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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 : 150.10-18s Introduction • LASER: a powerful tool – Coherence in space and time – Pulsed LASERs: high power into a short duration Electric (pulse) field Continuous Pulses time • Two goals for LASERs: – Reach UV-X wavelengths (1-100nm) – Generate shorter pulses (10-18s) 3 Outline -> How does the HHG allow to achieve shorter space and time scales? 1. Link time / frequency 2. Achieve shorter LASER pulse duration 3. HHG characteristics & semi-classical model 4. 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) = I0.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: E (t ) En (t ) a n cos( nt n ) n n • More spectral components => Shorter pulse • Spectral components not in phase (« chirp ») => Longer pulse 8 Phase of the spectral components Time Frequency d E (t ) E ( )e it Fourier E() E() ei ( ) 2 transform Phase of the ω component chirp + no chirp chirp - No chirp: Phase of each ω All the ω in phase Moment of arrival of each ω Electric field in minimum pulse function of time duration 9 Fourier limit • Link between the pulse duration and its spectral width Δt: pulse duration Fourier Δω: spectral width transform I(t) I(ω) • Fourier limit: Δω ∙ Δt ≥ ½ 1 t ω I(t) I(ω) • For a perfect Gaussian: Δω ∙ Δt = ½ 2 t ω I(t) 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 – Solution: reach shorter wavelengths ∙ • Problem: few LASERs below 200nm – 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=h2ν λ0 = 800nm λ0/2 = 400nm fundamental wavelength harmonic wavelength • Problems: – low-order harmonic generation (λ/2 or λ/3) – crystal: not below 200nm – 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) I(ω) 2nd HG (Harmonic Generation) ω ω 15 ω0 2ω0 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 200nm • Low-order harmonic generation: not sufficient • One of the best solutions: gas jet/cell High-order Harmonic Generation (HHG) in particular in gases λ 0 λ0/n 16 Part 3 HHG characteristics & Semi-classical model 17 Harmonic generation in gases Grating Gas jet LASER Number of photons source fundamental wavelength λ0 • Classical HG • Low efficiency Harmonic order n • Multiphotonic ionization of the gas: n ∙ hν0 -> h(nν0) LASER output => Low orders harmonic wavelengths λ0/n (New & Ward, 1967) 18 Increasing of LASER intensity • Energy : ε = 1J • Short pulse : Δt < 100fs I = ε/Δt/S > 1018 W/cm² • Focused on a small area : S = 100μm² Intensity Pulse length 1019 W/cm² λ ~ 800nm 100ns 1015 100ps 1013 100fs 1fs 109 Years 1967 1988 HHG 19 High-order Harmonic Generation (HHG) in gases Grating Gas jet « plateau » LASER Number of photons source « cutoff » fundamental wavelength λ0 • How to explain? • up to harmonic order 300!! • quite high output intensity Harmonic order n •Interest : • UV-X ultrashort-pulsed LASER output LASER source harmonic wavelengths λ0/n (Saclay & Chicago, 1988) 20 Semi-classical model in 3 steps - hn=Ip+Ek Elaser Ek x - Ip - - - - - 0t = 0 0t ~ /2 0t = 3/2 0t ~ 2 Electron of a gas atom 1 Tunnel 2 Acceleration in the 3 Recombination to Fundamental state ionization electric LASER field fundamental state P.B. Corkum PRL 71, 1994 (1993) K. Kulander et al. SILAP (1993) Periodicity T0/2 harmonics are separated by 20 Energy of the emitted photon = Ionization potential of the gas (Ip) + Kinetic energy won by the electron (Ek) 21 The cutoff law • Kinetic energy gained by the electron F(t) = qE0 ∙ cos(ω0t) & F(t) = m ∙ a(t) a(t) = (qE0/m) ∙ cos(ω0t) v(t) = (qE0/ω0m) ∙ *sin(ω0t)-sin(ω0ti)] ti: ionization time => v(ti)=0 Ek(t) = (½)mv²(t) ∝ I ∙ λ0² hνmax = Ip + Ekmax « plateau » • Maximum harmonic order Number of photons « cutoff » hνmax = Ip + Ekmax hν ∝ Ip + I ∙ λ0² • Harmonic order grows with: – Ionization potential of the gas – Intensity of the input LASER beam Harmonic order n – Square of the wavelength The cutoff law is proved by of the input LASER beam!! the semi-classical model 22 Electron trajectory Electron position x(ti)=0 x v(ti)=0 Different harmonic orders different trajectories different emission times te 0 1 Time (TL) If short traj. selected (spatial filter on axis) Harmonic order Short traj. Long traj. 21 Positive chirp of output LASER beam Chirp > 0 Chirp < 0 19 on attosecond timescale: 17 the atto-chirp 15 Mairesse et al. Science 302, 1540 (2003) 23 0 Emission time (t ) Kazamias and Balcou, PRA 69, 063416 (2004) Part 3 conclusion HHG characteristics gas jet/cell • Input LASER beam: I~1014-1015W/cm² ; λ=λ0 ; linear polarization λ0 λ0/n • Jet of rare gas: ionization potential Ip • Output LASER beam: train of odd harmonics λ0/n, up to order n~300 ; hνmax ∝ Ip + I.λ0² Number of photons E=hν hνmax = Ip+Ekmax Plateau Cutoff Order of the harmonic • Semi-classical model: – 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∙λ0² • 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 – λ0 ~ 800nm • One harmonic of the output LASER beam – Δt ~ femtosecond – λ0/n ~ some nanometers (UV or X wavelength) Intensity Intensity 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=λ0/n -> λ0 = 800nm ; order n~150 ; λ~5nm • Bandwidth: Δλ -> 25 harmonics i.e. Δλ~2nm E(t) • Fourier limit for a Gaussian: Δω ∙ Δt = ½ • Δω/ω = Δλ/λ ; ω = c/λ • Δω = c ∙ Δλ ∙ (n/λ0)² • Δt = (λ0/n)² ∙ (1/cΔλ ) • Δt ~ 10 ∙ 10-18s -> 10 attosecond pulses! Intensity ~ 10 fs Time T0/2 • If all harmonics in phase: T0/2N generation of pulses with Δt ~ T0/2N 27 Time Chirp of the train of harmonics • Problem: confirmation of the chirp predicted by the theory Emission times measured in Neon at λ0=800nm ; I=4 1014 W/cm2 T0/2 T0/2N Intensity ~ 10 fs • During the duration of the process (~10fs): – Generation of a distorted signal – No attosecond structure of the sum of harmonics 28 Time Solution: select only few harmonics 45 (Measurement in Neon) H25-33 (5) IR field (absolute value) 40 XUV pulse (H33 to H53) 20 H35-43 35 Intensity (arbitrary units) H45-53 150 as Intensity (a.u.) 30 15 25 130 as H55-63 10 20 15 + 5 10 5 0 0 250 500 750 1000 1250 1500 1750 2000 0 Time (as) 0 500 1 000 1 500 2 000 2 500 Time (as) Mairesse et al, 302, 1540 Science (2003) Mairesse et al, Science 302, 1540 (2003) Y. Mairesse et al. Science 302, 1540 (2003) 23 harmonics Δt=150 as (ΔtTF=50 as) Optimum spectral bandwith: 11 harmonics Δt=130 as (ΔtTF=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 10as 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 ~100as 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

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harmonic generation, high-order harmonics, Phys. Rev. Lett, Phys. Rev, phase matching, harmonic spectrum, harmonic emission, laser intensity, laser pulses, laser pulse

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posted: | 11/27/2010 |

language: | English |

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