21. Fourier Optics 21. Fourier Optics

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21. Fourier Optics 21. Fourier Optics

OBJECT

Time domain Space domain

SPECTRUM

Temporal frequency [ f_t : 1/sec ] Spatial frequency [ f_X : 1/m ] Fourier

Transformation

 Fourier transform 과 Optics가 무슨 관련이 있는가?

 Spatial Frequency 란 무엇인가?

 Fourier Optics 는 어떤 유용성이 있는가?

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Fourier Transforms in space-domain and time-domain Fourier Transforms in space-domain and time-domain

Spectrum

( ) ( )

ikx

f x g k e dk

g k f x e dx

 





( ) ( )

i t

f t g e d

g f t e dt



 



 





Object

2 _S k   f 2 f_t

  

1 1

S

f  m

    

Object in Time Object in Space

Angular Frequency

Frequency

Temporal Frequency Spatial Frequency

k 2

 

wave number

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Fourier Transforms Fourier Transforms

Fourier Transform pair : One dimension

Fourier Transform pair : Two dimensions

(4)

Fraunhofer diffraction and Fourier Transform Fraunhofer diffraction and Fourier Transform

Spectrum plane

Fraunhofer approximation

0 0

2

X

Y

X X

k k

r r

Y Y

k k

r r









 

   

 

   

 

The Fraunhofer pattern E_P is the Fourier transform of E_s !

Spatial angular frequency

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Remind! Diffraction under paraxial approx.

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Properties of 1D FT

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Properties of 1D FT - scaling -

There is an inverse scaling relation

between functions and their transforms

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Properties of 1D FT - translation (shifting) -

Shifting in the time -domain leads to phase delay in the frequency –domain (no shift in frequency -domain), so FT amplitude is unaltered.

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Properties of 1D FT - modulation -

Modulation in the time -domain

leads to frequency shifting in the frequency-domain

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Properties of 1D FT

- addition of two shifts - Properties of 1D FT

- addition of two shifts -

In Fourier optics,

this represents the interference for two-slit diffraction

Cosine modulation of the amplitude

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Properties of 1D FT

- convolution theorem - Properties of 1D FT

- convolution theorem -











h t g  h t  d t

g( ) ( ) ( ) ( )

 ^g ⁽ ^t ⁾ ^h ⁽ ^t ⁾  ^G ⁽ ^f ⁾ ^H ⁽ ^f ⁾

F  

Convolution in the time -domain leads to multiplication in the frequency -domain

Definition of convolution integral :

Convolution theorem

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

^g⁽^t⁾ ^h⁽^t⁾



^G⁽ ^f ⁾^H^*⁽ ^f ⁾

F  

Properties of 1D FT - correlation theorem -

Correlation in the time -domain leads to multiplication in the frequency -domain

Definition of correlation integral :

Correlation theorem











h t g  h  t d t

g( ) ( ) ( ) ( )

 

) ( ) ( )

) (

)

( ) (

) (

) ( )

( )

(

* 2

2 2

f H f G (f

H d e

g

t τ

; T dT

e e

T h g

d

dt e

t h

g d

dt e

d t h

g t

h t

g F

* f

j

f j fT j

ft j

 





 



















 







































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







 h t g  h t  d  t

g ( ) ( ) ( ) ( )











 h t g  h  t d  t

g ( ) ( ) ( ) ( ) Convolution

Correlation

g() h(-)

 

g()h(-) _

g()h(t-) _

0 t







 g  h t  d t

V( _i) ( ) ( _i )

t_i 

g(t) h(t)

t t

T/2

-T/2 -T/2 T/2

t V(t)

0 ^T

-T

g()h(-t) _

0 t

t V(t)

0 ^T

-T t_i

V(t_i)



 ~

i : t

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Properties of 1D FT - Parseval’s theorem -

Area under the absolute value squared of a function is equal to area under the absolute value squared of its transform

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Some frequently used functions

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Some frequently used functions

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Optics and 2D Fourier transform Optics and 2D Fourier transform

Superposition of plane waves

   

( , )

1

(

_x

,

_y

) (

_x

,

_y

)exp 2 (

_x _y

)

_x _y

g x y ^{= F}

^

G f f   

_^

G f f j  xf  yf df df

 

( , , ) (

_x

,

_y

)exp 2 (

_x _y

)

_x _y ^jkz

g x y z   

_^

G f f j  xf  yf df df e

g(x,y ;z=z₀)

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Optical spectrum analysis

- Fourier transformation by lens - Optical spectrum analysis

- Fourier transformation by lens -

X

Y

k k X f k k Y

f



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Example of Optical Fourier Transform

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How can a convex lens perform the FT How can a convex lens perform the FT

The phase at P(x_i) is leading that at the origin by

i i

i x

f j x

x

jk 



 



 





0

2 1

)

(sin 

  P(x_i)

_i

i

x

i

sin 

)

0

( x E

E

_i



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How can a convex lens perform the FT How can a convex lens perform the FT

f_o f_o

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Input placed against lens

Input placed in front of lens

Input placed behind lens

back focal plane

Fourier Transform with Lenses Fourier Transform with Lenses

이 경우에 대해 자세히 알아보자

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Fourier transforming property of a convex lens Fourier transforming property of a convex lens

The input placed in front of the lens

   

2 2

exp 1

2 2

, , exp

f l

k d

A j u

f f

U u U x y j xu y dxdy

j f f

 

 

 





   

 

       

 

    

 

 

If d = f,

f

  ^,

l

  ^, ^exp ²  

U u A U x y j xu y dxdy

j f f

  

 





 

    

 

 

Exact Fourier transform !

Input U_l(x,y)

Output U_f(u,v)

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Fourier and Inverse Fourier Transformation by Lenses



 _ 

( _x, _y) F f f

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4-f System and Optical Filtering

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Example of Spatial Filters

Band-pass filtering

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Example of Spatial Filters

Low-pass filtering

High-pass filtering

Band-pass filtering

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Example of Spatial Filters

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Example of Spatial Filters : Correlator

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System evaluation using the transfer function

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Fourier transform spectroscopy

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Fourier transform spectroscopy

21. Fourier Optics 21. Fourier Optics