beam quality. Thus, tapered fiber is a unique and easy way to implement fundamental mode regime propagation (and amplification) in a multimode fiber. Filling factor The efficiency of absorption of pumping energy in the fiber is an important parameter of a double-clad fiber laser. In many cases this efficiency can be approximated with where is the cross-sectional area of the cladding is the radius of the core (which is taken to be circular) is the absorption coefficient of pump light in the core is the length of the double-clad fiber, and is a dimensionless adjusting parameter, which is sometimes called the "filling factor"; . The filling factor may depend on the initial distribution of the pump light, the shape of the cladding, and the position of the core within it. The exponential behavior of the efficiency of absorption of pump in the core is not obvious. One could expect that some modes of the cladding (or some rays) are better coupled to the core than others; therefore, the "true" dependence could be a combination of several exponentials. Only comparison with simulations justifies this approximation, as shown in the figure above and to the right. In particular, this approximation does not work for circular fibers, see the initial work by Bedo et al., cited below. For chaotic fibers, approaches unity. The value of can be estimated by numerical analysis with propagation of waves, expansion by modes or by geometrical optics ray tracing, and values 0.8 and 0.9 are only empirical adjusting parameters, which provide good agreement of the simple estimate with numerical simulations for two specific classes of double-clad fibers: circular offset and rectangular. Obviously, the simple estimate above fails when the offset parameter becomes small compared to the size of cladding. The filling factor approaches unity especially quickly in the spiral-shaped cladding, due to the special boundary behavior of the modes of the Dirichlet Laplacian. Designers of double-clad fiber look for a reasonable compromise between the optimized shape (for the efficient coupling of pump into the core) and the simplicity of the manufacturing of the preform used to draw the fibers. The power scaling of a fiber laser is limited by unwanted nonlinear effects such as stimulated Brillouin scattering and stimulated Raman scattering. These effects are minimized when the fiber laser is short. For efficient operation, however, the pump should be absorbed in the core along the short length; the