High-order Harmonic Generation (HHG) in gases

by Benoît MAHIEU

1

Introduction• Will of science to achieve lower scales

– Space: nanometric characterization

– Time: attosecond phenomena (electronic vibrations)

Period of the first Bohr orbit : 150.10-18s

λ = c/ν

2

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

time

ContinuousPulses

3

Outline

-> How does the HHG allow to achieveshorter space and time scales?

1. Link time / frequency2. Achieve shorter LASER pulse duration3. HHG characteristics & semi-classical model4. Production of attosecond pulses

4

Part 1

Link time / frequencyt / ν (or ω = 2πν)

5

LASER pulses

• Electric field E(t)

• Intensity I(t) = E²(t)

• Gaussian envelop: I(t) = I0.exp(-t²/Δt²)

‹t›: time of the mean value

I(t)

Δt: width of standard deviation Δt = pulse duration

6

Spectral composition of a LASER pulse

FOURIERTRANSFORM

Pulse = sum of different spectral components

7

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

n

nnnn

n tatEtE )cos()()(

8

Phase of the spectral components

Time Frequency

2)()(

deEtE ti )()()( ieEE 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

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 widthFourier transform

t

I(t)

t

I(t)

ω

I(ω)

ω

I(ω)

t

I(t)

ω

I(ω)

1

2

310

Part 1 conclusionLink 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

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

• Problems: – low-order harmonic generation (λ/2 or λ/3)– crystal: not below 200nm– other solutions not so efficient

2 photons E=hν 1 photon E=h2ν

λ0 = 800nmfundamental wavelength

λ0/2 = 400nmharmonic wavelength

14

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

I(ω)

ωω0

I(ω)

2ω0ω

2nd HG (Harmonic Generation)

15

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

λ0 λ0/n

16

Part 3

HHG characteristics&

Semi-classical model

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Harmonic generation in gases

GratingGas jet

LASER source

fundamental

wavelength λ0

Harmonic order nN

umbe

r of p

hoto

ns

LASER output harmonic wavelengths λ0/n

• Classical HG

• Low efficiency

• Multiphotonic ionization of the gas: n h∙ ν0 -> h(nν0)=> Low orders

(New & Ward, 1967) 18

Increasing of LASER intensity

Pulse length

Intensity

Years

109

1013

1015

1019 W/cm²

• Energy : ε = 1J• Short pulse : Δt < 100fs I = ε/Δt/S > 1018 W/cm² • Focused on a small area : S = 100μm²

HHG

1fs

100fs

100ps

100ns

1967 1988

λ ~ 800nm

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High-order Harmonic Generation (HHG) in gases

GratingGas jet

LASER source

fundamental

wavelength λ0

Harmonic order nN

umbe

r of p

hoto

ns

LASER output harmonic wavelengths λ0/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

Semi-classical model in 3 steps

3 Recombination tofundamental state

t ~ 2

h=Ip+Ek

Acceleration in theelectric LASER field

t = 3/2

2Tunnelionization

t ~ /2

Elaser

K. Kulander et al. SILAP (1993)

P.B. Corkum PRL 71, 1994 (1993)

x

t = 0

Ip

1

Ek

Energy of the emitted photon =

Ionization potential of the gas (Ip) + Kinetic energy won by the electron (Ek)

Periodicity T0/2 harmonics are separated by 20

- - - -

-

-

-

Electron of a gas atomFundamental state

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

• 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!!

Harmonic order n

Num

ber o

f pho

tons

« plateau »

« cutoff »

hνmax = Ip + Ekmax

The cutoff law is proved by the semi-classical model 22

Electron trajectory

Positive chirp of output LASER beamon attosecond timescale:

the atto-chirp

Emission time (te)

x

Time (TL)0 1

Harmonicorder

15171921

0

Electron position

Different harmonic orders different trajectories

different emission times te

Long traj.Short traj.

If short traj. selected (spatial filter on axis)

Kazamias and Balcou, PRA 69, 063416 (2004)

Chirp > 0 Chirp < 0

Mairesse et al. Science 302, 1540 (2003)

x(ti)=0v(ti)=0

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Part 3 conclusionHHG characteristics

• Input LASER beam:I~1014-1015W/cm² ; λ=λ0 ; linear polarization

• 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

• 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

gas jet/cell

λ0 λ0/n

Number of photons E=hν

Order of the harmonic

Plateau

Cutoff

hνmax = Ip+Ekmax

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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– Δt ~ femtosecond– λ0/n ~ some nanometers (UV or X wavelength)

• -> Selection of one harmonic– Characterization of processes at UV-X scale and fs duration

Inte

nsity

Inte

nsity

TimeHarmonic order

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« 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• 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/c∙ Δλ )

• Δt ~ 10 10∙ -18s -> 10 attosecond pulses!

• If all harmonics in phase:generation of pulses with Δt ~ T0/2N

T0/2N

T0/2

Intensity

Time

~ 10 fs

E(t)

Time

27

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

T0/2N

T0/2

Intensity

Time

~ 10 fs

Emission times measured in Neon at λ0=800nm ; I=4 1014 W/cm2

28

H35-43

H45-53

0 500 1 000 1 500 2 000 2 5000

5

10

15

20

25

30

35

40

45

Inte

nsity

(ar

bitr

ary

units

)

Time (as)

H55-63

Y. Mairesse et al. Science 302, 1540 (2003)

Δt=150 as (ΔtTF=50 as)23 harmonics

+

H25-33 (5)

Optimum spectral bandwith:Δt=130 as (ΔtTF=120 as)11 harmonics

150 as

0 250 500 750 1000 1250 1500 1750 20000

5

10

15

20

In

tensity

(a.u

.)

Time (as)

IR field (absolute value) XUV pulse (H33 to H53)

130 as

Mairesse et al, 302, 1540 Science (2003)Mairesse et al, Science 302, 1540 (2003)

(Measurement in Neon)Solution: select only few harmonics

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Part 4 conclusionProduction of attosecond pulses

Shorter pulse -> wider bandwidth (Δω.Δt = ½) + no chirpi.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 ConclusionHigh-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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