ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

4: Fourier Optics

45

•Lecture 4–Outline

• Introduction to Fourier Optics

• Two dimensional transforms

• Basic optical layout

• The coordinate system

• Detailed example

• Isotropic low pass

• Isotropic high pass

• Low pass in just one dimension

• Phase contrast imaging

Pedrotti3, Chapter 21: Fourier Optics

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado 46

Properties of 2D FTs

( )∫ ∫∞

∞−

dydxyxf2

,Parseval’s thm ( )∫ ∫∞

∞−

yxyx dfdfffF2

,=

Each of these has a direct physical analog with optics.

( ) ( ) ( )yx

fyfxj

yx dfdfeffFyxf yx∫ ∫∞

∞−

+=

π2,,Definition ( ) ( ) ( )

dydxeyxfyxF yx fyfxj

∫ ∫∞

∞−

+−=

π2,,

( ) ( ) Real∈−± xfxfEven/odd function ( )Imaginary

Real ∈xfF↔

( ) Real∈xfReal function ( ) ( )xx fFfF −= ∗↔

( )dyyxf∫∞

∞−

,Projection slice thm ( )0,xfF↔

( ) ( )∫ ∫∞

∞−

−− ηξηξηξ ddyxgf ,, * ( ) ( )yxyx ffGffF ,, *

Correlation ↔

( ) ( )∫ ∫∞

∞−

−− ηξηξηξ ddyxgf ,, ( ) ( )yxyx ffGffF ,,Convolution ↔

( ){ }yxfR ,θ ( ){ }vuFR ,θRotation ↔

( )00 , yyxxf −− ( ) ( )yx fyfxjevuF 002

,+− π

Shift ↔

b

y

a

xf , ( )

yx fbfaFba ,Scaling ↔

( ) ( )yxgyxf ,, βα + ( ) ( )yxyx ffGffF ,, βα +Linearity ↔

•Lecture 4–Fourier optics

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado 47

General spatial filtering

Collimate Object FT Filter Inverse FT Output

fFT fFT fFT fFT

( ) ( )∫ ∫∞

∞−

−− ηξηξηξ ddyxgf ,,

( ) ( )FTFTFTFT F

y

Fx

F

y

Fx GF λλλλ

′′′′ ,,

( )yxf ,

1

( )FTFT F

y

FxF λλ

′′ ,

Thus the 2D object f(x,y) has been filtered with the 2D filter with impulse

response g(x,y).

Prepare input

Input mask

Take FT

Filter mask

Take inverse FT

“4F” processing system:

•Lecture 4–Fourier optics

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Coordinate system

48

So spatial frequency fx is related to

coordinate x´ by the scale factor F λ0

θλλ sin0=x

x x′

F

θ

0λ

θ

x

x

fF

F

Fx

0

0

sin

λ

λ

λ

θ

=

=

=′

0sin λθ=xf

or

•Lecture 4–Fourier optics

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Simple optical Fourier transforms

49

Focal length F = 100 mm

Laser wavelength λ0 = 632 nm

µm 200=xλµm 316

200

632.0000,100

=

×=′x

µm8.282

µm 2002

=

×=

= yx λλµm 223

8.282

632.0000,100

=

×=

′=′ yx

•Lecture 4–Fourier optics

Amplitude cosine, aka diffraction grating

Rotate object by 45o

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Low pass, sharp cutoff

50

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/500 µm-1

Filter cutoff position = 126.4 µm

Focal length = 100 mm

Laser wavelength = 632 nm

All plots show amplitude of E

•Lecture 4–Example

252.8 µm pinhole

Smoothed, but Gibbs ringing

due to sharp filter edges

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Low pass, smooth cutoff

51

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/500 µm-1

Filter cutoff position = 126.4 µm

Edge smoothing = 132 µm

Focal length = 100 mm

Laser wavelength = 632 nm

•Lecture 4–Example

Now just nicely smoothed

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

High pass, narrowband

52

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/1200 µm-1

Filter cutoff position = 52.6 µm

Edge smoothing = 58.2 µm

Focal length = 100 mm

Laser wavelength = 632 nm

•Lecture 4–Example

Sharp filter used for clarity

Note sharp edges, darkening of

large, uniform areas (~DC)

105.2 µm “dot”

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

High pass, wideband

53

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/300 µm-1

Filter cutoff position = 210.6 µm

Edge smoothing = 46.6 µm

Focal length = 100 mm

Laser wavelength = 632 nm

•Lecture 4–Example

Only edges remain. Almost a

“line drawing”

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Vertical low pass

54

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/200 µm-1

Filter cutoff position = 316 µm

Edge smoothing = 93.6 µm

Focal length = 100 mm

Laser wavelength = 632 nm

Horizontal lines at edges of eyes gone

Vertical lines above nose remain.

•Lecture 4–Example

632 µm horiz. slit

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Horizontal low pass

55

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/200 µm-1

Filter cutoff position = 316 µm

Edge smoothing = 93.6 µm

Focal length = 100 mm

Laser wavelength = 632 nm

Horizontal lines at edges of eyes remain.

Vertical lines above nose gone.

•Lecture 4–Example

632

µm

vert.

slit

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Simpler object

56

Original

Low-pass

High-pass

Low-pass, different

cutoff in x&y

•Lecture 4–Example

ECE 4606 Undergraduate Optics Lab

Robert R. McLeod, University of Colorado

Phase contrast

57

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/5000 µm-1

Filter cutoff position = 12.6 µm

Focal length = 100 mm

Laser wavelength = 632 nm

Phase has become amplitude.

Zernike won the 1953 Nobel in Physics for this.

•Lecture 4–Phase contrast

( )Ansel

Anselj

e maxπ−

Knife edge