# M squared

In laser science, the parameter **M ^{2}**, also known as the

**beam quality factor**, 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 TEM

_{00}(Gaussian) laser beam, M

^{2}is exactly one.

The M^{2} 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]

## Utility

M^{2} 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 M^{2} value far from unity.[1] Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M^{2} value close to unity.

The value of M^{2} is determined by measuring D4σ or "second moment" width. Unlike the beam parameter product, M^{2} is unitless and does not vary with wavelength.

## 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 M^{2} times the beam area but 1/M^{2} 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.

## Applications

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.

M^{2} determines how tightly a collimated beam of a given diameter can be focused: the diameter of the focal spot varies as M^{2}, and the irradiance scales as 1/M^{4}. For a given laser cavity, the output beam diameter (collimated or focused) scales as M, and the irradiance as 1/M^{2}. This is very important in laser machining and laser welding, which depend on high fluence at the weld location.

Generally, M^{2} 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.

## 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
- "Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios". ISO Standard.
**11146**. 2005. Cite journal requires`|journal=`

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