Dirichlet Laplacian

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Dirichlet Laplacian refers to the mathematical problems with the Helmholtz equation

~(\Delta + \lambda) \Psi =0 ~

where Δ is the Laplace operator; in the two-dimensional space,

\Delta= \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}

differentiates with respect to coordinates ~x and ~y~; ~\lambda is a real number (called eigenvalue), and ~\Psi is function of these coordinates.

The additional equation ~\Psi=0~ at the boundary of some domain corresponds to the Dirichlet boundary condition.

Fig.1. Spiral-shaped boundary of the domain (blue), its chunk (red), and 3 segments of a ray (green).
Fig.1. Spiral-shaped boundary of the domain (blue), its chunk (red), and 3 segments of a ray (green).

The Dirichlet Laplacian may arize from various problems of mathematical physics; it may refer to modes of at idealized drum, small waves at the surface of an idealized pool, as well as to a mode of an idealized optical fiber in the paraxial approximation. The last application is most practical in connection to the double-clad fibers; in such fibers, it is important, that most of modes of the fill the domain uniformly, or the most of rays cross the core. The poorest chape seems to be the circularly-symmetric cdomain[1][2] ,[3]. The modes of pump should not avoid the active core used in double-clad fiber amplifiers. The spiral-shaped somain happens to be especially efficient for such an application due to the boundary behavior of modes of Dirichlet laplacian.[4]

The therorem about boundary behavior of the Dirichlet Laplacian if analogy of the property of rays in geometrical optics (Fig.1); the angular momentum of a ray (green) increases at each reflection from the spiral part of the boundary (blue), until the ray hits the chunk (red); all rays (except those parallel to the optical axis) unavoidly visit the region in vicinity of the chunk to frop the excess of the angular momentum. Similarly, all the modes of the Dirichlet Laplacian have non-sero values in vicinity of the chunk. The normal component of the derrivative of the mode at the boundary can be interpreted as pressure; the pressure integrated over the surface gives the force. As the mode is steady-state solution of the propagation equation (with trivial dependence of the lingitudinal coordinate), the total force should be zero. Similarly, the angular momentum of the force of pressure should be also zero. However, there exist the formal proof, which does not refer to the analogy with physical system.[4]

[edit] References

  1. ^ S. Bedo; W. Luthy, and H. P. Weber (1993). "The effective absorption coefficient in double-clad fibers". Optics Communications 99: 331–335. doi:10.1016/0030-4018(93)90338-6. 
  2. ^ Leproux, P.; S. Fevrier, V. Doya, P. Roy, and D. Pagnoux (2003). "Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump". Optical Fiber Technology 7 (4): 324–339. doi:10.1006/ofte.2001.0361. 
  3. ^ A. Liu; K. Ueda (1996). "The absorption characteristics of circular, offset, and rectangular double-clad fibers". Optics Communications 132: 511–518. doi:10.1016/0030-4018(96)00368-9. 
  4. ^ a b Kouznetsov, D.; Moloney, J.V. (2004). "Boundary behavior of modes of Dirichlet laplacian". Journal of Modern Optics 51 (13): 1955–1962.