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# How to convert FWHM measurements to 1/e^2 halfwidths

Sometimes, manufacturers provide Gaussian beam data as Full Width Half Maximum (FWHM) measurements, but OpticStudio uses the 1/e^2 measure of beam width. This article describes how to convert FWHM measurements to 1/e^2 halfwidth measurements, which are used by Zemax.
Dan Hill
04/04/2007

# Introduction

For Gaussian beam size measurements, Zemax uses the 1/e^2 half-width point, which means the intensity has fallen to about 13.5% of the peak. However, often times manufacturer’s data sheets provide only Full Width Half Maximum (FWHM) measurements and not 1/e^2 half-widths.

# The relationship between the FWHM and 1/e^2 halfwidth of a gaussian beam

For a truly TEM00, rotationally symmetric & normalized Gaussian beam, there is a linear relationship between the FWHM and 1/e^2 values.

The intensity of a Gaussian beam goes as:

where w is the half width of the beam to the 1/e^2 intensity point at some distance from the waist along the propagation axis, and r is the radial distance from the center of the beam. The width, w, at some z position is given by:

where w0 is the waist radius at the 1/e2 point.

For a normalized Gaussian beam, we know that the FWHM is the point at which the beam reaches half of the peak intensity. As a result, our equation simplifies to:

The FWHM is the full-width of the beam at half of the maximum intensity, so we need to divide this value by 2 so that we can replace it with r, the radial size.

Simplifying, we get:

Taking the natural log of both sides, and bringing the constant to the other side of the equation yields:

Solving for w, the relationship between the FWHM and the 1/e2 intensity point becomes:

# Summary:

Gaussian beam FWHM data can easily be converted into the 1/e^2 value with the conversion factor calculated here.