### Chapter 5

```V. Fourier transform
5-1. Definition of Fourier Transform
* The Fourier transform of a function f(x) is
defined as
F  f ( x )  F ( u ) 



f ( x )e
 2  iux
The inverse Fourier transform, F
f (x)  F
f (x) 

1
F  f ( x )


F (u )e
2  iux
du
1
dx
3-D: the Fourier transform of a function f(x,y,z)

F (u, v, w ) 
 
f ( x, y , z )e
 2  i ( ux  vy  wz )

Note that

r  x xˆ  y yˆ  z zˆ
dxdydz

u  u uˆ  v vˆ  w wˆ
ux+vy+wz: can be considered as a scalar
 
product of r  u
if the following conditions are met!
xˆ  uˆ  1; xˆ  vˆ  0 ; xˆ  wˆ  0
yˆ  uˆ  0 ; yˆ  vˆ  1; yˆ  wˆ  0
zˆ  uˆ  0 ; zˆ  vˆ  0 ; zˆ  wˆ  1
 
 r  u  ux  vy  wz
Therefore,


F  f ( r )  F ( u ) 

u



  2  ir u 
f ( r )e
dr
the vector may be considered as a
vector in “Fourier transform space”
The inverse Fourier transform in 3-D space:
F
1


F ( u )  f ( r ) 



 2  ir u 
F (u )e
du
5-2. Dirac delta function
  for x  a
 (x  a)  
 0 for x  a



 ( x  a ) dx  1
Generalized function: the limit of a sequence
of functions
gn (x) 
n

e
 nx
2
n 
1

2



g n ( x ) dx  1
 : standard Gaussian width parameter
Gaussian Integration:
G 
Star from
G 
2



e
x
2
dx 





e
e
y
2
x
2
dx
dy 



2
e
2
( x  y )
dxdy
dxdy: integration over a surface 
change to polar coordinate (r, )
x  r cos  ; y  r sin 
x  y  r
2
G 
2
2
2
 
0
2

e
r
rdrd 
0
G ; G 
2
2
1/2

rd
r
dr




e


e
 nx
2
 nx
2
dx  ?
dx 

Let


n
 gn (x) 

e
y
2
dy
y 

n
e
 nx
2
dy 
nx
1


n




e
y
2
n dx
dy 
n
g n ( x ) dx  1
Consider sequence of function
g1 ( x ) 
1

e
x
2
g 3 ( x ); g 4 ( x );......
g2(x) 
2

g 256 ( x ) 
e

2 x
2
256

e
 256 x
2
a = 1/n
http://en.wikipedia.org/
wiki/Dirac_delta_functi
on
What happen when n = 
(a) g  ( 0 )  
(b) g  ( x  0 )  0
(c) the
width
of
the
center
peak
=
0

(d)  g  ( x ) dx  1

The sequence only useful if it appears as part
of an integral, e.g.
lim
n 



g n ( x ) f ( x ) dx  lim
n 



n

e
 nx
2
f ( x ) dx
Only f(0) is important
 f ( 0 ) lim
n 



n

e
 nx
2
dx  f ( 0 )
Dirac Delta Function: limit of Gaussian
distribution function
n
 ( x )  lim

n 
e
 nx
2
There are infinitely many sequences that can
be used to define the delta function
lim
n 



n

e
 nx
2
f ( x ) dx  f ( 0 ) 

lim
n 



n

e
 nx
2



dx  1 



 ( x ) f ( x ) dx
 ( x  x ' ) f ( x ) dx  f ( x ' )



 ( x ) dx
Dirac delta function is an even function

f (x) 


F (u )e
2  iux
du
F (u ) 



f ( x ' )e
 2  iux '
dx '


 2  iux '
2  iux


f (x)  
f ( x ' )e
dx ' e
du


 
 
f (x) 




f ( x' ) e
  
Note that
f (x) 



 ( x ' x ) 
y
 2  iux '
e
2  iux
du  dx '

e
 2  iu ( x '  x )
=
f ( x ' ) ( x '  x ) dx '



e
 2  iu ( x '  x )
du
 ( y) 



e
 2  iuy
du
Similar
F (u ) 



f ( x )e
 2  iux
dx


2  iu ' x
 2  iux


F (u )  
F (u ' )e
du ' e
dx


 
 


2i ( u ' u ) x

F (u )   F (u ' )  e
dx  du '
  


 F (u ) 



f ( u ' ) ( u '  u ) du '
 ( u ' u ) 
Let y  u '  u



e
2i ( u ' u ) x
 ( y) 




dx
e
2  iyx
dx
Compare to  ( y )   e  2  iuy du

 ( y )   ( y )
5-3. A number of general relationships may
be written for any function f(x)
real or complex.
Real Space
f(x)
f(-x)
f(ax)
f(x)+g(x)
f(x-a)
df(x)/dx
dnf(x)/dxn
Fourier Transform Space
F(u)
-F(-u)
F(u/a)/a
F(u)+G(u)
e-2iauF(u)
2iuF(u)
(2iu)nF(u)
Example
(1)
u
F { f ( ax )}  F  
a a
1
F { f ( ax )} 


f ( ax ) e

 2  iux
dx
Set X = ax
F { f ( au )} 
F { f ( au )} 

 2  iu

f ( X )e

1
a 
a
dX
a
 2i


f ( X )e
u
 F 
a a
1
X
u
a
X
dX
(2)
F { f ( x  a )}  e
F { f ( x  a )} 

 2  iau


F u 
f ( x  a )e
 2  iux
dx
Set X = x - a
F { f ( x  a )} 
F { f ( x  a )} 




e
f ( X )e


f ( X )e
 2  iu a

 2  iu ( X  a)
 2  iu ( X  a)


f ( X )e
 F { f ( x  a )}  e
F u 
dX
 2  iuX
F u 
 2  iau
d (X  a)
dX
(3)
F{
F{
df ( x )
dx
df ( x )
}  2  iuF u 

}


dx
df ( x )
F{
df ( x )
}
dx



 2  iux
dx
dx
f (x)  F
f (x) 
e

1
F  f ( x )


F (u )e
2  iux
du
d  
2  iu ' x
 e  2  iux dx
F
(
u
'
)
e
du
'

dx   
2  iu ' x
 
  2  iux
de
F{
}     F (u ' )
du '  e
dx


dx
dx




df ( x )
2  iu ' x
 2  iux


F{
} 
F ( u ' ) 2  iu ' e
du ' e
dx


 
 
dx
df ( x )

F{
df ( x )
dx


2i ( u ' u ) x

}   2  iu ' F ( u ' )  e
dx  du '
  


 ( u ' u )
F{
df ( x )
}
dx
F{
df ( x )
dx



2  iu ' F ( u ' ) ( u '  u ) du '
}  2  iuF ( u )
5-4. Fourier transform and diffraction
(i) point source or point aperture
A small aperture in 1-D:  (x) or  (x-a).
Fourier transform the function 
Fraunhofer diffraction pattern
For  (x):
F { ( x )} 



 ( x )e
 2  iux
 2  iu 0
dx  e



=1
=1
The intensity is proportional
 ( x ) dx
=1
| F (u ) |  1
2
For  (x-a):
F { ( x  a )} 



 ( x  a )e
 2  iux
dx
Set X = x-a
F { ( X )} 



e
 ( X )e
 2  iua
F { ( x  a )}  e



 2  iu ( X  a )
 ( X )e
dX
 2  iuX
dX
 2  iua
The intensity is proportional
| F (u ) |  1
2
The difference between the point source at
x = 0 and x = a is the phase difference.
(ii) a slit function
| x | b 2
 0 when
f (x)  
 1 when
| x | b 2
F { f ( x )}  F ( u ) 
F (u ) 
F (u ) 

b/2
e
 2  iux
b / 2
e
 2  iux
 2  iu



dx 
b/2

b / 2
e
f ( x )e
1
 2  iu
  iub
 2  iux

e
dx
b/2
b / 2
e
 iub
 2  iu

 2  iux
d (  2  iux )
sin(  ub )
u
c.f. the kinematic diffraction from a slit
c.f. the kinematic diffraction from a slit
Chapter 4 ppt p.28
~'
~  L b i ( kR   t )
E 
e
R
 kb sin  
sin 

2


kb sin 
2
F (u )  b
  ub 
 u 
sin(  ub )
 ub
kb sin 
sin 

2

2  b sin 

2

 b sin 

(iii) a periodic array of narrow slits

f (x) 
  ( x  na )
n  
F { f ( x )}  F ( u ) 



f ( x )e
 2  iux
dx
 
  2  iux
F ( u )      ( x  na )  e
dx

 n  



F (u ) 
 
n  

F (u ) 

n  


e
 ( x  na ) e
 2  iuna



 2  iux
dx

 ( x  na ) dx 

n  
e
 2  iuna


x
n

n  
1
1 x

F (u ) 

e
0
 2  iuna

n  

e
 2  iuna


n  
n0




e
2  iuna
n0

F (u ) 
 e
2  iua
n0
F (u ) 
 2  iua

n
n0
1
1 e

   e
n
2  iua

1
1 e
 2  iua
1
1
e
 2  iuna
1
Discussion
For e  2  iua
F (u ) 
1
1
1 e
F (u ) 
2  iua

1 e
 2  iua
(1  e
2  iua
1
1 e
 2  iua
1 e
)( 1  e
1
2  iua
 2  iua
1
 F (u )  0
)
=1
For
e
 2  iua
1
 F (u )  
It occurs at the condition
e
 2  iua
 1  cos( 2  ua )  i sin( 2  ua )
 2  ua  2  h
 ua  h
h: integer
In other words,

F (u ) 
  ( ua
 h)
h  
Note that  ( ax ) 
 (x)
|a |
  for x  0
 (x)  
 0 for x  0




 ( ax ) dx 





 (| a | x ) dx 


 (| a | x ) dx



 ( x )

dx
|a |
 ( x ) dx  1
Set

1
|a |
x | a | x
The Fourier transform of f(x) 


h 

F ( u )    ( ua  h )    a ( u  )


a


h  
h  
F (u ) 
1
a


h  
 (u 
h
a
)
where a > 0
Hence, the Fourier transform of a set of
equally spaced delta functions with a period
a in x space
 a set of equally spaced delta functions
with a period 1/a in u space
Similarly，a periodic 3-D lattice in real space;
(a, b, c)


 (r ) 


    ( x  ma , y  nb , z 
pc )
m   n   p  


F {  ( r )}  F ( u )
 
   

 m  


1
abc
n  
  2  iu  r 
dr
  ( x  ma , y  nb , z  pc ) e
p  







h
k
l
    ( u  a ) ( v  b ) ( w  c )
h   k   l  
This is equivalent to a periodic lattice in
reciprocal lattice (1/a 1/b 1/c).
(iv) Arbitrary periodic function

Fe
f (x) 
2  ihx / a
h
h  
F { f ( x )}  F ( u )
 
2  ihx / a   2  iux
    Fh e
e
dx


 h  




F 
h
h  



F 
h
h  



e
2  ihx / a


e
 2i ( u  h
h
 F  (u  a )
h
h  
e
 2  iux
a
)x
dx
dx
http://en.wikipedia.o
rg/wiki/Fourier_seri
es
Hence，the F(u) ；i.e. diffracted amplitude,
is represented by a set of delta functions
equally spaced with separation 1/a and each
delta function has “weight”, Fh, that is equal
to the Fourier coefficient.
5-5. Convolution
The convolution integral of f(x) and g(x) is
defined as
c( x)  f ( x)  g ( x) 



f ( X ) g ( x  X ) dX
examples：
(1) prove that f(x)  g(x) = g(x)  f(x)
c( x)  f ( x)  g ( x) 



f ( X ) g ( x  X ) dX
Set Y = x - X
f (x)  g(x) 
f (x)  g(x) 






f ( x  Y ) g (Y ) d ( x  Y )
f ( x  Y ) g (Y ) dY
 g(x)  f (x)
(2) Prove that
f (x)  (x) 
f (x)  (x)  f (x)



f ( X ) ( x  X ) dX

f ( x )   ( x )  f ( x )   ( x  X ) dX  f ( x )

(3) Multiplication theorem
If F { f ( x )}  F ( u ); F { g ( x )}  G ( u )
then
F { f ( x )  g ( x )}  F ( u )  G ( u )
(4) Convolution theorem (proof next page)
If F { f ( x )}  F ( u ); F { g ( x )}  G ( u )
then
F { f ( x )  g ( x )}  F ( u )  G ( u )
Proof: Convolution theorem



F { f ( x )  g ( x )}  
f ( X ) g ( x  X ) dX

 
 




 




 






f ( X )e
f ( X )e
f ( X )e
 2  iuX
g ( x  X ) dX e
 2  iu ( x  X )
 2  iuX
g ( x  X ) dX e
 2  iu ( x  X )
d (x  X )
 2  iu ( x  X )
d (x  X )
 2  iuX
= F(u)
 F (u )  G (u )
 e  2  iux dx

dX



g ( x  X )e
= G(u)
dx
Example：diffraction grating
n  ( N 1 ) / 2
f (x) 
a single slit or
ruling function
  ( x  na )  g ( x )
n   ( N 1 ) / 2
a set of N delta function
 n  ( N 1 ) / 2

F { f ( x )}  F    ( x  na )  g ( x ) 
 n   ( N 1 ) / 2

 n  ( N 1 ) / 2

F    ( x  na )  
 n   ( N 1 ) / 2


n  ( N 1 ) / 2
 2  iuna
e
n   ( N 1 ) / 2



 n  ( N 1 ) / 2
  2  iuna
 ( x  na )  e
dx
   n   (
N 1 ) / 2



 ( x  na ) dx

n  ( N 1 ) / 2
 2  iuna
e
n   ( N 1 ) / 2
n  ( N 1 ) / 2
 2  iuna

e
e
 iu ( N  1 ) a
n   ( N 1 ) / 2
e
e

 iu ( N  1 ) a

e
 2  iuna
n0
1 e
 2  iuNa
1 e
 iu ( N  1 ) a
n  N 1
e
 2  iua
  iuNa
e
sin(  uNa )
  iua
(e
(e
 iuNa
e
  iuNa
  iua
e
  iua
sin(  ua )
F g ( x )   G ( u )
F { f ( x )} 
sin(  uNa )
sin(  ua )
 G (u )
)
)
Supplement # 1
Fourier transform of a Gaussian function is
also a Gaussian function.
Suppose that f(x) is a Gaussian function
f (x)  e
F { f ( x )}  F ( u ) 




e
 iu 

  ax 

a 

Define 
2
e
  iu 


 a 
 ax 



e
2
2
2
2
a x
a x
e
 2  iux
dx
( ax  y ) 
2
2
dx
 iu
a
( ax )  2  iux  y
2
2
2 axy  2  iux
 y 
 iu
a
F (u ) 
F (u ) 



1
e
e

2
e
  iu 


a


d
a
 u 


 a 
2



a
e


F (u ) 

a
i  1
2
2
e
 u 


 a 
2
d
Chapter 5 ppt
page 5

2
Gaussian Function
in u space
Standard deviation is defined as the range of the
variable (x or u) over which the function drops
by a factor of e  1 / 2 of its maximum value.
f (x)  e
 x 
2
a x
1
2a

F (u ) 
 u 
2
2
e
2
e
1 / 2
 ax 
1
2
x
e
a
a
  x u 
a x
2
1
 u 


 a 
2
2
1
 u 

 
2
 a 
 u
2a
a
2

1
2

u
a

1
2
c.f
x  p ~ h
x  k ~ h
x  
h
2
k ~ h
 x   k ~ 2
f (a, x )  e
f (2, x )
f (8, x )
f (a, u ) 

 u
2

e
a
f (2, u )
f (8, u )
a
2
2
2
a x
2
Supplement #2
Consider the diffraction from a single slit
The result from a single slit
~'
~  L i ( kR   t )
E 
e
e

b / 2
R
b/2
 2  iz
sin 

dz
The expression is the same as Fourier transform.
F (u ) 



f ( x )e
 2  iux
dx
Supplement #3
Definitions in diffraction
Fourier transform and inverse Fourier transform
System 1
System
System
 F ( u )   f ( x ) e  2  iux dx



:

2  iux
f
(
x
)

F
(
u
)
e
du




 F ( u )   f ( x ) e  iux dx
 

2:
1 
iux
F ( u ) e du
 f (x) 

2  


1

 iux
F (u ) 
f
(
x
)
e
dx


2  
3: 

1
iux
 f (x) 
F ( u ) e du


2  
System 4
System 5
System 6
relationship among Fourier transform, reciprocal
lattice, and diffraction condition
System 1, 4
Reciprocal lattice
 





*
*
bc
ca
ab
*
a     ;b     ;c    
a  (b  c )
b  (c  a )
c  (a  b )

*
*
*
*
G hkl  h a  k b  l c
Diffraction condition


*
S   S  G hkl


*
k   k  2  G hkl
System 2, 3, 5, 6
Reciprocal lattice
 




*
2 b  c  *
2 c  a  *
2 a  b
a     ;b     ;c    
a  (b  c )
b  (c  a )
c  (a  b )

*
*
*
*
G hkl  h a  k b  l c
Diffraction condition


*
S   S  2  G hkl


*
k   k  G hkl
```