# Verifying the expression for a Gaussian laser beam distribution.

This section describes how to verify that our expression for a Gaussian laser beam distribution is accurate. This example is calculated using MATLAB: listing of MATLAB program.

## Generate a histogram of r_{1} values from a set of random numbers.

The computer uses a random number generator to generate a set of N=10,000 random numbers referred to as RND. This set of RND is then converted into a set of r_{1} values using the expression r1 = w*sqrt(-ln(RND)).

Make a histogram of this set of r_{1} values using the MATLAB expression [n,x] = hist(r1,nb) which describes the number (n) of r_{1} values which fall into the bin centered at x, where there are nb bins. The bin size is specified: dx = x(3) - x(2).

Normalize n by N to yield the fraction of the total in a bin, then normalize by dx to yield a probability density function, p(r_{1}). The integral of p(r_{1})dr_{1} over all r is approximated by the MATLAB expression sum(p*dx) which equals unity as it should.

Finally, for the sake of easy recognition by you, the reader, we divide p(r_{1}) by the term (2 pi r_{1}) to yield the Gaussian distribution of photon launch along a radial line extending out from the origin.

## Create analytic p(r_{1})/(2 pi r_{1})

For comparison, create an analytic expression for a Gaussian curve:

y = exp(-r^{2}/w^{2})/(pi*w^{2})
which is equivalent to p(r_{1})/(2 pi r_{1}).

## Plot the histogram and analytic expressions.

The following figure plots the normalized histogram (yellow circles) generated by the random number generator and the analytic expression (red line):

IN SUMMARY, the normalized histogram agrees with the analytic expression illustrating that the Monte Carlo expression, r_{1} = w*sqrt(-ln(RND)), is correct.

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