Electromagnetic wave propagation in Particle-In-Cell codes

Transcription

1 Electromagetic wave propagatio i Particle-I-Cell codes Remi Lehe Lawrece Berkeley Natioal Laboratory (LBNL) US Particle Accelerator School (USPAS) Summer Sessio Self-Cosistet Simulatios of Beam ad Plasma Systems S. M. Lud, J.-L. Vay, R. Lehe & D. Wikleher Colorado State U, Ft. Collis, CO, Jue, 016

2 Electromagetic Waves: Outlie 1 Numerical dispersio ad Courat limit Dispersio ad Courat limit i 1D Dispersio ad Courat limit i 3D Spectral solvers ad umerical dispersio Ope boudaries coditios Silver-Müller boudary coditios Perfectly Matched Layers

3 1D discrete propagatio equatio i vacuum Remider: 1D discrete Maxwell equatios i vacuum 1/ B `+1/ y `+1/ t 1 c +1 ` t ǹ = = ǹ +1 `+1/ ǹ ` 1/ t B = r E) (from 1 te = r B) These equatios ca be combied ito a propagatio equatio for : 1 c +1 ` ǹ t 1 1 ` ` c t = `+1/ `+1/ = t = ǹ ǹ +1 ` 1/ t 1/ `+1/ + + 1/ `+1/ ` 1/ ` ǹ 1 1/ ` 1/ t 1/ ` 1/ t 1D discrete propagatio equatio i vacuum 1 c +1 ` E ǹ 1 x + ` t = `+1 ǹ + ǹ 1 i.e. 1 t ǹ ǹ 3

4 1D dispersio relatio 1D discrete propagatio equatio i vacuum 1 c +1 ` E ǹ 1 x + ` t = E x`+1 E ǹ ǹ x + 1! Vo Neuma aalysis: assume the solutios of this equatio are of the form E 0 e ik i!t (propagatig wave), i.e. E ǹ ik ` i! t x = E 0 e Replacig this asat ito the discrete progagatio equatio yields e ik` c e i!(+1) t ik` i! t e c e i! t i!( 1) t + e t = e i! t e 1 (e c e i! t i! t/ +e i! t t = e e i! t/ ) t = (e ik / ik(`+1) ik ik` i! t e e ik / ) e ik` ik(` 1) + e +e ik 1D dispersio relatio 1! t c t si = 1 si k (istead of! = c k ) 4

5 c t apple! Numerical dispersio For c t apple, the discrete dispersio relatio 1! t c t si = 1 k si has real solutios!, for ay k:! = ± c t k t arcsi si Thus, the phase velocity v v = ± k =!/k is: c t arcsi t k si Numerical dispersio I a PIC code, the electromagetic waves propagate (i vacuum) at a velocity which depeds o k (ad o t, ), istead of propagatig at the speed of light: v = ±c 5

6 c t apple! Numerical dispersio v = k c t arcsi t k si Aimatio: c t =0.5 NB: k = /, = : shortest wavelegth supported by the grid. The shorter the wavelegth, the slower the propagatio. 6

7 c t>! Courat limit For c t>, the discrete dispersio relatio 1! t c t si = 1 k si has o real solutios!, for k close to /. The solutio! is imagiary ad the correspodig mode is ustable. Courat limit (a.k.a. CFL limit) Stadard EM-PIC codes are ustable for c t > (i 1D). Thus, practical use of electromagetic PIC codes is restricted to t apple /c. For a give spatial resolutio, thislimitshow fast a simulatio ca advace i time. Electrostatic PIC codes do ot have this limitatio! Ca be much faster tha EM-PIC codes to simulate a system over a give period of time, by takig large timesteps t. 7

8 Electromagetic Waves: Outlie 1 Numerical dispersio ad Courat limit Dispersio ad Courat limit i 1D Dispersio ad Courat limit i 3D Spectral solvers ad umerical dispersio Ope boudaries coditios Silver-Müller boudary coditios Perfectly Matched Layers

9 Dispersio ad Courat limit i 3D Derivatio of dispersio relatio Combie discrete Maxwell equatio! Discrete propagatio equatio! Vo Neuma aalysis! Numerical dispersio relatio Same process i 3D. The Vo Neuma aalysis assumes: E = E 0 e ik xx+ik y y+ik i!t 3D Numerical dispersio relatio si! t c t = si kx x x + si k y y y + si k istead of the physical dispersio! = c (k x + k y + k ) Courat limit (a.k.a CFL limit) i 3D c t apple 1 q 1 x + 1 y + 1 9

10 Numerical dispersio i 3D 3D Discrete dispersio relatio si! t c t = si kx x x + si k y y y + si k Velocity depeds o the wavelegth ad propagatio directio. Example: expadig electromagetic wave Physical Simulated Eve for t = t CFL : waves are slower tha c alog the mai axes. 10

11 Impact of umerical dispersio Aimatio: laser-wakefield acceleratio A short ad itese laser pulse, followed by a relativistic electro buch, eters a plasma (geerated from a gas jet). The laser pulse geerates a wake i the plasma, with electric fields that ca accelerate the electro buch. Simulatio with the Yee scheme (ad low resolutio): The laser is artificially slow (umerical dispersio) Thus the electro buch catches up with the laser very soo! 11

12 Electromagetic Waves: Outlie 1 Numerical dispersio ad Courat limit Dispersio ad Courat limit i 1D Dispersio ad Courat limit i 3D Spectral solvers ad umerical dispersio Ope boudaries coditios Silver-Müller boudary coditios Perfectly Matched Layers

13 Yee scheme Fiite-di erece i space ad time e.g. cotiuous @x discrete equatio : B = B 1/ t( ˆ@ x E y ˆ@y ) with ˆ@ x F i,j,` = F i+ 1,j,` F i 1,j,` x Physical Simulated Aisotropic Waves propagate slower tha c. 13

14 Pseudo-spectral solver Fourier trasform i space, fiite-di erece i time e.g. cotiuous equatio :! Fourier space Fiite di erece i time : ˆB = ik x Ê y ik y Ê x ˆB 1/ t ik x Êy ik y Ê x! Use backwards FFT to obtai B from ˆB Physical Simulated Isotropic Waves propagate faster tha c. 14

15 Aalytical pseudo-spectral solver (Haber et al., 1973) Fourier trasform i space, fiite-di erece i time e.g. cotiuous @x ik x Ê y ik y Ê Fourier space : ˆB Aalytical itegratio of the coupled Maxwell equatios i time: ˆB +1 =cos(kc t) ˆB si(kc t) kc ik x Ê y ik y Ê x k = q k x + k y + k! Use backwards FFT to obtai B +1 from ˆB +1 Physical Simulated Isotropic Waves propagate exactly at c. 15

16 Dispersio ad Courat limit: coclusios Electromagetic solvers have a maximum value for the timestep t (Courat limit), which depeds o the dimesio (ad the method of discretiatio) Below the Courat limit, waves may propagate at speeds that artificially di er from c (umerical dipersio). This ca have a strog impact i some physical situatios. Spectral solvers ca mitigate (or eve elimiate) umerical dispersio. 16

17 Electromagetic Waves: Outlie 1 Numerical dispersio ad Courat limit Dispersio ad Courat limit i 1D Dispersio ad Courat limit i 3D Spectral solvers ad umerical dispersio Ope boudaries coditios Silver-Müller boudary coditios Perfectly Matched Layers

18 Boudary coditios ad EM-PIC Remider: 1D discrete Maxwell equatios i vacuum 1/ `+1/ `+1/ t +1 ` t ǹ = ǹ +1 = c `+1/ ǹ ` 1/ 0 (N 1) N 0 1/ 1 3/ N 3/ N 1 N 1/ N The grid is fiite: For ` =0: ` 1/ For ` = N: `+1/ is udefied. is udefied.! Assumptios are eeded, for the value of 1/ ad N+1/. 18

19 Boudary coditios ad EM-PIC 0 (N 1) N 0 1/ 1 3/ N 3/ N 1 N 1/ N Typical assumptios Periodic: 1/ = N 1/ ad N+1/ = 1/ Dirichlet: 1/ =0ad N+1/ =0 Neuma: 1/ = 1/ ad N+1/ = N 1/ 0 =0ad@ N =0) 19

20 Boudary coditios ad EM-PIC Problem: Dirichlet ad Neuma boudary coditios reflect the EM waves. For may physical problems, we eed the boudaries to absorb the waves. Aimatio: Neuma boudary coditios This is because, physically, a outgoig wave does ot satisfy ( ) = 0 (Dirichlet) ( ) =0(Neuma) 0

21 Silver-Müller absorbig boudary (right-had side) 0 (N 1) N 0 1/ 1 3/ N 3/ N 1 N 1/ N The value of N+1/ should be chose so as to be cosistet with a outgoig wave. Physically, for a outgoig wave propagatig to the right (from Maxwell s equatio): Numerically, we ca express it as: (, t) = 1 c (, t) Because of staggerig: N N+1/ + N 1/ = 1 c N = 1 c +1 N + N 1

22 Silver-Müller absorbig boudary (right-had side) 0 (N 1) N 0 1/ 1 3/ N 3/ N 1 N 1/ N By combiig the equatios: N+1/ + N 1/ we obtai +1 N t N = 1 c +1 N = c + N N+1/ (right-propagatig wave) N 1/ (Maxwell equatio) Silver-Müller boudary coditio (right-had side) +1 c t N = 1 N + c t c t + c t + N 1/ See e.g. Bjor Egquist (1977)

23 Silver-Müller absorbig boudary (right-had side) Silver-Müller boudary coditio (right-had side) +1 c t N = 1 N + c t c t + c t + N 1/ Aimatio: Silver-Müller boudary coditios 3

24 Silver-Müller absorbig boudary (left-had side) 0 (N 1) N 0 1/ 1 3/ N 3/ N 1 N 1/ N By combiig the equatios: 1/ + 1/ we obtai t = 1 c = c / 1/ (left-propagatig wave) (Maxwell equatio) Silver-Müller boudary coditio (left-had side) +1 c t c t 0 = 1 0 c t + c t + 1/ 4

25 Silver-Müller absorbig boudary i 3D Maxwell equatio: +1 i+ 1,j,` i+ 1,j,` c t = B + 1 i+ 1,j+ 1,0 B + 1 i+ 1,j 1,0 y + 1 i+ 1,j,` i+ 1,j,` 1 Silver-Müller boudary coditio (left-had side) +1 i+ 1,j,0 = 1 c t c t + i+ 1,j,0 c t c t + B + 1 y i+ 1,j, 1 +c t B + 1 i+ 1,j+ 1,0 B + 1 i+ 1,j 1,0 + Similar equatios for the right-had side + Similar equatios for B x ad E y y 5

26 Silver-Müller absorbig boudary i 3D Limitatio I 3D, the Silver-Müller boudary coditios are oly well-adapted for waves i ormal icidece. The reflectio coe ciet R( ) quickly icreases with the agle of icidece. 6

27 Electromagetic Waves: Outlie 1 Numerical dispersio ad Courat limit Dispersio ad Courat limit i 1D Dispersio ad Courat limit i 3D Spectral solvers ad umerical dispersio Ope boudaries coditios Silver-Müller boudary coditios Perfectly Matched Layers

28 Perfectly Matched Layers (i D) Perfectly Matched Layers (Bereger, 1994) Surroud the simulatio box by additioal layers of cells, where the Maxwell equatios are modified so as to progressively damp the waves. I the t = y t E y = x t B x E y y I e.g. the right-had layer: y t = y t E y = x B 0 E y B = B x + B t B x x E y 0 B t B y y Modified Maxwell equatios: Artificial (uphysical) coductivity The B field is (artificially) split i two 8

29 Perfectly Matched Layers (i D) Aimatio with propagatig waves: Waves i ormal icidece are absorbed. Waves i graig icidece propagate as if they did ot feel the boudary. 9

30 Perfectly Matched Layers: ormal icidece Explaatio based o cotiuous equatios Trasverse EM wave propagatig alog x E =0 E y 6=0! B y =0 B = B x I the t = y t E y = x te y = t B x E y y E t B x E y x I the right-had layer: y t = y t E y = x B 0 E y B = B x + B t B x x E y 0 B t B y te y = x B 0 E t B x E y 0 B 30

31 Perfectly Matched Layers: ormal icidece There is a solutio (cotiuous i E y ad B )witho reflected wave. I the bulk (x t E y = x t B x E y Solutio: I the right-had layer (x t E y = x B 0 E t B x E y 0 B Solutio: E E y = E 0 cos(k(x B = E 0 c cos(k(x ct)) ct)) E y = E 0 cos(k(x ct))e 0 c x B = E 0 c cos(k(x ct))e 0 c x y x 0 x The wave is damped before reachig the ed of the outer layer. 31

32 Perfectly Matched Layers: graig icidece Trasverse EM wave propagatig alog y 6=0 E y =0! B x =0 B = B y I the t = y t E y = x t = t B x E y y E t B y x I the right-had layer: y E t = y t E y = x B 0 E y B = B x + B t B x x E y 0 B t B y t = y t B y The propagatio equatios are idetical i the bulk ad the outer layer. A wave i graig ididece does ot feel the boudary. 3

33 Ope boudary coditios: coclusio If o special care is take at the boudary, it will by default produce a reflected wave. Silver-Müller boudary coditios: Easy to implemet But oly cacels reflectio for waves at ormal icidece Perfectly Matched Layers Need extra layers of cells, where the Maxwell equatios are artificially modified. The aisotropic Maxwell equatios lead to proper behavior for waves with ay icidece agle. 33

34 Refereces Bereger, J.-P. (1994). A perfectly matched layer for the absorptio of electromagetic waves. J. Comput. Phys., 114(): Bjor Egquist, A. M. (1977). Absorbig boudary coditios for the umerical simulatio of waves. Mathematics of Computatio, 31(139): Haber, I., Lee, R., Klei, H., ad Boris, J. (1973). 34