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Study of Slow Light Surface Plasmons Study of Slow Light Surface Plasmons by Abraham Vázquez Guardado A Dissertation Submitted to the Program in Optics, Optics Department in Partial Fullfillment of the Requierements for the Degree of Master of Science in Optics at the National Institute for Astrophysics, Optics and Electronics July 2012 Santa María Tonantzintla, Puebla Advisors: Prof. José Javier Sánchez Mondragón1 Prof. Miguel Torres Cisneros2 1 INAOE 2 University of Guanajuato © INAOE 2012 All rights reserved The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole o in part. Study of Slow Light Surface Plasmons Estudio de Plasmones Superficiales Lentos Ing. Abraham Vázquez Guardado INAOE Coordinación de Óptica Tesis de Maestría en Ciencias Especialidad en Óptica Asesores: Dr. José Javier Sánchez Mondragón Dr. Miguel Torres Cisneros Sta. María Tonantzintla, Pue. México Julio 2012 Summary INAOE 2012 Summary In this thesis we study and analyze the Slow Light (SL) property of Surface Plasmon– Polaritions (SPPs) for three cases, for lossless metals, for lossy metals, and for this latter but considering an active dielectric. In the first case, we study the qualitative property of SL in SPPs in a metal/dielectric and dielectric/metal/dielectric waveguide. Despite it is an unrealistic approach, the study is suitable to understand the general problem. Then we proceed with the following case when metal losses are considered in both waveguides. It is here where we observe how the SL property of SPP is affected. In addition, propagation losses in the SL regime are considerably high such that even when a moderate high group index can be achieved the propagation length never surpass one micron. It is why the active medium is opted in order to reduce abortion losses in the metal. Finally, in the third case we study the effects an active dielectric medium has in the SL performance of SPPs in the metal/dielectric waveguide. We study two approaches, when the gain response is constant and when it is a gain distribution function. In the first one we can see that the SL property of SPP does not improve but only the propagation length. It means that the natural SL regime of SPPs does not change. On the second method, however, we observe that, due to the finiteness of the gain response, a new SL region is induced in the SPP dispersion curve even far from the natural SL region. In general, this theoretical study is with the aim to understand the property of SL in SPPs. In this work we introduce a new analysis for the dielectric gain medium that predicts that, if the gain response is considerably narrow, the SPPs can experience an induced SL behavior. This approach was not considered before in the study of amplified SL SPPs. Abraham Vázquez–Guardado i INAOE 2012 Summary Resumen En este trabajo de tesis estudiamos y analizamos la propiedad de Luz Lenta (LL) en Plasmones–Polaritones Superficiales (PPS). Tres casos de estudio se consideran: para metales sin pérdidas, para metales con pérdidas y para estos últimos pero considerando un dieléctrico activo. En el primer caso, estudiamos cualitativamente la propiedad de LL en PPS en las guías de ondas, metal/dieléctrico y dieléctrico/metal/dieléctrico. A pesar que éste es un enfoque no realista, el estudio es adecuado para entender el problema. Posteriormente, continuamos con el caso de un metales con pérdidas en ambas guías de ondas. Es aquí donde observamos cómo la propiedad de LL en PPS y la distancia de propagación se ven afectadas. Incluso cuando un índice de grupo moderadamente alto se puede conseguir la distancia de propagación nunca rebasa una micra. Es por ello que un medio activo es considerado para reducir las pérdidas por absorción en el metal. Finalmente, en el tercer case estudiamos los efectos que tiene un medio activo en el desempeño de PPS lentos en la guía de ondas metal/dieléctrico. Estudiamos dos enfoques, modelo de ganancia constante y modelo de ganancia finita. En el primer caso podemos observar que la propiedad de LL en PPS no mejora, sólo lo hace la distancia de propagación. Esto significa que el régimen natural de LL del PPS no se ve afectado. Sin embargo, en el segundo caso observamos que una nueva región de LL se induce en la curva de dispersión del PPS. Esta región puede estar incluso lejos de la región natural de LL. En general, este estudio teórico es con el objetivo de entender la propiedad de LL de los PPS. En este trabajo introducimos un análisis de ganancia en el dieléctrico nuevo que predice que si la respuesta de ganancia es lo suficientemente angosta, el PPS puede experimentar un comportamiento lento inducido, algo que no se había tocado antes. ii Abraham Vázquez–Guardado Acknowledgements INAOE 2012 Agradecimentos En el marco de apoyo y cooperación para la realización de este trabajo participaron distintas personas e instituciones. Quiero agradecer a mis padres y hermanos por haber sido un apoyo incondicional mientras realizaba mis estudios de maestría. Al Dr. José Javier Sánchez Mondragón, por su apoyo académico en el transcurso de mis estudios de maestría y trabajos adyacentes que se generaron. Al Instituto Nacional de Astrofísica Óptica y Electrónica, por ofrecer el posgrado de Maestría en Ciencias con Especialidad en Óptica, por medio del Departamento de Óptica, y por formarme y guiarme en esta travesía. De igual forma, agradezco al Consejo Nacional de Ciencia y Tecnología por varios motivos. Primero, por ser el órgano promotor de la ciencia y tecnología en el país a través de centros de investigación como el INAOE. Segundo, por brindarme beca de maestría. Finalmente, por brindarme la oportunidad de realizar una estancia de investigación en el extranjero, para la cual me proporcionó una beca mixta. Además, quiero agradecer al Prof. Robert W. Boyd, líder del Canada Excellence Research Chair in Quantum Nonlinear Optics, por brindarme la oportunidad de trabajar en su prestigiado grupo de investigación en la Universidad de Ottawa. De lo cual se forjo el tema principal de mi trabajo de tesis. Abraham Vázquez Guardado Sta. María Tonantzintla, Puebla Julio 2012 Abraham Vázquez–Guardado iii INAOE 2012 Acknowledgements iv Abraham Vázquez–Guardado Content INAOE 2012 Index Preface. 1! Chapter 1 Introduction. 3! Summary 3! Introduction 3! 1.1–Maxwell’s equations. 4! 1.2–The wave equations. 5! 1.3–Constitutive relations. 6! 1.4–Spectral representation of time–dependent fields. 8! 1.5–Time–harmonic fields. 9! 1.6–The space notation. 10! 1.7–Electric and magnetic field components. 10! 1.8–Boundary conditions. 12! 1.9–Complex permittivity function. 14! 1.10–Conservation of energy. 15! 1.11–The permittivity function of the free electron gas. 17! 1.12–Free electron gas dispersion and volume plasmons. 19! 1.13–Real metals and interband transitions. 21! 1.14–Group and phase velocities. 23! 1.15–Slow, fast, backward and stop light. 26! 1.16–Slow light mechanisms. 28! 1.16.1–The Kamers–Kronig relations and the susceptibility. 28! 1.16.2–Resonance in materials. 30! References. 34! Chapter 2 Slow Light Surface Plasmons. 37! Summary 37! Introduction 37! Abraham Vázquez–Guardado v INAOE 2012 Content 2.1–The wave equation for inhomogeneous media. 38! 2.2–Two–layer planar interface. 41! 2.2.1–Analysis for a generic Drude–like metal. 43! 2.3–Three–layer planar interface. 46! 2.4–Metal losses in SPPs performance. 50! References. 53! Chapter 3 Slow Surface Plasmons in amplifying media. 55! Summary 55! Introduction 55! 3.1–Dielectric with active media. 57! 3.2–Condition for bound modes. 59! 3.3–Constant gain model and condition for SPP lossless propagation. 62! 3.4–Finite gain model. 66! 3.4.1–Gaussian gain response model. 68! References. 74! Chapter 4 Results, discussions and further work. 77! 4.1–Results and discussions 77! 4.2–Gain requirements 81! 4.3–Future work 82! References 83! Appendix. 85! A.1–Equations of Chapter 1. 85! A1.1–Macroscopic Maxwell’s equations and constitutive equations. 85! A.1.2–The wave equations. 85! A.1.3–Electric and magnetic field components. 86! A.1.4–Boundary conditions. 88! A.1.5–Complex permittivity functions. 90! A.1.6–The permittivity function of the free electron gas. 91! A.1.7–The permittivity function of a metal with interband transitions. 92! vi Abraham Vázquez–Guardado Content INAOE 2012 A.1.8–Dispersion of volume Plasmon. 93! A.2–Equations of Chapter 2. 94! A.2.1–TE wave equation for the Two–Layer waveguide. 94! A.2.2–TM wave equation for the two–layer waveguide. 96! A.2.3–TE non–existence condition of SPPs. 97! A.2.4–SPP dispersion relation. 98! A.2.5–Dispersion relation in a three–layer system. 100! A.3–Equations of Chapter 3. 104! A.3.1–The constant gain model. 104! A.3.2–Condition for bound solutions. 105! A.3.3–Condition for lossless SPP propagation. 109! A.3.4–Gaussian model for the active medium. 112! A.3.4.1–Hilbert transform of the Gaussian. 113! A.3.4.2–Calculation of the group index 116! List of figures and tables 121! Abraham Vázquez–Guardado vii INAOE 2012 Content viii Abraham Vázquez–Guardado Preface INAOE 2012 Preface. In this master thesis we will study Surface Plasmon–Polaritons [SPPs] and its Slow Light [SL] property, the group index, in planar waveguides. First, we review the passive planar waveguide without considering losses in the metal. This represents an ideal case. Then the lossy case is studied. Both are carried out for the two and three–layer systems.
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