Characterising the quality of a laser beam and other optics in an optical system is critical when looking to understand the overall performance of the system.

There are a variety of methods that can be used to determine the performance of the laser, such as the M^{2} factor, the beam parameter product, and power in the bucket. The shape of the beam itself must also be considered. When analysing the overall optical system as well as individual components, as individual components, the Strehl ratio is used to compare its actual vs ideal performance. Using these methods gives a comprehensive understanding of the real performance of a laser system and can be used to predict the final performance of the system.

**The M**^{2} factor

^{2}factor

One of the most common ways to characterise beam quality is with the M^{2 }factor. This quantity describes the beam quality by comparing the shape of the beam to an ideal Gaussian beam. According to the ISO Standard 11146, the M^{2 }factor is dependent on the beam waist (w0), the divergence angle of the laser (θ), and the lasing wavelength (λ), such that:^{1}

Additionally, consider the divergence angle of a gaussian beam:

It is possible to show that the M^{2 }factor for a diffraction-limited gaussian beam is equal to 1 by substituting in for θ in equation 1. The equation simplifies as follows:

When applying this to the real-world performance of a laser, this value must be greater than or equal to 1 as M^{2 }values smaller than this are not possible. Lower M^{2} factors more efficiently utilise the beam’s power with their tighter focus. The M^{2} factor is also used when approximating the radius of the propagating beam.^{2}

When looking to use the M^{2} factor to understand the real-world beam quality of a laser, ISO 11146 required five beam radius measurements at varying positions on the optical axis in both the near and far field of the beam. The actual beam radius (w(z)) is related to the wavelength (λ), beam waist (w0) and M^{2} factor as follows:^{3}

**Power in the bucket**

The power in the bucket (PIB) technique for characterising beam quality is most frequently used with high-power laser systems. It is calculated by integrating the laser power over a specific “bucket”, which refers to a particular spot on the material’s surface being processed. The specifications of the ideal nearfield bucket shape are critical for making accurate comparisons to ideal scenarios. Additionally, the far-field bucket shape must be well defined. PIB is commonly reported in terms of horizontal and vertical beam quality, as follows:^{4}

Beam parameter product When evaluating the quality of a laser beam, the beam parameter product (BPP) is a valuable metric frequently used to characterise fibre or semiconductor lasers with large M^{2} factors. The BPP is defined as the product of the beam radius at the beam waist and the half-angle beam divergence. It is related to the M^{2} factor through the following equation:

Since M^{2} must be ≥1, the minimum value for BPP would be equal to λ/π. Thus, a larger BPP equates to worse beam quality.

**Circular vs elliptical beams**

The shape of the beam plays a critical role in determining the overall performance of the laser system. Most commonly, a laser beam is either circular or elliptical. Different beam shapes have different effects on the laser system’s performance. For example, elliptical beams have a larger focused spot size than circular beams, resulting in overall lower irradiance. When characterising elliptical beams, the axis with the larger divergence is defined as the fast axis, while the smaller axis is defined as the slow axis. It is possible to change the shape of the laser beam using a variety of optics. When minor adjustments need to be made to the beam shape, a cylindrical lens can be used to either stretch or shrink the beam along the desired axis.

**The Strehl ratio**

The Strehl ratio is used to analyse the performance of an optic or optical system by comparing its true performance with its ideal performance. It is defined as the ratio of the actual maximum focal spot irradiance of the optic from a point source to the ideal maximum irradiance from a theoretical diffraction-limited optic. This ratio is approximately related to the RMS transmitted wavefront error such that^{5}:

Here, S represents the optic’s Strehl ratio, and σ is defined as the optic’s RMS wavefront error reported in waves. When S is equal to 1, the optic is said to be ideal and completely free of aberrations. However, real-world lenses with a Strehl ratio greater than or equal to 0.8 are considered to be “diffraction-limited”.

**References**

1. International Organization for Standardization. (2005). ‘Lasers and laser related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios’ (ISO 11146).

2. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology, RP Photonics, October 2017, www.rp-photonics.com/ encyclopedia.html.

3. Hofer, Lucas. ‘M² Measurement’. DataRay Inc., 12 Apr. 2016, www.dataray.com/blogm2-measurement.html.

4. Strehl, Karl W. A. ‘Theory of the telescope due to the diffraction of light,’ Leipzig, 1894.

5. Mahajan, Virendra N. ‘Strehl ratio for primary aberrations in terms of their aberration variance.’ JOSA 73.6 (1983): 860-861.

**More information**

For more from Edmund Optics on assessing beam quality and laser system performance