The commonly used definition of beam quality includes far-field spot radius, far-field divergence angle, diffraction limit multiple U, Strehl ratio, factor M2 , power on target surface or loop energy ratio, etc.

Beam quality is an important parameter of laser. Two common expressions of beam quality are BPP and M2 which are derived based on the same physical concept and can be converted from each other. Laser beam quality is important because it is a key physical quantity to judge whether the laser is good or not and whether the precision processing can be carried out. For many kinds of single-mode output lasers, high-quality lasers usually have very high beam quality, corresponding to a very small M2, such as 1.05 or 1.1. Moreover, the laser can maintain good beam quality throughout its service life, and M2 value is almost unchanged. For laser precision machining, high quality beam is more conducive to shaping, so as to carry out flat top laser machining without damaging the substrate and without thermal effect. In practice, M2 is mostly used for solid and gas lasers, while BPP is mostly used for fiber lasers when labeling the specifications of lasers.

Laser beam quality is usually expressed by two parameters: BPP and M². M²is often written as M2. The following figure shows the longitudinal distribution of the Gaussian beam, where W is the beam waist radius and θ is the far-field divergence half angle.

Conversion of BPP and M2

BPP (Beam Parameter Product) is defined as waist radius W multiplied by far-field divergence half angle θ:

         BPP = W × θ

The far-field divergence half angle θ of Gaussian beam is:

        θ₀ = λ / πW₀

M² is the ratio of the beam parameter product to the beam parameter product of the fundamental mode Gaussian beam:

        M² =（W×θ）/（W₀×θ₀）= BPP /（λ / π）

It is not difficult to find from the above formula that BPP is independent of wavelength, while M² is also not related to laser wavelength. They are mainly related to the cavity design and assembly accuracy of the laser.

The value of M² is infinitely close to 1, indicating the ratio between the real data and the ideal data. When the real data is closer to the ideal data, the beam quality is better, that is, when M² is closer to 1, the corresponding divergence angle is smaller, and the beam quality is better.

Measurement of BPP and M2
Beam quality analyzer can be used to measure the beam quality. Beam quality also can be measured by using light analyzer with complex operation. Data are collected from different locations of the laser cross section and then are synthesized by a build-in program to produce M2. M2 can not be measured if there is misoperation or measurement error in the process of sampling. For high power measurements, sophisticated attenuation system is needed to keep the laser power within a measurable range and avoid any damage of the instrument detection surface.

The optical fiber core and numerical aperture can be estimated according to the figure above. For fiber lasers, the waist radius ω₀= fiber core diameter /2 = R, θ = sinα =α= NA (numerical aperture of fiber).

Summary of BPP, M2, and Beam Quality

The smaller BPP, the better laser beam quality.

For 1.08µm fiber lasers, M2 = 1, BPP = λ / π = 0.344 mm mrad

For 10.6µm CO₂ lasers, single fundamental mode M2 = 1, BPP = 3.38 mm mrad

Assuming that the divergence angles of two single fundamental mode lasers (or multi-mode lasers with same M2) are the same after focusing, the focal diameter of the CO₂ laser is 10 times that of the fiber laser.

The closer M2 is to 1, the better the laser beam quality is.

When the laser beam is in Gaussian distribution or near Gaussian distribution, the closer the M2 is to 1, the closer the actual laser is to the ideal Gaussian laser, the better the beam quality is.