1. Relativistic nonlinear optics in laser-plasma interaction Institute of Atomic and Molecular Sciences Academia Sinica, Taiwan National Central University, Taiwan Jyhpyng Wang National Taiwan University, Taiwan

2. Outline Relativistic nonlinearity in laser-plasma interaction Relativistic harmonic generation and optical rectification Relativistic induced birefringence Generation of intense few-cycle mid-infrared pulses

3. peak intensity: 10 20 W/cm 2 (10-  m focal spot) electric field: 3.2  10 13 V/m (50  Coulomb field in hydrogen ) 100-TW laser at Nat’l Central Univ.

4. Hamiltonian of an electron in a laser field vector potential scalar potential relativistic intensity: mass increase due to quivering motion: canonical momentum

5. Relativistic nonlinearity in laser-plasma interaction Relativistic effects on plasma refractive index Wave mixing mediated by plasma waves Relativistic nonlinearity of the Lorentz force relativistic self-phase modulation nonlinear force

6. Theoretical analysis of the electron motion Lorentz force Poisson’s Equation Continuity Equation normalized vector and scalar potentials : known laser field,, solution Phys. Rev. A 76, 063815 (2007)

7. Modification of the laser field Maxwell Equation 0-  source term optical rectification 1-  source term nonlinear refractive index n-  source term harmonic generation nonlinear source terms (functions of )

8. Harmonic generation and optical rectification Phys. Rev. A 80, 023802 (2009) Phys. Rev. A 76, 063815 (2007)

9. intensity dependence Relativistic second harmonic generation theory experiment density dependence 2nd harmonic beam profile fundamental beam profile E. Takahashi, et al, Phys. Rev. E 65, 016402 (2001)

10. Relativistic optical rectification theory transverse laser profile THz field particle-in-cell simulation longitudinal laser profile THz field

11. Relativistic induced birefringence Phys. Rev. A 83, 033801 (2011)

12. Two-beam interaction via plasma waves Maxwell Equation a and a' create plasma waves of k  k', which scatter a x into a x '. induced birefringence nonlinear source terms (functions of )

13. Comparison with particle-in-cell simulation theory simulation

14. Generation of few-cycle intense mid- infrared pulses Phys. Rev. A 82, 063804 (2010)

15. Nonlinear phase modulation in the bubble regime density modulation relativistic self-phase modulation modulation of refractive index laser field electron density

16. Ge-wafer photo-switch mid-IR pulse excitation pulse pinhole mid-IR pulse A. J. Alcock and P. B. Corkum, Can. J. Phys. 57, 1280 (1979)

17. Ge-wafer photo-switch mid-IR pulse excitation pulse pinhole mid-IR pulse A. J. Alcock and P. B. Corkum, Can. J. Phys. 57, 1280 (1979)

18. Temporal profile of the mid-IR pulse photo-switch gated transmission pump pulse: 205 mJ/42 fs excitation pulse: 500  J/38 fs plasma density: 4.1x10 19 cm -3 reconstructed temporal profile pulse duration X 4.6 ps 9.8 ps 5-mm Ge window X ～ 15 fs mid-IR energy (arb. units) intensity (arb. units) consistent with particle-in-cell simulation delay of excitation pulse with respect to mid-IR pulse (ps)

19. Comparing with simulation and theoretical estimation Simulation: mid-IR peak power in the bubble: > 0.5 TW Square of the electric field of the numerically filtered mid-IR pulse The mid-IR pulse is encapsulated in the low-density bubble, hence is not absorbed by the plasma. The wavelength-scale bubble ensures high spatial coherence. 2 － 20  m 6 － 10  m 2 － 6  m 10 － 20  m Estimation based on Fourier transform of the phase modulated pulse Measured energy: 3 mJ (conversion efficiency=1.5%)

20. Summary By solving the equation of motion for electrons under an intense laser field, one can obtain the nonlinear current density as the source of relativistic nonlinear optics. Low-order nonlinearity (nonlinear refractive index, harmonic generation, optical rectification, induced birefringence …) can be understood well from such analysis. The theory has been verified by experiments and 3-D particle-in-cell simulation.

21. Collaborators Core members of the 10-TW and 100-TW laser facilities Prof. Prof. Szu-yuan Chen, Academia Sinica, Taiwan Prof. Jiunn-Yuan Lin, National Chung-Cheng Univ., Taiwan Prof. Hsu-Hsin Chu, National Central Univ., Taiwan Theoretical Analysis Prof. Gin-yih Tsaur, Tunghai Univ., Taiwan Computer Simulation Prof. Shih-Hung Chen, National Central Univ., Taiwan

22. Thank you for your attention.