# High-order Harmonic Generation in Gases _HHG_

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```					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?

2.   Achieve shorter LASER pulse duration
3.   HHG characteristics & semi-classical model
4.   Production of attosecond pulses

4
Part 1

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
• 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

Questions?

Thanks to:
Pascal Salières (CEA Saclay)
Manuel Joffre (Ecole Polytechnique)
Yann Mairesse (CELIA Bordeaux)
David Garzella (CEA Saclay)
32

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