Quasinormal-Mode Expansion of the Scattering Matrix

Open Access

Quasinormal-Mode Expansion of the Scattering Matrix

Filippo Alpeggiani, Nikhil Parappurath, Ewold Verhagen, and L. Kuipers

Phys. Rev. X 7, 021035 – Published 5 June 2017

Abstract

It is well known that the quasinormal modes (or resonant states) of photonic structures can be associated with the poles of the scattering matrix of the system in the complex-frequency plane. In this work, the inverse problem, i.e., the reconstruction of the scattering matrix from the knowledge of the quasinormal modes, is addressed. We develop a general and scalable quasinormal-mode expansion of the scattering matrix, requiring only the complex eigenfrequencies and the far-field properties of the eigenmodes. The theory is validated by applying it to illustrative nanophotonic systems with multiple overlapping electromagnetic modes. The examples demonstrate that our theory provides an accurate first-principles prediction of the scattering properties, without the need for postulating ad hoc nonresonant channels.

Received 22 November 2016

DOI:https://doi.org/10.1103/PhysRevX.7.021035

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)

Electromagnetism Photonic crystals Photonics

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Filippo Alpeggiani^1,2, Nikhil Parappurath¹, Ewold Verhagen¹, and L. Kuipers^1,2

¹Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
²Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

Popular Summary

A scattering matrix is an essential mathematical tool for physicists working in numerous fields. It quantitatively describes how light or particles change when they scatter off one another. First developed to solve problems in quantum field theory, it now constitutes the basic machinery for calculating many key physical quantities, such as the interaction probability of atomic nuclei, electrical transmission through extremely small conductors, or to what degree nanoparticles absorb light. In this work, we demonstrate that the scattering matrix of a system can be fully determined from the knowledge of its quasinormal modes—the states that exist for a system coupled to its environment in the absence of any driving input. In this sense, the quasinormal modes represent the bare framework around which the frequency response of the system is built.

We validate our formalism with some illustrative examples from the field of nanophotonics, confirming that it embodies an effective and powerful tool for making predictions from first principles. In a natural way, our theory accounts for the effective interaction among different quasinormal modes that originate from the coupling to a common external environment. For this reason, it is directly applicable to any number of multiple overlapping modes and to any arbitrary configuration of input-output channels. Moreover, the theory does not require any ad hoc assumptions, such as the fitting of an additional nonresonant background.

By casting the scattering matrix in these terms, it provides a deeper physical understanding of a system than other approaches. In particular, the theory represents a key for deciphering, understanding, and engineering the scattering properties of complex optical systems.

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Issue

Vol. 7, Iss. 2 — April - June 2017

Subject Areas

Photonics

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Images

Figure 1
Schematic of $m$ ports coupled to $n$ quasinormal modes with amplitudes $a_{j}$ ( $j = 1, \dots, n$ ) and linked by a direct-coupling term $C$ . The notation $s_{p +}$ and $s_{p -}$ ( $p = 1, \dots, m$ ) refers to the amplitude of incoming and outgoing waves, respectively.
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Figure 2
Application of the theory to a photonic crystal slab composed of a square lattice of air holes etched in a suspended silicon membrane. (a) Real and imaginary part (log scale) of the quasinormal-mode complex eigenfrequencies, together with the corresponding symmetry of the modes (even, odd) by inversion with respect to the slab middle plane. (b) Transmission intensity computed by expanding the scattering matrix on the quasinormal modes (red solid line) compared with the exact result by the Fourier modal method (dashed line) [35].
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Figure 3
(a) Circles: real and imaginary part (log scale) of the quasinormal-mode eigenfrequencies of a square lattice of L-shaped patterned structures in a silicon membrane. The unit cell of the structure is represented in the inset ( $t = 0.4 a$ , $l_{1} = 0.6 a$ , and $l_{2} = 0.3 a$ ). Note that the structure is not symmetric by inversion along the $z$ axis. (b) Total transmittance $T$ and (c) cross-polarized transmittance $T_{x y}$ computed by expanding the scattering matrix on the quasinormal modes (solid line) compared with the exact result by the Fourier modal method (dashed line) [35].
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Figure 4
(a) Circles: real and imaginary part (log scale) of the complex eigenfrequencies of a square lattice (lattice constant $a = 200 nm$ ) of 60-nm-diameter metallic particles embedded in a 200-nm dielectric slab ( $ϵ = 12.1$ ). (b) Transmission intensity computed from the quasinormal-mode expansion of the scattering matrix (red solid line) compared with a finite-element frequency-by-frequency calculation (dashed line) [36]. The inset shows a close-up of the spectrum in the highlighted frequency region. The dots indicate the frequency-by-frequency calculation.
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Figure 5
(a) Absorption cross section of a multilayered spherical nanoparticle constitued of a dielectric core and alternating layers of a Drude metal and a dielectric ( $ϵ = 2.1$ ), as shown in the inset. The red solid line is obtained from the quasinormal-mode expansion, whereas the dashed curve is the exact result from generalized Mie theory. The values of the radii of the different layers, starting with the inner one, are $r_{1} = 0.012 λ_{p}$ , $r_{2} = 0.0186 λ_{p}$ , $r_{3} = 0.138 λ_{p}$ , $r_{4} = 0.18 λ_{p}$ , with all lengths being expressed in units of the plasma wavelength $λ_{p} = 2 π c / ω_{p}$ . (b) Real and imaginary part of the quasinormal-mode eigenfrequencies included in the expansion. All modes are TM polarized. Circles and crosses refer to $l = 1$ and $l = 2$ modes, respectively, where $l$ is the azimuthal number.
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