As leaders in the laser beam profiling business, we have worked with *M²* for decades. In this blog post we discuss *M²*, when to use it, and the way it is measured. *M²* is a very useful measurement for a certain subset of laser beams—those that are predominately Gaussian—and determines how tightly an actual laser beam can be focused in comparison to a theoretically perfect Gaussian beam. Sometimes, customers wish to use *M²* as a laser beam metric, even when it isn't the most appropriate metric to use for their application. Even if *M²* isn't the right diagnostic metric for your beam, we would love to work with you and design a custom metric to assess the quality of your beam.

*M²*

*M²* is one of the most common metrics used in laser beam diagnostics. Although *M²* is listed as a beam quality metric, a more correct definition would be a beam propagation factor (Siegmann 1998). The strict definition of *M²* is

Where *ω* is the real beam's beam waist, *θ* is the real beam's divergence angle, *ω*_{0} is an ideal beam's beam waist and *θ*_{0} is the ideal beam's divergence angle. Essentially, *M²* determines how closely a beam matches a perfect Gaussian. A perfect Gaussian is said to have *M²* = 1, whereas for all other beams, the presence of transverse modes (other than TEM_{00}) cause *M²* ≥ 1.

If the *M²* value of a beam is greater than 3, then *M²* is most likely not the best metric with which to measure the quality of the beam (Ross 2013). *M²* is used to determine the modal composition of the beam and compare the beam to a theoretically perfect Gaussian. If the beam is not close to a Gaussian, a different metric should be used.

### Ideal Gaussian Beam and Real Beam Widths

A mathematical understanding of a laser beam's propagation is required to understand the *M²* measurement. The electric field for a Gaussian beam can be found by solving the wave equation. Solving the wave equation with Cartesian coordinates gives the Hermite-Gaussian modes, whereas solving the wave equation with polar coordinates provides the equations for the Laguerre-Gaussian modes. In most laser applications a Gaussian TEM_{00} beam (in which the equation for the Hermite-Gaussian mode and Laguerre-Gaussian mode is equal) is desired. The irradiance of the TEM_{00} mode is given by

Where *I*_{0} is the peak irradiance at the beam waist, *w*_{0} is the beam radius at the beam waist, *r* is the radius in the transverse direction and *w*(*z*) is the beam radius at the *z* position in the propagation plane. *w*(*z*) can further be described by

Where *z _{r}* is the Raleigh range given by

and *λ* is the beam's wavelength. However, for a real laser beam the Raleigh range is actually given by

So that the above equation can be rewritten

Where we have also added a displacement *z*_{0} from the origin along z axis. This equation is plotted for two different *M²* values in Fig. 2.

### Experimentally Determining *M²*

To accurately quantify *M²*, the radius of the beam must be measured at multiple locations along the axis of propagation. For an ISO 11146 compliant measurement, the beam radius must be measured at five z positions in the near field and five z positions in the far field. Once the radius has been measured at multiple locations, a fitting algorithm can be used to determine *ω*_{0}, *z*_{0}, and *M²* using the equation above. Different devices can be used to measure the beam radius along the axis of propagation. Below we list a few devices offered by DataRay; however, for an in-depth look at our *M²* measurement devices, see our recent *M²* datasheet.

#### Cameras with Stages

For the best *M²* determination, a CCD or CMOS camera should be used. A camera records the beam's full two dimensional intensity profile with the pixel array, whereas scanning slits and knife edge sensors only record the summation of intensity along the axes. Therefore, the camera sensor is better able to detect subtleties in the beam's composition and provides the best beam characterization. The ISO 11146 standard discusses camera based measurements in great depth (see Fig 1.). DataRay offers several cameras, which, in conjunction with a translation stage, give accurate and reliable *M²* measurements.

#### Scanning Slit Profilers with Stages

Some beams—such as very small beams and far infrared beams—cannot be accurately profiled with a camera sensor. In these applications we recommend a scanning slit profiler mounted on a translation stage. DataRay offers the Beam'R2 with M2DU stage (see Figure 4).

*M²*calculations with the BeamMap2.

#### Instantaneous Scanning Slit Profilers

Almost all *M²* measurement systems utilize a translation stage to take measurements along the axis of propagation. DataRay offers an instantaneous *M²* measurement system: the BeamMap2. The BeamMap2 utilizes a rotating puck with slits for measuring the beam. However, the slits are placed at different locations in the z plane so that the beam radius is measured at multiple z planes in real time. An in depth explanation of the BeamMap2 and its real-time *M²* measurement can be found here.

### Conclusion

*M²* is an excellent way to determine the quality of a near-Gaussian beam. We've discussed the theory behind *M²* measurements in this blog post and outlined DataRay's systems for measuring *M²*. In future blog posts we will discuss important experimental considerations when measuring *M²*. If you have further questions on *M²*, or its measurement, feel free to contact us. Additionally, to see the full line of our beam profiling products visit our website.

### References

A.E. Siegman, "How to (Maybe) Measure Laser Beam Quality," in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Ross, T. Sean., and Society Of Photo-optical Instrumentation Engineers. Laser Beam Quality Metrics. SPIE / International Society for Optical Engineering, 2013. Print.