New efficient and elegant method to solve electromagnetic scattering problems

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Many optical scattering problems can in principle be solved by a perturbation approach in which the electromagnetic field is expressed as a so-called Born series whose terms are computed recursively. In most optical scattering problems however, the Born series is divergent, which renders the method useless. Recently, in a study by the Optics Group of TU Delft, the Padé approximation has been applied in conjunction with the perturbation approach to extract an approximation of the scattered field from the diverging Born series. Not only is this method semi-analytical, which often is preferable over fully numerical methods, but it is moreover computationally  very fast and inexpensive because  no large system of equation has to be solved as with conventional methods.

The method has successfully been applied to several test problems. An example of the results of the Padé method applied to a two-dimensional problem is illustrated in Fig. 1, where the scattering of a plane wave by an infinitely long cylinder made of silver is shown. The method is efficient for inverse scattering problems as well.

Ref: T. A. van der Sijs, O. El Gawhary, and H. P. Urbach, “Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Padé approximants”, Phys. Rev. Research, 2, 013308 (2020).

Figure 1: The  z-component of the electric field in the case of scattering by an infinitely long cylinder. The cylinder has radius 100 nm and is made of silver and the wavelength in vacuum is 400 nm. The Born series in this case is strongly divergent; the largest term used to calculate the Padé approximant of order 5 is 3.4×106 as large as the incident field. a) The modulus of the analytical solution. b) The modulus and c) the local absolute error of the most accurate Padé approximant (which is of order 5).
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