Polarization transverse-pattern dynamics in lasers and their classification based on the characterization of its singular points

ABSTRACT Vectorial Karhunen-Loeve modes were used for the description of the polarization transverse pattern dynamicsof the Zeeman multitransverse mode laser with large Fresnel number and anisotropic cavity. Stokes parameterdistribution of the laser field in the transverse plane define new 2D vector field. Catastrophe theory and Newtondiagram method were used for the classification of the laser field based on the characterization of the 2D vectorfield behaviour in the vicinity of its singular points in the transverse plane. The results of numerical experimentsshow that the regimes with rotating patterns are closely connected with A catastrophe functions of argument r = /x2 + y2, where x, y are the transverse coordinates. For the oscillating patterns the regimes can be classified using Newton diagrams which are also connected with corresponding polynomials.Keywords: laser, transverse patterns, polarization, Stokes parameters, vector field, catastropes, singularities 1. INTRODUCTION The problem of the description of the polarization transverse pattern dynamics in lasers with large number oftransverse modes is complicated enough.'5 For the classification of the structures of laser field its singularpoints (zeroes, saddle points, maxima and minima) can be However, in these works the used modelsof laser fields are far from reality. Another approach utilizing ierarchy of symmetry breaking was also used forclassification of theThere is an effective way to work with very complicated signals and processes based on Karhunen—Loeveprocedure.'°16 In latter paper the laser regimes with rotating patterns were investigated and it was shown thatthese regimes are connected with catastrophes A of x iy arguments, where x, y are transverse coordinates.In this work we also used Karhunen—Loeve modes (KL—modes), however, we added the laser regimes corre-sponded to oscillating patterns. We introduced vector modes of Karhunen—Loeve, transformed four-dimensionalvector field representing polarized laser transverse patterns to two-dimensional vector field representing trans-verse distribution of the Stokes parameters (2DSP), discussed some properties of this 2D vector field and tried