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Application of the Differential Method to Diffraction Gratings That Utilize Total Intemal Reflection Facets

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Application of the Differential Method to Diffraction Gratings That Utilize Total Intemal Reflection Facets Application of the Differential Method to Diffraction Gratings That Utilize Total Intemal Reflection Facets BY Michael Sem Dyck Smith A Thesis Subrnitted to the Faculty of Graduate Studies In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department of Physics University of Manitoba Winnipeg, Manitoba O August. 1998 National Wrary BibTithèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services seMces bibliographiques 395 WeUuigton Street 395. nre Wellington OttawaûN K1AW Ottiwa ON K1AW Canada canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire7prêter7 districbuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copy~@~tin this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantid extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othenivise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. THE UNIVERSITY OF MAMTOBA FACULTY OF GRADUATE STUDIES ***** COPYRIGHT PERMISSION PAGE A ThesidPractieum submitted to the Facnlty of Graduate Shidies of The University of Manitoba in pailial fulfrllment of the requirements of the degree Uïchael Sean Dyck Smith Permission has been granted to the Library of The University of Manitoba to lend or seii copies of th& thesis/practicum, to the National Library of Canada to microfilm thb thesis and to lend or sel1 copies of the film, and to Dissertations Abstracts International to pubtish an abstract of this thesidpracticum. The author reserves other pubiication rïghts, and neither this thesir/practicum nor extensive extraeu from it may be printed or otherwlre reproduced without the author's written permission. ABSTRACT Integrated optical grating devices with facets designed to take advantage of total intemal reflection have recently been demonswted. To date. anaiysis of these total internal reflection (TIR) gratings has been Limited to an elementary ray optics approach. This thesis presents the fmt analysis of these gratings based on the full electromagnetic theory of iight. The vaiidity of designing diffraction gratings with total internal reflection facets is demonstrated. Results indicate that the efficiency of the retro-reflected order of 20~order gratings etched in silica gass is enhanced by more than 11 dB for the TE mode when the TIR grating design is used in place of a similar echelle grating without facet metalization. Cornparisons are made between results found using the full electromagnetic theory of light and simple scalar wave approximations. qualitative agreement is found for the retro-reflected order, particularly for grating orden 15-25. ACKNOWLEDGEMENTS I wish to thank my advisor Dr. Ken McGreer and Tusfor the opportunity to do research in the exciting field of fiber optics. 1 appreciate Dr. McGreer's help in choosing a thesis topic and reading this thesis. 1 am grateful for the excellent facilities TRLnbs has provided as well as the financial assistance of TRLabs and NSERC. 1 thank Dr. Sergey Sadov for sharing his knowledge of diffraction @ng theory and for his assistance debugging my computer programS. 1 aIso thank fellow graduate students Heather Hnatiuk, Dan Jackson, and Dr. Zh?jian Sun for their help and discussions related to course work and my thesis. 1 wish to thank my family and friends for their interest and encouragement over the course of my studies. 1 am particularly grateful for my wife Valerie's understanding and constant support. My sister Alison haais0 been very encouraging. CONTENTS LIST OF FIGURES....................................... ..................................... ................................................. vi LIST OF TABLES............. .. ..................................................................................................................... ix 1 INTRODUCTION ......................................... ............................................ ................................................ 1 1.1 BACKGROUND................................................................................................................................... 1 1 .2 APPLICATION .................................................................................................................................... 3 1-3 ST.ATE.!IE!! QF PROBrbhi ................................................................................................................. ,C 1.4 OBJECTTVE........................................................................................................................................ 7 1.5 O~INEOFTHESIS ............ .. ......................................................................................................... 7 2.1 BASICGRATING THEORY .......................... .... .......................................................................... 9 2.2 SCAM WAVETHEORY ................................................................................................................. 12 2.2.1 Ra? Oprics ........................................................................................................................... 12 2-22 The Fresnel Equarions .......................................................................................................... 13 2.2.3 Fraunhofer Diffracrion ....................................................................................................... 14 2.3 ELEC~ROMAGN~ICTHEORY OF GRATINGS.......... .... .............................................................. 16 2.3.1 Norarion ................................................................................................................................ 16 2.3.2 Manvell 's Equarions ........................... .... ......................................................................... 17 2.3.3 The Helmholt= Equation ....................................................................................................... 18 2.3.4 BortndapCondirions ............................................................................................................ 19 2.3.5 General Die!ecrric Graring................................................................................................... 20 2.3.6 Rayleigh Expansions............................................................................................................. 23 2.3.7 Calcrilaring Efllcirncies ........................................................................................................ 27 2.3.8 Propagation in rhe Graring Region ..................... ............................................................. 29 2.4 THEDIFFERENTIAL METHOD ........................................................................................................... 31 2.4.1 Definirions ............................................................................................................................ 31 2.42 Calcularion of T and R.......................................................................................................... 32 .. 2.4.3 Calcularion of Efictenczes .................................................................................................... 34 2.4.4 Centering Real Orders in rhe Vecrors................................................................................... 35 24.5 ABriefNoreonTMpolarizarion ..................... ., ............... ,. .................................................. 36 3.1 CONSISTENCY OF RESULTS Wïïii PUBLISHED RESULTS ........................ ,...... .......................... 37 3.2 DETERMININGACCURXY OF RESULTS.......................................................................................... 30 3.2. / Dependency of accuraq on Nb,,,, ........................................................................................ 42 3.2.2 Dependency of accuracy on Nw, ....................................................................................... 43 4 RESULTS AND DISCUSSION ......................... .... ....................................................-.....................47 4.1 SCOPEOF STUDY............................................................................................................................ 47 3.2 THESCAUR WAVE APPROX~MATION ............................................................................................ 48 4.3 COMPARISONOF SCAUR WAVE APPROXIMATION AND DIFFERENTIALMETHOD RESULTSFOR RETRO-REFLECTEDE~C~CIES ............................................................................................................ 52 4.3.1 Retro-Rejlecred Enciencies as a Funcrion of Refracrive Index ......................................... 52 4.3.2 Retro-Reflected EB-ciencies as a Function of Grating Order .............................................. 60 4.4 RESULTSFOR THETIR GRA~G.................................................................................................... 65 4-41 TIR Gratutg Eficiencies as a Function of Refiacrive Index ................................................. 65 4.4.2 TIR Grating Eficiencies ai^ a Function of Grating Order.................. ... ..............................
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