Electromagnetic Waves: Outline. Electromagnetic wave propagation in Particle-In-Cell codes. 1D discrete propagation equation in vacuum

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1 Elecromageic Waves: Oulie Elecromageic wave propagaio i Paricle-I-Cell codes Remi Lehe Lawrece Berkele aioal Laboraor (LBL) umerical dispersio ad Coura limi Dispersio ad Coura limi i D Dispersio ad Coura limi i 3D Specral solvers ad umerical dispersio US Paricle Acceleraor School (USPAS) Summer Sessio Self-Cosise Simulaios of Beam ad Plasma Ssems S. M. Lud, J.-L. Va, R. Lehe & D. Wikleher Colorado Sae U, F. Collis, CO, 3-7 Jue, 6 Ope boudaries codiios Silver-Müller boudar codiios Perfecl Mached Laers D discree propagaio equaio i vacuum Remider: D discree Mawell equaios i vacuum / B `+/ `+/ + ` E ǹ c E ǹ + E ǹ B `+/ ` / B r E) r B) c These equaios ca be combied io a propagaio equaio for : c + ` ǹ ` ` c `+/ `+/ Eǹ ǹ + ` / / `+/ + + / `+/ ` / ǹ ` / ` / / ` / D dispersio relaio D discree propagaio equaio i vacuum c + ` E ǹ + ` E `+ E ǹ ǹ +! Vo euma aalsis: assume he soluios of his equaio are of he form E e ik i! (propagaig wave), i.e. E ǹ ik ` i! E e Replacig his asa io he discree progagaio equaio ields e ik` i!(+) e c ik` i! e c e i! i!( ) + e e i! e ik(`+) (e c e i! i! / +e i! e e i! / ) (e ik / ik ik` i! e e ik / ) e ik` ik(` ) + e +e ik D discree propagaio equaio i vacuum c + ` E ǹ + ` E `+ E ǹ + E ǹ i.e. E E ǹ 3 D dispersio relaio! c si k si (isead of! c k ) 4

2 c apple! umerical dispersio c apple! umerical dispersio For c apple, he discree dispersio relaio! c si k si has real soluios!, for a k:! ± c k arcsi si v k c arcsi k si Aimaio: c.5 Thus, he phase veloci v!/k is: v ± c k arcsi umerical dispersio k si I a PIC code, he elecromageic waves propagae (i vacuum) a a veloci which depeds o k (ad o, ), isead of propagaig a he speed of ligh: v ±c 5 B: k /, : shores wavelegh suppored b he grid. The shorer he wavelegh, he slower he propagaio. 6 c >! Coura limi Elecromageic Waves: Oulie For c >, he discree dispersio relaio! c si k si has o real soluios!, for k close o /. The soluio! is imagiar ad he correspodig mode is usable. Coura limi (a.k.a. CFL limi) umerical dispersio ad Coura limi Dispersio ad Coura limi i D Dispersio ad Coura limi i 3D Specral solvers ad umerical dispersio Sadard EM-PIC codes are usable for c > (i D). Thus, pracical use of elecromageic PIC codes is resriced o apple /c. For a give spaial resoluio, hislimishow fas a simulaio ca advace i ime. Elecrosaic PIC codes do o have his limiaio! Ca be much faser ha EM-PIC codes o simulae a ssem over a give period of ime, b akig large imeseps. 7 Ope boudaries codiios Silver-Müller boudar codiios Perfecl Mached Laers

3 Dispersio ad Coura limi i 3D Derivaio of dispersio relaio Combie discree Mawell equaio! Discree propagaio equaio! Vo euma aalsis! umerical dispersio relaio Same process i 3D. The Vo euma aalsis assumes: E E e ik+ik+ik 3D umerical dispersio relaio si! c si k si k + + si k i! umerical dispersio i 3D 3D Discree dispersio relaio si! c si k si k + + si k Veloci depeds o he wavelegh ad propagaio direcio. Eample: epadig elecromageic wave Phsical Simulaed isead of he phsical dispersio! c (k + k + k ) Coura limi (a.k.a CFL limi) i 3D c apple q Eve for CFL : waves are slower ha c alog he mai aes. Impac of umerical dispersio Elecromageic Waves: Oulie Aimaio: laser-wakefield acceleraio A shor ad iese laser pulse, followed b a relaivisic elecro buch, eers a plasma (geeraed from a gas je). The laser pulse geeraes a wake i he plasma, wih elecric fields ha ca accelerae he elecro buch. Simulaio wih he Yee scheme (ad low resoluio): The laser is arificiall slow (umerical dispersio) Thus he elecro buch caches up wih he laser ver soo! umerical dispersio ad Coura limi Dispersio ad Coura limi i D Dispersio ad Coura limi i 3D Specral solvers ad umerical dispersio Ope boudaries codiios Silver-Müller boudar codiios Perfecl Mached Laers

4 Yee scheme Fiie-di erece i space ad ime e.g. coiuous discree equaio : B B / wih ˆ@ F i,j,` Phsical F i+,j,` F ( ˆ@ E ˆ@ ) Aisoropic Waves propagae slower ha c. Pseudo-specral solver Fourier rasform i space, fiie-di erece i ime e.g. coiuous equaio :! Fourier space Fiie di erece i ime : ˆB! Use backwards FFT o obai B ik Ê ˆB / ik Ê ik Ê Isoropic ik Ê Waves propagae faser ha c. 3 4 Aalical pseudo-specral solver (Haber e al., 973) Fourier rasform i space, fiie-di erece i ime e.g. coiuous ik Fourier space : Aalical iegraio of he coupled Mawell equaios i ime: ˆB + cos(kc ) ˆB si(kc ) kc ik Ê! Use backwards FFT o obai B + from Phsical Simulaed ik Ê ˆB + q k k + k + k Isoropic Waves propagae eacl a c. Dispersio ad Coura limi: coclusios Elecromageic solvers have a maimum value for he imesep (Coura limi), which depeds o he dimesio (ad he mehod of discreiaio) Below he Coura limi, waves ma propagae a speeds ha arificiall di er from c (umerical dipersio). This ca have a srog impac i some phsical siuaios. Specral solvers ca miigae (or eve elimiae) umerical dispersio. 5 6

5 Elecromageic Waves: Oulie Boudar codiios ad EM-PIC Remider: D discree Mawell equaios i vacuum umerical dispersio ad Coura limi Dispersio ad Coura limi i D Dispersio ad Coura limi i 3D Specral solvers ad umerical dispersio Ope boudaries codiios Silver-Müller boudar codiios Perfecl Mached Laers / `+/ `+/ + ` ǹ ǹ + c B `+/ ǹ ` / ( ) / 3/ 3/ E / The grid is fiie: For ` : ` / is udefied. For ` : `+/ is udefied.! Assumpios are eeded, for he value of / ad +/. 8 Boudar codiios ad EM-PIC ( ) / 3/ 3/ E / Boudar codiios ad EM-PIC Problem: Dirichle ad euma boudar codiios reflec he EM waves. For ma phsical problems, we eed he boudaries o absorb he waves. Aimaio: euma boudar codiios Tpical assumpios Periodic: / / Dirichle: / ad +/ ad B +/ B / euma: / / ad +/ B / ad@ ) This is because, phsicall, a ougoig wave does o saisf ( ) (Dirichle) ( ) (euma) 9

6 Silver-Müller absorbig boudar (righ-had side) ( ) Silver-Müller absorbig boudar (righ-had side) ( ) / 3/ 3/ E / The value of +/ should be chose so as o be cosise wih a ougoig wave. Phsicall, for a ougoig wave propagaig o he righ (from Mawell s equaio): (, ) E(, ) c umericall, we ca epress i as: Because of saggerig: c E +/ + B / c + + E / 3/ 3/ E / B combiig he equaios: +/ + B / we obai E (righ-propagaig wave) c c B +/ / (Mawell equaio) Silver-Müller boudar codiio (righ-had side) + c + c c + c + B / See e.g. Bjor Egquis (977) Silver-Müller absorbig boudar (righ-had side) Silver-Müller boudar codiio (righ-had side) + c + c c + c + B / Aimaio: Silver-Müller boudar codiios Silver-Müller absorbig boudar (lef-had side) ( ) / 3/ 3/ E / B combiig he equaios: / + / we obai + c c B + + / / (lef-propagaig wave) (Mawell equaio) Silver-Müller boudar codiio (lef-had side) + c c c + c + B / 3 4

7 Silver-Müller absorbig boudar i 3D Silver-Müller absorbig boudar i 3D Mawell equaio: + + E i+,j,` i+ B,j,` c + B i+,j+, i+,j, + i+,j,`+ + i+,j,` Limiaio I 3D, he Silver-Müller boudar codiios are ol well-adaped for waves i ormal icidece. The reflecio coe cie R( ) quickl icreases wih he agle of icidece. Silver-Müller boudar codiio (lef-had side) + i+,j, c c i+ c +,j, c + B + i+,j, + B + B +c i+,j+, i+,j, + Similar equaios for he righ-had side + Similar equaios for B ad E 5 6 Elecromageic Waves: Oulie umerical dispersio ad Coura limi Dispersio ad Coura limi i D Dispersio ad Coura limi i 3D Specral solvers ad umerical dispersio Ope boudaries codiios Silver-Müller boudar codiios Perfecl Mached Laers Perfecl Mached Laers (i D) Perfecl Mached Laers (Bereger, 994) Surroud he simulaio bo b addiioal laers of cells, where he Mawell equaios are modified so as o progressivel damp he waves. I he B E I e.g. he righ-had E B E B B + E Modified Mawell equaios: Arificial (uphsical) coducivi The B field is (arificiall) spli i wo 8

8 Perfecl Mached Laers (i D) Perfecl Mached Laers: ormal icidece Eplaaio based o coiuous equaios Trasverse EM wave propagaig alog E Aimaio wih propagaig waves: Waves i ormal icidece are absorbed. Waves i graig icidece propagae as if he did o feel he boudar. E 6! B B B I he E B E E I he righ-had E B E B B + E B B E B 9 3 Perfecl Mached Laers: ormal icidece There is a soluio (coiuous i E ad B )wiho refleced wave. Perfecl Mached Laers: graig icidece Trasverse EM wave propagaig alog I he bulk ( E E Soluio: E E cos(k( B E c cos(k( c)) c)) I he righ-had laer ( E B E B Soluio: E E cos(k( c))e c B E c cos(k( c))e c E The wave is damped before reachig he ed of he ouer laer. 6 E! B B B I he E B E I he righ-had E B E B B + E B E 3 The propagaio equaios are ideical i he bulk ad he ouer laer. A wave i graig ididece does o feel he boudar. 3

9 Ope boudar codiios: coclusio Refereces If o special care is ake a he boudar, i will b defaul produce a refleced wave. Silver-Müller boudar codiios: Eas o impleme Bu ol cacels reflecio for waves a ormal icidece Perfecl Mached Laers eed era laers of cells, where he Mawell equaios are arificiall modified. The aisoropic Mawell equaios lead o proper behavior for waves wih a icidece agle. Bereger, J.-P. (994). A perfecl mached laer for he absorpio of elecromageic waves. J. Compu. Phs., 4():85. Bjor Egquis, A. M. (977). Absorbig boudar codiios for he umerical simulaio of waves. Mahemaics of Compuaio, 3(39): Haber, I., Lee, R., Klei, H., ad Boris, J. (973)