Updated to plot convolution of aberrated psf with an object
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Karthik
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@ -1,22 +1,37 @@
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%% Zernike Polynomials
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%% Compute and plot Zernike Polynomials
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resolution = 100; % Resolution
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NumberOfGridPoints = 100; % Resolution
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plotZernike(2, 0, resolution) % Defocus (Z2^0)
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plotZernike(2, 0, NumberOfGridPoints) % Defocus (Z2^0)
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%%
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plotZernike(2, 2, resolution) % Astigmatism (Z2^2)
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plotZernike(2, 2, NumberOfGridPoints) % Astigmatism (Z2^2)
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%%
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plotZernike(3, 1, resolution) % Coma (Z3^1, x-direction)
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plotZernike(3, 1, NumberOfGridPoints) % Coma (Z3^1, x-direction)
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%%
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plotZernike(3, -1, resolution) % Coma (Z3^-1, y-direction)
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plotZernike(3, -1, NumberOfGridPoints) % Coma (Z3^-1, y-direction)
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%%
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plotZernike(4, 0, resolution) % Spherical Aberration (Z4^0)
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plotZernike(4, 0, NumberOfGridPoints) % Spherical Aberration (Z4^0)
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%% Aberrated PSF
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C = [0.0, 0.0, 0.0, 0.0, 0.0]; % Zernike coefficients
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pupil_radius = 0.5; % Pupil radius (normalized)
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%% Compute and plot Aberrated PSF, Image
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NumberOfGridPoints = 1024; % Number of grid points per side
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PupilRadius = 0.010; % Radius of pupil [m]
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Length = 0.5; % Total size of the grid [m]
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GridSpacing = Length / NumberOfGridPoints; % Grid spacing [m]
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Wavelength = 421e-9; % Optical wavelength [m]
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ImageDistance = 0.7; % Image distance [m]
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plotAberratedPSF(C, pupil_radius, resolution);
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% Generate PSF
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C = [0.3, 0.2, 0.1, -0.1, 0.4]; % Zernike coefficients
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[X, Y, PSF] = generateAberratedPSF(C, PupilRadius, NumberOfGridPoints, GridSpacing);
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plotAberratedPSF(X, Y, PSF);
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% Generate object
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Object = generateObject(Wavelength, ImageDistance, NumberOfGridPoints, GridSpacing);
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% Convolve the object with the PSF to simulate imaging
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Image = convolveObjectWithPSF(abs(Object).^2, PSF, 1);
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%% Functions
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function Z = computeZernikePolynomials(n, m, r, theta)
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% Zernike polynomial function for radial and angular coordinates (r, theta)
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@ -50,13 +65,13 @@ function Z = computeZernikePolynomials(n, m, r, theta)
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end
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end
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function plotZernike(n, m, resolution)
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function plotZernike(n, m, NumberOfGridPoints)
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% n: radial order
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% m: azimuthal frequency
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% resolution: number of points for plotting
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% NumberOfGridPoints: number of points for plotting
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% Create a grid of (r, theta) coordinates
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[theta, r] = meshgrid(linspace(0, 2*pi, resolution), linspace(0, 1, resolution));
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[theta, r] = meshgrid(linspace(0, 2*pi, NumberOfGridPoints), linspace(0, 1, NumberOfGridPoints));
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% Calculate the Zernike polynomial for the given (n, m)
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Z = computeZernikePolynomials(n, m, r, theta);
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@ -79,16 +94,19 @@ function plotZernike(n, m, resolution)
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shading interp;
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end
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function [theta, r, PSF] = modelPSF(C, pupil_radius, resolution)
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function [X, Y, PSF] = generateAberratedPSF(C, PupilRadius, NumberOfGridPoints, GridSpacing)
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% C is the vector of Zernike coefficients [C_defocus, C_astigmatism, C_coma, C_spherical, ...]
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% resolution is the number of points for the grid (NxN grid)
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% pupil_radius is the radius of the pupil aperture
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% NumberOfGridPoints is the number of points for the grid (NxN grid)
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% PupilRadius is the radius of the pupil aperture
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% Create a grid of (r, theta) coordinates
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[theta, r] = meshgrid(linspace(0, 2*pi, resolution), linspace(0, 1, resolution));
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% Create a grid of (x, y) of pupil-plane coordinates in Cartesian space
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[X, Y] = meshgrid((-NumberOfGridPoints/2 : NumberOfGridPoints/2-1) * GridSpacing);
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% Convert (x, y) to polar coordinates (r, theta)
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[theta, r] = cart2pol(X, Y);
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% Pupil function: 1 inside the pupil radius, 0 outside
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P = double(r <= pupil_radius); % 2D mask
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P = generateCircularMask(X, Y, 2 * PupilRadius); % 2D mask
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% Wavefront error from Zernike polynomials
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W = C(1) * computeZernikePolynomials(2, 0, r, theta) + ... % Defocus (Z2^0)
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@ -98,26 +116,22 @@ function [theta, r, PSF] = modelPSF(C, pupil_radius, resolution)
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C(5) * computeZernikePolynomials(4, 0, r, theta); % Spherical Aberration (Z4^0)
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% Fourier transform of the pupil function with aberrations
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PSF = abs(fftshift(fft2(P .* exp(1i * W)))).^2; % Intensity distribution
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PSF = abs(calculateExtendedFFT2(P .* exp(-1i * 2*pi * W), GridSpacing)).^2;
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end
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function plotAberratedPSF(C, pupil_radius, resolution)
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function plotAberratedPSF(X, Y, PSF)
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% C: Zernike coefficients [C_defocus, C_astigmatism, C_coma_x, C_coma_y, C_spherical]
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% pupil_radius: Radius of the pupil aperture
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% resolution: Number of points for plotting
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% PupilRadius: Radius of the pupil aperture
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% NumberOfGridPoints: Number of points for plotting
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% Generate PSF using the updated modelPSF function
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[theta, r, PSF] = modelPSF(C, pupil_radius, resolution);
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% Convert polar to Cartesian coordinates for plotting
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[X, Y] = pol2cart(theta, r);
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PSF = PSF / max(PSF(:));
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figure(1)
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clf
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set(gcf,'Position',[50 50 950 750])
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surf(X, Y, PSF, 'EdgeColor', 'none');
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xlim([-0.05 0.05])
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ylim([-0.05 0.05])
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view(2); % 2D view
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axis equal tight;
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shading interp;
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colorbar;
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colormap jet;
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@ -125,3 +139,36 @@ function plotAberratedPSF(C, pupil_radius, resolution)
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xlabel('X', 'FontSize', 16);
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ylabel('Y', 'FontSize', 16);
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end
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function C = convolveObjectWithPSF(A, B, GridSpacing)
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N = size(A, 1);
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C = calculateExtendedInverseFFT2(calculateExtendedFFT2(A, GridSpacing) .* calculateExtendedFFT2(B, GridSpacing), 1/(N*GridSpacing));
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end
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function z = generateCircularMask(x, y, D)
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r = sqrt(x.^2+y.^2);
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z = double(r<D/2);
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z(r==D/2) = 0.5;
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end
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function G = calculateExtendedFFT2(g, Delta)
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G = fftshift(fft2(fftshift(g))) * Delta^2;
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end
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function g = calculateExtendedInverseFFT2(G, Delta_f)
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N = size(G, 1);
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g = ifftshift(ifft2(ifftshift(G))) * (N * Delta_f)^2;
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end
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function ret = generateRectangle(x, D)
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if nargin == 1, D = 1; end
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x = abs(x);
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ret = double(x<D/2);
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ret(x == D/2) = 0.5;
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end
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function obj = generateObject(Wavelength, ImageDistance, NumberOfGridPoints, GridSpacing)
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% Image-plane coordinates
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[U, V] = meshgrid((-NumberOfGridPoints/2 : NumberOfGridPoints/2-1) * ((Wavelength * ImageDistance) / (NumberOfGridPoints * GridSpacing)));
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obj = (generateRectangle((U-1.4e-4)/5e-5) + generateRectangle(U/5e-5)+ generateRectangle((U+1.4e-4)/5e-5)) .* generateRectangle(V/2e-4);
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end
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