Sochasc Maxwell Equaons n Phoonc Crsal Modelng and Smulaons Hao-Mn Zhou School of Mah Georga Insue of Technolog Jon work wh: Al Adb ECE Majd Bade ECE Shu-Nee Chow Mah IPAM UCLA Aprl 14-18 2008 Parall suppored b NSF
Oulne Inroducon & Movaon Drec Mehod A Sochasc Model Wener Chaos Expansons WCE Numercal Mehod based on WCE Smulaon resuls Concluson
Inroducon & Movaon Sochasc PDE s : Flud Dnamcs Engneerng Maeral Scences Bolog Fnance Soluons are no longer deermnsc. Man neres: sascal properes such as mean varance. Mul-scale srucures.
Inroducon & Movaon J J j j = δ Spaall ncoheren source: J j J Such as dffuse lgh n opcs.
Inroducon & Movaon Applcaons n sensng Raman specroscop for bo and envronmenal sensng Human ssue Phoonc Crsal specromeer n nano-scale Spaall ncoheren lgh
Phoonc Crsal Specromeer Oupu A Oupu B Oupu C Oupu D Oupu E Oupu F Incden Spaall Incoheren Feld Heerogeneous Phoonc Srucure Specrall Dverse Feld Deecors Mulplex mulmodal specromeer
Inroducon & Movaon a Inpu Source ncoheren a A J Oupu a B elecrc feld nens E 2 Phoonc Crsal desgned as he medum goal: model he ncoheren source and smulae oupu Frs sep n he desgn of Phoonc Crsal specromeers Opmal desgn of he shapes of Phoonc Crsals for larges band gap Kao- Osher-Yablonovch 05 Wave propagaon s governed b Maxwell equaons
Maxwell Equaon Maxwell equaons: H r E r = μ E r H r = ε r + J r E H = 0 = 0 Helmholz wave equaon: E r = με r E r 2 2 Er Hr Jr J r μ elecrc feld magnec feld Inpu ncoheren source
Helmholz wave equaon z-nvaran srucure mples wo ses of decoupled equaons Transverse Magnec TM E z H x H Transverse Elecrc TE H z E x E 3D space srucure reduces o 2D Helmholz TM wave equaon: E x + E x με x E x = μ J x xx PDE s are lnear.
Drec Mehod for PC Specromeer Incoheren proper mples he drec brue-force mehod. Inpu nonzero pon source p a and J p = 0 j. j Compue oupu elecrc feld a B Toal elecrc nens a B: Wh pon source? Non-pon sources such as plane waves p 2 p 2 p lead o coheren oupus. Pro: correc phscs lnear equaons + ncoheren oupus. Con: ver neffcen. J E p E x = E x. 2 p 2 B B E = E + j E E p j
A Sochasc Model E x + E x με x E x = μ J x xx a Spaall ncoheren source J x X V z A = V = sn? ω Sochasc model 0 exp T 0 X = dw W 2 10ax10a Phoonc Crsal as he smulaon medum Brownan Moon. More general: J x = f xdwx
Sochasc Helmholz Wave Equaons E x + E x με x E x = μ J x xx Curren dens s a sochasc source. Soluon for elecrc feld s random. Mone Carlo smulaon s slow and hard o recover he ncoheren properes Our sraeg: WCE.
Mone Carlo No man compuaonal mehods avalable. Tradonal mehods Mone Carlo MC smulaons Solve he equaons realzaon b realzaon. Each realzaon he equaons become deermnsc and solved b classcal mehods. The soluons are reaed as samples o exrac sascal properes.
Mone Carlo MC can be ver expensve: Has slow convergence governed b law of large numbers O and he convergence s no monoone. n s he number of MC realzaons Need o resolve he fne scales n each realzaon o oban he small scale effecs on large scales whle onl large scale sascs are of neress such as long me and large scale behavors. Hard o esmae errors Mus nvolve random number generaors whch need o be carefull chosen. 1 n
Wener Chaos Expansons Goal: Desgn effcen numercal mehods. Separae he deermnsc properes from randomness. Has beer conrol on he errors. Avod random number generaors all compuaons are deermnsc.
Wener Chaos Expansons u x dw Funcons depends on Brownan moon W. W conans nfnel man ndependen Gaussan random varables s me and/or spaal dependen. u x dw WCE: decompose b orhogonal polnomals smlar o a specral mehod bu for random varables.
Wener Chaos Expansons m s 2 an orhonormal bass of L 0 Y such as harmonc funcons n our compuaons. Y Defne ξ = whch are ndependen m s dws Gaussan. 0 Le ξ = ξ ξ consruc Wck s producs 1 2 T = α ξ Hα ξ. α =1 s a mul-ndex H ξ Herme polnomals. α
Wener Chaos Expansons Cameron-Marn1947: an can be decomposed as u x dw where u x dw u x Ta = α ξ α u x = E u x ξ T ξ. α α
Wener Chaos Expansons Sascs can be reconsruced from Wener Chaos coeffcens mean Eu x = u x 0 varance E 2 u x = α 2 u α Hgher order momens can be compued oo.
Wener Chaos Expansons Properes of Wck s producs: E T 0 ξ = 1 E ξ = 0 α E T α T α T β 0 = 1 0 α β α = β
Wener Chaos Expansons Wener Chaos expansons have been used n Nonlnear flerng Zaka equaon Loosk Mkulevcus & Rozovsk 97 Sochasc meda problems Mahes & Bucher 99 Theorecal sud of Sochasc Naver-Sokes equaons Mkulevcus & Rozovsk 02
Herme Polnomal Expansons A long hsor of usng Herme polnomals n PDE s conanng Gaussan random varables. Random flows: Orszag & Bssonnee 67 Crow & Canavan 70 Chorn 7174 Malz & Hzl 79 Sochasc fne elemen: Ghanem e al 9199. Specral polnomal chaos expansons: Karnadaks Su and collaboraors a collecon of papers and WCE for problems n flud: Hou Rozovsk Luo Zhou 04
WCE for sochasc Helmholz equaon Expand he source and elecrc feld: Take advanage of Equaon s lnear so elecrc feld has expanson ξ T J dw J = = =1 m dw ξ ξ T x E dw x E = = m V dw J ξ Onl Gaussan lnear = x E dw x E ξ
WCE for sochasc Helmholz equaon The sochasc equaon s convered no a collecon decoupled of deermnsc Helmholz equaons E x + E x με x E x = μ V m xx Sandard numercal mehods such as fne dfference me doman FDTD n our smulaon can be appled.
WCE for sochasc Helmholz equaon WCE coeffcens x are coheren! E The elecrc feld nens a oupu s compued b E = E = E. 2 p 2 2
WCE for sochasc Helmholz equaon The elecrc felds from pon sources can be recovered b p E = x m j E j x j Oher momens can be compued b pon source soluons n he sandard was. Under relave general condons WCE coeffcens deca quckl E p x E 1 O r r Relaed o he smoohness of he soluons.
Smulaon of a spaall ncoheren source Comparson of he drec mehod brue-force smulaon and he WCE mehod Convergence of he WCE mehod Exremel fas convergence for 10ax10a example
Smulaon of a spaall ncoheren source 3 For a 20ax10a example doubled szed 20ax10a Percenage error Number of coeffcens For 15 coeffcens he gan n smulaon me s 32.e. 32 mes faser smulaon Less han 1% error and more han one order of magnude faser smulaon Over 2 order of magnude faser smulaons for praccal phoonc crsals.
Concluson Proposed a sochasc model for ncoheren source Desgn a fas numercal mehod based on WCE o smulae he ncoheren source for phoonc crsals. The mehod can be coupled wh oher fas Maxwell equaons solvers. More han 2 order of magnude faser smulaons can be acheved for praccal srucures. The model and mehod are general and can be appled o oher pes of sochasc problems nvolvng ncoheren sources.