Abstract

A novel method is developed in this paper to characterize the band diagram and band modal fields of gyromagnetic photonic crystals that support topological one-way edge states. The proposed method is based on an integral equation formulation that utilizes the broadband Green’s function (BBGF). The BBGF is a hybrid representation of the periodic lattice Green’s function with imaginary extractions that has accelerated convergence and is suitable for broadband evaluations. The effects of the tensor permeability of the gyromagnetic scatterers are incorporated in a new formulation of surface integral equations (SIEs) with BBGF as the kernel that can be solved by the method of moments. The results are compared against Comsol simulations for various cases to demonstrate the accuracy and efficiency of the proposed method. Simulations results are illustrated and discussed for the modes of topological photonic crystals in relation to the physics of degeneracy, applied magnetic fields, and bandgaps.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Topological photonic crystals [1] have attracted increasing interests in recent years due to their peculiar one-way edge modes and potential applications in unidirectional wave guidance. One method to implement topological photonics is to use the magnetic field response of gyromagnetic materials placed in external DC magnetic fields to break the time-reversal symmetry [2,3]. Scattering from gyromagnetic scatterers and calculation of topological invariants are thus of interest to be examined [4]. Researchers have developed various numerical methods for periodic scatter analysis, including the plane wave expansion method [511], the multiple scattering method [12,13], the finite element method (FEM), the finite difference method in time domain (FDTD) [14,15], etc. Of special interest is an integral equation (IE) based method [1620] that is recently being developed to characterize the band structure of periodic scatterers. The method introduces a novel form of broadband representation of the periodic Green’s function, named the broadband Green’s function (BBGF). Combined with the method of moments, the surface integral equation (SIE) is converted into a linear eigenvalue problem of moderate size, leading to band structure within a broad frequency band simultaneously. The linear eigenvalue problem is defined on the basis of the surface currents projected on to the Floquet plane waves. The IE based method is expected to have better accuracy and associated with less number of unknowns. Key to the BBGF technique is a hybrid representation of the periodic lattice Green’s function, which extracts out a fixed wavenumber component of the Green’s function to accelerate the convergence rate of the spectral representation of the periodic Green’s function. Such mixed spatial-spectral domain representation is related to the well-known Ewald summation technique [21] and similar fast convergence representations [22]. However, the BBGF formulation is simpler and retains a separable wavenumber dependence as in its original spectral domain series, rendering it suitable for broadband evaluations. The hybrid representation of BBGF is also attractive as it can be readily generalized to represent the Green’s function including the periodic scatterers by replacing the Floquet planewaves with normalized modal fields [20,23,24], which can be subsequently used for dealing with scattering from bounded periodic arrays [25]. Similar techniques have also been applied to construct Green’s functions of irregular cavities effectively [2628].

In the original development of the hybrid representation of the lattice Green’s function [6,19], a real low wavenumber component is extracted out, leading to improved convergence of the Floquet plane wave expansion, yet the term being extracted out requires special techniques to be evaluated accurately. This technique has been applied to characterize the band diagram of periodic PEC scatterers [16] and dielectric scatterers [19]; both of them convert the SIEs into a linear eigenvalue problem, yielding the band diagram from eigenvalues and modal fields from eigenvectors. The seminal work in [16] and [19] demonstrates the efficiency and potential of the BBGF technique in band characterization.

Recently, it was found that the extracted term can be chosen at an imaginary wavenumber, at which the extracted term can be directly evaluated in exponentially converging spatial series [29]. The imaginary wavenumber extraction also eliminates the unnecessary real-axis pole in the remaining spectral series due to the extraction. Thus both spatial series and spectral series converge faster than using real low wavenumber extractions. In this paper, we adopt such new broadband representation of the periodic Green’s function in characterizing the photonic bands of periodic gyromagnetic scatterers placed in external DC magnetic fields. The external magnetic fields induce a tensor permeability of the gyromagnetic scatterers and break the time-reversal symmetry, leading to nonzero Chern number and the topological unidirectional edge modes [2,3].

The tensor permeability of the scatterer leads to three unknowns in the two coupled SIEs: the pilot electric field component along the axis, its normal derivative and its tangential derivative. In discretizing the SIEs, the roof-top basis function is used to represent the pilot field unknowns so that the tangential derivative is directly related to the pilot field component. The BBGF with an imaginary wavenumber extraction is applied to decouple the SIEs into a boundary integral part and a plane wave expansion part of fast convergence rate. The former rigorously accounts for field discontinuities across the boundary, while the latter is singularity free and efficient to be evaluated over a broad frequency range. Such hybrid representation of the SIEs is then converted into a linear eigenvalue problem of moderate size due to the fast convergence of the spectral expansions. The leading eigenvalues and eigenvectors are directly related to the band structure and modal fields of the photonic crystal, respectively, yielding effective broadband characterization of the photonic crystals. Modeling results show that for the periodic gyromagnetic scatterers under study, degeneracy between second and third band is lifted and a band gap is opened due to the applied DC magnetic fields. Comparisons with the Comsol simulations show good agreement and confirm the effectiveness of the proposed method. These results are also consistent with the physical discussions in [2,3] on the non-zero Chern numbers and unidirectional edge states.

Although IE-BBGF based method has been used to characterize bands of PEC [16] and dielectric periodic scatterers [19], we are not aware about its application in characterizing periodic gyromagnetic scatterers. Since tangential derivatives of the surface unknowns are involved in the SIEs, pulse basis is no longer valid. The roof-top basis function is thus chosen to represent the surface unknowns, relating the tangential derivatives directly to the surface unknowns. It is the first time that high order basis function and Galerkin’s method be used in complementary to the BBGF in characterizing the band structure. In the earlier work of the BBGF, pulse-basis function and point-matching test scheme have been applied [16,19].

The structure of this paper is organized as follows. In Section 2, the novel formulation to calculate the band diagram and modal fields using SIE and the BBGF is presented. In Section 3, several numerical results are discussed and compared against Comsol simulations to show its accuracy and efficiency. Finally, in section 4, the conclusions are drawn.

2. Formulating band diagram and modal fields using the BBGF

In this paper, we analyze the band of the topological photonic crystal proposed in [2] as illustrated in Fig. 1. In this design, a periodic array of circular gyromagnetic scatterers with permittivity $\varepsilon$ and tensor permeability $\overline {\overline {\mu }}$ are embedded in the air background with permittivity $\varepsilon _{0}$ and permeability $\mu _{0}$. The lattice is in the xOy plane, and the circular cylindrical scatters are of radius $r$. Let the primary lattice vectors be $\overline {a}_{1}$ and $\overline {a}_{2}$ with lattice constant $a$ and the unit cell area $\Omega =\left \vert \overline {a}_{1}\times \overline {a} _{2}\right \vert$. $S_{pq}$ denotes the boundary of the scatterer in the $(p,q)$-th cell and $A_{pq}$ is the domain of the scatterer in the $(p,q)$-th cell bounded by $S_{pq}$.

 

Fig. 1. The structure of the gyromagnetic photonic crystals under study. The radius of cylindrical scatterers $r=0.11a$, where $a$ is the lattice constant. The scatterers with permittivity $\varepsilon$ and permeability $\overline {\overline {\mu }}$ are surrounded by air with permittivity $\varepsilon _0$ and permeability $\mu _0$.

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Distinct from the regular dielectric scatterers studied in [19], the gyromagnetic scatterers are characterized by a permeability tensor when placed in an external DC magnetic field along the $z$-axis.

$$\overline{\overline{\mu}}=\left[ \begin{array} [c]{ccc} \mu & i\kappa & 0\\ -i\kappa & \mu & 0\\ 0 & 0 & \mu_{0} \end{array} \right]$$
When the external DC field vanishes, $\kappa =0$, and $\mu =\mu _{0}$, it simplifies to the special case studied in [19].

2.1 Surface integral equations and the discretization

To exhibit exotic topological wave behavior, the TMz (Ez) polarization is of interest. We denote the electric field outside scatterers $\overline {E}_{0}=\widehat {z}\psi _{0}$ and inside scatterers $\overline {E}_{1}=\widehat {z}\psi _{1}$. It can be shown that $\psi _{0}$ and $\psi _{1}$ satisfy the homogeneous Helmoholtz equations

$$\nabla^{2}\psi_{0}+\omega^{2}\varepsilon_{0}\mu_{0}\psi_{0}=0$$
$$\nabla^{2}\psi_{1}+\omega^{2}\varepsilon\widetilde{\mu}\psi_{1}=0$$
where $\widetilde {\mu }=\frac {\mu ^{2}-\kappa ^{2}}{\mu }$. Following the procedures outlined in [17], one can derive the extinction theorem,
$$\begin{aligned}-\int_{S_{00}}d\overline{\rho}^{\prime}\left( \psi_{0}\left( \overline{\rho }^{\prime}\right) \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{0}-g_{P} ^{0}\widehat{n}^{\prime}\cdot\nabla^{\prime}\psi_{0}\left( \overline{\rho}^{\prime }\right) \right) =\left\{ \begin{array} [c]{cc} \psi_{0}\left( \overline{\rho}\right) & \overline{\rho}\textrm{ outside scatterer}\\ 0 & \overline{\rho}\textrm{ inside scatterer} \end{array} \right.\end{aligned}$$
$$\begin{aligned}\int_{S_{00}}d\overline{\rho}^{\prime}\left( \psi_{1}\left( \overline{\rho }^{\prime}\right) \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{1}-g_{P} ^{1}\widehat{n}^{\prime}\cdot\nabla^{\prime}\psi_{1}\left( \overline{\rho}^{\prime }\right) \right) =\left\{ \begin{array} [c]{cc} \psi_{1}\left( \overline{\rho}\right) & \overline{\rho}\textrm{ inside scatterer}\\ 0 & \overline{\rho}\textrm{ outside scatterer} \end{array} \right.\end{aligned}$$
where $g_{P}^{0}$ is the periodic lattice Green’s function defined in the background media and $g_{P}^{1}$ is the lattice Green’s function defined in the scatterer,
$$g_{P}^{0}\left( k_{0},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right) =\sum_{\overline{R}}\left[ \exp\left( i\overline{k}_{i}\cdot\left( \overline{R} \right) \right) g(k_{0};\overline{\rho},\overline {\rho}^{\prime}+\overline{R})\right]$$
$$g_{P}^{1}\left( k_{1},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right) =\sum_{\overline{R}}\left[ \exp\left( i\overline{k}_{i}\cdot\left( \overline{R} \right) \right) g(k_{1};\overline{\rho},\overline {\rho}^{\prime}+\overline{R})\right]$$
where $k_{0}=\omega \sqrt {\varepsilon _{0}\mu _{0}}$ and $k_{1}=\omega \sqrt {\varepsilon \widetilde {\mu }}$ are the wavenumbers in the background and scatterers, respectively. $g\left ( k;\overline {\rho },\overline {\rho }^{\prime }\right ) =\frac {i}{4}H_{0}^{\left ( 1\right ) }\left ( k\left \vert \overline {\rho }-\overline {\rho }^{\prime }\right \vert \right )$ is the free-space Green’s function in the media of wavenumber $k$. $\overline {k}_{i}$ is the Bloch wavevector in the irreducible Brillouin zone . $\overline {k}_{i}=\beta _{1}\overline {b} _{1}+\beta _{2}\overline {b}_{2},0\leq \beta _{1},\beta _{2}\leq \frac {1} {2};$ $\overline {b}_{1}=2\pi \frac {\overline {a}_{2} \times \widehat {z}}{\Omega },\overline {b}_{2}=2\pi \frac {\widehat {z} \times \overline {a}_{1}}{\Omega }$ are the reciprocal lattice vectors. $\overline {R}=p_a\overline {a}_1+q_a\overline {a}_2$ for any integer $p_a$ and $q_a$.

Distinct from the scalar case studied in [19], the pilot fields $\psi _{0}$ and $\psi _{1}$ are connected through the boundary conditions

$$\psi_{0}=\psi_{1}$$
$$\frac{1}{\mu_{0}}\widehat{n}\cdot\nabla\psi_{0}=\frac{1}{\widetilde{\mu} }\widehat{n}\cdot\nabla\psi_{1}+i\eta\widehat{t}\cdot\nabla\psi_{1}$$
where $\widetilde {\mu }=\frac {\mu ^{2}-\kappa ^{2}}{\mu },\eta =-\frac {\kappa }{\mu ^{2}-\kappa ^{2}}$. $\widehat {n}$ represents the unit vector of the outward normal on the boundary of the scatterer, and $\widehat {t}$ represents the tangential unit vector on the boundary. The direction of $\widehat {t}$ is chosen such that $\widehat {n},\widehat {t},\widehat {z}$ forms a right hand triplet.

Substituting the boundary conditions into the extinction theorems leads to the surface integral equations (SIEs)

$$\begin{aligned}&\int_{S_{00}}d\overline{\rho}^{\prime} \psi_{1}\left( \overline{\rho }^{\prime}\right) \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{0}\\ &-\int_{S_{00}}d\overline{\rho}^{\prime}g_{P}^{0}\mu_{0}\left( \frac{1}{\widetilde{\mu}}\widehat{n}^{\prime}\cdot\nabla^{\prime}\psi _{1}\left( \overline{\rho }^{\prime}\right) +i\eta\widehat{t}^{\prime}\cdot\nabla^{\prime}\psi_{1}\left( \overline{\rho }^{\prime}\right) \right) =0,\quad \overline{\rho}\longrightarrow S_{00}^{-}\end{aligned}$$
$$\int_{S_{00}}d\overline{\rho}^{\prime}\left( \psi_{1}\left( \overline{\rho }^{\prime}\right) \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{1}-g_{P} ^{1}\widehat{n}^{\prime}\cdot\nabla^{\prime}\psi_{1}\left( \overline{\rho}^{\prime }\right) \right) =0,\quad\overline{\rho}\longrightarrow S_{00}^{+}$$

To solve the SIEs, define $u_{1}=\widehat {n}^{\prime }\cdot \nabla \psi _{1}\left ( \overline {\rho }^{\prime }\right )$ and represent the unknowns $\psi _{1}$ and $\widehat {n}\cdot \nabla \psi _{1}$ using roof-top basis function. Roof-top basis function is defined from $(t_{n-1} = t_{n}-\Delta t)$ to $(t_{n+1} = t_{n}+\Delta t)$

$$f_{n}(t)=\left\{ \begin{array} [c]{c} \frac{1}{\Delta t}\left( t-t_{n}+\Delta t\right) ;\qquad t_{n}-\Delta t\leqslant t\leqslant t_{n}\\ -\frac{1}{\Delta t}\left( t-t_{n}-\Delta t\right) ;\qquad t_{n}\leqslant t\leqslant t_{n}+\Delta t\\ 0;\qquad otherwise \end{array} \right.$$
Thus the derivation of $f_n\left (t\right )$ can be expressed as
$$f^{\prime}_{n}(t)=\left\{ \begin{array} [c]{c} \frac{1}{\Delta t} ;\qquad t_{n}-\Delta t\leqslant t\leqslant t_{n}\\ -\frac{1}{\Delta t} ;\qquad t_{n}\leqslant t\leqslant t_{n}+\Delta t\\ 0;\qquad otherwise \end{array} \right.$$
$\psi _{1}$ and $u_{1}$ are expanded as
$$\psi_{1}=\sum_{n}{\psi_{1n} f_{n}(t)}$$
$$u_{1}=\sum_{n}{u_{1n} f_{n}(t)}$$
And this leads to the tangential derivative component related to the field component through
$$\widehat{t}^{\prime}\cdot\nabla^{\prime}\psi_{1} = {\displaystyle\sum_{n}} \widehat{t}^{\prime}\cdot\nabla^{\prime}\left[ \psi_{1n}f_{n}\left(t^{\prime}\right) \right] = {\displaystyle\sum_{n}} \psi_{1n}f_{n}^{\prime}\left( t^{\prime}\right)$$

Then the method of moments (MoM) with Galerkin’s testing scheme is applied to convert Eq. (6) into matrix equations

$$\left( \overline{\overline{A}}_{0}-i{\mu}_{0}{\eta}\overline{\overline{C}}\right) \overline{\psi}_{1}-\frac{\mu_{0}}{\widetilde{\mu}}\overline{\overline{B}}_{0}\overline{u}_{1}=0$$
$$\overline{\overline{A}}_{1}\overline{\psi}_{1}-\overline{\overline{B}} _{1}\overline{u}_{1}=0$$

As the boundary $S_{00}$ is discretized into $N$ patches, the $\overline {\overline {A}}_{0},\overline {\overline {B}}_{0}, \overline {\overline {C}},\overline {\overline {A}}_{1},\overline {\overline {B}}_{1},$ are $N\times N$ matrices, and $\overline {\psi }_{1} ,\overline {u}_{1}$ are $N\times 1$ column vectors. Let $m,n=1,2,\ldots ,N.$ The matrix elements are, respectively,

$$\left[\overline{\overline{A}}_0\left( k_{0},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right)\right]_{mn}= {\displaystyle\int\limits_{S_{m}}} dtf_{m}(t) {\displaystyle\int\limits_{S_{n}}} dt^{\prime}\left[ \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{0}\right] f_{n}(t^{\prime})$$
$$\left[\overline{\overline{B}}_0\left( k_{0},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right)\right]_{mn}= {\displaystyle\int\limits_{S_{m}}} dtf_{m}(t) {\displaystyle\int\limits_{S_{n}}} dt^{\prime}g_{P}^{0}f_{n}(t^{\prime})$$
$$\left[\overline{\overline{C}}\left( k_{0},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right)\right]_{mn}=\int\limits_{S_{m}}dtf_{m}\left( t\right) \int\limits_{S_{n}}dt^{\prime} g_{P}^{0} f_{n}^{\prime}\left( t^{\prime}\right)$$
$$\left[\overline{\overline{A}}_1\left( k_{1},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right)\right]_{mn}= {\displaystyle\int\limits_{S_{m}}} dtf_{m}(t) {\displaystyle\int\limits_{S_{n}}} dt^{\prime}\left[ \widehat{n}^{\prime}\cdot\nabla^{\prime}g_{P}^{1}\right] f_{n}(t^{\prime})$$
$$\left[\overline{\overline{B}}_1\left( k_{1},\overline{k}_{i};\overline{\rho};\overline{\rho }^{\prime}\right)\right]_{mn}= {\displaystyle\int\limits_{S_{m}}} dtf_{m}(t) {\displaystyle\int\limits_{S_{n}}} dt^{\prime}g_{P}^{1}f_{n}(t^{\prime})$$
where $S_{n}$ represents the non-vanishing domain of the $n$-th basis function, ranging from $(t_{n}-\Delta t)$ to $(t_{n}+\Delta t)$.

2.2 BBGF with imaginary wavenumber extraction

It is in practice difficult to solve the eigen-frequencies directly from the matrix equations Eq. (10). In this and the next sections, we adopt the technique developed in [19] to convert the SIEs into a linear eigenvluae problem of a moderate size. A hybrid representation of the lattice Green’s function is the key to this conversion. Distinct from [19], where a real low wavenumber is chosen for the broadband representation of the Green’s function, we adopt the recent development in [29] and choose an imaginary wavenumber component for the extraction. The imaginary wavenumber extraction improves the convergence rate of both the spatial and spectral series.

Equation (4) provides the defination of the empty lattice Green’s function $g_{P}^{0}$ and $g_{P}^{1}$, but is difficult to evaluate due to its slow convergence. To improve the convergence, we follow the technique developed in [29] and decompose it into the Green’s function evaluated at a chosen fixed imaginary wavenumber and the remainder

$$g_{P}^{0}\left( k_{0},\overline{k}_{i};\overline{\rho},\overline{\rho}^{\prime }\right) =g_{P}^{0}\left( i\xi_{0},\overline{k}_{i};\overline{\rho},\overline{\rho }^{\prime}\right) +g_{B}^{0}\left( k_{0},i\xi_{0},\overline{k}_{i};\overline{\rho },\overline{\rho}^{\prime}\right)$$
where
$$g_{P}^{0}\left( i\xi_{0},\overline{k}_{i};\overline{\rho},\overline{\rho}^{\prime }\right) =\sum_{\overline{R}}\left[ \exp\left( i\overline{k}_{i}\cdot \overline{R} \right) g(i\xi_{0};\overline{\rho},\overline{\rho}^{\prime }+\overline{R})\right]$$
$$\begin{aligned}g_{B}^{0}\left( k_{0},i\xi_{0},\overline{k}_{i};\overline{\rho },\overline{\rho}^{\prime}\right) & =\frac{\left( \xi_{0}^{2}+k_{0}^{2}\right) }{\Omega}\sum_{\overline{K} }\frac{\exp\left( i\overline{K}\left( \overline{\rho}-\overline{\rho }^{\prime}\right) \right) }{\left( \left\vert \overline{K}\right\vert ^{2}-k_{0}^{2}\right) \left( \left\vert \overline{K}\right\vert ^{2}+\xi_{0} ^{2}\right) }\\ \overline{K} & =\overline{k}_{i}+\overline{G}\\ \overline{G} & =p_e\overline{b}_{1}+q_e\overline{b}_{2} \end{aligned}$$

Note that $g_{P}^{0}\left ( i\xi _{0},\overline {k}_{i};\overline {\rho },\overline {\rho }^{\prime }\right )$ is expanded in spatial series in Eq. (12b) with exponential convergence rate, while the reminder $g_{B}^{0}\left ( k_{0},i\xi _{0},\overline {k} _{i};\overline {\rho },\overline {\rho }^{\prime }\right )$ is expanded in spectral series in Eq. (12c) with asymptotic convergence rate of $1/\left \vert \overline {K}\right \vert ^{4}$.

In the hybrid representation of the periodic Green’s function in Eq. (12a), mixed spatial and spectral series are adopted, both with fast convergence rate. Moreover, the only wavenumber $k_{0}$ dependence lies in the rational factor $\left ( \xi _{0}^{2}+k_{0}^{2}\right ) /\left ( \left \vert \overline {K}\right \vert ^{2}-k_{0}^{2}\right )$ in the reminder $g_{B}^{0}$. This renders the hybrid representation suitable to be evaluated over a broad frequency range of interest. Thus, it is named the BBGF.

The remainder $g_B^0$ is then rewritten into a more symmetric form:

$$\begin{aligned}g_{B}^{0} & =\frac{1}{\Omega}\sum_{\overline{K}}\frac{\exp\left( i\overline{K}\left( \overline{\rho}-\overline{\rho}^{\prime}\right) \right) }{\left( \frac {1}{k_{0}^{2}+\xi_{0}^{2}}-\frac{1}{\left\vert \overline{K}\right\vert +\xi_{0}^{2} }\right) \left( \left\vert \overline{K}\right\vert ^{2}+\xi_{0}^{2}\right) ^{2}}\\ & =\sum_{\overline{K}}R_\alpha\left( i\xi_{0},\overline{\rho}\right) W_\alpha\left( k_{0},i\xi_{0}\right) R_\alpha^{\ast}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \end{aligned}$$
where
$$\begin{aligned}R_\alpha\left( i\xi_{0},\overline{\rho}\right) &=\frac{1}{\sqrt{\Omega}}\frac{\exp\left( i\overline{K}\overline{\rho}\right) }{\left( \left\vert \overline {K}\right\vert ^{2}+\xi_{0}^{2}\right) }\\W_\alpha\left( k_{0},i\xi_{0}\right) &=\frac{1}{\lambda\left( k_{0},i\xi_{0}\right) -D\left( i\xi_{0}\right) }\\ \lambda\left( k_{0},i\xi_{0}\right) &=\frac{1}{k_{0}^{2}+\xi_{0}^{2}}\\D\left( i\xi_{0}\right) &=\frac{1}{\left\vert \overline{K}\right\vert ^{2}+\xi_{0} ^{2}} \end{aligned}$$
And the normal derivative
$$\widehat{n}^{\prime}\cdot\nabla^{\prime}g_{B}^{0}=\sum_{\overline{R}}\left[R_\alpha\left( i\xi_{0},\overline{\rho }\right) W\left( k_{0},i\xi_{0}\right) Q_\alpha^{\ast}\left( i\xi_{0},\overline{\rho }^{\prime}\right)\right]$$
where
$$Q_\alpha\left( i\xi_{0},\overline{\rho}^{\prime}\right) =\widehat{n}^{\prime} \cdot\nabla^{\prime}R_\alpha\left( i\xi_{0},\overline{\rho}^{\prime}\right) =\left( \widehat{n}^{\prime}\cdot i\overline{K}\right) R_\alpha\left( i\xi_{0},\overline{\rho}^{\prime}\right)$$
Similarly, we have for $g_{P}^{1}$,
$$g_{p}^{1}\left( k_{1},\overline{k}_{i};\overline{\rho},\overline{\rho }^{\prime}\right) =g_{p}^{1}\left( i\xi_{1},\overline{k}_{i};\overline{\rho },\overline{\rho}^{\prime}\right) +g_{B}^{1}\left( k_{1},i\xi_{1} ,\overline{k}_{i};\overline{\rho},\overline{\rho}^{\prime}\right)$$
where $\xi _{1}=\left ( k_{1}/k_{0}\right )\xi _{0}=\sqrt {\varepsilon _{r} \mu _{r}}\xi _{0}$, $\varepsilon _{r}=\varepsilon /\varepsilon _{0}$, $\mu _{r}=\widetilde {\mu }/\mu _{0}$ and
$$g_{P}^{1}\left( i\xi_{1},\overline{k}_{i};\overline{\rho},\overline{\rho}^{\prime }\right) =\sum_{\overline{R}}\left[ \exp\left( i\overline{k}_{i}\cdot \overline{R} \right) g(i\xi_{1};\overline{\rho},\overline{\rho}^{\prime }+\overline{R})\right]$$
$$\begin{aligned}g_{B}^{1}\left( k_{1},i\xi_{1},\overline{k}_{i};\overline{\rho} ,\overline{\rho}^{\prime}\right) &=\frac{\left( \xi_{1}^{2}+k_{1}^{2}\right) }{\Omega}\sum_{\overline{K} }\frac{\exp\left( i\overline{K}\left( \overline{\rho}-\overline{\rho }^{\prime}\right) \right) }{\left( \left\vert \overline{K}\right\vert ^{2}-k_{1}^{2}\right) \left( \left\vert \overline{K}\right\vert ^{2}+\xi_{1} ^{2}\right) }\\ &= \sum_{\overline{K}}R_\alpha\left( i\xi _{1},\overline{\rho}\right) W_\alpha\left( k_{1},i\xi_{1}\right) R_\alpha^{\ast}\left( i\xi_{1},\overline{\rho}^{\prime}\right)\end{aligned}$$
$$W_\alpha\left( k_{1},i\xi_{1}\right) =\frac{1}{\lambda\left( k_{1},i\xi_{1}\right) -D_\alpha\left( i\xi_{1}\right) }$$
$$\lambda\left( k_{1},i\xi_{1}\right) =\frac{1}{k_{1}^{2}+\xi_{1}^{2}}=\frac {1}{\varepsilon_{r}\mu_{r}}\lambda\left( k_{0},i\xi_{0}\right)$$

2.3 Converting the SIEs into a linear eigenvalue problem

Using the hybrid representation of $g_{P}$, the impedance matrices $\overline {\overline {A}}$, $\overline {\overline {B}}$, and $\overline {\overline {C}}$ are rewritten into

$$\overline{\overline{A}}_{0}\left(k_0\right)=\overline{\overline{A}}_{0}(i\xi_{0})+ \overline {\overline{R}}\left( i\xi_{0},\overline{\rho}\right) \overline{\overline{W} }\left( k_{0},i\xi_{0}\right) \overline{\overline{Q} }^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right)$$
$$\overline{\overline{B}}_{0}\left(k_0\right)=\overline{\overline{B}}_{0}(i\xi_{0})+\overline {\overline{R}}\left( i \xi_{0},\overline{\rho}\right) \overline{\overline{W} }\left( k_{0},i\xi_{0}\right) \overline{\overline{R}}^{\dagger}\left( i\xi _{0},\overline{\rho}^{\prime}\right)$$
$$\overline{\overline{C}}\left(k_0\right)=\overline{\overline{C}}(i\xi_{0})+\overline{\overline{R} }\left( i\xi_{0},\overline{\rho}\right) \overline{\overline{W}}\left( k_{0},i\xi_{0}\right) \overline{\overline{R}}^{\dagger}_{C}\left( i\xi_{0} ,\overline{\rho}^{\prime}\right)$$
$$\overline{\overline{A}}_{1}\left(k_1\right)=\overline{\overline{A}}_{1}(i\xi_{1})+\overline {\overline{R}}\left( i\xi_{1},\overline{\rho}\right) \overline{\overline{W} }\left( k_{1},i\xi_{1}\right) \overline{\overline{Q}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right)$$
$$\overline{\overline{B}}_{1}\left(k_1\right)=\overline{\overline{B}}_{1}(i\xi_{1})+\overline {\overline{R}}\left( i\xi_{1},\overline{\rho}\right) \overline{\overline{W} }\left( k_{1},i\xi_{1}\right) \overline{\overline{R}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right)$$
where
$$\begin{aligned}\begin{array} [c]{c} \left[ \overline{\overline{R}}\left( i \xi,\overline{\rho}\right) \right] _{m\alpha}= {\displaystyle\int_{S_{m}}} dt\left[ f_{m}(t)R_{\alpha}\left( i\xi,\overline{\rho}\right) \right]\\ \left[ \overline{\overline{W}}\left( k,i\xi\right) \right] _{\alpha\alpha }=\frac{1}{\lambda\left( k,i\xi\right) -D_{\alpha}\left( i\xi\right) }\\ \left[ \overline{\overline{Q}}\left( i \xi,\overline{\rho}\right) \right] _{m\alpha}= {\displaystyle\int_{S_{m}}} dt\left[ \left( i{\overline{K}}\cdot{\widehat{n}^{\prime}}\right) f_{m}(t)R_{\alpha}\left( i\xi,\overline{\rho}\right) \right]\\ \left[ \overline{\overline{R}}_{C}\left( i\xi,\overline{\rho}\right) \right] _{m\alpha}=\int_{S_{m}}dt \left [f_{m}^{\prime}\left( t\right) R_{\alpha }\left( i\xi,\overline{\rho} \right)\right ] \end{array}\end{aligned}$$

In this hybrid representation, $\overline {\overline {A}}_{0}(i\xi _{0})$, $\overline {\overline {B}}_{0}(i\xi _{0})$, $\overline {\overline {A}}_{1}(i\xi _{1})$, $\overline {\overline {B} }_{1}(i\xi _{1})$, $\overline {\overline {C}}(i\xi _{0})$ are $N\times N$ matrices, $\overline {\overline {R}}$ and $\overline {\overline {Q}}$ are $N\times M$ matrices, and they are independent of the wavenumber $k$. $\overline {\overline {W}}$ and $\overline {\overline {D}}$ are $M\times M$ diagonal matrices with $\left [ \overline {\overline {D}} \left ( i\xi \right )\right ]_{\alpha \alpha }=D_{\alpha }\left ( i\xi \right )$. Here $N$ is the number of discretizations on the reference scatterer boundary $S_{00}$, while $M$ is the number of the Floquet plane waves used to truncate the remainder spectral series of the broadband Green’s function as denoted in Eqs. (12c) and (15c). The superscript † means Hermitian adjoint. Note $\overline {\overline {W}}$ is the only matrix that is dependent on the wavenumber $k$.

Substituting Eq. (16) into Eq. (10),

$$\begin{aligned}0 &= \left( \overline{\overline{A}}_{0}\left( i\xi_{0}\right) -i\mu _{0}\eta\overline{\overline{C}}\left( i\xi_{0}\right) \right) \overline{\psi}_{1}\\ &\quad +\overline{\overline{R}}\left( i\xi_{0},\overline{\rho }\right) \overline{\overline{W}}\left( k_{0},i\xi_{0}\right) \overline {\overline{Q}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \overline{\psi}_{1}\\ &\quad -i\mu_{0}\eta\overline{\overline{R}}\left( i\xi_{0},\overline{\rho }\right) \overline{\overline{W}}\left( k_{0},i\xi_{0}\right) \overline {\overline{R}}_{C}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \overline{\psi}_{1}\\ &\quad -\frac{\mu_{0}}{\widetilde{\mu}}\overline{\overline{B}}_{0}\left( i\xi _{0}\right) \overline{u}_{1}-\frac{\mu_{0}}{\widetilde{\mu}}\overline {\overline{R}}\left( i\xi_{0},\overline{\rho}\right) \overline{\overline{W} }\left( k_{0},i\xi_{0}\right) \overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \overline{u}_{1} \end{aligned}$$
$$\begin{aligned}0 &= \overline{\overline{A}}_{1}\left( i\xi_{1}\right) \overline{\psi} _{1}+\overline{\overline{R}}\left( i\xi_{1},\overline{\rho}\right) \overline{\overline{W}}\left( k_{1},i\xi_{1}\right) \overline{\overline{Q} }^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right) \overline{\psi }_{1}\\ &\quad -\overline{\overline{B}}_{1}\left( i\xi_{1}\right) \overline{u} _{1}-\overline{\overline{R}}\left( i\xi_{1},\overline{\rho}\right) \overline{\overline{W}}\left( k_{1},i\xi_{1}\right) \overline{\overline{R} }^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right) \overline{u} _{1} \end{aligned}$$
Define
$$\overline{b}=\overline{\overline{W}}\left( k_{0},i\xi_{0}\right) \left( \overline{\overline{Q}}^{\dagger}\left( i\xi_{0},\overline{\rho }^{\prime}\right) -i\mu_{0}\eta\overline{\overline{R}}_{C}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \right) \overline{\psi}_{1} -\overline{\overline{W}}\left( k_{0},i\xi_{0}\right)\frac{\mu_{0}}{\widetilde{\mu}}\overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \overline{u}_{1}$$
$$\overline{c}=\overline{\overline{W}}\left( k_{1},i\xi_{1}\right) \left[ \overline{\overline{Q}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime }\right) \overline{\psi}_{1}-\overline{\overline{R}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right) \overline{u}_{1}\right]$$
then
$$\begin{aligned}\left[ \overline{\overline{W}}\left( k_{0},i\xi_{0}\right) \right] ^{-1}\overline{b} &=\left[ \lambda\left( k_{0},i\xi_{0}\right) \overline{\overline{I}}-\overline{\overline{D}}\left( i\xi_{0}\right) \right] \overline{b}\\ &=\left( \overline{\overline{Q}}^{\dagger}\left( i\xi_{0},\overline{\rho }^{\prime}\right) -i\mu_{0}\eta\overline{\overline{R}}_{C}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \right) \overline{\psi}_{1}\\ &\quad-\frac{\mu_{0}}{\widetilde{\mu}}\overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) \overline{u}_{1} \end{aligned}$$
$$\begin{aligned}\left[ \overline{\overline{W}}\left( k_{1},i\xi_{1}\right) \right] ^{-1}\overline{c} &=\left[ \lambda\left( k_{1},i\xi_{1}\right) \overline {\overline{I}}-\overline{\overline{D}}\left( i\xi_{1}\right) \right] \overline{c}\\ &=\overline{\overline{Q}}^{\dagger}\left( i\xi_{1},\overline{\rho }^{\prime}\right) \overline{\psi}_{1}-\overline{\overline{R}}^{\dagger }\left( i\xi_{1},\overline{\rho}^{\prime}\right) \overline{u}_{1}\end{aligned}$$
which is
$$\begin{aligned}\lambda\left( k_{0},i\xi_{0}\right) \overline{b}&=\overline{\overline{D} }\left( i\xi_{0}\right) \overline{b}\\ &+\left( \overline{\overline{Q}} ^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime}\right) -i\mu_{0} \eta\overline{\overline{R}}_{C}^{\dagger}\left( i\xi_{0},\overline{\rho }^{\prime}\right) \right) \overline{\psi}_{1}\\ &\quad-\frac{\mu_{0}}{\widetilde{\mu }}\overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime }\right) \overline{u}_{1}\end{aligned}$$
$$\begin{aligned}\lambda\left( k_{1},i\xi_{1}\right) \overline{c}&=\frac{1}{\varepsilon_{r} \mu_{r}}\lambda\left( k_{0},i\xi_{0}\right) \overline{c}\\ &=\overline {\overline{D}}\left( i\xi_{1}\right) \overline{c}+\overline{\overline{Q} }^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime}\right) \overline{\psi }_{1}-\overline{\overline{R}}^{\dagger}\left( i\xi_{1},\overline{\rho }^{\prime}\right) \overline{u}_{1}\end{aligned}$$
where $\overline {\overline {I}}$ is the $M \times M$ identity matrix.

Using the defination of $\overline {b}$ and $\overline {c}$, the matrix of Eq. (18) becomes

$$\left( \overline{\overline{A}}_{0}\left( i\xi_{0}\right) -i\mu_{0} \eta\overline{\overline{C}}\left( i\xi_{0}\right) \right) \overline{\psi }_{1}-\frac{\mu_{0}}{\widetilde{\mu}}\overline{\overline{B}}_{0}\left( i\xi_{0}\right) \overline{u}_{1}+\overline{\overline{R}}\left( i\xi _{0},\overline{\rho}\right) \overline{b}=0$$
$$\overline{\overline{A}}_{1}\left( i\xi_{1}\right) \overline{\psi} _{1}-\overline{\overline{B}}_{1}\left( i\xi_{1}\right) \overline{u} _{1}+\overline{\overline{R}}\left( i\xi_{1},\overline{\rho}\right) \overline{c}=0$$
Next we express $\overline {\psi }_1,\overline {u}_1$ with $\overline {b},\overline {c},$
$$\left[ \begin{array} [c]{c} \overline{\psi}_{1}\\ \overline{u}_{1} \end{array} \right] =-\left[ \overline{\overline{Z}}\left( i\xi_{0}\right) \right] ^{-1}\left[ \begin{array} [c]{cc} \overline{\overline{R}}\left( i\xi_{0},\overline{\rho}\right) & \overline{\overline{0}}\\ \overline{\overline{0}} & \overline{\overline{R}}\left( i\xi_{1} ,\overline{\rho}\right) \end{array} \right] \left[ \begin{array} [c]{c} \overline{b}\\ \overline{c} \end{array} \right]$$
where
$$\overline{\overline{Z}}\left( i\xi_{0}\right) =\left[ \begin{array} [c]{cc} \overline{\overline{A}}_{0}\left( i\xi_{0}\right) -i\mu_{0}\eta \overline{\overline{C}}\left( i\xi_{0}\right) & -\frac{\mu_{0}} {\widetilde{\mu}}\overline{\overline{B}}_{0}\left( i\xi_{0}\right) \\ \overline{\overline{A}}_{1}\left( i\xi_{1}\right) & -\overline{\overline {B}}_{1}\left( i\xi_{1}\right) \end{array} \right]$$
where the $\overline {\overline {0}}$s are the zero matrices with appropriate dimensions. Reorganizing Eq. (21),
$$\begin{aligned} \lambda\left( k_{0},i\xi_{0}\right) \left[ \begin{array} [c]{c} \overline{b}\\ \overline{c} \end{array} \right] &=\left[ \begin{array} [c]{cc} \overline{\overline{D}}\left( i\xi_{0}\right) & \overline{\overline{0}}\\ \overline{\overline{0}} & \varepsilon_{r}\mu_{r}\overline{\overline{D}}\left( i\xi_{1}\right) \end{array} \right] \left[ \begin{array} [c]{c} \overline{b}\\ \overline{c} \end{array} \right] \\ &\quad +\left[ \begin{array} [c]{cc} \overline{\overline{Q}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime }\right) -i\mu_{0}\eta\overline{\overline{R}}_{C}^{\dagger}\left( i\xi _{0},\overline{\rho}^{\prime}\right) & -\frac{\mu_{0}}{\widetilde{\mu} }\overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime }\right) \\ \varepsilon_{r}\mu_{r}\overline{\overline{Q}}^{\dagger}\left( i\xi _{1},\overline{\rho}^{\prime}\right) & -\varepsilon_{r}\mu_{r} \overline{\overline{R}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime }\right) \end{array} \right] \left[ \begin{array} [c]{c} \overline{\psi}_{1}\\ \overline{u}_{1} \end{array} \right] \end{aligned}$$
Eliminating $\overline {\psi }_{1}$ and $\overline {u}_{1}$ using Eqs. (23) and (24), we obtain an eigenvalue problem of size 2M by 2M,
$$\overline{\overline{P}}\left[ \begin{array} [c]{c} \overline{b}\\ \overline{c} \end{array} \right] =\lambda\left( k_{0},i\xi_{0}\right) \left[ \begin{array} [c]{c} \overline{b}\\ \overline{c} \end{array} \right]$$
where
$$\begin{aligned}\overline{\overline{P}} =\left[ \begin{array} [c]{cc} \overline{\overline{D}}\left( \xi_{0}\right) & \overline{\overline{0}}\\ \overline{\overline{0}} & \varepsilon_{r}\mu_{r}\overline{\overline{D}}\left( \xi_{1}\right) \end{array} \right] -\overline{\overline{F}} \left[ \overline{\overline{Z}}\left( i\xi_{0}\right) \right] ^{-1}\left[ \begin{array} [c]{cc} \overline{\overline{R}}\left( i\xi_{0},\overline{\rho}\right) & \overline{\overline{0}}\\ \overline{\overline{0}} & \overline{\overline{R}}\left( i\xi_{1} ,\overline{\rho}\right) \end{array} \right]\end{aligned}$$
where
$$\begin{aligned}\overline{\overline{F}}=\left[ \begin{array} [c]{cc} \overline{\overline{Q}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime }\right) -i\mu_{0}\eta\overline{\overline{R}}_{C}^{\dagger}\left( i\xi _{0},\overline{\rho}^{\prime}\right) & -\frac{\mu_{0}}{\widetilde{\mu} }\overline{\overline{R}}^{\dagger}\left( i\xi_{0},\overline{\rho}^{\prime }\right) \\ \varepsilon_{r}\mu_{r}\overline{\overline{Q}}^{\dagger}\left( i\xi _{1},\overline{\rho}^{\prime}\right) & -\varepsilon_{r}\mu_{r} \overline{\overline{R}}^{\dagger}\left( i\xi_{1},\overline{\rho}^{\prime }\right) \end{array}\right]\end{aligned}$$
Note that in Eq. (25), the matrix $\overline {\overline {P}}$ is independent of the wavenumber $k$. Thus Eq. (25) is a linear eigenvalue problem from where all the eigenvalues and eigenvectors can be solved simultaneously. Knowing the eigenvalues $\lambda$, the mode wavenumbers are obtained from the relation of $\lambda \left ( k,i\xi \right ) =\frac {1} {k^{2}+\xi ^{2}}$. The modal surface currents distributions $\overline {\psi }_{1}$ and $\overline {u}_{1}$ are calculated from the eigenvectors through Eq. (23). The modal fields are then computed from the extinction theorems Eq. (3) from the modal currents.

3. Results and discussions

In this section, we calculate the band diagram and modal fields of the periodic structure as depicted in Fig. 1 with $r = 0.11a$, $\varepsilon = 15\varepsilon _0$. We consider the two cases with and without external DC magnetic field, respectively. When no external DC magnetic field is applied, $\kappa = 0$ and $\mu =\mu _0$, and this is the case when the scatterer is characterized by a scalar permeability; when external DC magnetic field is applied, $\kappa = 12.4\mu _0$ and $\mu = 14\mu _0$, and this is the case when the scatterer is characterized by a tensor permeability. In the calculations, the imaginary wavenumber $\xi$ is chosen to be $\pi /a$, and the spatial series and the spectral series in the hybrid representation of the lattice Green’s functions are truncated at $2$ and $10$, respectively, i.e., in Eq. (12b), $\overline {R} =p_a\overline {a}_{1}+q_a\overline {a}_{2}, -2 \leqslant p_a,q_a\leqslant 2$, and in Eq. (12c), $\overline {G} =p_e\overline {b}_{1}+q_e\overline {b}_{2}, -10 \leqslant p_e,q_e\leqslant 10$.

3.1 Distinguish physical modes from nonphysical modes

As indicated in the formulation, all the band eigenvalues are obtained simultaneously from the linear eigenvalue problem. It is found that the eigenvalues include both physical modes and nonphysical modes. This is consistent with the intrinsic limitation of the electric-field integral equation (EFIE) formulation, with the nonphysical modes corresponding to resonance modes that do not satisfy the extinction theorems. Thus, the extinction theorems of Eq. (3) can be used to identify physical modes out of nonphysical modes. Equation (3) indicates that for physical modes, the electric field calculated from the surface currents with $g_p^0$ is nonzero outside the scatterer and zero inside the scatterer, while the electric field calculated with $g_p^1$ is zero outside the scatterer and nonzero inside the scatterer. Figure 2 shows the fields both inside and outside the reference scatterer computed with the two extinction theorems at $M$ point with the normalized eigen-frequencies $f_n = 0.6003$ and $0.5932$, respectively. These results are corresponding to the tensor permeability case. It is clear that the results shown in Fig. 2(a) and Fig. 2(b) satisfy the extinction theorems while the results in Fig. 2(c) and Fig. 2(d) not, suggesting that 0.6003 is a physical mode while 0.5932 is nonphysical.

 

Fig. 2. Using extinction theorem to distinguish physical modes from spurious modes: modal fields corresponding to the two eigenvalues of $f_n = 0.6003$ and $0.5932$ calculated at $M$ point. Results are corresponding to scatterers with a tensor permeability. (a) Electric field calculated at $f_n = 0.6003$ using Eq. (3a); (b) electric field calculated at $f_n = 0.6003$ using Eq. (3b); (c) electric field calculated at $f_n = 0.5932$ using Eq. (3a); and (d) electric field calculated at $f_n = 0.5932$ using Eq. (3b). The field of (a) is zero inside the scatterer and nonzero outside the scatterer while that of (b) is zero outside and nonzero inside, suggesting $f_n = 0.6003$ is a physical mode. On the other hand, (c) and (d) indicate $f_n = 0.5932$ is a nonphysical spurious mode.

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3.2 Band diagram

The band diagrams of the physical modes derived from the BBGF approach are illustrated in Fig. 3, and compared with Comsol simulations.

In Fig. 3, solid lines represent the Comsol simulated results while scatter lines represent the results of BBGF. The band diagrams without and with external DC magnetic fields are reported in Figs. 3(a) and 3(b), respectively. It is noted that a band gap is opened between the second and the third bands with the application of the external DC magnetic fields. Such band gap is the region where topological unidirectional edge states are supported [2]. In both cases, the BBGF results are in excellent agreement with Comsol simulations, indicating the effectiveness of the proposed method in analyzing topological photonics.

 

Fig. 3. Band diagrams calculated by the BBGF method (scatter line) in comparison with Comsol simulations (soild line). (a) Without external DC magnetic field, $\mu =\mu _0$ and $\kappa =0$; (b) With external DC magnetic field, $\mu =14\mu _0$ and $\kappa =12.4\mu _0$. A band gap is opened between the second and third bands with external DC magnetic fields.

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3.3 Modal fields

The modal fields corresponding to the second and the third lowest band at $M$ point in the Brillouin zone are illustrated in Fig. 4 and Fig. 5 for the cases without and with the DC external magnetic fields, respectively. In the scalar permeability case, the two bands, labeled by $A$ and $B$, respectively, are degenerate at $M$ point, with $f_n = 0.6677$. Their corresponding modal fields are linked to each other through a rotation of $90^{\circ }$. The external DC magnetic fields have lifted the degeneracy at $M$ point. The two lifted bands labeled by $A'$ and $B'$ are associated with $f_n = 0.5271$ and $f_n = 0.6003$, respectively. It is noted that these two bands are corresponding to quite similar modal fields. In both cases, the BBGF derived modal fields are compared with Comsol simulations. It is noted that in Fig. 5 the results for the degeneracy-lifted modes are in good agreement. On the other hand, the modal fields in Fig. 4 for the degenerate modes are similar between BBGF and Comsol but not identical, a result of the non-uniqueness of the eigenvector basis corresponding to the same eigenvalue.

 

Fig. 4. The degenerate modal fields without external DC magnetic field at $A$ and $B$ points as indicated in Fig. 3(a). Results are compared with Comsol simulations. (a) BBGF calculated modal fields corresponding to $A$ with $f_n=0.6677$; (b) BBGF calculated modal fields corresponding to $B$ with $f_n=0.6677$; (c) Comsol simulated modal fields corresponding to $A$ with $f_n=0.6677$; (d) Comsol simulated modal fields corresponding to $B$ with $f_n=0.6677$.

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Fig. 5. The degeneracy-lifted modal fields with external DC magnetic field at $A'$ and $B'$ points as indicated in Fig. 3(b). Results are compared with Comsol simulations. (a) BBGF calculated modal fields corresponding to $A'$ with $f_n=0.5271$; (b) BBGF calculated modal fields corresponding to $B'$ with $f_n=0.6003$; (c) Comsol simulated modal fields corresponding to $A'$ with $f_n=0.5271$; (d) Comsol simulated modal fields corresponding to $B'$ with $f_n=0.6003$.

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3.4 CPU time analysis

The CPU time is recorded when running a homemade BBGF code in Matlab R2015b to compute the band diagram on a desktop with Intel Core i9-9700k CPU @ 3.60 GHz and 32 GB RAM. Since the computation for different $\overline {k}_{i}$’s are independent, the CPU time for one $\overline {k}_{i}$ is described. Table 1 illustrates the CPU time measured in seconds and in percentages for the tensor permeability case. It takes a total of ~235s to complete the band analysis at one $\overline {k}_{i}$, of which only ~0.8s is spent on the eigenvalue analysis; 226 seconds (96.2% of the CPU time) is spent on computing the impedance matrices, Eq. (23b), at the chosen fixed imaginary wavenumbers; another 8.2s is spent on computing the eigen-matrix $\overline {\overline {P}}$ of Eq. (26a). After these one-time calculations, all the eigenvalues are obtained simultaneously in ~0.8s through Eq. (25). On the other hand, Comsol takes ~1.6s to obtain the first 8 bands on the same computer. Note that our implementation of Eqs. (23b) and (26a) is not optimized, involving many unnecessarily repeated calculations. These calculations can be significantly reduced through parallel computing and code optimization since the evaluation of matrix elements are independent. A fully optimized and parallelized code has potential for about 300 times speed up (235/0.8 = 293.75). On the other hand, since the impedance matrix is only computed once in the proposed method, the time spent to compute multiple eigenvalues is far less than directly searching the roots of the determinant of the impedance matrices without using the broadband Green’s function approach. The latter is a nonlinear root-searching problem that requires calculating the impedance matrices multiple times.

Tables Icon

Table 1. CPU time analysis of the tensor case.

4. Conclusions

In this paper, a novel method is proposed to characterize the band diagram and modal fields of topological photonic crystals with gyromagnetic periodic scatterers, which is the first time that scattering from periodic gyromagnetic scatterers are analyzed using IE-BBGF method. The special material constitutive parameters lead to special boundary conditions and pose complexity in solving SIEs as there are three unknowns and only two surface integral equations. To solve these equations, roof-top basis function and Galerkin’s scheme are used to discretize the SIEs into matrix form. Using the hybrid representation of the BBGF with imaginary wavenumber extraction, the matrix equation is converted into a linear eigenvalue problem of moderate size. All the bands and modal fields are then computed simultaneously from the eigenvalues and eigenvectors, respectively. Comparisons against Comsol simulations indicate the accuracy and efficiency of the proposed method. The work also demonstrates the possibility to combine roof-top basis functions and the Galerkin’s approach with the BBGF formulation. This suggests that the BBGF can be used as a general formulation scheme in well-developed method of moments (MoM) techniques. It is also noted that the general concept of BBGF as illustrated in Eqs. (12) and (15) are immune to material dispersion. However, it will become difficult to convert the surface integral equations into a linear eigenvalue problem with material dispersion as the eigenvalues $\lambda (k_1,i\xi _1 )$ can be no longer related to $\lambda (k_0,i\xi _0 )$ through a simple linear coefficient as in Eq. (15e) for non-dispersive media.

Funding

Zhejiang University; National Natural Science Foundation of China (61901411).

Acknowledgments

This work was led by Principal Supervisor Dr. Shurun Tan.

Disclosures

The authors declare no conflicts of interest.

References

1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]  

2. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008). [CrossRef]  

3. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef]  

4. F. R. Prudêncio and M. G. Silveirinha, “First principles calculation of topological invariants of non-hermitian photonic crystals,” arXiv preprint arXiv:2003.01539 (2020).

5. K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef]  

6. K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990). [CrossRef]  

7. M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991). [CrossRef]  

8. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992). [CrossRef]  

9. M. Kafesaki and C. M. Soukoulis, “Historical perspective and review of fundamental principles in modeling three-dimensional periodic structures with emphasis on volumetric EBGs,” Metamaterials: Physics and Engineering Explorations, 4211–4238 (2006).

10. J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).

11. H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992). [CrossRef]  

12. K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993). [CrossRef]  

13. Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000). [CrossRef]  

14. S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996). [CrossRef]  

15. R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999). [CrossRef]  

16. L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015). [CrossRef]  

17. Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

18. S. Tan and L. Tsang, –Band structures and modal fields in topological acoustics: An integral equation formulation,” in 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (IEEE, 2019), pp. 1927–1928.

19. L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24(2), 945–965 (2016). [CrossRef]  

20. L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018). [CrossRef]  

21. P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. 369(3), 253–287 (1921). [CrossRef]  

22. M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005). [CrossRef]  

23. S. Tan and L. Tsang, “Green’s functions, including scatterers, for photonic crystals and metamaterials,” J. Opt. Soc. Am. B 34(7), 1450–1458 (2017). [CrossRef]  

24. W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019). [CrossRef]  

25. S. Tan and L. Tsang, “Scattering of waves by a half-space of periodic scatterers using broadband Green’s function,” Opt. Lett. 42(22), 4667–4670 (2017). [CrossRef]  

26. T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019). [CrossRef]  

27. M. Sanamzadeh and L. Tsang, “Broadband vector potential dyadic Green’s function and normal modes in 3-D cavity of irregular shape,” IEEE Trans. Micro. Theory Tech. (2020).

28. H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

29. S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019). [CrossRef]  

References

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  1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
    [Crossref]
  2. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
    [Crossref]
  3. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
    [Crossref]
  4. F. R. Prudêncio and M. G. Silveirinha, “First principles calculation of topological invariants of non-hermitian photonic crystals,” arXiv preprint arXiv:2003.01539 (2020).
  5. K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
    [Crossref]
  6. K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990).
    [Crossref]
  7. M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991).
    [Crossref]
  8. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
    [Crossref]
  9. M. Kafesaki and C. M. Soukoulis, “Historical perspective and review of fundamental principles in modeling three-dimensional periodic structures with emphasis on volumetric EBGs,” Metamaterials: Physics and Engineering Explorations, 4211–4238 (2006).
  10. J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).
  11. H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
    [Crossref]
  12. K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
    [Crossref]
  13. Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
    [Crossref]
  14. S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
    [Crossref]
  15. R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999).
    [Crossref]
  16. L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
    [Crossref]
  17. Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.
  18. S. Tan and L. Tsang, –Band structures and modal fields in topological acoustics: An integral equation formulation,” in 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (IEEE, 2019), pp. 1927–1928.
  19. L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24(2), 945–965 (2016).
    [Crossref]
  20. L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
    [Crossref]
  21. P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. 369(3), 253–287 (1921).
    [Crossref]
  22. M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005).
    [Crossref]
  23. S. Tan and L. Tsang, “Green’s functions, including scatterers, for photonic crystals and metamaterials,” J. Opt. Soc. Am. B 34(7), 1450–1458 (2017).
    [Crossref]
  24. W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
    [Crossref]
  25. S. Tan and L. Tsang, “Scattering of waves by a half-space of periodic scatterers using broadband Green’s function,” Opt. Lett. 42(22), 4667–4670 (2017).
    [Crossref]
  26. T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
    [Crossref]
  27. M. Sanamzadeh and L. Tsang, “Broadband vector potential dyadic Green’s function and normal modes in 3-D cavity of irregular shape,” IEEE Trans. Micro. Theory Tech. (2020).
  28. H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).
  29. S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019).
    [Crossref]

2019 (3)

W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
[Crossref]

T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
[Crossref]

S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019).
[Crossref]

2018 (1)

L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (1)

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

2014 (1)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

2008 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

2005 (1)

M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005).
[Crossref]

2000 (1)

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

1999 (1)

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999).
[Crossref]

1996 (1)

S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
[Crossref]

1993 (1)

K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

1992 (2)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

1991 (1)

M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991).
[Crossref]

1990 (2)

K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref]

K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990).
[Crossref]

1921 (1)

P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. 369(3), 253–287 (1921).
[Crossref]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

Chan, C. T.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref]

Chew, W. C.

W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
[Crossref]

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Dai, Q.

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Dai, Q. I.

W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
[Crossref]

Ding, K.-H.

T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
[Crossref]

L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
[Crossref]

Ewald, P. P.

P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. 369(3), 253–287 (1921).
[Crossref]

Fan, S.

S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
[Crossref]

Feng, Z.

Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

Fernandes, C. A.

M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005).
[Crossref]

Gan, H.

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Goertzen, A. L.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

Haus, J.

H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Ho, K.

K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref]

Inguva, R.

H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Joannopoulos, J.

S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).

Kafesaki, M.

M. Kafesaki and C. M. Soukoulis, “Historical perspective and review of fundamental principles in modeling three-dimensional periodic structures with emphasis on volumetric EBGs,” Metamaterials: Physics and Engineering Explorations, 4211–4238 (2006).

Leung, K.-M.

K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990).
[Crossref]

Li, E.-P.

Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

Liao, T.-H.

T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
[Crossref]

Liu, Y.

K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990).
[Crossref]

Liu, Z.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

Lu, L.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Maradudin, A.

M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991).
[Crossref]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).

Page, J. H.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

Plihal, M.

M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991).
[Crossref]

Prudêncio, F. R.

F. R. Prudêncio and M. G. Silveirinha, “First principles calculation of topological invariants of non-hermitian photonic crystals,” arXiv preprint arXiv:2003.01539 (2020).

Qiu, Y.

K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

Sanamzadeh, M.

M. Sanamzadeh and L. Tsang, “Broadband vector potential dyadic Green’s function and normal modes in 3-D cavity of irregular shape,” IEEE Trans. Micro. Theory Tech. (2020).

Sha, W. E.

W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
[Crossref]

Sheng, P.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005).
[Crossref]

F. R. Prudêncio and M. G. Silveirinha, “First principles calculation of topological invariants of non-hermitian photonic crystals,” arXiv preprint arXiv:2003.01539 (2020).

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Soukoulis, C. M.

K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref]

M. Kafesaki and C. M. Soukoulis, “Historical perspective and review of fundamental principles in modeling three-dimensional periodic structures with emphasis on volumetric EBGs,” Metamaterials: Physics and Engineering Explorations, 4211–4238 (2006).

Sözüer, H. S.

H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Tan, S.

S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019).
[Crossref]

L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
[Crossref]

S. Tan and L. Tsang, “Green’s functions, including scatterers, for photonic crystals and metamaterials,” J. Opt. Soc. Am. B 34(7), 1450–1458 (2017).
[Crossref]

S. Tan and L. Tsang, “Scattering of waves by a half-space of periodic scatterers using broadband Green’s function,” Opt. Lett. 42(22), 4667–4670 (2017).
[Crossref]

L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24(2), 945–965 (2016).
[Crossref]

S. Tan and L. Tsang, –Band structures and modal fields in topological acoustics: An integral equation formulation,” in 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (IEEE, 2019), pp. 1927–1928.

Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

Tanaka, M.

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999).
[Crossref]

Tsang, L.

T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
[Crossref]

S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019).
[Crossref]

L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
[Crossref]

S. Tan and L. Tsang, “Scattering of waves by a half-space of periodic scatterers using broadband Green’s function,” Opt. Lett. 42(22), 4667–4670 (2017).
[Crossref]

S. Tan and L. Tsang, “Green’s functions, including scatterers, for photonic crystals and metamaterials,” J. Opt. Soc. Am. B 34(7), 1450–1458 (2017).
[Crossref]

L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24(2), 945–965 (2016).
[Crossref]

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

S. Tan and L. Tsang, –Band structures and modal fields in topological acoustics: An integral equation formulation,” in 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (IEEE, 2019), pp. 1927–1928.

M. Sanamzadeh and L. Tsang, “Broadband vector potential dyadic Green’s function and normal modes in 3-D cavity of irregular shape,” IEEE Trans. Micro. Theory Tech. (2020).

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
[Crossref]

Wang, C.-F.

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Winn, J.

J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).

Xia, T.

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Ziolkowski, R. W.

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999).
[Crossref]

Ann. Phys. (1)

P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. 369(3), 253–287 (1921).
[Crossref]

Appl. Phys. Lett. (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61(4), 495–497 (1992).
[Crossref]

IEEE Trans. Antennas Propag. (1)

M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag. 53(1), 347–355 (2005).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Nature (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electron. 31(9/10), 843–855 (1999).
[Crossref]

Phys. Rev. B (5)

H. S. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

K.-M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62(4), 2446–2457 (2000).
[Crossref]

S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54(16), 11245–11251 (1996).
[Crossref]

M. Plihal and A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44(16), 8565–8571 (1991).
[Crossref]

Phys. Rev. Lett. (3)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

K. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref]

K.-M. Leung and Y. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65(21), 2646–2649 (1990).
[Crossref]

Prog. Electromagn. Res. (5)

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

S. Tan and L. Tsang, “Efficient broadband evaluations of lattice Green’s functions via imaginary wavenumber components extractions,” Prog. Electromagn. Res. 164, 63–74 (2019).
[Crossref]

L. Tsang, K.-H. Ding, and S. Tan, “Broadband point source Green’s function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method,” Prog. Electromagn. Res. 163, 51–77 (2018).
[Crossref]

T.-H. Liao, K.-H. Ding, and L. Tsang, “Broadband Green’s function with higher order low wavenumber extractions for an inhomogeneous waveguide with irregular shape,” Prog. Electromagn. Res. 164, 75–95 (2019).
[Crossref]

W. C. Chew, W. E. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” Prog. Electromagn. Res. 166, 147–165 (2019).
[Crossref]

Other (7)

M. Sanamzadeh and L. Tsang, “Broadband vector potential dyadic Green’s function and normal modes in 3-D cavity of irregular shape,” IEEE Trans. Micro. Theory Tech. (2020).

H. Gan, Q. Dai, T. Xia, W. C. Chew, and C.-F. Wang, “Broadband spectral numerical Green’s function for electromagnetic analysis of inhomogeneous objects,” IEEE Antennas and Wireless Propagation Letters, (2020).

Z. Feng, S. Tan, L. Tsang, and E.-P. Li, –Efficient characterization of topological photonics using the broadband Green’s function,” in 2019 Photonics & Electromagnetics Research Symposium-Fall (PIERS-Fall) (IEEE, 2019), pp. 1092–1099.

S. Tan and L. Tsang, –Band structures and modal fields in topological acoustics: An integral equation formulation,” in 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (IEEE, 2019), pp. 1927–1928.

M. Kafesaki and C. M. Soukoulis, “Historical perspective and review of fundamental principles in modeling three-dimensional periodic structures with emphasis on volumetric EBGs,” Metamaterials: Physics and Engineering Explorations, 4211–4238 (2006).

J. D. Joannopoulos, R. D. Meade, and J. Winn, “Photonic crystals: Molding the flow of light. 1995,” Appendix D, Princeton University of Press (1997).

F. R. Prudêncio and M. G. Silveirinha, “First principles calculation of topological invariants of non-hermitian photonic crystals,” arXiv preprint arXiv:2003.01539 (2020).

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Figures (5)

Fig. 1.
Fig. 1. The structure of the gyromagnetic photonic crystals under study. The radius of cylindrical scatterers $r=0.11a$, where $a$ is the lattice constant. The scatterers with permittivity $\varepsilon$ and permeability $\overline {\overline {\mu }}$ are surrounded by air with permittivity $\varepsilon _0$ and permeability $\mu _0$.
Fig. 2.
Fig. 2. Using extinction theorem to distinguish physical modes from spurious modes: modal fields corresponding to the two eigenvalues of $f_n = 0.6003$ and $0.5932$ calculated at $M$ point. Results are corresponding to scatterers with a tensor permeability. (a) Electric field calculated at $f_n = 0.6003$ using Eq. (3a); (b) electric field calculated at $f_n = 0.6003$ using Eq. (3b); (c) electric field calculated at $f_n = 0.5932$ using Eq. (3a); and (d) electric field calculated at $f_n = 0.5932$ using Eq. (3b). The field of (a) is zero inside the scatterer and nonzero outside the scatterer while that of (b) is zero outside and nonzero inside, suggesting $f_n = 0.6003$ is a physical mode. On the other hand, (c) and (d) indicate $f_n = 0.5932$ is a nonphysical spurious mode.
Fig. 3.
Fig. 3. Band diagrams calculated by the BBGF method (scatter line) in comparison with Comsol simulations (soild line). (a) Without external DC magnetic field, $\mu =\mu _0$ and $\kappa =0$; (b) With external DC magnetic field, $\mu =14\mu _0$ and $\kappa =12.4\mu _0$. A band gap is opened between the second and third bands with external DC magnetic fields.
Fig. 4.
Fig. 4. The degenerate modal fields without external DC magnetic field at $A$ and $B$ points as indicated in Fig. 3(a). Results are compared with Comsol simulations. (a) BBGF calculated modal fields corresponding to $A$ with $f_n=0.6677$; (b) BBGF calculated modal fields corresponding to $B$ with $f_n=0.6677$; (c) Comsol simulated modal fields corresponding to $A$ with $f_n=0.6677$; (d) Comsol simulated modal fields corresponding to $B$ with $f_n=0.6677$.
Fig. 5.
Fig. 5. The degeneracy-lifted modal fields with external DC magnetic field at $A'$ and $B'$ points as indicated in Fig. 3(b). Results are compared with Comsol simulations. (a) BBGF calculated modal fields corresponding to $A'$ with $f_n=0.5271$; (b) BBGF calculated modal fields corresponding to $B'$ with $f_n=0.6003$; (c) Comsol simulated modal fields corresponding to $A'$ with $f_n=0.5271$; (d) Comsol simulated modal fields corresponding to $B'$ with $f_n=0.6003$.

Tables (1)

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Table 1. CPU time analysis of the tensor case.

Equations (57)

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μ ¯ ¯ = [ μ i κ 0 i κ μ 0 0 0 μ 0 ]
2 ψ 0 + ω 2 ε 0 μ 0 ψ 0 = 0
2 ψ 1 + ω 2 ε μ ~ ψ 1 = 0
S 00 d ρ ¯ ( ψ 0 ( ρ ¯ ) n ^ g P 0 g P 0 n ^ ψ 0 ( ρ ¯ ) ) = { ψ 0 ( ρ ¯ ) ρ ¯  outside scatterer 0 ρ ¯  inside scatterer
S 00 d ρ ¯ ( ψ 1 ( ρ ¯ ) n ^ g P 1 g P 1 n ^ ψ 1 ( ρ ¯ ) ) = { ψ 1 ( ρ ¯ ) ρ ¯  inside scatterer 0 ρ ¯  outside scatterer
g P 0 ( k 0 , k ¯ i ; ρ ¯ ; ρ ¯ ) = R ¯ [ exp ( i k ¯ i ( R ¯ ) ) g ( k 0 ; ρ ¯ , ρ ¯ + R ¯ ) ]
g P 1 ( k 1 , k ¯ i ; ρ ¯ ; ρ ¯ ) = R ¯ [ exp ( i k ¯ i ( R ¯ ) ) g ( k 1 ; ρ ¯ , ρ ¯ + R ¯ ) ]
ψ 0 = ψ 1
1 μ 0 n ^ ψ 0 = 1 μ ~ n ^ ψ 1 + i η t ^ ψ 1
S 00 d ρ ¯ ψ 1 ( ρ ¯ ) n ^ g P 0 S 00 d ρ ¯ g P 0 μ 0 ( 1 μ ~ n ^ ψ 1 ( ρ ¯ ) + i η t ^ ψ 1 ( ρ ¯ ) ) = 0 , ρ ¯ S 00
S 00 d ρ ¯ ( ψ 1 ( ρ ¯ ) n ^ g P 1 g P 1 n ^ ψ 1 ( ρ ¯ ) ) = 0 , ρ ¯ S 00 +
f n ( t ) = { 1 Δ t ( t t n + Δ t ) ; t n Δ t t t n 1 Δ t ( t t n Δ t ) ; t n t t n + Δ t 0 ; o t h e r w i s e
f n ( t ) = { 1 Δ t ; t n Δ t t t n 1 Δ t ; t n t t n + Δ t 0 ; o t h e r w i s e
ψ 1 = n ψ 1 n f n ( t )
u 1 = n u 1 n f n ( t )
t ^ ψ 1 = n t ^ [ ψ 1 n f n ( t ) ] = n ψ 1 n f n ( t )
( A ¯ ¯ 0 i μ 0 η C ¯ ¯ ) ψ ¯ 1 μ 0 μ ~ B ¯ ¯ 0 u ¯ 1 = 0
A ¯ ¯ 1 ψ ¯ 1 B ¯ ¯ 1 u ¯ 1 = 0
[ A ¯ ¯ 0 ( k 0 , k ¯ i ; ρ ¯ ; ρ ¯ ) ] m n = S m d t f m ( t ) S n d t [ n ^ g P 0 ] f n ( t )
[ B ¯ ¯ 0 ( k 0 , k ¯ i ; ρ ¯ ; ρ ¯ ) ] m n = S m d t f m ( t ) S n d t g P 0 f n ( t )
[ C ¯ ¯ ( k 0 , k ¯ i ; ρ ¯ ; ρ ¯ ) ] m n = S m d t f m ( t ) S n d t g P 0 f n ( t )
[ A ¯ ¯ 1 ( k 1 , k ¯ i ; ρ ¯ ; ρ ¯ ) ] m n = S m d t f m ( t ) S n d t [ n ^ g P 1 ] f n ( t )
[ B ¯ ¯ 1 ( k 1 , k ¯ i ; ρ ¯ ; ρ ¯ ) ] m n = S m d t f m ( t ) S n d t g P 1 f n ( t )
g P 0 ( k 0 , k ¯ i ; ρ ¯ , ρ ¯ ) = g P 0 ( i ξ 0 , k ¯ i ; ρ ¯ , ρ ¯ ) + g B 0 ( k 0 , i ξ 0 , k ¯ i ; ρ ¯ , ρ ¯ )
g P 0 ( i ξ 0 , k ¯ i ; ρ ¯ , ρ ¯ ) = R ¯ [ exp ( i k ¯ i R ¯ ) g ( i ξ 0 ; ρ ¯ , ρ ¯ + R ¯ ) ]
g B 0 ( k 0 , i ξ 0 , k ¯ i ; ρ ¯ , ρ ¯ ) = ( ξ 0 2 + k 0 2 ) Ω K ¯ exp ( i K ¯ ( ρ ¯ ρ ¯ ) ) ( | K ¯ | 2 k 0 2 ) ( | K ¯ | 2 + ξ 0 2 ) K ¯ = k ¯ i + G ¯ G ¯ = p e b ¯ 1 + q e b ¯ 2
g B 0 = 1 Ω K ¯ exp ( i K ¯ ( ρ ¯ ρ ¯ ) ) ( 1 k 0 2 + ξ 0 2 1 | K ¯ | + ξ 0 2 ) ( | K ¯ | 2 + ξ 0 2 ) 2 = K ¯ R α ( i ξ 0 , ρ ¯ ) W α ( k 0 , i ξ 0 ) R α ( i ξ 0 , ρ ¯ )
R α ( i ξ 0 , ρ ¯ ) = 1 Ω exp ( i K ¯ ρ ¯ ) ( | K ¯ | 2 + ξ 0 2 ) W α ( k 0 , i ξ 0 ) = 1 λ ( k 0 , i ξ 0 ) D ( i ξ 0 ) λ ( k 0 , i ξ 0 ) = 1 k 0 2 + ξ 0 2 D ( i ξ 0 ) = 1 | K ¯ | 2 + ξ 0 2
n ^ g B 0 = R ¯ [ R α ( i ξ 0 , ρ ¯ ) W ( k 0 , i ξ 0 ) Q α ( i ξ 0 , ρ ¯ ) ]
Q α ( i ξ 0 , ρ ¯ ) = n ^ R α ( i ξ 0 , ρ ¯ ) = ( n ^ i K ¯ ) R α ( i ξ 0 , ρ ¯ )
g p 1 ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) = g p 1 ( i ξ 1 , k ¯ i ; ρ ¯ , ρ ¯ ) + g B 1 ( k 1 , i ξ 1 , k ¯ i ; ρ ¯ , ρ ¯ )
g P 1 ( i ξ 1 , k ¯ i ; ρ ¯ , ρ ¯ ) = R ¯ [ exp ( i k ¯ i R ¯ ) g ( i ξ 1 ; ρ ¯ , ρ ¯ + R ¯ ) ]
g B 1 ( k 1 , i ξ 1 , k ¯ i ; ρ ¯ , ρ ¯ ) = ( ξ 1 2 + k 1 2 ) Ω K ¯ exp ( i K ¯ ( ρ ¯ ρ ¯ ) ) ( | K ¯ | 2 k 1 2 ) ( | K ¯ | 2 + ξ 1 2 ) = K ¯ R α ( i ξ 1 , ρ ¯ ) W α ( k 1 , i ξ 1 ) R α ( i ξ 1 , ρ ¯ )
W α ( k 1 , i ξ 1 ) = 1 λ ( k 1 , i ξ 1 ) D α ( i ξ 1 )
λ ( k 1 , i ξ 1 ) = 1 k 1 2 + ξ 1 2 = 1 ε r μ r λ ( k 0 , i ξ 0 )
A ¯ ¯ 0 ( k 0 ) = A ¯ ¯ 0 ( i ξ 0 ) + R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) Q ¯ ¯ ( i ξ 0 , ρ ¯ )
B ¯ ¯ 0 ( k 0 ) = B ¯ ¯ 0 ( i ξ 0 ) + R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) R ¯ ¯ ( i ξ 0 , ρ ¯ )
C ¯ ¯ ( k 0 ) = C ¯ ¯ ( i ξ 0 ) + R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) R ¯ ¯ C ( i ξ 0 , ρ ¯ )
A ¯ ¯ 1 ( k 1 ) = A ¯ ¯ 1 ( i ξ 1 ) + R ¯ ¯ ( i ξ 1 , ρ ¯ ) W ¯ ¯ ( k 1 , i ξ 1 ) Q ¯ ¯ ( i ξ 1 , ρ ¯ )
B ¯ ¯ 1 ( k 1 ) = B ¯ ¯ 1 ( i ξ 1 ) + R ¯ ¯ ( i ξ 1 , ρ ¯ ) W ¯ ¯ ( k 1 , i ξ 1 ) R ¯ ¯ ( i ξ 1 , ρ ¯ )
[ R ¯ ¯ ( i ξ , ρ ¯ ) ] m α = S m d t [ f m ( t ) R α ( i ξ , ρ ¯ ) ] [ W ¯ ¯ ( k , i ξ ) ] α α = 1 λ ( k , i ξ ) D α ( i ξ ) [ Q ¯ ¯ ( i ξ , ρ ¯ ) ] m α = S m d t [ ( i K ¯ n ^ ) f m ( t ) R α ( i ξ , ρ ¯ ) ] [ R ¯ ¯ C ( i ξ , ρ ¯ ) ] m α = S m d t [ f m ( t ) R α ( i ξ , ρ ¯ ) ]
0 = ( A ¯ ¯ 0 ( i ξ 0 ) i μ 0 η C ¯ ¯ ( i ξ 0 ) ) ψ ¯ 1 + R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) Q ¯ ¯ ( i ξ 0 , ρ ¯ ) ψ ¯ 1 i μ 0 η R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) R ¯ ¯ C ( i ξ 0 , ρ ¯ ) ψ ¯ 1 μ 0 μ ~ B ¯ ¯ 0 ( i ξ 0 ) u ¯ 1 μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) W ¯ ¯ ( k 0 , i ξ 0 ) R ¯ ¯ ( i ξ 0 , ρ ¯ ) u ¯ 1
0 = A ¯ ¯ 1 ( i ξ 1 ) ψ ¯ 1 + R ¯ ¯ ( i ξ 1 , ρ ¯ ) W ¯ ¯ ( k 1 , i ξ 1 ) Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ψ ¯ 1 B ¯ ¯ 1 ( i ξ 1 ) u ¯ 1 R ¯ ¯ ( i ξ 1 , ρ ¯ ) W ¯ ¯ ( k 1 , i ξ 1 ) R ¯ ¯ ( i ξ 1 , ρ ¯ ) u ¯ 1
b ¯ = W ¯ ¯ ( k 0 , i ξ 0 ) ( Q ¯ ¯ ( i ξ 0 , ρ ¯ ) i μ 0 η R ¯ ¯ C ( i ξ 0 , ρ ¯ ) ) ψ ¯ 1 W ¯ ¯ ( k 0 , i ξ 0 ) μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) u ¯ 1
c ¯ = W ¯ ¯ ( k 1 , i ξ 1 ) [ Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ψ ¯ 1 R ¯ ¯ ( i ξ 1 , ρ ¯ ) u ¯ 1 ]
[ W ¯ ¯ ( k 0 , i ξ 0 ) ] 1 b ¯ = [ λ ( k 0 , i ξ 0 ) I ¯ ¯ D ¯ ¯ ( i ξ 0 ) ] b ¯ = ( Q ¯ ¯ ( i ξ 0 , ρ ¯ ) i μ 0 η R ¯ ¯ C ( i ξ 0 , ρ ¯ ) ) ψ ¯ 1 μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) u ¯ 1
[ W ¯ ¯ ( k 1 , i ξ 1 ) ] 1 c ¯ = [ λ ( k 1 , i ξ 1 ) I ¯ ¯ D ¯ ¯ ( i ξ 1 ) ] c ¯ = Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ψ ¯ 1 R ¯ ¯ ( i ξ 1 , ρ ¯ ) u ¯ 1
λ ( k 0 , i ξ 0 ) b ¯ = D ¯ ¯ ( i ξ 0 ) b ¯ + ( Q ¯ ¯ ( i ξ 0 , ρ ¯ ) i μ 0 η R ¯ ¯ C ( i ξ 0 , ρ ¯ ) ) ψ ¯ 1 μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) u ¯ 1
λ ( k 1 , i ξ 1 ) c ¯ = 1 ε r μ r λ ( k 0 , i ξ 0 ) c ¯ = D ¯ ¯ ( i ξ 1 ) c ¯ + Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ψ ¯ 1 R ¯ ¯ ( i ξ 1 , ρ ¯ ) u ¯ 1
( A ¯ ¯ 0 ( i ξ 0 ) i μ 0 η C ¯ ¯ ( i ξ 0 ) ) ψ ¯ 1 μ 0 μ ~ B ¯ ¯ 0 ( i ξ 0 ) u ¯ 1 + R ¯ ¯ ( i ξ 0 , ρ ¯ ) b ¯ = 0
A ¯ ¯ 1 ( i ξ 1 ) ψ ¯ 1 B ¯ ¯ 1 ( i ξ 1 ) u ¯ 1 + R ¯ ¯ ( i ξ 1 , ρ ¯ ) c ¯ = 0
[ ψ ¯ 1 u ¯ 1 ] = [ Z ¯ ¯ ( i ξ 0 ) ] 1 [ R ¯ ¯ ( i ξ 0 , ρ ¯ ) 0 ¯ ¯ 0 ¯ ¯ R ¯ ¯ ( i ξ 1 , ρ ¯ ) ] [ b ¯ c ¯ ]
Z ¯ ¯ ( i ξ 0 ) = [ A ¯ ¯ 0 ( i ξ 0 ) i μ 0 η C ¯ ¯ ( i ξ 0 ) μ 0 μ ~ B ¯ ¯ 0 ( i ξ 0 ) A ¯ ¯ 1 ( i ξ 1 ) B ¯ ¯ 1 ( i ξ 1 ) ]
λ ( k 0 , i ξ 0 ) [ b ¯ c ¯ ] = [ D ¯ ¯ ( i ξ 0 ) 0 ¯ ¯ 0 ¯ ¯ ε r μ r D ¯ ¯ ( i ξ 1 ) ] [ b ¯ c ¯ ] + [ Q ¯ ¯ ( i ξ 0 , ρ ¯ ) i μ 0 η R ¯ ¯ C ( i ξ 0 , ρ ¯ ) μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) ε r μ r Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ε r μ r R ¯ ¯ ( i ξ 1 , ρ ¯ ) ] [ ψ ¯ 1 u ¯ 1 ]
P ¯ ¯ [ b ¯ c ¯ ] = λ ( k 0 , i ξ 0 ) [ b ¯ c ¯ ]
P ¯ ¯ = [ D ¯ ¯ ( ξ 0 ) 0 ¯ ¯ 0 ¯ ¯ ε r μ r D ¯ ¯ ( ξ 1 ) ] F ¯ ¯ [ Z ¯ ¯ ( i ξ 0 ) ] 1 [ R ¯ ¯ ( i ξ 0 , ρ ¯ ) 0 ¯ ¯ 0 ¯ ¯ R ¯ ¯ ( i ξ 1 , ρ ¯ ) ]
F ¯ ¯ = [ Q ¯ ¯ ( i ξ 0 , ρ ¯ ) i μ 0 η R ¯ ¯ C ( i ξ 0 , ρ ¯ ) μ 0 μ ~ R ¯ ¯ ( i ξ 0 , ρ ¯ ) ε r μ r Q ¯ ¯ ( i ξ 1 , ρ ¯ ) ε r μ r R ¯ ¯ ( i ξ 1 , ρ ¯ ) ]

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