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 r₁ 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 r₁ values using the expression r1 = w*sqrt(-ln(RND)).

Make a histogram of this set of r₁ values using the MATLAB expression [n,x] = hist(r1,nb) which describes the number (n) of r₁ 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(r₁). The integral of p(r₁)dr₁ 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(r₁) by the term (2 pi r₁) to yield the Gaussian distribution of photon launch along a radial line extending out from the origin.

Create analytic p(r₁)/(2 pi r₁)

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

which is equivalent to p(r₁)/(2 pi r₁).

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, r₁ = w*sqrt(-ln(RND)), is correct.