Dirichlet Laplacian

Dirichlet Laplacian

Dirichlet Laplacian refers to the mathematical problems with the Helmholtz equation

~(Delta + lambda) Psi =0 ~

where Delta 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.


200px|thumb|right|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 cdomaincite journal| author=S. Bedo|coauthors= W. Luthy, and H. P. Weber
title=The effective absorption coefficient in double-clad fibers
journal=Optics Communications
volume=99| issue=| pages=331–335| year=1993
url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-46JGTGD-M5&_user=10&_coverDate=06%2F15%2F1993&_alid=550903253&_rdoc=1&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c8a4c3ecc3d9a4e9ecb84f96cfef0333
doi=10.1016/0030-4018(93)90338-6] cite journal|title=Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump
author= Leproux, P.| coauthors=S. Fevrier, V. Doya, P. Roy, and D. Pagnoux| journal=Optical Fiber Technology| url=http://www.ingentaconnect.com/content/ap/of/2001/00000007/00000004/art00361
volume=7 | year=2003 | issue=4 | pages=324–339|doi=10.1006/ofte.2001.0361] ,cite journal
title=The absorption characteristics of circular, offset, and rectangular double-clad fibers
author=A. Liu| coauthors= K. Ueda| url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-497C4YV-BW&_user=10&_coverDate=12%2F15%2F1996&_alid=550869877&_rdoc=3&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=3&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=688bbca25fdd98e29caadb676b003c1e
journal=Optics Communications| volume=132| year=1996| pages= 511–518
doi=10.1016/0030-4018(96)00368-9] .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 theboundary behavior of modes of Dirichlet laplacian.cite journal
title=Boundary behavior of modes of Dirichlet laplacian
author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=Journal of Modern Optics
url=http://www.metapress.com/content/be0lua88cwybywnl/?p=5464d03ba7e7440f9827207df673c804&pi=6
free=http://www.ils.uec.ac.jp/~dima/TMOP102136.pdf
volume=51 | year=2004 | issue=13 | pages=1955–1962]

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 thechunk (red); all rays (except those parallel to the optical axis) unavoidly visit the region in vicinity of the chunk to frop the excess of theangular momentum. Similarly, all the modes of the Dirichlet Laplacian have non-sero values in vicinity of the chunk. The normal component of the derrivativeof the mode at the boundary can be interpreted as pressure; the pressure integrated over the surface gives the force. As the mode is steady-statesolution 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.cite journal
title=Boundary behavior of modes of Dirichlet laplacian
author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=Journal of Modern Optics
url=http://www.metapress.com/content/be0lua88cwybywnl/?p=5464d03ba7e7440f9827207df673c804&pi=6
free=http://www.ils.uec.ac.jp/~dima/TMOP102136.pdf
volume=51 | year=2004 | issue=13 | pages=1955–1962]

References


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