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Using MATLAB to Compute Diffraction Patterns of Complex Apertures

Using MATLAB to Compute Diffraction Patterns of Complex Apertures. Stephen Schultz Department of Electrical and Computer Engineering Brigham Young University. Lens Focus. Fresnel becomes Fraunhofer at focal plane 2D Fourier transform Take 2D transform of complex structures Slits

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Using MATLAB to Compute Diffraction Patterns of Complex Apertures

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

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

  3. 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

  4. 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 • A=imread(‘slit’, ‘gif’)

  5. MATLAB Fourier Transform2. Computing DFT • Computing DFT • Use the two-dimensional FFT command E=fft2(A) • Puts center of the beam in the corners • Use the fftshift command to put it into the center E=fftshift(fft2(A)) A=imread(‘slit’, ‘gif’) fft2( A ) • fftshift( fft2( A ) )

  6. 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)

  7. 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

  8. 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 • f=p/N

  9. 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 x=Dx n • fx=f/Dx • Compare this to the analytic triangle function

  10. 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

  11. MATLAB Fourier Transform3. 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 M=64 M=256

  12. Spatial Resolution3. Spatial Coordinates • Spatial resolution, Dx, independent of number of pixels • Need larger image, W, without increasing slit size • Accomplished with zero padding

  13. 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(-lambda*f*N/(2*W), lambda*f*N/(2*W), N); y = linspace(-lambda*f*M/(2*W), lambda*f*M/(2*W), M); pcolor(x, y, I) • Most work done using graphics program

  14. Aperture Diffraction • Rectangular aperture • Slit size • Cassegrain telescope • Obscuration size • Struts

  15. Fourier Transform of a Letter(2. Fraunhofer Diffraction)

  16. Fourier Transform of a Word(2. Fraunhofer Diffraction)

  17. Pattern Recognition(2. Fraunhofer Diffraction)

  18. Optical Image Processing (2. Fraunhofer Diffraction) • Perform Fourier transform on an image • Filter the image (low pass or high pass filter) • Perform another Fourier transform

  19. 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

  20. Experimental Implementation(3. Experimental) • Requirement to make the implementation practical • Long focal length lenses • Easier focusing • Larger features (fx=x/lf) • Opaque masks • Transparencies transferred to chrome mask • Put various patterns on same mask • Spatial filtered red laser • Object is a simple transparency

  21. Implementation of 4F project(3. Experimental)

  22. 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

  23. Grid Filtering(3. Experimental) • Grid transparency • Slit image plane mask • Switch filter pattern by small shifts in mask 2.7 mm Grid with 110mm Filter Slit 2.7 mm Grid with 30mm Filter Slit

  24. Image Filtering(3. Experimental)

  25. Blurring of an Image(3. Experimental) Original Image Blurred Image

  26. Transition(3. Experimental) • Transition from image to spatial frequency Image off-focus Image at focus

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