Boundary behavior of modes of a Dirichlet Laplacian 
by  Dmitrii KOUZNETSOV and Jerome V. MOLONEY
Abstract.
The problem of uncoupled modes of pump in the double-clad fiber amplifier
is analyzed. While such modes avoid the core of the fiber,
we consider the Dirichlet Laplacian problem which neglects the core.
In this approach, boundary of the cladding is treated as an ideal mirror.
The laws of conservation established for the paraxial equation of diffraction
give certain integral relations for the derivatives of modes at the 
boundary. Such relations are formulated as theorems.
These theorems show the ways of non-traditional design of single-mode 
fiber amplifiers with multimode pump. In particular, the conservation of 
momentum can be applied in the design of the slab-pumped fiber amplifier.
The conservation of angular momentum predicts high efficiency of coupling 
of pump into a doped core embedded in a	spiral-shaped double-clad fiber.
Such predictions agree with results of numerical experiments published 
recently and allows the geometric optics interpretation. The theorems 
suggested give the keys to the novel design of fiber amplifiers.

%\pacs{02.30.Jr,03.65.Ge 42.55.Wd 42.81.-i 43.40.Dx 46.70.Hg. }
%\keywords{Dirichlet Laplacian, eigenfunctions, quantum biliard}