# Beam Divergence

Definition: a measure for how fast a laser beam expands far from its focus

German: Strahldivergenz

Units: °, mrad

Formula symbol: <$\theta$>

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/761

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The beam divergence (or more precisely the *beam divergence angle*) of a laser beam is a measure for how fast the beam expands far from the beam waist, i.e., in the so-called *far field*. Note that it is not a local property of a beam, for a certain position along its path, but a property of the beam as a whole. (In principle, one could define a local beam divergence e.g. based on the spatial derivative of the beam radius, but that is not common.)

A low beam divergence can be important for applications such as pointing or free-space optical communications. Beams with very small divergence, i.e., with approximately constant beam radius over significant propagation distances, are called *collimated beams*; they can be generated from strongly divergence beams with beam collimators.

Some amount of divergence is unavoidable due to the general nature of waves (assuming that the light propagates in a homogeneous medium, not e.g. in a waveguide). That amount is larger for tightly focused beams. If a beam has a substantially larger beam divergence than physically possibly, it is said to have a poor beam quality. More details are given below after defining what divergence means quantitatively.

## Quantitative Definitions of Beam Divergence

Different quantitative definitions are used in the literature:

- According to the most common definition, the beam divergence is the derivative of the beam radius with respect to the axial position in the far field, i.e., at a distance from the beam waist which is much larger than the Rayleigh length. This definition yields a
*divergence half-angle*(in units of radians), and further depends on the definition of the beam radius. For Gaussian beams, the beam radius is usually defined via the point with <$1/e^2$> times the maximum intensity. For non-Gaussian profiles, an integral formula can be used, as discussed in the article on beam radius. - Sometimes,
*full angles*are used instead, resulting in twice as high values. - Instead of referring to directions with <$1/e^2$> times the maximum intensity, as is done for the Gaussian beam radius, a full width at half-maximum (FWHM) divergence angle can be used. This is common e.g. in data sheets of laser diodes and light-emitting diodes. For Gaussian beams, this kind of full beam divergence angle is 1.18 times the half-angle divergence defined via the Gaussian beam radius (<$1/e^2$> radius).

As an example, an FWHM beam divergence angle of 30° may be specified for the fast axis of a small edge-emitting laser diode. This corresponds to a 25.4° = 0.44 rad <$1/e^2$> half-angle divergence, and it becomes apparent that for collimating such a beam without truncating it one would require a lens with a fairly high numerical aperture of e.g. 0.6. Highly divergent (or convergent) beams also require carefully designed optics to avoid beam quality degradation by spherical aberrations.

## Divergence of Gaussian Beams and Beams with Poor Beam Quality

For a diffraction-limited Gaussian beam, the <$1/e^2$> beam divergence half-angle is <$\lambda / (\pi w_0)$>, where <$\lambda$> is the wavelength (in the medium) and <$w_0$> the beam radius at the beam waist. This equation is based on the paraxial approximation, and is thus valid only for beams with moderately strong divergence. It also shows that the product of beam waist radius <$w_0$> and the divergence angle <$\theta$> (called the beam parameter product) is not changed by any optical system without optical aberrations which transforms a Gaussian beam into another Gaussian beam with different parameters.

A higher beam divergence for a given beam radius, i.e., a higher beam parameter product, is related to an inferior beam quality, which essentially means a lower potential for focusing the beam to a very small spot. If the beam quality is characterized with a certain *M*^{2} factor, the divergence half-angle is

As an example, a 1064-nm beam from a Nd:YAG laser with perfect beam quality (<$M^2 = 1$>) and a beam radius of 1 mm in the focus has a half-angle divergence of only 0.34 mrad = 0.019°.

## Spatial Fourier Transforms

For obtaining the far field profile a beam, one may apply a two-dimensional transverse spatial Fourier transform to the complex electric field of a laser beam (→ *Fourier optics*). Effectively this means that the beam is considered as a superposition of plane waves, and the Fourier transform indicates the amplitudes and phases of all plane-wave components. For propagation in free space, only the phase values change; it is thus easy to calculate propagation over large distances in free space, or alternatively in a homogeneous optical medium.

The width, measured e.g. as the root-mean-squared (r.m.s.) width, of the spatial Fourier transform can be directly related to the beam divergence. This means that the beam divergence (and in fact the full beam propagation) can be calculated from the transverse complex amplitude profile of the beam at *any* one position along the beam axis, assuming that the beam propagates in an optically homogeneous medium (e.g. in air).

## Measurement of Beam Divergence

For the measurement of beam divergence, one usually measures the *beam caustic*, i.e., the beam radius at different positions, using e.g. a beam profiler.

It is also possible to derive the beam divergence from the complex amplitude profile of the beam in a single plane, as described above. Such data can be obtained e.g. with a *Shack–Hartmann wavefront sensor*.

One may also simply measure the beam intensity profile at a location far away from the beam waist, where the beam radius is much larger than its value at the beam waist. The beam divergence angle may then be approximated by the measured beam radius divided by the distance from the beam waist.

See also: beam radius, laser beams, collimated beams, beam parameter product, beam quality, beam pointing fluctuations, beam profilers, free-space optical communications, spotlight 2007-07-11

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics AG. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

## Questions and Comments from Users

2020-05-21

What are some typical beam divergence values used for long distance satellite communications?

The author's answer:

That differs between different usage scenarios. For example, if you want to reach a satellite with a sender on Earth, you can use a relatively large optical system, e.g. a with a 1-m diameter, and then achieve a correspondingly small beam divergence of the order of a microradian. For optical communications between different satellites, or for the backlink of a satellite, you typically need to use smaller optics and thus have a correspondingly larger beam divergence.

The size of the required optics is often the limiting factor. Another factor may be the highly precise orientation required for working with low-divergence beams.

2020-08-26

Does the divergence depend on the medium, for example air vs. glass (no fiber optic)?

The author's answer:

Yes. A beam with a certain initial focus will expand faster in air than in glass, for example. Its wavelength is shorter in glass.

2021-03-04

What is the best way of accurately measuring the beam divergence of a large laser beam – for example a beam with w0 = 5 cm and around 120 µrad divergence (half angle) at 1500 nm?

The author's answer:

A direct measurement is of course difficult, since you would need to use a very large propagation distance. (The Rayleigh length would be 5.24 km.) However, I am afraid that any other approach, introducing some additional optics (e.g. a telescope or an interferometer) would introduce additional uncertainties. Therefore, it may be necessary to use that direct approach, somehow realizing that huge propagation distance of e.g. more than 15 km.

One might also think about where exactly and increased divergence could hurt you in your specific application. Maybe that would give you some useful hints.

2021-06-19

If I use an aperture of 15 mm × 10 mm (v × h) at the output coupler of an excimer having a 3 mrad × 1 mrad beam divergence, how big is the beam at a distance of 4 meters?

The author's answer:

A rough estimate: (3 mrad × 1 mrad) · 4 m = 12 mm × 4 mm for a tiny aperture; adding the initial aperture size, we arrive at 27 mm × 14 mm.

2021-12-20

If I know the beam diameter at two points, can I compute the beam waist from those measurements alone? For example if the beam diameter is 1.06 mm at 62 cm from the laser head and 1.54 mm at 90 cm can I compute the w0?

The author's answer:

You can estimate it that way, particularly if you also know the position of the beam waist (e.g. that it must be at a flat output coupler mirror). It would be better, however, to use more data points, reducing the impact of measurement errors.

2022-02-20

Is there a simple relationship between r.m.s. phase error (compared to flat) and M^{2}, divergence, BPP, or times diffraction-limited?

The author's answer:

No, it is not just the r.m.s. phase excursion because it also matters a lot how quickly the optical phase varies across the beam.

2022-03-16

How do I calculate the beam divergence of diode laser if I know the beam intensity profile at two different points separated by 50 cm, for example?

The author's answer:

Provided that the beam focus is outside these two points, and that the beam diameter at the focus is much smaller than at those points, you can calculate the beam divergence angle as the difference of the beam radius divided by the distance of 50 cm.

2022-03-26

How can I calculate the beam divergence from the beams size at the laser output and on the target at a known distance?

The author's answer:

If the beam radius at the target is much larger than at the laser output, the half-angle divergence is just the beam radius at the target divided by the distance.

2022-04-23

Would it be possible to estimate the M^{2} factor of the beam emitted from a laser diode by using its FWHM divergence in combination with its emitting area dimensions?

The author's answer:

Yes, for a rough estimate that can be used, although the exact beam radius will not be known and that kind of divergence value is not the variance-based value which is needed for <$M^2$>.

2022-06-14

Knowing the beam size at far field and the divergence angle, how to calculate the beam size at the laser exit?

The author's answer:

That can be done if you know the beam quality factor (easily derived from the equation in the article): <$w_0 = M^2 \lambda / (\pi \theta )$>.

2022-08-02

If I know the beam size at 2 different point such as at 1200 mm and 2 km, and the wavelength is also known, can I calculate the divergence and <$M^2$>? I am getting <$M^2 < 1$>.

The author's answer:

At least for the <$M^2$> calculation, you need the beam waist radius at the focus, which you cannot get from those data alone. Getting the divergence as such should be possible, assuming that the beam monotonously expands between the two points.

2022-08-16

Do you have experience with a beam which has different divergence before and after the waist in a homogeneous medium?

The author's answer:

No, and I think this is not possible. At least not for the common definition of beam divergence based on the D4σ method.

2022-11-07

As I am reading the section on spatial Fourier transforms, I had a thought: is the width of the spatial Fourier transform related to the linewidth of the laser? I.e., do higher Q resonators yield better beam quality? Or am I mixing things up here?

The author's answer:

The linewidth is related to a *temporal* Fourier transform – that's really a different issue.

2022-11-17

How can I calculate the beam divergence for a multimode fiber?

The author's answer:

That depends on the launch conditions, but should in any case not be substantially larger than its numerical aperture.

2023-01-30

I have a fiber coupled light-source with 105/125 μm fiber. I know that I wish to achieve a Beam Divergence of 1 × 1 mrad. What should be the fiber NA in order to get the Beam Divergence of 1 ×1 mrad?

The author's answer:

In order to get such a low divergence of light directly from the fiber, you would need to have an extremely small numerical aperture – far outside the practical range. The solution must therefore be different: using a suitable lens behind the fiber end.

2023-04-27

How to calculate the divergence angle of a Lorentzian beam?

What is the beam shape with the minimum divergence?

The author's answer:

You can calculate the far field distribution essentially by applying a Fourier transform to the Lorentzian shape. Then you can calculate the divergence angle, based on some chosen criterion (e.g. FWHM).

The answer to the second question depends on the beam divergence criterion. For example, when using second-moment based width definitions, a Gaussian shape leads to minimum divergence for a given beam waist diameter.

2024-01-22

I have the info of wavelength, beam quality in mm · mrad and minimum laser light cable diameter for laser cutting machine. May I suppose the minimum laser light cable diameter is the beam waist radius?

The author's answer:

Yes, that way you can estimate the beam divergence obtained when focusing such that you get into the fiber. Compare that with the numerical aperture to check whether the launch efficiency can be high.

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2020-04-16

If I know the emitting area (2 μm × 1 μm) of a laser diode; how to calculate the divergence angle for that?

The author's answer:

From the emitting area alone you cannot do that; the transverse shape of the complex amplitude profile is also relevant. For a rough estimate, you may assume a Gaussian beam with <$w$> = 1 μm for the wider direction, for example, assuming a single-mode profile.