Method and simulation device for wave-optically simulating laser propagation in a region delimited by a component, and corresponding computer program product and storage medium comprising same

Abstract

The invention relates to a computer-implemented method for wave-optically simulating the propagation of a laser beam (8) in a region (2) delimited by a component (1). The region (2) is divided into a free region (3) and an edge region (4) adjoining the component (1). In a corresponding simulation model, the electric field of the laser beam (8) is modelled by a unidirectional wave equation. In this simulation model, at least part of the wave equation in the edge region (4) is solved exactly by means of a finite difference method using matrix exponential functions of the finite difference matrix operator. The invention further relates to a corresponding computer program product, a computer-readable storage medium containing same and a simulation device for executing the method.

Description

[0001] METHOD AND SIMULATION DEVICE FOR WAVE-OPTICALLY SIMULATING LASER PROPAGATION IN AN AREA LIMITED BY A COMPONENT AND CORRESPONDING COMPUTER PROGRAM PRODUCT AND STORAGE MEDIUM TO WHICH

[0002] The present invention relates to a computer-implemented method for simulating the propagation of a laser beam. The invention also relates to a simulation device for carrying out the method. Furthermore, the invention relates to a corresponding computer program and a corresponding computer-readable storage medium.

[0003] Lasers can be used effectively in a wide variety of applications. However, corresponding laser systems can be highly complex structures these days. Optimizing such systems, for example with regard to efficiency, beam quality, robustness, or similar aspects, can therefore be difficult and costly. In principle, models and simulation methods could enable faster and simpler optimization than, for example, hardware-based experiments. However, detailed simulation methods, such as finite element methods (FEM), are not practical for larger laser systems or areas due to the associated computational effort, while highly simplified approaches that describe laser propagation everywhere as if in free space are not feasible.The laser does not provide sufficient accuracy in spatially limited beam paths and when the laser beam interacts with components and surfaces. Therefore, improvements are needed in this area.

[0004] The object of the present invention is to enable a particularly effective and efficient optimization of laser systems.

[0005] The problem is solved by the subject matter of the independent claims. Further possible embodiments of the invention are specified in the dependent claims, the description, and the drawings. Features, advantages, and possible embodiments set forth in the description for one of the subject matter of the independent claims are to be regarded, at least analogously, as features, advantages, and possible embodiments of the respective subject matter of the other independent claims, as well as of any possible combination of the subject matter of the independent claims, optionally in conjunction with one or more of the dependent claims.

[0006] The computer-implemented method according to the invention can be used for or in the wave-optical simulation of the propagation of laser radiation or a laser beam through or in a spatially limited area or system of a laser system by at least one component. This component can be, for example, a laser resonator, a beam guide, an amplifier, optics of the laser system, or the like. Accordingly, the at least one component can be, for example, a cavity, an amplifier tube, a housing, a vacuum tube, a mirror, a lens, an aperture or diaphragm, an optically linear or non-linear crystal, in particular interfaces of such a crystal at which, for example, laser radiation is reflected, or the like.Based on a corresponding simulation, the design and / or arrangement of at least one component and / or the laser system as a whole can then be optimized with regard to at least one given criterion or parameter.

[0007] In the method according to the invention, system data is provided or acquired that specifies the geometry of the area, i.e., for example, its shape and / or a position of the at least one component, and in particular also the geometry of the at least one component or at least a surface of the at least one component facing the area or bounding the area. The system data can thus specify geometric boundary conditions that limit the propagation or spread of the laser beam. Furthermore, the system data also specifies optical properties of the at least one component or of its surface bounding the area. In particular, the system data can specify or describe a complex refractive index of the component or of the surface of the component, i.e., of the material of the component. Optical properties of the component, i.e., in particular of the corresponding component surface, can influence the behavior of the laser beam or...The method according to the invention determines or influences at least the electric field of the laser beam at the surface of the component and thus ultimately also the propagation, i.e., the intensity distribution and / or the phase, of the laser beam. In the method according to the invention, the entire area fundamentally available for the propagation of the laser beam is divided into a free region and a boundary or edge region. In the free region, the laser beam can propagate freely. Therefore, effects that may arise from an interaction of the laser beam with the component can be negligibly small there, at least within a given numerical accuracy of the respective simulation. The boundary or edge region, on the other hand, borders directly on the at least one component.Accordingly, interactions can occur there, i.e., interactions between the laser beam and the component, such as diffraction effects, reflections, and / or interference effects, which can significantly influence the propagation of the laser beam. The free zone is therefore further away from the component or its surface than the edge zone. Accordingly, the free zone can also be referred to as the inner zone and the edge zone as the outer zone. The thickness of the edge zone, i.e., the distance to which the edge zone extends from the component towards the free zone, can, for example, be specified as a parameter in a corresponding simulation model for the method according to the invention. The thickness, i.e., the extent of the edge zone, can be fixed or, for example, predetermined or predefined depending on the respective wavelength of the laser beam.Furthermore, the thickness of the edge region can additionally or alternatively depend on a step size, particularly in the propagation direction, or on a number of calculation subunits, on the basis of which the inventive method performs the wave-optical simulation of laser propagation.

[0008] In the method according to the invention, in a predetermined or provided simulation model for simulating the propagation of the laser beam, at least the electric field of the laser beam is modeled, i.e., described or represented, by a pseudo-vector unidirectional wave equation. This wave equation can thus serve as the basis for the wave-optical simulation of the propagation of the laser beam. The electric field can be described or modeled generally, or at least in the free region, by two global polarization components. These global polarization components can, for example, be defined in a fixed, predetermined coordinate system. This coordinate system can, for example, be fixed to the system, i.e., rigid with respect to the respective laser system, a laser source, or an environment, such as a laboratory system. This allows for a simple description or...Modeling as well as a simple transfer of respective simulation results to a respective real laser system are made possible.

[0009] According to the invention, at least a part of the wave equation in the boundary region, and in particular also in the free region, is solved formally exactly in the simulation model using a predefined finite difference method (FDM) with matrix exponential functions of the finite difference matrix operator of the FDM used. An FDM is a numerical method for solving ordinary and partial differential equations. In particular, at least a part of a transverse diffraction, or a part of a corresponding transverse diffraction operator, and / or the reflection of the laser beam at the surface of the at least one component that defines the region can be solved or calculated.In other words, the transverse diffraction and / or reflection of the laser beam in the boundary region can be at least partially calculated using FDM, taking into account the geometric, i.e., spatial, boundaries of the region or the at least one bounding component. A corresponding finite difference matrix operator can thus be interpreted as a diffraction operator for describing or calculating the transverse diffraction of the laser beam. "Transverse" here can describe directions that are perpendicular to a predefined central longitudinal axis of a defined beam path or beam path running through the region, or perpendicular to a propagation direction of the laser beam, such as its central beam axis. The finite difference method can be part of the simulation model, i.e., included or implemented within it.

[0010] The formal or exact solution of the wave equation proposed here, using matrix exponential functions of the corresponding operator, enables a stable and fast calculation of the laser beam propagation. In particular, the method can be effectively and efficiently implemented or executed on GPUs, for example, using the CUDA interface or similar.

[0011] The propagation of the laser beam can be calculated, in particular, stepwise along the propagation direction or along the time axis, as will be explained in more detail elsewhere. The present invention can reduce the solution of a very large linear system of equations, such as would otherwise be required for simulating propagation, to comparatively simple multiplications of sparse matrices, which require comparatively little computational effort. This is computationally much more efficient and can at least largely avoid stability problems, for example, of the forward Euler method and undesirable parasitic oscillations that can occur in the Crank-Nicolson method.Thus, by applying the present invention, the propagation of a laser beam can be simulated with good accuracy, high speed, and efficiency even in component-constrained areas, i.e., with limited and readily available computing resources and a practical time expenditure. For example, a supercomputer is not necessary; a conventional laptop can suffice.

[0012] The present invention can be used, or form a basis for, supporting the development of laser systems and / or, for example, for analyzing and optimizing the alignment of components or caustics of a laser system, such as for improving output power, output beam quality, or output beam stability. Likewise, the present invention can enable an improved understanding of amplifier behavior, losses, or light leakage in laser amplifiers, and support the identification of remedial strategies. Furthermore, the present invention can be used to identify or assess potential parasitic laser paths or parasitic cavity lasing risks within the laser system, for example, based on the detection of depolarizations. Thus, the invention can, for example, enable or support improved sensitivity of an optical laser cavity or...to achieve a corresponding design and thereby improve the robustness of a corresponding laser system. The present invention can be applied or used, for example, for designing or laying out components and geometries and / or for determining the adjustment of components and / or for deterministically adjusting the adjustment or alignment of components and / or for determining or estimating whether predetermined target parameters, such as a target output power or a specific beam parameter along the beam path or the like, are likely to be achieved and / or for determining or estimating which changes or effects are likely to result from an adjustment or readjustment of a laser system, and / or for predictive maintenance. In one possible embodiment of the present invention, the laser beam is located in the free area or for the free area.whose electric field is modeled, i.e., described or represented, by two global polarization components. In the simulation model, a common operator is then used for both global polarization components—that is, the same operator is used to calculate the transverse diffraction of the laser beam, i.e., to solve the corresponding operator or corresponding part of the equation or model. In other words, the transverse diffraction in the free region can be calculated here in the global polarization basis of the electric field. This takes into account that in the free region, the influence of the component, i.e., a corresponding boundary or surface, on the polarization or propagation of the laser beam is negligibly small. Therefore, with the proposed design, a particularly simple calculation, i.e., simulation, of the transverse diffraction of the laser beam in the free region can be achieved.This can limit the overall computational effort required for the simulation.

[0013] In a further possible embodiment of the present invention, the laser beam or its electric field in the boundary region is modeled in the simulation model by s-polarized and p-polarized components with respect to the respective local surface of the at least one component that delimits the region. For example, as described elsewhere, the laser beam or its electric field can be described, in principle or in the free region, by two global polarization components, which can be projected onto the local s- and p-polarized components in the boundary region. This takes into account that the polarization can determine or influence the behavior of the laser beam at the component surface or a corresponding interaction of the laser beam with the component.By describing the process with locally s- and p-polarized components, this can be calculated—and thus simulated—particularly easily and consistently across different locations. This allows the behavior of the laser beam at the component surface to be simulated consistently and relatively easily, even if the component surface is not flat or if the area is bounded by several components positioned at an angle to each other.

[0014] In a possible further development of the present invention, the derivatives or differential quotients of the s-polarized and p-polarized components of at least the electric field of the laser beam are determined in the simulation model for the stencil scheme of the finite difference method, based on predefined continuity conditions for the electric and magnetic fields of the laser beam at the surface of the at least one component that bounds the region. Predefined impedance boundary conditions for the surface of the bounding component, for example Leontovich boundary conditions, can also be taken into account. This allows for an effective and efficient simulation of the propagation of the laser beam in the boundary region.

[0015] In a possible further development of the present invention, different individual operators for the s-polarized and p-polarized components are used in the simulation model for calculating the transverse diffraction and / or reflection of the laser beam in the boundary region, i.e., for solving the corresponding operator or corresponding equation or model part. This allows for the consideration of different polarization-dependent interactions of the laser beam with the component and, consequently, different influences on the propagation, intensity distribution, and / or phase of the laser beam in the boundary region. This enables a particularly accurate simulation of the laser beam propagation, which ultimately allows for a particularly efficient and effective optimization of the laser system.

[0016] In a further possible embodiment of the present invention, fields calculated in the simulation model for the s-polarized and p-polarized components—for example, the respective calculated transverse diffraction or the solutions of the corresponding operators or model parts—are projected back onto the global polarization components, i.e., into the global polarization basis, and combined with fields or solutions calculated for the free region. The corresponding partial results calculated in the respective local s- and p-polarization basis can thus first be expressed in or with the global polarization components, i.e., converted into these global polarization components or the corresponding polarization basis. This then enables a simple and consistent combination with the partial results for the free region.Thus, despite the initial decomposition and the different handling of the free region and the boundary region, propagation—that is, a unified and consistent solution for the entire region—can ultimately be provided. This allows for the provision of a simulation result that is particularly easy to use and understand for the aforementioned purposes. In a further possible embodiment of the present invention, the complete wave equation is solved in or by means of the simulation model using a split-operator method (SOP, also known as a split-step method). In this method, at least parts relating to the diffraction or reflection of the laser beam, other linear parts, and other nonlinear parts are solved individually, while keeping the other parts constant. The other linear parts can, for example, relate to a position-dependent refractive index.The other nonlinear components can, for example, relate to laser gain or laser loss. These other linear and nonlinear components or terms can be solved analytically. The Baker-Campbell-Hausdorff formula, for example, can be used or applied as a split-operator method or as part thereof. The embodiment of the present invention proposed here enables a particularly simple simulation or calculation of propagation, which is therefore also practically applicable to larger areas or geometries.

[0017] In a further possible embodiment of the present invention, thermal lensing effects are considered and modeled in the simulation model as a spatially dependent refractive index. The spatially dependent refractive index can, for example, be calculated from solutions of the nonlinear heat equation, particularly by means of explicit time discretization. By rigorously considering thermal lensing effects, the accuracy of the laser beam propagation simulation can be improved with a manageable effort. This ultimately allows for improved development or optimization of the laser system, enabling faster or more efficient processes.

[0018] In a further possible embodiment of the present invention, a polarization change is applied in the simulation model for non-perpendicular reflections of the laser beam, particularly at mirrors that are struck by the laser beam at an angle—especially a relatively small one—to the optical axis or to the normal perpendicular to the mirror surface. This allows the actual behavior of the laser beam, i.e., its real propagation, to be reproduced, or simulated, with particular accuracy. The present invention also relates to a computer program product comprising commands that, when executed by a computer, cause it to perform the method according to the invention. The computer program product according to the invention can therefore be, for example, an operating or computer program for a computer or a simulation device.

[0019] The present invention also relates to a computer-readable storage medium on which the computer program product according to the invention is stored.

[0020] The present invention also relates to a simulation device in which a simulation model for the method according to the invention, in particular the simulation model mentioned in connection with the method according to the invention, is implemented or stored for execution, in particular automatically or semi-automatically, by means of the simulation device. The simulation device according to the invention comprises means for carrying out the method. These means can, for example, include an input interface, a process device, such as a microprocessor, microchip, microcontroller, or GPU, i.e., a graphics processor or graphics accelerator, or the like, and a computer-readable data storage device, in particular the computer-readable storage medium according to the invention, or also a working memory. The simulation device can be configured to perform, during or after the execution of the method,The simulation results generated by the simulation model are to be made available in the data storage and / or via an output interface. The simulation device according to the invention can, for example, be a computer or a computer system.

[0021] Further features of the invention may become apparent from the following description of the figures and from the drawings. The features and combinations of features mentioned above in the description, as well as the features and combinations of features shown below in the description of the figures and / or in the figures themselves, can be used not only in the combinations specified, but also in other combinations or individually, without departing from the scope of the invention.

[0022] The drawing shows in Fig. 1 a partial schematic representation to illustrate an operator for the partial solution of a wave equation in a central free area of ​​a laser beam path;

[0023] Fig. 2 shows a partial schematic representation illustrating an operator for the partial solution of the wave equation for s-polarization in an outer boundary region of the laser beam guidance;

[0024] Fig. 3 shows a partial schematic representation illustrating an operator for the partial solution of the wave equation for p-polarization in the outer boundary region of the laser beam guidance;

[0025] Fig. 4 shows a partial schematic representation illustrating the laser intensity of a simulated propagation of a laser beam in the laser beam guide; and

[0026] Fig. 5 shows a partial schematic representation to illustrate a simulated propagation of the laser beam in a central free area of ​​the laser beam guidance and reflection at a component limiting the laser beam guidance.

[0027] Identical or functionally equivalent elements are marked with the same reference symbols in the figures.

[0028] In real laser systems, laser light or laser radiation does not propagate unhindered as it would in free space, but rather interacts with, for example, spatial boundaries, optical elements, interfaces, and the like. Taking this into account in a simulation can create a basis for particularly effective and efficient development and optimization. However, an accurate wave-optical simulation of a complete laser system can be challenging, as such a system can comprise many components and areas with varying linear and nonlinear properties. At least quasi-free-space propagation in some areas can be simulated with sufficient accuracy and reasonable computational effort using established methods. However, a complete simulation or model of a laser system would also need to include a description or...This includes modeling propagation in the vicinity of boundary or interface surfaces and interactions with geometric boundaries. The latter cannot be achieved with sufficient accuracy using conventional methods for simulating propagation in free space.

[0029] The following describes, with reference to the figures, a corresponding simulation model and simulation method, or framework, for simulating the propagation of laser radiation in spatially limited areas. Input data for the simulation model can include, for example, geometric data specifying the geometry of the laser system to be simulated (i.e., the shape and position of at least one component 1), material data describing at least one component 1 or other components of the laser system, laser system gain parameters, and / or similar data. Outputs from the simulation model can include an intensity profile or intensity distribution and / or a complex amplitude of the electric field, including the phase.If only a complex amplitude of the electric field is provided, the simulation model can determine the intensity profile or intensity distribution based on this.

[0030] Figure 1 shows a schematic representation of a section of a laser system with a first part of a simulation result. The section of the laser system is laterally bounded by at least one component 1. This component 1 thus defines a region 2 in which laser radiation or a laser beam 8 (see Figures 4 and 5) can propagate or be guided. Region 2 is subdivided into a central free area 3 and at least one edge region 4. The edge region 4 is located between the free area 3 and the component 1 and borders it. The geometrically, i.e., spatially, bounded region 2 can, for example, be located in an amplifier tube (e.g., made of ceramic or entirely or partially of quartz) or be an amplifier region between an inner electrode (e.g., made of ceramic) and an outer electrode (e.g., made entirely or partially of aluminum), or the like.The framework presented here can simulate the propagation of laser radiation even in such confined regions and thus—possibly in combination with known, conventional methods—complete the methods required for a full simulation of laser radiation propagation in or through a complete laser system. For the simulation, region 2 can be populated with or represented by a numerical grid.

[0031] In the simulation model presented here, propagation is initially calculated separately and independently for the free-space region 3 and the boundary region 4, respectively. Figure 1 schematically illustrates a free-space operator 5 and its output or application result. This can represent a solution to a wave equation describing propagation only for the free-space region 3 and only for one step along a propagation direction perpendicular to the plane of the drawing, i.e., the transverse plane. A corresponding part of the simulation model can therefore essentially describe or model the free-space propagation of the laser beam 8 sufficiently far from the boundary conditions. A corresponding contribution to the overall solution, i.e., the free-space operator 5 shown here, can asymptotically decrease to zero in the direction of the boundary conditions, i.e., in the plane of the drawing towards the boundary region 4.For free area 3, the propagation, i.e. the corresponding solution, can be calculated jointly for two global polarization components.

[0032] For the boundary region 4, the propagation is simulated or calculated separately for two different polarization components. Figure 2 shows a first boundary region operator 6 for s-polarization and its output or application result. This can represent a solution of the wave equation for describing the propagation only for the boundary region 4 and only for the same step along the propagation direction perpendicular to the drawing or transverse plane, and only for an s-polarized component with respect to the respective local surface of component 1 that bounds the region 2. Figure 3 shows a second boundary region operator 7 for p-polarization and its output or application result. This can also represent a solution of the wave equation for describing the propagation only for the boundary region 4 and only for the same step along the propagation direction perpendicular to the drawing or transverse plane.The propagation direction in the transverse plane is represented only for a p-polarized component with respect to the respective local surface of component 1 that bounds region 2. Corresponding parts of the simulation model thus describe the solution of the wave equation within the boundary region 4, which is influenced by the boundary conditions. The contribution in the boundary region 4 to the overall solution decreases from component 1 towards the free region 3 to zero. The solutions from Fig. 1 for the free region 3, from Fig. 2 for the s-polarized component in the boundary region 4, and from Fig. 3 for the p-polarized component in the boundary region 4—i.e., the free region operator 5, the first boundary region operator 6 for the s-polarization, and the second boundary region operator 7 for the p-polarization, or their respective outputs or application results—can be combined to obtain an overall solution.To obtain the overall intensity distribution for region 2 for each polarization of the laser beam 8.

[0033] Specifically, starting from Maxwell's equations, suitable approximations can first be made to obtain a unidirectional wave equation for the vectorial electric field of the laser radiation as a basis for the wave-optical simulation of the propagation. Using a carrier-envelope decomposition or a paraxial approximation, the laser beam 8 can initially be described or modeled as a complex electric field with two global polarization components, either within the initial free region 8 or within the free region 8. For example, the two global polarization components can be positioned vertically and horizontally in a laboratory frame, in which, for instance, the laser system can also be stationary.In a resulting (2+1)D model, with two dimensions spanning the transverse plane and a propagation dimension, which may be, for example, a direction perpendicular to the transverse plane or a time axis, the solution of the wave equation can propagate the laser beam 8 along the propagation dimension.

[0034] The coordinate types or coordinate systems used in the simulation model can be adapted to the specific geometry of the application and selected accordingly. For example, region 2 might be cylindrical. The simulation model can then be implemented entirely or partially in Cartesian and cylindrical coordinates. Within the limited region 2, at least part of the transverse diffraction operator can be solved using finite differences, taking boundary conditions into account. An isotropic finite difference support scheme can be used for this purpose in Cartesian coordinates. This reduces the anisotropy of the dispersion relations for different directions along a numerical grid.In cylindrical coordinates, the azimuthal part of the transverse diffraction operator can be solved in the Fourier domain using an angular spectrum method (ASM). This is possible here because, unlike the radial direction, there are no boundary conditions or limits in the azimuthal direction in the example under consideration. A complex wave field can be expanded into an infinite sum of plane waves of the same frequency but different propagation directions. Only the radial part, which describes a circularly bounded region—for example, the area bounded by a cylindrical electrode formed by component 1—is then solved using finite differences.

[0035] Materials at boundaries—for example, the surface of component 1 adjacent to boundary region 4—can be described by their complex refractive index. At the boundary, the global polarization components of the electric field of the laser beam 8 can be projected onto the local surface with respect to s- and p-polarized polarization components. Based on predefined continuity conditions for the electric and magnetic fields of the laser beam 8 at the boundary, i.e., the surface of component 1, the derivatives or differential quotients of the s- and p-polarized components of the electric field can be calculated via impedance boundary conditions. The derivatives or differential quotients calculated in this way can complete the finite difference support point scheme at the boundary or for boundary region 4.

[0036] There are various methods for solving the transverse diffraction operator in the finite-difference formalism. Traditionally, finite-difference discretizations of differential operators are solved using the forward Euler method, the backward Euler method, or the Crank-Nicolson method, resulting in linear sparse tridiagonal matrices or systems of equations. In contrast, this approach utilizes a formally exact solution of the wave equation using matrix exponential functions of the finite-difference matrix operator. This method reduces the solution of the linear sparse tridiagonal matrices or systems of equations to multiplications of sparse matrices. This requires significantly less computational effort and largely avoids the stability problems of the forward Euler method and parasitic oscillations that can occur in the Crank-Nicolson method.

[0037] To account for the boundary conditions, the transverse diffraction operators for the different polarizations, or the s- and p-polarization components, can be decomposed into inner and outer regions, i.e., the free region 3 and the boundary region 4. In the inner region, i.e., the free region 3, effects or influences of the interaction of the laser beam 8 or the electric field with the component 1 are negligible within the limits of the respective numerical accuracy. Therefore, a single operator can be used there to solve or calculate the transverse diffraction in the basis of the global polarization components. In the outer region, i.e., the boundary region 4, the laser beam 8 or the electric field is decomposed into the s- and p-polarized components – each with respect to the local surface of the component 1.For these two polarization components, the transverse diffraction is then solved or calculated separately using individual operators. The fields resulting for the boundary region 4 can then be projected back into the global polarization basis and combined or superimposed with the fields resulting for the free region 3. This yields the complete solution of the transverse diffraction operator over the entire geometry, i.e., for the entire region 2.

[0038] The remaining linear terms—such as a position-dependent refractive index for modeling thermal lensing effects—and nonlinear terms of the wave equation—such as those for modeling laser losses and laser gain—can be solved analytically. Finally, a split-operator method, specifically the Baker-Campbell-Hausdorff operator split, can be used to solve the complete wave equation, solving the diffraction, linear, and nonlinear parts sequentially while holding the other parts constant.

[0039] The vectorial wave-optical simulation model can be completed by integrating a polarization change for non-perpendicular reflections of the laser beam or the electric field, for example at mirrors, which are struck at a small angle, i.e., small deviations from the perpendicular.

[0040] The simulation model or corresponding program code can be implemented in MATLAB, particularly using CUDA-based computation, to achieve the shortest possible computation time. For example, practical experience has shown that a simulation of beam propagation along a beam path of approximately 50 m through a laser system with a practically useful grating spacing of, say, 50 pm can be executed or calculated in about 1 minute on a standard laptop computer. The computation or execution time can therefore be short enough to make parameter scans practical as well. In this context, for example, the propagation or...The intensity distribution or properties of an output beam of the laser system can be simulated for different values ​​or settings of an angle or position of the beam axis and / or a beam waist position and / or a beam diameter and / or a beam waist diameter and / or a beam quality or a diffraction index and / or a beam shape or a beam profile and / or the like.

[0041] Fig. 4 schematically and exemplarily depicts the laser beam 8 simulated according to the described method, i.e., with the described simulation model, within component 1 or between two components 1, for example, between two electrodes. Viewed in the plane of the drawing, the laser beam 8 can propagate in one direction, here designated as the y-direction, at least substantially unimpeded, while its propagation in the perpendicular direction, here designated as the x-direction, is limited by the at least one component 1. Corresponding boundary conditions lead to reflections at the at least one component 1, where the intensity drops to at least nearly zero. Furthermore, this results in interference fringes or an interference pattern in the x-direction.

[0042] It should be noted that the intensity in the region of the interface, i.e., the inner surface of the at least one component 1 facing region 2, can depend on the complex refractive index of the material of the at least one component 1, as well as on the polarization and the angle of incidence of the laser beam 8. Accordingly, the intensity in the boundary region 4 or at the at least one component 1 does not necessarily drop to almost zero in all examples and applications.

[0043] Fig. 5 schematically and exemplarily depicts the laser beam 8 simulated according to the described method, i.e., with the described simulation model, upon reflection from component 1. Resulting interference patterns are also visible here. The relative errors in the simulated, i.e., calculated, reflected power can be on the order of 10⁻¹⁰ for both the s-polarized and the p-polarized components. 4or better yet, lie. Figures 4 and 5 illustrate the ability of the simulation model to simulate the interaction of laser light with interfaces or components 1 in relatively large geometries.

[0044] The fundamental method described, i.e., a simulation model with the described structure, can also be applied to other applications or domains, meaning to other problems that are mathematically equivalent to the determining unidirectional wave equation—which can be expressed or interpreted as a general diffusion equation—or can be formulated accordingly. For example, it can be used to simulate heat transport or temperature development, where time can serve as a propagation dimension, i.e., as the z-coordinate of the wave equation. Similarly, it can be used, for example, to simulate radar systems or radar propagation, or similar phenomena.

[0045] In summary, light propagation in confined areas or media can be described or simulated using a split-step solution of the pseudo-vector unidirectional wave equation. Transverse diffraction, including consideration of geometric boundaries such as those of optical or beam-guiding elements or elements surrounding a beam path, is solved using finite differences by imposing or applying continuity conditions to the magnetic and electric fields via impedance boundary conditions. The formal solution of the wave equation using matrix exponential functions of the diffraction operator enables a stable, fast, and efficient calculation or simulation of the propagation. Accordingly, a simulation model is proposed here that incorporates component boundaries in the optical path, i.e., along the beam path, and interactions of the laser radiation with these components.This allows for a more accurate representation or simulation of the real-world effects within a laser system compared to previous approaches, which in turn can enable improved design or optimization of the laser system and / or its components. REFERENCE SYMBOL LIST.

[0046] 1 Component 2 Propagation area

[0047] 3 Outdoor area

[0048] 4 Edge area

[0049] 5 Free-space operator

[0050] 6 First boundary operator (for s-polarization) 7 Second boundary operator (for p-polarization)

[0051] 8 Laser beam

Claims

PATENT CLAIMS 1. Computer-implemented method for wave-optical simulation of the propagation of a laser beam (8) in a spatially limited area (2) of a laser system by at least one component (1) for optimizing a design and / or arrangement of the at least one component (1), wherein - Plant data are recorded which specify the geometry of the area (2) and optical properties of the at least one component (1), - the area (2) is subdivided into a free area (3) in which the laser beam (8) can propagate freely, and a boundary area (4) which borders the at least one component (1), - in a provided simulation model for simulating the propagation of the laser beam (8), the electric field of the laser beam (8) is modeled by a unidirectional wave equation, - in the simulation model at least a part of the wave equation in the boundary region (4) is solved exactly using a finite difference method with matrix exponential functions of the finite difference matrix operator.

2. Method according to claim 1, characterized in that in the simulation model the laser beam (8) in the free area (3) is modeled by two global polarization components and a common operator is used for both global polarization components to calculate the transverse diffraction of the laser beam (8).

3. Method according to one of the preceding claims, characterized in that in the simulation model the laser beam (8) in the boundary region (4) is modeled by s-polarized and p-polarized components with respect to the local surface of the at least one component (1).

4. Method according to claim 3, characterized in that the derivatives of the s-polarized and p-polarized components for the support point scheme of the finite difference method in the simulation model based on predefined continuity conditions for the electric and magnetic fields of the laser beam (8) at the surface of the at least one component (1) bounding the region (2) are determined.

5. Method according to claim 3 or 4, characterized in that different individual operators for the s-polarized component and the p-polarized component are used in the simulation model for calculating the transverse diffraction and / or the reflection of the laser beam (8) in the boundary region (4).

6. Method according to one of claims 3 to 5 and according to claim 2, characterized in that fields calculated in the simulation model for the s-polarized component and the p-polarized component are projected onto the global polarization components and combined with fields calculated for the free area (3).

7. Method according to one of the preceding claims, characterized in that the complete wave equation is solved in the simulation model using a split-operator method, wherein at least parts for diffraction, other linear parts and other nonlinear parts are solved individually while keeping the other parts constant.

8. Method according to one of the preceding claims, characterized in that thermal lensing effects are taken into account in the simulation model by means of a location-dependent refractive index.

9. Method according to one of the preceding claims, characterized in that a polarization change for non-perpendicular reflections of the laser beam (8) is applied in the simulation model.

10. Computer program product comprising instructions which, when executed by a computer, cause the computer to execute the method according to any of the preceding claims.

11. Computer-readable storage medium on which the computer program product according to claim 10 is stored.

12. Simulation device in which a simulation model for the method according to one of claims 1 to 9 is implemented for execution by means of the simulation device and comprises the means for execution of the method.

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