# Gaussian Beam Propagation and ABCD Law

1. Nov 10, 2011

### randy06

I am trying to evaluate the intensity profile of a Gaussian beam after passing through an optical system. Based on what I have read from this source http://www.ucm.es/info/euoptica/org/pagper/jalda/docs/libr/laserandgaussian_eoe_03.pdf and a few others, I should be able to find the intensity profile from the ABCD matrix (transfer matrix) of an optical system.

I wrote a matlab program to do this, and I have included the code at the bottom of this post. From what I understand the new beam waist after passing through the optical system should be at the focus of the system. This means that the intensity profile should be narrow at the focus and broader before and after the focus; however, my code does not show this.

Can someone tell me if my method is incorrect or where my code may be wrong. In my code just change the value of d to see that the width of the intensity profile grows regardless of the value of f. For right now I am just using a thin lens for simplicity.

Also, if someone knows of a better way to do this I would like to hear it.

Matlab code:

% units are meters

s = 4;
x=(-2*10^-s):(10^-(s+2)):(2*10^-s);

d = 0*10^-3; % z distance after lens
f = .5*10^-3; % focus of the lens

%ABCD matrix of thin lens with translation d
A = 1;
B = d;
C = 1+(-d/f);
D = 1;

z = 1*10^-3; % propagation distance from source to lens
lambda = 1*10^-6; % wavelength
k = 1/lambda;
w0 = 25*10^-6; % beam waist

zR = (pi*(w0^2))/lambda; % Rayleigh range
w = w0*((1+(z/zR)^2)^.5); % spot size
R = z*(1+(zR/z)^2); % radius of curvature of the beam's wavefront

q1 = 1/((1/R)-((i*lambda)/(pi*(w^2)))); % complex beam parameter before lens
q2 = (A*q1+B)/(C*q1+D); % complex beam parameter after lens

E = exp(-(((i*k)/(2*q2)))*x.^2); % electric field
I = abs((E.^2))/2; % intensity

figure(1)
plot(x,I)