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M2 Factor

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

German: M2-Faktor

Categories: general optics, optical metrology

Formula symbol: M2

Units: (dimensionless)

How to cite the article; suggest additional literature

The M2 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 [4], it is defined as the beam parameter product divided by λ / π, 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

beam divergence for non-ideal laser beam

where w0 is the beam radius at the beam waist and λ the wavelength. A laser beam is often said to be “M2 times diffraction-limited”.

A diffraction-limited beam has an M2 factor of 1, and is a Gaussian beam. Smaller values of M2 are physically not possible. A Hermite–Gaussian beam, related to a TEMnm resonator mode, has an M2 factor of (2n + 1) in the x direction, and (2m + 1) in the y direction [1].

The M2 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.

Beam Quality Calculations

Center wavelength:
M2 factor: calc (from BPP)
Beam parameter product: calc (from M2)
Beam waist radius: calc (using the BPP)
Divergence half-angle: calc (using the BPP)

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

For not circularly symmetric beams, the M2 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 M2 factor is fairly low for the fast axis and much higher for the slow axis.

According to ISO Standard 11146 [4], the M2 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 sensors, which require the characterization of the beam only in a single plane.

Note that the M2 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 M2 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 M2 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 [4] for details.

Errors in M2 Measurements

Unfortunately, essential details of the ISO 11146 standard are often not observed in M2 measurements, with the result that wrong M2 values are obtained or even published. Some frequently made mistakes in measurements based on the beam caustic are explained in the following:

When different instruments deliver different M2 values, this may easily be caused by such errors, rather than by the instruments themselves.


[1]A. E. Siegman, “New developments in laser resonators”, Proc. SPIE 1224, 2 (1990)
[2]A. E. Siegman, “Defining, measuring, and optimizing laser beam quality”, Proc. SPIE 1868, 2 (1993)
 [3]X. Luo et al., “Power content M2-values smaller than one”, Appl. Phys. B 98 (1), 181 (2010)
[4]ISO Standard 11146, “Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios” (2005)

(Suggest additional literature!)

See also: beam quality, beam parameter product, beam divergence, brightness, Gaussian beams, Spotlight article 2007-06-11
and other articles in the categories general optics, optical metrology

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