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# M squared

In laser science, the parameter **M²** is the ratio of the beam parameter product (BPP) of an actual beam to that of an ideal Gaussian beam at the same wavelength. It is often referred to as the **beam quality factor**, since its value can be used to quantify the degree of variation the actual beam is from such an ideal beam. M² is a better guide to beam quality than Gaussian appearance, however, since there are many cases in which the 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. For a single mode TEM_{00} Gaussian laser beam, M² is exactly one.

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.

The quality of a beam is important for many applications. In fiber-optic communications beams with an M^{2} close to 1 are required for coupling to single-mode optical fiber. Laser machine shops care a lot about the M^{2} parameter of their lasers because the beams will focus to an area that is M^{2} times larger than that of a Gaussian beam with the same wavelength; in other words, the fluence scales as 1/M^{2}. The general rule of thumb is that M^{2} increases as the laser power increases. It is difficult to obtain excellent beam quality and high average power (100 W to kWs) due to thermal lensing in the laser gain medium.

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]} It is especially useful for determining the degree of beam divergence of real laser beams and the minimum focussed spot size.

The value of M² can be determined by measuring D4σ or "second moment" width. Unlike the beam parameter product, M² is unitless and has no variance with wavelength.

## See also

## References

**^**Siegman,, A. E. (October 1997). "How to (Maybe) Measure Laser Beam Quality" (PDF). Archived from the original (PDF) on June 4, 2011. Retrieved Feb 8, 2009. Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California**^**"ISO Standard 11146, Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios". 2005.