Resonant-state expansion of light propagation in nonuniform waveguides

A rigorous approach for precise and efficient calculation of light propagation along nonuniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoing-wave boundary conditions, form a natural basis for expansion of the local electromagnetic field. Using such an expansion at fixed frequency, we convert the wave equation for light propagation in a nonuniform waveguide into an ordinary second-order matrix differential equation for the expansion coefficients depending on the coordinate along the waveguide. We illustrate the method on several examples of nonuniform planar waveguides and evaluate its efficiency compared to the aperiodic Fourier modal method and the finite element method, showing improvements of one to four orders of magnitude. A similar improvement can be expected also for applications in other fields of physics showing wave phenomena, such as acoustics and quantum mechanics.

Figure 1

(a) Sketch of the scattering geometry and the considered planar dielectric waveguide with a rectangular hole. (b) Electric field amplitude for excitation with WG mode 1 at $\hslash \omega =3\mathrm{eV}$ from the left, on a linear color scale as given, overlaid with the WG outline. (c) Relative power transmission $T_{ij}$, from incoming left WG mode $j$ to outgoing right WG mode $i$, as function of the photon energy $\hslash \omega$.

Figure 2

(a) RS wave numbers—poles of the GF of the BWG in the complex $k$ plane. The red line splits the $k$ plane into physical and unphysical half planes, according to the cut $\mathrm{\Gamma }$ in the $\xi$ plane. (b) Poles on the physical (circles) and unphysical (crosses) Riemann sheet and the cut (red line) of the GF in the complex $\xi$ plane, calculated for a photon energy $\hslash \omega$ of 3 eV ($\omega a\approx 3.04$).

Figure 3

Relative error of the S matrix $\widehat{S}$ versus computational time on a CPU Intel Core i7-5830 K. Data is shown for WG-RSE, a-FMM and ComSol, and $\hslash \omega$ of 1, 3, and 5 eV, as labeled. The basis size $N$ is indicated for the 3 eV WG-RSE data.

Figure 4

As Fig. 1 but for a double hole as sketched in (a). The structure has mirror symmetry with respect to the plane $x=0$. (b), (c) Electric field amplitude for excitation with WG mode 1 (b) and 3 (c), at $\hslash \omega =4.783\mathrm{eV}$ from the left, on a linear color scale as given, overlaid with the WG outline. (d) Relative power transmission $T_{ij}$, from left WG mode $j$ to right WG mode $i$, as function of the photon energy.

Figure 5

As Fig. 1, but filling the hole in the waveguide with gold. The discrete energies used are the ones tabulated in Ref. [31].

Figure 6

As Fig. 3, but filling the hole in the waveguide with gold.

Figure 7

As Fig. 6, but using as reference ${\widehat{S}}_0$ the ComSol solutions with $m=6$ (see Appendix pp6).

Figure 8

As Fig. 1, but filling the hole in the waveguide with $\epsilon =2.6$, and creating a cavity structure made of two 100 period Bragg mirrors of this perturbation with a period of $2L=1800\mathrm{nm}$, surrounding a cavity of length $2L$. (a) Schematic of the structure. (b) Electric field amplitude for excitation with WG mode 1 propagating from the left, on a linear color scale as given, overlaid with the WG outline. Data outside the stop-band at $\hslash \omega =1.23\mathrm{eV}$, in the stop band at $\hslash \omega =1.245\mathrm{eV}$, and at the cavity resonance $\hslash \omega =1.24585\mathrm{eV}$. (c) Relative power transmission $T_{11}$, reflection $R_{11}$, and loss $L_1=1-T_{11}-R_{11}$, as function of the photon energy $\hslash \omega$. (d) As (c), but restricting the S-matrix calculation to the WG mode.

Figure 9

As Fig. 3, but for the waveguide cavity structure and the three photon energies outside the stop-band at 1.23 eV, in the stop band at 1.245 eV, and on resonance at 1.24585 eV, see Fig. 8. The reference $2\times 2$ scattering matrix ${\widehat{S}}_0$ was calculated using the WG-RSE with the largest basis considered, $N=20000$.

Figure 10

Physical Riemann $\xi$ sheet. Symbols mark poles of the Green's function $G(x,x^{\prime };\xi )$ of the BWG. The cut $\mathrm{\Gamma }$ starting from the branch point at $\xi ={\omega }^2$ is shown by a red line. Green curve marks the path of integration.

Figure 11

Cut weight ${\sigma }_{\pm }$ along the cut $\mathrm{\Gamma }$ for a photon energy of $\hslash \omega =3\mathrm{eV}$. Other parameters as in Fig. 1 of the main text.

Figure 12

Relative error of the in-plane wave vector ${\varkappa }_j$ for the layer with the hole, for the WG modes ($j=1$: solid, $j=2$: dashed, $j=3$: dotted) versus basis size $N$ (a) and CPU time (b), for different photon energies $\hslash \omega$ as indicated. The approximate convergence scalings $\propto N^{-2.5}$ and $\propto t^{-1}$ are indicated.

Figure 13

Schematic of the structure used in ComSol for Fig. 3 of the main text. Blue indicates the waveguide, green indicates PMLs, vertical red lines mark ports. We use one port per waveguide mode.