Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov

Phys. Rev. B 96, 045129 – Published 20 July 2017

Abstract

We present an analytical formulation for the normalization of resonant states at oblique incidence in one- and two-dimensional periodic structures with top and bottom boundaries to homogeneous space, such as photonic crystal slabs and arrays of nanoantennas. The normalization is validated by comparing the resonant state expansion using one and two resonant states with numerically exact results. The predicted changes of resonance frequency and linewidth due to perturbations of refractive index or geometry can be used to study resonantly enhanced refractive index sensing as well as the influence of disorder. In addition, the normalization is essential for the calculation of the Purcell factor.

Received 21 March 2017
Revised 2 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.045129

Physics Subject Headings (PhySH)

Photonics Plasmons Polaritons

Bloch wave theory Bloch-Floquet theorem Classical electromagnetism Plane-wave expansion Rigorous coupled-wave analysis

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. Weiss^*, M. Schäferling, and H. Giessen

4th Physics Institute and Research Center SCoPE, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

N. A. Gippius

Skolkovo Institute of Science and Technology, Novaya Street 100, Skolkovo 143025, Russia

S. G. Tikhodeev

Department of Physics, Lomonosov Moscow State University, Moscow 119991, Russia and A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova Street 38, Moscow 119991, Russia

W. Langbein and E. A. Muljarov

Cardiff University, School of Physics and Astronomy, The Parade, Cardiff CF24 3AA, United Kingdom

^*t.weiss@pi4.uni-stuttgart.de

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Issue

Vol. 96, Iss. 4 — 15 July 2017

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Images

Figure 1
(a) Schematic of the considered one-dimensional photonic crystal slab defining the relevant geometrical parameters. All parameters have been chosen following Ref. [33]. The periodic layer consists of materials with refractive indices of 1.5 and 2.5, with the former being also the refractive index of the substrate. The superstrate is vacuum. (b) Photonic crystal slabs exhibit resonances that decay radiatively by emission to the far field. Consequently, the resonant field distribution grows exponentially to the exterior, as depicted here for the square of the electric field $E_{m}$ of a transverse-electric (TE) mode at $k_{x} = 5.236 μ m^{- 1}$ and a resonance energy of $1891.4 - 12.6 i$ meV integrated over one unit cell in the $x$ direction.
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Figure 2
The normalization of radiative resonances contains a volume and a surface contribution. The normalization volume spans in a lateral direction over one unit cell and contains at least the outermost interfaces to the homogenous top and bottom half spaces, as indicated by $V_{0}$ in panel (a). Increasing the volume of integration for the normalization, e.g., by adding a volume of height $Δ z$ in the substrate region, results in an additional volume contribution $I_{m}^{Δ V}$ that diverges for $Δ z \to \infty$ . Panel (b) displays the magnitude of the correctly normalized resonant electric field distribution in one unit cell for the same resonance as in Fig. 1. For this normalized field, the black solid line in panel (c) depicts $| I_{m}^{Δ V} |$ . This divergence is compensated by the modification $Δ S_{m}$ of the surface term, as indicated by the red solid line.
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Figure 3
Comparison of first-order perturbation theory (black solid and blue dashed lines) and numerically exact results (blue squares and black dots) for two transverse-magnetic (TM) resonances at $k_{x} = 0.2 μ m^{- 1}$ . The resonance energies are $2736.6 - 0.2 i$ meV and $2737.5 - 5.2 i$ meV; the corresponding normalized resonant electric field distributions can be seen in panels (a) and (b), respectively. Panels (c) and (d) show the resulting resonance energy and linewidth as a function of the refractive index in the high-index material of the periodic layer. The results of the first-order perturbation theory have been derived for a reference index of 2.5.
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Figure 4
Resonance energy (a) and linewidth (b) of the resonances depicted in Fig. 3 as a function of the in-plane momentum $k_{x}$ . The black dotted line in panel (a) denotes the Rayleigh anomaly (RA) due to the opening of the $(- 1)$ diffraction order in the substrate. As seen in panels (c) and (d), the predictions from the first-order perturbation theory (black solid and blue dashed lines) for the real and imaginary parts of $\partial E / \partial n$ ( $E = ℏ ω$ ) agree well with the numerically calculated derivatives $Δ E / Δ n$ (blue squares and black dots) for changes of the refractive index in the high-index material of the periodic layer.
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Figure 5
(a) The geometry of Fig. 1 is modified by shifting the low-index region of the periodic layer in every second unit cell by a distance $s$ . Thus, a superperiodicity is introduced, which allows coupling of the $(- 2^{'})$ and the $(1^{'})$ fundamental transverse-electric resonances (thin black dotted lines) at the border of the first Brillouin zone of the superperiodic system at $k_{x} = π / 2 P \approx 5.236 μ m^{- 1}$ . Using the resonant state expansion (black solid and dashed lines) for these two resonances taken at $s = 0$ allows for predicting the behavior of the resonance energies [panels (c) and (d)] and linewidths [panels (c) and (e)] of the coupled modes (red dots), shown here as a function of shift $s$ for $k_{x} = π / 2 P$ [panels (b) and (c)] and in-plane momentum $k_{x}$ for $s = 5$ nm [panels (d) and (e)].
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