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If the complex field distribution of a monochromatic field is known in one plane perpendicular to the beam direction, the field distribution in any other plane can be computed numerically, and the <$M^2$> could be obtained from that. As a technically simpler solution, one can directly compute <$M^2$> from the field distribution in one plane based on a few integrals [3].

Angleofdivergenceformula

If you know everything about the wavefront aberrations of the beam, how to pack all that information into <$M^2$>, which is just a simple number? That cannot work.

You seemed to expect that the shape of the far field should be similar to that of the focus, but that is not always the case. For a Gaussian beam, it would be so, but a Gaussian intensity profile alone doesn't tell whether it is indeed a Gaussian beam.

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You wrote in a previous answer that the M2 value is not enough to estimate the fiber coupling efficiency in a single mode fiber. Could you please elaborate a little? If I know that the intensity profile is indeed Gaussian and I know all its parameters, and I also know everything about the wavefront aberrations of the beam, shouldn't I be able in principle to calculate the coupling efficiency with the M2?

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Is it relevant if the beam is divergent or convergent right before the imaging lens? Could moving the measurement setup along the beam axis produce different results?

Beam divergence anglecalculator

Would it be possible to compute the fiber coupling efficiency into a monomode fiber of a beam with a given M² factor (assuming perfect lenses/imagery system with no aberrations)?

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Laserbeam divergenceand spot size

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?

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Beam divergenceof laser

In your case, the donut shape in the far field might result either from the mode shape of the SMF or from some distortions caused by the optical system.

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Unfortunately, essential details of the ISO 11146 standard are often not observed in <$M^2$> measurements, with the result that wrong <$M^2$> values are obtained or even published. Some frequently made mistakes in measurements based on the beam caustic are explained in the following:

Not really. The <$M^2$> factor does not contain sufficient information for that. At most, you may make some estimate, based on a certain assumption on the type of beam profile.

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The <$M^2$> factor of a laser beam limits the degree to which the beam can be focused for a given beam divergence angle, which is often limited by the numerical aperture of the focusing lens. Together with the optical power, the beam quality factor determines the brightness (more precisely, the radiance) of a laser beam.

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No, you would need the phase properties, not just the intensity profile. For example, the intensity profile could be nicely Gaussian despite a large <$M^2$> value.

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Beam divergence angleformula

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No, this does not matter as long as you can properly measure the beam waist ratio and the beam divergence. It is of course a good sanity check for measurements.

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For not circularly symmetric beams, the <$M^2$> factor can be different for two directions orthogonal to the beam axis and to each other. This is particularly the case for the output of diode bars, where the <$M^2$> factor is fairly low for the fast axis and much higher for the slow axis.

Gaussianbeam divergence angle

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 <$M^2$>.

Is it also possible to estimate <$M^2$> without knowing the divergence, from the intensity distribution in or immediately around the beam focus?

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When different instruments deliver different <$M^2$> values, this may easily be caused by such errors, rather than by the instruments themselves.

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Ask RP Photonics for any advice concerning laser beam quality, e.g. requirements for specific applications or measurement techniques. You may also get a training course for your team on issues like laser beams, beam quality and other aspects of lasers.

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A diffraction-limited beam has an <$M^2$> factor of 1, and is a Gaussian beam. Smaller values of <$M^2$> are physically not possible. A Hermite–Gaussian beam, related to a TEMnm resonator mode, has an <$M^2$> factor of <$(2n + 1)$> in the <$x$> direction, and <$(2m + 1)$> in the <$y$> direction [1].

Note that the <$M^2$> factor, being a single number, cannot be considered as a complete characterization of beam quality. The actual quality of a beam for a certain application can depend on details which are not captured with such a single number.

What isbeam divergence

According to ISO Standard 11146 [6], the <$M^2$> factor can be calculated from the measured evolution of the beam radius along the propagation direction (i.e. from the so-called caustic). See the article on beam quality for more details. A number of rules have to be observed, e.g. concerning the exact definition of the beam radius and details of the fitting procedure. Alternative methods are based on wavefront sensors, e.g. Shack–Hartmann wavefront sensors, which require the characterization of the beam only in a single plane.

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Lowdivergencelaser

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I am using an <$M^2$> system to measure the far-field divergence of a nearly-collimated beam. After the imaging lens, I am seeing a donut-shaped pattern before and after focus. The center of the beam is dark and there is one bright outer ring. At focus, the beam looks Gaussian. Do you know if this might be due to diffraction at the aperture or some characteristic of my collimating lens? The source is from a single-mode fiber, and the lens is aspheric.

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Detailed simulations are not possible without knowing the complete beam profile. The <$M^2$> factor alone says relatively little about the beam. Basically, you can only predict the evolution of beam radius, but not of the full beam profile. For certain purposes, however, one constructs a beam profile which has the required <$M^2$> factor and is hopefully somewhat representative of such profiles.

where <$w_0$> is the beam radius at the beam waist and <$\lambda$> the wavelength in the medium (e.g. air). A laser beam is often said to be “<$M^2$> times diffraction-limited”.

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The M2 factor (M squared factor), also called beam quality factor or beam propagation factor, is a common measure of the beam quality of a laser beam. According to ISO Standard 11146 [6], it is defined as the beam parameter product divided by <$\lambda / \pi$>, the latter being the beam parameter product for a diffraction-limited Gaussian beam with the same wavelength. In other words, the half-angle beam divergence is

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Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

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A collimated beam is light with weak divergence, meaning it's a flow of photonics that move in parallel to one another, without dispersing.

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The encyclopedia covers this topic well: with a long overview article on laser material processing and specialized articles on

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The concept of the <$M^2$> factor not only allows one to quantify the beam quality with a single number, but also to predict the evolution of the beam radius with a technically very simple extension of the Gaussian beam analysis: one simply has to replace the wavelength with <$M^2$> times the wavelength in all equations. This is very convenient for, e.g., designing the pump optics of diode-pumped lasers. Note, however, that this method works only when the D4σ method for obtaining the beam radius is used, which is suitable also for non-Gaussian beam shapes; see again ISO Standard 11146 [6] for details.

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