basic definitions nice

8/13/2019 Basic Definitions Nice

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Convolution & Autocorrelation

Pulse Widths & The Uncertainty Principle:

Parseval's Theorem:

Convolution & the Convolution Theorem

The Shah function

Trains of pulses and laser modes

Autocorrelation

The Autocorrelation Theorem

The FT of a fields autocorrelation is its spectrum

Cant obtain the intensity from its autocorrelation

2 21( ) ( )

2f t dt F d

1v t

http://frog.gatech.edu/talks.html/
http://frog.gatech.edu/talks.htmlhttp://frog.gatech.edu/talks.html


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The Pulse Width

There are many definitions of the"width" or length of a wave or pulse.

The effective width is the width of a rectangle whose heightand

areaare the same as those of the pulse.

Effective width Area / height:

Advantage: Its easy to understand.

Disadvantages: The Abs value is inconvenient.

We must integrate to .

1( )

(0)efft f t dt

f

t

f(0)

0

teff

t

t

(Abs value is

unnecessary

for intensity.)


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The rms pulse width

The root-mean-squared widthor rms width:

Advantages: Integrals are often easy to do analytically.

Disadvantages: It weights wings even more heavily,

so its difficult to use for experiments, which can't scan to )

1/ 2

2 ( )

( )

rms

t f t dt

t

f t dt

t

t

The rms width is the second-order moment.


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The Full-Width-

Half-Maximum

Full-width-half-maximum

is the distance between the

half-maximum points.

Advantages: Experimentally easy.

Disadvantages: It ignores satellite

pulses with heights < 49.99% of the

peak!

Also: we can define these widths in terms of f(t) or of its intensity,|f(t)|2.

Define spectralwidths ( ) similarly in the frequency domain (t ).

t

tFWHM

1

0.5

t

tFWHM


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The Uncertainty Principle

The Uncertainty Principle says that the product of a function's widths

in the time domain (t) and the frequency domain () has a minimum.

(Different definitions of the widths and the Fourier Transform yield

different constants.)

1 1 (0)( ) ( )exp( [0] )(0) (0) (0)

1 1 2 (0)( ) ( )exp(

(0) (0) (0)

Ft f t dt f t i t dt f f f

fF d F i d

F F F

(0) (0)2

(0) (0)

f Ft

F f 2t 1t

Combining results:

or:

Define the widths

assumingf(t) and

F()peak at 0:

1 1( ) ( )

(0) (0)t f t dt F d

f F


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The Time-Bandwidth Product

For a given wave, the product of the time-domain width (t) and

the frequency-domain width () is the

Time-Bandwidth Product (TBP)

t

TBP

A pulse's TBP will always be greater than the theoretical minimum

given by the Uncertainty Principle (for the appropriate width definition).

The TBP is a measure of how complex a wave or pulse is.

Even though every pulse's time-domain and frequency-domain

functions are related by the Fourier Transform, a wave whose TBP is

the theoretical minimum is called "Fourier-Transform Limited."


7/29

The coherence time (tc= 1/)

indicates the smallest temporal

structure of the pulse.

In terms of the coherence time:

TBP = t = t / tc

= about how many spikes are in the pulse

A similar argument can be made in the frequency domain, where the

TBP is the ratio of the spectral width to the width of the smallest

spectral structure.

The Time-Bandwidth Product is a

measure of the pulse complexity.

t

tc

I(t)A complicated

pulse

time


8/29

Temporal

andSpectral

Shapes


9/29

Parsevals TheoremParsevals Theorem says that the

energy is the same, whether you

integrate over time or frequency:

Proof:

2 21( ) ( )

2

f t d t F d

2

( ) ( ) *( )

1 1( exp( ) *( exp( )2 2

1 1( ) *( ') exp( [ '] ) '

2 2

1 1( ) *( ') [2 ')] '

2 2

f t dt f t f t dt

F i t d F i t d dt

F F i t dt d d

F F d d

21 1

( ) *( ) ( )2 2

F F d F d


10/29

Time domain Frequency domain

f(t)

|f(t)|2

F()

|F()|2

t

t

Parseval's Theorem in action

The two shaded areas (i.e., measures of the light pulse energy) are

the same.


11/29

The Convolution

The convolution allows one function to smear or broaden another.

( ) ( ) ( ) ( )

( ) ( )

f t g t f x g t x dx

f t x g x dx

changing variables:

xt - x


12/29

The convolution

can be performedvisually.

Here, rect(x) *rect(x) = (x)


13/29

Convolution with a delta function

Convolution with a delta function simply centers the function on the

delta-function.

This convolution does not smear outf(t). Since a devices performance

can usually be described as a convolution of the quantity its trying to

measure and some instrument response, a perfect device has a delta-

function instrument response.

( ) ) ( ) ( )

( )

f t t a f t u u a du

f t a


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The Convolution TheoremThe Convolution Theorem turns a convolution into the inverse FT of

the product of the Fourier Transforms:

Proof:

{ ( ) ( )} = ( ) ( )f t g t F G wF

{ ( ) ( )} ( ) ( ) exp( )

( ) ( ) exp( )

( ){ ( exp( )}

( ) exp( ) ( ( (

f t g t f x g t x dx i t dt

f x g t x i t dt dx

f x G i x dx

f x i x dx G F G

F


15/29

The Convolution Theorem in action

2{ ( )}

sinc ( / 2)xk

F

{rect( )}sinc( / 2)

xk

F

rect( ) rect( ) ( )x x x

2sinc( / 2) sinc( / 2) sinc ( / 2)k k k


16/29

The symbol III is pronounced shahafter the Cyrillic character III, which is

said to have been modeled on the Hebrew letter (shin) which, in turn,

may derive from the Egyptian a hieroglyph depicting papyrus plants

along the Nile.

The Shah Function

The Shah function, III(t), is an infinitely long train of equally spaced

delta-functions.

t

III( ) ( )

m

t t m


17/29

The Fourier Transform of the Shah

Function

If = 2n, where nis an integer,

the sum diverges; otherwise,

cancellation occurs. So:

{III( )} III(t F

)exp( )

)exp( )

exp( )

m

m

m

t m i t dt

t m i t dt

i m

t

III(t)

F {III(t)}

2


18/29

The Shah Function

and a pulse train

( )m

f t mT

( ) III( / ) ( )E t t T f t

wheref(t)is the shape of each pulse and Tis the time between

pulses.

Set t /T = m ort = mT

An infinite train of identical pulses

(from a laser!) can be written:

( / ) ( )m

t T m f t t dt


19/29

An infinite train of identical pulses can be written:

E(t) = III(t/T) *f(t)

wheref(t)represents a single pulse and Tis the time between pulses.The Convolution Theorem states that the Fourier Transform of aconvolution is the productof the Fourier Transforms. So:

The Fourier Transform of an Infinite Train of Pulses

( )

III( / ) (2

E

FT

If this train of pulses results from a single pulse bouncing back andforth inside a laser cavity of round-trip time T. The spacing between

frequencies is then = /Tor = 1/T.


20/29

The Fourier Transform of a Finite Pulse Train

A finitetrain of identical pulses can be written:

( ) {III( / ) ( )} ( )E t t T g t f t

( ) {III( / 2 ) ( )} ( )E T G F

whereg(t)is a finite-width envelope over the pulse train.


21/29

A lasers frequencies are often called longitudinal modes.Theyre separated by 1/T= c/2L.

Which modes lase depends on the gain profile.

Frequency

Int

ensity

Here,

additional

narrowband

filtering hasyielded a

single mode.

Laser Modes


22/29

The 2D generalization of the Shah function:

The Bed of Nails function

We wont do anything with this function, but I thought you might like

this colorful image Can you guess what its Fourier transform is?


23/29

The Central Limit Theorem

The Central Limit Theorem says:

The convolut ion of the convolut ion o f the convolut ion etc .

appro aches a Gauss ian.

Mathematically,

f(x)*f(x)*f(x)*f(x)*... *f(x) exp[(-x/a)2]

or:f(x)*n exp[(-x/a)2]

The Central Limit Theorem is why nearly everything has a Gaussian

distribution.


24/29

The Central Limit Theorem for a square function

Note that P(x)*4 already looks like a Gaussian!


25/29

The autoconvolution of a functionf(x)is given by:

Suppose, however, that prior to multiplication and integration we do not reverse oneof the two component factors; then we have the integral:

which may be denoted byf f. A single value off fis represented by:

The Autocorrelation

The shaded area is the value of the autocorrelation

for the displacementx. In optics, we often define the

autocorrelation with a complex conjugate:

( ) ( )f f f t f t t dt

( ) ( )f t f t t dt

*( ) ( )f t f t t dt


26/29

The Autocorrelation Theorem

The Fourier Transform of the autocorrelation is the spectrum!

Proof: 2

( ) *( ) ( )f t f t t dt f t

F F

*

* *

( ) *( ) exp( ) ( ) *( )

( ) exp( ) ( )

( ) exp( ) ( ) ( ) ( ) exp( )

( ) exp( ) *( )

f t f t t dt i t f t f t t dt dt

f t i t f t t dt dt

f t i y f t y dy dt f t F i t dt

f t i t dt F

F

2( ) *( ) ( )F F F


27/29

The Autocorrelation Theorem in action

2( ) sinc ( / 2)t F

rect( ) sinc( / 2)t F

2

sinc( / 2)

sinc( / 2)

sinc ( / 2)

rect( ) rect( )

( )

t t

t

rect( )t

( )t

sinc( / 2)

2sinc ( /2)

t

t


28/29

The Autocorrelation Theorem for a light wave field

The Autocorrelation Theorem can be applied to a light wave field,

yielding an important result:

2

2

( ) *( { ( )}

(

E t E t t dt E t

E

F F

Remarkably, the Fourier transform of a light-wave fields autocorrelationis its spectrum!

This relation yields an alternative technique for measuring a light waves

spectrum.

This version of the Autocorrelation Theorem is known as the Wiener-

Khintchine Theorem.

= the spectrum!


29/29

The Autocorrelation Theorem for a light

wave intensity

The Autocorrelation Theorem can be applied to a light wave intensity,yielding a less important, but interesting, result:

Many techniques yield the intensity autocorrelation of an ultrashort

laser pulse in an attempt to measure its intensity vs. time (which is

difficult).

The above result shows that the intensity autocorrelation is not

sufficient to determine the intensityit yields the magnitude, but not

the phase, of

2

( ) ( ) ( )I t I t t dt I t

F F

{ ( )}I tF

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