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

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

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

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

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

z Out of phase Coherence length

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

L Efficiency (  ) Depletion

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

Wave vectors

L Efficiency (  ) The lengths of the problem

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

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

Wave vectors

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

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

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

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

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

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

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

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

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

Time Tunneling Acceleration in the continuum Recombination Attosecond pulse train

Time Harmonic spectrum Energy Is this always true?

Generation of short light pulses