Is there a simple relationship between r.m.s. phase error (compared to flat) and M2, divergence, BPP, or times diffraction-limited?

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?

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.

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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 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).

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.

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.

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.

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.

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?

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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.

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?

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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.

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.

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?

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.

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Would it be possible to estimate the M2 factor of the beam emitted from a laser diode by using its FWHM divergence in combination with its emitting area dimensions?

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.

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 <$M^2 < 1$>.

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.

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.

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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.

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.

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

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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.

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?

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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.)

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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.

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The amount of prism my doctor gave me was sort of a best guess. I only need the prism correction for near vision but for distance I'm fine.

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).

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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.

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 M2 factor, the divergence half-angle is

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?

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

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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.

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 <$M^2$>.

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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.

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.

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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.

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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.

The Siemens Star is a radial starburst pattern commonly used to adjust internal focus settings in camera lens and for measuring camera resolution. This notebook produces a Siemens Star test chart in the form in [Scalable Vector Graphics (svg)[(https://en.wikipedia.org/wiki/Scalable_Vector_Graphics) that can be rendered by most modern web browsers. The idea for this notebook was motivated by this post on a forum at Digital Photography Review (DPReview).

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°.

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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!