WHY????

Ultrashort laser pulses

(Very) High field physics

Highest peak power, requires highest concentration of energy

E L I

Create … shorter pulses (attosecond)Create x-rays (point source)Imaging

High fields high nonlinearities high accuracy

F=ma0~ 31 Å

1015 W/cm2, 800 nm

20

Electrons ejected by tunnel ionization can be re-captured by the next half optical cycle of opposite sign. The interaction of the returning electron with the atom/molecule leads to high harmonic generation and generation of single attosecond pulses.

-1

0

1

To do this you need to control a single cycle

Resolve very fast events

- “Testing” Quantum mechanics

Probing chemical reactions

Pump probe experiments

All applicatons require propagation/manipulation of pulses

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0

1

MANIPULATION OF THIS PULSE

Chirped pulse

LEADS TO THIS ONE:

Propagation through a medium with time dependent index of refraction

Pulse compression: propagation through wavelength dependent index

Why do we need the Fourier transforms?

Construct the Fourier transform of

“Linear” propagation in frequency domain

“Non-Linear” propagation in time domain

Actually, we may need the Fourier transforms (review)

0

Properties of Fourier transforms

Shift

Derivative

Linear superposition

Specific functions: Square pulse Gaussian Single sided exponential

Real E(E*(-

Linear phase

Product Convolution

Derivative

Description of an optical pulse

Real electric field:

Fourier transform:

Positive and negative frequencies: redundant information Eliminate

Relation with the real physical measurable field:

Instantaneous frequency

0 z

t t

What is important about the Carrier to Envelope Phase?

Slowly Varying Envelope Approximation

Meaning in Fourier space??????

Forward – Backward Propagation

Maxwell Equation

s = t – n/c Zr = t + n/c z

20 0

02

n nE P

z c t z c t t

2 2 2 20

02 2 2 2

nE P

z c t t

No scattering

i s i rE e e FΕ Ε

No coupling between EF & EB

22 2

024

cE P

r s n s r

22 2i s i s

0 F2

ce 2 i P e

r s r 4 n s r

F FΕ Ε

No linear assumption

Slowly varying envelope

22

02

22

0

2

1

2

F

F

ci P

r n

Pn

FΕ

22

2F FP P

t

Study of linear propagation

(Maxwell second order)

Solution of 2nd order equation

22

02

( ) ( , ) 0E zz

0( )P E 0( ) (1 ( ))

( )( , ) ( , ) ik zE z E 0 e

( ) ( )2 20k

Propagation through medium

No change in frequency spectrum

To make F.T easier shift in frequencyExpand k value around central freq l

l

( )( , ) ( , ) lik zz 0 e ε ε

( )1( , ) ( ,0) ( )

2lik z i tt z e e d

z

Z=0

1( , ) ( , ) ( )

2i tE t z E z e d

Dispersion includedk real

10

gz v t

ε ε

Study of linear propagation

Expansion orders in k(Material property

l

l

2| 22

1( , ) ( ,0) (1 | ( ) ) ( )

2l

dkiik z i td d k

t z e e e i z dd

ε ε

II)( )( , ) ( , ) lik zz 0 e ε ε

ll| ( )| ( )( , )

22

2 l

1 d kdk i zi z ik z2d d0 e e e

ε

l

l

| ( )( , ) ( | ( ) ) l

dk 2i z 2 ik zd2

1 d k0 e 1 i z e

2 d

ε

22

2

( ) 1( ) ( )

2ixtt

x x e d xt

ε ε

2 2

2 2

10

2g

i d k

z v t d t

ε ε ε

Second

Study of linear propagation

Propagation in the time domain

PHASE MODULATION

n(t)or

k(t)

E(t) = (t)eit-kz

(t,0) eik(t)d (t,0)

DISPERSION

n()or

k()() ()e-ikz

Propagation in the frequency domain

Retarded frame and taking the inverse FT:

PHASE MODULATION

DISPERSION

Application to a Gaussian pulse

Evolution of a single pulse in an ``ideal'' cavity

Dispersion

Kerr effect

Kerr-induced chirp

20 0

02

n nE P

z c t z c t t

2 2 2 20

02 2 2 2

nE P

z c t t

22

2F FP P

t

Study of propagation from second to first order

From Second order to first order (the tedious way)

( ) ( )kz kz

2 2 2 20 i t i t

02 2 2 2

ne P e

z c t t

2 2 22

2 2 2 2 2

22

0 0 02

1 2ik 2ik

c z c t c t z

P i P Pt t

01 i cP

z c t 2

(Polarization envelope)

Pulse duration, Spectral width

Two-D representation of the field: Wigner function

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

Time TimeF

requ

ency

Fre

quen

cy

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

Time TimeF

requ

ency

Fre

quen

cyGaussian Chirped Gaussian

Wigner Distribution

Wigner function: What is the point?

Uncertainty relation:

Equality only holds for a Gaussian pulse (beam) shape free of anyphase modulation, which implies that the Wigner distribution for aGaussian shape occupies the smallest area in the time/frequencyplane.

Only holds for the pulse widths defined as the mean square deviation