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1 Retrospective Theses and Dissertations Iowa State Universit Capstones, Theses and Dissertations 7 The stud of electromagnetic wave propagation in photonic crstals via planewave based transfer (scattering) matri method with active gain material applications Ming Li Iowa State Universit Follow this and additional works at: Part of the Condensed Matter Phsics Commons, and the Optics Commons Recommended Citation Li, Ming, "The stud of electromagnetic wave propagation in photonic crstals via planewave based transfer (scattering) matri method with active gain material applications" (7). Retrospective Theses and Dissertations This Dissertation is brought to ou for free and open access b the Iowa State Universit Capstones, Theses and Dissertations at Iowa State Universit Digital Repositor. It has been accepted for inclusion in Retrospective Theses and Dissertations b an authorized administrator of Iowa State Universit Digital Repositor. For more information, please contact digirep@iastate.edu.

2 The stud of electromagnetic wave propagation in photonic crstals via planewave based transfer (scattering) matri method with active gain material applications b Ming Li A dissertation submitted to the graduate facult in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Condensed Matter Phsics Program of Stud Committee: Kai-Ming Ho, Major Professor Gar Tuttle Jianwei Qiu Joseph Shinar Joerg Schmalian Iowa State Universit Ames, Iowa 7 Copright Ming Li, 7. All rights reserved.

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4 To m grand father and grand mother ii

5 iii Table of Contents Chapter. General introduction. Histor of photonic crstal. Comparison between crstal and photonic crstal 3 Chapter. The planewave based transfer (scattering) matri method - core algorithms 5. Mawell s Equations 5. Fourier space epansion 8.3 Transforming partial differential equations to linear equations.4 Fourier space Mawell s Equations for uniform medium 7.5 Building the transfer matri and scattering matri 9.6 Using scattering matri for various applications.6. Spectrum from S matri 3.6. Band structure from S matri Field mode profile from S matri 7 Chapter 3. Interpolation for spectra calculation 3 3. Wh interpolation works? An eample of application of interpolation Origin of Lorentzian resonant peaks 4 Chapter 4. Higher-order planewave incidence Planewave incidence Comparison between oblique incidence and fied k value incidence Higher-order incidence C v Group Higher-order planewave and its smmetr Possible propagation modes for higher-order incidence Eample of application of higher-order incidence 66 Chapter 5. Perfectl matched laer used in TMM Motivation of introducing perfectl matched laer Theor of perfectl matched laer and Z ais PML 75

6 iv 5.. Background of PML Performance of simple parameter approach Two strategies to improve the performance Application of PML to periodic D waveguide Perfectl matched laer for X, Y ais and its application to D waveguide Analtical solutions of D dielectric slab waveguide Numerical results of TMM with side PMLs PML application eample: dispersive sub- λ aluminum grating 89 Chapter 6. TMM etension to curvilinear coordinate sstem Transform into curvilinear coordinate Curved waveguide simulation 97 Chapter 7. Application of TMM to diffractive optics 4 7. Finding phase b TMM 4 7. Confirmation b Snell s Law Case stud: bo spring structures for electromagnetic wave deflection Geometr of bo spring structures Simulation results of bo spring structures Chapter 8. TMM algorithm with active gain material etension 6 8. Rate equation, the starting point 6 8. Defining the electric field dependent dielectric constant for gain material Gain-TMM algorithm for laser device simulation 8.4 D DBR laser, an eample of GTMM application D woodpile photonic crstal laser, an eample of GTMM application 6 Chapter 9. Microwave eperiments for woodpile photonic crstal cavities 3 9. Instrument setup for microwave eperiments 3 9. Resonant frequenc and Q value for fied length cavit Effects of cavit size on resonant frequencies 36 Chapter. Future developments and applications of TMM 4. Go beond planewave basis the localized light orbital 4. Future applications of photonic crstal concepts 4

7 v Acknowledgments Si ears have been passed since m first da at the lovel mid-west small town Ames, and twelve ears since m first da at Department of Phsics, Xiamen Universit. It is so long and lonel journe to reach the goal that it is mission impossible without m famil s financial and spiritual supports. I am grateful in heart to m grandparents, parents and aunt. During m PhD stud at Iowa State Universit, m adviser Dr. Kai-Ming Ho gives me enormous help and valuable advices not onl in academic research but also in personal life. I have learnt a lot from his deep understanding and sharp vision at photonic crstal research fields, as well as his broad knowledge at other areas. Here I would like to sincerel thank him. Also during m PhD stud, I learnt a lot from m stud committee faculties: Dr. Joerg Schmalian taught me four graduate courses which made the foundation of m phsics knowledge; Dr. Gar Tuttle taught me one graduate course and man microwave eperiments skills; Dr. Joseph Shinar s OLED presentations gave me a lot of hints for applications of m research; and Dr. Jianwei Qiu s advices made me think deeper and more fundamental in the theor of m research. I would like to thank m stud committee members for their help. I would also like to thank Dr. Dave Turner for valuable discussion and help at parallel computation. Sometimes doing research is frustrating, and ou will need buddies to back ou up. Dr. Jiangrong Cao (Canon USA) and Dr. Xinhua Hu (Ames Lab) are two such good buddies to discuss with. Their help and friendship are ver important to me; and the TMM/GTMM package will not be possible to finish without their help. Last I would like to thank all m friends at Ames who make m life not so boring, especiall m girl friend Ruiue who alwas encourages me and supports m stud.

8 vi This work was performed at Ames Laborator under Contract No. DE-AC-7CH358 with the U.S. Department of Energ. The United States government has assigned the DOE Report number IS-T 889 to this thesis.

9 vii Abstract In this dissertation, a set of numerical simulation tools are developed under previous work to efficientl and accuratel stud one-dimensional (D), two-dimensional (D), D slab and three-dimensional (3D) photonic crstal structures and their defects effects b means of spectrum (transmission, reflection, absorption), band structure (dispersion relation), and electric and/or magnetic fields distribution (mode profiles). Further more, the lasing propert and spontaneous emission behaviors are studied when active gain materials are presented in the photonic crstal structures. Various phsical properties such as resonant cavit qualit factor, waveguide loss, propagation group velocit of electromagnetic wave and light-current curve (for lasing devices) can be obtained from the developed software package. First, the planewave based transfer (scattering) matri method (TMM) is described in ever detail along with a brief review of photonic crstal histor (Chapter and ). As a frequenc domain method, TMM has the following major advantages over other numerical methods: () the planewave basis makes Mawell s Equations a linear algebra problem and there are mature numerical package to solve linear algebra problem such as Lapack and Scalapack (for parallel computation). () Transfer (scattering) matri method make 3D problem into D slices and link all slices together via the scattering matri ( S matri) which reduces computation time and memor usage dramaticall and makes 3D real photonic crstal devices design possible; and this also makes the simulated domain no length limitation along the propagation direction (ideal for waveguide simulation). (3) It is a frequenc domain method and calculation results are all for stead state, without the influences of finite time span convolution effects and/or transient effects. (4) TMM can treat dispersive material (such as metal at visible light) naturall without introducing an additional computation; and meanwhile TMM can also deal with anisotropic material and magnetic material (such as perfectl matched laer) naturall from its algorithms. (5) Etension of TMM to deal with

10 viii active gain material can be done through an iteration procedure with gain material epressed b electric field dependent dielectric constant. Net, the concepts of spectrum interpolation (Chapter 3), higher-order incident (Chapter 4) and perfectl matched laer (Chapter 5) are introduced and applied to TMM, with detailed simulation for D, D, and 3D photonic crstal eamples. Curvilinear coordinate transform is applied to the Mawell s Equations to stud waveguide bend (Chapter 6). B finding the phase difference along propagation direction at various XY plane locations, the behaviors of electromagnetic wave propagation (such as light bending, focusing etc) can be studied (Chapter 7), which can be applied to diffractive optics for new devices design. Numerical simulation tools for lasing devices are usuall based on rate equations which are not accurate above the threshold and for small scale lasing cavities (such as nano-scale cavities). Recentl, we etend the TMM package function to include the capacit of dealing active gain materials. Both lasing (above threshold) and spontaneous emission (below threshold) can be studied in the frame work of our Gain-TMM algorithm. Chapter 8 will illustrate the algorithm in detail and show the simulation results for 3D photonic crstal lasing devices. Then, microwave eperiments (mainl resonant cavit embedded at laer-b-laer woodpile structures) are performed at Chapter 9 as an efficient practical wa to stud photonic crstal devices. The size of photonic crstal under microwave region is at the order of centimeter which makes the fabrication easier to realize. At the same time due to the scaling propert, the result of microwave eperiments can be applied directl to optical or infrared frequenc regions. The sstematic TMM simulations for various resonant cavities are performed and consistent results are obtained when compared with microwave eperiments. Besides scaling the eperimental results to much smaller wavelength, designing potential photonic crstal devices for application at microwave is also an interesting and important topic.

11 i Finall, we describe the future development of TMM algorithm such as using localized functions as basis to more efficientl simulate disorder problems (Chapter ). Future applications of photonic crstal concepts are also discussed at Chapter. Along with this dissertation, TMM Photonic Crstal Package User Manual and Gain TMM Photonic Crstal Package User Manual written b me, Dr. Jiangrong Cao (Canon USA) and Dr. Xinhua Hu (Ames Lab) focus more on the programming detail, software user interface, trouble shooting, and step-b-step instructions. This dissertation and the two user manuals are essential documents for TMM software package beginners and advanced users. Future software developments, new version releases and FAQs can be tracked through m web page: In summar, this dissertation has etended the planewave based transfer (scattering) matri method in man aspects which make the TMM and Gain-TMM software package a powerful simulation tool in photonic crstal stud. Comparisons of TMM and GTMM results with other published numerical results and eperimental results indicate that TMM and GTMM is accurate and highl efficient in photonic crstal device simulation and design.

12 Chapter. General introduction. Histor of photonic crstal The histor of human development is the histor of how people can utilize and control the properties of materials, nature or man made. At the ver beginning age of civilization, human being learnt how to use and manipulate with the mechanical propert of stone to make stereotpe tools for everda life. We call this period Stone Age. Later on, people studied how to get metal and allo from ore. Metal and allo have better mechanical properties and the application of those materials towards agriculture made the human societ development possible. Even more recent invention of steam locomotive was based on how to improve the mechanical movement which made modern civilization possible. In most of the mankind histor, we are improving on how to control or utilize the mechanical properties of materials. Although the electrical and optical properties were noticed b us long time ago, the theoretical stud of fundamental electrical and optical phenomena is not done until around two hundred ears ago due to the tin size of electron, photon and atomic structure. Mawell introduced his famous equations at 864 to sstematicall describe the behavior of electromagnetic wave. In 96, Schrödinger published his quantum mechanics paper to describe how electrons behave. In the middle of th centur, with the efforts of both theoretical and eperimental phsicists, we can control the motion of electrons b introducing defects into pure crstals or semiconductors. After we had the abilit to control the electrical properties, the electrical engineering industr development is possible and it has profound impact on our dail life. -5

13 Nature crstal is a periodic arrangement of atoms which gives periodic potential to electrons inside the crstal. Block s theor can be applied to Schrödinger s equation and the solution of Schrödinger s equation reveals the possible was to control the motion of electrons. However there are no nature available "photonic" crstals to provide a similar wa to control photon or electromagnetic wave as crstal to electrons. Can ou image it that it is not until 987, more than ears after Mawell s Equation, that the concept of photonic crstal was introduced b Eli Yablonovitch 6 and Sajeev John 7? And it is onl after three ears for the concept to be finall confirmed b K.M. Ho 8 and coworker. But after the 987 concept breakthrough and 99 concept confirmation, both theor and eperiment are booming based on the idea of photonic crstal. Various applications, such as low-loss waveguide and high Q resonant cavit, are proposed in recent ears and some are close to the stage of mass production. The application of photonic crstal devices will be tremendous in people s everda life. Even the simplest wa to control light propagation, the internal refraction, has alread changed the entire communication industr via the invention of optical fiber. With the new concept of photonic crstal, the better qualit photonic crstal fibers are on the market. Now let s look back what people were doing after the 987 concept breakthrough. A lot of phsicist both from theoretical and eperimental joined the research field of photonic crstal. But at the first a few ears, people were struggling to prove the photonic crstal concept b obtaining consistent results of eperimental and numerical simulation. Eperimental phsicists first adopted the cut-and-tr method which basicall depends on the luck of the proposed geometr structure. But after man tedious works, the so called full band gap structures were still illusions. At the same time, theoretical phsicists adopted the method to solve scalar wave functions for electrons to stud electromagnetic wave. Due to the vector nature of electromagnetic wave and Mawell s Equations, the scalar wave approaches failed. At 99, Kai-Ming Ho 9 and coworker introduced planewave epansion methods to solve the vector Mawell s Equations and successfull predicted that diamond structure will have full

14 3 band gap. Later on, Eli Yablonovitch made the first photonic crstal at microwave region based on the predicted diamond structure. The concept of photonic crstal then was firml established and numerous fabrication techniques and numerical simulation methods were introduced in this fast developing research field. Now it is alread twent ears after the 987 concept breakthrough, photonic crstal has been studied intensivel. However, with the eception of photonic crstal fibers, ver few concepts have been able to pass from the scientific research stage to high throughputs mainstream products. Besides the challenges in manufacturing, one of the main reasons behind this situation is the lack of efficient and versatile numerical computation tools for photonic crstal devices simulation, especiall three-dimensional structures with defects. Our planewave based transfer (scattering) matri method is proved to be an efficient and accurate numerical simulation tool for photonic crstal through this thesis via various structures and applications.. Comparison between crstal and photonic crstal The status of photons in photonic crstal and the status of electrons in crstal have man similarities: both sstems are eigenvalue problem; the geometr periodicit in both sstems leads to the application of Bloch Theorem which leads to the concept of band and band structure; in both sstems the introduction of defects makes possible of controlling the corresponding electric or optical properties. Based on those similarities, man concepts and strategies in quantum mechanics and solid state phsics can be borrowed to the research field of photonic crstal, such as reciprocal lattice and Brillouin zone. However, there are two major differences between electron and photon: () there is no interaction between photons (for linear optics) while there are electron-electron interaction; () there is no characteristic length for photons and the band structure can be scaled to an length scale while there is a nature length scale for electrons. -

15 4 In principle, photonic crstals are periodic arrangement of dielectric material in one direction, two directions or all three directions in space, and we call them D, D or 3D photonic crstal correspondingl. In general cases periodic structures do not guarantee the eistence of full photonic band gap in which no propagating modes eist for an directions. In later part of this thesis, we use planewave based transfer (scattering) matri method to stud the spectrum, band structure and mode profiles for various photonic crstal structures. With the etension of transfer (scattering) matri method to active gain materials, lasing and spontaneous emission with the present of photonic crstal background can be studied. References:. John D. Joannopoulos, Photonic crstal Molding the Flow of Light, Princeton Universit Press, 995. Steven G Johnson, Photonic crstal The road from theor to practice, Kluwer Academic Publishers, 3. Neil W. Ashcroft and N. David Mermin, Solid State Phsics, Brooks Cole Press, M. Born and E. Wolf, Principles of Optics, 7th edition, Cambridge Universit Press, J. D. Jackson, Classical Electrodnamics, 3rd edition, Wile & Sons Press, 4 6. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Phsics and Electronics," Phs. Rev. Lett. 58, 59 (987) 7. S. John, "Strong Localization of Photons in Certain Disordered Dielectric Superlattices", Phs. Rev. Lett. 58, 486 (987) 8. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Eistence of a photonic gap in periodic dielectric structures", Phs. Rev. Lett. 65, 35 (99)

16 5 Chapter. The planewave based transfer (scattering) matri method - core algorithms This chapter contains ver detailed derivation of how to get the scatter matri (S matri) for general 3D structures from the Mawell s Equations and acts like a literature review part. Most of the content has been published b Dr. Zhi-Yuan Li at a set of journal papers which are listed at the reference part of this chapter. After the S matri is read, transmittance, reflectance and absorptance can be obtained directl through the scattering matri algorithm. Photonic crstal band structure and electric and magnetic field distribution (mode profile) at an given location can also be obtained with a few more steps. The first several sections on how to get the S matri ma be looked through quickl and those sections of how to use calculated S matri to get spectrum, band structure or mode profile ma be focused in detail first. Then topics on how to get the S matri can be revisited in more detail. -. Mawell s Equations The most general case of macroscopic Mawell s Equations in Gaussian Unit is Eq. (.) and (.) where field vectors E, D, H, and B are function of time and space. In most of this thesis, we onl deal with passive and non-magnetic material, i.e. there are no free charge and no free current (Eq. (.3)) and μ () r =. With this simplification we can get the Mawell s Equations onl involve E and H field (Eq. (.4)). Further more, with the assumption that E and H field are harmonic in time (Eq. (.5)), we can separate the space variable and time variable and focus on the space variation of E and H field at given frequenc (Eq. (.6)). Now we introduce wave vector k (Eq. (.7)) and the first two Mawell s Equations becomes Eq. (.8). The angular frequenc or wave vector is then acting like one input parameter and ever angular frequenc follows the identical calculation procedure. This is

17 6 the feature of frequenc domain method: there will be no relation across different frequencies and no time dependence which will lead to the stead status solutions. B E = c t D 4π H= + J c t c D = 4πρ B = (.) D= ε ε () r E B= μμ() r H μ( r) = μ =, ε = c =.997 cmis (.) J =, ρ = (.3) H E = c t ε () r E H = c t ε () re= H = (.4) E= E() r e H= H() r e iωt iωt (.5) ω Er () = i Hr () c ω Hr () = i ε () rer () c ε ( rer ) ( ) = Hr ( ) = (.6)

18 7 k ω π f π = = = (.7) c c λ Er () = ik Hr () Hr () = ik ε () rer () (.8) Eq. (.8) are actuall vector differential equations and there are altogether si equations when we write the field vector out with respect to their components (Eq. (.9)). E E E z = z + = E z E z E = ikh ikh ikh z H z H = ikε () r E z Hz H, + = ikε () r E z H H = ikε () r Ez (.9) We can epress and H z and E z in term of E, E, E z from Eq. (.9) and get equations involving H and E, H (Eq. (.)) and eliminate E, H and H onl (Eq. (.)) H z H E z z E ( E = ) ik H H = ( ) ik ε () r (.) Equation set (.) is our starting point to solve the macroscopic Mawell s Equations at periodic dielectric medium (i.e. photonic crstal structures) via the planewave based transfer (scattering) matri method. To solve this partial differential equation set, our approach is to epand the field components and dielectric function into reciprocal space (Fourier space) which makes a set of difficult partial differential equations into relativel easier linear algebra problems; or we call it planewave based approach.

19 8 E H H = [ ( )] + ikh z ik ε ( r) E H H = [ ( )] ikh z ik ε () r H E E = [ ( )] ikε ( r) E z ik H E E = [ ( )] + ikε ( r) E z ik (.). Fourier space epansion Now let s suppose electromagnetic wave is propagating along Z ais towards a trunk of photonic crstal. Inside the photonic crstal, the dielectric distribution at an XY plane (perpendicular to Z ais) is periodic in both X and Y directions (for 3D photonic crstal) or uniform in one direction and periodic in the other direction (for D photonic crstal) or uniform in both X and Y directions (for D photonic crstal). Actuall uniform distribution at an direction is equivalent to have arbitrar periodicit along this direction, and in principle the dielectric distribution functions have double periodicit along X and Y directions in an Z ais positions (Figure -). Then the XY unit cell is defined as the minimum repeat area in the XY plane, and if the dielectric constant is uniform along X or Y ais, the length of XY unit cell along that direction can be arbitrar. The edge of unit cell is defined as the lattice constant ( a, a ) with the lattice points represent b R = ma + ma where m and m are integers. The reciprocal lattice points are then defined as G= nb + nb where n and n are integers and ( b, b ) the reciprocal lattice constant. With the requirement of e igir = or GR= i Nπ (Block s Theorem), b = π / a and b = π / a can be obtained. Then an periodic function f() r = f( r+ R) can be epressed as f ( ) = f e ig i r r where f G is the Fourier coefficient G

20 9 for each reciprocal lattice G. Or write in more detailed wa at Eq. (.). Here we assume a and a are along X and Y ais respectivel (orthogonal lattice constant), but in general a and a can be along an directions. X Y D D 3D Figure -: Dielectric distribution function of double periodicit along X and Y directions Z π π i( n + n ) a a Gn, n n= n= f(, ) = f e π π a a i( n + n ) a a ( π ) fg = f (, ) e dd n, n aa (.) The reciprocal of dielectric distribution function / ε( r ) is a periodic function and can be epressed in Fourier space (Eq. (.3)). In the real calculation, the sum over m and n must be truncate to finite terms, for eample: N m N, N n N with N = (N + )(N + ) called the total number of planewave. When a plane electromagnetic wave is incident from the left hand side on a photonic crstal slab with incident wave vector k = ( k, k, kz). The electromagnetic field at an arbitrar point r can be written into the superposition of plane waves with vector E mn and H mn the unknown epansion coefficients (Eq. (.4)); or epressed in term of scalar field components E, E, H, H and truncated to finite total planewave numbers at Eq. (.5). mn, mn, mn, mn,

21 = ε( r+ R) ε( r) = ε () r = ε mn N N N π π i( m + n ) a a ε mne m= N n= N N m= N n= N ε i( Gmn, + Gmn, ) mne π π a ( m n ) a i + a a ( π ) = e aa ε (, ) π π G = m, G = n mn, mn, a a dd (.3) Er () = E () ze m= n= Hr () = H () ze m= n= mn mn i( kmn, + kmn, ) i( kmn, + kmn, ) π kmn, = k + Gmn, = k + m a π kmn, = k + Gmn, = k + n a (.4) N N E () r = E () z e m= N n= N N N mn, E () r = E () z e m= N n= N N N mn, H ( r) = H ( z) e m= N n= N N N mn, H () r = H () z e m= N n= N mn, i( kmn, + kmn, ) i( kmn, + kmn, ) i( kmn, + kmn, ) i( kmn, + kmn, ) (.5)

22 .3 Transforming partial differential equations to linear equations With Eq. (.3) and (.5) read, we plug them into the Mawell s Equations for each component (Eq. (.)). Here, we will derive the first equation at Eq. (.) to Fourier space; the other three equations at Eq. (.) can be derived similarl. First, we write out a set of equations of partial derivative of E, H, H in Fourier space (Eq. (.6)) and with the last two equations at Eq. (.6) we can obtain Eq. (.7). E z H H = = = i= j= i= j= i= j= E z ij, i( kij, + kij, ) e i( kij, + kij, ) ik H e ij, ij, i( kij, + kij, ) ik H e ij, ij, (.6) H H i( kij, + kij, ) = i ( kij, Hij, kij, Hij, ) e (.7) i= j= H H = ik ε ( r) ik i( Gij, + Gij, ) i( kij, + kij, ) ε e ij i( k H k H ij, ij, ij, ij, ) e i= j= i= j= = ε e k H k H e ij mn, mn, mn, mn, k i= j= m= n= i( Gij, + Gij, ) i( kmn, + kmn, ) ( ) mn ij mn ij (,,,, ) i( ( k, + G, ) + ( k, + G, ) ) = ε k H k H e ij mn mn mn mn k i= j= N m= n= (.8)

23 k + G = k = k mn, ij, ( m+ i)( n+ j), m n, k + G = k = k mn, ij, ( m+ i)( n+ j), m n, with : m = m + i, n = n + j < m< < m < < n< < n < (.9) With Eq. (.7) and Eq. (.3) read, we can rewrite the term under the partial derivative of right hand side of the first Mawell s Equation (Eq. (.)) into Fourier space notation (Eq. (.8)). With redefined indices of Eq. (.9), Eq. (.8) can be rewritten as Eq. (.). H H ik ε ( r) i ( ) ( k ) + k mn, mn, k H k H e = ε ij mn, mn, mn, mn, k i ( ) ( k ) + k mn, mn, k H k H e = ε ij ( m i)( n j), ( m i)( n j), ( m i)( n j), ( m i)( n j), k i j m n = = = = i N j N m n = = = = (.) = mn k i j m n = = = = ε i ( ) ( k + k ), mn, k H k H e ij ( m i)( n j), ( m i)( n j), ( m i)( n j), ( m i)( n j), m i = i, n j = j, i = m i, j = n j < i< < i < < j< < j < (.) H H ik ε ( r) = ε k i = j = m= n= = ε k i= j= m= n= i( kmn, + kmn, ) ( m i )( n j )( kij H, ij k H, ij, ij, ) e i( kmn, + kmn, ) ( )( )( k H k H m i n j ij, ij, ij, ij, ) e (.) With further redefined indices at Eq. (.), we can reach Eq. (.). So the right hand side of the first Mawell s Equations (Eq. (.)) can be epressed as Eq. (.3). Finall, according to the first equation at Eq. (.), we can get the relation between E ij, and H ij,,

24 3 H ij,, ε ij (Eq. (.4)). Similarl, all four equations at Eq. (.) can be transformed into the i, j components equation set (Eq. (.5)) with δ, = for i = m, and δ, = otherwise. im jn im jn H H + kh k ε ( ) i i r i i( k + k ) mn, mn, = k ε mn, ( m i)( n j)( k H k H ij, ij, ij, ij, ) e k i= j= m= n= + ik H i( k e + k ) ij, ij, ij, i= j= i i( k + k ) ij, ij, = k ε ij, ( i m)( j n)( k H k H mn, mn, mn, mn, ) e k i= j= m= n= + ik H i( k e + k ) ij, ij, ij, i= j= (.3) E ik = H k H k + i k H (.4) ij, ij, ε i m, j n( mn, mn. mn, mn, ) ij, z k mn Now we define two column vectors (Eq. (.6)) to represent the unknown coefficients of E ij,, ij, E, H ij, and H ij, at Eq. (.5). Then the equation set of unknown coefficients (Eq. (.5)) can be epressed in a concise format (Eq. (.7)) where matri T and T are defined at Eq. (.8). E E H ik = ε ( H k H k ) + ik H ij, ij, i m, j n mn, mn. mn, mn, ij, z k mn ik = ε ( H k H k ) ik H ij, ij, i m, j n mn, mn. mn, mn, ij, z k mn z H ik = δ ( k E k E ) ik ε E k ij, ij, i k mn im, jn mn, mn. mn, mn, i m, j n mn. mn ij, ij, = δim, jn( kmn, Emn. kmn, Emn, ) + ik εi m, j nemn. z k mn mn (.5)

25 4 E = (..., E, E,...) ij, ij, H = (..., H, H,...) ij, ij, T T (.6) E = TH, H = TE z z (.7) k ε k k ε k + k δ ij, mn i ij, i m, j n mn, ij, i m, j n mn, im, jn = k k ε k k δ k ε k ij, i m, j n mn, im, jn ij, i m, j n mn, T T k δ k k δ k k ε ij, mn i ij, im, jn mn, ij, im, jn mn, i m, j n = k k δ k + k ε k δ k ij, im, jn mn, i m, j n ij, im, jn mn, (.8) If we use finite planewave number: N i N, N j N, we can define several planewave related variables: N = N = (N + )(N + ), N = N and N = 4N. The t dimension of the unknown coefficient vector E and H is N p ; and the dimension of matri T and T is Np Np. From Eq. (.7), we can get a single nd order differential equation (Eq. (.9)) with dimension of matri P and Q Np Np. As Q is a matri of dimension Np Np, the eigenvalue and eigenvector of Q can be found b linear algebra through Eq. (.3). With the eigenvalue β calculated, Eq. (.9) can be epressed b Eq. (.3) with each element of the unknown coefficients in vector E function of coordinate z. p t s t E = ( TT ) E = PE = QE z with : P = T T = Q (.9) QE = β E (.3) z E+ β E = (.3) z E l + βl El = (.3)

26 5 + iβlz -iβlz + E ( z) = E e + E e = E ( z) + E ( z) l l l l l (.33) Tpicall, for a matri of dimension Np Np, there are will be N p eigenvalues and of eigenvectors. So β and E will have N p set N p different set and we distinguish them b a subscript l with l = [, N p ] (Eq. (.3)). The solution of each eigenvector (from Eq. (.3)) has two propagating modes and can be written as Eq. (.33). The N p eigenvalues and set of eigenvector of Q can be arranged as Eq. (.34) and the Np Np matri S is defined. N p S = E E E E ij,, ij,, ij,, N p ij,, N p E E E E ij,, ij,, ij,, N p ij,, N p N p (.34) β = ( β β β β ) N p N p N p Now we have a bunch of information inside the matri S. First electric field components and E can be obtained through the Fourier coefficients (i.e. E N p element column vector E ). While E satisfies Eq. (.9) and matri Q s elements are function of Fourier components of ε and /ε (i.e. the real space structure of the macroscopic medium) with Q s eigenvalues and eigenvectors listed at Eq. (.34). In general, an field Fourier coefficient at vector E is the superposition of all the eigenvectors and there are two propagating directions for each eigenvector (Eq. (.33)). So E can be epressed in term of the superposition of two set of + propagation eigenvectors (Eq. (.35)) or in a more concise form Eq. (.36). Here E ( z) and E ( z) are both N p vectors of unknown coefficients which represents the weight of each of N p sets of eigenmodes or eigenvectors. Those coefficients are kept unknown until we have an initial incident condition for z =, for eample a planewave incident is defined as

27 6 one of the eigenmodes in the uniform medium. With the initial incident condition set, those coefficients can be obtained for other locations in stead of the incident surface and in turn the field vector can be obtained through those coefficients and the eigenmodes (eigenvectors) of the medium b transfer or scattering matri which will be discussed later. E E E E ij,, ij,, ij,,n ij,,n E ij, + ( ) ij,, ij,, ij,,n ij,,n l E ij, N p E = = E E E E E z + E E E E ij,, ij,, ij,,n ij,,n E E E E E ( z) ij,, ij,,n ij,,n l ij,, N p N p (.35) + + E E = S( E + E ) = ( S, S) E (.36) The magnetic field s Fourier coefficient vector H (Eq. (.6)) can be epressed in term of E + and E via the first equation of Eq. (.7) after one step transformation (Eq.(.37)). Here in the equation, the operation of for each column. T β means the l th column of T will multiple β l E H = T E = T ( S, S) z z E + + E E = it β ( S, S) = ( T, T) E E + (.37) Then we can combine Eq. (.36) and (.37) to get a more concise epression of relation between field Fourier coefficient and the weight of eigenmodes in Eq. (.38).

28 7 + Ez ( ) S S E ( z) H( z) = T T E ( z) (.38) We agree it is a length and boring derivation to get Eq. (.38), but there are still a few steps from here to the final transfer (scattering) matri algorithm. + iβh + E ( z+ h) e E ( z) = h (.39) iβ E ( z+ h) e E ( z) The left hand side and right hand side of Eq. (.38) are for the same z position. If the macroscopic medium is uniform along z ais, then the eigenmodes (eigenvectors) (i.e. S or T ) remain the same and weight of the each eigenmode will remain the same but with additional phase factor which is illustrated at Eq. (.39). We usuall set the incident surface position as the z = and the incident wave vector is a set of numbers, for eample the zero order planewave incidence onl have i =, j = elements nonzero and all other elements are set to zero..4 Fourier space Mawell s Equations for uniform medium At previous section, we discussed in detail on how to get the relation (Eq. (.38)) between the field Fourier coefficients ( E and H ) and the weight of eigenmodes ( E + and E ) for an XY double-periodic structures while uniform along z ais direction. Now in this section, we are going to deal with an easier case, the structure of XY plane is uniform medium. For uniform medium, we do not need to do Fourier transformation to the dielectric function. The Mawell s Equations (Eq. (.8)) for uniform medium can be written as Eq. (.4) where ε is the dielectric constant of the uniformed medium. Now with one step further (Eq. (.4)), we

29 8 can get the vector wave equation epressed as Eq. (.4) where n = ε, the refractive inde of the uniform medium. Then the planewave solution can be found from Eq. (.4). H() r = Er ( ) ik Er () = ikεe() r ik (.4) ( Er ()) kεer () ( ) ( i ) = Er = Er Er () () () idr () = iεer () = ε ier () = (.4) ( ) ( nk ) Er + Er = () () Hr () = E() r ik ω k k k k c = = + + z β ij = nk kij, kij, (.4) If the same definition of column vector of E and H (Eq. (.6)) is used, then the relation of E and H are epressed at Eq. (.43), with S unit matri and T block diagonal matri. + + E( z) S S E ( z) S S Ω ( z) = H( z) T T E ( z) T T Ω ( z) S T ij = I = k β ij, ij, ij, ij ij k k k β k + β k k ij, ij ij, ij, (.43) After the above relation is obtained, we can go to the net stage of building transfer and scattering matri along the propagation direction for uniform or XY double periodicit structures.

30 9.5 Building the transfer matri and scattering matri The transfer (scattering) matri method is based on the linkage between neighborhood tangential field vectors. For an arbitrar 3D problem, we can alwas cut the 3D problem to D slices along the propagation direction and assume the dielectric distribution is onl a function of coordinate and within each D slice (i.e. the dielectric material along z direction is constant). Some structures can satisf this assumption eactl and naturall, for eample: the laer-b-laer woodpile structure in which each laer has the same dielectric distribution along z direction. Some structure can approimatel satisf this assumption if the slices are thin enough along the z direction, for eample: those structures with continuous change dielectric distribution along z direction. At Figure - the scheme of transfer matri method is illustrated. For each D slices, we assume there are two infinitel thin air (or an other uniform medium) films around it. These artificial air films will make no impact on the problem because the thickness of them is set to be zero. The purpose of those artificial air films is to connect tangential component of the electromagnetic field throughout the whole structure. And in turn get the transfer matri and scattering matri of the whole structure. The fact that the tangential components of electric and magnetic field are continues is the ke to get the field vector connected between neighbors. At the air films and within the slices of left hand side of slice i (position zi ), we have the boundar condition matched at Eq. (.44). At the air films and within the slices of right hand side of slice i (position z i ), we have the boundar condition matched at Eq. (.45). Also there is a connection between the vector within the slice s left and right boundar Eq. (.46) in which the thickness of the slice is h.

31 Z + Ω + Ωn Ω Ωn + Ωi zi z i + Ωi Ωi Ωi Slice i Figure -: Scheme of transfer (scattering) matri method ( ) ( ) S S S S E z + + Ωi i i i i T T = T T Ω i i i E z i i ( z ) ( z ) + + S S i i Ei i S S Ωi T T = T T i i Ei i Ωi (.44) (.45) + iβh + Ei ( zi+ ) e Ei ( zi) = iβ h Ei ( zi+ ) e Ei ( zi) (.46) with : z = z + h i+ i From the above three equations, we can eliminate E + i and E i, and get a relation between the left air film and the right air film at Eq. (.47). We can define T i as the transfer matri for

32 the i th slice. And the overall transfer matri ( T matri) for the whole structure is given b Eq. (.49). + iβh + Ω a a e a a i Ωi = a a a a iβh Ω e i Ωi = ( + i i ), = a S S T T a a (.47) = ( i i ), = a S S T T a a T t t a a e a a i i iβh t t a a e a a = = i i i iβh + + Ωi Ωi = Ti Ω Ω i i (.48) T TT T iii T (.49) = n n- n- But the T matri has been proved to be numericall unstable when the structure along the propagation direction is thick, which is due to the fact that the evanescent wave components in the planewave epansion will increase eponentiall if entire T matries are multiplied. In other word, the eponential increase term accumulation makes the poor numerical stabilit for T matri. One solution to the T matri problem is adopting the scattering matri (S matri) method epressed at Eq. (.5). The overall S matri can be found b connecting S matri at each slice through an iteration algorithm. Suppose the first n S matri is S n- and the S matri for the n th slice is n s, then the S matri for the total n slices will be n S which is given b Eq. (.5). In the real calculation, we first set S matri to be I, the identit matri, which represents S matri of an air slice, then use the iteration algorithm to get the total

33 S matri of the first slices and I. Then appl the iteration algorithm to all the other slices to get the total S matri for the whole structure. p = [ a e a a e a ] iβh iβh p = a e a iβh t = a e a [ a e a a e a ] iβh iβh iβh t = a i i s s p t + pt pt + p t Si = i i = s p s t + p t pt + p t (.5) + + Ωi Ωi = Si Ωi Ωi S = s [ I S s ] S n n n n n S = s + s S [ I s S ] S n n n n n n n S = s + S s [ I S s ] S n n n n n n n S = S [ I s S ] s n n n n n (.5) Finall, we obtained a numerical stable scattering matri S for the whole structure. And the S matri has the internal electrodnamics properties of this particular structure. Tile now, we are onl transforming the Mawell s Equations with certain dielectric distribution. There are still no initial conditions or boundar conditions (ecept the XY double periodicit) applied to our structure. For different purpose, such as spectrum, band diagram or mode profile, we can appl desired initial condition and boundar condition to the S matri..6 Using scattering matri for various applications In this section, we will discuss several direct applications from the calculated S matri to get the spectrum, the band structure and the electric and magnetic field distributions (mode

34 3 profiles). Detailed derivations are supplied with actual calculation eamples from published results. More applications can be added to this transfer (scattering) matri scheme for future development..6. Spectrum from S matri Now let s review Figure - and suppose that the S matri for the whole structure is known. Then we have relation Eq. (.5) which connects the field column vector at left and right side of the photonic crstal structure. To get the spectrum, we need a boundar condition: the + incident electromagnetic wave E can be epressed b Ω (for a simple planewave incident onl the center element ( i =, j = ) is set to and all the other elements are set to ) and Ω n is set to zero because there are no propagating waves to the structure from left hand side. With this boundar condition and the knowledge of S matri, we can get the column vector + Ω n and Ω which represent the transmission and reflection field component vectors. + + Ωn Ω = S Ω Ωn E = S E, E = S E t r (.5) After the transmission and reflection coefficient are obtained, we can get the electric field and magnetic field. The transmittance or reflectance rate is the ratio of energ flu of transmission wave or reflection wave toward incident wave. The energ flu is proportional to the Ponting vector P= E H. The ratio of Ponting vector magnitude gives the transmittance or reflectance. The absorptance is defined as one minus the transmittance minus the reflectance. Those relations are summarized at Eq. (.53) with the summation of ij for lateral wave vector k + k k in which all the modes are propagation mode. ij, ij,

35 4 k E H = k t t t H ij βij H ij E kz E T = E = E R = E = r r r H ij ij H ij E kz A= T R E E β (.53) To show an eample of our transfer (scattering) matri calculation result, we select one tet book structure: laers of alternating slabs, one has dielectric constant 3 and the other has dielectric constant, both has thickness.5a where a is the lattice constant. Figure -3 is our transfer (scattering) matri result which shows a band gap between normalized frequencies around.5 to.5 which is consistent with the tet book result.. Transmission/Reflection Rate Transmission Reflection Normalized Frequenc (ωa/πc) Figure -3: Spectrum of D photonic crstal

36 5.6. Band structure from S matri There are several methods to get band structure from the S matri, Dr. Zhi-Yuan Li mentioned three different schemes. Here I onl discuss the one which is used in m transfer (scattering) matri simulation package. To get the photonic band structure, we need impose a periodic boundar condition along the stacking direction (i.e. the selected wave propagation direction) to simulate infinite structure,. According to Bloch s theorem, the relation of field components at position r and position r+ R satisfies Eq. (.54) with R the periodicit. u( + ) = e ik i r R R u( r) (.54) If we take the desired direction alone a 3 as the role of R to calculate the band diagram alone this particular direction, then the S matri between point r and point r+ R can be obtained according previous sections. After the S matri is read, we can write out the field components at position r and r+ a 3 as in Eq. (.56) from the Bloch s theorem. Rearrange Eq. (.55) and Eq. (.56), we can get Eq. (.57) which is a standard generalized eigenproblem A = λb with A and B are both square matrices. + + Ω Ω = S (.55) Ω Ω + + Ω Ω 3 e = ikia Ω Ω (.56) + + S Ω ikia I S 3 Ω e S I = S Ω Ω (.57)

37 6 In this approach, normalized frequenc ω and lateral Bloch s vector k and k are given eplicitl as input parameters, while k z is left to be determined; or in function form: k = f( ω, k, k ). To find k z, we adopted a widel used on shell approach: for z 3 propagating mode, k z must be a real number which impl that the eigenvalue of i must be a comple number of unit modulus. In the calculation, first we solved the eigenvalue problem Eq. (.57), then pick up all the eigenvalues of module equals one, and then get k z from corresponding eigenvalues. e ik a When we solve the band diagram b this approach, the material to build the photonic crstal can be dispersive which due to the fact that each solution is for individual frequenc point. One disadvantage of this approach is that: for each direction along the Brillouin Zone, we need calculate S matri and solve eigenvalue problem individuall which means we need to cut the 3D photonic crstal through different directions to get a complete band diagram. To illustrate the usage of our simulation package, we repeat the calculation of D photonic crstal with alternating material slab: one laer with dielectric constant 3 and thickness.a and the other laer with dielectric constant and thickness.8a where a is the lattice constant. Consistent result is obtained when compared with our TMM results...8 Frequenc (ωa/πc) Wave vector (ka/π) Figure -4: Band diagram of D photonic crstal

38 7.6.3 Field mode profile from S matri One other propert we are interested in is the field mode profile across the photonic crstal structures. To get the electromagnetic field distribution at an intermediate plane within the photonic crstal structure, we need to use the two-side S matri formulation. For an given intermediate plane Z a, the whole photonic crstal structure is divided into two parts: the front part and the back part with each part have one S matri: S for front part and S for back part. Figure -5 gives a sketch of the two-side S matri scheme. We can find that the arbitrar intermediate field vectors E + a and E a are related to the E, Et and E r through Eq. (.58) from which we can get Eq. (.59). z a E E r E a + E a Et S S Figure -5: Two-side S matri scheme E S S E + a = Er S S Ea + Et S S Ea = Ea S S (.58)

39 8 E = S E + S E + a a E = S E + S E E E r t = S E + a = S E + a a a (.59) Combining first and forth equation of Eq. (.59), we can get the first equation of Eq. (.6) which is a standard linear algebra problem ( A E + a, we can use the second equation of Eq. (.6) to calculate = B) and can be solved easil. After getting E a. ( I S S ) Ea = S E E + = S E + a a (.6) E + a and E a are the column vectors of E ± ij, for the desired slices, or the plane perpendicular to the propagation direction. Then we can use Eq. (.6) to calculate the electric field of and components which are at the infinite thin air film adjacent to the desired plane. According to the boundar condition, the tangential component is continuous, the electric field of and components at the photonic crstal structure are the same as in the infinite thin air film. N N + ( ) mn, mn, E () r = E + E e m= N n= N N N + ( ) mn, mn, E () r = E + E e m= N n= N N N + ( ) mn, z mn, z E ( r) = E + E e z m= N n= N i( kmn, + kmn, ) i( kmn, + kmn, ) i( kmn, + kmn, ) (.6) To get electric field z component at the air film, we need use Mawell s Equations id= i ε E= which leads to i E =. So E ij, z can be found easil b Eq. (.6).

40 9 E = k E + k E (.6) ( ) ij, z ij, ij, ij, ij, kij, z But Ez( r ) inside the photonic crstal structure is not same as in the infinite air film. To get correct Ez( r ) inside the photonic crstal structure we need appl the boundar condition of the continuit of D z field: Dz = Dz εez = εez Ez = ε ε Ez. But we can not appl this condition directl in real space; instead we need to appl it to the Fourier space. Here we use the Fourier epansion of ε ij instead of ε for this problem. Finall we can get ij the electric field z component vector as the last equation of (.63), then we can appl the last equation of (.6) to get the electric field distribution of Ez( r ) inside the photonic crstal structures. For magnetic field H, we can first tr to obtain the H field vector of, components at air film through the E field vector b Eq. (.64). Then the z component can be figured out b i H = or Eq. (.65). Then follow the similar procedure as Eq. (.6) to get the magnetic field distribution inside the photonic crstal of all three components. ij = ε = E ij PhC ij, z ij mn ( z) e ij, z i( kij, + kij, ) i m, j n mn, z i( kij, + kij, ) i( kij, + kij, ) air E ( z) e ε e ε E air ( z) e i( kij, + kij, ) E z = ε E z PhC ( ) air ( ) ij, z i m, j n mn, z mn ij ij (.63) + + Hij, kij, kij, kij, βij Eij, = + H ij, kβ ij kij, βij kij, k + ij, E + ij, Hij, kij, kij, kij, βij Eij, = H ij, kβ ij kij, βij kij, k ij, E + ij, (.64)

41 3 H = k H + k H (.65) ( ) ij, z ij, ij, ij, ij, kij, z Now I am going to close this review chapter of introduction for transfer (scattering) matri method. It is onl the core knowledge of this numerical tool. There are a lot of other aspects to make this method a good simulation tool, such as appling structure smmetr, introducing perfectl matched laer boundar condition instead of periodic boundar condition, pre calculation analsis and post calculation analsis, various input waves instead zero order plane wave (for eample higher-order plane wave, Gaussian wave, waveguide eigenmode etc.) and the fleibilit to deal with different materials (such as dispersive, anisotropic, or magnetic materials). Some of those topics will be discussed in this thesis and some are not. It is still a developing method, and new concepts and strategies are ver likel to be introduced soon. References:. John D. Joannopoulos, Photonic crstal Molding the Flow of Light, Princeton Universit Press, 995. Steven G Johnson, Photonic crstal The road from theor to practice, Kluwer Academic Publishers, 3. J. D. Jackson, Classical Electrodnamics, 3rd edition, Wile & Sons Press, 4 4. M. Born and E. Wolf, Principles of Optics, 7th edition, Cambridge Universit Press, Z. Li and L. Lin, "Photonic band structures solved b a plane-wave-based transfermatri method", Phs. Rev. E 67, 4667, (3) 6. L. Lin, Z. Li and K. Ho, "Lattice smmetr applied in transfer-matri methods for photonic crstals ", J. Appl. Phs. 94, 8 (3)

42 3 7. Z.Y. Li and K.M. Ho, "Application of structural smmetries in the plane-wave-based transfer-matri method for three-dimensional photonic crstal waveguides", Phs. Rev. B 68, 457 (3) 8. Z.Y. Li and K.M. Ho, "Light propagation in semi-infinite photonic crstals and related waveguide structures", Phs. Rev. B 68, 55 (3) 9. Z.Y. Li and K.M. Ho, "Analtic modal solution to light propagation through laer-blaer metallic photonic crstals", Phs. Rev. B 67, 654 (3). Z.Y. Li and K.M. Ho, "Bloch mode reflection and lasing threshold in semiconductor nanowire laser arras", Phs. Rev. B 7, 4535 (5)

43 3 Chapter 3. Interpolation for spectra calculation One of the most important applications of photonic crstal is introducing point defects (i.e. cavit) into the pure photonic crstal structure to make a photonic crstal resonant cavit which can be widel used in solid state laser (acting as the resonant cavit) and communication industr. The advantage of photonic crstal resonant cavit compared with conventional resonant cavit is: first due to the present of photonic crstal background, the Q value can be ver large; second the size of the resonant cavit can be ver small and compact. Those two properties is the ke to improve the solid state laser performance and properties. And how to efficientl and accuratel get the eact resonant peak frequenc, the Q value and electric field mode profiles are essential for laser resonant cavit design. The planewave based transfer (scattering) matri method is a frequenc domain method and spectra can be calculated at an arbitrar resolutions, i.e. arbitrar small or large frequenc steps. There are mabe several resonant peaks inside the photonic band gap frequenc region, and we have no idea where those resonant modes located and what is the Q value before the whole spectra is obtained. To get all possible resonant modes, ver high spectra resolution is required, ideall continues spectra are preferred. The calculation for each frequenc data point is almost identical, so increase the resolution will increase the calculation time linearl. We can not get continues spectra due to the calculation time limitation. Tpicall, getting around less than frequenc data points in the band gap range of the spectra is acceptable in term of time consumption for 3D structures. For a tpical transmission spectrum through a GaN woodpile photonic crstal structure with the band gap ranges from around.5 μ m to.6 μ m, if the resonant mode of Q value is around, and resonant frequenc.55 μ m, then we need at least the spectrum to have a

44 33 resolution of μ m or 88 data points to barel see this resonant peak. A Q value of, is onl moderate in photonic crstal cavities, recentl S. Noda reported a photonic crstal cavit with Q value as large as,,. To get a wa out of this challenge, we introduced interpolation into our planewave based transfer (scattering) matri method. In this chapter, the theoretical foundation is discussed with one eample of application of this concept to the woodpile laer-b-laer photonic crstal cavities. 3. Wh interpolation works? As we have alread known that all the spectra from Mawell s Equations are Lorentzian shape which can be determined b several parameters, one naturall asked question is can we use as few as possible spectrum data points to get an acceptable estimate of those parameters and in turn get the analtical form of the spectrum? There are two tpes of methods to answer the above question, one is interpolation and the other is regression. Regression is a statistic wa to figure out the unknown parameter b minimizing the fitting error. Tpicall a large set of data is required to get good results. In our case, we know the spectra are Lorentzian and we have to use the so called non-linear regression strateg. It ma help a little bit in the analsis with adequate data points, but this is not we want to do. Can we just use a few data points while still get accurate results? The answer goes to interpolation. Interpolation uses onl ver few data points but with the knowledge of what the function form (with a few undecided parameters). It just like solve a function with unknown variables: for eample if there are unknown parameters ( ab, ) in one function = f( a, b, ) with the form f ( ab,, ) eplicit, we can onl use two set of (, ) to get the unknown parameter. But this is onl true for those two set of (, ) are eactl acquired from eperiments or calculation. Error is usuall introduced for eperiments and certain numerical simulations. The biggest difference between regression and interpolation is: the interpolated function will go through each data points while the regressed function ma or ma not go through each

45 34 data points. There are two ke points to use interpolation for the planewave based transfer (scattering) matri method spectra data points: first we know the eact spectra functions form (Lorentzian shape); second each data points are accurate enough. Now let s start from the most general form of the superposition of multiple peak Lorentzian functions Eq. (3.) with {,,, } a b c d total 4m independent unknown parameters. With a few i i i i steps, the multiple peak Lorentzian function can be written in term of rational function Eq. (3.) with total of 4m + unknown parameters. With one additional redundanc parameter, we can adopt a widel used, robust and efficient interpolation / etrapolation algorithm for diagonal rational functions. The Bulirsch-Stoer algorithm of the Neville tpe produces the so called diagonal rational function with the degrees of numerator and denominator equal (if m is even) or with the degree of the denominator larger b one (if m is odd). One other advantage of using the diagonal rational function interpolation is that in real transmission spectra, there are ma be some non-lorentzian feature which can be handled b rational function instead of Lorentzian function (detailed discussion will be on later sections). m b i f( ) = ai + i= ( ci) + d (3.) i m a i ac i i + ( ac i i + ad i i + b) i f( ) = i= c i + di m P3, i + P4, i+ P 5, i = i= + P, i+ P, i = m i p i i= m m + i i= q i (3.)

46 35 3. An eample of application of interpolation This section is modified from a paper published at Optics Letters Vol. 3. No. (page 6) at Januar 6 with title: "High-efficienc calculations for three-dimensional photonic crstal cavities", b M. Li, Z. Li, K. Ho, J. Cao, and M. Miawaki. Eperimental and numerical studies of photonic crstals (PC) have been eperiencing eponential growth for more than a decade and numerous scientific and engineering advances have been made. However, with the eception of PC fibers, ver few concepts have been able to pass from the scientific research stage to high throughputs mainstream products. Besides the challenges in manufacturing, one of the main reasons behind this situation is the lack of efficient and versatile numerical simulation tools for PC structures, especiall defective three-dimensional (3D) PC structures which can be used for waveguides,3,4 and resonant cavities. 5,6 There are alread established numerical simulation methods. 7,8,9 However, serious consideration of the trade-off between the targeted result accurac (e.g. resolutions etc.) and the projected computation time is still a dail dilemma faced b researchers working on 3D PC devices. In this letter, we present methods that can minimize this trade-off for cavit embedded 3D PCs. The approach includes a planewavebased transfer matri method (TMM) and a robust rational function interpolation/etrapolation implementation., A significant increase in speed with high numerical accurac for modeling 3D PC cavit modes was demonstrated based on this approach. When an incident electromagnetic wave is directed toward a slab of 3D PC, the transmission rate through it should be eponentiall attenuated across the whole frequenc range of the directional band gap, which contains the full photonic band gap. 3 When there is a cavit mode in the 3D PC slab, the resonant transmission through the cavit will result in a

47 36 Lorentzian shape peak in the transmission power spectrum. The peak frequenc corresponds to the cavit mode s frequenc, and the ratio between the peak frequenc and the FWHM (full-width half-maimum) of the peak corresponds to the mode s Q value. 4,5 The stationar electromagnetic field distribution through out the volume is the cavit mode shape, when the incident light is set at the cavit mode s resonant frequenc. Therefore, one can in principle characterize ever aspects of individual cavit modes based on such transmission calculations. At first sight, however, since we do not know the number of cavit modes and their frequenc positions, it appears that for high Q cavit modes corresponding sharp transmission spectral peaks, a large number of frequencies have to be calculated to resolve all the modes in the band gap. 5 Unit Cells 5 Unit Cells Cladding Laers z L: Cavit Length Y X Cavit Embedded Figure 3-: A 5-b-5 sized PC cavit supercell structures (left) and crosssection of the cavit laer (right) To solve this practical challenge, an interpolation/etrapolation strateg was deploed. Unlike the comple frequenc domain root searching approach emploed before (e.g. reference 6, and similar method has been used in D slab PC cavit structures), we utilized a full global deterministic real frequenc domain rational function interpolation/etrapolation formalism: the Stoer-Bulirsch algorithm which does not require eplicit initial conditons., A sum of Lorentzian peaks which is the general form that an

48 37 resonant transmission spectrum should follow is simpl a sub-class of diagonal rational functions. Therefore, ideall one can analticall etract ever spectral detail at an resolution, once the spectral values at M=4N+ frequenc points are known, for a spectrum that contains equal or less than N resonant peaks. The sampling of those M frequenc points can be arbitrar. This fact eliminates the great practical challenge of searching narrow bandwidth resonant peaks across the wide bandwidth span of a photonic band gap. As a demonstration, we performed numerical simulation for a 3D laer-b-laer PC structure 7 illustrated in Figure 3-. The refractive inde of the rod material is set at 3.5. Both w/a and d/a ratios are.39, where w is the width of each rod in the - plane, d is the thickness of each rod along the z-orientation, and a is the pitch between rods in each laer or is referred as lattice constant in some literatures. The left side of Figure 3- is a 3D illustration of a 5-b-5 sized supercell embedded with an optical cavit, where the supercell size is 5a 5a in the - plane. In this stud, to characterize the numerical influences of finite supercell sizes, we calculated multiple supercell sizes, including 3-b-3, 4-b-4, 5-b-5 and 6-b-6. All structures in this stud (ecept otherwise specified) have laers along the z- orientation and the optical cavit is formed b removing a section of rod in eactl one lattice constant (a) in the th laer, which is illustrated b a cross-section diagram shown on the right side of Figure 3-. The transmission spectrum for the 3-b-3 unit cells with a z-oriented incident beam (both - and - polarizations) at discrete sampling frequencies was calculated b TMM, shown as the scattered square smbols in Figure 3-. Firstl, 3 evenl spaced frequenc points were calculated, and the estimated error term from the interpolation of those 3 points indicated more data points were required around normalized frequenc.44 and.445. Then 8 additional frequenc points were calculated around.44 and.445, and interpolation repeated with a total of frequenc points. The -polarization incident does not show an resonant feature throughout the whole directional band gap. The solid blue line in Figure 3-

49 38 is the result of the, output points for the -polarization from the data interpolation. The green line in Figure 3- is the estimated error term generated b the interpolation routine itself., This estimated error term not onl provides a direct gauge of the reliabilit of each interpolation, but also provide a direct indication of where to add more input data points if necessar f = / Confirmation Points.6.4. Q=9, Transmission Rate ,-Point High Resolution Spectrum Build-in Error Term for Interpolation Original points b TMM Normalized Frequenc (a/λ) Figure 3-: The z-directional transmission spectrum (blue line) across the full bandgap range with its estimated error term (green line). The upperleft inset shows a zoom-in linear plot of the Lorentzian resonant transmission peak, with its peak frequenc and Q value labeled. The upperright inset shows a zoom-in view of an asmmetric transmission feature, with 6 confirmation TMM frequenc points (purple crosses).

50 39 The upper-right inset of Figure 3- is a zoom-in view of a sharp and non-lorentzian feature near the normalized frequenc It also shows 6 independent confirmation frequenc points (purpule crosses) calculated b TMM. The numericall perfect match between these 6 confirmation data points and the,-point high resolution spectrum (blue line) is a direct proof of the accurac of the rational function interpolation procedure. This non- Lorentzian feature is not a localized cavit mode which is confirmed b its mode shape calculated in TMM. 8 Namel, its mode shape reveals strong concentration of the electromagnetic field at the two air/pc interfaces, instead of the cavit itself. Although we don t have definite analsis of this specific resonance et, these non-lorentzian resonant features are not rare and are also observed in other simpler grating sstems. 9 Such numericall stable and high accurac performance of the interpolation routine is partiall due to the fact that TMM is a frequenc domain calculation method. Unlike time domain simulation methods (e.g. Finite-Difference Time-Domain (FDTD)), the individual power spectrum data point calculated from frequenc domain methods is the stationar result after the evolution of infinitel long time, without the influences of finite time span convolution effects and/or transient effects. This is the reason wh such interpolation can numericall work across a wide bandwidth (e.g. covering the full band gap range in one run), while methods such as the Pade approimation used in conjunction with FDTD programs can onl cover much smaller bandwidth for each run in order to correct the convolution effects. The upper-left inset of Figure 3- shows a zoom-in view of the Lorentzian peak near normalized frequenc.44, which corresponds to a localized cavit mode of interest. Plotted in linear scale at the spectral resolution Δf=6-7, the eact details of the cavit mode are resolved and characterized as its resonant frequenc f =.4494 and Q value of.98 4 as labeled in the inset. To characterize the influences of the finite supercell sizes prescribed b TMM, we also calculated the transmission spectra when increasing the supercell size from 3 3 unit cells

51 4 through 6 6 unit cells. Figure 3-3 shows high resolution (Δf=6-7 ) spectra for the 4 different supercell sizes. The cavit mode properties (resonant frequenc and Q value) etracted from the spectra (in Figure 3-3) are shown in Figure 3-4 (a). For supercell sizes larger than 4 4 unit cells, the cavit mode resonant frequenc has alread stabilized within an uncertaint range of ±.%. Also shown in Figure 3-4(a) is the Q value which stabilizes between 4 and 5 4 even quicker than the resonant frequenc. Due to the periodic boundar condition used in TMM, the change of the super cell size is a change of the cavit laer s specific geometr. Figure 3-4 (a) shows that the Q value is not sensitive to the specific geometries of optical cavities embedded in 3D PCs. This result is in agreement with results reported in reference 6 and : the Q values of cavities with different sizes and shapes embedded in the same 3D PC structure do not change too much Unit Cells Power Spectrum Normalized Frequenc (a/λ) Figure 3-3: The spectra for varing supercell sizes

52 4 This is quite different from the usual behavior of D membrane PC cavities where radiation loss into the light cone plas a pivotal role in the cavit loss.,3 In 3D laer-b-laer PC cavities, the most significant factor determining the Q values is the number of cladding laers along the z-orientation. 6, Figure 3-4 (b) shows the Q value of cavit mode increase monotonicall and almost eponentiall as the number of cladding laers increasing along the z-orientation. Figure 3-4: (a) Etracted from the spectra in Figure 3-3, the cavit resonant mode frequenc and Q value var and stabilize, as supercell size increases. (b) The cavit mode resonant frequenc and Q value are plotted as functions of total number of laers along the z direction. Our method uses relativel moderate computer resources. For instance, the whole high resolution spectrum through the 5-b-5 structure shown in Figure 3-3 can be obtained in less than three hours with a 4-CPU (Intel Xeon TM 3. GHz) computer cluster. The interpolation takes less than one second. Another advantage is that increasing the total number of laers along the z-direction does not increase the computation effort ver much due to the repeating laer structure pattern. In conclusion, we have proposed and numericall demonstrated a highl efficient numerical modeling method for 3D PC cavit structures, based on a combination of TMM and rational function interpolation, which can be used at other D or 3D PC structures.

53 4 3.3 Origin of Lorentzian resonant peaks Although it is well known that the resonant peaks of electromagnetic wave inside photonic crstal are Lorentzian form. It is not trivial from first sight; actuall it is the propert of general wave equations. When there is interference between continuous modes and localized modes, more complicate Fano peaks ma be present. Both Lorentzian peaks and Fano peaks can be interpolated through our rational function interpolation discussed in the first section of this chapter. In this section, detailed derivation of the origin of Lorentzian resonant peaks is discussed. We start from general wave propagation case. The loss (i.e. lifetime, qualit factor, Q) of a resonant mode can be understood as the summation of the coupling of this resonant mode to all radiation modes outside the cavit (Eq. (3.3)) where n is summarizing over all radiation modes, which include the outward propagating waveguide modes, if waveguides are used in the vicinit of the cavit. = (3.3) Q Q tot n n In a coupling Q eperiment, the right hand side of Eq.(3.3) can be divided into two parts which are epressed at Eq. (3.4) where the first term stands for all of the injection channels (modes) being used in the coupling Q eperiment, and the second term stands for all other radiation modes, where no injection are presented. c = + (3.4) Q Q Q tot n= n n ~ c n

54 43 So, let s simplif the notations and rewritten as Eq. (3.5) where Q c and channels coupling Q and the Q for rest of the radiation modes. Q rec are injection c = + = + (3.5) Q Q Q Q Q tot n= n n ~ c n c rec Therefore, according to the general definition of Q, without an injection, the energ in a cavit resonant mode decas with time (Eq. (3.6)) where Wt () is the energ in the resonant mode, ω is the central frequenc of the resonant mode itself with the combined effect of all coupling loss mechanisms. t Q W() t = W() t e ω tot (3.6) t= It can be proven that the power spectrum coupled into the cavit mode will be Eq.(3.7) which is a Lorentzian line with peak value epressed at Eq. (3.8) and FWHM epressed at Eq. (3.9). S( ω) = (3.7) Q ( ) c Qres ω ω ( + ) + 4 Qc Qres ω S = S( ω) = (3.8) ω ω 4 ma = Q c Qres ( + ) Qc Qres ω Δ ω = ω ( + ) (3.9) Qc Qres Qtot Now let s go back to prove Eq. (3.7) b starting from the general oscillation problem epressed at Eq. (3.) where S + is the injection channels amplitude. S + is the normalized power flow delivered b this supermode, for eample in the unit of Watt. Then we use letter a to represent the amplitude of the resonant mode we are coupling to. Or, it s the peak value on the resonant mode profile. And it s normalized that a equals the total energ in the mode in the unit of Joule. Eq. (3.) is a loss oscillator (/ τ tot ), driven b an eternal source

55 44 ( κ S+ ) where κ is the coupling strength between the injection channels and the resonant mode. When the driven term is turned on at frequenc ω with the time dependence of the solution of equation Eq. (3.) is found at Eq. (3.) j t e ω, da dt = jω a a+ κ S+ (3.) τtot a = κ = κ S+ j( ω ω ) + τtot τ c (3.) Then Eq. (3.) can be simplified to Eq. (3.) given the arbitrar constant phase difference between S+ ( t) and at () can be fied to make the constant κ be a real number. Simultaneousl, the phase constant for S ( t) is also fied due to this choice. With the energ conservation relation (Eq. (3.3)) and Eq. (3.4), we can obtain Eq. (3.5). da dt = + (3.) jω a τ a τ S+ tot c da + dt τ res S S = + a (3.3) S t = S t + a t (3.4) () β + () τ () c S t = S t + a t (3.5) () + () τ () c Finall with Eq. (3.) and Eq. (3.5), the reflection coefficient of the resonator sstem can be written as Eq. (3.6) and the power spectrum coupled into the cavit sstem can be found as Eq. (3.7), i.e. we proved Eq. (3.7). S S + (/ τ c) (/ τres) j( ω ω ) = (/ τc) + (/ τres) + j( ω ω ) (3.6)

56 45 S S+ Q ( ) c Qres ω ω ( + ) + 4 Qc Qres ω (/ τ ) (/ τ ) j( ω ω ) (/ τc) + (/ τres) + j( ω ω ) c res S( ω) = = = (3.7) References:. J. C. Knight, T. A. Birks, P. St. J. Russel, and D. M. Atkin, "All-silica single-mode fiber with photonic crstal cladding," Opt. Lett., 547 (996). A. Mekis, J.C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J.D. Joannopoulous "High transmission through sharp bends in photonic crstal waveguides," Phs. Rev. Lett. 77, 3787 (996) 3. SY Lin, E Chow, V Hietala, PR Villeneuve, and JD Joannopoulos, "Eperimental demonstration of guiding and bending of electromagnetic waves in a photonic crstal," Science 8, 74 (998) 4. A. Chutinan, and S. Noda, "Highl confined waveguides and waveguide bends in three-dimensional photonic crstal," Appl. Phs. Lett. 75, 3739 (999) 5. E. Özba, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, "Defect structures in a laer-b-laer photonic band-gap crstal," Phs. Rev. B 5, 396 (995) 6. M. Okano, A. Chutinan, and S. Noda, "Analsis and design of single-defect cavities in a three-dimensional photonic crstal," Phs. Rev. B 66, 65 () 7. J. B. Pendr, "Photonic Band Structures," J. Mod. Optic. 4, 9 (994) 8. A. Taflove, and S. C. Hagness, Computational Electrodnamics, Artech House, MA, 9. M. G. Moharam and T. K. Galord, "Rigorous coupled-wave analsis of planargrating diffraction, " J. Opt. Soc. Am 7 8 (98). L. L. Lin, Z. Y. Li and K. M. Ho, "Lattice smmetr applied in transfer-matri methods for photonic crstals," J. Appl. Phs. 94, 8 (3). W. H. Press, Numerical Recipes, Cambridge Universit Press, Cambridge, UK, 99

57 46. J. Stoer and R. Bulirsch, Introduction to Numerical Analsis, Springer-Verlag, New York, E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Phsics and Electronics," Phs. Rev. Lett. 58, 59 (987) 4. R. B. Adler, L. J. Chu, R. M. Fano, Electromagnetic Energ Transmission and Radiation, John Wile & Sons, New York, H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, New Jerse, Ph. Lalanne, J.P. Hugonin, and J.M. Gerard, "Electromagnetic stud of the qualit factor of pillar microcavities in the small diameter limit", Appl. Phs. Lett. 84, 476, (4) 7. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas and M. Sigalas, "Photonic band gaps in three dimensions: new laer-b-laer periodic structures," Solid State Commun. 89, 43 (994) 8. Ming Li, J.R. Cao, X. Hu, M. Miawaki, and K.M. Ho, "Fano Peaks in the Band Gap of 3D Laer-b-laer Photonic Crstal", to be published. 9. S. Fan, and J.D. Joannopoulos, "Analsis of Guided Resonances in Photonic Crstal Slabs", Phs. Rev. B 65, 35 (). S. De and R. Mittra, "Efficient computation of resonant frequencies and qualit factors of cavities via a combination of the Finite-Difference Time-Domain technique and the Pade approimation," IEEE Microw. Guided W. 8, 45 (998). S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, S. Noda, "Control of Light Emission b 3D Photonic Crstals, " Science 35, 7 (4). O. Painter, K. Srinivasan, J. D. O Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crstal slab waveguides," J. Opt. A-Pure Appl. Op. 3, S6 () 3. K. Srinivasan and O. Painter, "Momentum space design of high-q photonic crstal cavities," Opt. Epress, 67 ()

58 47 Chapter 4. Higher-order planewave incidence Photonic crstal itself is a structure with high smmetries due to its repeating pattern. The planewave input ma also have certain smmetries. Then the solution of the Mawell s Equations via the planewave based transfer (scattering) matri method ma have smmetr properties and it can lead to degenerac phenomena. For eample if we use the zero order plane wave incidence to ecite photonic crstal cavit with certain smmetr, not all the resonant modes can be ecited. In this chapter, we appl the general group theor to our method to stud this problem and introduce the concept of higher-order incidence as a solution. One detailed eample of the higher-order incidence is applied towards our classical laer-b-laer woodpile photonic crstal cavit arra. 4. Planewave incidence As discussed at the section.6. to get spectra from the calculated S matri, we must use certain incidence wave as input. And this incidence wave is represented b a column vector E of N p elements (Eq. (.5)). The simplest and widel used incidence is the zero order planewave incidence with two polarizations s polarization (or called e polarization) and p polarization (or called h polarization). The zero order e polarization is defined as E, =, E, = and all other elements are zero at vector E. The zero order h polarization is defined as E, =, E, = and all other elements are zero at vector E. Even with zero order incidence towards three dimensional photonic crstals, the relationship between the incident angles ( θ, ϕ ), polarization angle ( α ) and the phsical position of photonic crstal is usuall comple. To make those relations clear, we define the convention used in our method in this section.

59 48 First, the phsical real space coordinate is defined at Figure 4-; the rod of the laer-b-laer woodpile photonic crstal structure has its rods parallel to X or Y ais according to different laers. The origin point O should be at the center of the first laer of the photonic crstal, but to illustrate more detail, I shift the photonic crstal to the first quadrant of XY plane.) The Z ais is the cladding laer (through origin O and point out the paper in Figure 4-). In the illustration figure the woodpile laer-b-laer photonic crstal structure has all its laers with Z ais position larger than zero. Figure 4-: Coordinate sstem (Real object) Second, define the incident angle based on the geometr coordinate ( XYZ ) illustrated at Figure 4-. We adopt the convention widel used at various phsics sstems: the azimuthal

60 49 angle ( θ, ϕ ) is defined in Figure 4-. The wave vector k defines the incident direction. Line Ok projects on XOY plane giving OB. Z k θ Y O ϕ B X Figure 4-: Coordinate sstem (XYZ ais) Third, for each incident direction k, there are two independent polarization directions for the electric field vector. To uniquel define the two independent polarization directions, we first define an incident plane. The incident plane is the plane which contains the vector k and ais OZ which is the normal direction of the first photonic crstal laers (or the XOY plane). In Figure 4-, the incident plane is plane ZOBk. (Note: incident plane is alwas perpendicular to XOY.) Now, we can define the two independent polarization direction: e which is perpendicular to the incident plane and e which is in the incident plane and perpendicular to e and k. In fact e, e and k are orthogonal to each other and composes a right hand Cartesian coordinate. After we define the two independent polarization angles, we can define the polarization direction when the polarization angle equals to α where α is rotating counter-clockwise from the e ais (in Figure 4-3). Note: the k direction points out of e O ' e plane. We can use e, e and α to epress an polarizations via eα = (cosα) e + (sinα) e. The h polarization means its electrical field is parallel to e. And, the e polarization means its electrical field is parallel to e.

61 5 e (i.e. the e polarization case) O ' α e (i.e. the h polarization case) Figure 4-3: Polarization angle Fourth, now we have defined the incident angle ( θ, ϕ ) and polarization angle α and the relationship between them to the phsical position of photonic crstal at real space coordinate ( XYZ ). Now we can epress an incidence in term of the components at the XYZ coordinate sstem. We can realize this b two steps; first find the three components of e and e b Eq. (4.) and then appl eα = (cosα) e + (sinα) e. Finall we can get the, z, components of arbitrar planewave incidence e α (Eq. (4.)) at the XYZ coordinate sstem. e = ( sin ϕ) ˆ+ (cos ϕ) ˆ (4.) e ˆ ˆ = ( cosϕ cos θ) + ( sinφ cos θ) + (sin θ) zˆ e α, e α, e α, z = sinϕ cosα cosϕ cosθ sinα = cosϕ cosα sinφ cosθ sinα = sinθ sinα (4.)

62 5 Now, to get a feeling or what is reall looks like, here I also post some real pictures: the pencil represents the incident light, at the end of the pencil, we can see there are three arrows: the red arrow is e, the red arrow with black head is e and the blue arrow is electric field direction for a certain polarization angle α. Figure 4-4: Incident angle (Real object) Here is some special case of the incident angle: Figure 4-5 shows the normal incident case in which the origin O and θ =. O ' are at the same position; Figure 4-6 shows a special case where For each pair of incident angle, there are two independent polarizations can be chosen according to the convention of D photonic crstal: TE or TM mode. But the definition of TE and TM is not clear for 3D or D photonic crstal, even for D photonic crstal, people usuall have different conventions for different geometr cases. In our planewave based

63 5 transfer (scattering) matri algorithm, the incident electric magnetic wave is defined b incidence angles ( θ, ϕ ) and polarization angle ( α ). θ =, ϕ =, α Y E e Z X e α Z O O ' Photonic crstal plane: XOY EM polarization plane e and e Figure 4-5: Special case, normal incident θ =, ϕ, α e Y Z E α ϕ X O e Figure 4-6: Special case, theta =

64 53 The normal incidence e and h polarization have ver high smmetr, for eample rotate 8 degree and X ais mirror smmetr and Y ais mirror smmetr which will lead to mode degenerac and normal incidence calculation onl will miss some modes. Although oblique incidence can break an geometr related smmetr and can ecite all possible modes in the spectra, the interpolation results is bad due to the reason we are going to discuss in the net section. The wa to keep the good interpolation result of normal incidence while etracting all possible modes is our recentl introduced higher-order normal incidence. Under some circumstances oblique incidence is prohibit (for eample: the smmetric transfer matri method which utilize the smmetr propert of the geometr and input plane wave to reduce memor usage and calculation time dramaticall), and higher-order normal incidence is the onl choice to etract all possible resonant modes from the planewave based transfer (scattering) matri algorithm. 4. Comparison between oblique incidence and fied k value incidence As we mentioned in the last section, the spectra interpolation result from data points of oblique incidence is bad but the spectra interpolation result from data points of higher-order normal incidence is good. In this section, we eplain wh the oblique incidence interpolation is bad and introduce the theoretical foundation of fied k value incidence. Let s take a tpical point defect embedded in the photonic crstal for eample. The planewave based transfer (scattering) matri method adopts super cell configuration, i.e. the periodic boundar condition for both X and Y directions (Figure 4-7). The relationship between the super cell configuration and actual finite size photonic crstal with cavit embedded can be justified as: when the size of the super cell is large enough that the interaction across neighborhood cavit is negligible, then the result of the super cell approimates the result of the finite photonic crstal with cavit.

65 54 Y Figure 4-7: Illustration of the concept of super cell X The related etended Brillouin Zone of a super cell cavit embedded photonic crstal is conceptuall illustrated at Figure 4-8 with the purple region representing the band gap and green line representing the defect mode. Tpicall, the defect mode (cavit mode) resonant frequenc is slight changed while the k value various. The frequenc at Γ points is corresponding to the normal incident case. If a few normal incidence data points is used to interpolate the spectrum, we are going to get good interpolation results with the resonant frequenc eactl as the frequenc at Γ point which is confirmed b the detailed eample at section 3.. But as we mentioned at last section that modal degenerac ma happen due to the smmetr configuration while the

66 55 oblique incidence can etract all possible modes. However the oblique incidence has bad convergence on spectra interpolation. Figure 4-8: Etended Brillouin Zone for super cell configuration: the ais is frequenc and ais is the k vales. If we look close to the etended Brillouin Zone (Figure 4-8) with the dotted line represents an oblique incidence for fied angle, then we can realize that for oblique incidence, the resonant frequenc is not actuall one fied value but a small range of frequenc. Tring to interpolate several data points belong to different resonant frequencies (although the difference is ver small) into fied peak Lorentzian curve is for sure not a good approach and the bad performance of oblique incidence is eplained. One wa to preserve the good interpolation performance while breaking the smmetr related modal degenerac is using the fied non-zero k value normal incidence. We still use the normal incidence (set the azimuthal angle θ to zero), but when we calculate the transfer matri and scattering matri we use the nonzero fied k value. The dashed line of Figure 4-8 is eactl this situation. In this case, the resonant frequenc is one fied value which is slight

67 56 different to the Γ points resonant value. Because the non-smmetric propert of the nonzero fied k, all possible resonant modes can be etracted while all data points belong to one fied resonant peak so the interpolation works well. One directl application of the fied k value is to obtain a detailed variation of resonant frequenc verses the k value or the detailed dispersion relation for photonic crstal devices. One eample of this application is discussed at section 4.4. But how about if someone wants eactl all the Γ point resonant values while there is indeed model degenerac? The answer is we can use higher-order incidence when the super cell configuration is used. 4.3 Higher-order incidence Usuall onl the zero order planewave is propagation mode in vacuum and higher-order planewaves are evanescent mode below a frequenc threshold (detailed eplanation at later part of this section). But for higher-order planewave incidence, there eists propagation modes within the band gap frequenc range which is the most interested frequenc range to researchers. In this section, the concept of higher-order propagation planewave is discussed in ver detail with smmetr consideration. We suppose the photonic crstal cavit has X ais mirror smmetr and Y ais mirror smmetr (i.e. belongs to C v group); and we are going to appl higher-order normal incidence to break this smmetr while still take advantage of the normal incidence and smmetr properties of photonic crstal C v Group Tpical smmetries involved in photonic crstal are X mirror smmetr and/or Y mirror smmetr. For structures with both X and Y mirror smmetr, the belong to C v group. The most important propert of C v group is that there are 4 irreducible representations: A,

68 57 A, B and B. And the electric field and magnetic field eigen mode should belong to one of the irreducible representations. There are 4 operations for C group: EC ˆ, ˆ ˆ ˆ, σ, σ defined at Eq. (4.3). v ( ) = ( z) ( ) = ( ) ( z) = ( z) ( z) = ( z) Eˆ,, z,, Cˆ,, z,, z ˆ σ,,,, ˆ σ,,,, (4.3) l P = χ ˆ RPR (4.4) h R To determine which representation a function belongs, we need to appl the project operator (Eq. (4.4)) into that function for each representation. If the result is zero, then this function is not belonging to this representation, and if the result is nonzero, then this function is belonging to this representation. For an functions, it should belong to at least one of four representations. In Eq. (4.4), h is the group size (i.e. how man operations, for C v it is 4); χ is the number in the character table of the group; P ˆR is the operators and l is the dimension of the group. The character table of C v group is listed at Table 4-. Table 4-: Character table of C v group Ê Ĉ A A - - B - - B - - ˆ σ ˆ σ

69 Higher-order planewave and its smmetr The order of planewave is defined b the inde ( i, j ) in the column vector of incidence (Eq. (4.5) here n cl is the refractive inde of cladding material). For an order, the e polarization (e-pol.) has Eij, =, Eij, =, and the h polarization (h-pol.) has Eij, =, Eij, =. The corresponding magnetic field components can be found through electric field components via Eq. (4.6). i(, +, ) E z = E e e (,, ) i(, +, ) E z = E e e (,, ) ij ij ij, ij, k k β z ij ij i ij k k β z ij ij i ij ω with : βij = k kij, kij, ; k = ncl c (4.5) H, ij kij, kij, kij, β ij E ij, = H ij, kβ ij kij, βij kij, k ij, E + ij, (4.6) Now let s consider the zero order ( i, j ) = = incidence, and consider air cladding, i.e. n =. Then the electric field X and Y components for both e-pol. and h-pol. can be found cl through Eq. (4.5) and epressed in Eq. (4.7). And the corresponding magnetic field can be found through the magnetic components which can be calculated through Eq. (4.6), and the result is epressed at Eq. (4.8). e-pol h-pol e-pol h-pol ikz ( ) ( ) i ( ) = ( ) = : E,, z = e ; E,, z = :,, ;,, E z E z e ( ) ( ) ikz ( ) ( ) :,, ;,, kz H z = H z = e i : E,, z = e ; E,, z = kz (4.7) (4.8)

70 59 H H Ez (,, z) = ikε (4.9) After obtaining the electric field X and Y components, the Z components can be found b Eq. (4.9). For zero order normal incidence, the electric field Z components turn out to be zero for both e-pol. and h-pol.. So the zero order normal incidence e-pol. kz () = ˆ ( e i ) there is onl X component; and the zero order normal incidence h-pol. ikz () = ˆ ( e ) there is onl Y component. Er, Er, Now let s analsis the ( i, j ) = = order incidence with k, = π / a, k, = (here a is the lattice constant along the Y direction of the super cell). According to Eq. (4.5), we have ( / ) ( / ) k k, k, c a β = = ω π and the corresponding X and Y electric field components of both e-pol. and h-pol. can be found through Eq. (4.5) and is epressed in Eq.(4.). The X and Y magnetic field components of both e-pol. and h-pol. is epressed in Eq. (4.). The electric field Z components can be found through Eq. (4.9) with the alread known X and Y components of magnetic field (Eq. (4.)). And it turns out to be Eq. (4.) with C, C and C 3 three nonezero constants. e-pol h-pol π i a iβz : E ( z,, ) = e e ; E ( z,, ) = ( ) ( ) :,, ;,, π i a E z = E z = e e iβz (4.) e-pol ( ) ( ) :,, ;,, π i a iβz ( ) ( ) π i a H z = H z = Ce e iβz h-pol: H,, z = C e e ; H,, z = (4.)

71 6 e-pol ( ) ( z) : z,, : Ez,, = 3 π i a E z = C e e iβz h-pol (4.) The similar procedure can be applied to all other orders; and the all three components of electric field can be found for each order. After we get the electric field for each order, we can appl the group theor analsis discussed at section 4.3. to find the smmetr properties of the electric field of each order. Let s start with the simplest zero order ( i, j ) kz = = incidence of e-pol.: () = ˆ ( e i ) Er. Appl the project operator (Eq. (4.4)) to Er () to each of the four irreducible representations. If the result is zero then Er () is not belonging to this representation and if the result is none zero, then Er ( ) is belonging to this representation. Er ( ) should belong to at least one of the representations. B the definition of the four smmetr operations at Eq. (4.3), we can get the smmetr transformed Er () epressed at Eq. (4.3). To test whether Er () belongs to A irreducible representation, we need appl the projection operator to Er ( ) with the coefficient in the character table of C group s first line. Then we get ( A P ) = which v means Er () does not belong to A irreducible representation. The same procedure can be applied to A, B and B irreducible representations and result is put at Eq. (4.4). The conclusion is that the zero order e-pol. incidence belongs to B representation. Eˆ Er () = ˆ ikz ( e ) ikz ˆ ( e ) ikz ( e ) ikz ˆ( e ) Cˆ Er () = ˆ σ Er () = ˆ ˆ σ Er () = (4.3) ikz The similar procedure can be applied the zero order incidence with h-pol.: Er () = ˆ ( e ) with the four smmetr operation listed at Eq. (4.5) and the projection operation to each

72 6 irreducible representation at Eq. (4.6). And the result indicates that the zero order h-pol. incidence belongs to B representation. ( A ) P E C ( A ) P E C ( B ) P E C ( B ) P E C ( ˆ ˆ σ ˆ σ ) Er () = Er () + ˆ Er () + Er () + Er () = 4 Er () = ( ˆEr () + ˆ ˆ ˆ Er () σer () σ Er ()) = 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () + σer () σ Er ()) 4 Er () = ( ˆEr () ˆ Er () ˆ σer () + ˆ σ Er ()) = 4 Eˆ Er () = ˆ Cˆ Er () = ˆ σ Er () = ˆ σ Er () = ˆ ( A ) P E C ( A ) P E C ( B ) P E C ( B ) P E C ikz ( e ) ikz ˆ ( e ) ikz ˆ( e ) ikz ( e ) ( ˆ ˆ σ ˆ σ ) Er () = Er () + ˆ Er () + Er () + Er () = 4 Er () = ( ˆEr () + ˆ ˆ ˆ Er () σer () σ Er ()) = 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () + σer () σ Er ()) = 4 Er () = ( ˆEr () ˆ Er () ˆ σer () + ˆ σ Er ()) 4 (4.4) (4.5) (4.6) Now we have the information of the zero order incidence of both polarizations: e-pol. belongs B representation and h-pol. belongs to B representation. The phsics meaning of the above information is: the zero order e-pol. incidence can onl ecite resonant modes belong to B representation and an other possible modes of A, A or B can not be ecited b the zero order e-pol. incidence (or in other words, there will be no such resonant mode in the spectrum of zero order e-pol. incidence); the zero order h-pol. incidence can onl ecite resonant modes belong to B representation and an other possible modes of A, A or B

73 6 can not be ecited b the zero order h-pol. incidence (or in other words, there will be no such resonant mode in the spectrum of zero order h-pol. incidence). In actual photonic crstal cavit, the resonant modes ma belong to an of the 4 irreducible representations, so we need analsis higher-order incidence to make sure all of four irreducible representations are covered. Now let s do the similar analsis to the i =, j = higher-order incidence with the electric field epressed in Eq. (4.7). B appling the 4 smmetr transformation operation Ê, Ĉ, ˆ σ and ˆ σ and using the projection operator, we can get the results that e-pol. belongs to A and B irreducible representations (Eq. (4.8)) and h-pol. belongs to A and B irreducible representations (Eq. (4.9)). So higher-order ( i, j ) = = incidence with e-pol. and h-pol. can cover all four representations and all possible resonant modes can be ecited if both h-pol. and h-pol. of ( i, j ) = = order incidence are used. e-pol. ( ) ( ) π π i i a iβz a iβz ˆ e-pol: E r = e e Ce e zˆ h-pol: E r = e ( A ) P E C ( A ) P E C ( B ) P E C ( B ) P E C π i a iβz e ˆ ( ˆ ˆ σ ˆ σ ) Er () = Er () + ˆ Er () + Er () + Er () 4 Er () = ( ˆEr () + ˆ Er () ˆ σer () ˆ σ Er ()) = 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () + σer () σ Er ()) 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () σer () + σ Er ()) = 4 (4.7) (4.8)

74 63 h-pol. ( A ) P E C ( A ) P E C ( B ) P E C ( B ) P E C ( ˆ ˆ σ ˆ σ ) Er () = Er () + ˆ Er () + Er () + Er () = 4 Er () = ( ˆEr () + ˆ Er () ˆ σer () ˆ σ Er ()) 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () + σer () σ Er ()) = 4 Er () = ( ˆEr () ˆ ˆ ˆ Er () σer () + σ Er ()) 4 (4.9) Further analsis shows that higher-order incidence ( i =, j = ) has the same results as higher-order incidence ( i =, j = ). Similarl, the higher-order incidence ( i =, j = ± ) covers A and B irreducible representations for e-pol. and A and B irreducible i, j can cover representations for h-pol. Each polarizations of an higher-order with ( ) all four irreducible representations. Those results are summarized at Table 4- with i. Table 4-: Irreducible repressions coved b high order incidence (, ) (, ± i) ( ± i,) ( ± i, ± i) e h e h e h e h Rep. B B A B A B A B A B all 4 all Possible propagation modes for higher-order incidence As we mentioned at the beginning of this section, not all the higher-order planewave are propagation mode for all frequenc ranges. We need pa close attention to the wave vector k z epressed in Eq. (4.). When k z is real number, it is propagation mode; and when k z is imaginar number, it is evanescent mode which will deca eponentiall with respect to the propagating distance. The resonant cavit modes can onl be ecited b propagation modes. ω k = β = k k k ; k = n (4.) c z ij ij, ij, cl

75 64 For zero order incidence ( i, j ) = = kij, = kij, =, kz = k and it is real number for an frequenc. So zero order incidence is propagating mode for all frequenc range. Now let s = = order incidence for eample: kij, =, kij, = π / a and k z is real number take ( i, j ) onl for nclω / c π / a or ω πc ( n a ) / cl. There is a cut-off frequenc for propagation modes. Below that cut-off frequenc, it is evanescent wave and can not be used to ecite resonant cavit modes. Luckil, the frequenc range we are interested in (within the band gap range) is usuall above this cut-off frequenc. Based on our eperience, the first band gap lower frequenc edge is usuall above. normalized frequenc. If we set the normalized cut-off frequenc to be., then the cut-off frequenc ω takes the larger value of ω in ωa / π c. and ωa / π c.. Here A and A is the lattice constant of the super cell, while a and a are the length of the unit cell along X and Y direction. Usuall we have A = ma and A = ma with m and n integers (for eample: m = 5, n = 5 ; we call it 55 super cell). Now let s setup a lower frequenc boundar ( c in Eq. (4.)) and suppose all calculation is above this boundar. Based on this lower frequenc boundar, we can find which higher order incidence are propagation modes and which order are evanescent modes. c a = ω, ω = c π (4.) π c c a π π k = i ; k = j (4.) ij, ij, ma na For order (, ) i j incidence with m n super cell configuration, k ij, and k ij, can be epressed in Eq. (4.). With our pre-set lower frequenc boundar c, the order ( i, j ) incidence is propagation mode onl if the relation of Eq.(4.3) is satisfied. For fied mn, and a, a, the maimized order propagation incidence can be determined.

76 65 ω cl ij, ij, n k + k or c π c π π ncl i + j a ma na or n c i + ( / ) cl m n a a j (4.3) If we draw all the higher-order ( i, j ) into a diagram with k and k as the X and Y ais and keep a = a and m = n, the are isolated points of square grid. All the grid points within the circle with center at origin and radius of mnclc are propagation modes, which is illustrated at Figure 4-9: Propagation modes in k space. k (π/l) L=5 a (,) (,) (,) (,) k (π/l) Light-line with low frequenc edge Figure 4-9: Propagation modes in k space All the above discussion is based on normal higher-order incidence with fied incidence wave vector k = k =, i.e. the Γ point at Figure 4-8. In case of other fied incidence o wave vector k, k, the propagation mode is still with in the circle with radius mnclc, but o the center of the circle is now ( k, k o ).

77 Eample of application of higher-order incidence This section is modified from a paper published at Optics Letters Vol. 3, No. 3 (page 3498) at December 6 with title: "Higher-order incidence transfer matri method used in three-dimensional photonic crstal coupled-resonator arra simulation", b M. Li, X. Hu, Z. Ye, K. Ho, J. Cao, and M. Miawaki. Photonic crstals (PCs), periodic dielectric media, can inhibit electromagnetic (EM) wave propagation in certain frequenc ranges called photonic band gaps (PBGs)., Introducing point or line defects into PCs can create highl localized defect modes within the PBGs, 3 resulting in resonant cavities of high qualit factor Q and low loss waveguides. 4,5,6 Recentl, coupled PC cavit arras have received much attention due to their potential application in integrated optical circuits. 7,8,9, However, most of these structures are based on twodimensional (D) PCs or D PC slab. There eist few studies 8 on resonant cavit arras in three-dimensional (3D) PCs due to difficulties in fabrications and numerical simulations. In this letter, we theoreticall stud periodic resonant arras in 3D PCs b using the planewave based transfer-matri methods (TMM), with higher-order planewave incidence and rational-function interpolation 3 techniques. As an eample, both the qualit factor and dispersion relation are obtained ver efficientl for a resonant cavit arra based on the laer-b-laer woodpile PCs. An interesting ultra-slow negative group velocit is observed in this structure. To our knowledge, this is the first time that dispersions are calculated for coupled resonant cavit arras in 3D woodpile PCs in all directions. The 3D laer-b-laer woodpile 4 PC is composed of 5 laers (in the z direction) of square dielectric rods of refractive inde n =. 4 and width.35a, with the cladding material at both ends along z direction of refractive inde n cl. Here the lattice constant a is the distance between two neighbored rods (shown in Figure 4-(a)). The cavities are located in the 3th

78 67 laer and of double periodicities of 5a in both and directions. Each cavit is created b filling a volume of a b a b.35a with the rod material ( n =. 4 ) (shown in Figure 4- (b)). Figure 4-: The X-Y-plane-periodic arra of 3D woodpile PC (with cavit in the 3th laer): (a) the 5-b-5 super cell of total 5 laers along z direction with dielectric cladding of refractive inde n cl (the cladding is not shown); (b) top view of the 3th laer with the cavit of volume of a b a b.35a and n =.4; and (c) the irreducible Brillouin zone of (q, q ). The 3D woodpile PC is composed of dielectric square rods of n =.4 and width.35a, where a is the distance between two neighbored rods (i.e. the lattice constant). The related reciprocal lattice ( G = iπ /(5 a), G = jπ /(5 a) ) and irreducible Brillouin zone of the Bloch wave-vector ( q, q ) for this periodic cavit arra are shown in Figure 4-(b) and Figure 4- (c), respectivel. Since the resonant cavit arra has -ais and - ais mirror smmetr, the cavit modes belong to the C v group with four irreducible representations (A, A, B, and B ) and the character table of the C v group is listed in Figure 4- (c). 5 We consider the incidence of planewaves with wave-vector ( k / q G, k q G, kz ( k k k) ) = + = + = upon the above resonant cavit arra, where

79 68 k = ncl / c with c the speed of light and n cl the cladding substrate refractive inde. ω k + k < k and k + k > k corresponds to propagating and evanescent waves, respectivel (shown in Figure 4- (a) and (b)). In our TMM algorithm,, the Bloch boundar condition with ( q, q ) is used for a 5 5 supercell. Figure 4-: (a) Incidence of a planewave with wave-vector ( k = q + G, k q + G = and ( ) / k z = k k k ) upon the 3D PC cavit arra (periodic in the X-Y plane); solid (dashed) arrows stand for the propagating (evanescent) waves. (b) The reciprocal lattice for the 5-b-5 super cell PC cavit arra; the G points inside (outside) the dashed circle (with radius k = ncl ω / c and center (-q, -q ), shown for ω a /( πc) =. 43, n cl =. and q = q = ) stand for the propagating (evanescent) waves. (c) The C v group character table for the PC cavit arra with E, C, and smmetr operation. 5 (d) The irreducible representations for the e- and h- polarized iπ / 5a = jπ / 5a. incident planewaves of order (i, j), i.e. = ( ) and ( ) The electric field coefficient vector ( ij,, ij, ) G G G E E of planewaves with = iπ /(5a ), G = jπ /(5 ) (named as order ( i, j ) in the following) can be transferred a from laer to laer in the z direction b a transfer matri. Here i, j = N,...,,... N and a favorable convergence is found using N = 5 for the present 5 5 supercell. Previousl 3, normal incidence with k = k = (i.e. order (,) with q q = ) was used to stud the PC cavities. However, not all the cavit modes could be ecited with (,) incidence with q q = (as shown in Figure 4-) because of group smmetr considerations. To ecite = =

80 69 all the cavit modes of PC cavities, we performed higher-order ( i, j ) planewave incidences with both e- and h- polarizations defined b E, E and E, E ij, = ij, = ij, = ij, = respectivel, where the superscript inde represents the electric field components before entering the photonic crstal structure. The incidence of planewaves with q q = is firstl considered. The irreducible = representations that the order ( i, j ) could cover can be found through the projection operation, 5 and the results are shown in Figure 4- (d). Clearl, if the zero order ( i = j = ) (i.e. normal incidence) is used alone, onl the B or B representation is ecited (for e- and h- polarizations, respectivel) and an resonant modes belonging to A or A representation will not be ecited. Actuall for this particular photonic crstal cavit arra structure there is indeed one resonant mode (mode A) belongs to A representation at normalized frequenc.43948; and the other resonant mode (mode B) belongs to B representation at normalized frequenc The mode shape profiles for both modes are also calculated b TMM with higher-order incidence and are illustrated in Figure 4-3 (a). Based on the projection operation 5, an higher order ( i, j ) incidence with i and j will cover all the representations and both resonant modes can be ecited. To find the frequenc and qualit factor of the resonant modes, the nearl continuous transmission spectra are obtained accuratel from the rational function interpolation of individual frequencies ranging in the first band gap 3. Figure 3 shows the transmission results for the planewave incidence of the first four orders. Although the transmission amplitude for different orders and polarizations varies dramaticall, the resonant frequenc and Q value for each resonant mode are practical identical: f A =.43948, Q A = 9 and f B =.44763, 5 Q = 69 (the relative difference of f and Q for different incidences is less than and B 4, respectivel). The ecited resonant peaks for each order ( i, j ) incidence agree well with the above group theor analsis. For eample, the group theor analsis indicates that the (,) order incidence with e-polarization covers the irreducible representations of A and

81 7 B, and hence the mode B (belonging to B ) instead of mode A (belonging to A ) appears in the calculated transmission spectrum for the (, ) incidence. There is also a third resonant mode found b TMM at normalized frequenc a /( πc) = ω. Figure 4-: Transmission spectra for the planewaves incidence of q q = and order (i, j) upon the 3D PC cavit arra with e-pol and h- = pol defined b E, E and E, E respectivel. The ij, = ij, = ij, = ij, = resonant mode A and B are also labeled. The first band gap in the z direction (i.e. the (, ) incidence) opens from ω a /( πc) =. 395 to ω a /( πc) =. 55. There is the third resonant mode at normalized frequenc ωa / πc =. which is not shown in this figure. ( ) 4635 The transmission spectra are calculated for planewave incidence with different ( q, q ). We note that all the cavit modes can be ecited for the case of ( q, q ) due to the broken smmetr of non-zero q and q. The frequencies of resonant cavit modes can be etracted from the spectra and the related dispersion relations are illustrated at Figure 4-3 (b). The

82 7 slopes of the dispersions are ver flat, indicating ultra-slow group velocities in the woodpile PC cavit arra structure. The average group velocit (calculated b finding the slope of the straight line connecting two high smmetr points) of mode A in the Γ-X(X') direction is 5 6 υ g =.c= 3.6 m/ s ( υ g =.48c=.4 m/ s) and that of mode B in the Γ-X' 6 direction is υ g =.56c=.7 m/ s. It is interesting that a negative group velocit of υ = = can be achieved for mode B in the Γ-X direction. 5 g.9c.7 m/ s Figure 4-3:(a) Electric field mode profiles for the cavit modes A and B at point ( q = q = ); (b) dispersion relation of both cavit modes in the 3D PC cavit arra with black curve corresponds for -X-M and red dashed curve for -X'-M. Please note -X-M and -X'-M represent different (q, q ) direction as shown at Figure 4-(c).

83 7 In conclusion, we have developed a higher-order planewave incidence concept for the planewave based transfer matri method with rational function interpolation algorithm to efficientl simulate three-dimensional photonic crstal devices. As an eample, the dispersion relations and qualit factors were calculated for the resonant cavit arra embedded in laer-b-laer photonic crstal structure. An interesting ultra-slow negative group velocit is observed in this structure. One other advantage of TMM is that there is no limitation on the length of photonic crstal structure along the propagation direction making the TMM ideal for waveguide simulation. One other direct application of this paper is to find out the wave guide loss b means of finding the resonant Q value for different ( k, k ). References:. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Phsics and Electronics," Phs. Rev. Lett. 58, 59 (987). S. John, "Strong Localization of Photons in Certain Disordered Dielectric Superlattices, " Phs. Rev. Lett. 58, 486 (987) 3. J.D. Joannopoulos, R.D. Meade and J.N. Winn, Photonic Crstal, Princeton Universit Press, New Jerse, E. Özba, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, "Defect structures in a laer-b-laer photonic band-gap crstal," Phs. Rev. B 5, 396 (995) 5. A. Chutinan, and S. Noda, "Highl confined waveguides and waveguide bends in threedimensional photonic crstal," Appl. Phs. Lett. 75, 3739 (999) 6. M. Okano, A. Chutinan, and S. Noda, "Analsis and design of single-defect cavities in a three-dimensional photonic crstal," Phs. Rev. B 66, 65 ()

84 73 7. A. Yariv, Y. Xu, R.K. Lee and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analsis," Optics Lett. 4, 7 (999) 8. M. Baindir, B. Temelkuran and E. Ozba, "Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonic Crstals, " Phs. Rev. Lett. 84, 4 () 9. H. Altug and J. Vuckovic, "Two dimensional coupled photonic crstal resonator arras, " Appl. Phs. Lett. 84, 6 (4). H. Altug and J. Vuckovic, "Eperimental demonstration of the slow group velocit of light in two-dimensional coupled photonic crstal microcavit arras," Appl. Phs. Lett. 86, (5). Z.Y. Li and L.L. Lin, "Photonic band structures solved b a plane-wave-based transfermatri method," Phs Rev. E 67, 4667 (3). L. L. Lin, Z. Y. Li and K. M. Ho, "Lattice smmetr applied in transfer-matri methods for photonic crstals," J. Appl. Phs. 94, 8 (3) 3. M. Li, Z.Y. Li, K.M. Ho, J.R. Cao and M. Miawaki, "High-efficienc calculations for three-dimensional photonic crstal cavities," Optics Lett. 3, 6 (6) 4. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas and M. Sigalas, "Photonic band gaps in three dimensions: new laer-b-laer periodic structures," Solid State Commun. 89, 43 (994) 5. V. Heine, Group theor in quantum mechanics, Pergamom Press, NY, 96

85 74 Chapter 5. Perfectl matched laer used in TMM In numerical simulations, absorption boundar condition is ver useful to define finite calculation domain which is etremel important in finite-difference time-domain method (FDTD). And numerous approaches have been introduced b FDTD eporters, in which the perfectl matched laer (PML) absorbing boundar condition,3,4,5 can be naturall adapted to our planewave based transfer (scattering) matri method (TMM). As mentioned in previous chapters, TMM s ke assumption is periodic boundar condition. In order to achieve finite size isolated calculation domain, we must put a finite thickness absorption boundar such as perfectl matched laer, which will absorb all the incoming waves and act like infinite boundar enclosed a finite size domain. One other application of perfectl matched laer is to put it at the end of the wave guide to simulate infinite long waveguide structures. The mode profile of one dimensional dielectric waveguide and the optical properties of subwavelength aluminum grating with semi-infinite substrate are eamples of PML applied to TMM which illustrate the accurac and power of the application of perfectl matched laer. 5. Motivation of introducing perfectl matched laer The intrinsic boundar condition used in TMM is Bloch periodic boundar condition. For perfect photonic crstals (PC) simulation, the periodic boundar condition reveals the periodic nature of photonic crstal and TMM returns the properties (such as spectrum, band structure and mode profile) for the infinitel large PC structures. For more interested defective PC structures, the supercell concept and periodic boundar condition (i.e. an infinite arra of defects) are adopted to approimate a single defect embedded in the infinite or finite large PC. The convergence of increasing the size of the supercell for a resonant cavit within laer-b-laer woodpile PC has been studied at section 3.. Periodic boundar

86 75 conditions are also used in FDTD and other popular numerical methods to simulate infinitel large periodic PC structures. In eperiments and real applications the PC structures are all finite in size, and it is usuall important to stud the defect structure within a finite size PC. The perfectl matched laer absorption boundar condition first introduced in FDTD can be adapted to TMM to simulate finite size PC structures. Although the performance of PML has been etensivel studied in FDTD, there are ver few reports on PML applications to TMM method. In this chapter the theoretical foundation and detailed benchmark results of perfectl matched laer will be discussed along with several application eamples. 5. Theor of perfectl matched laer and Z ais PML 5.. Background of PML The concept of PML was first introduced at 994 b J.P. Berenger to improve the performance of absorption boundar conditions in FDTD methods. Later Sacks discussed in his 995 paper that an anisotropic uniaial material with both permittivit (dielectric constant) and magnetic permeabilit diagonal tensor can act as perfect match interface. The uniaial PML is directl derived from the Mawell Equations to realize the perfectl matched conditions. Such material can absorb all the incoming electromagnetic energ for all incidence angles, i.e. there is absolutel no reflection and transmission through this medium. Although we can not find this kind of material in nature or man-made material tile now, but we can use phsical parameters to describe this special medium. Our planewave based transfer (scattering) matri method can treat anisotropic material naturall, which makes the anisotropic uniaial material our ideal candidate to realize the perfectl matched absorbing condition. 6,7

87 76 The illustration of PML of two polarizations e-pol (or called s wave) and h-pol (or called p wave) for Z -ais is at Figure 5- (a) and Figure 5- (b). The propagation direction is along Z ais, and the interface ( XY plane) is perpendicular to Z ais. The isotropic region at the left side of the interface has dielectric constant ε and magnetic permeabilit μ. The anisotropic PML region right to the interface will be perfectl matched to the isotropic region if the dielectric constant tensor ε and magnetic permeabilit tensor μ are given b Eq. (5.) with s z = a+ bi, an comple number. For ideal Z -ais PML, there will be no reflection for planewave of an incidences at all frequencies and the transmitted wave in the PML is eponential etenuated b factor ep( α) with α bz cos θ / λ (with b imaginar part of s z, z thickness of PML, λ wavelength, and θ incident angle). s wave E H θ k Interface X z 3 H E s wave E H p wave 3 R PML T Y (a) p wave H E θ k Interface X z R (b) PML T 3 (c) X 3 Figure 5-: (a) e-pol (s wave) oblique incidence upon Z-ais PML, (b) h-pol (p wave) oblique incidence upon Z-ais PML: E for electric field and H for magnetic field. (c) 33 supercell illustration at XY plane of PC structure with a defect located at the center with s and p wave shown for normal incidence; three PML regions are labeled: for X-ais side PML, for Y- ais side PML, and 3 for XY corner PML.

88 77 ε = ε s ; μ = μ s sz s = sz / sz (5.) In FDTD simulation with finite spatial sampling, to get the ideal PML performance, fine tuning of s z is required and sometimes multiple laers of different s z are used, such as polnomial-graded PML. But in our TMM method, for the Z ais perfectl matched laer, a single laer with thickness half of the unit cell and s = 4 + 4i is good enough (detail at section 5..). z 5.. Performance of simple parameter approach The following are the performance of the Z -ais perfectl matched laer of s = 4 + 4i with thickness of.5a ( a the unit cell lattice constant) for the normal incidence case ( θ = ). At Figure 5-, the reflection rate of a long frequenc range ( to normalized frequenc) is plotted: for both e and h polarizations the reflection rate is below -5dB and ecellent match condition is observed. At Figure 5-3, the transmission rate after the.5a thick perfectl matched laer is plotted: for both e and h polarizations the transmission is decreasing eponentiall when the normalized frequenc increases. z The larger the normalized frequenc, the shorter the wave length and more number of wavelength in the.5a thickness, so the more absorption (less transmission) from the perfectl matched laer is observed. But even with normalized frequenc., the transmission is less than %. To make better performance of the perfectl matched laer at lower frequenc range, we can increase the thickness of laer. With the increase of the thickness, the reflection rate remains the same while the transmission rate is decreased eponentiall.

89 78 E- E- TMM h-polarization TMM e-polarization Reflection Rate E-4 E-6 E-8-5dB E Normalized Frequenc (a/λ) Figure 5-: Reflection from perfect match interface Transmission Rate TMM h-polarization TMM e-polarization Normalized Frequenc (a/λ) Figure 5-3: Transmission rate after the perfect match laer In the formulation, the result is independent of incident angles; but in the real work numerical simulation, the performance of the perfectl matched laer is indeed depending on the

90 79 incidence angle and there are a lot of studies on how to improve the large angle incidence perfectl matched laer performance in FDTD research fields. Fortunatel, our TMM method does not require much attention on that issue. Although, the performance do var when the incidence angle changes, the overall performance is still ecellent: at Figure 5-4 the reflection rate is calculated for incidence angle range from to 89 degree for normalized frequenc.4 and perfectl matched laer thickness.5a and s z = 4+ 4i. Even for ver high incidence angle 89 degree, the refection is below 9 db for both polarizations. The transmission rate is also calculated for various angles and the result is plotted at Figure 5-5 for normalized frequenc.4 and perfectl matched laer thickness.5a and s = 4 + 4i. We can see that the performance is OK for smaller incidence angle z (for eample around. transmission rate at 8 degree). But the transmission is larger than.5 when the incidence angle approach to 9 degree. Although the electromagnetic wave mabe in a distribution of incidence angles and high incidence angle components is rare, we still need to be ver careful when appl the perfectl matched laer and assume the transmission is purel zero case. E-9 e-pol 89-degrees, R=-9dB E- h-pol Reflection Rate E- E- E-3 E-4 E-5 E-6 E-7 89-degrees, R=-dB E-8 E-9 E Incident Angle θ (degree) Figure 5-4: Reflection rate vs. incidence angle

91 8 e-pol h-pol Transmission Rate Incident Angle θ (degrees) Figure 5-5: Transmission rate vs. incidence angle 5..3 Two strategies to improve the performance Two strategies are studied to improve the performance of PML in term of attenuating the transmission: one is to increase the imaginar part of s z and the other is to increase the thickness of the PML. Both approaches have the same effect to increase of the attenuation factor α bz cos θ / λ. The transmittance and reflectance of both methods are illustrated at Figure 5-6 with both cases: double the thickness of PML and double the imaginar part of s z. As shown Figure 5-6 (a), the performance of PML is improved for all incident angles, but when the incident angle approaches to 9 degree, the transmittance is alwas approaching to %; while the perfectl matched condition (no reflection) still remains valid for all angles (Figure 5-6 (b)). Further studies such as grading of PML are required to further improvement of the transmittance attenuation performance. Even without grading, the reflectance is alread below which is much better than the complicated grading PML in FDTD.

92 8 (a) Transmittance (b) Reflectance - Increase PML thickness Change s z -3 Increse PML thickness Change s z T R Z Incident Angle (degree) T R Z Figure 5-6: Two approaches to improve the performance of Z-ais PML -- double the thickness of PML and double the imaginar part of s z : (a) transmittance is attenuated more for ever incidence angle but still approaches to % as incidence angle approaches 9 degree, (b) reflectance is still perfectl matched for all incidence angles. Onl s wave shows, p wave presents the same behaviors Application of PML to periodic D waveguide Now we can appl the Z direction perfectl matched laer to one dimensional waveguide structure illustrated at Figure 5-7: the blue region represents dielectric material ( ε = 5.76 ) and the background is air; the width of the waveguide is d =.5a and the length of the waveguide is L= 3a with a the lattice constant along direction. The planewave is incident from left and propagating along Z direction. We choose the normalized frequenc a / λ =.4.

93 8 Figure 5-7: D wave guide (without perfect match laer) z The planewave based TMM adopts the periodic boundar condition at Y direction. Due to the air-dielectric material interface termination at the end of the waveguide, a standing wave is formed which is shown clear in the mode profile plot from TMM (Figure 5-8). Those results are tpical for finite length waveguide. While FDTD software can also do this task; but when it comes to infinite waveguide case, FDTD has some difficulties. We can simulate infinite long D waveguide b putting a laer of PML at the end of the waveguide and the mode profile distribution of electric field is plotted at Figure 5-9. The standing wave feature disappeared and guided wave presents within the waveguide slab. The PML has thickness of.5a with s = 4 + 4i. There are ver little energ penetrated into the PML and energ is z absorbed eponentiall as it propagates into the PML. The detailed comparison of TMM results with the analtical results will be discussed in section Plot of E (,z) incident ε=5.76 Figure 5-8: Electric field distribution within the unit cell of D waveguide (without putting perfect match laer at the end).

94 83 UPML z air incident ε=5.76 UPML z GaN UPML z air Figure 5-9: Electric field distribution within the unit cell of D waveguide (without putting perfect match laer at the end). 5.3 Perfectl matched laer for X, Y ais and its application to D waveguide To get perfectl matched in the X or Y boundar within the XY plane (for eample eliminating the crosstalk between neighborhood in plane supercells), we can adopt the similar relation as the Z -ais PML: to match X boundar Eq. (5.) should be used; to match Y boundar Eq. (5.3) should be used; and to match the corner of XY boundar the product of Eq. (5.) and Eq. (5.3) should be used. Figure 5- (c) illustrates three tpes of XY plane PMLs applied to a 33 supercell structure with a defect embedded at center. It is trick to get direct performance benchmark of the X - or Y -ais side PML because in TMM we can onl collect spectra information in the propagation direction ( Z -ais), while the structure is periodic along the X and Y ais direction. To get an idea of the performance of X or Y side PML, the indirect approach of electric field distribution comparison with analtical results will be utilized. / s s = s (5.) s

95 84 s s = / s (5.3) s Now we appl those perfectl matched laer at X -ais and Z -ais direction to D waveguide to see the difference between the numerical calculation and analtical results. But before the comparison, we need first derive the analtical results of the D infinite waveguide Analtical solutions of D dielectric slab waveguide The one dimensional dielectric slab waveguide is discussed in man tet books, such as Optical Waves in Crstals b Yariv and Yeh 8. Here in this section, I outline the ke steps of the derivation and list the result for our later comparison purpose. The geometr of the D wave guide is illustrated in Figure 5-: the thickness of the wave guide is d and centered at = ; the waveguide is uniform along and z direction; the dielectric constant at region I and III is ε while the dielectric constant at region II is ε ; the guided wave will propagate along z direction. d/ -d/ I II III z Figure 5-: D wave guide analtical derivation illustraton

96 85 The Mawell s Equations Eq. (.6) needs to be solved in this particular structure. As along the propagation direction the whole dielectric structure is homogeneous, so the solution can be epressed as Eq. (5.4). And the wave equation through the Mawell s Equations can be written as Eq. (5.5) which are valid in each of the three regions (within each region ε is constant instead of function of coordinate). E(, z, ) = E( e, ) iβ z H(, z, ) = H( e, ) iβ z (5.4) ω E(, z, ) = ε E( z,, ) = kn E( z,, ) c ω H(, z, ) = ( z,, ) kn ε H = H( z,, ) c (5.5) There is no variation along direction, so an derivative with respect to is zero. So plug Eq. (5.4) to Eq. (5.5) we can get the relations for all three regions (Eq. (5.6)). The magnetic field H ( ) satisfies the same equation set with E ( ) replaced b H ( ). I : E( ) + ( kn β ) E( ) = II : E( ) + ( kn β ) E( ) = III : E( ) + ( kn β ) E( ) = (5.6) There are two different modes in this case: TE mode and TM mode. For TE mode, and H z are non-zero; for the TM mode, H, E and E, H E z are non-zero. We are going to solve these two modes separatel. For guided mode, we need the electric field magnitude to vanish as ± which implies kn < β < kn.

97 86 Now let s take TM mode into detail consideration which has can take H, E and E z non-zero. We H (Eq.(5.7)) as the independent variable and E and E z can be derived accordingl b Eq.(5.8). i( z t) (,,, ) = ( ) H z t H e β ϖ β E( ) = H( ) ωε( ) i Ez( ) = H( ) ωε( ) (5.7) (5.8) The solution of H ( ) to the three regions can be epressed as Eq. (5.9) with k = β n k, k = β n k, k = ω / c and ABC,, undetermined constant. The solution of Ez( ) can be found from Eq. (5.8) and epressed at Eq. (5.). Using the continuities of H( ) and Ez( ), we can obtain the following relations: A Bcos( k d / α) C Bcos k d / α Ak / n = Bk / n sin k d /+ α and = +, = ( + ), ( ) / = / sin ( /+ ), which is simplified and listed at Eq. (5.). Ck n Bk n k d α I H ( ) = Ae k ( + d /) ( α ) II H ( ) = Bcos k + III H ( ) = Ce k ( d/) (5.9) I ik Ez ( ) = Ae ωn k ( + d/) II ik Ez ( ) = Bsin k+ ωn III ik Ez ( ) = Ce ωn ( α ) k ( d /) (5.)

98 87 ( ) ( α ) ( α ) ( α ) k / n = k / n tan k d / k + k = n n k, α = nπ / A= Bcos k d / + C = Bcos k d / + (5.) For the lowest TM mode of relation is given b k / n k / n tan ( k d /) n = (with, B, A C cos ( k d / ) field components epressed at Eq. (5.). α = = = = ), the dispersion = and ( ) k + k = n n k. Finall we get the ( ) H ( ) = Ce, H ( ) = cos k, H ( ) = Ce I k ( + d /) II III k ( d /) Cβ β β E ( ) = e, E ( ) = cos ( k ), E ( ) = e ωn ωn ωn I k ( + d /) II III k ( d /) I ik k ( + d /) II ik ( / III i k k d ) E ( ) = e, E ( ) = sin ( k ), E ( ) = e z z z ωn ωn ωn (5.) Similar approach can be applied to the TM mode and E, Hand H z at three regions can be found Numerical results of TMM with side PMLs After we get the analtical results for the D infinitel long dielectric waveguide, detailed quantified comparison with TMM calculation can be done. The mode profile of D dielectric waveguide with z ais onl PML has been plotted at Figure 5-9. Due to the periodic boundar condition along ais, this result is a set of infinite parallel dielectric waveguide arra. Although the interaction between neighborhood waveguide is small, the result at Figure 5-9 is still not identical to the analtical solution. To eliminate the cross talk between neighborhood waveguide, the side wall PMLs are added as shown at Figure 5-. To get detailed comparison, we take laer #48 shown as blue dashed line at Figure 5- and plotted at Figure 5- which is for TE mode with E non zero. Without the -sidewall PML, the

99 88 magnitude of electric field component is larger than the analtical result at the edge of the unit cell ( =±.5a ) due to the interaction between neighborhood waveguides; the results within the waveguide s dielectric material region is ver close. With the -sidewall PML, the performance is improved and the TMM calculation results are more consistent with analtical results at the edge of the unit cell boundar. UPML air Laer #48 UPML +z air UPML z air incident n =.4 UPML z GaN UPML z air UPML air UPML +z air Figure 5-: Mode profile of D dielectric waveguide with side PML..9 n air =. n air =. Analtical Solution Without X-sidewall UPMLs With X-sidewall UPMLs Normalized E () n GaN = Figure 5-: Detailed comparison of electric field magnitude with analtical solutions

100 PML application eample: dispersive sub- λ aluminum grating As the eperimental technique improved in recent ears, the proposed future devices used in optics and laser are getting smaller and smaller. Those recentl proposed devices are so small that sometimes much smaller than the wavelength of interested electromagnetic wave. TMM can be applied directl to sub- λ components such as resonant cavit, waveguide, grating etc. In this section, the sub- λ aluminum grating is studied with the application of PML to simulate infinite thick substrate at visible light frequenc. The geometr of the grating is illustrated at Figure 5-3. It is periodic along direction and uniform along direction. The incident planewave is from top and propagates along the z direction with azimuthall incident angle θ = 7. The aluminum width is.68 μ m, and the air width is. μ m. So the lattice constant along direction is.7 μ m and the filling ration is 4 %. The thickness of the grating is.36 μ m. The substrate is made b SiO with refractive inde n sub =.55. We calculate two cases of different substrate thickness: one with finite thickness of.5 mm and the other with infinite thickness b appling PML at the end of the substrate. The refractive inde of dispersive aluminum is listed at Table 5- with n = n+ ik. z Al Air Substrate Figure 5-3: Geometr sub-wavelength of metal grating

101 9 Table 5-: refractive inde of Aluminum at visible wavelength wavelength n k wavelength n k ( μ m ) ( μ m ) There are two polarizations for this case: TE polarization (s wave) has the electric field vector perpendicular to the incident plane and TM polarization (p wave) has the electric field vector in the incident plane. First the sub- λ grating with finite thickness of.5 mm is calculated; the reflection rates of both polarizations are plotted at Figure 5-4 (a). For TE mode, the reflection is ver high (around 9%) and it is almost a constant in the whole visible light frequenc range. But for TM mode, it is quiet different. The most obvious different feature compared with TE mode is the oscillations in the whole spectrum which is due to the interference with the strong reflection from the interface of the substrate end to the air. The reflection rate of TM mode change a lot within the visible light range and at around.45μ m the reflection is almost zero. This can be used as the polarization light splitter to separate different polarizations.

102 9. (a) Finite substrate (b) Infinite substrate.8 s wave p wave s wave p wave Reflectance.6.4 θ R T SiO Y Z X θ R T SiO Y Z Z-ais PML X Wavelenght (μm) Wavelenght (μm) Figure 5-4: Reflection rate of finite thickness sub-wavelength grating Reflectance of sub-wavelength grating for both s and p planewave with incident angle 7 degree at visible frequencies: (a) with finite SiO substrate, and (b) with infinite SiO substrate b appling Z-ais PML at the end of substrate. Net, we did a calculation on infinite substrate case. Here we used the z ais PML discussed at section 5. with s = 4 + 4i. The reflection rate for both TE and TM polarization are z plotted at Figure 5-4 (b). For TE mode, there is not much difference compared with the finite thickness substrate case. But for TM mode, the oscillation disappears for the infinite thickness of substrate. This is eas to eplain because for infinite substrate, there will be no reflection from other end of the substrate and hence no interference. At wavelength of.45μ m, the TM mode s reflection rate is nearl zero, and it is perfect for polarization beam splitter. To get better understanding of the behaviors at wavelength of.45μ m, the electric magnitude mode profiles are plotted for both TE and TM waves at Figure 5-5. The TE mode is E dominated, and strong reflection occur at the front of the grating which

103 9 corresponding to the high reflectance. On the other hand, the TM mode is E dominated and the electromagnetic energ can propagate through the grating and substrate. (a) X (b)..4.5 Air Al Air Al Z.8. SiO s wave E SiO p wave E Figure 5-5: Electric field mode profiles of sub-wavelength grating for the infinite substrate case at wavelength 4.5um: (a) TE mode (s wave) incidence, E is dominate and electromagnetic energ is highl reflected, (b) TM mode (p wave) incidence, E is dominate and electromagnetic energ is % transmitted. One of our recent research shows that a coordinate transformation can be applied to TMM to simulate curved waveguide structure. The result of the coordinate transformation is ver similar to the PML we applied in this chapter ecept that the parameter of s z is a real function of coordinate. 9 Although PML is an artificial material first introduced for numerical simulation purpose, researchers are enthusiastic to seek PML-like materials or structures of controllable electric permittivit and magnetic permeabilit., With those two parameters well controlled, EM wave cloaking, or invisible material can be achieved. TMM can also be used in the design of such permittivit and permeabilit controllable materials just like the implementation of PML absorption boundar condition.

104 93 References:. Taflove and S.C. Hagness, Computational Electrodnamics (Artech Houses, ). J.P. Berenger, "A perfectl matched laer for the absorption of electromagneticwaves", J. Computational Phsics, 4, 85 (994) 3. J.P. Berenger, "Perfectl matched laer for the FDTD solution of wave-structure interaction problems", J. Computational Phsics, 7, 363, (996) 4. Z.S. Sacks, D.M. Kingsland, R. Lee and J.F. Lee, "A perfectl matched anisotropic absorber for use as an absorbing boundar condition", IEEE Trans. Antennas and Propagation, 43, 46 (995) 5. S.D. Gedne, "An anisotropic perfectl matched laer-absorbing medium for the truncation of FDTD lattices", IEEE Trans. Antennas and Propagation, 44, 63 (996) 6. Z.Y. Li and K.M. Ho, "Application of structural smmetries in the plane-wave-based transfer-matri method for three-dimensional photonic crstal waveguides", Phs. Rev. B 68, 457 (3) 7. Z.Y. Li and K.M. Ho, "Bloch mode reflection and lasing threshold in semiconductor nanowire laser arras", Phs. Rev. B 7, 4535 (5) 8. Yariv and P. Yeh, Optical waves in crstal, (Wile, 984) 9. Z. Ye, X. Hu, M. Li and K.M. Ho, "Propagation of guided modes in curved nanoribbon waveguides", Appl. Phs. Lett. 89, 48 (6). X. Hu, C.T. Chan, J. Zi, M. Li and K.M. Ho, "Diamagnetic response of metallic photonic crstals at infrared and visible frequencies", Phs. Rev. Lett. 96, 39 (6). S.A. Cummer, B.I. Popa, D. Schurig, D.R. Smith and J. Pendr, "Full-wave simulations of electromagnetic cloaking structures", Phs. Rev. E 74, 366 (6)

105 94 Chapter 6. TMM etension to curvilinear coordinate sstem The planewave based transfer (scattering) matri method is developed in curvilinear coordinates to stud the guided modes in curved nanoribbon waveguides. The problem of a curved structure is transformed into an equivalent straight structure with spatiall-dependent tensors of dielectric constant and magnetic permeabilit. We investigate the coupling between the eigenmodes of the straight part and those of the curved part when the waveguide is bent. We show that curved sections can result in strong oscillations in the transmission spectrum similar to the recent eperimental results in reference. This chapter is modified from a paper published at Applied Phsics Letters 89, 48 (6) with title: "Propagation of guided modes in curved nanoribbon waveguides", b Z. Ye, X. Hu, M. Li, K. Ho and P. Yang. 6. Transform into curvilinear coordinate In previous chapter, the planewave based transfer matri method (TMM) is mainl used in Euclidian coordinate sstem (, z)., The unit cells for an applicable structures are all rectangular in plane and straight at the propagation direction ( z ais). Although nonorthogonal lattice can be taken in the plane; it is not possible to deal with an structure that has an curvature along the propagation direction ( z ais). To stud the properties of curved photonic crstal structures, we need to re-develop the TMM algorithm in curvilinear coordinate sstem ( ', ', s) following previous work on curvilinear coordinate sstem. The relation between the Euclidian coordinate sstem and the curvilinear coordinate sstem is illustrated at Figure 6-. For Euclidian coordinate sstem,, z, are perpendicular parewisel and the propagation direction is z ais; for curvilinear coordinate sstem, s is

106 95 the propagation direction and ais is alwas pointing to the center of the curvature of radius R. The relation between the Euclidian sstem and curvilinear sstem can be epressed 3 3 b Eq. (6.). Let (,, ) be the coordinates of Euclidian sstem and (,, ) q q q be the coordinates of curvilinear sstem. The new coordinate sstem can be characterized b its covariant basis vector a i and metric g ij defined b Eq. (6.). In our curvilinear sstem case, the covariant basis vector and metric is epressed at Eq. (6.) and Eq. (6.). z ( a ) ( b) s ' ' R Figure 6-: Coordinate sstem (a) Euclidian (b) curvilinear s s = 'cos + R cos R R = ' s s z = 'sin + Rsin R R (6.)

107 96 3 ai =,, i i i q q q g k = q q k ij i j (6.) a a a ' ' s ' s s = cos,, sin R R = (,, ) ' s s ' s s = sin + sin,, cos + cos R R R R R R (6.3) g ij = ' R (6.4) Now we can appl the coordinate transformation toward Mawell s Equations in terms of differential equations relating to the transverse components of fields (Eq.(6.5) with some parameters defined at Eq. (6.6)). E H H s ' ik ε ' ' ' ' ' = [ ( )] ' ' ' = [ ( )] 3 3 ' E' H' H ' = [ ( )] ik μ H s ' ik ε ' ' H E E s ' ik μ ' ' ik μ H ik ε E ' ' H' E' E ' = [ ( )] + ik ε E s ' ik μ ' ' ' (6.5) ε = ε = εα, μ = μ = μα ε μ ' ε3, μ3, α = = = α α R (6.6)

108 97 So the arc structure can be viewed as a straight one with effective ε and μ tensors depending on the transverse coordinate ' with the relation displaed at Eq. (6.7). Giving a close look toward Eq. (6.7), we can notice that the have the same formation as the uni-aial perfect match laer discussed at reference but with the elements ε, ε, ε3, μ, μ, μ 3 function of coordinate. One other difference between perfect match laer and curvilinear coordinate transformation is that: for PML, each element is a comple number in which the imaginar part plas the role of absorption; for curvilinear coordinate transformation, each element is a real number. The curved part of the waveguide is perfectl matched with the straight part in the s direction, but not it the ', ' directions. ε μ ε = ε, μ = μ (6.7) ε 3 μ 3 When the curved waveguide is effectivel considered as a straight one with ε and μ given b Eq. (6.6) and Eq. (6.7), we can appl the well-developed planewave TMM to calculate the dispersion relations and eigenmode profiles in the curved waveguide. Semiconductor nanowires can nanoribbons have man promising optoelectronic applications such as waveguide,3, lasers 4-6, optical switches, and sensors Curved waveguide simulation With the knowledge of the curvilinear coordinate transformation for TMM, we can stud the curved waveguide structure in detail. The geometr used in this section is similar to the eperiment done in reference. The refractive inde of the background, square waveguide, substrate is: n =., n =., n =.5 respectivel. The cross-section of the square nanoribbon waveguide is 36 nm( ') 5 nm( '), shown as Figure 6-.

109 98 The dispersion relations can be calculated for both the straight waveguide (a) and the arc one with R = μm (b) which are shown in Figure 6-3; the insets give the E distribution for the first mode and E distribution for the second mode at 6 nm. 5 nm n = n =. 36nm n =.5 Figure 6-: Geometr illustration of nanoribbon waveguide (a).5 k a/(π)..5 (b)..5 k a/(π) k s a/(π) Figure 6-3: Dispersion relations of straight (a) and curved waveguide (b) with the supercell lattice constant a = μm. The dashed line is the light line in substrate: ks = nk, ( k= π / λ ). The left inset is the E distribution of the first mode and the right inset is the E distribution of the second mode at λ = 6nm.

110 99 The guided modes move downwards for a bent waveguide and the modal field shifts outwards from the center of curvatures ( ' direction) 8,9. The first and fourth modes are E polarized, and the second and third modes are E polarized. Modes after the fifth mode are not as highl polarized as the first four modes. We consider the transmission coefficients for a curved waveguide joint between two straight waveguides. Here we onl present the self transmission coefficients of guided eigenmode i( i =,,... ), which is defined as the ratio of the transmission energ flu of particular mode i and the incident energ flu (onl mode i incident). The reason for doing this is that in propagation along the straight waveguide, we epect some guided modes are much more sensitive to waveguide imperfections (such as sidewall roughness) than others. These sensitive modes are much more likel to disappear during propagation. We start from a simple "U" shape structure, made of two semi-infinite straight waveguides connected b a semicircular waveguide (see the inset in Figure 6-4 (a)). First we set R = μm. The self transmission of the first si guided eigenmodes is shown in Figure 6-4 (a). One can see regular fluctuations in transmission like in reference but with much weaker amplitude. The reflection is ver small ( < 4, not shown here). That agrees well with the above analsis of ε and μ given b Eq. (6.6) and Eq. (6.7). Then we tr an "L" structure, made of two semi-infinite straight waveguides connected b a quarter-circular waveguide with R = μm. The result is shown in Figure 6-4 (b). The amplitude of transmission fluctuation is even weaker; but the position of the transmission peaks and bottoms of the first four modes are about the same compared to Figure 6-4 (a). It is interesting that the first two E polarized modes (the st and 4th modes) have almost the same period, and the first two E polarized modes (the nd and 3rd modes) also do so.

111 We tested structures with different curvatures and different span lengths. We found that: (i) the amplitude of transmission fluctuation decreases as R is increased and (ii) the period of transmission fluctuation is onl related to the span length of the arc part and decreases as we etend the arc part. The first rule is natural to understand. Smaller radius of curvature enlarges the perturbation to the sstem, causing the transmission to fluctuate more intensivel (see Figure 6-4 (c), where R is set to be 4μ m ). The second rule can be eplained b mode conversion. Let us begin from a simple model. Suppose there are two modes in the waveguide marked i and j for the straight part and i ' and j ' for the curved part. If the arc part has length L, we can write down the self-transmission of mode i when the reflection is ver small: ' j ( ) ' ikil ik L 4 4 ' ii ' i ' i ij ' j ' i ii ' ij ' ii ' ii ' t e t + t e t = t + t + t t cos Δ k L where t, t, t, t are the convention coefficients from mode i to i ', i ' to i, i to ii' ii ' ij' ji ' respectivel, and * * ii' ii ' ij' ji ' Δ k = k k ; ' ' ' j i j ' and j ' to i t = t, t = t. Because Δ k ' is not ver sensitive to R, the period of transmission fluctuation is mainl related to the span length L of the arc part. The conversion between modes with the same polarization is much stronger than between different polarization modes. So the st, 4th modes have similar self-transmission periods and the nd, 3rd modes also have similar periods. However, higher order modes are not highl polarized in either or direction. The conversion rates between different polarized modes are not small. So their self-transmissions do not show fluctuations as regular as the first four modes (see the bottom plots of Figure 6-4 (a) and (b)). Our numerical results suggest a possible eplanation for the strong regular oscillation in the output spectrum observed b Law et al. The observed oscillations can be caused b a rippling section of nanoribbon (as shown in Figure 6-5 inset), which has a small bending radius for the curved part. We calculate the structure in Figure 6-5 inset for different parameters, and one of the results is shown in Figure 6-5. The parameters are: R = 3μm, L = μm, L = μm and periods of fluctuations (Figure 6-5 inset onl shows period). The self- 5

112 transmission diagram ehibit similar strong oscillations in the 3rd, 4th and 5th modes as in reference.. (a) Self-Transmission (b) Self-Transmission (c) Self-Transmission Wavelength(nm) R= μm 3 R=μm R= 4μm Figure 6-4: Self-transmission of the first si modes numbered from to 5 for: (a): U structure of R = μm, (b): L structure of R = μm, and (c): U structure of R = 4μm. Mode and mode 3 have the similar pattern; while mode and mode 4 have the similar pattern.

113 Figure 6-5: Self-transmission of the first si modes for the structure in the inset but of two periods of fluctuation, when R = 3μm, L = μm and L = μm. Inset: illustration of a waveguide containing a rippling section. 5 In summar, we have developed an improved TMM method in curvilinear coordinates to stud curved nanoribbon waveguides. Our method can be applied to an shape of curved waveguides. From our results we can etract and eplain two rules concerning the period and multitude of the transmission fluctuations. We finish b calculating a rippling waveguide structure and obtain oscillations in transmission similar to those observed in eperiments. References:. M. Law, D.J. Sirbul, J.C. Goldberger, R.J. Sakall, and P. Yang, "Nanoribbon Waveguides for Subwavelength Photonics Integration", Science 35, 69 (4). M. Skorobogati, S. Jacobs, S. Johnson, Y. Fink, "Geometric variations in high inde-contrast waveguides, coupled mode theor in curvilinear coordinates", Opt. Epress,, 7 () 3. C.J. Barrelet, A.B. Gretak, and C.M. Lieber, "Nanowire Photonic Circuit Elements", Nano Lett. 4, 98 (4)

114 3 4. M.H. Huang, S. Mao, H. Feick, H.Q. Yan, Y.Y. Wu, H. Kind, E. Weber, R. Russo, and P.D. Yang, "Room-temperature ultraviolet nanowire nanolasers", Science 9, 897 () 5. X. Duan, Y. Huang, R. Agarwal, and C.M. Lieber, "Single-nanowire electricall driven lasers", Nature (London) 4, 4 (3) 6. A.V. Masolv and C.Z. Ning, "Reflection of guided modes in a semiconductor nanowire laser", Appl. Phs. Lett. 83, 37 (3) 7. H. Kind, H. Yan, B. Messer, M. Law, and P. Yang, "Nanowire ultraviolet photodetectors and optical switches", Adv. Mater. (Weinheim, Ger.) 4, 58 () 8. V. Van, P.P. Absil, J.V. Hrniewicz, and P.T. Ho, "Propagation loss in single-mode GaAs-AlGaAs microring resonators: Measurement and model", J. Lightwave Technol. 9, 734 () 9. Y.A. Vlasov and S.J. McNab, "Losses in single-mode silicon-on-insulator strip waveguides and bends", Opt. Epress,6 (4)

115 4 Chapter 7. Application of TMM to diffractive optics Besides the magnitude of the electromagnetic wave transmission or reflection rate, the planewave based transfer (scattering) matri method (TMM) can also calculate the phase information of the electric or magnetic field. When the phase at ever point of a plane is determined b the geometr (shape of a structure) and composition (material with different refractive indices), the properties of electromagnetic wave or light (such as bending, focusing) can be determined. Photonic crstal structure can be designed to meet certain purpose in controlling the behaviors of electromagnetic wave. For eample, dielectric grating (D photonic crstal) has been proposed to appl to camera lens to eliminate chromatic aberrations. In this chapter, TMM is developed to calculate the phase difference and in turn to design novel photonic crstal structures for various applications. 7. Finding phase b TMM The spectrum can be obtained b finding the Ponting vector through Eq. (.53) at Section.6.. Before the Ponting vector is calculated, TMM has alread obtained the electric field in term of comple vector ( E, E ) for transmission and reflection wave b Eq. (.5). The t r information of the comple electric field vectors is used to calculate the spectrum, and the phase factor information of this comple electric field vectors is used to obtain the phase of the electric field in the range of ( π, π ). Let s denote the imaginar and real part of the E t or respectivel and the mode (or absolute value) of E t or phase of the electric field ( E or t E r ) can be found b Eq. (7.). E r at Eq. (.5) to be Im and Re E r at Eq. (.5) to be Ab, then the

116 5 Im ϕ = arcsin if Im and Re Ab Im ϕ = π arcsin if Im and Re < Ab -Im ϕ = π + arcsin if Im < and Re < Ab -Im ϕ = arcsin if Im < and Re Ab (7.) Cautions must be aware when calculating the phase of photonic crstal devices, because we onl have the information of phase within the range of ( π, π ) ; and there is no information of how man π has alread passed. In the real situation, those π periods are essential to determine the phase difference and we have to recover this information. One wa to recover the π period is to first calculate one repeat pattern along the propagation direction (one geometr period along z ais); and if the phase of this one geometr period is less than π then the phase from TMM is the real phase of one geometr period. For multiple geometr periods, we can simpl multipl the number of geometr periods and the real phase of one geometr period. Please note we can not cut one geometr period into small portions to calculate the phase of each portion individuall to get the phase of one geometr period. B doing so, we have destroed the internal properties of the structure. One other approach to get the real phase of arbitrar periods is to start calculate the phase from ver low frequenc (around zero normalized frequenc) which correspond to ver long wavelength. For ver long wavelength, the phase difference between the two ends of the structure is less than π. B increasing the frequenc graduall, we can track each jump of phase which indicates the number of periods (or π s). Then we can obtain the real phase of an frequenc. B the wa, the phase change of a uniform medium with refractive inde n, normalized frequenc α and thickness t (in the unit of normalized length) can be calculated as: ϕ = kt= αn( π) t/ c which can be used as a rough reference value for the phase. z

117 6 7. Confirmation b Snell s Law The law of refraction (Snell s Law) is a good eample to confirm our idea of modeling the propagation properties b finding the phase difference through TMM. The law of refraction can be used in an interface between two uniformed medium with the mathematical form Eq. (7.) with n and n the refractive inde of those two uniformed medium and α and α the incident angle and refraction angle. n sinα = n sinα (7.) One of the simplest was to steer the electromagnetic wave is to use an incline of dielectric material with the incline angle θ and refraction inde n ; and the background is air with refractive inde n. Now let s suppose the electromagnetic wave incidents verticall as shown at Figure 7-. The deflection angle is defined as the variation of refracted wave with respect to the incidence wave (shown as α at Figure 7-). Given fied incline angle θ and background refractive inde n, the deflection angle α is a function of refractive inde n. According to law of refection, we have n sinθ = n sin( θ + α) ; and the deflection angle α can be epressed as Eq. (7.3). For a set of parameter of: n = 3., n =. and θ =, the deflection angle is α =.396. Here we have the deflection angle twice as large as the incline angle. B changing the incline angle, we can reach certain range of deflection angles. But there is several shortage of this simplest design: first it is not eas to change the incline angle θ for solid inclines (high refractive inde liquid ma solve this problem); the transmission rate is low due to the contrast of the two mediums (i.e. strong reflection occurs at the dielectric material air interface). n n α = sin sinθ θ (7.3)

118 7 α θ n θ θ n Figure 7-: Deflection of angle of a simple incline b Law of refraction On the other hand, we can deal the same problem b using the phase difference approach. The same structure is re-plotted at Figure 7- with the incline s length L and height d. The deflection angle α can be epressed as α = π / β θ. The angle β can be determined from the relation of wave vector k AB and k which can be calculated from the phase difference (Eq. (7.4)). From Eq. (7.4), we can get the epression of k AB in term of k at Eq. π / β = sin n sinθ. Finall, (7.5). Then angle β can be found through Eq. (7.6). So ( ) we get the same result α = sin ( n sinθ) θ as the law of refraction. d A k AB k β θ α n = k L n θ B Figure 7-: Deflection of angle of a simple incline b phase difference approach. The deflection angle α can be determined b the phase difference of arbitrar two horizontal locations.

119 8 Δ ϕ = k AB AB AB AB Δ ϕ = kd= knd (7.4) kab Δϕ AB = = k sin n θ (7.5) d /sinθ ( k k ) ( n θ ) β = cos / = cos sin (7.6) AB With the idea of phase difference, we confirm the result of the law of refraction. It is eas to calculate the phase difference in uniform medium, but this is not true for photonic crstal structures. Our TMM method can calculate the phase difference of an structures for an given frequenc and in turn find the deflection angle. At the same time the transmission rate can be also figured out via TMM. We can appl this approach to our bo spring case (detail at net section) to find the phase difference between the compressed and un-compressed part and then figure out the deflection angle for certain geometr configuration. 7.3 Case stud: bo spring structures for electromagnetic wave deflection In this section, we are going to use the concept of phase difference from TMM to determine the deflection angle of electromagnetic wave through a photonic crstal structure. Our purpose is to easil control the direction of electromagnetic wave propagation b adjusting the photonic crstal structure, mean while the transmission rate is still acceptable. As we mentioned in last section, the law of refraction used in two uniform medium interface is lack of fleibilit to change the deflection angle and the transmission through the interface is affected b the high refractive inde contrast. One advantage of the two uniform medium interface is that the response of the reflection angle is broad band (the deflection angles are

120 9 the same for an frequenc if the material is non-dispersive). For photonic crstal structures, the frequenc responses is usuall non linear; and it ma onl work for a short frequenc range, for eample in the vicinit of band gaps. In certain application, a narrow bandwidth is still acceptable. There are several was to control and adjust the response of photonic crstal structure, for eample: mechanical, electrical or optical response. The most direct wa to change the photonic crstal structure is mechanicall change its shape and geometr. One candidate for such purposes is bo springs which are easil modified b different comparison rates. The bo spring photonic crstal has been studied b other researchers and the eistence of band gap has been confirmed. 3, Geometr of bo spring structures Bo spring is a rather complicate photonic crstal structure. The bo spring is periodic in plane and there are mabe overlaps across neighborhood springs. Figure 7-3 shows an eample of overlapped 55 period bo springs. The bo spring can be compressed to achieve desired properties. The most compressed situation is the overlap of a set of bo rings and the most uncompressed situation is a set of parallel rods in plane. Figure 7-3: 3D view of 55 periods of bo spring

121 The top view and side view of the bo spring photonic crstal structure are shown at Figure 7-4. The overlap detail can be observed. Along the z direction there are four equivalent segments which have the same lift angle (shown at Figure 7-5) and length. Those four equivalent segments are perpendicular to each other. The lattice constant along or direction is the length between two neighbor rods (shown at Figure 7-5). (a) (b) Figure 7-4: Top view (a) and side view (b) of the bo spring photonic crstal structure. Overlap feature can be observed in detail. For one period along the z direction, there are four equivalent segments. Horizontal Projection Lattice constant Lift angle ¼ vertical length Figure 7-5: Detail of one segment along the z direction

122 For the calculation of uncompressed bo spring photonic crstal structures, we use the following parameters: lattice constant (in plane rod to rod distant) a ; rod length.6a ; rod radius.6a ; lift angle ; refractive inde of the rod material n = Simulation results of bo spring structures First, we studied the band structure of uncompressed bo spring (parameters listed on the end of last section) and three cases of compressed bo spring with the C/ C ratio.,.4 and.6 where C is the length of one period compressed bo spring along z direction and C is the length of one period uncompressed bo spring along z direction. The band structures of all four cases are plotted at Figure ωa/πc.6.4. Band-C Band-C Band-C Band-C3. Γ Z X RM A Figure 7-6: Band structure for uncompressed and compressed bo spring photonic crstals. With the information of each case (uncompressed and compressed), we can stud the relation between deflection angle ( β ) and compressed angle (θ ) based on the illustration of Figure 7-7. The left side is the uncompressed bo spring and right ride is the compressed bo spring with compression ration C/ C and the bo spring is compressed graduall between the left

123 side and right side and form an incline. The length between the left and right bo spring is set to a and the vertical length is 5 periods. β θ ϕ K d K β θ 5a 5a-5Δc C a Ci Figure 7-7: Illustration of how to deflect electromagnetic wave through uncompressed and compressed bo spring photonic crstal. Then we appl the strateg develop at section 7. to get the deflection angle b phase difference at left and right side. Here, we assume we can calculate the phase for each bo spring individuall without the interaction across neighborhood bo springs and then get the phase difference of left and right side. We first calculate the phase for the uncompressed bo spring (left side) and then calculate the phase of the compressed bo spring (right side). Then the phase difference is obtained just like the procedure to confirm the Snell s Law (Eq. (7.7) with α = ka /π the normalized frequenc).

124 3 π ϕ = β θ 5a 5( a Δc) 5Δc θ = tan = tan a a Δφ a π Δ φ = φ φi, kd p = Δ φ, sokp =, d=, k= α d cosθ a k p Δφ/ d Δφ cos θ /( a) β = cos cos cos k = = α π / a α π / a Δφ cosθ = cos α π (7.7) After the phase difference obtained b TMM, we can get the relationship of deflection angle for different normalized frequenc at fied compression ratio. The result of compression ration.6 at right side is shown at Figure 7-8 (in this result, the distance between the uncompressed and compressed spring is one lattice constant). The deflection angle is oscillating and remains small for a large frequenc range below the first band gap. But ver large deflection angle can be achieved at the vicinit of first band gap (Figure 7-9); at the same time the transmission rate is still acceptable (Figure 7-): at normalized frequenc a / λ =.675, the deflection angle is 6.6 and the transmission of both uncompressed and compressed bo spring is around 7%. 6. deflection angle (degree) Not compressed Compressed 6% deflection angle Large defleciton angle occurs with frequenc close to the first band gap. Detail will show on net slide Transmission Normalized Freq. (a/λ) Figure 7-8: Deflection angle for all frequenc with transmission for both uncompressed and compressed case.

125 4 deflection angle (degree) a/λ= Normalized Freq. (a/λ) Figure 7-9: Detailed deflection angle at the vicinit of first band gap..8 Not compressed compressed 6% Transmission.6.4 a/λ= Normalized Freq. (a/λ) Figure 7-: Detailed transmission at the vicinit of first band gap The non-linear effect at the photonic band edge makes possible the large deflection angle. But the bandwidth for this large deflection angle is quite small. The deflection angle can be positive (deflected to the left) or negative (deflected to the right) which corresponds to normal refraction and negative refraction respectivel. In this eample, although we have found the deflection angle and transmission rate as functions of normalized frequenc, the design of device of certain function is far from completed. Optimization is needed to get large bandwidth and more fleible deflection angle responses. But the idea of analzing

126 5 phase at the position of plane from TMM can be generalized and applied to future device design and simulation. References:. M. Born and E. Wolf, Principles of Optics, 7th edition, Cambridge Universit Press, d=63 webpage accessed October 7 3. O. Toader and S. John, "Square spiral photonic crstals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps", Phs. Rev. E 66, 66 () 4. O. Toader and S. John, "Proposed Square Spiral Microfabrication Architecture for Large Three-Dimensional Photonic Band Gap Crstals", Science, 9, ()

127 6 Chapter 8. TMM algorithm with active gain material etension Photonic crstal cavit has been long proposed to be a good candidate for low-threshold laser, while current available numerical methods based on rate equations are not efficient and not accurate for simulation such small scale devices and for situations well above the threshold. Our previous chapters discussed the planewave based transfer (scattering) matri method (TMM) mainl focus on passive material and the simulation results such as spectra, mode profiles for cavit embedded photonic crstal are onl valid for cold cavit. However, our TMM algorithm is not limited to the passive material onl; in this chapter detailed algorithm etension with capabilit for direct modeling active gain material is included along with several simulated active photonic crstal devices. The major advantage of Gain material TMM (GTMM) is to provide more precise lasing spectra, mode shape information and lightcurrent (L-I) curve due to the fact that optical solution in GTMM is obtained in hot cavit condition which is self-consistent with laser pumping intensit. Eventuall, GTMM can be used to solve comple 3D photonic crstal laser and light emitting diode (LED) structures. 8. Rate equation, the starting point Rate equation is the fundamental simulation technique used in laser and LED device design which connects the electric propert and optical propert together b a set of differential equations. The gain medium laer is tpicall embedded in the middle of the structure; Figure 8- shows a schematic picture of a diode laser. The middle gain laer has width w, thickness d, length L, and a refractive inde higher than the top and bottom laers, supporting a guiding mode with average width of d / Γ. When a current with densit of J is injected downward, the middle laer can present a gain factor g to amplif light.,,3

128 7 I=J W L d Γ z d w L L out Figure 8-: Schematic of a photonic crstal laser To simplif the derivation and show the main idea of how rate equation works, we use the simple mode rate equation (Eq. (8.)) with quantum dot gain relation (Eq. (8.)) with the average carrier densit N, the average photon densit P, and gain factor g. The inject current I and the light output L out are given b Eq. (8.3). The phsical meanings of η i, q,τ, v g, α c, A, N etc. are listed and eplained at Table 8- with a set of tpical values of a diode laser. dn ηi J = τ N υg g P dt q d dp Γ N β = + υ g ( Γ g α c) P dt τ s (8.) g = A ( N N ) (8.) I = J w L L = P E Γ V τ out p c p (8.3) dn dp = = (8.4) dt dt

129 8 Table 8-: Phsical meanings of parameters in rate equation with tpical numerical values for a diode laser Phsical meaning Tpical numerical values Vacuum speed of light: c 3 cm/s Single electron charge: q.6-9 C Plank s constant: h J s Differential gain: A.5-6 cm Transparent carrier densit: N cm -3 Optical group inde: n g 3.5 Effective velocit: v g v g = c/n g Facet reflectivit: R, R R = R = (n g -) (n g +) - Absorption loss: α cm - Facet loss: α r α r = (/L)Ln[/(R R )] Total optical loss: α c α c =α r +α Carrier spontaneous emission lifetime: τ s 3-9 s Carrier non-radioactive deca lifetime: τ nr 3-6 s Total carrier lifetime: τ τ =(τ - s + τ - nr ) - Photon out-coupling lifetime: τ p τ p =/( v g α r ) Lasing wavelength in vacuum: λ.3-4 cm Thickness of active laer: d. -5 cm Frequenc: ν ν = c/λ Single photon energ: E p E p = hν = hc/λ Confinement factor: Γ.5 Spontaneous emission factor: β -4 Cavit width: w 5-4 cm Cavit length: L.5 cm Cavit active volume: V c d w L

130 9 At the stead state (i.e. the constant wave output with a single lasing frequenc), the average carrier densit and average photon densit for fied inject currents should not change (Eq. (8.4)). For a given inject current, the stead state quantities can be obtained from Eq. (8.) to Eq. (8.4). With large inject current approimation ( g α Γ ), the results are epressed at Eq. (8.5). c E pηiα r Lout ( I Ith) qα c ( ) I = α Γ A + N q V η τ th c c i dl di out E pηα i r = qα c (8.5) The first equation of Eq. (8.5) indicates there is a lasing threshold, and the third equation of Eq. (8.5) indicates one additional carrier will generate η i α r /α c photon when the inject current is above the threshold. The plot of the first equation of Eq. (8.5) illustrates the relation between the output light intensit (in unit of Watt) and input current (in Ampere) which is usuall called the L-I curve. 8. Defining the electric field dependent dielectric constant for gain material In the single mode rate equation algorithm (section 8.), the second equation of Eq. (8.) is used to describe the light in the cavit b using averaged parameters α c and Γ. Those two parameters are difficult to define if the gain material has comple structure and the dimension of the cavit is comparable with lasing wavelength (such as D PC slab cavit laser and 3D photonic crstal laser). Even if the definitions of averaged parameters are given, the calculations can onl be performed for the cold cavit case without gain (i.e. below the threshold).

131 To avoid the difficulties, we will not use the second equation of Eq. (8.) with averaged parameters to describe the light in laser cavit. Instead, we will use rigorous GTMM formulism and electric field dependent dielectric constant for gain materials. The dielectric constant of gain material can be written as Eq. (8.6) with ε ini the dielectric constant when the electric field at the gain material is zero. ε c ω = ( εini i g) (8.6) With the first equation of Eq. (8.), Eq. (8.) and Eq. (8.4), we can epress the average carrier densit and gain factor in Eq. (8.7). With photon densit P in term of electric field and single photon energ (Eq. (8.8)), the gain factor can be further rewritten as Eq. (8.9). ητ N = g = i q d J Aτυ gpn + Aτυ P A( ητ i q d J N) + Aτυ P + g g (8.7) ε / P= E E p (8.8) g g = + a E g = AητJ /( qd) A N i a = Aτυ ε /( E ) g p (8.9) To consider the spontaneous emission, the second equation of Eq. (8.) is rewritten as Eq. (8.), and the gain factor with spontaneous emission can be epressed as Eq. (8.). It should be mentioned that the spontaneous emission factor β is the coupling factor between the total emission of dipole and the lasing cavit mode and depends on the eact coordinates and polarization of dipole. Spatiall dependent β should be used if we would like to know

132 the eact behavior below the lasing threshold. But for simplicit, we will use an averaged β for the gain materials. dp = υg ( Γ g' αc) P dt β g' = g+ N υτp g s (8.) g b+ b E g' = + β + a E E ( + a E ) b = EητJ/( τ qdυ ε) b = ANτ/ τ p i s g s (8.) 8.3 Gain-TMM algorithm for laser device simulation In last section, we have epressed the dielectric constant of gain medium and gain factor in term of electric field intensit. Various photonic crstal lasers have been proposed recentl. 4,5,6,7,8 Now in this section the gain transfer (scattering) matri algorithm (GTMM) is presented with iterations to simulate laser devices. Same as the TMM algorithm assumed, periodic boundar condition is applied to the XY plane and the planewave in defined as the usual wa. We use N p vector A i and B i to represent the electric field Fourier coefficient at each slice i, and from our previous chapters the outgoing waves can be obtained from the incidence wave b the scattering matri S as epressed at Eq. (8.). The schematic of GTMM structure is presented at Figure 8-. The lasing status is when there is no light input while there is still continues light output. An + A = S B B n+ (8.)

133 A B z i n A n+ B n+ Figure 8-: Schematic of Gain-TMM with the whole photonic crstal structure divided into n slices and the outgoing planewaves are found b the S matri and incidence planewaves. Mathematicall to simulate a laser problem, we set solving When () A = ( A,,,) and B + = (,,) () A for a given output L, b Eq. (8.3) where p = for all evanescent waves. out right () A is obtained, we can use the TMM algorithm to calculate the field distribution. With the calculated field distribution, the new dielectric constant for gain medium can be obtained through Eq. (8.6). In turn, with the updated dielectric constant of gain medium a new S matri can be found. With the new S matri, a new j () A can be obtained. n M ( j) out, right = j n+ j= L p A A = A S ( j) () n+ j (8.3) Repeat the iteration, until the stead A () is finall achieved for given inject current I and output power L out, right. Then we will calculate A (), the incident planewave, for various L out, right and incident frequencies. When the inject current is above threshold ( I Ith () A can be found for non-zero out, right > ), a zero L at certain incident frequenc which indicates a stead finite light output eists for zero input light or lasing. The frequenc where a zero A () is found is the lasing frequenc and the light output power for certain current makes the L-I curve. The detailed procedures will be illustrated through an eample of D DBR laser at net section.

134 3 8.4 D DBR laser, an eample of GTMM application The multilaer D DBR cavit structure is illustrated at Figure 8-3. Based on the mentioned parameters at the figure caption, there will be a resonant mode at.98μ m which is obtained from TMM spectrum calculation and the results are also shown at Figure 8-3. w d Z - Transmittance Wave length (μm) Figure 8-3: Schematic of D DBR laser with a multilaer structure (AB) 4 EGE(BA) 4, where G is the Gain material laer. The refractive indices and the laer thicknesses are as following: n AEG,, = 3.56, n B = 3.77, and da = 64.49nm db = 79.63nm, de = 37.78nm, dg = nm. The transition rate shows μ. a single resonant peak at.98 m As mentioned at last section, for fied incident frequenc and light output with the presence of active gain material inside the structure, the required light input can be obtained through the iteration process until a stead light input value is reached. B varing the light output

135 4 value with fied inject current and incident frequenc, the light output vs. light input graph can be obtained. Three light output vs. light input relations are plotted at Figure 8-4: basicall there are two saddle points for each curves. Above threshold the saddle point located at the larger light output region reaches zero which corresponds to lasing, while below threshold, the saddle point located at the smaller light output region reaches zero which corresponds to spontaneous emission, and just at the threshold those two saddle points come together and reaches zero at lasing frequenc. For light output vs. light input relations, if the incident frequenc is not the lasing frequenc, neither of those two saddle points reaches zero, i.e. no lasing or spontaneous emission are found. Above threshold (I=5.E-8) - Below threshold (I=.5E-8) Just at threshold (I=.6E-8) -4 Light input (W) Light output (W) Figure 8-4: Light input vs. light out graph at lasing frequenc for three different inject currents: above threshold, below threshold, and just at threshold. For each inject current, b iteration for various incident frequencies, eventuall there will a saddle point reaches zero which corresponds one (light output, current) point located at the L-I curve. When the inject current is above threshold, it is lasing case, otherwise it is a

136 5 spontaneous case. The light output intensit is several orders larger at lasing case than the spontaneous emission. The L-I curve of the D DBR laser is illustrated at Figure above threshold case: I=5E-8 A Light output power (W) below threshold case: I=5E-9 A Just at threshold case: I=.6E-8 A Inject current (A) Figure 8-5: Light output power vs. inject current (L-I curve) for D DBR laser. Three tpical inject currents are selected to show the difference across above threshold, below threshold and just at threshold with corresponding light input vs. light out graphs shown at Figure 8-4. The lasing frequenc is found to be almost the same as the resonant frequenc of passive cavit with little dependence on inject currents. This is reasonable because of the small thickness of the gain medium laer and the small change of the dielectric constant of the gain medium (for the D DBR case, the imaginar part of dielectric constant is onl around -.3). But to show the little difference of lasing frequenc, the D DBR lasing frequenc for various inject currents are present at Figure 8-6.

137 6.E+ Lasering Frequenc (normalized).e+.e+.e+.e+.e+.e+.e+.e+.e+ below threshold case: I=5E-9 A ω = L Just at threshold case: I=.6E-8 A ω L = above threshold case: I=5E-8 A ω = L.E+.E+.E-8.E-8 3.E-8 4.E-8 5.E-8 Inject Current (A) Figure 8-6: Lasing frequenc for various inject current, here unit of normalized frequenc: μm / λ ω L is in 8.5 3D woodpile photonic crstal laser, an eample of GTMM application To test the performance and the capacit of GTMM, we applied it to a 3D photonic crstal laser. The gain medium is located at the center of the cavit laer embedded in the classical laer-b-laer woodpile photonic crstal structure. The cross sections of 3-b-3 super cell of the woodpile structure are illustrated at Figure 8-7 (a) with the gain medium laer has the same geometr as the cavit laer. From the TMM spectra results, there are two resonant modes (for the passive cavit) inside the band gap from two different polarizations (shown on Figure 8-7 (b)). We are going to focus on the -polarization mode which has resonant frequenc 98nm with Q value around 8 and transmittance around 5%.

138 7 Figure 8-7: (a) Cross sections of the 3D woodpile photonic crstal laser (ABCD) EGE(BCDA) composed of rectangular dielectric rods in air, where G is the gain medium laer. The structure has a periodicit of 3a along both X and Y directions, where a = nm. The rods have thickness da,b,c,d =.3a, d G = nm, d E = nm and width w =.5a, and refractive inde of (b) Transmission spectra for normal incidence of two polarized. The insect shows the photonic crstal laser structure with red laer the gain medium laer. With the assumption of uniform pumping current in the gain medium, the total lasing output and lasing frequenc can be found for different inject currents and the results are shown at Figure 8-8 (a). Above threshold, the hot cavit electric field mode profile shows strong spatial dependence which in turn causes the strong spatial hole burning effects of nonuniform imaginar part of gain material dielectric constant and non-uniform gain factor at different locations of the gain medium laer (Figure 8-8 (b)). Due to the large change of imaginar part of dielectric constant at gain medium laer, the deviation of lasing frequenc form the passive cavit resonant frequenc is large and strongl depends on the inject current.

139 8 Figure 8-8: (a) light output power vs. inject current (L-I curve) of 3D woodpile laser; (b) electric field distribution and imaginar part of dielectric constant inside the gain medium laer when lasing at inject current 4nA. The L-I curve, lasing mode profile and lasing frequenc chirping effect can be calculated from the gain material transfer (scattering) matri method. The D DBR laser case confirms the accurac of GTMM and the 3D woodpile laser case shows the power of GTMM. In the future, the GTMM algorithm can be applied efficientl and accuratel to simulate and design future photonic crstal laser devices. References:. L.A. Coldren and S.W. Corzine, Diode Lasers and Photonic Integrated Circuites, John Wile and Sons, New York, 995. A. Yariv, Quantum Electronics, John Wile and Sons, New York, 989