ivan bazarov cornell physics department / classe

Fourier Optics P3330 Exp Optics FA’20161

Fourier Optics

Ivan BazarovCornell Physics Department / CLASSE

Outline• 2D Fourier Transform• 4-f System• Examples of spatial frequency filters• Phase contrast imaging• Matlab FFT


Properties of 2D Fourier Transforms

f(x, y) =

ZZF (f

x

, f

y

)ei2⇡(xfx+yf

y

)df

x

df

y

F (fx

, f

y

) =

ZZf(x, y)e�i2⇡(xf

x

+yf

y

)dxdy

Definitions:

Linearity:

Scaling:

Shift:

↵f(x, y) + �g(x, y) ! ↵F (fx

, f

y

) + �G(fx

, f

y

)

f

⇣x

a

,

y

b

⌘ ! |ab|F (af

x

, bf

y

)

f(x� x0, y � y0) ! F (fx

, f

y

)e�i2⇡(x0fx+y0fy)


Properties of 2D Fourier Transforms (contd.)

Rotation:

Convolution:

Parseval’s theorem:

Slice theorem:

R

✓

{f(x, y)} ! R

✓

{F (fx

, f

y

)}

ZZf(x, y)g(x� x, y � y)dxdy ! F (f

x

, f

y

)G(fx

, f

y

)

ZZ|f(x, y)|2dxdy =

ZZ|F (f

x

, f

y

)|2dfx

df

y

Zf(x, y)dy ! F (f

x

, 0)


4-f System

FFT FFT FFT FFT

Collimate Mask FT Filter IFT Out

“4-f system”Prepare input 1Input mask

f(x, y)

Take FT F

✓x

0

�FFT,

y

0

�FFT

◆ Filter mask F

✓x

0

�FFT,

y

0

�FFT

◆G

✓x

0

�FFT,

y

0

�FFT

◆

Inverse FTZZ

f(x, y)g(x� x, y � y)dxdy

2D object f(x,y) has been filtered with 2D filter with impulse response g(x,y)


Coordinate system after the lens: spatial frequency converter

Thus, the spatial frequency fx is related to coordinate x’ by scaling factor Fl


Simple Fourier Transform

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


Low pass filter (sharp cutoff)ECE 4606 Undergraduate Optics Lab


Low pass, sharp cutoff

50

Multiplied by

=

REAL SPACE FOURIER SPACE

Filter cutoff frequency = 1/500 µm-1

Filter cutoff position = 126.4 µmFocal length = 100 mmLaser 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



Low pass, sharp cutoff

50

Multiplied by

=




All plots show amplitude of E


252.8 µm pinhole

Smoothed, but Gibbs ringing due to sharp filter edges


Low pass filter (smooth cutoff)ECE 4606 Undergraduate Optics Lab


Low pass, smooth cutoff

51

Multiplied by

=



Filter cutoff position = 126.4 µmEdge smoothing = 132 µmFocal length = 100 mmLaser wavelength = 632 nm


Now just nicely smoothed



Low pass, smooth cutoff

51

Multiplied by

=



Filter cutoff position = 126.4 µmEdge smoothing = 132 µmFocal length = 100 mmLaser wavelength = 632 nm


Now just nicely smoothed


High pass (narrow band)ECE 4606 Undergraduate Optics Lab


High pass, narrowband

52

Multiplied by

=



Filter cutoff position = 52.6 µmEdge smoothing = 58.2 µmFocal length = 100 mmLaser wavelength = 632 nm


Sharp filter used for clarity

Note sharp edges, darkening of large, uniform areas (~DC)

105.2 µm “dot”



High pass, narrowband

52

Multiplied by

=





Sharp filter used for clarity

Note sharp edges, darkening of large, uniform areas (~DC)

105.2 µm “dot”


High pass (high band)ECE 4606 Undergraduate Optics Lab


High pass, wideband

53

Multiplied by

=





Only edges remain. Almost a “line drawing”



High pass, wideband

53

Multiplied by

=





Only edges remain. Almost a “line drawing”


Vertical low pass (“smeers” vertically)ECE 4606 Undergraduate Optics Lab


Vertical low pass

54

Multiplied by

=



Filter cutoff position = 316 µmEdge smoothing = 93.6 µmFocal length = 100 mmLaser wavelength = 632 nm

Horizontal lines at edges of eyes goneVertical lines above nose remain.


632 µm horiz. slit


Horizontal low pass (“smeers” horizontally)ECE 4606 Undergraduate Optics Lab


Horizontal low pass

55

Multiplied by

=



Filter cutoff position = 316 µmEdge smoothing = 93.6 µmFocal length = 100 mmLaser wavelength = 632 nm

Horizontal lines at edges of eyes remain.Vertical lines above nose gone.


632

µm vert. slit


Simpler objects



Simpler object

56

Original

Low-pass

High-pass

Low-pass, different cutoff in x&y


A side note: this is how bandpass filters look like in the frequency (focus) domain.


Phase contrast imagingECE 4606 Undergraduate Optics Lab


Phase contrast

57

Multiplied by

=




Phase has become amplitude.Zernike won the 1953 Nobel in Physics for this.

•Lecture 4–Phase contrast

( )Ansel

Anselj

e maxπ−

Knife edge


Some MATLAB tips: 2D functions

• Create a matrix that evaluates 2D Gaussian: exp(-p/2(x2+y2)/s2)– >>ind = [-32:1:31] / 32;– >>[x,y] = meshgrid(ind,-1*ind);– >>z = exp(-pi/2*(x.^2+y.^2)/(.25.^2));– >>imshow(z)– >>colorbar

x’

y’

x

y

64x64 matrix

-1

0

31/32-1


FFT “splits” low frequencies

1D Fourier Transform– >>l = z(33,:);– >>L = fft(l);

sample #

10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

l

10 20 30 40 50 600

2

46

8

10

12

10 20 30 40 50 60-4

-2

0

2

4

sample #

phas

e (L

)ab

s (L

)

1 33 64

u0 fNfN/2

1 33 64

x [Matlab]0 xmax

low frequencies (aka shift theorem)


Use FFTSHIFT prior to/after FFT or FFT2

Use fftshift for 2D functions– >>smiley2 = fftshift(smiley);


If your FFT looks jagged…

f = zeros(30,30);f(5:24,13:17) = 1;imshow(f,'InitialMagnification','fit')

F = fft2(f);F2 = log(abs(F));imshow(F2,[-1 5],'InitialMagnification','fit');colormap(jet); colorbar

Discrete Fourier Transform Computed Without Padding


Apply zero-padding!

F = fft2(f,256,256);imshow(log(abs(F)),[-1 5]); colormap(jet); colorbar

Discrete Fourier Transform With Padding

“wrong” low frequency location!

% Apply FFTSHIFTF = fft2(f,256,256);F2 = fftshift(F);imshow(log(abs(F2)),[-1 5]); colormap(jet); colorbar

Normal FT look


Links/references

http://ecee.colorado.edu/~mcleod/teaching/ugol/lecturenotes/Lecture%204%20Fourier%20Optics.pdf

http://www.medphysics.wisc.edu/~block/bme530lectures/matlabintro.ppt

http://www.mathworks.com/help/images/fourier-transform.html

ivan bazarov cornell physics department / classe

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