TRANSFER MATRIX APPROACH TO PROPAGATION OF ANGULAR PLANE WAVE SPECTRA ...

106 Pages·2011·1.72 MB·English

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. 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

ii
○c Copyright by
Han Li
All rights reserved
2011

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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.

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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.

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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
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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
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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
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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
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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
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