Transcript High-order Harmonic Generation in Gases (HHG)

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
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
(pulse)
Electric
field
Continuous
Pulses
time
• Two goals for LASERs:
– Reach UV-X wavelengths (1-100nm)
– Generate shorter pulses (10-18s)
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Outline
-> How does the HHG allow to achieve
shorter space and time scales?
1.
2.
3.
4.
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 / ν (or ω = 2πν)
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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)
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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:
E(t )   En (t )   a n cos(nt  n )
n
n
• More spectral components => Shorter pulse
• Spectral components not in phase (« chirp ») => Longer pulse
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Phase of the spectral components
Time
E (t )   E ( )e
Phase of each ω
Frequency
d
2
Fourier
transform
chirp +
no chirp
it
E()  E() ei ( )
Phase of the
ω component
chirp -
No chirp:
All the ω in phase
Moment of
arrival of each ω
Electric field in
function of time
minimum pulse
duration
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Fourier limit
•
Link between the pulse duration and its spectral width
Δt: pulse duration
Fourier
transform
Δω: spectral width
I(ω)
I(t)
• Fourier limit: Δω ∙ Δt ≥ ½
1
t
I(t)
•
For a perfect Gaussian: Δω ∙ Δt = ½
ω
I(ω)
2
t
I(t)
ω
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
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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?
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Classical harmonic generation
• In some materials, with a high LASER intensity
2 photons E=hν
λ0 = 800nm
fundamental wavelength
1 photon E=h2ν
λ0/2 = 400nm
harmonic wavelength
• Problems:
– low-order harmonic generation (λ/2 or λ/3)
– crystal: not below 200nm
– other solutions not so efficient
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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)
2nd
ω0
ω
HG
I(ω)
(Harmonic Generation)
2ω0
ω
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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
λ
(HHG) in particular in gases
0
gas jet/cell
λ0/n
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Part 3
HHG characteristics
&
Semi-classical model
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Harmonic generation in gases
Grating
LASER
source
fundamental
wavelength λ0
• Classical HG
• Low efficiency
• Multiphotonic ionization
of the gas: n ∙ hν0 -> h(nν0)
=> Low orders
Number of photons
Gas jet
Harmonic order n
LASER output
harmonic wavelengths λ0/n
(New & Ward, 1967)
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Increasing of LASER intensity
• Energy : ε = 1J
• Short pulse : Δt < 100fs
• Focused on a small area : S = 100μm²
Pulse length
I = ε/Δt/S > 1018 W/cm²
Intensity
1019 W/cm²
λ ~ 800nm
100ns
1015
100ps
1013
100fs
1fs
109
1967
1988
HHG
Years
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High-order Harmonic Generation
(HHG) in gases
Grating
Gas jet
LASER
source
fundamental
wavelength λ0
• How to explain?
• up to harmonic order 300!!
• quite high output intensity
•Interest :
• UV-X ultrashort-pulsed
LASER source
Number of photons
« plateau »
« cutoff »
Harmonic order n
LASER output
harmonic wavelengths λ0/n
(Saclay & Chicago, 1988)
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Semi-classical model in 3 steps
-
Elaser
-
x
Ip
-
-
0t = 0
0t ~ /2
Electron of a gas atom
Fundamental state
1 Tunnel
ionization
-
Ek
-
-
0t = 3/2
2 Acceleration in the
electric LASER field
hn=Ip+Ek
0t ~ 2
3 Recombination to
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 =
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Ionization potential of the gas (Ip) + Kinetic energy won by the electron (Ek)
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
• Maximum harmonic order
 hνmax = Ip + Ekmax
hν ∝ Ip + I ∙ λ0²
• Harmonic order grows with:
– Ionization potential of the gas
– Intensity of the input LASER beam
– Square of the wavelength
of the input LASER beam!!
Number of photons
« plateau »
« cutoff »
Harmonic order n
The cutoff law is proved by
the semi-classical model
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Electron trajectory
Electron position
x
x(ti)=0
v(ti)=0
Different harmonic orders
 different trajectories
 different emission times te
1
0
Time (TL)
If short traj. selected (spatial filter on axis)
Harmonic
order
Short traj.
Chirp > 0
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Long traj.
Chirp < 0
Positive chirp of output LASER beam
on attosecond timescale:
the atto-chirp
Mairesse et al. Science 302, 1540 (2003)
0
Emission time (t )
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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
• 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
Plateau
λ0/n
hνmax = Ip+Ekmax
Cutoff
• Semi-classical model:
Order of the
harmonic
– 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
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Part 4
Production of attosecond pulses
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Temporal structure of one harmonic
• Input LASER beam
– Δt ~ femtosecond
– λ0 ~ 800nm
• One harmonic of the output LASER beam
Intensity
Intensity
– Δt ~ femtosecond
– λ0/n ~ some nanometers (UV or X wavelength)
Harmonic order
Time
• -> Selection of one harmonic
– Characterization of processes at UV-X scale and fs duration
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« Sum » of harmonics without chirp:
an ideal case
•
•
•
•
•
•
•
Central wavelength: λ=λ0/n -> λ0 = 800nm ; order n~150 ; λ~5nm
E(t)
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 )
Time
~ 10 fs
Δt ~ 10 ∙ 10-18s -> 10 attosecond pulses! Intensity
T0/2
• If all harmonics in phase:
generation of pulses with Δt ~ T0/2N
T0/2N
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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
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Time
Solution: select only few harmonics
(Measurement in Neon)
H25-33 (5)
IR field (absolute value)
45
40
XUV pulse (H33 to H53)
H35-43
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Intensity (a.u.)
Intensity (arbitrary units)
20
30
25
20
15
H45-53
150 as
15
130 as
H55-63
10
+
5
10
0
5
0
250
500
500
1 000
1000 1250 1500 1750 2000
Time (as)
0
0
750
1 500
Time (as)
23 harmonics
Optimum spectral bandwith:
11 harmonics
2 000
2 500
Mairesse
Mairesseetetal,
al,Science
302, 1540
302,Science
1540 (2003)
(2003)
Y. Mairesse et al. Science 302, 1540 (2003)
Δt=150 as (ΔtTF=50 as)
Δt=130 as (ΔtTF=120 as)
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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
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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
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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)
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