Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Pad\'e approximants

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Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Padé approximants

T. A. van der Sijs, O. El Gawhary, and H. P. Urbach

Phys. Rev. Research 2, 013308 – Published 13 March 2020

Abstract

Electromagnetic scattering is the main phenomenon behind all optical measurement methods where one aims to retrieve the shape or physical properties of an unknown object by measuring how it scatters an incident optical field. Such an inverse problem is often approached by solving, several times, the corresponding direct scattering problem and trying to find the best estimate of the object which is compatible with a set of measurements. Despite the existence of numerical methods, a powerful way to solve those direct problems would be to use a perturbation approach where the field is expressed as a series, known as the Born series. The advantage of a perturbation approach stems from the fact that each term of the series has a clear physical meaning and can unveil much more about the scattering process than a purely numerical approach can offer. This method is however unpractical under so-called strong-scattering conditions because the corresponding Born series strongly diverges. In this work, we will show how to solve this problem by employing Padé approximants and how to treat electromagnetic problems well beyond the weak-scattering regime. This approach can represent an important building block to the application of the Born series to direct and inverse problems, with potential applications in superresolution, optical metrology, and phase retrieval.

Received 19 November 2019
Accepted 31 January 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013308

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Classical optics Interference & diffraction of light Metrology

Atomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

T. A. van der Sijs¹, O. El Gawhary^1,2,*, and H. P. Urbach¹

¹Optics Research Group, Imaging Physics Department, Delft University of Technology, Van der Waalsweg 8, 2628 CH Delft, Netherlands
²VSL Dutch Metrology Institute, Thijsseweg 11, 2629 JA Delft, Netherlands

^*o.elgawhary@tudelft.nl

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Vol. 2, Iss. 1 — March - May 2020

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Optics

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Figure 1
(a) Geometry of the 1D electromagnetic problem studied. The scatterer has relative permittivity $ɛ_{r}$ and thickness $d$ . (b) Absolute value of the analytic solution of the total field $U (z)$ . Each curve corresponds to a different medium: $ɛ_{r} = 4$ , blue-solid line, $ɛ_{r} = 4 + i$ , red-dashed line, $ɛ_{r} = 4 + 4 i$ black-dot dashed line. Also, $z_{1} = 1 μ m$ and $d = 500$ nm.
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Figure 2
Some perturbation orders of Eq. (6), for the case where $ɛ_{r} = 4 + i, d = 500$ nm and $λ = 800$ nm. The series clearly diverges for any value of $z$ .
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Figure 3
Approximation of the total field as obtained through Padé rational functions. (a) shows the approximation of the solution of the scattering problem by using the Padé approximants. The figure shows the absolute value of the $P_{6}^{6} (z)$ rational polynomial for the three different cases given in Fig. 1. Comparing the plots to Fig. 1, it is evident that the Padé approximants provide an accurate estimate of the solution. (b) shows the convergence to the analytical solution as reached by Padé rational functions for a medium with $ɛ_{r} = 4$ . The Padé approximants $P_{1}^{1} (z), P_{3}^{3} (z)$ and $P_{6}^{6} (z)$ are shown.
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Figure 4
Geometry of the 2D problem. The incident plane wave $U_{i}$ represents either the $z$ component of the electric field, $E_{z}$ , or the $z$ component of the magnetic field $H_{z}$ . In this work, we will only consider the TM polarization case (namely the $E_{z}$ case).
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Figure 5
Amplitude of the $ℓ$ -th term of the Born series, from $0 th$ to $17 th$ order, for cylinders of different sizes and materials. In all cases, the terms show an exponential growth.
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Figure 6
Analytical solution, the best Padé approximant, the error $e_{abs, N}$ of that approximant and the relative error per Padé order for four different configurations. (a)–(d) concern a silver cylinder with $R = 100 nm$ . (a) shows the analytical solution, (b) the Padé approximant $P_{5}^{5}$ , (c) the absolute error $e_{abs, 5} (r)$ , and (d) the relative energy error per Padé order. (e)–(h) concern a silicon cylinder with $R = 100 nm$ . (e) shows the analytical solution, (f) the Padé approximant $P_{4}^{4}$ , (g) the absolute error $e_{abs, 4} (r)$ , (h) the relative energy error per Padé order. (i)–(l) concern a silver cylinder with $R = 400 nm$ . (i) shows the analytical solution, (j) the Padé approximant $P_{12}^{12}$ , (k) the absolute error $e_{abs, 12} (r)$ , and (l) the relative energy error per Padé order. (m)–(p) concern a silicon cylinder with $R = 400 nm$ . (m) shows the analytical solution, (n) the Padé approximant $P_{11}^{11}$ , (o) the absolute error $e_{abs, 11} (r)$ , and (p) the relative energy error per Padé order. The black dashed circle shows the boundary of the cylinder.
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Physical Review Research

Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Padé approximants

T. A. van der Sijs, O. El Gawhary, and H. P. Urbach

Phys. Rev. Research 2, 013308 – Published 13 March 2020

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