focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to beam quality than Gaussian appearance because there are many cases in which a beam can look Gaussian, yet have an M2 value far from unity. Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M2 value close to unity. The quality of a beam is important for many applications. In fiber-optic communications beams with an M2 close to 1 are required for coupling to single-mode optical fiber. M2 determines how tightly a collimated beam of a given diameter can be focused: the diameter of the focal spot varies as M2, and the irradiance scales as 1/M4. For a given laser cavity, the output beam diameter (collimated or focused) scales as M, and the irradiance as 1/M2. This is very important in laser machining and laser welding, which depend on high fluence at the weld location. Generally, M2 increases as a laser's output power increases. It is difficult to obtain excellent beam quality and high average power at the same time due to thermal lensing in the laser gain medium. Multi-mode beam propagation Real laser beams are often non-Gaussian, being multi-mode or mixed-mode. Multi-mode beam propagation is often modeled by considering a so-called "embedded" Gaussian, whose beam waist is M times smaller than that of the multimode beam. The diameter of the multimode beam is then M times that of the embedded Gaussian beam everywhere, and the divergence is M times greater, but the wavefront curvature is the same. The multimode beam has M2 times the beam area but 1/M2 less beam intensity than the embedded beam. This holds true for any given optical system, and thus the minimum (focussed) spot size or beam waist of a multi-mode laser beam is M times the embedded Gaussian beam waist. See also Laser beam profiler Strehl ratio References Laser science