This article explains how to compute the effective width of an arbitrary POP beam.

**Authored By Nam-Hyong Kim**

## Introduction

Computing the size and location of a Physical Optics beam on a surface is a very frequent requirement. If beams are known in advance to be simple Gaussians, then parameters like *1/**e**2* width or full-width-half-maximum intensity points can be used to characterize the beam size.

However, if a beam is not Gaussian, or has been aberrated or truncated by propagation through an optical system, intensity points may not be useful indicators of beam width. This is particularly true when subsidiary maxima are present in the beam. The *second moment* of the beam is a general and universally applicable method of computing the 'effective width' of any beam, Gaussian or not, coherent or incoherent.

## The first and second moments of a beam

The following analysis follows that of Siegman [1]. Consider a beam with irradiance distribution *I(x,y)*. The irradiance of a complex amplitude distribution is defined as *I=E.**E**** where *E**** represents the complex conjugate of the field. The first moment of this arbitrary beam is defined as:

where *c _{x}* is called the beam centroid and represents the 'center of mass' of the beam on the optical surface. This provides the location of the 'center' of the beam. A similar equation applies in y.

The second moment of the beam is given by:

where *c _{x}* is the centroid defined above. The beam width is then defined as W

_{x}=2σ

_{x}.

Note that if the beam is perfectly Gaussian its second moment is the 1/e^{2} point. The second moment, however, provides a meaningful measure of beam width whether the beam is Gaussian or not.

The POPD optimization operand can be used to report beam width value in the merit function editor. For detailed information about the POPD operand, refer to the Optimization chapter of the user manual.

Data items 23 and 24 return the X and Y beam widths respectively. The Physical Optics calculation must be configured appropriately and have its settings saved before POPD can be used. For example:

This will configure the POP calculation. Then, insert a POPD operand into the Merit Function Editor with the Data parameter set to 23 or 24. The value of the POPD operand is the semi-diameter of the beam using the second moment criteria on the specified surface:

## Common misconceptions

Do not confuse the pilot beam size with the effective beam size. Although the two are related if the beam is a Gaussian beam, the effective width is more general. As the beam propagates and is truncated, aberrated, develops secondary maxima or demonstrates any non-Gaussian behavior, the pilot beam and effective beam width will not give the same values. The primary purpose of the pilot beam is to control the Physical Optics Propagation algorithms.

The pilot beam will only be a good indicator of the real beam size if the real beam is a good Gaussian.

## References

[1] A. E. Siegman, Lasers, University Science Books (1986), R. Herloski, S. Marshall, and R. Antos

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