Verfahren zur Analyse und Synthese quasioptischer Komponenten und Systeme (English shop)

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The fundamental problem of quasi-optical beam shaping is the targeted wavefront conversion of the electromagnetic field. Quasi-optical components such as gratings, phase elements and mirrors, or their combined arrangement into a quasi-optical system, serve to perform this wavefront conversion. The present work compiles methods that are suitable both for the analysis and for the synthesis of these components. Furthermore, a synthesis method is presented with which the above-mentioned problem of beam shaping can be solved.

Specifically, the following topics are treated in the analysis and synthesis parts of this work: complex point sources, Rayleigh-Sommerfeld diffraction integral, method of discrete singularities, physical optics, coupled wave method, Li’s inversion rules, Gerchberg-Saxton algorithm, analytical beam shaping, BFGS method, LBFGS method.

Complex point sources are suitable for modelling aperture radiators. With complex point sources, analytically exact solution formulae are available that are valid throughout the entire space. The Rayleigh-Sommerfeld diffraction integral and its numerical implementation are treated, since calculating the wave propagation between two planes of a quasi-optical system is a fundamental task. The method of discrete singularities is presented because it permits a highly accurate calculation of mirror systems in the two-dimensional case. However, with the computing power available nowadays it is at present still unsuitable for use in synthesis in the three-dimensional case. Physical optics is therefore treated as an alternative. Although it is an approximate method, smooth mirrors in the three-dimensional case can be calculated quite accurately with it. The coupled wave method is expedient for calculating the important class of rectangular gratings. Both perfectly conducting and dielectric gratings are treated here. In particular, the treatment of dielectric gratings in H-polarisation has for decades been subject to convergence problems, which can, however, be remedied by following the aforementioned inversion rules of Li. In order to give an overview of the synthesis methods known from optics, the Gerchberg-Saxton algorithm for phase retrieval and analytical beam shaping are examined. Concluding the synthesis part, the BFGS method and its more memory-efficient variant, the LBFGS method, are treated, since these two quasi-Newton methods are used in the synthesis method presented.

In addition, the following new methods are developed in this work:

Fast far field approximation combined with physical optics. The fast far field approximation is a method that was originally devised to accelerate the evaluation of integral equations in scattering problems – similar to the fast multipole methods. In this work, the fast far field approximation is combined with physical optics in order to accelerate the analysis of three-dimensional mirrors. Ultimately, this achieves the goal of significantly reducing the computation time of the presented synthesis method in the design of dual-mirror systems.

Integral method for volume gratings. This method permits the analysis of gratings with arbitrary profiles consisting of an inhomogeneous material. It is based on a coupled system of integral equations, where the kernels of the occurring integrals are of Picard type. On account of this particular property, recursion formulae are presented in this work with which these integrals can be calculated with almost linear computational complexity as well as with linear memory requirements. This method is thus ideally suited to solving iteratively the system of equations resulting from a numerical implementation. Furthermore, a powerful preconditioner is presented for E-polarisation.

Synthesis method for quasi-optical beam shaping. The presented method makes it possible to determine the surface functions of quasi-optical components and systems in such a way that a targeted wavefront conversion of the electromagnetic field can be carried out. For this purpose, scalar products and their induced norms are introduced on all surfaces of interest, and a target functional is defined on the output plane or output surface. The dependencies of the electromagnetic fields between these surfaces can be captured by integral operators, which themselves in turn depend on the analysis method applied. In order to minimise the target functional, the analytical gradient with respect to the surface functions to be determined is derived by means of the calculus of variations. The decisive step here lies in reformulating the first variation into directional derivatives, or scalar products of partial derivatives and variations of the surface functions. Here the partial derivatives form the components of the analytical gradient and exist in each case on the surfaces to be determined. With the derivatives determined in this way, one possesses the valuable information as to how the surface must be changed at each point so that the value of the target functional can be reduced. In a subsequent numerical implementation on a computer, the surface functions and analytical gradients can consequently be discretised arbitrarily finely. However, it is also necessary to find a gradient method that can cope with the very large number of variables to be optimised. The LBFGS method, presented in detail, is well suited to this. In order to control the smoothness of the optimised surface functions, two types of smoothing are performed. By means of a low-pass filtering of the surfaces, these can be smoothed globally to a certain extent, while a moving average filtering allows the surfaces to be smoothed locally. The latter variant offers the advantage that only those regions of the surfaces need to be smoothed in which the radius of curvature falls below a previously defined limit.

In Chapter 5, Quasi-optical beam shaping, the synthesis method is first applied by way of example to ideal phase gratings, which are designed as beam splitters for the far field in both the one-dimensional and the two-dimensional case. The efficiencies achieved by the splitters are generally well above 90%. Subsequently, it is discussed how rectangular gratings can be dimensioned as beam splitters and star couplers with very high efficiencies. For this purpose, the dielectric rectangular gratings are optimised with an evolution strategy, while the target functions of the perfectly conducting rectangular gratings considered, which depend only on the height and the width of the groove, are evaluated graphically. The quasi-optical 3 dB hybrid is likewise discussed. It is shown how this can be realised by means of a grating or, in the simplest case, by means of a dielectric plate. Finally, within the framework of the presented synthesis method, the analytical gradient for a dual-mirror system is derived. This dual-mirror system constitutes a quasi-optical beam shaper, which can accordingly be designed in general with the aid of the synthesis method.

As an important special case of a beam shaper, this dual-mirror system has been designed several times, taking into account the problem of element spacing, as a quasi-optical power combiner or power divider at an operating frequency of 150 GHz. A vectorial field measurement system is used to verify the designed dual-mirror systems. Representative of the various designs, the calculated and measured field patterns of a 3 × 3 power combiner are compared, whereby very good agreement can be established.