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
Rr
 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 jt dv

(inverse Fourier transform)

The component with frequency has a complex

amplitude F(), given by

F ( ) f (t )e jt dt

(Fourier transform)
 Useful Fourier relations in optics
between t and , and between x and .

f (t ) 1 F ( ) ( )

f (t ) (t ) F ( ) 1
sin( / 2)
f (t ) 1 / 2 t / 2 F ( ) 2

0 elsewhere

sin[( 0 ) ]
f (t ) cos(0 t ) t F ( ) 2
2 2
[( 0 ) ]
0 elsewhere 2
 Useful Fourier relations in optics
between t and , and between x and .

t2

2 2
f (t ) e 2
F () e

f (t ) 1 nT t nT sin( ) sin(
NT
)
2 2 2 2
n 0,1, .....N 1 F ( )
T
sin( )
0 elsewhere 2 2
Position or time Angle or frequency
 2

2 2 Angle or frequency
Position or time

F ( ) f (t )e jt dt


j j
e 2
e 2
E0 2 e jt
dt Eo

2 j

sin
E0 2

2
Application of Fourier relation: Single- slit diffraction
Single slit diffraction

X=a/2 a

X=0

= x sin

a a
sin( k sin ) sin( )

E ( ) eikx sin dx a 2 a
a a
k sin
2


a
 The applications of the Fourier relation:

-Spatial harmonics and angles of propagation

f (t ) 1 / 2 t / 2 sin( / 2)
F ( ) 2
0 elsewhere

f ( x) 1 a / 2 x a / 2 F ( k ) 2a
sin( ka / 2)
0 elsewhere ka
?
Frequency, time, or position
 N
2
N

0 Time 2
Frequency
N
E E0 e j (0 i )t ji (t )
i 0

jNt
(t ) : time independent 1 e
E E0 e j0t
1 e jt
N
2

sin t
E E* E
2 2

sin t
2
 N
2
N

0 Time 2
Frequency
N
E E0 e j (0 i )t ji (t )
i 0

jNt
(t ) : time independent 1 e
E E0 e j0t
1 e jt
N
2

sin t
E E* E
2 2

sin t
2
Mode-locking
 N

NX

x0 Angle
Position x X

NX
2

sin
EE E
* 2
X
sin

Diffraction grating, radio antenna array

 The applications of the Fourier relation:
NT
f (t ) 1 nT t nT sin( ) sin( )
2 2
F ( ) 2 2
n 0,1, .....N 1
T
0 elsewhere sin( )
2 2
1/x

k
k=2 / x 2x
kz
(8)

Finite number of elements

-Graded grating for focusing

-Fresnel lens
Fourier transform between two focal planes of a lens

f(x,y) g(x,y)

d f
Focal
Z=0 A B plane
 The use of spatial harmonics
for analyses of arbitrary field pattern

Consider a two-dimensional complex electric field at z=0 given by

j 2 x x j 2 y y
f ( x, y ) Ae
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
k=2 / x 2x
kz

j 2 x x j 2 y y
f ( x, y) Ae
Thus by decomposing a spatial distribution of electric field into
spatial harmonics, each component can be treated separately.

j 2 x x j 2 y y jk z z
U ( x, y, z ) Ae

1
k z 2 ( x2 y2 )1 / 2
2
1/x

k
k=2 / x 2x
kz
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
j 2 x x j 2 y y
U ( x, y,0) Ae
j 2 x x j 2 y y jk z d
U ( x, y, d ) Ae
j 2 x x j 2 y y
Ae H ( x , y )
1
k z 2 ( x2 y2 )1 / 2
2

1/x
1
j 2 ( 2
2 1/ 2
) d
H ( x , y ) e
x y

2
k
k=2 / x 2x
kz
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

j 2 x x j 2 y y jk z d
U ( x, y, d ) Ae
j 2 x x j 2 y y
Ae H ( x , y )
1
k z 2 ( x2 y2 )1 / 2
2

1/x
1
j 2 ( 2
2 1/ 2
) d
H ( x , y ) e
x y
2
k
k=2 / x 2x
kz
Source
E E

z=0 z=0
To generalize:

1/x
k2 2x k1x
k1
k2z

(k 2 k1 ) x 2 x N

Grating momentum
 Stationary gratings vs. Moving gratings

Deflection Deflection + Frequency shift

The small angle approximation (1/ <<)
for the H function

1 d d 2
2 ( ) d 2
2 2 1/ 2
(1 )
2 1/ 2
2 (1 )

2 x y
2
=
jkd jd ( x y )
2 2
H ( x , y ) e e

A correction factor for the

transfer function for the
plane waves
 F(x) H(x)F(x)

z=0
z
 F(x) H(x)F(x)

z=0 z

F ( x,0) f ( x )e 2 x x d x

F ( x) H ( x ) f ( x )e 2 x x d x

jkx jd ( x y )
2 2
H ( x , y ) e e

Express F(x,z) in =x/z

 F(x) H(x)F(x)

z=0 z

F ( x,0) f ( x )e 2 x x d x

F ( x) H ( x ) f ( x )e 2 x x d x

jkx jd ( x y )
2 2
H ( x , y ) e e

Express F(x,z) in =x/z

 The effect of lenses
Converging lens

A lens is to introduce a quadratic phase shift to the

wavefront given by
.
x2 y2
j
f
e
 Fourier transform using a lens

f(x,y) g(x,y)

d f
Focal
Z=0 A B plane

j 2 ( x x y y )
f ( x, y ) F ( x , y )e d x d y
 f(x,y) g(x,y)

d f
Focal
Z=0 A B plane
j 2 ( x x y y )
f ( x, y ) F ( x , y )e d x d y
x2 y2
j
f jd ( x2 y2 ) j 2 ( x x y y )
U B ( x, y , z B ) e jkd
e e F ( x , y )e
( x x0 ) 2 ( y y 0 ) 2
j
A( x , y )e f

j ( d f )( x2 y2 )
A( x , y ) e jkd
e F ( x , y )
f(x,y) g(x,y)

f f
Focal
Z=0 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
Rr
 Holography
:

Recording of full information of an optical image,

including the amplitude and phase.

E E* E
2
Amplitude only:

Amplitude and phase ( E ( x ) e i ( x ) E R ) ( E ( x ) e j ( x ) E R )

E 2 E R 2 EER cos ( x)
2
 A simple example of
recording and reconstruction:

k1 k2

d

2 sin( )
2
 A simple example of
recording and reconstruction:

k1 k2

d

2 sin( )
2

d

2 sin( )
2
k1 k2

/2

d
?
2 sin( )
2
 Another example:
Volume hologram

k1 k2

d

2 sin( )
2
Volume grating

d

2 sin( )
2
 k1

d

2 sin( )
2
k1
k2

k1

d

2 sin( )
2
 D

A C

Bragg condition

AB BC CD 2d cos
 D

A C

B
Another example:
Image reconstruction of a point
illuminated by a plane wave.

Writing
Reading
 E(x,y)

Er

Recorded pattern

jk x x 2
E r E ( x, y ) e Er2 Er E ( x, y )e jkx x Er E ( x, y )e jkx x E 2 ( x, y )
Recorded pattern

jk x x 2
E r E ( x, y ) e 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