Finite Element Calculation of Photonic Band Structures for Frequency Dependent Materials

Abstract

Band structure calculation of frequency dependent photonic crystals has important applications. The associated eigenvalue problem is nonlinear and the development of convergent numerical methods is challenging. In this paper, we formulate the band structure problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operators. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. Then a spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

Appendix

We present some preliminaries on holomorphic Fredholm operator functions and the abstract approximation theory for the associated eigenvalue problems (see, e.g., [3, 13, 22, 23]). Let X, Y be complex Banach spaces and $\Omega \subset \mathbb {C}$ be compact and simply connected. Denote by $\mathcal {L}(X,Y)$ the space of bounded linear operators from X to Y.

Definition 7.1

An operator $A \in \mathcal {L}(X, Y)$ is said to be Fredholm if

• 1. The range of A, denoted by $\mathcal {R}(A)$, is closed in Y;

• 2. The null space of A, denoted by $\mathcal {N}(A)$, and the quotient space $Y/\mathcal {R}(A)$ are finite-dimensional.

The index of A is the integer defined by

$$\begin{aligned} \text {ind}(A) = \dim \mathcal {N}(A) - \dim (Y/\mathcal {R}(A)). \end{aligned}$$

Definition 7.2

Let E be a Banach space and $\Omega \subset \mathbb C$ be an open set. A function $f: \Omega \rightarrow E$ is called holomorphic if, for each $w \in \Omega$,

$$\begin{aligned} f'(w) := \lim _{z \rightarrow w} \frac{f(z)-f(w)}{z-w} \end{aligned}$$

exists.

Let $T: \Omega \rightarrow \mathcal {L}(X, Y)$ be a holomorphic operator function on $\Omega$. Denote by $\Phi _0(\Omega ,\mathcal {L}(X,Y))$ the set of holomorphic Fredholm operator functions of index zero [13]. Assume that $T \in \Phi _0(\Omega ,\mathcal {L}(X,Y))$, i.e., for each $\omega \in \Omega$, $T(\omega ) \in \mathcal {L}(X, Y)$ is a Fredholm operator of index zero. The eigenvalue problem is to find $(\omega ,u)\in \Omega \times X,~u\ne 0$, such that

$$\begin{aligned} T(\omega )u=0. \end{aligned}$$

(7.1)

The resolvent set $\rho (T)$ and the spectrum $\sigma (T)$ of T with respect to $\Omega$ are, respectively, defined as

$$\begin{aligned} \rho (T)=\{\omega \in \Omega : T(\omega )^{-1} \in \mathcal {L}(Y,X) \}\quad \text {and} \quad \sigma (T)=\Omega \backslash \rho (T). \end{aligned}$$

Throughout the paper, we assume that $\rho (T)\ne \emptyset$. Then the spectrum $\sigma (T)$ has no cluster points in $\Omega$ and every $\omega \in \sigma (T)$ is an eigenvalue [22].

To approximate the eigenvalues of T, we consider operator functions $T_n\in \Phi _0(\Omega ,\mathcal {L}(X_n,Y_n))$, $n\in \mathbb {N}$, such that the following properties hold [3, 22].

• (b1) There exist Banach spaces $X_n,Y_n$, $n\in \mathbb {N}$, and linear bounded mappings $p_n\in \mathcal {L}(X,X_n)$, $q_n\in \mathcal {L}(Y,Y_n)$ such that

  $$\begin{aligned} \lim \limits _{n \rightarrow \infty }\Vert p_n v\Vert _{X_n}=\Vert v\Vert _X,\, v\in X,\quad \lim \limits _{n \rightarrow \infty }\Vert q_n v\Vert _{Y_n}=\Vert v\Vert _Y,\, v\in Y. \end{aligned}$$

• (b2) The sequence $\{T_n(\cdot )\}_{n \in \mathbb {N}}$ satisfies

  $$\begin{aligned} \Vert T_n(\omega )\Vert < \infty \quad \text {for all } \omega \in \Omega , n \in \mathbb {N}. \end{aligned}$$

• (b3) $\{T_n(\omega )\}_{n \in \mathbb {N}}$ approximates $T(\omega )$ for every $\omega \in \Omega$, i.e.,

  $$\begin{aligned} \lim _{n \rightarrow \infty } \Vert T_n(\omega )p_n x - q_n T(\omega ) x\Vert _{Y_n} = 0 \quad \text {for all } x \in X. \end{aligned}$$

• (b4) Let $\omega \in \Omega$ and $\{x_n\} \subset X_n, n\in \mathbb {N}$ be bounded. For any subsequence $\{x_n\}$, $n\in N \subset \mathbb N$,

  $$\begin{aligned} \lim _{\mathbb {N} \ni n\rightarrow \infty } \Vert T_n(\omega ) x_n - q_n y\Vert _{Y_n}=0 \end{aligned}$$

  for some $y\in Y$, there exists a subsequence $N' \subset N$ and an element $x\in X$ such that

  $$\begin{aligned} \lim \limits _{N' \ni n\rightarrow \infty } \Vert x_n-p_n x\Vert _{X_n}=0. \end{aligned}$$

If the above conditions are satisfied, one has the following abstract approximation result (see Section 2 of [23] or Theorem 2.10 of [3]).

Theorem 7.3

Assume that (b1)-(b4) hold. For any $\omega \in \sigma (T)$ there exists $n_0\in \mathbb {N}$ and a sequence $\omega _n\in \sigma (T_n), n\ge n_0$, such that $\omega _n\rightarrow \omega$ as $n\rightarrow \infty$. Furthermore, letting $G(\omega )$ be the generalized eigenspace of $\omega$ and $r_0$ be the maximum rank of eigenvectors associated to $\omega$, it holds that

$$\begin{aligned} |\omega _n-\omega |\le C \varepsilon _n ^{1/{r_0}}, \end{aligned}$$

where

$$\begin{aligned} \varepsilon _n=\max \limits _{|\eta -\omega |\le \delta } \max \limits _{v\in G(\omega )}\Vert T_n(\eta ) p_n v-q_n T(\eta )v\Vert _{Y_n}, \end{aligned}$$

for sufficiently small $\delta > 0$.