The Dirac delta function
The Fourier
Transform
Some FT examples
exp(iw0 t)
cos(w0 t)
Some Fourier Transform theorems
t
w
t
w
Complex conjugate: f*(t)
Shift: f(t–a )
Scale: f(at )
Sum: f(t) + g(t)
Derivative: f'(t)
Modulation: f(t)cos(w0 t)
The intensity and phase and the spectral
intensity and phase
The 2D Fourier Transform
t
w
The relative importance of intensity and
phase
The Dirac delta function
Unlike the Kronecker delta-function, which is a function of two
integers, the Dirac delta function is a function of a real variable, t.
 if t  0
 (t )  
 0 if t  0
(t)
t
The Dirac delta function
 if t  0
 (t )  
 0 if t  0
It’s best to think of the delta function as the limit of a series of
peaked continuous functions.
fm(t) = m exp[-(mt)2]/√p
(t)
f3(t)
f2(t)
f1(t)
t
Dirac -function Properties

t
  (t ) dt  1
-


-
-
  (t - a) f (t ) dt    (t - a) f (a) dt  f (a)

 exp(iwt ) dt  2p  (w 
-

 exp[i(w - w ')t ]
-
(t)
dt  2p  (w - w '
The Fourier Transform
and its Inverse

F (w ) 

f (t ) exp( -iw t ) dt
-

1
f (t ) 
2p

-
F (w ) exp(iw t ) dw
The Fourier Transform of (t) is 1.

  (t ) exp(-iwt ) dt  exp(-iw [0])  1
-
(t)
1
w
t

And the Fourier Transform of 1 is 2p(w):
 1exp(-iwt ) dt  2p  (w 
-
2p(w)
1
t
w
The Fourier transform of exp(iw0 t)

F
exp(iw0 t )





exp(iw0 t ) exp( -i w t ) dt
-
exp(-i [w - w0 ] t ) dt  2p  (w - w0 )
-
exp(iw0t)
Im
0
Re
0
F {exp(iw0t)}
t
t
0
w0
w
The function exp(iw0t) is the essential component of Fourier analysis.
It is a pure frequency.
The Fourier transform of cos(w0 t)

F
cos(w0t )
1

2
1

2




 cos(w t ) exp(-i w t ) dt
0
-
exp(i w0 t )  exp(-i w0 t ) exp(-i w t ) dt
-
 exp(-i [w - w ]t) dt
0
-

1
2

 exp(-i [w  w ]t) dt
0
-
 p  (w - w0 )  p  (w  w0 )
F {cos(w0t )}
cos(w0t)
0
t
-w0
0 w0
w
Scale Theorem
F { f (at )}  F (w /a) / a
The Fourier transform
of a scaled function, f(at):

Proof:

F { f (at )} 
f (at ) exp(-iw t ) dt
-
Assuming a > 0, change variables: u = at

F { f (at )} 

f (u) exp(-iw [ u /a]) du / a
-


 f (u) exp(-i [w /a] u) du / a
-
 F (w /a) / a
If a < 0, the limits flip when we change variables, introducing a
minus sign, hence the absolute value.
F(w)
f(t)
The Scale
Theorem
in action
The shorter
the pulse,
the broader
the spectrum!
This is the essence
of the Uncertainty
Principle!
Short
pulse
t
w
t
w
t
w
Mediumlength
pulse
Long
pulse
The Fourier
Transform of a
sum of two
functions
F(w)
f(t)
G(w)
g(t)
F {af (t )  bg (t )} 
aF { f (t )}  bF {g (t )}
t
f(t)+g(t)
Also, constants factor out.
w
t
t
w
F(w) +
G(w)
w
Shift Theorem
The Fourier transform of a shifted function, f (t - a ) :
F
 f (t - a)  exp(-iwa)F (w)
Proof :

F
 f  t - a   
f (t - a) exp( -iwt ) dt
-
Change variables : u  t - a


f (u ) exp( -iw[u  a]) du
-

 exp(-iw a)  f (u ) exp( -iwu )du
-
 exp(-iw a) F (w )
Application of the Shift Theorem
Suppose we’re measuring the spectrum of E(t), but a small fraction
of its irradiance, say e, takes a longer path to the spectrometer.
The extra light has the field, √e E(t–a), where a is the extra path.
The measured spectrum is:
S (w )  F {E (t )  e E (t - a)}
2
e E(t-a)
Spectrometer
E(t)
Performing the Fourier Transform and
using the Shift Theorem:
E(t)
 E (w )   e exp(-iw a) E (w )
2
2
2
 E (w ) 1   e exp(-iw a)
2
 E (w ) 1   e cos(w a) - i  e sin(w a)
 E (w )
2
2
1   e cos(w a)   e sin(w a) 
2
2
Application of the Shift Theorem (cont’d)
 E (w )
2


1  2  e cos(w a)  e cos2 (w a)   e sin 2 (w a) 


 E (w )
2
1  2  e cos(w a)  e
Neglecting e compared to √e and 1:
 E (w )
2
1  2  e cos(w a) 
The contaminated spectrum
will have ripples with a period
of 2p/a.
And these ripples will have a
surprisingly large amplitude.
If e = 1% (a seemingly small amount), these ripples will have an
amazingly large amplitude of 2√e = 20%!
The Fourier Transform of the complex
conjugate of a function
F
*
*
f
(
t
)

F
  (-w )
Proof:
F
*
f
 (t ) 


f * (t ) exp( -iwt )dt
-


   f (t ) exp( iwt )dt 
 -


*


   f (t ) exp( -i[ -w ]t ) dt 
 -

 F * ( -w )

*
Derivative Theorem
The Fourier transform of a derivative of a function, f’(t):
F { f '(t )}  iw F (w )
Proof:

F { f '(t )} 

f '(t ) exp(-iw t ) dt
-
Integrate by parts:

 -  f (t ) [( -iw ) exp( -iw t )] dt
-
 iw


f (t ) exp( -iw t ) dt
-
 iw F (w 
The Modulation Theorem: The Fourier
Transform of E(t) cos(w0 t)
F
E (t ) cos(w0t )


F
1
2


1
2





E (t ) cos(w0t ) exp( -i w t ) dt
-
E (t ) exp(i w0t )  exp(-i w0t )  exp(-i w t ) dt
-
E (t ) exp(-i [w - w0 ] t ) dt

-
E (t ) cos(w0t ) 
1
2

 E(t) exp(-i [w  w ]t ) dt
0
-
1
E (w - w0 ) 
2
F
1
E (w  w0 )
2
E(t)cos(w0t )
If E(t) = D(t), then:
-w0
0
w0
w
The Fourier transform and its inverse are
symmetrical: f(t) F(w) and F(t) f(w) (almost).
If f(t) Fourier transforms to F(w), then F(t) Fourier transforms to:


F (t ) exp( -i w t ) dt
-
Rearranging:
1
 2p
2p


F (t ) exp(i[ - w ] t ) dt
-
Relabeling the integration variable from t to w’, we can see that we
have an inverse Fourier transform:
 1
 2p 
 2p


-

F (w ) exp(i[ - w ]w ) dw 

 2p f (-w )
This is why it is often said that f and F are a “Fourier Transform Pair.”
Fourier Transform Magnitude and Phase
As with any complex quantity, we can decompose f(t) and F(w) into
their magnitude and phase.
f(t) can be written:
f(t) = Mag{f(t)} exp[ -i Phase{f(t)}]
where Mag{f(t)}2 is often called the intensity, I(t), and Phase{f(t)}
is called the phase, f(t). They’re the same quantities we’re used to
for light waves.
Analogously,
F(w) = Mag{F(w)} exp[ -i Phase{F(w)}]
The Mag{F(w)}2 is called the spectrum, S(w), and the Phase{F(w)}
is called the spectral phase, j(w).
Just as both the intensity and phase are required to specify f(t), both
the spectrum and spectral phase are required to specify F(w), and
hence f(t).
Calculating the Intensity and the Phase
It’s easy to go back and forth between the electric field and the
intensity and phase.
The intensity:
I(t) = |E(t)|2
The phase:
 Im[ E (t )] 
f (t )  - arctan 

 Re[ E (t )] 
Equivalently,
f(t) = -Im{ ln[E(t) ]
}
Im
E(ti)
-f(ti)
Re
Intensity and Phase of a Gaussian
The Gaussian is real, so its phase is zero.
Time domain:
A Gaussian
transforms
to a Gaussian
Frequency domain:
So the spectral phase
is zero, too.
The spectral phase of a time-shifted pulse
Recall the Shift Theorem: F
Time-shifted
Gaussian pulse
(with a flat phase):
So a time-shift
simply adds some
linear spectral
phase to the
pulse!
 f (t - a)  exp(-iw a)F (w)
What is the spectral phase anyway?
The spectral phase is the abs phase of each frequency in the wave-form.
All of these
frequencies have
zero phase. So
this pulse has:
w1
w2
j(w) = 0
w3
Note that this
wave-form sees
constructive
interference, and
hence peaks, at
t = 0.
w4
w5
w6
0
t
And it has
cancellation
everywhere else.
Now try a linear spectral phase: j(w) = aw.
By the Shift Theorem, a linear spectral phase is just a delay in time.
And this is what occurs!
w1
j(w1) = 0
w2
j(w2) = 0.2 p
w3
j(w3) = 0.4 p
w4
j(w4) = 0.6 p
w5
j(w5) = 0.8 p
w6
j(w6) = p
t
The spectral phase distinguishes a light
bulb from an ultrashort pulse.
Complex Lorentzian
and its Intensity
and Phase
a
Real
component
0
0
Imaginary
component
w
Real part
Imag part
1
1 a - iw
a
-w
L(w ) 

 2
2 i 2
a  iw
a  iw a - iw
a w
a w2
1
Intensity(w )  2
a  w2
 Im[ L(w )] 
Phase(w )  - arctan 
  arctan(w / a)
 Re[ L(w )] 
Intensity and Phase of a decaying
exponential and its Fourier transform
Time domain:
(solid)
Frequency domain:
Light has intensity and phase also.
We usually extract out
the carrier frequency.
A light wave has the time-domain electric field:
E (t )  Re


I (t ) exp i(w0t - f (t )) 
Intensity
Equivalently, vs. frequency:
The minus
signs are just
conventions.
Phase
(neglecting the
negative-frequency
component)
E (w )  S (w ) exp  -ij (w )
Spectrum
Spectral
Phase
Knowledge of the intensity and phase or the spectrum and spectral
phase is sufficient to determine the light wave.
Fourier Transform with respect to space
If f(x) is a function of position,

F (k )   f ( x) exp(-ikx) dx
-
x
F {f(x)} = F(k)
We refer to k as the “spatial frequency.”
k
Everything we’ve said about Fourier transforms between the t and w
domains also applies to the x and k domains.
The 2D Fourier Transform
F
=
(2){f(x,y)}

f(x,y)
= F(kx,ky)
f(x,y) exp[-i(kxx+kyy)] dx dy
If f(x,y) = fx(x) fy(y),
then the 2D FT splits into two 1D FT's.
But this doesn’t always happen.
x
F
y
(2){f(x,y)}
A 2D Fourier Transform: a square function
f(x,y)
Consider a square function in the xy
plane:
f(x,y) = rect(x) rect(y)
The 2D Fourier Transform splits into
the product of two 1D Fourier
Transforms:
F {f(x,y)} = sinc(kx/2) sinc(ky/2)
This picture is an optical determination
of the Fourier Transform of the
square function!
x
F
y
(2){f(x,y)}
Fourier Transform Magnitude and Phase
Rick
Linda
Pictures
reconstructed
using the
spectral phase
of the other
picture
Mag{F [Linda]}
Phase{F [Rick]}
Mag{F [Rick]}
Phase{F {Linda]}
The phase of the Fourier transform (spectral phase) is much
more important than the magnitude in reconstructing an image.