Sometimes, manufacturers provide Gaussian beam data as Full Width Half Maximum (FWHM) measurements. This article describes how to convert FWHM measurements to 1/e^2 halfwidth measurements, which are used by Zemax.

Authored By: Dan Hill

Published On: April 4, 2007

**The Relationship Between the FWHM and 1/e-Squared Halfwidth of a Gaussian Beam**

This article is also available in Japanese.

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.

For a truly TEM_{00}, rotationally symmetric & normalized Gaussian beam, there is a linear relationship between these two 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 *w*_{0} is the waist radius at the 1/e^{2} 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:

or,

Solving for *w*, the relationship between the FWHM and the 1/e^{2} intensity point becomes: