In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can be focused to a small spot. A Gaussian beam has the lowest possible BPP, , where is the wavelength of the light. The ratio of the BPP of an actual beam to that of an ideal Gaussian beam at the same wavelength is denoted M2 ("M squared"). This parameter is a wavelength-independent measure of beam quality. 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. Laser machine shops care a lot about the M2 parameter of their lasers because the beams will focus to an area that is M4 times larger than that of a Gaussian beam with the same wavelength and D4σ waist width; in other words, the fluence scales as 1/M4. The rule of thumb is that M2 increases as the laser power increases. It is difficult to obtain excellent beam quality and high average power (100 W to kWs) due to thermal lensing in the laser gain medium. Measurement There are several ways to define the width of a beam. When measuring the beam parameter product and M2, one uses the D4σ or "second moment" width of the beam to determine both the radius of the beam's waist and the divergence in the far field. The BPP can be easily measured by placing an array detector or scanning-slit profiler at multiple positions within the beam after focusing it with a lens of high optical quality and known focal length. To properly obtain the BPP and M2 the following steps must be followed: Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest). Measure the D4σ widths at 5 axial positions at least one Rayleigh length away from the waist. Fit the 10 measured data points to , where and is the second moment of the distribution in the x or y direction (see section on D4σ beam width), and is the location of the beam waist with second moment width of . Fitting the 10 data points yields M2, , and . Siegman showed