4: fourier optics - university of colorado...
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![Page 1: 4: Fourier Optics - University of Colorado Boulderecee.colorado.edu/~mcleod/teaching/ugol/lecturenotes/Lecture 4... · ECE 4606 Undergraduate Optics Lab Robert R. McLeod, University](https://reader035.vdocument.in/reader035/viewer/2022062223/5ab6c3507f8b9a7c5b8dfc23/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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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
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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”
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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”
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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
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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
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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
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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