placed there. One example of modes is shown in the figure below and to the right. Although some of modes show scarring and wide voids, none of these voids cover the core. The property of DCFs with spiral-shaped cladding can be interpreted as conservation of angular momentum. The square of the derivative of a mode at the boundary can be interpreted as pressure. Modes (as well as rays) touching the spiral-shaped boundary transfer some angular momentum to it. This transfer of angular momentum should be compensated by pressure at the chunk. Therefore, no one mode can avoid the chunk. Modes can show strong scarring along the classical trajectories (rays) and wide voids, but at least one of scars should approach the chunk to compensate for the angular momentum transferred by the spiral part. The interpretation in terms of angular momentum indicates the optimum size of the chunk. There is no reason to make the chunk larger than the core; a large chunk would not localize the scars sufficiently to provide coupling with the core. There is no reason to localize the scars within an angle smaller than the core: the small derivative to the radius makes the manufacturing less robust; the larger is, the larger the fluctuations of shape that are allowed without breaking the condition . Therefore, the size of the chunk should be of the same order as the size of the core. More rigorously, the property of the spiral-shaped domain follows from the theorem about boundary behavior of modes of the Dirichlet Laplacian. Although this theorem is formulated for the core-less domain, it prohibits the modes avoiding the core. A mode avoiding the core, then, should be similar to that of the core-less domain. Stochastic optimization of the cladding shape confirms that an almost-circular spiral realizes the best coupling of pump into the core. Tapered Tapered double-clad fiber (T-DCF) is a fiber whose outer and inner claddings and core diameters vary smoothly with length. The core at narrow side of T-DCF supports propagation of the fundamental mode only, whereas at the wide side the core is able to guide many modes. However, it was shown experimentally that light launched into the narrow end of a T-DCF propagates into the wide core without any changes of mode content. As a result, at the wide (substantially multimode) end of T-DCF light propagates only in the lowest-order mode with excellent