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Fourier theory
We know a lot about waves at a single w:n( )w , vp( )w , R( ), w absorption( )w …
Analyze arbitrary in terms of these, because Fourier told us that….
,E r t
How does a complicated optical pulse reflect off a surface?
,E r t
Fourier Series Example
4 4 4 4( ) sin 2 sin 6 sin 10 sin 14 ...
3 5 7f t t t t t
2 2 6 6 10 102 2 2 2 2 2( ) ...
3 3 5 5i t i t i t i t i t i ti i i i i i
f t e e e e e e
f(t)
cn or c(w)
Fourier Theory Summary
1( ) ( )
2i tf t f e d
1( ) ( )
2i tf f t e dt
0( ) im tm
m
f t c e
0
0
1( )
Tm ti
mc f t e dtT
Fourier Series: (f periodic or defined over (0,T)
Fourier Transforms: (f nonperiodic, all time)
( ) ( ) i t
m
f t c e
0
1( ) ( )
Ti tc f t e dt
T with cn c(w)
and mw0 w
Optic’s choice of sign for f(t): ( ), to matchi t i kx te e
http://phet.colorado.edu/en/simulation/fourier
Discrete vs continuous f(w)
Fourier Transform Example
f(t)
cn or c(w)
2( ) 1 cos 2
ic
FTs you should get to know!
FTs you should get to know!
Gaussian
Gaussian doesn’t have “ringing” in the FT!
Widths in t, w
“Uncertainty principle” in QM:related to time-frequency widths in
waves
101 waves
11 or approx: t O t
Uncertainty principle
E(t)
N functions added, equally spaced in frequency
11 waves 101 waves: Dw is much wider
Power spectrum
11 waves 101 waves
2f
Inverse FT: Does the same power spectrum give the same f(t)?
1( ) ( )
2i tf t f e d
Power spectrum of cos, sin
Uncertainty principle
101 waves
Why is it an inequality?
11 or approx: t O t
http://phet.colorado.edu/en/simulation/fourier
Importance of phase in f(w)
11 waves 101 waves
( ) a
1 Im[ ( )]( ) tan
Re[ ( )]
f
f
Try: Linear phase function:
( ) ( )iag e g Dt is the same.Pulse is shifted
( ) 0
( ) from adding closely spaced cos( ( ))i if t t
Importance of phase
11 waves 101 waves
2( ) a Try: Quadratic dependence:
2
( ) ( )iag e g
( ) 0
( ) from adding closely spaced cos( ( ))i if t t
Dt is much bigger for the same !Dw
Importance of phase
11 waves 101 waves
( ) rand
1 Im[ ( )]( ) tan
Re[ ( )]
f
f
Random dependence:
( ) 0
Dt is infinite (noise)
Dt is much bigger for the same !Dw
Summary: Importance of phase
11 waves 101 waves
Why is it an inequality?
11 or approx: t O t
f(t) changes greatly with phase ( ). f w The shortest is had only for ( ) = f w constant or linear.
All others will make
Dw comes entirely from |f(w)|, which has no phase information.
1 t
1 t
Carrier frequency-envelope principle
11 waves 101 waves
The FT f(w) is the FT of ___ centered at ____. The width Dw is the width of ____
Optical pulses are often a steady (“carrier”) wave at multiplied by an envelope function
( ) ( ) cos( )
( ) ( )sin( )
( ) ( ) i t
f t g t t
f t g t t
f t g t e
Which pulse f(t) will have f(w) centered around the highest frequencies?
a) b) c)
Which f(t) will have the greatest width Dw in f(w) around its central frequency?
a) b) c)
Compare the “ringing” in the FT of
rectangular pulse envelope
triangular pulse envelope
Gaussian pulse envelope
sinc pulse envelope
Fourier theory
1 1( ) [ ( )] ( )
2i tf t FT f f e d
1( ) [ ( )] ( )
2i tf FT f t f t e dt
Fourier theory and delta functions
( )ot t
( ) ( )of t t t du
( )ot t du
( )FT t
( )oFT t t
1( ) [ ( )] ( )
2i tf FT f t f t e dt
1 1( ) [ ( )] ( )
2i tf t FT f f e d
Fourier theory and delta functions
FT oi te
-1FT FTo ot t t t
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