Rainer Hartmann

Calculations of Eigenpolarization in Nd:YAG Laser Rods Due to Thermally Induced Birefringence

Laser crystals like Nd:YAG are widely used in laser resonators. The stress-induced birefringence of such laser crystals has a strong influence on beam quality, power, and polarization of the output laser beam. Detailed simulations using 3-D finite-element analysis and a 2-D Jones matrix analysis were performed to analyze these effects. The finite-element analysis is used to calculate stress and birefringence depending on the cut direction of the crystal and its rotation. Eigenvalues and eigenvectors of the electromagnetic field inside the resonators are calculated by Jones matrix analysis. The analysis includes resonators with Brewster plates. Output power and beam quality are calculated by dynamic multimode analysis. Simulation results are presented for [100]-cut, [110]-cut, and [111]-cut Nd:YAG crystals.

Characterisation of birefringence of [111]-cut crystal rod using side-pumping and crystal rotation

Birefringence influences the beam quality and output power of high power solid-state lasers. Inhomogeneous distribution of the thermal field inside the laser crystal rod leads to thermal strains and birefringence, due to the photoelastic effect. Analytical models have used the plane stress and plane strain assumption for an axial sym- metric pumped crystal. This leads in case of an [111]-cut to an axially symmetric birefringence pattern. However, numerical calculations of birefringence show a threefold symmetry pattern due to the anisotropic behavior of the photoelastic tensor. This disturbs the ideal use of a radial or azimuthal polarised beam. We analyzed a laser rod pumped at three sides with threefold symmetry, in order to reduce the effect of birefringence. Simulation results show birefringence is affected by rotation of the crystal around its [111]-axis. Smallest birefringence can be obtained by an optimal rotation with respect to the edges of the crystal. Therefore the output beam of this laser device is more suitable for generating radial or azimuthal polarisations.

A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension

Analysis of thermal depolarization compensation using full vectorial beam propagation method in laser amplifiers

d

Analysis of birefringence effects in laser crystals by full vectorial beam propagation method

. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

We developed a complex physical model for simulating laser amplifiers to numerically analyze birefringence effects. This model includes pump configuration, thermal lensing effects, birefringence, and beam propagation in the laser amplifier. Temperature, deformation, and stress inside the laser crystal were calculated using a three-dimensional finite element analysis (FEA). The pump configuration is simulated using a three-dimensional ray tracing or an approximation based on super-Gaussian functions. Our simulations show the depolarization of a linearly polarized electromagnetic wave in a cylindrical laser crystal. These simulations were performed using a three-dimensional full vectorial beam propagation method (VBPM). Stress induced birefringence can be compensated well for moderate pumping powers. High power amplification requires sensitive alignment. Our simulation technique calculates the influence of the photo-elastic effect inside the laser crystal accurately. Detailed knowledge about beam waist and depolarization is needed to develop compensation techniques for high power output beams with low depolarization losses.

Modeling and simulation of ultra-short pulse amplification

Modern laser technology demands powerful numerical tools to predict the efficiency of laser configurations. Birefringence has a strong influence on the beam quality and output power of a laser amplifier. We developed a complex physical model for simulating laser amplifiers and analyzing the birefringence effects. This model includes pump configuration, thermal lensing effects, birefringence, and beam propagation in the laser amplifier. The pump configuration is simulated using a complete three-dimensional ray tracing or by an approximation based on super-Gaussian functions. For an accurate modeling of the thermal lensing effect, the deformation of the end faces and the polarization dependent index of refraction was taken into account. Temperature, deformation and stress inside the laser crystal were calculated by a three-dimensional finite element analysis (FEA). In particular, the refractive index was accurately calculated by considering its temperature dependency and the photo elastic effect. This refractive index was used in the simulation of laser beam propagation through an amplifier. These simulations were performed by a complete three-dimensional vectorial beam propagation method (VBPM). The advantage of VBPM is that it can be applied to a polarization dependent index of refraction. This is important when taking into account the birefringence obtained by the photo elastic effect inside the laser crystal. The beam propagation method is based on finite elements on block structured grids as well as a Crank-Nicolson approximation in the propagation direction (FE-BPM). Reflecting boundaries were eliminated by introducing a perfect matching layer (PML). Simulation results show that a complete three-dimensional simulation model was useful in analyzing and optimizing high power laser amplifiers. The value of our model lies in the fact that it can take into account the crystal cut direction. Based on this the birefringence for simulating the laser beam quality and output power can be calculated.

Simulation of polarization effects in solid state lasers and amplifiers

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.

Efficient Ritz-Galerkin Discretization of PDEs with Variable Coefficients in arbitrary Dimensions using Sparse Grids

Ultra-short pulses with high average power are required for a variety of technical and medical applications. Single, multi-pass, and regenerative amplifiers are used, in order to increase the power of ultra-short lasers. Typical laser crystals for such amplifiers include Ti:Sapphire or Yb:YAG laser crystals. Difficulties in the amplification of ultra-short pulses include gain narrowing effects and dispersion effects in the laser crystal. In particular, these complications arise, when a pulse stretcher is needed before amplification of the laser beam. We present a technique to model and simulate the amplification of ultra-short pulses. This technique allows to model both gain narrowing effects and decrease of beam quality caused by amplification of the laser beam. This requires a detailed 3-dimensional simulation of population inversion. Gain narrowing effects are taken into account by analyzing the gain of the spectrum of the laser beam. It is important to distinguish amplifiers with one or only two passes and a regenerative amplifier. These two different kind of amplifiers are modeled by different approaches. A regenerative amplifier is modeled by a set of time dependent rate equations. However, a single pass amplifier is modeled by a set of spatial dependent rate equations. In both cases, a system of rate equations arises from spectral discretization of the laser beam. Detailed simulation results are presented.

A Sparse Grid Discretization with Variable Coefficient in High Dimensions

Polarization effects in laser crystals like Nd:YAG have a strong influence on output power and beam quality of laser resonators and laser amplifiers. Simulation techniques to analyze these effects are presented.