Basic Definitions Nice
Transcript of 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
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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."
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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
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Temporal
andSpectral
Shapes
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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
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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.
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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
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The convolution
can be performedvisually.
Here, rect(x) *rect(x) = (x)
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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
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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
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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
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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
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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
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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.
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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.
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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
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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?
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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.
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The Central Limit Theorem for a square function
Note that P(x)*4 already looks like a Gaussian!
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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
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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
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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
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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!
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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