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In laser science, the parameter **M ^{2}**, 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 TEM

_{00}(Gaussian) laser beam, M

^{2}is exactly one. Unlike the beam parameter product, M

^{2}is unitless and does not vary with wavelength.

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]}

## Measurement

There are several ways to define the width of a beam. When measuring the beam parameter product and M^{2}, 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]}

M^{2} can be measured by placing an 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 M^{2} the following steps must be followed:^{[3]}

- Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest).
- Measure the D4σ widths at 5 axial positions at least one Rayleigh length away from the waist.
- Fit the 10 measured data points to ,
^{[4]}

- Here is half of the beam width and is the location of the beam waist with width . Fitting the 10 data points yields M
^{2}, , and . 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.

- Here is half of the beam width and is the location of the beam waist with width . Fitting the 10 data points yields M

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 M^{2}. The procedure above gives a more accurate result in practice, however.

## 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 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.

## 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.

## See also

## References

- ^
^{a}^{b}^{c}^{d}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. CS1 maint: discouraged parameter (link) 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=`

(help)**^**ISO 11146-1:2005(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams."**^**See Siegman (1997), p. 9. There is a typo in the equation on page 3. Correct form comes from equations on page 9.