Fourier theory
How does a complicated optical pulse E  r , t )
reflect off a
surface?
We know a lot about waves at a single w:
n(w), vp(w), R(w), absorption(w)…
Analyze arbitrary E  r , t ) in terms of these, because
Fourier told us that….
Fourier Series Example
cn or c(w)
f(t)
4
 4
f (t )    sin  2 t )  
 
 3

 4
sin
6

t


)

 5



 4
sin
10

t


)

 7



 sin 14 t )  ...

 2i 
 2i 
 2i  i 6 t  2i  i 6 t  2i  i10 t  2i  i10 t
f (t )     ei 2 t    e i 2 t   
e

e

e

e
...




  
 
 3 
 3 
 5 
 5 
Fourier Theory Summary
Fourier Series: (f periodic or defined over (0,T)


f (t ) 
cm e
1
cm 
T
 imw0t
m 

f (t ) 

c(w )e
 iwt
m 
T

f (t )e  i mw0t dt
0
with cn  c(w) c (w )  1
and mw0  w
T
T

f (t )e  iwt dt
0
Fourier Transforms: (f nonperiodic, all time)
f (t ) 
1
2


f (w )e iwt d w
f (w ) 

Optic’s choice of sign for f(t):
e
 iwt
1
2


f (t )e  iwt dt

, to match e
i ( kx wt )
Discrete vs continuous f(w)
http://phet.colorado.edu/en/simulation/fourier
Fourier Transform Example
cn or c(w)
f(t)
c(w ) 
1  cos w 2 ) )

w
2 i

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
t w  O 1)
1
or approx: t 
w
101 waves
Uncertainty principle
N functions added,
equally spaced in
frequency
11 waves
E(t)
101 waves:
w is much wider
Power spectrum
f w )
2
Inverse FT:
Does the same power spectrum give the same f(t)?
11 waves
f (t ) 
1
2

 f (w )e

Power spectrum of cos, sin
 iwt 101 waves
dw
Uncertainty principle
t w  O 1)
1
or approx: t 
w
Why is it an inequality?
101 waves
http://phet.colorado.edu/en/simulation/fourier
Importance of phase in f(w)
f (t ) from adding closely spaced cos(wi t   (wi ))
Im[ f (w )]
 (w )  tan
Re[ f (w )]
1
 (w )  aw
 (w )  0
11 waves
101 waves
Try: Linear phase function:
g (w )  eiaw g (w )
t is the same.
Pulse is shifted
Importance of phase
f (t ) from adding closely spaced cos(wi t   (wi ))
 (w )  0
11 waves
Try: Quadratic dependence:
2

(
w
)

a
w
iaw 2
g (w )  e g (w )
101 waves
t is much bigger for the same w!
Importance of phase
Im[ f (w )]
 (w )  tan
Re[ f (w )]
1
11 waves
 (w )  0
Random dependence:
 (w )  rand
101 waves
t is infinite (noise)
t is much bigger for the same w!
Summary: Importance of phase
t w  O 1)
1
or approx: t 
w
Why is it an inequality?
w comes entirely from |f(w)|, which has no phase
information.
11 waves
101 waves 1
f(t) changes greatly with phase w). The shortest t 
w
is had only for w)  constant or linear.
1
All others will make t 
w
Carrier frequency-envelope principle
Optical pulses are often a steady (“carrier”) wave at
multiplied by an envelope function
f (t )  g (t ) cos(w t )
f (t )  g (t ) sin(w t )
w
11 waves
f (t )
 g (t )e
 iw t
101 waves
The FT f(w) is the FT of ___ centered at ____.
The width w is the width of ____
w
Which pulse f(t) will
have f(w) centered
around the highest
frequencies?
a) b) c)
Which f(t) will have the
greatest width w 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
f (w )  FT [ f (t )] 
1
f (t )  FT [ f (w )] 
1
2
1
2


f (t )e  iwt dt




f (w )e iwt d w
Fourier theory and delta functions
 (t  to )



f (w )  FT [ f (t )] 


2
 (t  to )du 
f (t )  FT 1[ f (w )] 

1
f (t ) (t  to )du 
FT  (t )  
FT  (t  to )  
1
2


f (t )e  iwt dt




f (w )e iwt d w
Fourier theory and delta functions
  t  to )  FT
FT e
 iwo t
-1
FT  t  t ) 
 
o