STUDIES OF PHOTONIC CRYSTALS WITH PLANE-WAVE EXPANSION METHOD THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Sarjana in Physics at the Institut Teknologi Bandung by YUDISTIRA VIRGUS 10204031 DEPARTMENT OF PHYSICS FACULTY OF MATHEMATICS AND NATURAL SCIENCE INSTITUT TEKNOLOGI BANDUNG 2008
STUDIES OF PHOTONIC CRYSTALS WITH PLANE-WAVE EXPANSION METHOD THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Sarjana in Physics at the Institut Teknologi Bandung by YUDISTIRA VIRGUS 10204031 Certified and approved by Alexander A. Iskandar. Ph.D NIP: 131875315 Supervisor DEPARTMENT OF PHYSICS FACULTY OF MATHEMATICS AND NATURAL SCIENCE INSTITUT TEKNOLOGI BANDUNG 2008
For my Mom The one who always keeps my spirit high I was born not knowing and have only had a little time to change that here and there. Richard Feynman, Letter to Armando Garcia J Not to know is bad. Not to wish to know is worse. Nigerian Proverb Le savant n étudie pas la nature parce que cela est utile; il l étudie parce qu il y prend plaisir, et il y prend plaisir parce qu elle est belle. Si la nature n était pas belle, elle ne vaudrait pas la peine d être connue, la vie ne vaudrait pas la peine d être vécue. The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and life would not be worth living. Henri Poincaré, Science and Method
ABSTRACT Dear Sir and Madam, will you read my book, it took me years to write, will you take a look? John Lennon and Paul McCartney, Paperback Writer In this Thesis, we numerically investigate the band structure of photonic crystals with plane-wave expansion method based on earlier works. We also compare the result for one-dimensional photonic crystals with the analytical result from transfer matrix method. First, we analyze the optical properties of multilayer periodic structures, one-dimensional photonic crystals, by means of Bloch s theorem and transfer matrix method. We focus on how to calculate the band structure of the systems analytically and show that there are photonic band gaps, i.e., a range of frequencies where no electromagnetic waves can propagate within the crystal. Such structures have been studied in great details because the structure response can be derived analytically, for example using standard transfer matrix method, and thus do not require powerful computational techniques. Furthermore, these results are easily tested since such structures can be easily fabricated with standard methods. iv
Next, we analyze the band structure of photonic crystals using plane-wave expansion method. This numerical method is based on the Fourier expansion of the electromagnetic field and the dielectric function. Generally, it involves intensive computations with thousands of plane waves. We use MATLAB to implement the plane-wave expansion method. First, we try to calculate the band structure of one-dimensional photonic crystal with this method and compare its results with the analytical result. Furthermore, two-dimensional photonic crystal will be studied using the plane-wave expansion method. Finally, we investigate a two-dimensional photonic crystal with a defect using a supercell approach. By introducing a defect to the crystal structure, the utility of photonic crystals is enhanced because the creation of localized frequency states within the photonic band gap. We also investigate the electric-field patterns of the defect modes that appear within the photonic band gap. v
ACKNOWLEDGEMENT It is said of a lonely man that he does not appreciate the life of society. This is like saying he hates hiking because he dislikes walking in thick forest on a dark night. Sebastian Roch Nicolas Chamfort I would like to thank my God for His blessing in my life. He is always there every time I need Him. Never a single second, He leaves me although I often forget Him. He teaches me a lot of things that the world has never told me before. I may never understand His unceasing love for me but I am happy that He loves me this way. There is no doubt that my parents always support me to reach my dreams. There were a lot of sufferings in their past lives, however; they kept giving me the best. They gave me the best school, the best love, the best protection and they just gave me the best parents. No son could ask better. In addition, I would like to thank my sisters, Vinda and Neshia, for their loves and supports for me. I would like to thank my advisor, Alexander Iskandar, for his advices and helps through the time I worked on my thesis. It would be impossible for me to finish my thesis without those precious advices. In addition, I would also thank Prof. Tjia May On for numerous useful questions about my thesis. He could vi
always see any small error in my works. That was really impressed me, since I could not see those errors myself. I would like to thank Erik Winfree for giving me a chance to work with him on SURF program. At that time, I really felt that I was a world class researcher. I really missed that time. I was also thankful to Rizal Hariadi and Nadine Dabby for being my co-mentor for SURF program. They taught me a lot of things about being a good researcher. I would also like to thank the rest of the DNA lab members; Georg, David, Dave, Eliza, Rob, Ton, etc. I would like to give my special thanks for Bernad Yurke, Bernie, for spending his time asking any science question from me. He gave me an unforgettable moment when we almost set the lab on fire. I am grateful to Caltech for supporting me with the SURF program. I am glad to have my dormitory friends who can always cast away my boredom. I love the moments that we play together. I am also happy for having my college friends. They always helped me with the technical problems I had when I was in ITB. Finally, I would like to thank Hendra Kwee, my senior in high school, physics Olympiad team, and college; for his explanation about the supercell method. It was probably impossible for me to finish my thesis on time without his explanation. Actually, there are still a lot of people whom I would like to thank for their support for me. However, it is just impossible to include all of them on these few pages. vii
TABLE OF CONTENTS Abstract.... Acknowledgement... iv vi Table of Contents.... viii Chapter I Introduction 1 1.1 Background 1 1.2 Outline of the Thesis.. 5 Chapter II Basic Electromagnetic Theory... 8 2.1 Maxwell s Equations and Boundary Conditions... 5 2.4.1. Maxwell s Equations 5 2.4.2. Boundary Conditions... 11 2.2 Wave Equations and Monocromatic Plane Waves... 13 2.3 Energy Density and Energy Flux of Electromagnetic Waves 16 2.4 Reflection and Refraction of Plane Waves.... 20 2.4.1. Snell s Law and Fresnel s Formulas 20 2.4.2. Reflection and Transmission of s Wave (TE Wave) 23 2.4.3. Reflection and Transmission of p Wave (TM Wave). 24 2.4.4. Reflectance and Transmittance 25 viii
2.4.5. Total Internal Reflection...... 28 2.4.6. Brewster Angle........ 32 Chapter III Optics of One-Dimensional Photonic Crystals. 33 3.1 Optics of a Single Homogeneous and Isotropic Layer. 34 3.1.1. Electromagnetic Treatment..... 34 3.1.2. Reflectance, Transmittance, and Absorptance... 38 3.1.3. Thick Layers and Spectral Averaging. 39 3.2 Matrix Formulation for Isotropic Layered Media.... 40 3.2.1. 2 2 Matrix Formulation for a Thin Film.. 40 3.2.2. 2 2 Matrix Formulation for Multilayer System... 43 3.2.3. Transmittance and Reflectance...... 46 3.3 Optics of Periodic Layered Media.... 47 3.3.1. Periodic Layered Media..... 47 3.3.2. Bloch Waves and Band Structures.. 51 3.3.3. Reflectance and Transmittance.... 56 Chapter IV Plane-Wave Expansion Method... 65 4.1 Crystal Structure and Reciprocal Lattice.. 66 4.1.1. Periodic Arrays of Atoms... 66 4.1.2. Reciprocal Lattice Vectors.. 71 4.1.3. Brillouin Zones.... 72 4.2 Eigenmodes of Photonic Crystals...... 74 4.2.1. Wave Equations... 74 ix
4.2.2. Electromagnetism as an Eigenvalue Problem.. 77 4.2.3. General Properties of a Hermitian Operator. 80 4.2.4. Electromagnetic Energy and Variational Principle.. 81 4.2.5. Why Not Use the Electric Field?. 83 4.2.6. Bloch s Theorem for Solving Eigenvalue Problems 84 4.2.7. Proof s of Bloch Theorem... 88 4.2.8. Eigenvalue Problems in Two-Dimensional Crystals... 89 4.2.9. Fourier Expansion of Dielectric Functions.. 92 4.2.10. Scaling Law and Time Reversal Symmetry 95 4.3 The Implementation of Plane-Wave Expansion Method...... 97 4.4 One-Dimensional Photonic Crystal... 101 4.4.1. Normal Incidence...... 101 4.4.2. Non-Normal Incidence... 105 4.5 Two-Dimensional Photonic Crystal... 109 4.5.1. In-Plane Propagation... 109 4.5.2. Square Lattice Case... 110 4.5.3. Mode Field Distribution... 113 Chapter V Defect Modes in Two-Dimensional Photonic Crystals... 118 5.1 General Properties... 119 5.2 Shifting Property of Fourier Transform... 120 5.3 Square Lattice Without Defect... 121 5.4 Defect Modes in Square Lattice... 128 x
5.5 Mode Field Distribution of Defect Modes... 133 Chapter VI Conclusion and Future Perspectives... 142 6.1 Conclusion... 142 6.2 Future Perspectives... 143 Appendix Proof of Chebyshev s Identity... 145 Bibliography... 151 Biographical Note... 156 xi