Generalized coupled-mode formalism in reciprocal waveguides with gain, loss, anisotropy, or bianisotropy

In anisotropic or bianisotropic waveguides, the standard coupled-mode theory fails due to the broken link between the forward- and backward-propagating modes, which together form the dual mode sets that are crucial in constructing coupled mode equations. We generalize the coupled-mode theory by treating the forward- and backward-propagating modes on the same footing via a generalized eigenvalue problem that is exactly equivalent to the waveguide Hamiltonian. The generalized eigenvalue problem is fully characterized by two operators, i.e., $(\overline{\mathit{L}},\overline{\mathit{B}})$, wherein $\overline{\mathit{L}}$ is a self-adjoint differential operator, while $\overline{\mathit{B}}$ is a constant antisymmetric operator. From the properties of $\overline{\mathit{L}}$ and $\overline{\mathit{B}}$, we establish the relation between the dual mode sets that are essential in constructing coupled-mode equations in terms of forward- and backward-propagating modes. By perturbation, the generalized coupled-mode equation can be derived in a natural way. Our generalized coupled-mode formalism (GCMF) can be used to study the mode coupling in waveguides that may contain gain, loss, anisotropy, or bianisotropy. We further illustrate how the generalized coupled theory can be used to study the modal coupling in anisotropy and bianisotropy waveguides through a few concrete examples.

Figure 1

(a) The schematic of elliptical waveguide with perfect electric conductor boundary with long axis $a={\lambda }_0$ and short axis $b=0.5{\lambda }_0$, where ${\lambda }_0$ is vacuum wavelength. (b) The real (red line) and imaginary (black line) part of effective modal indices, calculated from full-wave simulation using finite element method, as a function of ${\varepsilon }_r$. The two purple dashed lines in (b) indicate the range of ${\varepsilon }_r$, where GCMT and GCMF are applied. In (c) and (d), the real and imaginary parts of effective modal indices $n_{\text{eff}}$ as a function of increased magnitude of anisotropy $\mathrm{\Delta }\varepsilon$ for forward/backward-propagating modes, calculated from GCMF (red open circles), GCMT [43], (blue open squares), and full-wave simulation (gray line) are shown. In (e) and (f) [and (g) and (h)], the real/imaginary parts of $Ex$ obtained from full-wave simulation is shown for the modes $\varphi$ [and $\psi ]$, respectively, at ${\varepsilon }_r=-0.5$ marked by solid circle in (b).

Figure 2

Effective modal indices $n_{\text{eff}}$ as function of the magnetoelectric coupling ($\mathrm{\Delta }\chi$) that accounts for the bianisotropy. The cross section of the waveguide is elliptical, with long (short) axis $a=0.25{\lambda }_0$ ($b=0.2{\lambda }_0$). In (a), the $n_{\text{eff}}$ of forward/backward-propagating x-polarized mode is denoted by black/red solid circles. The gray solid lines are calculated from full-wave simulations. (b) shows modal coefficients of the hybridized mode upon a small perturbation, as $\mathrm{\Delta }\chi$ varies. The real part of x component of forward/backward normalized electric field is shown in (c) and (d), and the vector plots of the real part of in-plane electric field are also shown in (c) and (d) indicated by the arrows, the length of which is proportional to the magnitude of the vector field. (e), (f) show the imaginary part of field, and the other setting is same to (c) and (d). In (c)–(f), the field profiles are obtained from COMSOL at $\mathrm{\Delta }\chi =0.4$.

Figure 3

Effective mode indices $n_{\text{eff}}$ versus perturbation $\mathrm{\Delta }\varepsilon$ using GCMF, GCMT, and full-wave simulation. The structure is a double elliptical waveguide with long axis $a=0.3{\lambda }_0$, short axis $b=0.1{\lambda }_0$, and the center distance $d=0.75{\lambda }_0$. The red/blue/gray markers represent the results from GCMF/GCMT/full wave simulations, respectively. The (a), (b) and (c), (d) represent forward/backward-propagating modes $n_{\text{eff}}$, respectively. (a), (c) and (b), (d) represent real/imaginary parts of $n_{\text{eff}}$, respectively. The normalized $\mathrm{Re}\left(e_x\right)$ at $\mathrm{\Delta }\varepsilon =0.03$ calculated by GCMF are shown in (e)–(h), where (e), (f) for $\mathrm{Re}\left(e_x\right)$ of forward-propagating mode 1/2, and (g),(h) for $\mathrm{Re}\left(e_x\right)$ of backward-propagating mode 1/2.

Figure 4

The symmetry relations between the forward- and backward-propagating modes. $A$ represents a point on the reference plane of the cross section for the forward-propagating mode, and $A^{\prime }$ represents the corresponding point on the reference plane for the backward-propagating mode. In (a) and (b), $A$ and $A^{\prime }$ share the same coordinates in $x\text{−}y$ plane $[\mathit{r}(x,y)={\mathit{r}}^{\prime }(x,y)]$, while in (c), $A$ and $A^{\prime }$ are symmetric about the center $[\mathit{r}(x,y)=-{\mathit{r}}^{\prime }(x,y)]$. $\mathit{E}$ and $\mathit{H}$ stands for the transverse component of electromagnetic field, and asterisk $*$ stands for the operation of complex conjugate. The middle mirror represents the symmetry plane of the forward- and backward-propagating modes.