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+% Illustrates the use fo LaguerreGaussianE.m, decompose.m and recompose.m by
+% defining an off-center Guassian beam (Fig. 1, Col. 1) and recomposing it 
+% in a basis of Laguerre Gaussians defined about the center on the figure.
+% The recomposed beam is shown in Fig. 1, Col. 2, where we have used the
+% first 40 Laguerre Gaussian modes.  Figure 1, Col. 3 shows the
+% difference between the recomposed beam and the original.  Figure 2 shows
+% the magnitude of the coefficients of the various modes in the
+% decomposition.
+
+
+ploton=[1 1];
+overlaponly=0; showfigure=0;
+
+clear domain;
+
+screensize=0.0125;
+nptsr=500;
+nptstheta=100;
+accuracy=0.001;
+n=400;
+
+[rmesh,thetamesh,xmesh,ymesh]=polarmesh([0,screensize,nptsr],[0 2*pi nptstheta],'lin');
+domain(:,:,1)=rmesh; domain(:,:,2)=thetamesh;
+
+w=0.001;
+R=-1e3;
+lambda=0.795e-6;
+q=q_(w,R,lambda);
+
+%deltax=1.5*w;
+%deltay=1.5*w;
+deltax=0;
+deltay=0;
+wfactor=1;
+
+mask_R = (w/10);
+mask = 1.0*((xmesh.^2+ymesh.^2) < mask_R^2);
+%mask = 1.0;
+
+if overlaponly
+    z1=LaguerreGaussianE([0,2,q_(w,R,lambda),lambda],xmesh,ymesh,'cart');
+    z2=LaguerreGaussianE([0,2,q_(w,R,lambda),lambda],xmesh,ymesh,'cart');
+    a=overlap(z1,conj(z2),domain,rmesh)
+    if showfigure
+        figure(4);
+        subplot(221); h=pcolor(xmesh,ymesh,abs(z1).^2); shg; colorbar; axis square; set(h,'edgecolor','none');
+        subplot(222); h=pcolor(xmesh,ymesh,abs(z2).^2); shg; colorbar; axis square; set(h,'edgecolor','none');
+        subplot(223); h=pcolor(xmesh,ymesh,angle(z1)); shg; colorbar; axis square; set(h,'edgecolor','none');
+        subplot(224); h=pcolor(xmesh,ymesh,angle(z2)); shg; colorbar; axis square; set(h,'edgecolor','none');
+    end
+    return
+end
+
+z_before_mask=LaguerreGaussianE([0,0,q_(w*wfactor,R,lambda),lambda],xmesh+deltax,ymesh+deltay,'cart');
+zin=mask.*z_before_mask;
+clear tmat tmat_in
+tmat_in(1:n,1,1)=1;
+tmat_in(1:n,1,2)=0;
+[coeffs,tmat]=decompose(zin,domain,'lg',n,[q,lambda,accuracy]);
+%[coeffs,tmat]=decompose(zin,domain,'lg',tmat_in,[q,lambda,accuracy]);
+disp(' '); disp('horizontal');
+dispmat(abs(coeffs(:,:,1)));
+disp(' '); disp('vertical')
+dispmat(abs(coeffs(:,:,2)));
+
+%[rmesh,thetamesh,xmesh,ymesh]=polarmesh([0,screensize/10,nptsr],[0 2*pi nptstheta],'lin');
+%domain(:,:,1)=rmesh; domain(:,:,2)=thetamesh;
+zout=recompose(domain,'lg',coeffs,[q,lambda,accuracy]);
+
+
+if ploton(1)==1
+    figure(1);
+    subplot(331);
+    h=pcolor(xmesh,ymesh,abs(zin).^2); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('original intensity');    
+    subplot(332);
+    h=pcolor(xmesh,ymesh,abs(zout).^2); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('recomposed');
+    subplot(333);
+    h=pcolor(xmesh,ymesh,abs(zout).^2-abs(zin).^2); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('difference');
+    subplot(334);
+    h=pcolor(xmesh,ymesh,real(zin)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('original real part');
+    subplot(335);
+    h=pcolor(xmesh,ymesh,real(zout)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('recomposed');
+    subplot(336);
+    h=pcolor(xmesh,ymesh,real(zout)-real(zin)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('difference');
+    subplot(337);
+    h=pcolor(xmesh,ymesh,imag(zin)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('original imaginary part')
+    subplot(338);
+    h=pcolor(xmesh,ymesh,imag(zout)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('recomposed');
+    subplot(339);
+    h=pcolor(xmesh,ymesh,imag(zout)-imag(zin)); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+    title('difference');
+end
+
+if length(ploton)>=2 & ploton(2)==1
+    figure(2);
+    coeffplotmat=[coeffs(:,end:-1:2,2),coeffs(:,:,1)];
+    ps=[-size(coeffs(:,:,2),1)+1:size(coeffs(:,:,1),1)-1];
+    ms=[0:size(coeffs(:,:,2))-1];
+    [psmesh,msmesh]=meshgrid(ps,ms);
+    h=pcolor(psmesh,msmesh,log10(abs(coeffplotmat))); axis square; colorbar; drawnow; shg;
+    title('Log_{10} of coefficients of the modes in the decomposition');
+    xlabel('m'); ylabel('p');
+end
+
+
+[rmesh,thetamesh,xmesh,ymesh]=polarmesh([0,screensize/10,nptsr],[0 2*pi nptstheta],'lin');
+domain(:,:,1)=rmesh; domain(:,:,2)=thetamesh;
+zout_magn=recompose(domain,'lg',coeffs,[q,lambda,accuracy]);
+figure(3)
+title('recomposed');
+h=pcolor(xmesh,ymesh,abs(zout_magn).^2); set(h,'edgecolor','none'); axis square; colorbar; drawnow; shg;
+
+
+