Beam parameter product (BPP)

A lens can make a beam narrower only by making it diverge faster, and the product of the two, the waist radius $w_0$ times the far-field half angle $\theta$, stays fixed. That product is the beam parameter product,

and it measures how well a beam can be focused and delivered. It carries the same information as the beam quality factor $M^2$, but in absolute units, so beams at different wavelengths can be compared by how small a spot they make at a given working distance. Industrial laser specifications, particularly for fiber-delivered cutting and welding lasers, quote the BPP in mm·mrad.

Typical values

A diffraction-limited Gaussian beam ($M^2 = 1$) has a BPP of 0.339 mm·mrad at 1064 nm, 0.328 mm·mrad at 1030 nm and 3.37 mm·mrad at 10.6 µm, so a CO₂ laser with perfect beam quality still has ten times the BPP of a near-infrared solid-state or fiber laser. A beam leaving a multimode delivery fiber has a BPP of at most the core radius times its numerical aperture, reached when the beam fills the fiber: 11 mm·mrad for a 100 µm core with an NA of 0.22, equivalent to an $M^2$ of about 32 at 1070 nm. High-power diode laser stacks run from tens to hundreds of mm·mrad, which is why they are more often used to pump other lasers than to process materials directly.

Conventions

The BPP is normally defined with the waist radius and the half angle. Some datasheets use the diameter and the full angle, which multiplies the number by four; the unit is the same, so the convention must be checked before two values are compared. ISO 11146 defines both widths by second moments, which for a Gaussian coincide with the $1/e^2$ radius and half angle.

Conservation

The BPP of a beam cannot be reduced by lenses, mirrors or fibers without discarding power; it is the one-dimensional form of the conservation of étendue. A fiber or an aberrated lens can increase it, and so can combining beams side by side. Reducing it needs a lossy element, such as a spatial filter or an aperture, or a nonlinear or coherent process.

Measurement

The BPP is measured together with $M^2$ by recording the beam width at several positions through a focus and fitting the hyperbolic caustic, following ISO 11146; the procedure is described in the M² measurement article. For a beam leaving a fiber, the near-field spot on the fiber face and the far-field angle can be measured separately and multiplied.

References: ISO 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios; A. E. Siegman, Proc. SPIE 1224, 2 (1990).