Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Hig he r-o rde r Me tho ds fo r Simula ting L ig ht Pro pa g a tio n a nd L ig ht-Ma tte r I nte ra c tio n in Na no -Pho to nic Syste ms K urt Busc h I nstitut für T he o re tisc he F e stkö rpe rphysik Unive rsitä t K a rlsruhe , 76128 K a rlsruhe , Ge rma ny DF G-Ce nte r fo r F unc tio na l Na no struc ture s a nd K a rlsruhe Sc ho o l o f Optic s & Pho to nic s phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Ac kno wle dg me nts L a sha T ke she la shvili Mic ha e l K ö nig Je ns Nie g e ma nn Ja n Gie se le r Ma rtin Po to tsc hnig * K a i Sta nnig e l phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n L ine a r, no nline a r a nd q ua ntum o ptic a l pro b le ms in na no -pho to nic syste ms invo lve multiple time a nd le ng th sc a le s T his re q uire s a c c ura te , sta b le , a nd e ffic ie nt so lve rs fo r line a r a nd no nline a r Ma xwe ll’ s e q ua tio n a nd So lito n c o llisio n in a c o llisio n in a So lito n c o uple d syste ms fib e r Bra g g g ra ting fib e r Bra g g g ra ting phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n: Sta nda rd Appro a c he s F inite -E le me nt-Me tho d F DT D-Me tho d Disc re tiza tio n o n unstruc ture d g rids Disc re tiza tio n o n Ye e -g rid 2 nd o rde r in spa c e a nd time Hig he r-o rd e r in spa c e F re q ue nc y-d o ma in pre fe rre d E ffic ie nt a nd e a sy to imple me nt phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n: Do no t trust Co mpute rs I phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n: Do no t trust Co mpute rs I I phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n: Do no t trust Co mpute rs I I I phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Mo tiva tio n: Do no t trust Co mpute rs I V Silicon rods in air Two-level atom (initially excited) phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Outline T he K rylo v-Sub spa c e / Disc o ntinuo us Ga le rkin Appro a c h – Ho w the me tho d wo rks a nd pe rfo rms – Adva nc e d spa tia l disc re tiza tio n E xte nsio n to No nline a r & Co uple d Syste ms – L a wso n-T ra nsfo rma tio n a nd Ro se nb ruc k-Wa nne r so lve rs – Pe rfo rma nc e E xa mple s a nd Applic a tio ns – Spo nta ne o us e missio n in pho to nic c rysta ls – Pla smo nic struc ture s phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Outline T he K rylo v-Sub spa c e / Disc o ntinuo us Ga le rkin Appro a c h – Ho w the me tho d wo rks a nd pe rfo rms – Adva nc e d spa tia l disc re tiza tio n E xte nsio n to No nline a r & Co uple d Syste ms – L a wso n-T ra nsfo rma tio n a nd Ro se nb ruc k-Wa nne r so lve rs – Pe rfo rma nc e E xa mple s a nd Applic a tio ns – Spo nta ne o us e missio n in pho to nic c rysta ls – Pla smo nic struc ture s phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de T he K rylo v-Sub spa c e Me tho d Ma xwe ll e q ua tio ns in Sc hrö ding e r fo rm A fo rma l so lutio n o f is g ive n b y phot onics.t fp.uni-kar lsr uhe.de
� Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de T he K rylo v-Sub spa c e Me tho d Disc re tiza tio n o f a nd (e .g . o n a Ye e -Grid ) Ve ry la rg e b ut spa rse ma trix Ma trix-Ve c to r-Pro d uc ts a re fe a sa b le We do no t re q uire the full ma trix , o nly its a c tio n o n a ve c to r: We do no t wa nt a ny re stric tio ns o n the pro pe rtie s o f the ma trix (suc h a s ske w-symme try e tc .) phot onics.t fp.uni-kar lsr uhe.de
� Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de T he K rylo v-Sub spa c e Me tho d Build up the Krylov Subspa c e Ortho -no rma lize the b a sis b y Arno ldi-me tho d Ortho g o na l Ba sis Ob ta in pro je c tio n o f o nto he numb e r o f b a sis ve c to rs c a n b e sma ll (m ~ 10) T phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de T he K rylo v-Sub spa c e Me tho d T he ke y a ppro xima tio n the n is Wo rks fo r a rb itra ry ma tric e s T he a c c ura c y o f the me tho d is a t le a st Me mo ry usa g e : ( m +1)/ 2 re la tive to F DT D J. Nie g e ma nn, L . T ke she la shvili, a nd K . Busc h, J. Co mput. T he o r. Na no sc i. 4 , 627 (2007) phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Co mpa riso n o f Pe rfo rma nc e (1D) T he me tho d a llo ws muc h la rg e r time ste ps Krylov (m=4) Krylov (m=8) Krylov (m=16) Krylov (m=32) FDTD phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Co mpa riso n o f Pe rfo rma nc e (2D) I n a 2D syste m, the e ffe c t is e ve n mo re pro no unc e d ) Krylov (m=16) 2 ) Krylov (m=8) FDTD 4 3 = = m m ( ( v o v l y o r K l y r K phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de I mpo rta nt Add -Ons - Via ADE s Dispe rsive Ma te ria ls – Drude -, L o re ntz-, De b ye -Mo de l – Se llma ie r-type Mo de ls So urc e s Ope n Syste ms: Co mple x fre q ue nc y shifte d PML s phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Ma te ria l Dispe rsio n via ADE s All typic a l a na lytic dispe rsio n re la tio ns (Drude , L o re ntz, De b ye ) c a n b e imple me nte d via ADE s. E xpe rime nta l dispe rsio n fitte d b y c o mb ine d (multiple ) L o re ntz- o r Drude -te rms. E xa mple : (Sing le L o re ntz-te rm) phot onics.t fp.uni-kar lsr uhe.de
� � Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Adva nc e d Spa tia l Disc re tiza tio n With the K rylo v-sub spa c e me tho d , a c c ura c y o f time -inte g ra tio n c a n b e c hose n a rbitra rily Pro b le m: E rro r fro m the spa tia l disc re tiza tio n is limiting the to ta l a c c ura c y Hig he r o rde r ste nc ils Still o nly 2nd o rde r in the pre se nc e o f b o unda rie s Po ssib le so lutio n: Ad a ptive g rid re fine me nt a ro und b o unda rie s phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Unstruc ture d Grid in 1D Ada pt the g rid so the po int de nsity is hig he r a ro und the ma te ria l b o unda rie s n=1 n=1.5 phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Unstruc ture d Grid Pe rfo rma nc e (1D) 4 th -o rde r ste nc il a nd a da ptive g rids: 4 th o rde r is ma inta ine d in the pre se nc e o f ma te ria l b o unda rie s FDTD N=399 Krylov (m=8) Krylov (m=32) Krylov (m=64) Krylov (m=16) Krylov (m=4) N=799 N=1599 N=3199 K . Busc h e t a l., physic a sta tus so lidi (b ) 244 , 3479 (2007) phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Unstruc ture d Grids in 2D/ 3D Disc o ntinuo us Ga le rkin finite e le me nt te c hniq ue (b o rro we d fro m hydro dyna mic s) phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Re sults o n Unstruc ture d Grids (2D) phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de Outline T he K rylo v-Sub spa c e / Disc o ntinuo us Ga le rkin Appro a c h – Ho w the me tho d wo rks a nd pe rfo rms – Adva nc e d spa tia l disc re tiza tio n E xte nsio n to No nline a r & Co uple d Syste ms – L a wso n-T ra nsfo rma tio n a nd Ro se nb ruc k-Wa nne r so lve rs – Pe rfo rma nc e E xa mple s a nd Applic a tio ns – Spo nta ne o us e missio n in pho to nic c rysta ls – Pla smo nic struc ture s phot onics.t fp.uni-kar lsr uhe.de
� � Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de E xte nsio n to No nline a r Syste ms T he me tho d c a n b e e xte nde d to no nline a r syste ms L a wso n-T ra nsfo rma tio n: H is the line a r pa rt o f the no nline a r syste m Ro se nb ruc k-Wa nne r so lve rs: H is the Ja c o b ia n o f the no nline a r syste m phot onics.t fp.uni-kar lsr uhe.de
Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de E xte nsio n to No nline a r Syste ms L a wso n-T ra nsfo rma tio n phot onics.t fp.uni-kar lsr uhe.de
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