  1  1  M     D0  DPDll l  Ds . (4.4-4)  M 21 M 22   l1 

For multilayer structures, the calculation of M ij is formidable.

Therefore, it is logical to use software such as MATLAB for the calculations.

If light is incident from the left side and propagating through a multilayer, it has been shown that the reflection and transmission coefficients are given by

 B0 r  , A0   (4.4-5)  A' t  s .  A0

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  , M11   (4.4-6)  1 t  .  M11

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

6 thickness dmi 1.5 10 . Assume that the incident angle   0 , the

semi-infinite incident medium has refractive index n0 1 , the semi-infinite

substrate medium has refractive indexns  1, and that there are 7 periods, with

each period comprising two layers with n12 2.0, n 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

generating the figure can be found in Appendix A under the file, multilayerstructureplanewave.m and layeryehTE.m.

1.4

1.2

0.8

0.6

0.4 absolute value of electricamplitude

0.2

0 0 0.5 1 1.5 2 2.5 propagation distance(m) -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 n12 2.0, n 1.5 , each layer with a thickness

6 dmi 1.5 10 . 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

6 thickness dmi 1.5 10 . Assume that the incident angle   0 , the

semi-infinite incident medium has refractive index n0 1 , the semi-infinite

substrate medium has refractive indexns  1, and that there are 7 periods with

each period including two layers with n12 2.0, n 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

1.2

0.8

0.6

0.4 absolute value of electric amplitude

0.2

0 0 0.5 1 1.5 2 2.5 propagation distance(m) -5 x 10

Figure 4.9. Absolute value of electric field amplitude for the 7 period

structure, with each period including two layers with n12 2.0, n 1.5 ,

6 each layer with a thickness dmi 1.5 10 . 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

is expected to be no reflection on any optical boundary. In Figure 4.10, each

6 period is composed of 2 layers, each layer with a thickness dmi 1.5 10 .

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 n12 1, n 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 0 0.5 1 1.5 2 2.5 propagation distance(m) -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 n12 2.0, n 1.5 , each

6 layer with a thickness dmi 1.5 10 .

0.95

0.9

0.85

0.8 electric amplitude

0.75

0.7

0.65 0 0.5 1 1.5 2 2.5 propagation distance(m) -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 n12 2.0, n 1.5 , each

6 layer with a thickness dmi 1.5 10 .

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

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.

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.

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  pp 2.25,  1 and NIM layers of n  2.25 ,

n 1. Each layer has a thickness of ddpn  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 15m. The incident plane wave has a

wavelength of  11.11m 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.

(a)

(b)

incidence angles i  0, / 6 simulated using the TMM technique and FEM, respectively.

5.3 Comparison of TMM and FEM for TM plane wave incidence

The dynamical matrix derived in Equation (3.3-6) cannot be used in the

TM case because it pertains to the electric field. Here the H field is assumed to be the initial field to propagate in the multilayer structure [16].

Figure 5.2. Layered structure composed of materials with refractive indices with thickness .

Consider the TM case (p-wave) shown in Figure 5.2. Unlike the previous case, the magnetic field is tangential to the interface. The dynamical and the propagation matrices for the magnetic field H will be derived instead of the electric field E. The magnetic field in layer can be written as:

  '    '   H eHikl . r  eeikl . r it  Aeikl . r  Beeikl . r it   ll  l l  z  0  Hy   (5.3-1)    '    '  H  eikl1. r  H  eikl1. r eit  Ae' ikl1. r  Be' ikl1. r eit z  0  l1 l1   l1 l1 

Imposing the continuity of Ex and Hy ,:

  HHHlll11  H l (5.3-2 a,b)   EEElx lx l1, x  E l 1, x

From Maxwell’s equations,:

jjHHyy  E  Hx ˆˆ   zˆ  ExEzx  z ˆ (5.3-3)   zx

Upon using Equations (5.3-3), (5.3-2b) and setting kl,l1 z  k l ,l1 cos l,l1 ,

and k'l,l1 z  k' l,l1 cos l,l1 , it follows that

    HHHll l11  H l   (5.3-4 a,b) cosHH  cos  H   H   lllll1111 ll  l  

Equations (5.3-4) can be written in matrix form as:

HH   11ll 1 1 1   cos cos  cos   cos    llll l1111 l l l   HHll 1 

(5.3-5)

Therefore the dynamical matrix for TM case can be defined as:

11  D  , l 1,2,... lTM llcos  cos . (5.3-6) ll ll

Since it is more conventional to monitor electric fields, Maxwell’s equations can now be used to obtain

 E  sin H , zy c 0 (5.3-7)  EHxy cos , 0c 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.

(a)

(b)

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.

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 Ekmxm()represents the complex amplitude of a plane wave (m) with a transverse propagation constant kxm , which corresponds to an angle of propagation

 k  2 1 xm  m  sin  ,k0  . (5.4-1)  k0  

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

 jkxm x Emmxmx()xEkek   ( )  (5.4-2) m which represents the beam profile. Each plane wave of amplitude

Ekmxm() E mm () 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.

Spatial domain Spatial k frequency x domain

k z 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  pp 2.25,  1,  n 1.44 ,

n 1, and for beam waist w0   . The magnitude squared of the electric field

as it propagates inside the metamaterial structure using TMM and FEM are

shown.

(a)

(b)

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.

(a) (b)

(a)

(b) (c)

Figure 5.7. Initial and transmitted Gaussian profile (using TMM and FEM) after propagation in the structure defined in Section 3.2 but with

 p  2.25, p  1,  n 1.44 , n  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.

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

 pp2.25, 1 and NIM layers of  n   1.44 ,1n   . 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  pp2.25, 1 , and n   1.44 ,1n   , 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.

(a)

(b)

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 layers of  n 1.44 , n 1 . The MATLAB code can be found in Appendix A under wave.m and layeryehH.m.

(a) (b)

2 Figure 5.9. |Hy| as it propagates inside the metamaterial structure of the same parameters as in Figure 5.8(a) using (a) TMM and (b) FEM

2 respectively. |Ex| the magnitude squared of the x-component of the electric field using (c) TMM and (d) FEM respectively.

(a)

(b) (c)

2 Figure 5.10. (a) |Hy| during propagation inside the metamaterial structure

with p  2.25,  p 1,  n 1.44 , n 1, and for beam waist w0   , and

2 2 for TM incidence, (b) |Ex| , (c) |Ez| . 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.

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.

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.

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

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.

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.

[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.

[14] P. Yeh, “Optical waves in layered Media,” Sec 3.1, 1998.

[15] P. Yeh, “Optical waves in layered Media,” Sec 4.1, 1998.

[16] P. P. Banerjee, H. Li, R. Aylo and G. Nehmetallah, “Transfer matrix approach to propagation of angular plane wave spectra through metamaterial multilayer structures,” Proc. SPIE 8093, 80930-1-5 (2011).

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);

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

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]'; % Indices x1=n*h; % 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;

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;…

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)'; % Indices x1 = n*h; % 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');

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]…

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;

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))).

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)

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

end figure plot(x,real(E).^2 ) xlabel('propagation distance(m)'); ylabel('electric amplitude'); grid on;……………………………………….

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;

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)) …………………….

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;

while check==0

if x(s)

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…

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))];

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…

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));

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 …

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));

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 …

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));

% 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…