Christian Hellmann

Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation

The propagation of electromagnetic fields between non-parallel planes based on a spectrum-of-plane-wave analysis is discussed and formulations for an efficient numerical implementation are presented in detail. It is shown that with the help of interpolation techniques, the numerical implementation can be done with the standard uniform fast Fourier transform (FFT) of easy access. Different interpolation techniques are numerically examined, and it turns out that the use of cubic interpolation, together with the uniform FFT, brings both significantly increased computational efficiency and high simulation accuracy. Apart from the aspect of computational efficiency, all formulations in this work are generalized in a fully vectorial manner in comparison to previous works.

Resonator modeling by field tracing: a flexible approach for fully vectorial laser resonator modeling

Nowadays lasers cover a broad spectrum of applications, like laser material processing, metrology and communications. Therefore a broad variety of different lasers, containing various active media and resonator setups, are used to provide high design flexibility. The optimization of such multi-parameter laser setups requires powerful simulation techniques. In literature mainly three practical resonator modeling techniques can be found: Rigorous techniques, e.g. the finite element method (FEM), approximated solutions based on paraxial Gaussian beam tracing by ABCD matrices and the Fox and Li algorithm are used to analyze transversal resonator modes. All of these existing approaches have in common, that only a single simulation technique is used for the whole resonator. In contrast we reformulate the scalar Fox and Li integral equation for resonator eigenmode calculation into a fully vectorial field tracing operator equation. This allows the flexible combination of different modeling techniques in different subdomains of the resonator. The work introduces the basic concepts of field tracing in resonators to calculate vectorial, transversal eigenmodes of stable and unstable resonators.

Modeling the propagation of ultrashort pulses through optical systems

The propagation of harmonic fields through arbitrary optical components is the fundamental task in optical modeling. Unified optical modeling by field tracing uses different techniques for different components in order to ensure the best compromise between simulation effort and accuracy. This approach can be extended to non-harmonic fields. With a set of harmonic fields modeling partial coherence of stationary sources is enabled. The same approach can be applied to model the propagation of fully coherent ultrashort pulses through optical systems, which may include for instance lenses, gratings and micro-optical components. For that we can rely on field tracing with its numerous sophisticated propagation techniques for a single harmonic field. Methods to reduce frequency domain sampling are presented. They allow a convenient pulse modeling in practice. Several examples are presented using ultrashort pulse modeling with VirtualLab™.

Comparison of modelling techniques for multimode fibers and its application to VCSEL source coupling

Ray tracing and split-step method are the most efficient techniques to model multi-mode fiber. In this work, we also propose a geometrical optics based approach, which is beyond ray tracing. This approach, which is mathematically based on Runge-Kutta methods, handles not only ray information but light field information, e.g. amplitude and polarization. Then we discuss and compare the different techniques by the example of coupling of a VCSEL source into a multi-mode fiber.

Simulation, eigenmode analysis, and tolerancing for stable laser resonators

Recently the importance of numerical simulations for the design of laser resonators has grown considerably. This applies in particular if the alignment of components within the resonator is crucial for its stability. In such cases a tolerance analysis is required that can be done most efficiently using numerical simulation tools. In this paper, we introduce a computer model for resonators based on components and their combination using absolute or relative positioning. We show that this approach is the basis for tolerancing and sensitivity analysis. Further we discuss the concepts of field tracing and unified optical modeling that allow the coupling of several propagation methods within one modeling task. For laser resonators this involves in particular free space propagation methods as the Fresnel integral, geometrical optics and split step beam propagation methods. The primary goal is to provide a fully vectorial simulation as accurate as required and as fast as possible. This approach covers in particular general eigenmode models and general geometries including micro-structured surfaces that can be used for additional beam control as it is shown in the examples.

Algorithm for the propagation of electromagnetic fields through etalons and crystals

We investigate the propagation of general electromagnetic fields through optical layer structures made of either isotropic or anisotropic media, by using the spectrum-of-plane-waves analysis together with the S-matrix method. We also develop an algorithm based on the fast Fourier transform technique, with a numerically efficient sampling rule. By using this algorithm in combination with other system modeling techniques, we present a few simulation examples, such as field propagation through an isotropic Fabry-Perot etalon, as well as uniaxial crystal slabs with arbitrary orientation and optic axis direction.

Customized homogenization and shaping of LED light by micro cells arrays

The energy-efficient use of LED light requires the development of compact illumination systems for the customized homogenization and shaping of partially-coherent LED light. Therefore a design concept which is based on arrays of aperiodic micro structures, namely cells, for primary or secondary optics is introduced. Each cell of the array deflects locally the light into predefined directions and results in a light spot in the target plane. The light spots of all array cells together form the desired light pattern. The performance of three different cell geometries (linear gratings, micro prisms andmicromirrors) on the homogenization and shaping ofmonochromatic as well as white light LEDs is demonstrated. For the realistic evaluation of the illumination system an LED model including power spectrum, polarization, spatial and temporal coherence is chosen. Furthermore wave-optical effects like diffraction at the cell apertures are taken into account. For the grating cells arrays a rigorous analysis of the diffraction efficiencies is included.

Approximate solution of Maxwell’s equations by geometrical optics

Ray optics has constituted the fundament of optical modeling and design for more than 2000 years. In recent decades, the introduction of ray tracing software has brought a powerful optical design technology to everybody dealing with optics and photonics. However, with the development and availability of advanced light sources, the capability to produce micro and nano structures, the need for high NA systems, and a boost in the variety of applications and related demands on optical functions, the limitations of ray optics become obvious more often. Optical modeling based on physical optics is required and is the logical next step in the development of optical design. This requires a generalization of ray tracing and its connection with diffractive modeling techniques.

Fully-vectorial simulation and tolerancing of optical systems for wafer inspection by field tracing

The simulation, design and tolerancing of optical systems for wafer inspection is a challenging task due to the different feature sizes, which are involved in these systems. On the one hand light is propagated through macroscopic lens systems and on the other hand light is diffracted at microscopic structures with features in the range of the wavelength of light. Due to this variety of scale plenty of different physical effects like refraction, diffraction, interference and polarization have to be taken into account for a realistic analysis of such inspection systems. We show that all of these effects can be included in a system simulation by field tracing, which combines physical and geometrical optics. The main idea is the decomposition of the complex optical setup in a sequence of subdomains. Per subdomain a different approximative or rigorous solution of Maxwell’s equations is applied to propagate the light. In this work the different modeling techniques for the analysis of an exemplary wafer inspection system are discussed in detail. These techniques are mainly geometrical optics for the light propagation through macroscopic lenses, a rigorous Fourier Modal Method (FMM) for the modeling of light diffraction at the wafer microstructure and different free-space diffraction integrals. In combination with a numerically efficient algorithm for the coordinate transformation of electromagnetic fields, field tracing enables position and fabrication tolerancing. As an example different tilt tolerance effects on the polarization state and image contrast of a simple wafer inspection system are shown.

Non-sequential Optical Field Tracing

Optical field tracing methods generalize ray tracing methods by considering harmonic fields instead of ray bundles. This allows the smooth combination of different modeling techniques in different subdomains of the system. Based on tearing and interconnecting ideas, the paper introduces the basic concepts of non-sequential field tracing and derives the corresponding operator equations and a solution formula for the simulation task. The evaluation requires the solution of local Maxwell problems (tearing) and the continuity of the solution across boundaries is achieved along with the convergence of the iterative procedure (interconnecting). The number of local problems to be solved is optimized by a newly introduced light path tree algorithm. Finally some examples for the selection of local Maxwell solvers and numerical results are presented.