Docstoc

High-order Harmonic Generation in Gases _HHG_

Document Sample
High-order Harmonic Generation in Gases _HHG_ Powered By Docstoc
					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 it          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 20


                                   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

				
DOCUMENT INFO