*M*^{2} Factor

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a parameter for quantifying the beam quality of laser beams

Alternative term: beam quality factor

Categories: general optics, optical metrology

Units: (dimensionless)

Formula symbol: <$M^2$>

The *M*^{2} 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

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”.

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 TEM_{nm} resonator mode, has an <$M^2$> factor of <$(2n + 1)$> in the <$x$> direction, and <$(2m + 1)$> in the <$y$> direction [1].

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.

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*.

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.

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.

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.

## Errors in <$M^2$> Measurements

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:

- The beam radius is measured with a simple criterion, not based on the full intensity profile, although the beam profiles are not all close to Gaussian. Only for nearly Gaussian beam shapes, such simple measurement methods are allowed. For others, the D4σ method based on the second moment of the intensity distribution must be used.
- The beam is focused too tightly, so that the beam waist is too small to measure its beam radius precisely. For example, a CCD camera has a limited spatial resolution; it cannot be used for precise measurements if the beam diameter corresponds only to a few pixels.
- Background subtraction, a sensitive issue for the second-moment method, is not correctly done. Camera images can exhibit some background intensity level, which may either really be belonging to the laser beam (and should not be removed then) or is an artifact which must be removed. If such a background results from ambient light, the most reliable measure is to switch this off, or to carefully shield it with a black tube in front of the camera. (Subtracting a fixed level for all images is problematic since ambient light levels may change, e.g. when somebody moves in the room.) The background issue is particularly serious when the beam size is only a fraction of the camera's sensitive area.
- The beam intensity on a camera is too high or too low. If it is too high, the center pixels may be saturated, so that the beam intensity at the center is underestimated and the measured beam radius is too large. For too low intensities, intensity background issues may become more severe.
- The beam radii are not measured sufficiently far from the focus. In order to properly judge the beam divergence, the ISO 11146 standard demands that about half of the measurement points must be more than two effective Rayleigh lengths away from the beam focus (whereas the other half of the points is close to the focus, i.e., within one Rayleigh length). This may be difficult in practice when the beam waist is made relatively large, leading to a long Rayleigh length and correspondingly large space requirements for a correct measurement.

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

## Calculation of <$M^2$> Factor From Complex Field Distribution in a Plane

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].

## More to Learn

Encyclopedia articles:

Blog articles:

- The Photonics Spotlight 2007-06-11: “Beam Quality Measurements Can Easily Go Wrong”

### Bibliography

[1] | A. E. Siegman, “New developments in laser resonators”, Proc. SPIE 1224, 2 (1990); https://doi.org/10.1117/12.18425 |

[2] | A. E. Siegman, “Defining, measuring, and optimizing laser beam quality”, Proc. SPIE 1868, 2 (1993); https://doi.org/10.1117/12.150601 |

[3] | H. Yoda, P. Polynkin and M. Mansuripur, “Beam quality factor of higher order modes in a step-index fiber”, J. Lightwave Technol. 24 (3), 1350 (2006); https://doi.org/10.1109/JLT.2005.863337 |

[4] | X. Luo et al., “Power content M^{2}-values smaller than one”, Appl. Phys. B 98 (1), 181 (2010); https://doi.org/10.1007/s00340-009-3623-8 |

[5] | C. M. Mabena et al., “Beam quality factor of aberrated Laguerre–Gaussian optical beams”, Opt. Express 31 (16), 26435 (2023); https://doi.org/10.1364/OE.493594 |

[6] | 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

2022-04-07

How to take into account the M^{2} factor in simulations in Zemax or other software?

The author's answer:

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.

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 <$M^2$>.

2022-05-18

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)?

The author's answer:

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.

2023-06-09

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.

The author's answer:

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.

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.

2023-06-23

You wrote in a previous answer that the M^{2} 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 M^{2}?

The author's answer:

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.

2023-07-07

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

The author's answer:

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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2021-06-07

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?

The author's answer:

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.