Using MATLAB to Compute Diffraction Patterns of Complex Apertures

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Using MATLAB to Compute Diffraction Patterns of
Complex Apertures
Stephen SchultzDepartment of Electrical and
Computer EngineeringBrigham Young University
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Lens Focus

• Fresnel becomes Fraunhofer at focal plane
• 2D Fourier transform
• Take 2D transform of complex structures
• Slits
• Multiple slits
• Letters
• Pictures
• Fingerprints

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Fourier Transform with MATLAB

• Built-in Fast Fourier Transform (FFT)
• Y fft2(X)
• Performs a Discrete Fourier Transform (DFT)
• Uses the Cooley-Tukey algorithm
• Fastest with sizes 2n
• Process
• Discretize the aperture
• Compute DFT
• Relate integer parameter to spatial coordinates

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MATLAB Fourier Transform1. Aperture
discretization

• Create gray scale image in graphics program
• Black is blocking
• White is transmitting
• Gray scale would be variable amplitude
• Save as a gif file (In illustrator)
• Save for Web and Devices
• Set the image size (faster with 2n pixels)
• No transparency, not interlaced
• Read file into MATLAB
• Aimread(slit, gif)

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MATLAB Fourier Transform2. Computing DFT

• Computing DFT
• Use the two-dimensional FFT command
• Efft2(A)
• Puts center of the beam in the corners
• Use the fftshift command to put it into the
  center
• Efftshift(fft2(A))

Aimread(slit, gif)
fft2( A )
fftshift( fft2( A ) )
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Convert FFT to Continuous Fourier Transform
(FT)3. Spatial Coordinates

• FFT is a Discrete Fourier Transform (DFT)
• We need to relate the DFT to the FT
• We will do this with a 1D analysis and then
  extend it to 2D by just replacing the FFT command
  with FFT2
• Start with a sampled version of the g(x)

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Convert FFT to Continuous Fourier Transform
(FT)3. Spatial Coordinates

• Use an FFT, which is a discrete Fourier Transform
  (DFT)
• MATLAB assumes that it starts at zero rather than
  x so you need to shift it using the fftshift
  command

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Convert FFT to Continuous Fourier Transform
(FT)3. Spatial Coordinates

• Convert the DFT to a discrete time Fourier
  transform (DTFT)
• The index is scaled by the number in the array
• fp/N

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Convert FFT to Continuous Fourier Transform
(FT)3. Spatial Coordinates

• Convert the DTFT into a continuous Fourier
  transform (FT)
• Use the sampling period scaling factor xDx n
• fxf/Dx
• Compare this to the analytic triangle function

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Convert FFT to Continuous Fourier Transform
(FT)3. Spatial Coordinates

• Summary of spatial coordinates
• Relate to discrete frequency
• Relate to spatial frequency
• Fraunhofer diffraction relationship
• Spatial coordinates

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MATLAB Fourier Transform 3. Spatial Coordinates

• For the FFT the number of frequency points equals
  the initial spatial points
• The spatial extent of the diffraction pattern
• More pixels causes the analysis plane to be
  bigger

M64
M256
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Spatial Resolution 3. Spatial Coordinates

• Spatial resolution, Dx, independent of number of
  pixels
• Need larger image, W, without increasing slit
  size
• Accomplished with zero padding

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Summary of MATLAB Code (2. Fraunhofer
Diffraction)

• Fairly simple MATLAB coding (9 lines of code)
• A imread('slit','gif')
• M,N size(A)
• lambda 600e-9
• f 16.5e-3
• W 4e-3
• I (fftshift(fft2(A))).2
• x linspace(-lambdafN/(2W),
  lambdafN/(2W), N)
• y linspace(-lambdafM/(2W),
  lambdafM/(2W), M)
• pcolor(x, y, I)
• Most work done using graphics program

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Aperture Diffraction

• Rectangular aperture
• Slit size
• Cassegrain telescope
• Obscuration size
• Struts

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Fourier Transform of a Letter (2. Fraunhofer
Diffraction)
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Fourier Transform of a Word (2. Fraunhofer
Diffraction)
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Pattern Recognition(2. Fraunhofer Diffraction)
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Optical Image Processing (2. Fraunhofer
Diffraction)

1. Perform Fourier transform on an image
2. Filter the image (low pass or high pass filter)
3. Perform another Fourier transform

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Experimental Implementation(3. Experimental)

• Helps solidify what is happening
• Complimentary to simulations
• No approximations but few aperture variations
• Easy to look at transition between regimes
• Students learn sensitivities of optical image
  processing
• Alignment
• Opaqueness

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Experimental Implementation(3. Experimental)

• Requirement to make the implementation practical
• Long focal length lenses
• Easier focusing
• Larger features (fxx/lf)
• Opaque masks
• Transparencies transferred to chrome mask
• Put various patterns on same mask
• Spatial filtered red laser
• Object is a simple transparency

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Implementation of 4F project (3. Experimental)
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Fourier Plane Mask(3. Experimental)

• Chrome on glass
• Multiple patterns on the same mask
• Sizes from 10-370 mm
• Patterns
• Rectangular slits
• Rectangular blocks
• Circular holes
• Circular blocks

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Grid Filtering(3. Experimental)

• Grid transparency
• Slit image plane mask
• Switch filter pattern by small shifts in mask

2.7 mm Grid with 30mm Filter Slit
2.7 mm Grid with 110mm Filter Slit
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Image Filtering (3. Experimental)
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Blurring of an Image (3. Experimental)
Original Image
Blurred Image
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Transition (3. Experimental)

• Transition from image to spatial frequency

Image off-focus
Image at focus