1. ______ APPLICATION TO WAKEFIELD ACCELERATORS EAAC Workshop – Elba – June 2013 14 juillet 2016 | PAGE 1 CEA | 10 AVRIL 2012 X. Davoine 1, R. Lehe 2, A. Lifschitz 2, V. Malka 2 1 CEA, DAM, DIF, F-91297 Arpajon, France 2 LOA, 91761 Palaiseau, France NEW ALGORITHMS FOR CYLINDRICAL PIC CODE

2. Objectives Accurate simulations are needed, fast simulation are useful 3D PIC simulations are required 2D simulations only provide qualitative understanding of the physics Numerical schemes have to be improved to reduce the numerical errors 1 Development of reduced PIC codes, such as Calder-Circ 1, are useful to run fast simulations Outline Presentation of Calder-Circ 1 (cylindrical mesh) Charge conservation with cylindrical mesh New Maxwell solver to minimize numerical Cerenkov radiation Form-factor with cylindrical mesh 14 juillet 2016 EAAC | June 2013| PAGE 2 1 A. Lifschitz et al, JoCP 228, 1803 (2009)

3. Calder-Circ: a quasi-axisymmetric PIC code for LWFA 14 juillet 2016 EAAC | June 2013| PAGE 3

4. Calder-Circ: a quasi-axisymmetric PIC code for LWFA Cylindrical coordinates are used for the fields A Fourier decomposition is used in the poloidal direction The physic of LWFA is quasi-axisymmetric: Only a few modes are needed, the higher modes can be neglected Only a few “2D” modes are calculated: the simulation is faster 14 juillet 2016 EAAC | June 2013| PAGE 4 A. Lifschitz et al, JoCP 228, 1803 (2009)

5. Calder-Circ: similar to a 3D simulation but 50x faster 14 juillet 2016 EAAC | June 2013| PAGE 5 50x faster Laser and plasma parameters λ0 = 0.8 μm τ0 = 30 fs w0 = 9 μm a0 = 5 ne = 0.007 nc

6. Charge conservation with a cylindrical geometry 14 juillet 2016 EAAC | June 2013| PAGE 6

7. 2 methods can be used to calculate the current: The current can be calculated from the particles velocities and then projected on the grid: The current density can be obtained from the charge conservation equation 1 : Charge conservation with a cylindrical geometry 14 juillet 2016 EAAC | June 2013| PAGE 7 1 Esirkepov, JoCP 135 144 (2001) J = qv

8. 2 methods can be used to calculate the current: The current can be calculated from the particles velocities and then projected on the grid: The current density can be obtained from the charge conservation equation 1 : Charge conservation with a cylindrical geometry 14 juillet 2016 EAAC | June 2013| PAGE 8 1 Esirkepov, JoCP 135 144 (2001) J = qv

9. Charge conservation with a cylindrical geometry 14 juillet 2016 EAAC | June 2013| PAGE 9 Inversion of the operator Adaptation to cylindrical geometry of the methods proposed by Esirkepov 1 : 1 Esirkepov, JoCP 135 144 (2001)

10. Charge conservation error ~ numerical precision 14 juillet 2016 EAAC | June 2013| PAGE 10 Error on the Poisson equation without the charge conservation scheme for typical LWFA case: Error with the charge conservation scheme: 10 -16 > 10 -3 Propagation length [mm]

11. Improved Maxwell solver Suppression of the numerical Cerenkov radiation 14 juillet 2016 EAAC | June 2013| PAGE 11

12. The numerical Cerenkov radiation can degrade the electron beam properties Improved Maxwell solver Suppression of the numerical Cerenkov radiation 14 juillet 2016 EAAC | June 2013| PAGE 12 1 R. Lehe et al., PRSTAB 16 021301 (2013) New Maxwell solver can reduce the numerical Cerenkov radiation 1

13. Development of a new scheme to solve the Maxwell equations (R. Lehe - LOA) Improved Maxwell solver Suppression of the numerical Cerenkov radiation 14 juillet 2016 EAAC | June 2013| PAGE 13 LWFA simulation with the previous scheme LWFA simulation with the new scheme

14. Development of a new scheme to solve the Maxwell equations (R. Lehe - LOA) Improved Maxwell solver Suppression of the numerical Cerenkov radiation 14 juillet 2016 EAAC | June 2013| PAGE 14 LWFA simulation with the new scheme

15. Form-factor with cylindrical mesh 14 juillet 2016 EAAC | June 2013| PAGE 15

16. Linear form-factor are not fully satisfying 14 juillet 2016 EAAC | June 2013| PAGE 16 x r K-1KK+1 x S K (x) 1 0 j k dx dr k+1 j+1 14 juillet 2016

17. Linear form-factor are not fully satisfying 14 juillet 2016 EAAC | June 2013| PAGE 17 x r K-1KK+1 x S K (x) 1 0 j k dx dr k+1 j+1 14 juillet 2016 J.P. Verboncoeur, JoCP 174, 421 (2001) Density profile calculated with the exact volume Density profile calculated with the corrected volume

18. j-1jj+1 r S j (r) 1 0 Proposed form-factor to use the exact cell volumes 14 juillet 2016 EAAC | June 2013| PAGE 18 A quadratic spline should be use in the radial direction

19. Recent development in Calder-Circ: New form-factor Charge conservation algorithm New Maxwell solver to reduce the numerical Cerenkov radiation Next developments: Form-factor at 2 nd and 3 rd order Ionization by fields Interest: Better modeling of LWFA Possibility to model plasma with higher density (near critical density ?) Conclusion 14 juillet 2016 EAAC | June 2013| PAGE 19