# Fresnel propagation using matlab

1. Jul 4, 2008

### Raze2dust

Hi!

Didnt know where to post this..
I was trying to fresnel propogate a spherical beam using the formula

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

[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

2. Jul 4, 2008

### Raze2dust

hmm i guess too long for anyone to read fully..

suppose I have a point source at z=z0. What will be the equation of the Spherical wave which originates from there?

I tried using $$\frac{(z-z0)}{r^{2}}e^{jkr}$$

when is this valid?

3. Jul 5, 2008

### Raze2dust

hmm....will moving this to advanced physics forum help?