I should preface this by saying I am not much of a programmer or physicist, so please bear with me.
I am attempting to simulate the propagation of gaussian Bessel beam (as produced by an axicon) using Fourier propagation (Fresnel propagation?) but I am having some problems.
My code is below, as well as the function I found to do the propagation. The first part of the code does a simple approximation of a gaussian Bessel beam and works fine but lacks some of the characteristic features (diverging into a ring beam, reconstruction following occlusions, etc). For this reason, I wanted to take a cross section of the intensity profile (figure 2) and propagate that forward using the fourier method.
Figure three shows the result of propagation of only ten microns (very short compared to the expected length of the Bessel region (figure 1). For such a distance I would have expected almost no change to the intensity cross section.
I don't know what I am doing wrong, perhaps this method is simply unsuitable to this problem, though it seemed to me like a generic method for beam propagation in free space.
If anyone has some guidance, I would be most appreciative. Thanks.
clear;clf;n = 1.44; %index of refraction for the axicona = 5; %axicon angle alphab = asind(n*sind(a))-a; %the beta anglebt = 0; %bessel orderf1 = 75;f2 = 8;mag = f2/f1;z_cone = tand(a)*3;w = 3; %guassian beam diameter in mmy = 1064; %wavelength in nmI_0 = 0.1; %guassian beam intensityP = 8; %powerk = 2*3.14/(y*10^(-6)); %wavenumber in mmlambda = y/(10^6);z_r = w/2*cosd(b)/sind(b)*mag^2; %bessel length estimate in mm in airr_0 = 2.4048/(k*sind(b))*mag; % bessel radius in mm in airZ_0 = 1/2*z_r; %cross section position for xy plot, taken at some fraction of z_rimax = 512; %1000 good but slowjmax = 512; %1000 good but slowspanr = 0.03;%round(20*r_0*1000)/1000; %mmspanz = round(20*z_r)/10; %mm 0.8I = zeros(imax,jmax);ii=[0:1:(imax-1)];spanxy=spanr; %round(100*spanr/sqrt(2))/100;xy=I; %defining another matrix to hold an xy field perpendicular to the z axis%k = 2*3.14/(y*10^(-6)); %wavenumber in mm%z_r = w/(2*(n-1)*tand(a))*mag^2;%r_0 = 2.4048/(k*sind(b))*mag;z = (ii)*spanz/imax; %a vector containing all z valuescof1 = (8*P*k*sind(b)/w)*(z/z_r); %parts equation for approximate bessel beam, not dependent on r so taken out of loopexpon1 = exp(-2*z.^2/z_r^2); %another partfor j = 0:(jmax-1); r = spanr/2-(j)*spanr/jmax; bessel01 = (besselj(bt,r*2.4048/r_0))^2; II(j+1,:)=cof1.*expon1.*bessel01;endZ_0index = round(Z_0*imax/spanz);cof1xy = cof1(Z_0index);expon1xy = expon1(Z_0index);test = cof1xy*expon1xy;for u = 0:(imax-1); x = -spanxy/2+u*spanxy/imax; for v = 0:(jmax-1); y = spanxy/2-v*spanxy/imax; r = sqrt(x^2+y^2); xy((u+1),(v+1)) = test*((besselj(bt,r*2.4048/r_0))^2); endend%plot 1, approximationlabelsx = [0:spanz/10:spanz];labelsy = [(-spanr/2)*-1000:(spanr/10)*-1000:(spanr/2)*-1000];maxI = max(max(II./(1)));E = sum(sum(II));subplot(2,2,1), imshow(II./17.6, hot) %why divide by 17.6? just to reduce over saturation. for visual purposesaxis("tic")set(gca,'xTick',1:imax/10:imax);set(gca,'yTick',0:jmax/10:jmax);set(gca,'xTickLabel',labelsx);set(gca,'yTickLabel',labelsy);xlabel("Z-axis (mm)");ylabel('Radial-axis (\mum)')title('Spatial Intensity')H=colorbar;title(H,'Intensity (%)')set(H, 'yticklabel', [0,20,40,60,80,100])%plot 2, intensity cross section (seed for propagation)labelsx = [(-spanxy/2)*1000:(spanxy/10)*1000:(spanxy/2)*1000];labelsy = [(-spanxy/2)*-1000:(spanxy/10)*-1000:(spanxy/2)*-1000];subplot(2,2,2), imshow(xy/17.6, hot)axis("tic")set(gca,'xTick',1:imax/10:imax);set(gca,'yTick',0:imax/10:imax);set(gca,'xTickLabel',labelsx);set(gca,'yTickLabel',labelsy);xlabel('X-axis (\mum)');ylabel('Y-axis (\mum)')title('Spatial Intensity')H=colorbar;title(H,'Intensity (%)')set(H, 'yticklabel', [0,20,40,60,80,100])dz = (spanz-Z_0)/(imax-Z_0index);zz = [Z_0:(spanz-Z_0)/(imax-Z_0index):spanz]; %actual z positions for the propagated regionzz_adjusted = zz(1,:)-Z_0; %adjusted to set z_0 equal to zero for the purpose of the propagation equationzpmax = imax-Z_0index;L = spanxy;[M,~]=size(xy);dx=L/M;% Frequency coordinates sampling%fx=-1/(2*dx):1/L:1/(2*dx)-1/L;%invoke fourier propagationu2 = propTF(xy,spanxy,lambda,0.01);%plot 3, propagated intensity profilelabelsx = [(-spanxy/2)*1000:(spanxy/10)*1000:(spanxy/2)*1000];labelsy = [(-spanxy/2)*-1000:(spanxy/10)*-1000:(spanxy/2)*-1000];%subplot(2,2,3), imshow(h(:,:,100)*10,hot) %imshow(U(:,:,10)/17.6, hot)subplot(2,2,3), imshow(u2/17.6,hot) %imshow(U(:,:,10)/17.6, hot)axis("tic")set(gca,'xTick',1:imax/10:imax);set(gca,'yTick',0:imax/10:imax);set(gca,'xTickLabel',labelsx);set(gca,'yTickLabel',labelsy);xlabel('X-axis (\mum)');ylabel('Y-axis (\mum)')title('Spatial Intensity')H=colorbar;title(H,'Intensity (%)')set(H, 'yticklabel', [0,20,40,60,80,100])function[u2]=propTF(u1,L,lambda,z)%Fresnel propagation using the Transfer function% Based on Computational Fourier Optics by Voelz %% Assuming uniform sampling and presents reflections on the boundaries% %PARAMETERS% % u1 - Complex Amplitude of the beam at the source plane% L - Sidelength of the simulation window of the source plane% lambda - Wavelength % z - Propagation distance % u2 - Complex Amplitude of the beam at the observation plane% Input array size[M,~]=size(u1); % Sampling inteval sizedx=L/M; % Frequency coordinates sampling fx=-1/(2*dx):1/L:1/(2*dx)-1/L; % Momentum/reciprocal Space [FX,FY]=meshgrid(fx,fx);% Transfer functionH=exp(-1j*pi*lambda*z*(FX.^2+FY.^2)); H=fftshift(H);% Fourier transform of the transfer fucntionU1=fft2(fftshift(u1));% Convolution of the systemU2=H.*U1;% Fourier transform of the convolution to the observation planeu2=ifftshift(ifft2(U2)); endI tried increasing the resolution and/or the side length of the simulation window, on the off chance it was from some interactions with the simulation boundary, but didn't see a qualitative difference for short propagation distances (for large z values it does seem to be interacting with the boundaries, but its already deviated from the expected output long before then).