21. Fourier Optics21. Fourier Optics

OBJECT

Time domain

Space domain

SPECTRUM

Temporal frequency [ ft : 1/sec ]

Spatial frequency [ fX : 1/m ]

FourierTransformation

Fourier transform 과 Optics가 무슨 관련이 있는가?

Spatial Frequency 란 무엇인가?

Fourier Optics 는 어떤 유용성이 있는가?

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

Spectrum

( ) ( )

( ) ( )

ikx

ikx

f x g k e dk

g k f x e dx

( ) ( )

( ) ( )

i t

i t

f t g e d

g f t e dt

Object

2 Sk f2 tf

1 1 Sfm

Object in Time Object in Space

Angular Frequency

Frequency

Temporal Frequency Spatial Frequency

2k

wave number

Fourier Transforms Fourier Transforms Fourier Transform pair : One dimension

Fourier Transform pair : Two dimensions

Fraunhofer diffraction and Fourier Transform Fraunhofer diffraction and Fourier Transform

Spectrumplane

Fraunhoferapproximation

0 0

0 0

2

2

X

Y

X Xk kr r

Y Yk kr r

The Fraunhofer pattern EP is the Fourier transform of Es !

Spatial angularfrequency

Remind! Diffraction under paraxial approx.Remind! Diffraction under paraxial approx.

Properties of 1D FTProperties of 1D FT

Properties of 1D FT- scaling -

Properties of 1D FT- scaling -

There is an inverse scaling relationbetween functions and their transforms

Properties of 1D FT- translation (shifting) -

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.

Properties of 1D FT- modulation -

Properties of 1D FT- modulation -

Modulation in the time -domain leads to frequency shifting in the frequency-domain

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

Properties of 1D FT- convolution theorem -

Properties of 1D FT- convolution theorem -

dthgthtg )()()()(

)()()()( fHfGthtgF

Convolution in the time -domain leads tomultiplication in the frequency -domain

Definition of convolution integral :

Convolution theorem

)()()()( * fHfGthtgF

Properties of 1D FT- correlation theorem -Properties of 1D FT

- correlation theorem -

Correlation in the time -domain leads tomultiplication in the frequency -domain

Definition of correlation integral :

Correlation theorem

dthgthtg )()()()(

)()())(

)( )()(

)()(

)()()()(

*2

22

2

2

fHfG(fHdeg

tτ; TdTeeThgd

dtethgd

dtedthgthtgF

*fj

fjfTj

ftj

ftj

dthgthtg )()()()(

dthgthtg )()()()(

ConvolutionCorrelation

g() h(-)

g()h(-)

g()h(t-)

0 t

dthgtV ii )()()(

ti

g(t) h(t)

t tT/2-T/2 T/2-T/2

t

V(t)

0 T-T

g()h(-t)

0 t

t

V(t)

0 T-T ti

V(ti) ~:it

Properties of 1D FT- Parseval’s theorem -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

Some frequently used functionsSome frequently used functions

Some frequently used functionsSome frequently used functions

Optics and 2D Fourier transformOptics and 2D Fourier transform

Superposition of plane waves

1( , ) ( , ) ( , )exp 2 ( )x y x y x y x yg x y G f f G f f j xf yf df df = F

( , , ) ( , )exp 2 ( ) jkzx y x y x yg x y z G f f j xf yf df df e

g(x,y ;z=z0)

Optical spectrum analysis- Fourier transformation by lens -

Optical spectrum analysis- Fourier transformation by lens -

X

Y

Xk kf

Yk kf

Example of Optical Fourier Transform

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

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

iii xfxjxjk

0

12)(sin

P(xi)

i

iix sin

0)( ExE i

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

fo fo

Input placed

against lens

Input placed

in front of lens

Input placed

behind lens

back focal plane

Fourier Transform with LensesFourier Transform with Lenses

이 경우에 대해 자세히 알아보자

Fourier transforming property of a convex lensFourier transforming property of a convex lensThe input placed in front of the lens

2 2exp 1

2 2, , expf l

k dA j uf f

U u U x y j xu y dxdyj f f

If d = f, 2, , expf lAU u U x y j xu y dxdy

j f f

Exact Fourier transform !

InputUl(x,y)

OutputUf(u,v)

Fourier and Inverse Fourier Transformation by Lenses

( , )x yF f f

( , )x yF f f

4-f System and Optical Filtering

Example of Spatial Filters Band-pass filtering

Example of Spatial Filters

Low-pass filtering

High-pass filtering

Band-pass filtering

Example of Spatial Filters

Example of Spatial Filters : Correlator

System evaluation using the transfer function

Fourier transform spectroscopy

Fourier transform spectroscopy