Transcript Slide 1

1. Introduction to nonlinear optics
2. High-order harmonic generation in gases
Attosecond pulse generation
Introduction to nonlinear optics
Polarization
induced by a laser
field
Second harmonic
generation
P   0 (  (1) E   ( 2) E 2   (3) E 3  )
linear
response
P NL  2 0dE 2
nonlinear
response
First demonstration of second-harmonic generation
P.A. Franken (1961)
The second-harmonic beam was very weak because the process
was not phase-matched.
First demonstration of second-harmonic generation
The actual published results…
Introduction to nonlinear optics
Generate field =
solution of a wave
equation
Fundamental
Harmonic
generation
2nd harmonic
2
2 NL
1

E
1

P
2
 E 2 2  2
c t
c t 2
E ( z, t )  12 E1 ( z, t )eit ik z  cc
k 
n
c
2it  2 ik z
1
P2NL
(
z
,
t
)

P
(
z
,
t
)
e
 cc

2 2
Different phase velocity
E2 ( z, t )  12 E2 ( z, t )e 2it ik2 z  cc
e
e
z
Lcoh k  
Lcoh 
2 it  2 ik z

4 n
2 it ik2  z
Coherence
length
Out of
phase
n(2 )  n( )
Refractive
index
Phase-matching second-harmonic generation

Frequency
2
no (2 )  ne ( )
Refractive index
Using birefringence
ne
no

Frequency
2
Efficiency (h)
h  L2
Depletion
h  sin 2 (L / Lcoh )
Lcoh
L
Dependence of SHG intensity on length
Large k
Small k
The SHG intensity is sharply maximized if k = 0.
Wave vectors
3
1  2  3
  
k1  k2  k3
2
1

k3

k2

k1
The lengths of the problem
Efficiency (h)
h  L2
hLe
2  L / Labs
Labs
h  sin 2 (L / Lcoh )
Lcoh
Lamp
L
 L2 Fq ( L)
Phase
Fgen(z)- Fpol(z)
Dipole phase
40
 i I (z )
Dispersion
kz
z
-1 cm
1 cm
Dispersion
free electrons
Focusing
1
q tan (2 z / b)
-40
Intensity, pressure, focusing, many parameters!
Asymmetry before/after the focus
Fgen(z)- Fpol(z)
40

-1 cm
1 cm
Lcoh ( z )   /
[F gen ( z )  F pol ( z )]
z
Lcoh ( z )  Lcoh ( z, t )
Localized in space and in time!
Wave vectors
3
1  2  3
  
k1  k2  k3
2
1

k3

k2

k1
Generation of short light pulses
2.7 fs
1 eV
30 eV
2 cycles
2 
T

 c
XUV!
Generation of short light pulses
Fourier
Transform
0.1 eV
Frequency
10 eV
Time
   0.4
Broad
bandwidth!
Strong-Field Atomic Physics
I
The electron can
tunnel through the
distorted Coulomb
barrier
Interaction with the core
III
III
The electron wave packet
interacts with the remaining
core
II
The electron is accelerated
by the field, and may return
to the atomic core
High-Order Harmonic Generation in Gases
3
5
7
.
.
(2q  1)
Multiphoton
Plateau
Cut-off
Ferray et al., J. Phys. B 21, L31 (1988)
High-Order Harmonic Generation in Gases
Semi-classical three-step model
I
The electron tunnels through the
distorted Coulomb barrier
II
The free electron is accelerated
by the field, and may return to the
atomic core
III
The electron recombines with the atom,
emitting its energy as an XUV photon
High-Order Harmonic Generation in Gases
Electron dynamics
Electrons
Atom
Several bursts per
half laser cycle
Group delay
dispersion
Field
High-Order Harmonic Generation in Gases
3
5
Plateau
7
(2q  1)
Photons
.
.
H37
Cut-off
H43
H31
H49
H53
50
III
60
70
Energy (eV)
80
The electron recombines with the atom,
emitting its energy as an XUV photon
High-Order Harmonic Generation in Gases
Laser
Atomic
Medium
Gas cell
with rare
gas
Titanium-Sapphire, 800 nm
1 kHz, 2 mJ, 35 fs pulses
Tunneling
Acceleration in the
continuum
Recombination
Time
Time
Attosecond pulse
train
Harmonic spectrum

Attosecond pulse train

0
20
Frequency domain
Time domain

L

2L
Energy
Harmonic spectrum
=20eV
2

Time
Attosecond pulse train
2 200
as

Energy
Broad spectrum
Time
Single attosecond pulse
Energy
Is this always true?
Time
Generation of short light pulses