# ISO 11146 Beam Size Definitions¶

**Scott Prahl**

**Mar 2021**

A laser beam will usually consist of a superposition of several modes. A single mode beam radius is easily described by \(1/e^2\) points. This, of course, fails for any beam shape other than Gaussian. ISO 11146 is intended to provide a simple, consistent way of describing the width of a beam.

This notebook summarizes the mathematical definitions.

*If* `laserbeamsize`

*is not installed, uncomment the following cell (i.e., delete the initial #) and execute it with* `shift-enter`

. *Afterwards, you may need to restart the kernel/runtime before the module will import successfully.*

```
[1]:
```

```
#!pip install --user laserbeamsize
```

```
[2]:
```

```
import numpy as np
import matplotlib.pyplot as plt
try:
import laserbeamsize as lbs
except ModuleNotFoundError:
print('laserbeamsize is not installed. To install, uncomment and run the cell above.')
print('Once installation is successful, rerun this cell again.')
```

```
[3]:
```

```
lbs.draw_beam_figure()
```

The total power \(P\) is obtained by integrating the irradiance \(E(x,y)\) over the entire beam

## Center of beam¶

The center of the beam can be found by

## Variance¶

A useful parameter characterizing a general two-dimensional distribution \(E(x,y)\) is the variance in the \(x\) and \(y\) directions

In general, \(\sigma_x \ne \sigma_y\). For example, in semiconductor lasers the height and width of the emitting aperture differ. Such beams are called *astigmatic*.

Now, the quantities \(\sigma_x^2\) and \(\sigma_y^2\) will always be positive, but \(\sigma_{xy}^2\) can be negative, zero, or positive.

## Beam Radius and \(D4\sigma\) or D4sigma¶

For a Gaussian distribution centered at (0,0) with \(1/e^2\) radius \(w\) we find

This leads to the definition of the beam radius definition as adopted by ISO 11146,

## \(D4\sigma\) or D4sigma¶

The \(D4\sigma\) beam diameter is a simple rearrangement of the above equation in which it is noted that twice the radius or the beam diameter is equal to \(4\sigma\)

### Relationship to FWHM¶

Sometimes it is the full width at half maximum (FWHM) value that is desired or known. In general, there is no direct relationship between the \(1/e^2\) radius \(w\) and the FWHM. However, in the special case of a Gaussian beam, there is.

### Major and minor axes of an elliptical beam¶

The \(x\)-axis diameter \(d_x=2w_x\) is given by

and similarly \(d_y=2w_y\) is

except if \(\sigma_x^2=\sigma_y^2\) in which case

and

### The tilt angle of the ellipse \(\phi\)¶

This is measured as a positive angle counter-clockwise from the \(x\)-axis (see figure above).

where we use `np.arctan2(numerator,denominator)`

to avoid division by zero when \(\sigma_x^2=\sigma_y^2\)

### Ellipticity and Eccentricity¶

The ellipticity \(e\) is

and the eccentricity is