Regularization strategy for determining laser beam quality parameters

• Published in: Journal of inverse and ill-posed problems Vol. 23; no. 6; pp. 657 - 671
• Main Authors: Reginska, Teresa, Reginski, Kazimierz
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.12.2015; Walter de Gruyter GmbH
• Subjects: 35R25; 65D10; 65M30; 78A60; axial profile of laser beam; Beams (radiation); Cauchy problem for the Helmholtz equation; Dirichlet problem; discrepancy principle; Ill posed problems; Laser beams; Lasers; Mathematical models; Operators; optimal convergence rate; Regularization; Regularization methods; smoothing and curve fitting; Wavelengths
• ISSN: 0928-0219; 1569-3945
• DOI: 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 and 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.