2. High-order harmonic generation in gasesAttosecond pulse generation

1. Introduction to nonlinear optics

Polarization induced by a laser

field

)( 3)3(2)2()1(0 EEEP

linear response

nonlinear response

Introduction to nonlinear optics

202 dEPNL

Second harmonic generation

First demonstration of second-harmonic generation

P.A. Franken (1961)

The second-harmonic beam was very weak because the process was not phase-matched.

The actual published results…

First demonstration of second-harmonic generation

ccetzPtzP ziktiNL

2222

12 ),(),(Harmonic

generation

ccetzEtzE zikti

2222

12 ),(),(

Different phase velocity

Introduction to nonlinear optics

ccetzEtzE zikti ),(),( 12

1Fundamental

2nd harmonic

2

2

22

2

22 11

t

P

ct

E

cE

NL

Generate field = solution of a wave

equation

c

nk

kLcoh

z

ziktie 22

ziktie 22

nLcoh

4

Out of phase

Coherence length

Phase-matching second-harmonic generation

(2 ) ( )n n

2Frequency

Ref

ract

ive

inde

x

2Frequency

Ref

ract

ive

inde

x

ne

on(2 ) ( )o en n

Using birefringence

L

Eff

icie

ncy

()

cohL

2L

)/(sin 2cohLL

Depletion

Dependence of SHG intensity on length

Large k Small k

The SHG intensity is sharply maximized if k = 0.

321 kkk

12 321

3

1k3k

2k

Wave vectors

L

Eff

icie

ncy

()

cohL

2L

)/(sin 2cohLL

absLLeL /2 absL

The lengths of the problem

ampL

)(2 LFL q

Phase

Dispersion kz

z

gen(z)- pol(z)

Dipole phase

)(zIi

Dispersion free electronsFocusing

)/2(tan 1 bzq

-1 cm 1 cm

40

-40

Intensity, pressure, focusing, many parameters! Asymmetry before/after the focus

gen(z)- pol(z)

Localized in space and in time!

-1 cm 1 cm

40

z

zzzL polgen

coh

)]()([

/)(

),()( tzLzL cohcoh

321 kkk

12 321

3

1k3k

2k

Wave vectors

2.7 fs

2 cycles

Generation of short light pulses

cT

2 XUV!

1 eV

30 eV

Generation of short light pulses

Frequency Time

4.0

Broad bandwidth!

0.1 eV

10 eV

Fourier Transform

The electron can tunnel through the distorted Coulomb barrier

Strong-Field Atomic Physics

I

III

Interaction with the core

The electron wave packet interacts with the remaining core

IIThe electron is accelerated by the field, and may return to the atomic core

III

Ferray et al., J. Phys. B 21, L31 (1988)

Multiphoton

Plateau

Cut-off

High-Order Harmonic Generation in Gases

3

7

5

)12( q

.

.

Semi-classical three-step model

IIThe free electron is acceleratedby the field, and may return to theatomic core

III The electron recombines with the atom, emitting its energy as an XUV photon

The electron tunnels through thedistorted Coulomb barrierI

High-Order Harmonic Generation in Gases

High-Order Harmonic Generation in Gases

Electron dynamics

Group delay dispersion

Several bursts per half laser cycle

Atom

FieldElectrons

Short

Long

Plateau Cut-off

High-Order Harmonic Generation in Gases

3

7

5

)12( q

.

.

50 60 70 80

H53

H49

H43H37

H31

Photo

ns

Energy (eV)

IIIThe electron recombines with the atom, emitting its energy as an XUV photon

High-Order Harmonic Generation in Gases

AtomicMedium

Laser

Titanium-Sapphire, 800 nm 1 kHz, 2 mJ, 35 fs pulses

Gas cell with rare gas

Time

Time

Tunneling

Acceleration in the continuum

Recombination

Attosecond pulse train

Broad spectrum Single attosecond pulse

2200

Energy Time

as

Harmonic spectrum Attosecond pulse trainEnergy Time

L22L Time domainFrequency domain

=20eV

Attosecond pulse train

Time

0

Harmonic spectrum

Energy

02

Is this always true?

Generation of short light pulses