High-order%20Harmonic%20Generation%20(HHG)%20in%20gases

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High-order Harmonic Generation (HHG) in gases

• by Benoît MAHIEU

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Introduction

• Will of science to achieve lower scales
• Space nanometric characterization
• Time attosecond phenomena (electronic
  vibrations)

? c/?
Period of the first Bohr orbit 150.10-18s
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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 (10-18s)

Electric field
Continuous
Pulses
time
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Outline

• -gt 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

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Part 1

• Link time / frequency
• t / ? (or ?  2p?)

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LASER pulses

• Electric field E(t)
• Intensity I(t) E²(t)
• Gaussian envelop I(t) I0.exp(-t²/?t²)

I(t)
t time of the mean value
?t width of standard deviation ?t pulse
duration
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Spectral composition of a LASER pulse
Fourier transform
Pulse sum of different spectral components
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Effects of the spectral composition

• Fourier decomposition of a signal
• Electric field of a LASER pulse
• More spectral components gt Shorter pulse
• Spectral components not in phase ( chirp ) gt
  Longer pulse

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Phase of the spectral components

• Time

• Frequency

Fourier transform
Phase of the ? component
chirp no chirp
chirp -
No chirp minimum pulse duration
Phase of each ?
All the ? in phase
Moment of arrival of each ?
Electric field in function of time
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Fourier limit

• Link between the pulse duration and its spectral
  width
• Fourier limit ?? ?t ½
• For a perfect Gaussian ?? ?t ½

Fourier transform
?t pulse duration
?? spectral width
I(?)
I(t)
1
t
?
I(t)
I(?)
2
t
?
I(t)
I(?)
3
t
?
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Part 1 conclusionLink time / frequency

• A LASER pulse is made of many wavelengths inside
  a spectral width ??
• Its duration ?t is not  free  ?? ?t ½
• ?? ?t ½ Gaussian envelop pulse  limited
  by Fourier transform 
• If the spectral components ? are not in phase,
  the pulse is lengthened there is a chirp
• Shorter pulse -gt wider bandwidth no chirp

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Part 2

• Achieve shorter LASER pulse duration

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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?

at ?800nm Pulse cant be shorter than period!
T2,7 fs
at ?80nm (? c T)
T270 as
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Classical harmonic generation

• In some materials, with a high LASER intensity
• Problems
• low-order harmonic generation (?/2 or ?/3)
• crystal not below 200nm
• other solutions not so efficient

BBO crystal
2 photons Eh?
1 photon Eh2?
?0 800nmfundamental wavelength
?0/2 400nmharmonic wavelength

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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(?)
?
I(?)
I(?)
2nd HG (Harmonic Generation)
?
?
?0
2?0
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Part 2 conclusion Achieve shorter LASER pulse
duration

• Pulse duration is limited by optical periodgt
  Reach lower optical periods ie UV-X LASERs
• Technological barrier below 200nm
• Low-order harmonic generation not sufficient
• One of the best solutionsHigh-order Harmonic
  Generation(HHG) in particular in gases

gas jet/cell
?0
?0/n
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Part 3

• HHG characteristics
• Semi-classical model

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Harmonic generation in gases
Grating
Gas jet
LASER source fundamental wavelength ?0
Number of photons

• Classical HG
• Low efficiency
• Multiphotonic ionization of the gas n h?0 -gt
  h(n?0)
• gt Low orders

Harmonic order n
LASER output harmonic wavelengths ?0/n
(New Ward, 1967)
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Increasing of LASER intensity

• Energy e 1J
• Short pulse ?t lt 100fs I e/?t/S gt 1018
  W/cm²
• Focused on a small area S 100µm²

Intensity
Pulse length
1019
W/cm²
? 800nm
100ns
1015
100ps
1013
100fs
1fs
109
Years
1967
1988
HHG
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High-order Harmonic Generation (HHG) in gases
Grating
Gas jet
 plateau 
LASER source fundamental wavelength ?0
 cutoff 
Number of photons

• How to explain?
• up to harmonic order 300!!
• quite high output intensity
• Interest
• UV-X ultrashort-pulsed LASER source

Harmonic order n
LASER output harmonic wavelengths ?0/n
(Saclay Chicago, 1988)
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Semi-classical model in 3 steps
hnIpEk
-
-
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 Ionization
potential of the gas (Ip) Kinetic energy won by
the electron (Ek)
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The cutoff law

• Kinetic energy gained by the electron
• F(t) qE0 cos(?0t) F(t) m a(t)
• a(t) (qE0/m) cos(?0t)
• v(t) (qE0/?0m) sin(?0t)-sin(?0ti)ti
  ionization time gt v(ti)0
• Ek(t) (½)mv²(t) ? I ?0²
• Maximum harmonic order
• h?max Ip Ekmax
• h? ? Ip I ?0²
• Harmonic order grows with
• Ionization potential of the gas
• Intensity of the input LASER beam
• Square of the wavelengthof the input LASER
  beam!!

h?max Ip Ekmax
 plateau 
 cutoff 
Number of photons
Harmonic order n
The cutoff law is proved by the semi-classical
model
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Electron trajectory
Electron position
x(ti)0 v(ti)0
x

• Different harmonic orders
• different trajectories
• different emission times te

Time (TL)
0
1
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
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Chirp gt 0
Chirp lt 0
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Mairesse et al. Science 302, 1540 (2003)
0
Emission time (te)
Kazamias and Balcou, PRA 69, 063416 (2004)
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Part 3 conclusionHHG characteristics
gas jet/cell

• Input LASER beamI1014-1015W/cm² ??0
  linear polarization
• Jet of rare gasionization potential Ip
• Output LASER beamtrain of odd harmonics ?0/n,
  up to order n300 h?max ? Ip I.?0²
• 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 -gt gain of Ek ? I?0²
• Recombination of the electron with the atom -gt
  photoemission EIp Ek
• Explain the properties of the output beam -gt
  prediction of an atto-chirp

?0
?0/n
Number of photons Eh?
h?max IpEkmax
Plateau
Cutoff
Order of the harmonic
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Part 4

• Production of attosecond pulses

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Temporal structure of one harmonic

• Input LASER beam
• ?t femtosecond
• ?0 800nm
• One harmonic of the output LASER beam
• ?t femtosecond
• ?0/n some nanometers (UV or X wavelength)
• -gt Selection of one harmonic
• Characterization of processes at UV-X scale and
  fs duration

Intensity
Intensity
Harmonic order
Time
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 Sum  of harmonics without chirp an ideal case

• Central wavelength ??0/n -gt ?0 800nm
  order n150 ?5nm
• Bandwidth ?? -gt 25 harmonics i.e. ??2nm
• Fourier limit for a Gaussian ?? ?t ½
• ??/? ??/? ? c/?
• ?? c ?? (n/?0)²
• ?t (?0/n)² (1/c?? )
• ?t 10 10-18s -gt 10 attosecond pulses!
• If all harmonics in phasegeneration of pulses
  with ?t T0/2N

E(t)
Time
10 fs
Intensity
T0/2
T0/2N
Time
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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 ?0800nm I4
1014 W/cm2
T0/2
T0/2N
Intensity
10 fs
Time
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Solution select only few harmonics
(Measurement in Neon)
H25-33 (5)

Mairesse et al, 302, 1540 Science (2003)
Mairesse et al, Science 302, 1540 (2003)
Y. Mairesse et al. Science 302, 1540 (2003)
Optimum spectral bandwith
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Part 4 conclusionProduction of attosecond pulses

• Shorter pulse -gt wider bandwidth (??.?t 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 gt
  pulse is lengthened
• Solution Selection of some successive
  harmonics gt Generation of 100as pulses

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General ConclusionHigh-order Harmonic Generation
in gases

• One solution for two aims
• Achieve UV-X LASER wavelengths
• Generate attosecond LASER pulses
• Characteristics
• High coherence -gt interferometric applications
• High intensity -gt study of non-linear processes
• Ultrashort pulses
• Femtosecond one harmonic
• Attosecond selection of successive harmonics
  with small chirp
• In the futureimprove the generation of
  attosecond pulses

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Thank you for your attention!

• Questions?

Thanks to Pascal Salières (CEA Saclay) Manuel
Joffre (Ecole Polytechnique) Yann Mairesse
(CELIA Bordeaux) David Garzella (CEA Saclay)