Quasinormal mode expansion of optical far-field quantities

Felix Binkowski, Fridtjof Betz, Rémi Colom, Martin Hammerschmidt, Lin Zschiedrich, and Sven Burger

Phys. Rev. B 102, 035432 – Published 23 July 2020

Abstract

Quasinormal mode (QNM) expansion is a popular tool to analyze light-matter interaction in nanoresonators. However, expanding far-field quantities such as the energy flux is an open problem because QNMs diverge with an increasing distance to the resonant systems. We introduce a theory to compute modal expansions of far-field quantities rigorously. The presented approach is based on the complex eigenfrequencies of QNMs. The divergence problem is circumvented by using contour integration with an analytical continuation of the far-field quantity into the complex frequency plane. We demonstrate the approach by computing the angular resolved modal energy flux in the far field of a nanophotonic device.

Received 24 March 2020
Revised 6 May 2020
Accepted 22 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035432

Physics Subject Headings (PhySH)

Electromagnetism Nanoantennas Nanophotonics

Optical microcavities

Electromagnetic field calculations

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Felix Binkowski ¹, Fridtjof Betz¹, Rémi Colom¹, Martin Hammerschmidt ², Lin Zschiedrich ², and Sven Burger ^1,2

¹Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany
²JCMwave GmbH, Bolivarallee 22, 14050 Berlin, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 102, Iss. 3 — 15 July 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
One-dimensional resonator defined by layers with different refractive indices, where $n_{2} > n_{1}$ . Electric field solutions, $E (x, ω)$ and $E^{\circ} (x, ω)$ , are obtained by solving the Helmholtz equation with a source term corresponding to incoming plane waves with unit amplitude. Only scattered fields (a.u.) outside the resonator are shown. (a) Diverging field $E (x, {\tilde{ω}}_{k, Δ}) = A e^{i (n_{1} {\tilde{ω}}_{k, Δ} / c) | x |}$ , where ${\tilde{ω}}_{k}$ is a resonance pole of $E (x, ω)$ and ${\tilde{ω}}_{k, Δ} = {\tilde{ω}}_{k} + Δ {\tilde{ω}}_{k}$ is a frequency close to ${\tilde{ω}}_{k}$ . (b) Illustration of resonance poles and integration contours corresponding to the RPE for the energy flux density given by Eq. (2). The analytical continuation of the energy flux density has resonance poles with negative and with positive imaginary parts. (c) Nondiverging field $E^{\circ} (x, {\tilde{ω}}_{k, Δ}) = B e^{- i (n_{1} {\tilde{ω}}_{k, Δ} / c) | x |}$ . (d) Constant product $E (x, {\tilde{ω}}_{k, Δ}) \cdot E^{\circ} (x, {\tilde{ω}}_{k, Δ})$ , which relates to the energy flux density.
Reuse & Permissions
Figure 2
Circular Bragg grating resonator with localized light source. (a) Geometry with an illustration of the electric field intensity (a.u.) of the QNM corresponding to the eigenfrequency ${\tilde{ω}}_{2}$ ; see Table 1. The gallium arsenide (GaAs) grating has a thickness of $240 nm$ and consists of an inner disk with a radius of $550 nm$ and 10 rings with a width of $340 nm$ and a periodicity of $500 nm$ . The grating is placed on a silicon dioxide ( ${SiO}_{2}$ ) layer with a thickness of $240 nm$ , which is coated from below with a gold (Au) layer of $300 nm$ thickness. The light source is modeled by a dipole emitter placed at the center of the inner disk. The dipole radiates at the frequency $ω_{0}$ and is oriented in $x$ direction. (b) Radiation diagram at $ω_{0} = 2 π c / (1360 nm)$ for the total modal expansion $s_{tot} (θ)$ computed by Eq. (2) and for the quasiexact solution of the energy flux density $s (θ)$ . The quantities are evaluated at $r = 1 m$ and $φ = 90^{\circ}$ , which corresponds to the $y z$ plane. (c) Modal decomposition of the radiation diagram for the contributions ${\tilde{s}}_{2} (θ)$ , ${\tilde{s}}_{3} (θ)$ , and ${\tilde{s}}_{4} (θ)$ .
Reuse & Permissions
Figure 3
Modal expansions of Purcell enhancement and PCE for the resonator with a localized light source shown in Fig. 2. Eigenfrequencies ${\tilde{ω}}_{1}, \dots, {\tilde{ω}}_{9}$ are considered; see Table 1. (a) Modal expansion of the Purcell enhancement. The contributions ${\tilde{Γ}}_{1} (λ_{0}), \dots, {\tilde{Γ}}_{4} (λ_{0})$ correspond to the eigenfrequencies ${\tilde{ω}}_{1}, \dots, {\tilde{ω}}_{4}$ , respectively. The remaining modal contributions are added to the remainder of the expansion $\sum_{k = 5}^{9} {\tilde{Γ}}_{k} (λ_{0}) + Γ_{r} (λ_{0})$ . The term $Γ_{r} (λ_{0})$ includes also modal contributions corresponding to eigenfrequencies outside the integration contour $C_{r}$ . (b) Modal expansion of the PCE. Total modal expansion, $η_{tot} (λ_{0}) = \sum_{k = 1}^{9} {\tilde{η}}_{k} (λ_{0}) + η_{r} (λ_{0})$ , single modal contributions, ${\tilde{η}}_{1} (λ_{0}), \dots, {\tilde{η}}_{4} (λ_{0})$ , and the sum of other contributions, $\sum_{k = 5}^{9} {\tilde{η}}_{k} (λ_{0}) + η_{r} (λ_{0})$ .
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics