There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In this PHD thesis the scalar Fox and Li algorithm for the dominant transversal resonator eigenmode calculation is generalized to a fully vectorial field tracing concept. Therefore Fox and Li’s scalar integral equation is reformulated to a set of coupled operator equations. The introduction of a field tracing round trip operator concept shows that, in principle, any modeling technique which can be formulated to operate for electromagnetic fields can be used to simulate light propagation through the different subdomains of the resonator. This allows a flexible, fast, and accurate simulation of the fully vectorial dominant transversal resonator eigenmode. Several examples are presented to demonstrate the flexibility and accuracy of the field tracing approach.

Simulation of continuous-wave solid-state laser resonators using field tracing and a fully vectorial fox-li algorithm

Correct use of the photoelastic effect for the description of thermally induced refractive index change is discussed and the analytical relations between thermo-optic coefficients at zero stresses and zero strains are found for all classes of cubic crystals. These relations may be useful for the investigation of thermal effects in very promising sesquioxide class m3 laser crystals. An accepted set of elasto-optical coefficients of the YAG crystal and an alternative one found in the literature were used in numerical simulations. Significant differences in the calculated thermo-optic coefficients and induced birefringence are found using different sets of these coefficients. Misunderstandings related with the so-called photoelastic coefficients are resolved and new expressions for these coefficients are found. It is shown that the incorrect use of these coefficients for different pump beam distributions can lead to significant discrepancies for thermally induced birefringence. It is also shown that common use of the generalized thermo-optic coefficients significantly overestimates the values of optical power of thermal lenses when they are applied to the laser rods with lengths several times longer than their diameter.

On the use of photoelastic effect and plane strain or plane stress approximations for the description of thermal lensing

The computer-aided design of high quality mono-mode, continuous-wave solid-state lasers requires fast, flexible and accurate simulation algorithms. Therefore in this work a model for the calculation of the transversal dominant mode structure is introduced. It is based on the generalization of the scalar Fox and Li algorithm to a fully-vectorial light representation. To provide a flexible modeling concept of different resonator geometries containing various optical elements, rigorous and approximative solutions of Maxwell's equations are combined in different subdomains of the resonator. This approach allows the simulation of plenty of different passive intracavity components as well as active media. For the numerically efficient simulation of nonlinear gain, thermal lensing and stress-induced birefringence effects in solid-state active crystals a semi-analytical vectorial beam propagation method is discussed in detail. As a numerical example the beam quality and output power of a flash-lamp-pumped Nd:YAG laser are improved. To that end we compensate the influence of stress-induced birefringence and thermal lensing by an aspherical mirror and a 90° quartz polarization rotator.

Fully vectorial laser resonator modeling of continuous-wave solid-state lasers including rate equations, thermal lensing and stress-induced birefringence.

Powerful transversally diode-pumped Nd:YAG laser with low depolarization losses

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.

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

Thermal lensing e ects and birefringence in laser crystals strongly in uence beam quality of high power solid state lasers. Particularly, the e ects of birefringence are important in laser ampli ers using Nd:YAG crystals. To study these e ects, a simulation tool was developed, which allows to calculate birefringence in laser crystals for di erent kind of crystal cuts and pumping con gurations. The photoelastic e ects were accurately calculated by FE method. Simulation results are presented for di erent crystals and pumping. We also show that birefringence is not radially symmetric for a 111-cut Nd:YAG crystal, even when the pumping con guration is radially symmetric.

Simulation of birefringence in laser crystals

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.

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

The thermal lensing effect in the crystal influences the stability of solid-state lasers. The deformation of the end faces of the crystal and the temperature dependence of the refraction index causes this effect. The photoelastic effect produced by thermal induced stress in the crystal has also an influence. For high power lasers and for lasers with a radially polarized laser beam the analysis of the photoelastic effect is important. We use FE model to calculate the deformation, heat and stress distribution in the laser crystal with high accuracy in this work. Using the calculated stress distribution we also simulated the refraction index anisotropy with respect to the crystal orientation. We can study stability of laser resonators for both radially and azimuthally polarized laser beam, as a result of our simulation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Stability analysis of solid‐state lasers regarding thermal lensing effect

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.

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

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