# Beam Radius

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a measure of the transverse extension of a light beam

Units: m

Formula symbol: <$w$>

The definition of the radius of a laser beam with a flat-top profile is trivial, but most light beams have other transverse shapes. A frequently obtained shape is the Gaussian one, where the transverse intensity variation is described with the following equation:

$$I(r,z) = \frac{P}{{\pi w{{(z)}^2}/2}}\;\exp \left( { - 2\frac{{{r^2}}}{{w{{(z)}^2}}}} \right)$$where the beam radius <$w$> is the distance from the beam axis where the optical intensity drops to <$1/e^2$> (≈ 13.5%) of the value on the beam axis. At this radius, the electric field strength drops to <$1/e$> (≈ 37%) of the maximum value.

The usual formula symbol of the beam radius is <$w$>. Some authors use <$\omega$> instead, but that may be confused with an angular frequency.

The *beam diameter* is generally defined as twice the beam radius – no matter what the particular definition of beam radius is. For Gaussian beams, the FWHM beam diameter is ≈1.18 times the Gaussian beam radius (<$1/e^2$> value).

## Definition of Beam Radius for Arbitrary Profile Shapes

For arbitrary (possibly not Gaussian) beam shapes, several different definitions are common. It is possible to still use the <$1/e^2$> intensity criterion, or a full width at half-maximum (FWHM), or a radius including 86% of the beam energy, etc. The problem with this type of definitions is essentially that the result does not depend on, e.g., how quickly the intensity decays in the wings of the profile. To illustrate this, Figure 1 shows two intensity profiles which have the same FWHM width, although the dashed curve is clearly wider in a meaningful sense. In the case of complicated intensity patterns, it is even more obvious that an FWHM definition cannot be appropriate.

### ISO Standard 11146

For such reasons (and another reason, which is discussed below), the recommended definition is that of ISO Standard 11146, based on the second moment of the intensity distribution *I(x,y)*. For example, the beam radius in the <$x$> direction is

where the coordinates <$x$> and <$y$> must be taken to be relative to the beam center, i.e., such that the first moments vanish. The method is also called the *D4σ method* because for the beam *diameter*, one obtains 4 times the standard deviation of the intensity distribution.

One can define the beam radius <$w_y$> in an analogous fashion, just replacing <$x^2$> with <$y^2$> in the upper integral.

For Gaussian beams, the D4σ method gives the same result as the <$1/e^2$> method, whereas for other beam shapes there can be significant deviations. The D4σ method should be used particularly when trying to predict the evolution of the beam radius for not diffraction-limited beams. It has been shown that the usual rules for Gaussian beam propagation with a certain *M*^{2} factor then correctly describe this evolution, whereas errors occur when using beam radii defined in some other way. This is important to observe e.g. when designing the pump optics of a diode-pumped laser because clearly non-Gaussian beam shapes can occur.

Disadvantages of the second-moment method are that the beam radius calculation is somewhat complicated (it usually requires numerical code), and that the result is quite sensitive to the intensity of the outer parts of the profile. For example, it is easily compromised by some vertical offset in the measured intensity distribution (e.g. caused by ambient light) and by noise of the camera. A detector with high dynamic range is therefore required. Often one also uses special smoothing techniques for reducing measurement errors.

## Effective Beam Area

In the context of laser-induced damage, one often uses an *effective beam area*, which is defined as the optical power divided by the maximum intensity, and is considered to be π times the *effective beam radius* squared. For a Gaussian beam, that effective beam radius is smaller than the Gaussian beam radius by a factor square root of 2.

## Measurement of Beam Radius

For the measurement of beam radii, various methods are applied. Obviously, a first step should always be to determine which definition of beam radius is to be used.

Some of the definitions and related measurement methods are applicable only to Gaussian beams. For example, this is mostly true for beam profiles using the knife edge or slit method, i.e., measuring the time-dependent transmitted optical power when a knife edge or a slit is moved through the beam. That method can also be applied with quite simple equipment, e.g. a knife edge mounted on a translation stage and an optical power meter, but the procedure is relatively cumbersome.

For arbitrary beam shapes, one usually uses camera-based beam profilers, with which the beam radius according to ISO 11146 can usually be determined quite quickly. Note, however, that this method often requires the use of an optical attenuator in order to get into the limited dynamic range of the camera, and obviously one must take care that the beam profile is not distorted by the attenuator. Further, not all devices can be applied to laser pulses, or only in certain parameter regions.

## More to Learn

Encyclopedia articles:

Blog articles:

- The Photonics Spotlight 2007-07-11: “What is a Beam Width, Beam Size, and a Beam Waist?”

### Bibliography

[1] | H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5 (10), 1550 (1966); https://doi.org/10.1364/AO.5.001550 |

[2] | P. A. Bélanger, “Beam propagation and the ABCD ray matrices”, Opt. Lett. 16 (4), 196 (1991); https://doi.org/10.1364/OL.16.000196 |

[3] | The Photonics Spotlight 2020-09-30: Beam radius and beam quality of laser pulses |

[4] | ISO Standard 11146, “Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios” (2005) |

## Questions and Comments from Users

2023-04-12

Is the Mode Field Diameter (MFD) of a single-mode fiber an accurate estimate of the waist diameter at the exit of the fiber?

Furthermore, is it reasonable to assume that in this situation the beam focus is at the exit facet of the fiber?

The author's answer:

The beam profile directly at the fiber output is identical to that in the fiber, so the answer to the first question is yes.

For a single-mode fiber, the answer to the second question is also yes. Modes have flat wavefronts (neglecting propagation losses), so you are at the focus.

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2021-12-08

When dealing with laser beams at the exit of a fiber, is it correct to use the fiber diameter as <$w_0$>? If not, what value should be used?

The author's answer:

Usually, you have a fiber core which is much smaller than the total cross-section of the fiber. As a rough estimate, you may assume the core diameter to be <$2 w_0$>, if it is a single-mode fiber. If it is not, such estimates will not be sensible; you would need more information.