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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
Fourier transform pairs
46
•Lecture 4–Fourier optics
From Goodman, Introduction to Fourier Optics, p. 14
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 47
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 48
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
49
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
50
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
51
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
52
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
53
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
54
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
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 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
56
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
57
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
58
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