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Mathematical Work on the Foundation of Jones-Mueller Formalism and Its Application to Nano Optics

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Mathematical Work on the Foundation of Jones-Mueller Formalism and Its Application to Nano Optics Mathematical work on the foundation of Jones-Mueller formalism and its application to nano optics Ertan Kuntman ADVERTIMENT. La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del servei TDX (www.tdx.cat) i a través del Dipòsit Digital de la UB (diposit.ub.edu) ha estat autoritzada pels titulars dels drets de propietat intelꞏlectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei TDX ni al Dipòsit Digital de la UB. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a TDX o al Dipòsit Digital de la UB (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA. La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del servicio TDR (www.tdx.cat) y a través del Repositorio Digital de la UB (diposit.ub.edu) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio TDR o al Repositorio Digital de la UB. No se autoriza la presentación de su contenido en una ventana o marco ajeno a TDR o al Repositorio Digital de la UB (framing). Esta reserva de derechos afecta tanto al resumen de presentación de la tesis como a sus contenidos. En la utilización o cita de partes de la tesis es obligado indicar el nombre de la persona autora. WARNING. On having consulted this thesis you’re accepting the following use conditions: Spreading this thesis by the TDX (www.tdx.cat) service and by the UB Digital Repository (diposit.ub.edu) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized nor its spreading and availability from a site foreign to the TDX service or to the UB Digital Repository. Introducing its content in a window or frame foreign to the TDX service or to the UB Digital Repository is not authorized (framing). Those rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it’s obliged to indicate the name of the author. Mathematical work on the foundations of Jones-Mueller formalism and its application to nano optics Ertan Kuntman Departament de Física Aplicada Universitat de Barcelona Supervisors: Dr. Oriol Arteaga Barriel and Dr. Adolf Canillas Biosca This dissertation is submitted for the degree of Doctor of Philosophy in Physics October 2019 This thesis has been supported by an APIF-Universitat de Barcelona (UB) grant. The work has been partially funded by projects from Ministerio de Ciencia, Innovación del Min- isterio de Economía, Industria y Competitividad (MINECO) (CTQ2013-47401-C2-1-P, CTQ2017-87864-C2-1-P, EUIN2017- 88598) and from European Commission (PIIF-GA-2012-330513). Table of contents Acknowledgment1 Resumen en Español3 Summary5 Preface7 1 Background9 2 Introduction 15 2.1 Representations of deterministic optical media states and their properties . 16 2.2 Coherent linear combination of states . 18 2.3 Coherent combinations of states in a Young’s double slit experiment . 19 2.4 Quaternion formulation of optical media states . 20 2.5 Outer product of vector states and the density matrix . 21 2.6 Applications of the formalism: Interacting optical systems at the nano scale 22 2.7 Depolarization effects and decomposition of depolarizing Mueller matrices 26 3 List of Publications 29 3.1 Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions . 31 3.2 Vector and matrix states for Mueller matrices of nondepolarizing optical media 39 3.3 Formalism of optical coherence and polarization based on material media states 46 3.4 Mueller matrix polarimetry on a Young’s double-slit experiment analog . 57 3.5 Light scattering by coupled oriented dipoles: Decomposition of the scattering matrix . 61 3.6 Quaternion algebra for Stokes–Mueller formalism . 73 vi Table of contents 4 Discussion and Conclusions 81 Appendix: list of vectors and matrix states 89 Bibliography 93 Acknowledgment This thesis is a summary of five years of work and it would not have been possible to finishit without the help of many people. I would like to acknowledge some of them to emphasize my gratitude. First of all, my supervisor of this thesis, Oriol Arteaga. He introduced me to this field of physics and his work was the main influence for me. Without hishelpand contribution this work would not have been possible. I am also very grateful for his patience during those periods when my own motivation was low. I would also like to thank my thesis advisor Adolf Canillas. He has been a great influence for me in this field, and helped me throughout my teaching experience at the faculty. I must also mention my father, Mehmet Ali Kuntman, who had many intellectual contributions. His creativity, knowledge and vision has been an invaluable source of inspiration throughout this thesis, without which I would not have been able to finish it. I must also acknowledge his moral support during this period, he is a great mentor in my life and I am incredibly lucky to have him. I would also like to thank my beloved mother, Nihal Akata Kuntman, and my dear brother, Ahmet Kuntman, who never stopped supporting me in life. I would like to thank Razvigor Ossikovski, Jordi Sancho-Parramon, Tanzer Çakır and Joan Antó for their collaboration in this work. During the years that I have spent in Barcelona I have met many important people in my life. Firstly, I would like to thank Martín Girón Rodríguez for three great years that we spent together. I have learned so much from him and he made me a complete person in many ways. I am really lucky to know him. Also my close friends who taught me Spanish and helped me through everything that I have been through, such as Annalisa Albanese, Bruna Amoros, Georgios Domparas, Lesly Ponce, Vermeer Grange, Yuanzhen Liu. My great flatmates: Çaglar Küçük, Joshua Cawkill, Gwendolly Guadarrama and the amazing twins Oriol and Joel Pocostales, for always making me feel at home and introducing me to so many amazing aspects of life. Andrew Lavelle and Killian Dunne for their thought-provoking conversations during our incredible trips to the acid rock. Finally, Leticia Belazquez and Gary Amseian for their friendship and teaching me so many aspects of medicine. Resumen en Español La matriz de Jones y la matriz de Mueller-Jones (es decir la matriz de Mueller no despo- larizante) son los elementos matemáticos centrales del cálculo de óptica de polarización. En esta tesis discutimos otras formas que pueden usarse para representar las propiedades de polarización de los sistemas deterministas. Investigamos cuatro formas diferentes que interpretamos como los estados de los sistemas ópticos deterministas. El estado vectorial h j i es el elemento central de nuestro formalismo. La combinación paralela coherente de sistemas ópticos deterministas puede expresarse, más convenientemente, como una combinación lineal de estados vectoriales. En otras palabras, cualquier sistema óptico no despolarizante puede considerarse como una combinación coherente de otros sistemas deterministas que sirven como sistemas basé. Sin embargo, los estados vectoriales no son adecuados para representar la combinación en serie de sistemas ópticos, porque los vectores h no se pueden multiplicar j i como h h h . Observamos que existe un estado de matriz complejo Z que imita todas j 1ij 2ij 3i las propiedades de las matrices de Jones, incluida la propiedad de multiplicación de estados. Las matrices Z también son similares a las matrices de Mueller no despolarizantes por la relación M = ZZ∗. Mostramos que las matrices Z transforman la matriz de Stokes S en otra † matriz de Stokes S0 de acuerdo con la relación S0 = ZSZ , donde la matriz S corresponde al vector Stokes s y S corresponde al vector Stokes transformado s ( s = M s ). j i 0 j 0i j 0i j i Las matrices Z también transforman los vectores de Stokes, s , en vectores complejos j i s˜ según s˜ = Z s . Se puede demostrar que, a diferencia de un vector s , un vector s˜ j i j i j i j i j i contiene la fase introducida por el sistema óptico. Además observamos que los vectores h j i y las matrices Z son representaciones diferentes de estados cuaternión, h. Los estados de cuaternión se pueden agregar o multiplicar para producir nuevos estados de cuaternión, por lo tanto, son adecuados para representar cualquier combinación coherente de sistemas ópticos deterministas. Los resultados muestran que los estados de cuaternión transforman cuaternion † de Stokes, s, en un cuaternión de Stokes, s0, de acuerdo con el esquema s0 = hhssshh . Las formulaciones de matriz Z y cuaternión son especialmente útiles para describir la aparición de efectos de despolarización. El enfoque de matriz de densidad es más conveniente cuando queremos encontrar los componentes originales de una matriz de Mueller despolarizante. 4 Table of contents Las matrices de densidad asociadas con los sistemas ópticos deterministas (puros) se definen en términos de vectores h como H = h h .
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