Quasimodal expansion of electromagnetic fields in open two-dimensional structures

A quasimodal expansion method (QMEM) is developed to model and understand the scattering properties of arbitrary shaped two-dimensional open structures. In contrast with the bounded case which has only a discrete spectrum (real in the lossless media case), open resonators show a continuous spectrum composed of radiation modes and may also be characterized by resonances associated to complex eigenvalues (quasimodes). The use of a complex change of coordinates to build perfectly matched layers allows the numerical computation of those quasimodes and of approximate radiation modes. Unfortunately, the transformed operator at stake is no longer self-adjoint, and classical modal expansion fails. To cope with this issue, we consider an adjoint eigenvalue problem whose eigenvectors are biorthogonal to the eigenvectors of the initial problem. The scattered field is expanded on this complete set of modes leading to a reduced order model of the initial problem. The different contributions of the eigenmodes to the scattered field unambiguously appears through the modal coefficients, allowing us to analyze how a given mode is excited when changing incidence parameters. This gives physical insights to the spectral properties of different open structures such as nanoparticles and diffraction gratings. Moreover, the QMEM proves to be extremely efficient for the computation of local density of states.

Figure 1

Sketch of the studied structures and notations.

Figure 2

Guided modes, continuous spectrum, and leaky modes in an open waveguide.

Figure 3

Loci of the eigenfrequencies in the complex $\omega$ plane. Theoretical continuous spectrum (blue dashed line) is well approximated by discrete eigenvalues corresponding to PML modes (blue circles). The leaky modes unveiled by shifting the continuous spectrum in the complex plane have frequencies represented by red squares.

Figure 4

Geometry and mesh of the structure (a) and field maps $\text{Re}\left(E_z\right)$ for the eigenmodes 1 (b), 2 (c), and 3 (d).

Figure 5

Coupling coefficients $P_n$ as a function of $\lambda$ and ${\theta }_0$ for the modes 1 (top) and 2 (bottom). Horizontal dashed lines indicate the resonant wavelength.

Figure 6

Absorption cross section as a function of $\lambda$ for different incident angles ${\theta }_0$.

Figure 7

Absorption cross-section curves computed with QMEM as a function of $\lambda$ for ${\theta }_0={143}^{\circ }$, for different values of the number $M$ of modes retained in the expansion. The thick red curve corresponds to the reference values computed by solving the diffraction problem.

Figure 8

Electric field at $\lambda =10.2\mu \mathrm{m}$ and ${\theta }_0={143}^{\circ }$ calculated by solving the diffraction problem (a) and by the QMEM (b) with 50 modes.

Figure 9

Local density of states at $\lambda =10.2\mu \mathrm{m}$ calculated by solving the diffraction problem (a) and by the QMEM (b) with $M=500$ modes.

Figure 10

Setup of the problem for the lamellar grating. (a) Sketch of the studied diffraction grating. Parameters are $w=0.1\mu \mathrm{m}$, $h^g=3\mu \mathrm{m}$, ${\varepsilon }_{g^{\prime }}=16$, ${\varepsilon }^{-}=4.84$, ${\varepsilon }^{+}=1$, and $d=3\mu \mathrm{m}$; all materials are nonmagnetic ($\mu =1$.) (b) Computational cell for the FEM calculations. Top and bottom PML have stretching coefficient ${\zeta }^{+}={\zeta }^{-}=e^{i\frac{\pi }{4}}$ and their thicknesses are ${\widehat{h}}^{\pm }={\lambda }^{\mathrm{ref}}/\sqrt{{\varepsilon }^{\pm }}$. The thicknesses of the substrate and superstrate are $h^{\pm }={\lambda }^{\mathrm{ref}}/10$, with ${\lambda }^{\mathrm{ref}}=14\mu \mathrm{m}$. We apply quasiperiodicity conditions on the lateral boundaries with $\alpha =0\text{rad}{\text{m}}^{-1}$ or $\alpha ={10}^5\text{rad}{\text{m}}^{-1}$ and Neumann homogeneous boundary conditions on the outward boundaries of the PML. Maximum mesh element size is set to be ${\lambda }^{\mathrm{mesh}}/\left[20\sqrt{\left|\text{Re}\left(\varepsilon \right)\right|}\right]$, where ${\lambda }^{\mathrm{mesh}}=11\mu \mathrm{m}$.

Figure 11

Spectrum of the problem and leaky modes for $\alpha =0\text{rad}{\text{m}}^{-1}$. Top: eigenfrequencies in the complex plane (blue crosses) and theoretical curves of the continuous spectrum (dashed red and black curves for the substrate and the superstrate, respectively). The inset shows the position of the eigenvalues corresponding to the six leaky modes studied. Bottom: real part of $H_z$ for these six leaky modes.

Figure 12

Coupling coefficients $P_n$ as a function of the incident wavelength $\lambda$ (each curve is labeled by the corresponding mode index). The dashed vertical lines correspond to the position of the resonant wavelength ${\lambda }_n$ associated with each leaky mode.

Figure 13

Comparison between direct problem and QMEM. (a) and (b): reflection and transmission coefficients in the zeroth order $R_0$ (dashed blue curve) and $T_0$ (solid red curve). (c) and (d): relative integrated error ${\mathfrak{E}}_{\mathrm{int}}^{\mathrm{r}}$ (dashed dotted black curve), absolute errors on transmission ${\mathfrak{E}}_T^{\mathrm{a}}$ (solid red curve) and reflection ${\mathfrak{E}}_R^{\mathrm{a}}$ (dashed blue curve).