Complex representation of the electric field

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

Slowly Varying Envelope Approximation

Pulse duration, Spectral width

-6 -4 -2 0 2 4 6

-1

0

1

-20 -10 0 10 20

Delay (fs)

Chirped pulse

z

t

z = ctz = vgt

A propagating pulse

t

A Bandwidth limited pulse

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

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

Frequency and phase - CEP

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

Slowly Varying Envelope Approximation

Meaning in Fourier space??????

Robin K Bullough Mathematical Physicist

Robin K. Bullough (21 November 1929-30 August 2008) was a British Mathematical Physicist famous for his role in the development of the theory of the optical soliton.

J.C.Eilbeck J.D.Gibbon, P.J.Caudrey and R.~K.~Bullough, « Solitons in nonlinear optics I: A more accurate description of the 2 pi pulse in self-induced transparency »,Journal of Physics A: Mathematical, Nuclear and General,6: 1337--1345, (1973)

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