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Didnt know where to post this..

I was trying to fresnel propogate a spherical beam using the formula

[tex]U(x,y)=\frac{e^{jkz}e^{\frac{jk(x^{2}+y^{2})}{2z}}}{j\lambda z}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{U(\xi,\eta)e^{\frac{jk(\xi^{2}+\eta^{2})}{2z}}}e^{-j\frac{2\pi}{\lambda z}(x\xi+y\eta)} d\xi d\eta[/tex]

[Ref. Intro. to Fourier Optics by Goodman pg 67]

which is basically the Fourier Transform of the product of the complex field just to the right of the aperture and a quadratic phase exponential, aside from the multiplicative factors.

I useed the following matlab code..

clc;

clf

clear;

%Parameters

N=512;

L=0.001;

dx=2*L/N;

[x y]=meshgrid(-L:dx:L-dx,-L:dx:L-dx);%plane1

x0=0;y0=0;z0=0;

[X Y]=meshgrid(-L:dx:L-dx,-L:dx:L-dx);%plane2 for fresnel prop

lambda=532*10^(-9);

k=2*pi/lambda;

%sigma=j*k/(2*z);

z=0.1;

z2=0.2;

%r=sqrt((x-x0).^2+(y-y0).^2+(z-z0)^2);

psi1=zeros(N);

psi1(128:384,128:384)=exp(j*k*z); %sigma ignored since its too large

F1=psi1.*exp(j*k*(x.^2+y.^2)/(2*(z2-z)));

FFT1=fftshift(fft2(F1));

F2=exp(j*k*(z2-z))*exp(j*k*(X.^2+Y.^2)/(2*(z2-z)))/(j*lambda*(z2-z));

U=FFT1;

figure(1)

I=psi1.*conj(psi1);

imagesc(I);

axis square;

colormap(gray);

figure(2)

I=U.*conj(U);

imagesc(I);

axis square;

colormap(gray);

where I have used a square aperture at z=0.1 m and tried to propogate it till z=0.2 m

But the propogated image, instead of being diffracted more is coming out to be focussed..

I would be grateful for any help

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# Fresnel propagation using matlab

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