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

2. Introduction Will of science to achieve lower scales – Space: nanometric characterization – Time: attosecond phenomena (electronic vibrations) Period of the first Bohr orbit : 150.10 -18 s λ = c/ν 2

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

4. 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

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

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

7. Spectral composition of a LASER pulse F OURIER TRANSFORM Pulse = sum of different spectral components 7

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 8

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

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

11. 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

12. Part 2 Achieve shorter LASER pulse duration 12

13. 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

14. Classical harmonic generation 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 2 photons E=hν1 photon E=h2ν λ 0 = 800nm fundamental wavelength λ 0 /2 = 400nm harmonic wavelength 14

15. Dispersion / Harmonic generation Difference between: – Dispersion: separation of the spectral components of a wave – Harmonic generation: creation of a multiple of the fundamental frequency I(ω) ω ω ω0ω0 2ω02ω0 ω 2 nd HG (Harmonic Generation) 15

16. 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: High-order Harmonic Generation (HHG) in particular in gases gas jet/cell λ0λ0 λ 0 /n 16

17. Part 3 HHG characteristics & Semi-classical model 17

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

19. Increasing of LASER intensity Pulse length Intensity Years 10 9 10 13 10 15 10 19 W/cm² Energy : ε = 1J Short pulse : Δt 10 18 W/cm² Focused on a small area : S = 100μm² HHG 1fs 100fs 100ps 100ns 19671988 λ ~ 800nm 19

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

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

22. The cutoff law Kinetic energy gained by the electron  F(t) = qE 0 ∙ cos(ω 0 t)&F(t) = m ∙ a(t)  a(t) = (qE 0 /m) ∙ cos(ω 0 t)  v(t) = (qE 0 /ω 0 m) ∙ [sin(ω 0 t)-sin(ω 0 t i )] t i : ionization time => v(t i )=0  E k (t) = (½)mv²(t) ∝ I ∙ λ 0 ² Maximum harmonic order  hν max = Ip + E k max hν ∝ Ip + I ∙ λ 0 ² Harmonic order grows with: – Ionization potential of the gas – Intensity of the input LASER beam – Square of the wavelength of the input LASER beam!! Harmonic order nNumber of photons « plateau » « cutoff » hν max = Ip + E k max The cutoff law is proved by the semi-classical model 22

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

24. Part 3 conclusion HHG characteristics Input LASER beam: I~10 14 -10 15 W/cm² ; λ=λ 0 ; linear polarization Jet of rare gas: ionization potential Ip Output LASER beam: train of odd harmonics λ 0 /n, up to order n~300 ; hν max ∝ Ip + I.λ 0 ² 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∙λ 0 ² Recombination of the electron with the atom -> photoemission E=Ip + Ek – Explain the properties of the output beam -> prediction of an atto-chirp gas jet/cell λ0λ0 λ 0 /n Number of photons E=hν Order of the harmonic Plateau Cutoff hν max = Ip+Ek max 24

25. Part 4 Production of attosecond pulses 25

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

27. « Sum » of harmonics without chirp: an ideal case Central wavelength: λ=λ 0 /n -> λ 0 = 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/λ 0 )² Δt = (λ 0 /n)² ∙ (1/cΔλ ) Δt ~ 10 ∙ 10 -18 s -> 10 attosecond pulses! If all harmonics in phase: generation of pulses with Δt ~ T 0 /2N T 0 /2N T 0 /2 Intensity Time ~ 10 fs E(t) Time 27

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 T 0 /2N T 0 /2 Intensity Time ~ 10 fs Emission times measured in Neon at λ 0 =800nm ; I=4 10 14 W/cm 2 28

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

30. 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

31. 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

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 ) 32