SHORT PULSES

AS INTRODUCTION TO FOURIER TRANSFORMS

SPACE TIME ANALOGY:

What applies to pulses in time can be transposed tobeams in space

In time: dispersionIn space: diffraction

Pulse description --- a propagating pulse

A Bandwidth limited pulse No Fourier Transform involved

Actually, we may need the Fourier transforms (review)

Construct the Fourier transform of

Pulse Energy, Parceval theorem

Slowly Varying Envelope Approximation

Complex representation of the electric field

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

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)

A Bandwidth limited pulse

Some (experimental) displays of electric field versus time

-20 -10 0 10 20

Delay (fs)

Chirped 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

Construct the Fourier transform of

Pulse Energy, Parceval theorem

Poynting theorem

Pulse energy

Parceval theorem

Intensity?

Spectral intensity

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

Instantaneous frequency

In general one chooses:

And we are left with

0 2-2 44

Time (in optical periods)

-1

1

0

-1

Field (Field)7

0 2-2 44

Time (in optical periods)

1

0

-1

Field(Field)7

Frequency and phase – CEP – is it “femtonitpicking”?

0

Frequency and phase – CEP – is it “femtonitpicking”?

Slowly Varying Envelope Approximation

Meaning in Fourier space??????

0 2-2 44 0 2-2 44

Time (in optical periods)

-1-1

0 2-2 44 0 2-2 44

Time (in optical periods)

1

0

-1

1

0

-1

1

0

-1

Two pulses of 2.5 optical cycle. The blue line is the electric field.

The green dotted line is the seventh power.

T

Traditional CEP measurement through high order nonlinear interaction

High order effects depend on the CEP

The CEP – how to “measure” it?

G.G. Paulus et al, Phys. Rev. Lett. 91, 253004 (2003)

Pulse description --- a propagating pulse

A Bandwidth limited pulse No Fourier Transform involved

Actually, we may need the Fourier transforms (review)

Construct the Fourier transform of

Pulse Energy, Parceval theorem

Frequency and phase – CEP – is it “femtonitpicking”?

Slowly Varying Envelope Approximation

Complex representation of the electric field

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:

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

time0

Electric fieldamplitude

Pulse broadening, dispersion

z = ct

Spectral phase

z = v2t(slow)

z = v1t(fast)

Spectral phase

time

Electric fieldamplitude

z = v1t(fast)

z = v2t(slow)

time

E(t)

z = ct

Pulse broadening, dispersion

Broadening andchirping

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

Application to a Gaussian pulse