Tight-Focusing of Short Intense Laser Pulses in Particle-in-Cell Simulations of Laser-Plasma Interaction

Transcription

1 16/05/2017, CTU in Prague Tight-Focusing of Short Intense Laser Pulses in Particle-in-Cell Simulations of Laser-Plasma Interaction Bc. Petr Valenta Supervisors: doc. Ing. Ondrej Klimo, Ph.D. Dr. Stefan Andreas Weber Date: Page:

2 Content: 1) Goals & guidelines 2) Motivation 3) Particle-in-cell method 4) Paraxial approximation 5) Maxwell consistent approach 6) Implementation 7) Evaluation 8) Results 9) Conclusion 10) References Date: Page: 2

3 1) Goals & guidelines: Date: Page: 3

4 2) Motivation: Ultra-short laser pulses with intensities >10 21 W/cm 2 provide an unprecedented capability for basic research as well as a broad range of groundbreaking applications in diverse fields. Laser intensities can be increased by amplifying the input energy. This approach comes at great cost since it requires a higher level of complexity for the laser chain. A more effective route would be focusing to a sub-wavelength level. Strong nonlinearities of the basic equations governing laser-plasma interaction - numerical simulation codes have become an indispensable tool (accompanying the experiments for analysis and interpretation) In order to simulate tightly-focused pulses, laser fields at boundaries have to be consistent with Maxwell equations. Source: [1] Date: Page: 4

5 3) Particle-in-cell method: Date: Page: 5

6 3) Particle-in-cell method: Main computational cycle of particle-in-cell method: 1) Integrate the equations of motion (usually Leap- Frog scheme) 2) Particle weighting (interpolate charge and current densities to grid) 3) Solve the field equations (FDTD solver on staggered grid) 4) Field weighting (interpolate fields to particle positions) Date: Page: 6

7 3) Particle-in-cell method: Code EPOCH (Extendable PIC Open Collaboration project) [2]: multi-dimensional, relativistic, electromagnetic PIC code for plasma physics simulations Explicit, 2 nd order of accuracy written in FORTRAN and parallelized using MPI, dynamic load balancing FDTD field solver (using H. Ruhl scheme [3]) Relativistic particle pusher (Birdsall & Landon type [4], Villasenor & Buneman current weighting [5]) Instrumented to enable in situ visualization of the EM fields using ParaView Catalyst Date: Page: 7

8 4) Paraxial approximation: Date: Page: 8

9 4) Paraxial approximation: Date: Page: 9

10 5) Maxwell consistent approach: Several solutions to overcome drawbacks of paraxial approximation already proposed: 1) higher order approximations for Gaussian beams (complicated, not easy to implement, explicit analytical solution does not have to exist for other beam types) 2) Focusing geometry with perfectly reflecting mirrors to introduce tightly focused beam into the simulation domain (restricted to specific shapes) 3) Direct evaluation of Stratton-Chu integrals (computationally expensive) 4) Simple and efficient algorithm for a Maxwell consistent calculation of the EM fields at the boundaries of the simulation domain - Laser Boundary Conditions (LBC) Algorithm can describe any kind of laser pulses, in particular tightly focused, arbitrarily shaped and polarized ones - the solution of the Maxwell s equations is performed in a frequency space Calculations adapted from the work of I. Thiele et al. from CELIA, Bordeaux (2016) [6] Date: Page: 10

11 5) Maxwell consistent approach: Date: Page: 11

12 5) Maxwell consistent approach: Date: Page: 12

13 6) Implementation: 2D version, written in C++, object oriented to be easily extended to 3D Linked into EPOCH as a static library (in order not to disturb the code, added support for CMake) Parallelized using hybrid techniques (OpenMP + MPI computation time in most cases negligible in comparison with the main simulation) Fourier transforms can be computed using Intel MKL library, FFTW library or without any library (compile-time option) Computed fields dumped into shared files using binary coding (speed up output, save disk storage) Only transverse component of electric field (Ex) passed to the EPOCH at each time step (no significant slowdown or memory overhead), other fields computed by EPOCH All new parameters needed for tight-focusing (w0, focus distance, etc.) may be specified via input file, implementation works generally regardless the number of lasers in the simulation or boundaries that they are attached to Date: Page: 13

14 7) Evaluation: Test simulation parameters: Date: Page: 14

15 7) Evaluation: Transverse (Ey) and longitudinal (Ex) electric laser fields at focus in the case of paraxial approx. (a), (b) and Maxwell consistent approach (c), (d) In the case of paraxial approx. Ex and Ey reveal strong distortions and asymmetry w.r.t focus, the corresponding amplitude is significantly lower Date: Page: 15

16 7) Evaluation: Transverse and longitudinal slices of the transverse electric laser field (Ey) at focus in the case of paraxial approx. (a), (b) and Maxwell consistent approach (c), (d) In the case of paraxial approx. strong sidewings in the beam profile (a), asymmetry of the field in the longitudinal line-out (b) Date: Page: 16

17 7) Evaluation: Time evolution of transverse (Ey) (a) and longitudinal (Ex) (b) electric laser field at the boundary according to the Maxwell consistent approach (a) Spot size parameter according to Maxwell consistent approach (b) Dependency of the absolute focal point shift on the beam waist for the laser beam initialized using the paraxial approx. Date: Page: 17

18 7) Evaluation: Evaluation of the beam symmetry - transverse (a) and longitudinal (b) slice of the transverse electric laser field (Ey) at the front (blue) and rear (red) boundary (a) Transverse (Ey) electric field amplitude w.r.t. distance from focus (b) Maximal instantaneous laser intensity with respect to the distance from focus Date: Page: 18

19 7) Evaluation: Transverse (Ey) and longitudinal (Ex) electric laser fields at focus in the case of paraxial approx. (a), (b) and Maxwell consistent approach (c), (d) for the beam with w0 = 5λ One can clearly see, that there is no significant difference between the shapes of the electric field components Date: Page: 19

20 7) Evaluation: Transverse (a) and longitudinal (b) slice of the transverse (Ey) electric field at focus according to paraxial (red) and Maxwell consistent (blue) approach for the beam with w0 = 5λ for the beams focused to a spot with the size comparable to a center laser wavelength, paraxial approximation leads to a shifted location of the focus, asymmetric laser field profiles with distortions and lower amplitude the propagation of tightly focused Gaussian laser beams prescribed at boundaries according to the Maxwell consistent approach has been proven to be correct the paraxial approximation can be safely used when the beam size at focus is about one order of magnitude larger than the center laser wavelength Date: Page: 20

21 8) Simulations of focused laser beams: 2D3V large-scale PIC simulations of tightly focused p-polarized Gaussian laser beams interacting with solid targets at normal incidence have been performed using code EPOCH Identified the effects of the laser beam focal spot size on the laser-matter interaction results Domain: size 15 x 40 μm with N x = 1875 (δx = 10 nm), N y = 5000 (δy = 10 nm), T = 150 fs (δt = fs), in total: 9.6e6 cells, 4125 iterations Plasma: Solid target made of e -, H + with thickness = 2.0 μm, density = 100 n c, T e = T i = 100 ev, 2000 electrons per cell, 100 ions per cell, in total: 2e9 computational particles, CPML boundary conditions, no collisions, no QED Laser: λ = 0.8 μm, T = 30 fs (in FWHM), w 0 = 0.5, 1.0, 2.0, 4.0 μm, simulations with const. intensity (1e20 W/cm 2, 1e21 W/cm 2 ) or const. energy (30 mj), p-polarization, focal spot 8 μm from left boundary 1) E = const. (E = 30 mj) w 0 = 0.5 μm: absorption = % w 0 = 1.0 μm: absorption = % w 0 = 2.0 μm: absorption = 8.31 % w 0 = 4.0 μm: absorption = 6.19 % 2) I = const. (I = 1e20 W/cm 2 ) w 0 = 0.5 μm: absorption = % w 0 = 1.0 μm: absorption = % w 0 = 2.0 μm: absorption = % w 0 = 4.0 μm: absorption = % Date: Page: 21

22 8) Simulations of focused laser beams: The kinetic energy Ek of particles normalized to the laser pulse energy EL in time for the case of simulations with the laser intensity I = 1e20 W/cm 2 and with the beam waist (a) w0 = 0.5 μm, (b) w0 = 2.0 μm and for the case of simulations with the laser intensity I = 1e21 W/cm 2 and with the beam waist (c) w0 = 0.5 μm, (d) w0 = 2.0 μm Date: Page: 22

23 8) Simulations of focused laser beams: The phase space of the parallel component of the momentum (px) and the x-coordinate of all electrons in the simulation domain at the time t = 100 fs for the case of simulations with I = 1e20 W/cm 2 and with (a) w0 = 0.5 μm, (b) w0 = 2.0 μm and for the case of simulations with I = 1e21 W/cm 2 and with (c) w0 = 0.5 μm, (d) w0 = 2.0 μm Date: Page: 23

24 8) Simulations of focused laser beams: Two types of trajectories of randomly chosen electron samples for the case of simulations with I = 1e20 W/cm 2 and with (a), (b) w0 = 0.5 μm and (c), (d) w0 = 2.0 μm In the case of w0 = 0.5 μm, there is 59 % of electrons with the trajectories of the first type and 41 % of the second type, for the simulation of w0 = 2.0 μm, there is 22 % of electrons with the trajectories of the first type and 78 % of the second type Date: Page: 24

25 8) Simulations of focused laser beams: The spatial distribution of the mean kinetic energy of electrons at the time t = 80 fs for the case of simulations with the laser intensity I = 1e21 W/cm 2 and with the beam waist (a) w0 = 0.5 μm, (b) w0 = 2.0 μm Perpendicular component of the current density (Jy) at the time t = 100 fs for the case of simulations with I = 1e21 W/cm 2 and with (a) w0 = 0.5 μm, (b) w0 = 2.0 μm Date: Page: 25

26 8) Simulations of focused laser beams: A contour lines of ion critical density for two different beam waists at the time t = 100 fs for the case of simulations with (a) I = 1e20 W/cm 2 and (b) I = 1e21 W/cm 2 the plasma in the vicinity of incoming laser beam expands rapidly due to the space charge induced by electrons ejected into vacuum the absorption processes which take place when the laser pulse is obliquely incident on a plasma density gradient may also significantly contribute to the overall laser energy absorption the incidence angles for the laser intensity I = 1e21 W/cm 2 are about 20 and 1.5 for w0 = 0.5 μm and w0 = 2.0 μm, respectively Date: Page: 26

27 8) Simulations of focused laser beams: Phase space of parallel (px) and perpendicular (py) components of momentum of all electrons in the simulation domain at the time t = 100 fs for the case of simulations with I = 1e20 W/cm 2 and with (a) w0 = 0.5 μm, (b) w0 = 2.0 μm and for the case of simulations with I = 1e21 W/cm 2 and with (c) w0 = 0.5 μm, (d) w0 = 2.0 μm Date: Page: 27

28 8) Simulations of focused laser beams: Angular distribution of electrons in the whole simulation domain at the time t = 100 fs for the case of simulations with I = 1e20 W/cm 2 and with (a) w0 = 0.5 μm, (b) w0 = 2.0 μm and for the case of simulations with I = 1e21 W/cm 2 and with (c) w0 = 0.5 μm, (d) w0 = 2.0 μm Date: Page: 28

29 8) Simulations of focused laser beams: Electron energy distribution in the whole simulation domain for four different beam waists at the time t = 100 fs for the case of simulations with the laser intensity (a) I = 1e20 W/cm 2 and (c) I = 1e21 W/cm 2 Ion energy distribution in the whole simulation domain for four different beam waists at the time t = 150 fs for the case of simulations with the laser intensity (b) I = 1e20 W/cm 2 and (d) I = 1e21 W/cm 2 Date: Page: 29

30 9) Conclusion: The main differences have been observed between the cases of the focal spot larger than the center laser wavelength and the focus of sub-wavelength level effects of the laser focal spot size: the laser energy absorption efficiency sharply increases the direction of hot electrons moving forward is given by the ratio between the transverse and the longitudinal component of ponderomotive force, which increases as the focal spot size decreases - there is a larger amount of hot electrons spreading in the transverse direction with respect to the direction of incoming laser beam a significant cloud of hot electrons in front of the target has been observed, the space charge induced by hot electrons ejected into vacuum causes a rapid expansion of plasma in the vicinity of the focal spot there is a strong electric current along the front surface of the target which compensates the charge unbalance - this affects the electron trajectories and absorption processes the energy distribution functions of electrons are quantitatively different and the temperature of hot electrons is significantly higher the ion acceleration efficiency is almost independent of the focus Date: Page: 30

31 10) References: [1] P. Gibbon, Short Pulse Laser Interactions with Matter, Imperial College Press, 2005 [2] T. D. Arber et al. Contemporary particle-in-cell approach to laser-plasma modelling. Plasma Physics and Controlled Fusion, 57(11):113001, [3] H. Ruhl et al. The plasma simulation code: A modern particle-in-cell code with loadbalancing and GPU support [4] C. K. Birdsall, A. B. Langdon. Plasma Physics via Computer Simulation. Series in Plasma Physics. CRC Press, [5] J. Villasenor, O. Buneman. Rigorous charge conservation for local electromagnetic field solvers. Computer Physics Communications, 69(2-3), [6] I. Thiele, S. Skupin, R. Nuter. Boundary conditions for arbitrarily shaped and tightly focused laser pulses in electromagnetic codes. J. Comp. Phys. 321, 1110, 2016 Date: Page: 31

32 Thank you for your attention Date: Page: