1. 2. High-order harmonic generation in gases Attosecond pulse generation 1. Introduction to nonlinear optics

2. Polarization induced by a laser field linear response nonlinear response Introduction to nonlinear optics Second harmonic generation

3. First demonstration of second-harmonic generation P.A. Franken (1961) The second-harmonic beam was very weak because the process was not phase-matched.

4. The actual published results… First demonstration of second-harmonic generation

5. Harmonic generation Different phase velocity Introduction to nonlinear optics Fundamental 2nd harmonic Generate field = solution of a wave equation

6. z Out of phase Coherence length

7. Phase-matching second-harmonic generation Frequency Refractive index Frequency Refractive index Using birefringence

8. L Efficiency (  ) Depletion

9. Dependence of SHG intensity on length Large  kSmall  k The SHG intensity is sharply maximized if  k = 0.

10. Wave vectors

11. L Efficiency (  ) The lengths of the problem

12. Phase Dispersion z  gen (z)-  pol (z) Dipole phase Dispersion free electrons Focusing -1 cm 1 cm 40 -40 Intensity, pressure, focusing, many parameters! Asymmetry before/after the focus

13.  gen (z)-  pol (z) Localized in space and in time! -1 cm 1 cm 40 

14. Wave vectors

15. 2.7 fs 2 cycles Generation of short light pulses XUV! 1 eV 30 eV

16. Generation of short light pulses Frequency Time Broad bandwidth! 0.1 eV 10 eV Fourier Transform

17. The electron can tunnel through the distorted Coulomb barrier Strong-Field Atomic Physics I

18. III Interaction with the core The electron wave packet interacts with the remaining core II The electron is accelerated by the field, and may return to the atomic core III

19. Ferray et al., J. Phys. B 21, L31 (1988) Multiphoton Plateau Cut-off High-Order Harmonic Generation in Gases....

20. Semi-classical three-step model II The free electron is accelerated by the field, and may return to the atomic core III The electron recombines with the atom, emitting its energy as an XUV photon The electron tunnels through the distorted Coulomb barrier I High-Order Harmonic Generation in Gases

21. Electron dynamics Group delay dispersion Several bursts per half laser cycle Atom Field Electrons Short Long

22. Plateau Cut-off High-Order Harmonic Generation in Gases.... III The electron recombines with the atom, emitting its energy as an XUV photon

23. High-Order Harmonic Generation in Gases Atomic Medium Laser Titanium-Sapphire, 800 nm 1 kHz, 2 mJ, 35 fs pulses Gas cell with rare gas

24. Time Tunneling Acceleration in the continuum Recombination Attosecond pulse train

25. Time Harmonic spectrum Energy Is this always true?

26. Generation of short light pulses