Numerical modelling of caustics of solid state disk lasers

This paper concerns mathematical modelling of caustics of solid state laser YAG type with the disk active medium. The heat source model is developed on the basis of interpolation algorithms using geostatistical kriging method. The real laser beam power distribution and caustics are taken into account in the model. Measurements of laser beam power distribution and diameter of the laser beam spot for different focusing are performed using UFF100 analyzer. Yb:YAG laser emitted by Trumpf laser head D70 is used in the experiment. Presented results include the comparison of experimentally determined laser power intensity distribution and caustic with results obtained by developed interpolation model.


Introduction
Changes in the focal length of focusing lens in the optical system of solid state laser changes not only the laser spot diameter but also the laser beam power intensity distribution. Beam divergence which is a measure for how fast the beam expands far from the beam waist is usually defined by the measurement of beam causticthe beam radius at different positions. The measurement of laser beam power distribution is performed by using laser profilers [1][2][3].
In mathematical and numerical models of laser beam heat source power distribution only specific laser beam spot diameter is adopted, omitting the beam caustics. Generally, the distance from beam focusing position as a technological parameter influencing laser beam intensity distribution is usually neglected. Consequently, the laser beam intensity distribution models assumed in numerical analysis significantly differ from real Yb:YAG laser profile, obtained through experimental research [4,5].
This paper presents interpolation algorithms allowing a precise description of Yb:YAG laser power intensity distribution and its caustic. Elaborated models take into account the real laser power distribution obtained in experimental research made using TruDisk 12002 laser. Kriging method [6][7][8] is used in this study in the form of point Kriging for the interpolation of laser beam heat source power distribution.

Experimental research
The light is pumped to the laser beam head by optical fibers with core diameter d LLK . After the exit from the fiber, the beam is divergent by angle  LLK to diameter d 0 . The laser head is equipped with collimator lens having focal length f c = 200 mm and focusing lens with focal length f og = 400 mm. After focusing of laser light by focusing length by angle  f the spot diameter (laser beam focus diameter) is equal to d og . The following equations can be used to determine average diameter of the laser beam focus, Rayleigh length (Z R ) as well as the actual diameter d(z) depending on beam focusing position (z).
where λ is a wavelength of laser radiation. The measurement of beam power distribution and caustics is performed using beam profiler Prometec UFF100 (Fig. 2). UFF100 is a beam profiler for high power lasers. It measures laser beam profile due to hollow needle. Laser beam is flowing through rotating needle and is transported to the detector. Detector signals are transformed into digital signals and processed later by computer software. The power density distribution of the laser beam in a plane perpendicular to the axis of beam propagation (assumed to be z-axis) for different measurement planes is analyzed. The analysis is performed at the continuous power 900W of the laser beam. The position of measurement plane changes with respect to the theoretical distance from the beam focusing position in the range of ± 10 mm. For each testing planes, the radius of the beam in two perpendicular axes (w x and w y ) is defined from which an average radius of the laser beam spot is calculated (w) in accordance with standard PN-EN ISO 11146. Figure 3 presents experimentally obtained percentage distribution of Yb:YAG laser beam power for chosen beam focusing (z=0 and ± 5 mm). Figure 4 illustrates the beam caustic w x , w y and average radius w.
where w i are weight coefficients assigned to particular observations,   i i y x f , is the real value of the function (variable) at the measured point, n is the number of sampling points that are considered in estimating of the variable within the circle of radius r k from estimated point.
Coefficients w i are calculated on the basic of Kriging system of equations, as follows:  S at h  ∞, while C 0 is a function discontinuity (nugget effect). Coefficients S and C 0 are estimated using empirical semivariogram and sample variance Var, described by the following relations: is the number of basic points in a measurement set, f is the average value of the function in a measurement set.
Finally, after solving the system of equations the heat source power distribution for interpolation grid is calculated as follows: 4 Results and discussion  show the power distribution of the heat source obtained by Kriging method for interpolation grid step Δh=0.02 mm taking into account experimental data for z = 0 and at the distance from beam focusing position z=10mm respectively. In all figures the comparison of modelled percentage power intensity distribution with experimentally obtained distribution is illustrated in central axes of the heat source (x = 0 and y = 0).