# What is M Squared? See What Determines the Size of Your Beam Waist

M² or Beam Propagation Ratio, is a value that indicates how close a laser is to being a single mode TEM00 beam, which in turn determines how small a beam waist can be focused. For the perfect Gaussian TEM00 condition the M² equals 1.

For a laser beam propagating through space, the equation for the divergence θ0 of an unfocused beam is given by:

θ0= M²4λ/ΠD0 (D0 is the waist diameter of the laser beam)

Figure 1. Characteristics of a laser beam as it passes through a focusing lens

For a pure Gaussian TEM00 beam M² equals 1, and thus has no impact on the calculation. The calculation of the minimal beam spot after the lens is then: d0 = 4λ/πθ (θ is the beam divergence after the lens) Again with M² equal to 1, the focused spot is diffraction limited. For real beams, M² will be greater than 1, and thus the minimum beam waist will be larger by the M² factor.

### How is M² measured?

M² cannot be determined from a single beam profile measurement. The ISO/DIS 11146 requires that M² be calculated from a series of measurements as shown in figure 1. M² is measured on real beams by focusing the beam with a fixed position lens of known focal length, and then measuring the characteristics of the artificially created beam waist and divergence.

To provide an accurate calculation of M², it is essential to make at least 5 measurements in the focused beam waist region, and at least 5 measurements in the far field, two Rayleigh ranges away from the waist area. The multiple measurements ensure that the minimum beam width is found. In addition, the multiple measurements enable a "curve fit" that improves the accuracy of the calculation by minimizing measurement error at any single point. An accurate calculation of M² is made by using the data from the multiple beam width measurements at known distances from a lens, coupled with the known characteristics of the focusing lens.

### M² Measurement Solutions

Ophir-Spiricon have a number of solutions for the measurement of M² ranging from simple manual processes to fully automated dedicated instruments, depending on the frequency of the need to measure M² of lasers and laser systems. We have a system that will meet most needs, whether for research and development of new laser systems, manufacturing quality assurance, or maintenance and service of existing systems.

Figure 2. Focusing laser beam

Use the focal spot size calculator to see how M-Squared affects the size of the beam waist.

### Have questions? Require assistance?

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