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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 23, Issue 6


Regularization strategy for determining laser beam quality parameters

Teresa Regińska / Kazimierz Regiński
Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/jiip-2014-0084


A simplified model of a laser beam leads to an ill-posed Cauchy problem for the Helmholtz equation on an infinite strip. In the case of large wave numbers, the problem corresponds to an operator equation with an unbounded operator. The first problem considered concerns optimality of a spectral type regularization method for reconstructing the radiation field from measurements given only on a part of the boundary. The optimal order of convergence, previously known for particular cases, is proved for an arbitrary wave number and for nonzero Dirichlet and Neumann conditions under a priori and a posteriori choice of a regularization parameter. In the second part, the regularized solutions are employed to describe the geometrical properties of the beam and, in particular, to find approximate beam quality parameters such as the waist of the axial profile of the beam and its position. The mathematical considerations are preceded by a section where the main notions of laser beam optics are introduced and briefly explained.

Keywords: Cauchy problem for the Helmholtz equation; regularization; optimal convergence rate; discrepancy principle; axial profile of laser beam; smoothing and curve fitting

MSC: 35R25; 65D10; 65M30; 78A60

About the article

Received: 2014-11-27

Revised: 2015-02-18

Accepted: 2015-03-03

Published Online: 2015-05-21

Published in Print: 2015-12-01

Funding Source: Polish Ministry of Science and Higher Education

Award identifier / Grant number: N N515 524938

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 657–671, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0084.

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