Highorder Harmonic Generation HHG in gases by Benot

High-order Harmonic Generation (HHG) in gases by Benoît MAHIEU 1

Introduction • Will of science to achieve lower scales – Space: nanometric characterization λ = c/ν – Time: attosecond phenomena (electronic vibrations) Period of the first Bohr 2 orbit : 150. 10 -18 s

Introduction • LASER: a powerful tool – Coherence in space and time – Pulsed LASERs: high power into a short duration (pulse) Electric field Continuous Pulses time • Two goals for LASERs: – Reach UV-X wavelengths (1 -100 nm) – Generate shorter pulses (10 -18 s) 3

Outline -> How does the HHG allow to achieve shorter space and time scales? 1. 2. 3. 4. Link time / frequency Achieve shorter LASER pulse duration HHG characteristics & semi-classical model Production of attosecond pulses 4

Part 1 Link time / frequency t / ν (or ω = 2πν) 5

LASER pulses • Electric field E(t) • Intensity I(t) = E²(t) I(t) ‹t›: time of the mean value Δt: width of standard deviation Δt = pulse duration • Gaussian envelop: I(t) = I 0. exp(-t²/Δt²) 6

Spectral composition of a LASER pulse FOURIER TRANSFORM Pulse = sum of different spectral components 7

Effects of the spectral composition • Fourier decomposition of a signal: • Electric field of a LASER pulse: • More spectral components => Shorter pulse • Spectral components not in phase ( « chirp » ) => Longer pulse 8

Phase of the spectral components Time Frequency Fourier transform chirp + no chirp Phase of each ω Phase of the ω component No chirp: All the ω in phase Moment of arrival of each ω Electric field in function of time minimum pulse duration 9

Fourier limit • Link between the pulse duration and its spectral width Δt: pulse duration Fourier transform Δω: spectral width I(ω) I(t) • Fourier limit: Δω ∙ Δt ≥ ½ 1 t I(t) • For a perfect Gaussian: Δω ∙ Δt = ½ ω I(ω) 2 t I(t) ω I(ω) 3 t 10 ω

Part 1 conclusion Link time / frequency • A LASER pulse is made of many wavelengths inside a spectral width Δω • Its duration Δt is not « free » : Δω ∙ Δt ≥ ½ • Δω ∙ Δt = ½: Gaussian envelop – pulse « limited by Fourier transform » • If the spectral components ω are not in phase, the pulse is lengthened: there is a chirp • Shorter pulse -> wider bandwidth + no chirp 11

Part 2 Achieve shorter LASER pulse duration 12

Need to shorten wavelength • Problem: pulse length limited by optical period at λ=800 nm Pulse can’t be shorter than period! T=2, 7 fs – Solution: reach shorter wavelengths at λ=80 nm (λ = c ∙ T) T=270 as • Problem: few LASERs below 200 nm – Solution: generate harmonic wavelengths of a LASER beam? 13

Classical harmonic generation • In some materials, with a high LASER intensity 2 photons E=hν 1 photon E=h 2ν λ 0 = 800 nm fundamental wavelength λ 0/2 = 400 nm harmonic wavelength • Problems: – low-order harmonic generation (λ/2 or λ/3) – crystal: not below 200 nm – other solutions not so efficient 14

Dispersion / Harmonic generation Difference between: – Dispersion: separation of the spectral components of a wave I(ω) ω – Harmonic generation: creation of a multiple of the fundamental frequency I(ω) 2 nd HG ω0 ω (Harmonic Generation) I(ω) 2ω0 ω 15

Part 2 conclusion Achieve shorter LASER pulse duration • Pulse duration is limited by optical period => Reach lower optical periods ie UV-X LASERs • Technological barrier below 200 nm • Low-order harmonic generation: not sufficient • One of the best solutions: High-order Harmonic Generation λ (HHG) in particular in gases 0 gas jet/cell λ 0/n 16

Part 3 HHG characteristics & Semi-classical model 17

Harmonic generation in gases Grating LASER source fundamental wavelength λ 0 • Classical HG • Low efficiency • Multiphotonic ionization of the gas: n ∙ hν 0 -> h(nν 0) => Low orders Number of photons Gas jet Harmonic order n LASER output harmonic wavelengths λ 0/n (New & Ward, 1967) 18

Increasing of LASER intensity • Energy : ε = 1 J • Short pulse : Δt < 100 fs • Focused on a small area : S = 100μm² Pulse length I = ε/Δt/S > 1018 W/cm² Intensity 1019 W/cm² λ ~ 800 nm 100 ns 1015 100 ps 1013 100 fs 109 1967 1988 HHG Years 19

High-order Harmonic Generation (HHG) in gases Grating Gas jet LASER source fundamental wavelength λ 0 • How to explain? • up to harmonic order 300!! • quite high output intensity • Interest : • UV-X ultrashort-pulsed LASER source Number of photons « plateau » « cutoff » Harmonic order n LASER output harmonic wavelengths λ 0/n (Saclay & Chicago, 1988) 20

Semi-classical model in 3 steps - Elaser Ip - - w 0 t = 0 w 0 t ~ p/2 Electron of a gas atom Fundamental state 1 Tunnel ionization - - Ek - - w 0 t = 3 p/2 2 Acceleration in the electric LASER field hn=Ip+Ek w 0 t ~ 2 p 3 Recombination to fundamental state P. B. Corkum PRL 71, 1994 (1993) K. Kulander et al. SILAP (1993) Periodicity T 0/2 harmonics are separated by 2 w 0 Energy of the emitted photon = 21 Ionization potential of the gas (Ip) + Kinetic energy won by the electron (Ek)

The cutoff law • Kinetic energy gained by the electron Ø F(t) = q. E 0 ∙ cos(ω0 t) & F(t) = m ∙ a(t) Ø a(t) = (q. E 0/m) ∙ cos(ω0 t) Ø v(t) = (q. E 0/ω0 m) ∙ [sin(ω0 t)-sin(ω0 ti)] ti: ionization time => v(ti)=0 Ø Ek(t) = (½)mv²(t) ∝ I ∙ λ 0² hνmax = Ip + Ekmax • Maximum harmonic order Ø hνmax = Ip + Ekmax hν ∝ Ip + I ∙ λ 0² • Harmonic order grows with: – Ionization potential of the gas – Intensity of the input LASER beam – Square of the wavelength of the input LASER beam!! Number of photons « plateau » « cutoff » Harmonic order n The cutoff law is proved by the semi-classical model 22

Electron trajectory Electron position x x(ti)=0 v(ti)=0 Different harmonic orders Þ different trajectories Þ different emission times te 1 0 If short traj. selected (spatial filter on axis) Harmonic order Short traj. Chirp > 0 21 19 17 15 Time (TL) Long traj. Chirp < 0 Positive chirp of output LASER beam on attosecond timescale: the atto-chirp Mairesse et al. Science 302, 1540 (2003) 0 Emission time (t ) 23 Kazamias and Balcou, PRA 69, 063416 (2004)

Part 3 conclusion HHG characteristics gas jet/cell • Input LASER beam: I~1014 -1015 W/cm² ; λ=λ 0 ; linear polarization λ 0 • Jet of rare gas: ionization potential Ip • Output LASER beam: train of odd harmonics λ 0/n, up to order n~300 ; hνmax ∝ Ip + I. λ 0² Number of photons E=hν Plateau λ 0/n hνmax = Ip+Ekmax Cutoff • Semi-classical model: Order of the harmonic – Understand the process: • Tunnel ionization of one atom of the gas • Acceleration of the emitted electron in the electric field of the LASER -> gain of Ek ∝ I∙λ 0² • Recombination of the electron with the atom -> photoemission E=Ip + Ek – Explain the properties of the output beam -> prediction of an atto-chirp 24

Part 4 Production of attosecond pulses 25

Temporal structure of one harmonic • Input LASER beam – Δt ~ femtosecond – λ 0 ~ 800 nm • One harmonic of the output LASER beam Intensity – Δt ~ femtosecond – λ 0/n ~ some nanometers (UV or X wavelength) Harmonic order Time • -> Selection of one harmonic – Characterization of processes at UV-X scale and fs duration 26

« Sum » of harmonics without chirp: an ideal case • • Central wavelength: λ=λ 0/n -> λ 0 = 800 nm ; order n~150 ; λ~5 nm E(t) Bandwidth: Δλ -> 25 harmonics i. e. Δλ~2 nm Fourier limit for a Gaussian: Δω ∙ Δt = ½ Δω/ω = Δλ/λ ; ω = c/λ Δω = c ∙ Δλ ∙ (n/λ 0)² Δt = (λ 0/n)² ∙ (1/cΔλ ) Time ~ 10 fs Δt ~ 10 ∙ 10 -18 s -> 10 attosecond pulses! Intensity T 0/2 • If all harmonics in phase: generation of pulses with Δt ~ T 0/2 N 27 Time

Chirp of the train of harmonics • Problem: confirmation of the chirp predicted by theory Emission times measured in Neon at λ 0=800 nm ; I=4 1014 W/cm 2 T 0/2 N Intensity ~ 10 fs • During the duration of the process (~10 fs): – Generation of a distorted signal – No attosecond structure of the sum of harmonics 28 Time

Solution: select only few harmonics (Measurement in Neon) H 25 -33 (5) H 35 -43 150 as H 45 -53 130 as H 55 -63 + Mairesse et al, Science 302, 1540 (2003) Mairesse et al, 302, 1540 Science (2003) Y. Mairesse et al. Science 302, 1540 (2003) 23 harmonics Optimum spectral bandwith: 11 harmonics Δt=150 as (Δt. TF=50 as) Δt=130 as (Δt. TF=120 as) 29

Part 4 conclusion Production of attosecond pulses Shorter pulse -> wider bandwidth (Δω. Δt = ½) + no chirp i. e. many harmonics in phase • Generation of 10 as pulses by addition of all the harmonics? • Problem: chirp i. e. harmonics are delayed => pulse is lengthened • Solution: Selection of some successive harmonics => Generation of ~100 as pulses 30

General Conclusion High-order Harmonic Generation in gases • One solution for two aims: – Achieve UV-X LASER wavelengths – Generate attosecond LASER pulses • Characteristics – High coherence -> interferometric applications – High intensity -> study of non-linear processes – Ultrashort pulses: • Femtosecond: one harmonic • Attosecond: selection of successive harmonics with small chirp • In the future: improve the generation of attosecond pulses 31

Thank you for your attention! Questions? Thanks to: Pascal Salières (CEA Saclay) Manuel Joffre (Ecole Polytechnique) Yann Mairesse (CELIA Bordeaux) David Garzella (CEA Saclay) 32