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High-order Harmonic Generation (HHG) in gases

High-order Harmonic Generation (HHG) in gases. by Benoît MAHIEU. Introduction. Will of science to achieve lower scales Space : nanometric characterization Time: attosecond phenomena ( electronic vibrations). λ = c/ ν. Period of the first Bohr orbit : 150.10 -18 s. Introduction.

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High-order Harmonic Generation (HHG) in gases

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

  2. Introduction • Will of science to achievelowerscales • Space: nanometriccharacterization • Time: attosecondphenomena (electronic vibrations) λ = c/ν Period of the first Bohr orbit : 150.10-18s

  3. Introduction • LASER: a powerfultool • Coherence in space and time • PulsedLASERs: high power into a short duration (pulse) • Two goals for LASERs: • Reach UV-X wavelengths (1-100nm) • Generateshorter pulses (10-18s) Electric field Continuous Pulses time

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

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

  6. LASER pulses • Electric field E(t) • Intensity I(t) = E²(t) • Gaussian envelop: I(t) = I0.exp(-t²/Δt²) I(t) ‹t›: time of the mean value Δt: width of standard deviation Δt = pulse duration

  7. Spectral composition of a LASER pulse Fourier transform Pulse = sum of different spectral components

  8. 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

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

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

  11. Part 1 conclusionLink time / frequency • A LASER pulse is made of manywavelengthsinside a spectral widthΔω • ItsdurationΔt is not « free »: Δω ∙ Δt ≥ ½ • Δω ∙ Δt = ½: Gaussianenvelop – pulse « limited by Fourier transform » • If the spectral components ω are not in phase, the pulse islengthened: thereis a chirp • Shorter pulse -> widerbandwidth + no chirp

  12. Part 2 Achieve shorter LASER pulse duration

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

  14. Classicalharmonicgeneration • In some materials, with a high LASER intensity • Problems: • low-order harmonic generation (λ/2 or λ/3) • crystal: not below 200nm • other solutions not so efficient BBO crystal 2 photons E=hν 1 photon E=h2ν λ0 = 800nmfundamental wavelength λ0/2 = 400nmharmonic wavelength

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

  16. Part 2 conclusionAchieveshorter LASER pulse duration • Pulse durationislimited by opticalperiod=> Reachloweropticalperiodsie UV-X LASERs • Technologicalbarrierbelow 200nm • Low-orderharmonicgeneration: not sufficient • One of the best solutions:High-orderHarmonicGeneration(HHG) in particular in gases gas jet/cell λ0 λ0/n

  17. Part 3 HHG characteristics & Semi-classical model

  18. Harmonicgeneration in gases Grating Gas jet LASER source fundamentalwavelengthλ0 Number of photons • Classical HG • Lowefficiency • Multiphotonicionization of the gas: n ∙ hν0 -> h(nν0) • => Loworders Harmonicorder n LASER output harmonicwavelengthsλ0/n (New & Ward, 1967)

  19. 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

  20. High-orderHarmonicGeneration (HHG) in gases Grating Gas jet « plateau » LASER source fundamentalwavelengthλ0 « cutoff » Number of photons • How to explain? • up to harmonicorder 300!! • quitehigh output intensity • Interest : • UV-X ultrashort-pulsed LASER source Harmonicorder n LASER output harmonicwavelengthsλ0/n (Saclay & Chicago, 1988)

  21. Elaser Ek w0t = 3p/2 w0t ~ 2p w0t ~ p/2 Tunnelionization 2 Acceleration in theelectric LASER field 3 Recombination tofundamental state Semi-classical model in 3 steps hn=Ip+Ek - - Ip - - - - - w0t = 0 1 Electron of a gas atom Fundamental state P.B. Corkum PRL 71, 1994 (1993) K. Kulander et al. SILAP (1993) Periodicity T0/2  harmonics are separated by 2w0 Energy of the emitted photon = Ionizationpotential of the gas (Ip) + Kineticenergy won by the electron (Ek)

  22. The cutofflaw • Kineticenergygained 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 • Maximum harmonicorder • hνmax = Ip + Ekmax hν ∝ Ip + I ∙ λ0² • Harmonicordergrowswith: • Ionizationpotential of the gas • Intensity of the input LASER beam • Square of the wavelengthof the input LASER beam!! hνmax = Ip + Ekmax « plateau » « cutoff » Number of photons Harmonicorder n The cutoff law is proved by the semi-classical model

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

  24. Part 3 conclusionHHG characteristics gas jet/cell • Input LASER beam:I~1014-1015W/cm² ; λ=λ0 ; linearpolarization • Jet of rare gas:ionizationpotentialIp • Output LASER beam:train of oddharmonicsλ0/n, up to order n~300 ; hνmax ∝ Ip + I.λ0² • Semi-classical model: • Understand the process: • Tunnel ionization of one atom of the gas • Acceleration of the emittedelectron in the electricfield of the LASER -> gain of Ek ∝ I∙λ0² • Recombination of the electronwith the atom -> photoemission E=Ip + Ek • Explain the properties of the output beam -> prediction of an atto-chirp λ0 λ0/n Number of photons E=hν hνmax = Ip+Ekmax Plateau Cutoff Order of the harmonic

  25. Part 4 Production of attosecond pulses

  26. Temporal structure of one harmonic • Input LASER beam • Δt ~ femtosecond • λ0 ~ 800nm • One harmonic of the output LASER beam • Δt ~ femtosecond • λ0/n ~ somenanometers (UV or X wavelength) • -> Selection of one harmonic • Characterization of processesat UV-X scale and fsduration Intensity Intensity Harmonicorder Time

  27. « Sum » of harmonicswithoutchirp: an ideal case • Central wavelength: λ=λ0/n -> λ0 = 800nm ; order n~150 ; λ~5nm • Bandwidth: Δλ -> 25 harmonics i.e. Δλ~2nm • 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! • If all harmonics in phase:generation of pulses with Δt ~ T0/2N E(t) Time ~ 10 fs Intensity T0/2 T0/2N Time

  28. Chirp of the train of harmonics • Problem: confirmation of the chirp predicted by the theory • During the duration of the process (~10fs): • Generation of a distorted signal • No attosecond structure of the sum of harmonics Emission times measured in Neon at λ0=800nm ; I=4 1014 W/cm2 T0/2 T0/2N Intensity ~ 10 fs Time

  29. 150 as 130 as H35-43 H45-53 H55-63 23 harmonics 11 harmonics Δt=130 as (ΔtTF=120 as) Δt=150 as (ΔtTF=50 as) Solution: select only few harmonics (Measurement in Neon) H25-33 (5) + Mairesse et al, Science 302, 1540 (2003) Mairesse et al, 302, 1540 Science (2003) Y. Mairesse et al. Science 302, 1540 (2003) Optimum spectral bandwith:

  30. Part 4 conclusionProduction 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

  31. General ConclusionHigh-orderHarmonicGeneration 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

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

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