Nonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016 12 High Harmonic Generation 12.1 Atomic units 12.2 The three step model 12.2.1 Ionization 12.2.2 Propagation 12.2.3 Recombination 12.3 Attosecond pulses 12.4 The intensity challenge 12.4.1 The necessity of short drive pulses 12.4.2 Quantum diffusion 12.4.3 Propagation effects phase matching 1
12. High Harmonic Generation HHG: The three step model 12.1 Atomic units Hydrogen Atom + and h = 1 2
= E a Perturbative Nonlinear Optics: E-field in units of E a : Susceptibilities of order 1 χ (n) =χ (n-1) /E a χ (2) ~10-12 /(V/m) χ (3) ~10-23 /(V/m) 2 3
12.2 The Three Step Model 12.2.1 First Step - Ionization Multiphoton Ionization ) Keldysh Parameter: γ = ω 2 E >1 I p I P 4
Tunneling Regime Barrier-Suppression Regime Dominant when: Ponderomotive energy: Quasi-static approximation: Emission probability linear in field 5
Ionization rates Figure 12.6: Static ionization rate for Hydrogen on a linear and logarithmic scale. Figure 12.7: Static ionization rate for Helium on a linear and logarithmic scale. 6
12.2.2 Propagation Figure 12.8: Electron Trajectories 7
Trajectory with max. kinetic energy E k = 8
12.2.3 Recombination Schroedinger Equation in Dipole Approximation Atomic Hamiltonian Wavefunction of partially ionized atom Dipolmoment 9
Dipol acceleration and Ehrenfest Theorem After some calculations 10
12.2.3 Recombination Ideal sinusoidal single cycle pulse E(t) = E0 sin ωt. Secant hyperbolic pulse with 5fs FWHM duration and a max.field amplitude of 0.12au Figure 12.10-13: Simulated HHG spectra for hydrogen excited by Ti:sapphire pulses (800 nm, corresponding to ω = 0.057au). 11
Long and Short Trajectories Figure 12.11: Kinetic energy of long and short trajectories. 12
HHG from Multiple Cycle Fields Electric field 0.1 0.0-0.1 a Dipole radiation 0.1 b 0.0-0.1 Ground state population 1.000 0.998 0.996 0.994 c 0 2 4 time (multiples of period) 13
HHG Experiment and Theory Efficiency (a) 1E-4 1E-5 1E-6 1E-7 1.0 Experiment λ 0 =400 nm He Ne Ar Ar (b) 1E-4 1E-5 1E-6 1E-7 1.0 Theory λ 0 =400 nm Ar (c) Experiment λ 0 =800 nm 1E-6 1E-7 1E-8 1.0 Ar (d) 1E-6 1E-7 1E-8 1.0 Theory λ 0 =800 nm Ar HHG Spectrum [Normalized] 0.5 0.5 0.0 0.0 0.0 0.0 1.0 Ne 1.0 Ne 1.0 Ne 1.0 Ne 0.5 0.0 1.0 0.5 0.0 He 30 40 50 60 70 80 Photon Energy (ev) 0.5 0.0 1.0 0.5 He 0.0 30 40 50 60 70 80 Photon Energy (ev) 0.5 0.5 0.0 1.0 0.5 0.0 He 40 60 80 100 120 Photon Energy (ev) 0.5 0.5 0.0 1.0 0.5 0.0 He 40 60 80 100 120 Photon Energy (ev) a) Experimental results for HHG driven by 400 nm, 1 mj and 26 fs driver pulses with beam waist 30 μm: top row, efficiencies for Ar (50 mbar), Ne (300 mbar) and He (2 bar) using a 2 mm long nozzle; remaining rows, the respective normalized HHG spectra. b) Simulation results for Ar (2.5 10 14 W/cm 2 ), Ne (5.3 10 14 W/cm 2 ) and He (8.5 10 14 W/cm 2 ) for the same interaction parameters like in (a). c) Experimental results for HHG driven by 800 nm, 35 fs driver pulses with beam waist 40 μm: top row, efficiencies for Ar (50 mbar, 0.6 mj), Ne (300 mbar, 2 mj) and He (2 bar, 2 mj) using a 2 mm long nozzle; remaining rows, the respective normalized HHG spectra. d) Simulation results for Ar (1.2 10 14 W/cm 2 ), Ne (3.2 10 14 W/cm 2 ) and He (7.4 10 14 W/cm 2 ) for the same interaction parameters like in (c)
12.3 Attosecond Pulses Figure 12.14: Neighborhood of the most energetic trajectory, which is responsible for the highest frequency radiation emitted. Figure 12.15: (a) Amplitude of radiated HHG field for the same parameters as Fig. 13.9. Note the chirp. 15
Figure 12.16: The same as Fig. 13.11a, before and after high-pass filtering. 16
Figure 13.17: Ionization of helium in the presence of a linearly polarized electric field of a laser pulse with 800nm wavelength and a peak intensity 4 1015W/cm2: (a)electric field; (b)fraction of ionized electrons; (a)instantaneous ionization rate. The thin and the thick lines represent pulses of durations of 50fs and 5fs FWHM, respectively.[1] Other Difficulties: Quantum Diffusion Phase Matching 18
Quantum Diffusion Figure 12.18: Transverse quantum diffusion and its effect on single-atom HHG yield: The 3D surface plots show snapshots of the electron wavefunction, when the recol- liding electron wavepacket returns to its parent ion. The longitudinal and transverse dimensions are both scaled to 1 Bohr radius = 0.53 Å. As the driving laser wave- length (optical period) is increased (here from 0.8 μm to 1.3 μm), the electron wavefunction more strongly spreads transversely due to quantum diffusion, which rapidly reduces the single-atom HHG yield. [24] 19
Phase Matching 20
Other EUV and Hard X-ray sources Thorsten Uphues 21
Synchrotron 22
Relativistic electrons 23
Relativistic electrons radiate in a narrow forward cone 24
Three types of radiation 25
Bending magnet radiation 26
Bending magnet radiation 27
Bending magnet radiation revisited 28
Bending magnet summary 29
Undulator 30
Undulator radiation
Undulator radiation 32
1
2
Undulator radiation 3
4
5
Free-Electron Lasers 6
Incoherent versus coherent Radiation 7
FLASH Layout 8
Power versus undulator length 9
10