Updated script to compute the effect of polarization
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Karthik
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f6981b9233
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@ -30,6 +30,7 @@ wo_val = 100;
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% Set beam waists equal for simplicity
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wx1 = wo; wz1 = wo;
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wx2 = wo; wz2 = wo;
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%% Rotation matrix and k-vectors
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% Rotation matrix
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R = @(theta) [1 0 0; 0 cos(theta) -sin(theta); 0 sin(theta) cos(theta)];
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@ -43,44 +44,63 @@ RotatedCoords2 = @(theta) R(-theta) * [x; y; z];
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%% Define E fields
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% Generic orthogonal basis given a wavevector
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getPolarizationBasis = @(kvec) deal( ...
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simplify( cross(kvec, [1;0;0]) ), ...
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simplify( cross(kvec, cross(kvec, [1;0;0])) ) ...
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);
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% Get k-vectors
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k1vec = k1(theta);
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k2vec = k2(theta);
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% Get orthonormal basis for each polarization
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[e1_a, e1_b] = getPolarizationBasis(k1vec);
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[e2_a, e2_b] = getPolarizationBasis(k2vec);
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% Normalize them (optional if you want unit vectors)
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e1_a = simplify(e1_a / norm(e1_a));
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e1_b = simplify(e1_b / norm(e1_b));
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e2_a = simplify(e2_a / norm(e2_a));
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e2_b = simplify(e2_b / norm(e2_b));
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% Construct rotated polarization vectors using gamma
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e_pol_1 = cos(gamma) * e1_a + sin(gamma) * e1_b;
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e_pol_2 = cos(gamma) * e2_a + sin(gamma) * e2_b;
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coords = [x; y; z];
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rot1 = RotatedCoords1(theta);
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rot2 = RotatedCoords2(theta);
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% Polarization vector
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e_pol = cos(gamma)*[0; 0; 1] + sin(gamma)*[1; 0; 0];
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E1 = sqrt((2 * P) / (pi * wx1 * wz1)) * ...
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e_pol .* exp(1i * (k1(theta).' * coords)) * ...
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e_pol_1 .* exp(1i * (k1(theta).' * coords)) * ...
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exp(-(rot1(1)^2 / wx1^2) - (rot1(3)^2 / wz1^2));
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E2 = sqrt((2 * P) / (pi * wx2 * wz2)) * ...
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e_pol .* exp(1i * (k2(theta).' * coords)) * ...
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e_pol_2 .* exp(1i * (k2(theta).' * coords)) * ...
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exp(-(rot2(1)^2 / wx2^2) - (rot2(3)^2 / wz2^2));
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Efield = simplify(E1 + E2);
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IntensityVector = simplify(1/2 * real(conj(Efield) .* Efield)); % 3-component
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IntensityScalar = simplify(1/2 * sum(abs(Efield).^2)); % Scalar total intensity (with polarization interference)
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%% Intensity expression
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Intensity = simplify(1/2 * real(conj(Efield) .* Efield)); % 3-component
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%% ================ Plot lattice =================== %%
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%% ================ Plot lattice in Y-Z plane =================== %%
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% Define parameters
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theta_val = 10 * pi / 180; % 10 degrees in radians
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gamma_val = pi/2.0; % tilt of linear polarization
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theta_val = deg2rad(10); % half-angle between beams
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gamma_val = deg2rad(90); % tilt of linear polarization
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% Extract z-component of intensity at x = 0
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Iplane_z = simplify(subs(Intensity(3), x, 0));
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Iplane_z = simplify(subs(IntensityVector(3), x, 0));
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% Convert to function
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Iplane_func = matlabFunction(Iplane_z, 'Vars', {y, z, theta, wo, lambda, P, gamma});
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Iplane_z_func = matlabFunction(Iplane_z, 'Vars', {y, z, theta, wo, lambda, P, gamma});
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% Grid for y and z
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[ygrid, zgrid] = meshgrid(linspace(-1000, 1000, 500), linspace(-100, 100, 300));
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[ygrid, zgrid] = meshgrid(linspace(-1000, 1000, 500), linspace(-100, 100, 500));
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% Evaluate intensity
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Ivals = Iplane_func(ygrid, zgrid, theta_val, wo_val, lambda_val, P_val, gamma_val);
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Ivals = Iplane_z_func(ygrid, zgrid, theta_val, wo_val, lambda_val, P_val, gamma_val);
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% Normalization
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Ivals = Ivals / max(Ivals(:));
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@ -88,19 +108,17 @@ Ivals = Ivals / max(Ivals(:));
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% Plotting
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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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set(gcf,'Position',[50 50 900 700])
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contourf(ygrid, zgrid, Ivals, 200, 'LineColor', 'none');
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colormap('turbo');
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colorbar;
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% Preserve physical aspect ratio
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pbaspect([1 1 1]); % Set plot box aspect ratio to 1:1:1
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axis tight;
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axis tight
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xlabel('y [µm]', 'FontSize', 12);
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ylabel('z [µm]', 'FontSize', 12);
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title(['I_{plane}(y, z) at x = 0, \theta = ' num2str(rad2deg(theta_val)) '^\circ'], 'FontSize', 14);
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set(gca, 'FontSize', 12, 'Box', 'on');
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%% ================ Plot Potentials of lattice =================== %%
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% ================ Plot 1-D intensity profiles of lattice =================== %%
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% Find indices closest to zero in y and z grids:
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[~, idx_y0] = min(abs(ygrid(1,:))); % y=0 along columns
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@ -121,21 +139,132 @@ zvec = zgrid(:, 1);
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% Plot -Iprop/2 along y
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figure(2);
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(yvec, -Iprop_cut/2, 'LineWidth', 2);
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set(gcf,'Position',[50 50 900 700])
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plot(yvec, Iprop_cut, 'LineWidth', 2);
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title('Profile at x=0, z=0');
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xlabel('y [\mum]');
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ylabel('Depth');
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ylabel('Intensity');
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grid on;
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set(gca, 'FontSize', 12, 'Box', 'on');
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% Plot -Ivert/2 along z
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figure(3);
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(zvec, -Ivert_cut/2, 'LineWidth', 2);
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set(gcf,'Position',[50 50 900 700])
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plot(zvec, Ivert_cut, 'LineWidth', 2);
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title('Profile at x=0, y=0');
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xlabel('z [\mum]');
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ylabel('Depth');
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ylabel('Intensity');
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grid on;
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set(gca, 'FontSize', 12, 'Box', 'on');
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%% Plot scalar intensity to see effect from change in polarization
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theta_val = deg2rad(10); % half-angle between beams
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Iplane_scalar = simplify(subs(IntensityScalar, x, 0));
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Iplane_scalar_func = matlabFunction(Iplane_scalar, 'Vars', {y, z, theta, wo, lambda, P, gamma});
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% Prepare figure
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figure(4);
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clf
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set(gcf,'Position',[50 50 900 700])
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hold on
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% Define gamma values to compare
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gammas = linspace(0, pi/2, 10); % from 0 (p) to pi/2 (s)
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% Find y = 0 index
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[~, idx_y0] = min(abs(ygrid(1,:))); % column index
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% Colors for curves
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colors = turbo(length(gammas));
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% Preallocate contrast array
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contrasts = zeros(size(gammas));
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for i = 1:length(gammas)
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gamma_val = gammas(i);
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% Evaluate intensity
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Ivals = Iplane_scalar_func(ygrid, zgrid, theta_val, wo_val, lambda_val, P_val, gamma_val);
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Ivals = Ivals / max(Ivals(:)); % normalize
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% Extract z-cut at y = 0
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Icut = Ivals(:, idx_y0);
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% Compute contrast
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Imax = max(Icut);
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Imin = min(Icut);
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contrasts(i) = (Imax - Imin) / (Imax + Imin);
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% Plot
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plot(zvec, Icut, 'LineWidth', 2, 'DisplayName', ...
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['\gamma = ' num2str(rad2deg(gamma_val), '%0.1f') '^\circ'], ...
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'Color', colors(i, :));
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end
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% Finalize figure
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xlabel('z [\mum]', 'FontSize', 12);
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ylabel('Normalized Intensity', 'FontSize', 12);
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title('Intensity cut along z at y=0 for varying \gamma', 'FontSize', 14);
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legend('Location', 'best');
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grid on
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box on
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set(gca, 'FontSize', 12);
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% --- New Figure for Contrast vs Gamma ---
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figure(5);
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clf
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set(gcf,'Position',[50 50 900 700])
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plot(rad2deg(gammas), contrasts, '-o', 'LineWidth', 2);
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xlabel('\gamma [degrees]', 'FontSize', 14);
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ylabel('Fringe Contrast', 'FontSize', 14);
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title(['Contrast vs. Polarization Angle \gamma, \theta = ' num2str(rad2deg(theta_val)) '^\circ'], 'FontSize', 16);
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grid on
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box on
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set(gca, 'FontSize', 14);
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%%
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% --- Contrast vs Theta for fixed gamma = 0 and gamma = pi/2 ---
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% Define range of theta values (in radians)
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theta_vals = linspace(deg2rad(1), deg2rad(45), 50); % e.g., from 1° to 45°
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% Fixed gammas for contrast comparison
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gamma_fixed = [0, pi/2];
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contrast_vs_theta = zeros(length(gamma_fixed), length(theta_vals));
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for g_idx = 1:length(gamma_fixed)
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gamma_val = gamma_fixed(g_idx);
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for t_idx = 1:length(theta_vals)
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theta_val_local = theta_vals(t_idx);
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% Evaluate intensity on grid
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Ivals = Iplane_scalar_func(ygrid, zgrid, theta_val_local, wo_val, lambda_val, P_val, gamma_val);
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Ivals = Ivals / max(Ivals(:)); % normalize
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% Extract z-cut at y=0
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Icut = Ivals(:, idx_y0);
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% Compute contrast
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Imax = max(Icut);
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Imin = min(Icut);
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contrast_vs_theta(g_idx, t_idx) = (Imax - Imin) / (Imax + Imin);
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end
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end
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% Plot contrast vs theta for the two fixed gammas
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figure(6);
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clf
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set(gcf,'Position',[50 50 900 700])
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plot(rad2deg(theta_vals), contrast_vs_theta(1,:), '-o', 'LineWidth', 2, 'DisplayName', '\gamma = 0^\circ (p-pol)');
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hold on
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plot(rad2deg(theta_vals), contrast_vs_theta(2,:), '-s', 'LineWidth', 2, 'DisplayName', '\gamma = 90^\circ (s-pol)');
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xlabel('\theta [degrees]', 'FontSize', 14);
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ylabel('Fringe Contrast', 'FontSize', 14);
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title('Contrast vs. Half-angle \theta for fixed polarizations \gamma', 'FontSize', 16);
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legend('Location', 'southwest');
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grid on
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box on
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set(gca, 'FontSize', 14);
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