Fourier relations in Optics
Near field
Far field
Frequency
Pulse duration
Frequency
Coherence length
Beam waist
Beam divergence
Spatial dimension
Angular dimension
Focal plane of lens
The other focal plane
Huygens’ Principle
E(R)
E(r)
  


E (r )e jk ( R r ) 
E ( R)  
  dr
Rr
Fourier theorem
A complex function f(t) may be decomposed as a
superposition integral of harmonic function of all
frequencies and complex amplitude

f (t )   F ( )e jt dv

(inverse Fourier transform)
The component with frequency  has a complex
amplitude F(), given by

F ( )   f (t )e jt dt

(Fourier transform)
Useful Fourier relations in optics
between t and , and between x and .
f (t )  1
F ( )   ( )
f (t )   (t )
F ( )  1
f (t )  1
0
 / 2  t   / 2
elsewhere
f (t )  cos( 0 t )
0


t 

2
2
elsewhere
F ( )  2
sin(  / 2)


sin[(    0 ) ]
2
F ( )  

[(   0 ) ]
2
Useful Fourier relations in optics
between t and , and between x and .
f (t )  e
f (t )  1
0

t2

F ()  e
2


 nT  t 

 nT
2
2
n  0,1, .....N  1
elsewhere
 2 2

NT
sin(  ) sin( 
)
2
2
F ( ) 

T

sin(  )
2
2
Position or time
Angle or frequency
2



2

2
Angle or frequency
Position or time

F ( )   f (t )e  jt dt


 E0  2 e

 E0
 j t
dt  Eo
2
sin

2

2
e
j

e
 j
2
j

2
Application of Fourier relation:
Single- slit diffraction
Single slit diffraction
X
X=a/2
a

X=0
= x sin

E ( )   e ikx sin dx  a

a
a
sin(  )
sin  )

2
a
a
a
k sin 

2

sin( k
 

a
The applications of the Fourier relation:
-Spatial harmonics and angles of propagation
f (t )  1
0
f ( x)  1
0
 / 2  t   / 2
elsewhere
F ( )  2
 a / 2  x  a / 2
elsewhere
sin( ka / 2)
F ( k )  2a
ka
sin(  / 2)

?
Frequency, time, or position
N
2
N
0
Frequency
Time

N
E  E0  e j (0 i ) t  ji (t )
i 0
 (t ) : time independent
jNt
1

e
E  E0 e j 0 t
1  e jt
N 

sin
t

2
2
E  E*  E 
 
 sin
t 
2 

2
2

N
2
N
0
Frequency
Time

N
E  E0  e j (0 i ) t  ji (t )
i 0
 (t ) : time independent
jNt
1

e
E  E0 e j 0 t
1  e jt
N 

sin
t

2
2
E  E*  E 
 
 sin
t 
2 

Mode-locking
2
2

N

NX
x0
Position
Angle
x
 NX
sin

2

E  E*  E 
X
 sin








X
2
Diffraction grating, radio antenna array
The applications of the Fourier relation:
f (t )  1
0


 nT  t 


 nT
2
2
n  0,1, .....N  1
elsewhere
1/x
NT
sin(  ) sin( 
)
2
2
F ( ) 

T

sin(  )
2
2
k
k=2 /
x
kz
(8)
Finite number of elements
2x
-Graded grating for focusing
-Fresnel lens
Fourier transform between two focal planes of a lens
g(x,y)
f(x,y)
d
Z=0
f
A
B
Focal
plane
The use of spatial harmonics
for analyses of arbitrary field pattern
Consider a two-dimensional complex electric field at z=0 given by
f ( x, y )  Ae
 j 2 x x  j 2 y y
where the ’s are the spatial frequencies in the x and y directions.
The spatial frequencies are the inverse of the periods.
1/x
k=2 /
k
x
2x
kz
f ( x, y )  Ae
 j 2 x x  j 2 y y
Thus by decomposing a spatial distribution of electric field into
spatial harmonics, each component can be treated separately.
U ( x, y, z )  Ae
k z  2 (
1
2
 j 2 x x  j 2 y y  jk z z
  x2   y2 )1 / 2
1/x
k=2 /
k
x
kz
2x
Define a transfer function (multiplication factor) in free space for
the spatial harmonics of spatial frequency x and y to travel from
z=0 to z=d as
U ( x, y,0)  Ae
U ( x, y, d )  Ae
 Ae
k z  2 (
1

2
 j 2 x x  j 2 y y
 j 2 x x  j 2 y y  jk z d
 j 2 x x  j 2 y y
H ( x , y )
  x2   y2 )1 / 2
H ( x , y )  e
 j 2 (
1/x
1
2
2 1/ 2




) d
x
y
2

k=2 /
k
x
kz
2x
Define a transfer function (multiplication factor) in free space for
the spatial harmonics of spatial frequency x and y to travel from
z=0 to z=d as
U ( x, y, d )  Ae
 Ae
k z  2 (
1

2
 j 2 x x  j 2 y y  jk z d
 j 2 x x  j 2 y y
H ( x , y )
  x2   y2 )1 / 2
H ( x , y )  e
 j 2 (
1/x
1
2
2 1/ 2




) d
x
y
2

k=2 /
k
x
kz
2x
Source
E
z=0
E
z=0
To generalize:
1/x
k2
k1
2x k1x
k2z
 
(k 2  k1 )  xˆ  2 x N
“Grating momentum”
Stationary gratings vs. Moving gratings
Deflection
Deflection + Frequency shift
The small angle approximation (1/ <<)
for the H function
2 (
1

2
  ) d  2
2
x
2 1/ 2
y
d

(1   )
2 1/ 2
 2
d

(1 
2
2
)
=
H ( x , y )  e
2
2
 jkd  jd ( x  y )
e
A correction factor for the
transfer function for the
plane waves
F(x)
H(x)F(x)
D
z=0
z 
H(x)F(x)
F(x)
D
z 
z=0
F ( x,0)   f ( x )e 2 x x d x
F ( x)   H ( x ) f ( x )e 2 x x d x
H ( x , y )  e
2
2
 jkx  jd ( x  y )
e
Express F(x,z) in =x/z
H(x)F(x)
F(x)
D
z 
z=0
F ( x,0)   f ( x )e 2 x x d x
F ( x)   H ( x ) f ( x )e 2 x x d x
H ( x , y )  e
2
2
 jkx  jd ( x  y )
e
Express F(x,z) in =x/z
The effect of lenses
Converging lens
f
A lens is to introduce a quadratic phase shift to the
wavefront given by
.
e
x2  y2
j
f
Fourier transform using a lens
g(x,y)
f(x,y)
d
Z=0
f
A
B
f ( x, y )   F ( x , y )e
Focal
plane
 j 2 ( x x  y y )
d x d y
g(x,y)
f(x,y)
d
f
Z=0
A
B
f ( x, y )   F ( x , y )e
U B ( x, y , z B )  e
 jkd
e
x2  y2
j
f
 A( x , y )e
A( x , y )  e
 jkd
e
 j 2 ( x x  y y )
 jd ( x2  y2 )
Focal
plane
d x d y
F ( x , y )e
 j 2 ( x x  y y )
( x  x0 ) 2  ( y  y 0 ) 2
j
f
e
j ( d  f )( x2  y2 )
F ( x , y )
g(x,y)
f(x,y)
f
Z=0
f
Focal
plane
x y
g ( x, y )  h f F ( , )
f f
x y
hl  ( j / f ) F ( , )
f f
Huygens’ Principle
E(R)
E(r)
  


E (r )e jk ( R r ) 
E ( R)  
  dr
Rr
Holography
:
Recording of full information of an optical image,
including the amplitude and phase.
Amplitude only:
Amplitude and phase
E  E*  E
2
( E ( x ) e i ( x )  E R )  ( E ( x ) e  j ( x )  E R )
 E  ER  2 EER cos  ( x)
2
2
A simple example of
recording and reconstruction:

k2
k1
d


2 sin( )
2
A simple example of
recording and reconstruction:

k2
k1
d


2 sin( )
2
d


2 sin( )
2
k1
k2
/2
?
d


2 sin( )
2
Another example:
Volume hologram

k2
k1
d


2 sin( )
2
Volume grating
d


2 sin( )
2
k1
d


2 sin( )
2
k1
k2
k1
d


2 sin( )
2
D
A
d
C

B
Bragg condition
  AB  BC  CD  2d cos
D
A
d
C

B
Another example:
Image reconstruction of a point
illuminated by a plane wave.
Writing
Reading
E(x,y)
Er
Recorded pattern
E r  E ( x, y ) e
 jk x x 2
 Er2  Er E ( x, y )e  jkx x  Er E ( x, y )e jkx x  E 2 ( x, y )
Recorded pattern
E r  E ( x, y ) e
 jk x x 2
 Er2  Er E ( x, y )e  jkx x  Er E ( x, y )e jkx x  E 2 ( x, y )
Diffracted beam when illuminated by ER
Ediffracted 
Er2e  jkz  Er E ( x, y)e  jkx x  jkz z  Er E ( x, y)e jkx x  jkz z  E 2 ( x, y)e  jkz