Example of how an optical spectral line is mapped into the propagation angles, and the corresponding spatial projection, for non-paraxial regime (high angles)

NonParaxialExample.m

%Example of how an optical spectral line is mapped into the propagation angles, and the corresponding spatial projection, for non-paraxial regime (high angles)

% In short, the radial components of frequencies and angles are related with radial_frequencies = sin(radial_angles), 
% and the radial components of the spatial projection are related with radial_angles=atan(radial_spatial_projections).

% For small angles these trigonotmetric relations are just proportional relations (in paraxial aproximation).

% I will be surprises id there are not mistakes/bugs. Comments to @miguelscilight.


u=(-1:0.01:1);
v=(-1:0.01:1);

[U,V]=meshgrid(u,v);


%%
ro=sqrt(U.^2+V.^2); %magnitude of spectral component
phi=atan(U./V)+(V>=0)*pi; %angle of spectral component



h = figure;

filename = 'nonParaxial.gif';

%%
tol=0.025;
ScreenSize=5; %screen size in meters

flag=false;
for theta=pi*( [-0.2:0.02:0.2 0.2:-0.02:-0.2]); %ilumination / grating angle
    frec_angle=sin(theta); % spatial frequency associated to this angle
    
    subplot(131)
    F=ro<1.*(abs(V-frec_angle)<tol); % line spectrum

    imagesc(u,v,F);

    shg;
    axis image
    
    title('Frequency Spectrum')
    xlabel('Normalized spatial frequency \nu_x\lambda');
    ylabel('Normalized spatial frequency \nu_y\lambda');
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %ANGULAR SPECTRUM.
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    u2=(pi/2)*(-1:0.01:1);
    v2=(pi/2)*(-1:0.01:1);
    [U2,V2]=meshgrid(u2,v2);

    ro2=sqrt(U2.^2+V2.^2);
    phi2=atan(U2./V2)+(V2>=0)*pi;


    subplot(132)
    % the spatial frequencies are related by the angles of propagation
    %by appying sin(.) to the radial component of the angle (ro2)
    
        
    F2=(ro2<pi/2).*(abs(sin(ro2).*cos(phi2)+frec_angle)<tol);

    imagesc(u2,v2,F2);
    shg;
    axis image
   
    title('Angular Spectrum')
    xlabel('\theta_x [rad]');
    ylabel('\theta_y [rad]');
    

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %Projection in the Screen
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    u3=ScreenSize*(-1:0.01:1);
    v3=ScreenSize*(-1:0.01:1);
    [U3,V3]=meshgrid(u3,v3);

    ro3=sqrt(U3.^2+V3.^2);
    phi3=atan(U3./V3)+(V3>=0)*pi;

    % the radial component of the angles of propagation are related with the projection in the
    % screen by applying atan(.) to the radial component of the projection
    % (ro3). Afterwards we can appy sin() to find the spatial frequency
    % component.
    
    %Here a distance of 1 meter is asummed, d
    d=1;
    
    subplot(133)
    F3=(abs(sin(atan(ro3/d)).*cos(phi3)+frec_angle)<tol);

    imagesc(u3,v3,F3);
    shg;
    axis image
    title('Screen projection @ 1m (cropped)[m]')
    xlabel('x [m]');
    ylabel('y [m]');
    pause(0.1);
    
    frame = getframe(h); 
    im = frame2im(frame); 
    [imind,cm] = rgb2ind(im,256); 
    if (~flag)
        imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime', 0.1); 
        flag=true;
    else    
        imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime', 0.1); 
    end
    
end