Springer optical properties of photonic crystals sakoda; springer; 2001

professor Kazuaki Sakoda - ~ ~ ~ k k a i University ~ ~ Institute ~ for Electronic ~ ~Science~ c h ~ ~ r12tWest h 6, Kita-ku Sapporo 060 0812 To Fumi and Midori ISSN 0342-4111 I S B N 3.540-41199-2 Springer-Verlag B e r l i n Heidelberg N e w Y o r k ~ i b , a , v r ~ u l l g r r a~r a c a l u g i r r g ~ i n ~ ~ u b l i~~ a ~ti ri n Sakoda, Kazuaki, 1957~ Optical properties of photonic crystals Kazuaki Sakoda p m - (Springer series in optical science$, ISSN 034%-4ln; 80) Includes bibliagraphical references and index ISBN 3.540411992 (acid~freepaper) L Photons Crystal optics I Title 11 Springer series in optical sciences; v 80 Q C ~ ~ ) I P ~St4 W 2001 ~ R ' g - d ~ l l (10-0661~8 parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember 9.1965, in its current verrion, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecutioll under the Gerrr~arrCopyright Ldw Springer-Verlag Berlin Heidelberg New York a member of Bertelsmannspringer Science+R~~rin~sr Mpdir GmhH http:/im.ipringer.de Sprhnger-Verlag Berlin Heidelberg zoo1 Printed in Germany Prinfedon a n d ~ f r r cpaper SPIN 10881818 1711111 Springer Series in Kazuaki Sakoda OPTICAL SCIENCES ÿ he Springer Series m Optical Sciences, under the leadership ot Editor-in-Chief williarn T ~hoder, Georgia Institute of Technology, USA, and Georgia Tech Lorraine, France, provides an expanding selection ofresearch monographs in all majorareas ofoptics: lasers and quantum optics, ultrafast p h e ~ nomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest With this broad coverageoftopics, the series is ofuse to all researchscientistsandengineerr who need up-to-date reference books The editors encourage prospective authors to correspond with them in advance ofsubmitting a manuscript Submission of manuscripts should be made to the Editor-in-Chiefor one of the ~ d i t o r ssee also http:/lr?vwspringerde/phy~/b~~k~Iopticalscienc~/ Optical Properties of Photonic Crystals Editor-in-Chief William T Rhodes Georgia Tech Lorraine 2-3, rue Marconi $7070 Metr, France Phone: +33 (387) lo 3922 Fax: t33 (387) ro 3940 E-mail: wrhodes@georgiatech~metz.fr U R L http:/lwwwgeorgiatechhrnetz.fi http://users.ece.gatech.edu/-wrhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA ,0332-0250 Phone: +I 404 894 %929 F u : +L 404 894 4641 E-mail: bill.rhodes@ece.gatech.edu URL: http:llwww.ece.gatech.edu/prohIesi 1wrhodeslindex.htm With 95 Figures and 28 Tables Editorial Board Toshimitsu Asakura Ferenc Krausz Hokkai-Gakuen University Faculty ofEngineering 1-1, MinamiG26, Nishi n, Chuo-ku Sapporo, Hokkaido 064-0926, Tapan E~maikasakura@eli.hokkai-S-u.ac.jp Vienna University of Technology Photonics Institute Gusshausstrasse 271387 ,040 Wien, Austria Phone: +43 (I) 58801 387n Fax: +43 (1) 5880138799 E-mail: ferenc.krausz@tuwien.ac.at URL: http:/linfo.tuwien.ac.atlphatanikl harnelKrauszlCVhtm Karl-Heinz Brenner Chair af Optoelectronics University of Mannheim Institute of Computer Engineering B6,26 6813, Mannheim, Germany Phone: +49 (621) 181 2700 Fax: +49 (621) 181 2695 E-mail: brennereuni-mannheim.de URL: http:l/wvw.ti.uni~mannheim,del-i~e Theodor W Hansch Man-Planck-lnstitut f"r Quantenoptik Hans-Kopfermann-Strasse I 81748 Garching, Germany Phone: +49 (89) 2180 3211 or +49 (89) 32905 702 Fax: +49 (89) 32905 zoo E-mail: t.whaensch@physik,uni.muenchen.de URL:http:l/m.mpq.mpg.del-haensch Horst Weber Technische Universit5t Berlin Optischcs lnstitut Strasse des 17 luni 135 1062) Berlin, Germany Phone; +49 (30) 314 23585 F u : +49 0 ) 314 27850 E-mail: weber@physik.tu-berlin.de URL: http:l/wulv.ph~ik.t~~berlin,deIinstituteI OllWeberlWebhome.htm Springer ~ Preface Tlie interaction between the mdiat,ion field and matter is the most fundamental soilrce of dynamics in nature It brings about the absorption and emission of photons, elastic and inelastic light scattering, the radiative lifeti~neof electronic excitcd states, and so on Thc bugc amount of energy carried from the sun by photons is the source of all activities of creatures on the earth The absorption of photons by chlorophylls and the successive electronic excitation initiate a series of chemical reactions that are known as photosynthesis, which support all life on the earth Radiative energy is also the main source of all nleteorological phenomena The fundamentals of the radiation field and its interaction with matter were clarified by classical electromagnetism and quantum electrodynamics These theories, we believc, explain all electromagnetic phenomena They not only provide a firm basis for contemporarv physics but also generate a vast range of technological applications Thrsc include television, radar, optical lasers, light-emitting diodes, solar cells, and microwave telecorr~rnu~~ications etc Now the interaction between the radiation field and matter is so fundamemt,al that it may seem universal and invariant But in fact it is controllable This discovery has been the great motivating force of intensive investigations in optical physics during the last three decades In this book, I will show how it is controlled using photonic crystals, a remarkable inm:ntion realized by the combination of optical physics and contemporary microfabrication techniques I will also show how the controlled radiation field alters the optical properties of atoms and molecules embedded in t,he photonic crystals and what kinds of new phenomena and new physics are expected t o manifest t hemselves This book was written to serve as a comprehensive textbook covering the optical properties of photonic cryst,als It deals not ullly will, (Ire properties of the radiation field inside the photonic crystals but also t,heir peculiar optical response to external fields Only an elementary knowledge of electromagnetism, quantum mechanics solid-stntc physics, and complex analysis is required of the readcr Therefore, undergraduate students in physics, applird physics optics electronics, and electrical engi~~eering in the final year shoilld be ahle to read this book withont difficl~lt,~ Since tlir main recent dcvrl- VlII Preface oprnents such as t.he enhancement of stimulated e~nission.second harmonic generation and quadrature-phase squeezing are also treated in a dctailed and ~n~derstandable manner this book also provides important ideas for graduate students and researchers in this field would like to thank Professor Eiichi Hanamura and Professor Kuon Inoile who gave me a wonderful introdnct.ion to this excit.ing field I would also like to thank Professor Kmno Ohtaka and Professor Josepf~W Haus n h o gave me many suggestions on important problems I am grateful to Professor Sajeev John for giving me the opportunity to visit his laboratory in Toronto University in the summer of 1998 During that period I learned much ahout the qrlant.urn optics of photonic crystals I am also grateful to Dr Kurt Busch who helped me to confirm that the group theory worked well for the fcc st.ruct~~re I acknowledge Professor T n h Koda for his continuous encouragement since I was an undergraduate student of Tokyo University I deeply acknowledge Professor Toshimitsu Asakura and Dr Claus Ascheron for giving me thc opport.unity to write this hook I am grateful to many of my graduate students for their efforts in numerical calculations In particular Ms Hitomi Shiroma-Hirata Ms Noriko Kawai and Mr Takunori Ito did many of the calcnlatioms that are shown in this hook I am also grateful to Dr Tetsuyuki Ochiai who made the main contribution to the study of photonic crystal slabs prcscntcd in Chapter Sapporo February 2001 Kazuakz Sakoda ! Introduction E i g e n m o d e s of P h o t o n i c C r y s t a l s 13 2.1 Wave Equations and Eigenvalue Problems 13 2.2 Eigenvalue Problems in Twc+Dimensional Crystals 19 2.3 Scaling Law and Time Reversal Symmetry 21 2.4 Photonic Band Calculation 23 2.4.1 Fourier Expansion of Dielectric Rinct.ions 23 2.4.2 Some Examples 26 2.5 Phase Velocity, Group Velocity., and Encrgy Velocity 30 2.6 Calculation of Group Velocity 32 2.7 Complete Set of Eigenfunctions 34 2.8 Retarded Green's Function 39 S y m m e t r y of E i g e n m o d e s 3.1 Group Theory for TwwDimensional Crystals 3.2 Classification of Eigenmodes in the Square Lattice 3.3 Classification of Eigenmodes in the Hexagonal Lattice 3.4 Group Theory for Three-Dimensional Crystals 3.5 Classification of Eigenmodes in the Simple Cubic Lattice 3.6 Classification of Eigcnmodcs in thc fcc Latticc 43 43 55 57 62 65 75 Transmission S p e c t r a 81 4.1 Light Transmission and Bragg Reflection 81 4.2 Field Equations 83 4.2.1 E Polarization 8:l 4.2.2 H Polarization 85 Foriricr Transform of the Dielectric Function 87 4.3.1 Square Lattice 87 4.3.2 Hexagonal Lattice 89 4.4 Some Examples g l 4.4.1 SquareLattice 91 4.4.2 Hexagonal Latticp 91 4.5 Refraction Laxv for Photonic Crystals 95 O p t i c a l R e s p o n s e of P h o t o n i c C r y s t a l s 99 5.1 Solutions of Inhomogeneous Equations 99 5.2 Dipole Radiation 102 5.3 Stimulated Emission 105 5.4 Sum-Frequency Gcr~cratio~i 109 5.4.1 Three-Dimensional Case 109 5.4.2 Two-Dimensional Case 112 5.5 SHG in the Square Lattice 116 5.6 Free Indiiction Decay 121 Defect M o d e s i n P h o t o n i c C r y s t a l s 125 6.1 General Properties 125 6.2 Principle of Calculation 128 6.3 Point Defects in a Square Lattice 131 6.4 Point Defects in a Hexagonal Lattice 134 6.5 Line Defects in a Square Lattice 142 6.6 Dielectric Loss and Quality Factor 146 B a n d Calculation w i t h F r e q u e n c y - D e p e n d e n t Dielectric C o n s t a n t s 151 7.1 Principlc of Calculation 151 7.2 hlodificd Plane iVaves in hIetallic Crystals 154 7.3 Surface Plasmon Polaritons 161 7.3.1 Plasmun Polnritons on Flat Surface 162 7.3.2 Plasnlon Resonance on a Metallic Cylinder 165 7.3.3 Symn~ctrvof Plasmon Polaritons 169 7.3.4 Plasrrion Bands in a Square Lattice 171 P h o t o n i c C r y s t a l S l a b s 177 8.1 Eigcnmodcs of Uniform Slabs 177 Symmetry of Eigenmodrs 181 8.3 Photonic Band Structure and Trallslnission Spectra 183 8.4 Q11;rlity Factor 185 Low- Threshold Lasing D u e to Group- Velocity A n o m a l y 189 Enhanced Stimulated Elnission 189 9.2 Lasing Threshold 193 9.2.1 -4llalytical Expression 191 9.2.2 Numerical Estimation 195 10 Q u a n t u m O p t i c s in P h o t o n i c C r y s t a l s 10.1 10.2 10.3 10.4 201 Quantization of the Electromagnetic Field 201 Qua