In laser science, the parameter M², also known as the beam propagation ratio or beam quality factor is a measure of laser beam quality. It represents the degree of variation of a beam from an ideal Gaussian beam.^[1] It is calculated from the ratio of the beam parameter product (BPP) of the beam to that of a Gaussian beam with the same wavelength. It relates the beam divergence of a laser beam to the minimum focussed spot size that can be achieved. For a single mode TEM₀₀ (Gaussian) laser beam, M² is exactly one. Unlike the beam parameter product, M² is unitless and does not vary with wavelength.

The M² value for a laser beam is widely used in the laser industry as a specification, and its method of measurement is regulated as an ISO standard.^[2]

Measurement

There are several ways to define the width of a beam. When measuring the beam parameter product and M², one uses the D4σ or "second moment" width of the beam to determine both the radius of the beam's waist and the divergence in the far field.^[1]

M² can be measured by placing an array detector or scanning-slit profiler at multiple positions within the beam after focusing it with a lens of high optical quality and known focal length. To properly obtain M², the following steps must be followed:^[3]

1. Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest).
2. Measure the D4σ widths at 5 axial positions at least one Rayleigh length away from the waist.
3. Fit the 10 measured data points to
   $${\displaystyle W^{2}(z)=W_{0}^{2}+M^{4}\left({\frac {\lambda }{\pi W_{0))}\right)^{2}(z-z_{0})^{2},}$$

   
   where ${\displaystyle W(z)}$ is half of the ${\displaystyle {\text{D4))\sigma (z)}$ beam width, and ${\displaystyle z_{0))$ is the location of the beam waist with width ${\displaystyle W_{0))$.^[4] Fitting the 10 data points yields M², ${\displaystyle z_{0))$, and ${\displaystyle W_{0))$. Siegman showed that all beam profiles — Gaussian, flat-top, TEMxy, or any shape — must follow the equation above provided that the beam radius uses the D4σ definition of the beam width.^[1] Using other definitions of beam width does not work.

In principle, one could use a single measurement at the waist to obtain the waist diameter, a single measurement in the far field to obtain the divergence, and then use these to calculate the M². The procedure above gives a more accurate result in practice, however.

Utility

M² is useful because it reflects how well a collimated laser beam can be focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to beam quality than Gaussian appearance because there are many cases in which a beam can look Gaussian, yet have an M² value far from unity.^[1] Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M² value close to unity.

The quality of a beam is important for many applications. In fiber-optic communications beams with an M² close to 1 are required for coupling to single-mode optical fiber.

M² determines how tightly a collimated beam of a given diameter can be focused: the diameter of the focal spot varies as M², and the irradiance scales as 1/M⁴. For a given laser cavity, the output beam diameter (collimated or focused) scales as M, and the irradiance as 1/M². This is very important in laser machining and laser welding, which depend on high fluence at the weld location.

Generally, M² increases as a laser's output power increases. It is difficult to obtain excellent beam quality and high average power at the same time due to thermal lensing in the laser gain medium.

Multi-mode beam propagation

Real laser beams are often non-Gaussian, being multi-mode or mixed-mode. Multi-mode beam propagation is often modeled by considering a so-called "embedded" Gaussian, whose beam waist is M times smaller than that of the multimode beam. The diameter of the multimode beam is then M times that of the embedded Gaussian beam everywhere, and the divergence is M times greater, but the wavefront curvature is the same. The multimode beam has M² times the beam area but 1/M² less beam intensity than the embedded beam. This holds true for any given optical system, and thus the minimum (focussed) spot size or beam waist of a multi-mode laser beam is M times the embedded Gaussian beam waist.