Computation & Design for Nanophotonics by Ardavan Oskooi S.M., Massachusetts Institute of Technology (2008) B.A.Sc., University of Toronto (2004) Submitted to the Department of Materials Science and Engineering in partial fulfillment of the requirements for the degree of Doctor of Science in Materials Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2010 c Massachusetts Institute of Technology 2010. All rights reserved. Author............................................. ................. Department of Materials Science and Engineering May 10, 2010 Certified by......................................... ................. Steven G. Johnson Associate Professor of Mathematics Thesis Supervisor Certified by......................................... ................. Yoel Fink Associate Professor of Materials Science Thesis Supervisor Accepted by......................................... ................ Christine Ortiz Chair, Department Committee on Graduate Theses 2 Computation & Design for Nanophotonics by Ardavan Oskooi Submitted to the Department of Materials Science and Engineering on May 10, 2010, in partial fulfillment of the requirements for the degree of Doctor of Science in Materials Science and Engineering Abstract The versatility of computational design as an alternative to design by nanofabrication has made computers a reliable design tool in nanophotonics. Given that almost any 2d pattern can be fabricated at infrared length scales, there exists a large number of degrees of freedom in nanophotonic device design. However current designs are ad- hoc and could potentially benefit from optimization but there are several outstanding issues regarding PDE-based optimization for electromagnetism that must first be addressed: continuously and accurately deforming geometric objects represented on a discrete uniform grid while avoiding staircasing effects, reducing the computational expense of large simulations while improving accuracy, resolving the breakdown of standard absorbing boundary layers for important problems, finding robust designs that are impervious to small perturbations, and finally distinguishing global from local minima. We address each of these issues in turn by developing novel subpixel smoothing methods that markedly improve the accuracy of simulations, demonstrate the failure of perfectly matched layers (PML) in several important cases and propose a workaround, develop a simple procedure to determine the validity of any PML implementation and incorporate these and other enhancements into a flexible, free software package for electromagnetic simulations based on the finite-difference time- domain (FDTD) method. Next we investigate two classes of design problems in nanophotonics. The first involves finding cladding structures for holey photonic- crystal fibers at low-index contrasts that permit a larger class of materials to be used in the fabrication process. The second is the development of adiabatic tapers for coupling to slow-light modes of photonic-crystal waveguides that are insensitive to manufacturing and operational variability. Thesis Supervisor: Steven G. Johnson Title: Associate Professor of Mathematics Thesis Supervisor: Yoel Fink Title: Associate Professor of Materials Science 3 Education is Not the Filling of a Pail but the Lighting of a Fire. - William Butler Yeats 4 Acknowledgments MIT is a special place. Little did I know during my orientation visit one frosty spring week six years ago how this incredible place would alter the course of my life. MIT is teeming with revolutionary ideas dreamt up by the world’s most brilliant individuals. I met the remarkable Steven G. Johnson during my first week at MIT who had re- cently started as an Assistant Professor. I was fortunate to become the first graduate student in his group and have benefited tremendously from the experience ever since. From the initial months of working together and throughout, I have continued to be awed by the depth and breadth of his scientific repertoire. His insatiable curiosity, meticulousness, tenacity to see things through to the end, ability to make connections among disparate fields and profound commitment to sharing the fruits of knowledge make him an exceptional researcher; one who has set the bar for my own professional career. He is always brimming with new research ideas and is never afraid to tackle the toughest of problems with his quintessential clarity of thought. His “Mathemat- ical Methods in Nanophotonics” remains my favorite class of all time; the lectures are the stuff of legend. Steven cares deeply about his students. He spends an inordi- nate amount of time mentoring each of us and is never too tired to discuss science. Many times when I was stuck on research and feeling discouraged, a trip to Steven’s office made all the difference to my rejuvenation. His mastery in problem solving is unparalleled and I affectionately refer to him as the “Oracle” for his singular ability to solve any problem, no matter how complex. Steven has consistently challenged me to become a better scientist. He has been a wonderful mentor, advisor and friend. Thanks for putting up with such a stubborn student, Steven. There are three individuals who have left an indelible mark on my MIT experience. I am grateful to have gotten to know each of them well during my time here. I met Wenhao Liu during the fateful summer before MIT at the IBM Almaden Research Center in San Jose, CA. Wen’s charisma, wonder of the world around him and passion for technology are one of a kind. Happy memories abound: two years as roommates at the Sidney-Pacific Graduate Residence, trips to NYC, late nights 5 in Chinatown, hiking in NH, exploring Boston’s nightlife, and star-gazing on the MA coast. Wen has been a constant fixture of my life outside of work. I met Alejandro Rodriguez-Wong in my first semester who was then a precocious and enthusiastic undergraduate. We have spent countless hours discussing physics, debating contemporary issues, and playing practical jokes on one another in CMT. Alejandro’s strong work ethic and devotion to research set an excellent example for all. He is a diligent, gregarious and compassionate human being who relishes helping out those around him. His ebullient presence always livened our work environment. We share many unforgettable memories together. I met Monika Schleier-Smith serendipitously on a warm summer evening in my third year while walking home from work. Monika is a supremely talented individual and she dedicates herself to everything she does with utmost zeal. She is a gifted scientist, musician and athlete with an endearing personality. I will always remember with fondness our bike rides, walks and runs around Cambridge and Boston, rowing on the Charles on the 4th of July, evenings sharing a meal, movies at the LSC, trips to the BSO, and the many hours in conversation. She has a special place in my heart. I would also like to thank Professor John Joannopoulos for all his wisdom, support and guidance. JJ combines technical mastery, formidable communication skills, an effective management style with a vision for the future. He is the epitome of a great scientist. My sincere gratitude to Dr. Peter Bermel with whom I spent untold hours administering our group’s computers. Peter’s proficiency and affability made working together enjoyable. Thank you to all the talented members of the JDJ/SGJ/MS group, especially Dr. Aristeidis Karalis, Alexander McCauley and Hila Hashemi. I dedicate this thesis to my parents, Fereydoun and Haydeh, who had the courage to leave our homeland during a very turbulent time in its history. They managed to raise two young children on their own in a foreign land and provided us with every- thing we needed. Their unyielding love and encouragement has been an everlasting source of inspiration. I am eternally grateful for all the sacrifices they have made. I look back and smile. The best is yet to come. 6 Contents 1 Introduction 25 2 Sub-pixel smoothing for dielectric media 31 2.1 Summary ................................. 31 2.2 Overview.................................. 31 2.3 Designing subpixel smoothing algorithms with perturbationtheory. 33 2.4 Analysisofsmoothingperturbation . .. 36 2.5 Stable FDTD field-update implementation . .. 44 2.6 Numerical performance of methods for isotropic media . ...... 45 2.7 Numerical performance of methods for anisotropic media ....... 49 2.8 Field singularities at sharp corners . ... 52 2.9 Conclusion................................. 53 3 The failure of perfectly matched layers 55 3.1 Summary ................................. 55 3.2 Overview.................................. 56 3.2.1 VariousPMLFormulations. 57 3.2.2 PMLsinPhotonicCrystals . 58 3.3 PMLsversusAdiabaticabsorbers . 60 3.4 BriefreviewofPML ........................... 61 3.4.1 Mathematicalformulation . 61 3.4.2 Absorptionprofile. .. .. 63 3.5 Adiabatic theorems in electromagnetism . ... 64 7 3.6 FailureofPML .............................. 65 3.6.1 Homogeneous & inhomogeneous media . 65 3.6.2 Backward-wavestructures . 68 3.6.3 PMLs&adiabaticabsorbers. 69 3.7 Smoothness&Reflection . 72 3.7.1 Numericalresults . .. .. 73 3.7.2 Analysis .............................. 78 3.7.3 Adiabatic theorems in discrete systems . 82 3.8 TowardsBetterAbsorbers . 82 3.8.1 Smoothness & C functions ................... 87 ∞ 3.8.2 Balancing round-trip & transition reflections . .... 88 3.9 Conclusion................................. 90 4 A simple validation scheme for perfectly matched layers 93 4.1 Summary ................................