Pulse description --- a propagating pulse

A Bandwidth limited pulse No Fourier Transform involved

Fourier transforms review

Slowly Varying Envelope Approximation

OPTICS OF SHORT PULSES

with minimum of equations, maximum of analogies and hand waving.

time0

Electric fieldamplitude

Many frequencies in phase construct a pulse

A Bandwidth limited pulse

FREQUENCY

Time and frequency considerations: stating the obvious

TIME

E

A Bandwidth limited pulse

FREQUENCY

The spectral resolution of the cw wave is lost

TIME

E

A Bandwidth limited pulse

z

t

z = ctz = vgt

A propagating pulse

t

A Bandwidth limited pulse

We may need the Fourier transforms (review)

0

Shift

Derivative

Linear superposition

Specific functions: Square pulse Gaussian Single sided exponential

Real E(E*(-

Linear phase

Product Convolution

Derivative

Properties of Fourier transforms

Construct the Fourier transform of

0

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

We have to return to Maxwell's propagation equation:

In frequency

How to correctly propagate an ultrashort pulse without phase and group velocity

It is only if That the pulse propagates unchanged at velocity n/c

Group velocity is a concept that is clearly related to the SVEA

Maxwell’s equations, linear propagation

Propagation of the complex field

Maxwell’s equations, nonlinear propagation

Pulse broadening, dispersion

Maxwell’s equations, linear propagation

Dielectrics, no charge, no current:

Medium equation:

can be a tensor birefringence

In a linear medium:

Maxwell’s equations, nonlinear propagation

Maxwell’s equation:

Since the E field is no longer transverse

Gadi Fibich and Boaz Ilan PHYSICAL REVIEW E 67, 036622 (2003)

Is it important?

Only if

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 broadening, dispersion

Solution of 2nd order equation

22

02

( ) ( , ) 0E zz

0( ) (1 ( ))

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

( ) ( )2 20k

0( )P E 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 εz

Z=0

1( , ) ( , ) ( )

2i tE t z E z e d

1

0gz v t

ε ε

Study of linear propagation

Expand k to first order, leads to a group delay:

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

( )( , ) ( , ) lik zz 0 e ε εll

| ( )| ( )( , )

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

Study of linear propagation

Propagation in dispersive media: the pulse is chirped and broadening

Propagation in nonlinear media: the pulse is chirped

Combination of both: can be pulse broadening, compression,Soliton generation

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

Townes’soliton

Eigenvalue equation (normalized variables. Solution of type:

2D nonlinear Schroedinger equation

Normalization: and

Soliton equation in space

In space:

In time

Back to linear propagation: Gaussian pulse

-6 -4 -2 0 2 4 6

-1

0

1

-20 -10 0 10 20

Delay (fs)

Pulse propagation through 2 mm of BK7 glass

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

A Bandwidth limited pulse

Some (experimental) displays of electric field versus time

-6 -4 -2 0 2 4 6

-1

0

1

-20 -10 0 10 20

Delay (fs)

How was this measured?

A Bandwidth limited pulse

Some (experimental) displays of electric field versus time

-20 -10 0 10 20

Delay (fs)

Chirped pulse

Construct the Fourier transform of

Pulse Energy, Parceval theorem

Poynting theorem

Pulse energy

Parceval theorem

Intensity?

Spectral intensity