Hig he r-o rde r Me tho ds fo r Simula ting L ig ht Pro pa g a tio - - PowerPoint PPT Presentation

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Kur t Busch, Univer sit t 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

SLIDE 1

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 2

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 3

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo tiva tio n

So lito n So lito n c o llisio n in a c o llisio n in a fib e r Bra g g g ra ting fib e r Bra g g g ra ting

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 c o uple d syste ms

SLIDE 4

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo tiva tio n: Sta nda rd Appro a c he s

F DT D-Me tho d

Disc re tiza tio n o n Ye e -g rid 2nd o rde r in spa c e a nd time E ffic ie nt a nd e a sy to imple me nt

F inite -E le me nt-Me tho d

Disc re tiza tio n o n unstruc ture d g rids 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

SLIDE 5

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo tiva tio n: Do no t trust Co mpute rs I

SLIDE 6

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo tiva tio n: Do no t trust Co mpute rs I I

SLIDE 7

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo tiva tio n: Do no t trust Co mpute rs I I I

SLIDE 8

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Silicon rods in air

Two-level atom (initially excited)

Mo tiva tio n: Do no t trust Co mpute rs I V

SLIDE 9

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 10

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 11

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 12

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

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

f the ma trix (suc h a s ske w-symme try e tc .)

SLIDE 13

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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 Ob ta in pro je c tio n o f o nto T he numb e r o f b a sis ve c to rs c a n b e sma ll (m~ 10)

Ortho g o na l Ba sis

SLIDE 14

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

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

T he K rylo v-Sub spa c e Me tho 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)

SLIDE 15

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

FDTD Krylov (m=4) Krylov (m=8) Krylov (m=16) Krylov (m=32)

SLIDE 16

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

FDTD K r y l

( m = 4 ) Krylov (m=8) Krylov (m=16) K r y l

( m = 3 2 )

SLIDE 17

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

I mpo rta nt Add -Ons - Via ADE s

Dispe rsive Ma te ria ls

– Drude -, L

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

SLIDE 18

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

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

re ntz- o r Drude -te rms.

E xa mple : (Sing le L

re ntz-te rm)

SLIDE 19

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 20

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 21

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de FDTD Krylov (m=4) Krylov (m=8) Krylov (m=16) Krylov (m=32) Krylov (m=64) N=399 N=799 N=1599 N=3199

Unstruc ture d Grid Pe rfo rma nc e (1D)

4th-o rde r ste nc il a nd a da ptive g rids: 4th 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

K . Busc h e t a l., physic a sta tus so lidi (b ) 244, 3479 (2007)

SLIDE 22

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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)

SLIDE 23

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Re sults o n Unstruc ture d Grids (2D)

SLIDE 24

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 25

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 26

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 27

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

E xte nsio n to No nline a r Syste ms

With a sta nda rd E ule r sc he me , o ne o b ta ins the “ L

a wson- E ule r Sc he me ”:

I n pra c tic e , we use a 4th-o rde r Rung e -K utta sc he me inste a d o f E ule r: “ L

a wson4”

Ro se nb ruc k-Wa nne r so lve r pro po se d b y Ho c hb ruc k a nd L ub ic h:“ Hoc hbruc k4”

SLIDE 28

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Pe rfo rma nc e Co mpa riso n

Dispe rsio n-fre e 1D syste m with K e rr-No nline a rity

F D T D L a wson4 Rung e -Kutta Hoc hbruc k4

M. Po to tsc hnig e t a l., sub mitte d (2007)

SLIDE 29

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.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

SLIDE 30

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Mo difie d Ra dia tio n Dyna mic s

SLIDE 31

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Se mi-c la ssic a l De sc riptio n

F ull Ha milto nia n: F ie ld is tre a te d c la ssic a lly via Ma xwe ll’ s E q ua tio ns with po la riza tio n

SLIDE 32

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

I ntro duc ing the de nsity ma trix ρ whic h o b e ys

I nitia l po pula tio n diffe re nc e

T1: Relaxation T2: Dephasing γ : Dipole moment

Se mi-c la ssic a l De sc riptio n

SLIDE 33

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Se mi-c la ssic a l De sc riptio n

Ma xwe ll-Bo c h-E q ua tio ns in 1D

SLIDE 34

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Spo nta ne o us E missio n in 2D PhCs

J. Nie g e ma nn e t a l., in pre pa ra tio n

Corresponding bandstructure Corresponding bandstructure

SLIDE 35

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Spo nta ne o us E missio n in 2D PhCs

J. Nie g e ma nn e t a l., in pre pa ra tio n

SLIDE 36

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Spo nta ne o us E missio n in 2D PhCs

J. Nie g e ma nn e t a l., in pre pa ra tio n

SLIDE 37

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Metallic cylinder (r=25nm, Ag) Gaussian pulse

Re so na nc e s o f Pla smo nic Struc ture s

SLIDE 38

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Metallic cylinder (r=25nm, Ag) Gaussian pulse

Re so na nc e s o f Pla smo nic Struc ture s

SLIDE 39

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Re so na nc e s o f Pla smo nic Struc ture s

SLIDE 40

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Re so na nc e s o f Pla smo nic Struc ture s

Co urte sy o f Ma rtin Husnik

SLIDE 41

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

M. Husnik e t a l., sub mitte d

Re so na nc e s o f Pla smo nic Struc ture s

SLIDE 42

Kur t Busch, Univer sit ät Kar lsr uhe, kur t @t fp.physik.uni-kar lsr uhe.de phot onics.t fp.uni-kar lsr uhe.de

Summa ry a nd Outlo o k

Hig he r-o rde r time -do ma in simula tio n o f the Ma xwe ll e q ua tio ns using K rylo v-sub spa c e me tho ds So urc e s, UPML s/ CPML a nd dispe rsive ma te ria ls (Se llme ie r-type ) via a uxilia ry fie lds Disc o ntino us Ga le rkin te c hniq ue fo r “c o nfo rma l” spa tia l disc re tiza tio n E xte nsio n to no nline a r a nd c o uple d syste ms’ dyna mic s F uture wo rk:

– Pa ra lle liza tio n –

Applic a tion to c omple x photonic syste ms