Efficient and quantitative analysis of photon density of states for two-dimensional photonic crystals with omnidirectional light propagation

Omnidirectional light propagation in two-dimensional (2D) photonic crystals (PCs) has been investigated by extending the formerly developed 2D finite element analysis (FEA) of in-plane light propagation in which the corresponding band structure (BS) and photon density of states (PDOS) of 2D PCs with complex geometry configurations had been calculated more accurately by using an adaptive FEA in real space for both the transverse electric (TE) and transverse magnetic (TM) modes. In this work, by adopting a wave-guiding theory under the consideration of translational symmetry, the omnidirectional PDOS corresponding to both the radiative and evanescent waves can be calculated efficiently based on the in-plane dispersion relations of both TE and TM modes within the irreducible Brillouin zone. We demonstrate that the complete band gaps shown by previous work considering only the radiative modes will be closed by including the contributions of the evanescent modes. These results are of general importance and relevance to the spontaneous emission by an atom or to dipole radiation in 2D periodic structures. In addition, it may serve as an efficient approach to identifying the existence of a complete photonic band gap in a 2D PC instead of using time-consuming three-dimensional BS calculations.

Figure 1

Schematics of omnidirectional light propagation in 2D PCs for (a) a triangular lattice and (b) a square lattice with $k_z=k·\text{sin}\theta$ (black solid line) and the in-plane light propagation with $k_z=0$ (blue dashed line), where $\theta$ is the off-plane incident angle.

Figure 2

Comparisons of 3D total PDOS calculated by the FEM (black solid line) and the PWEM (green open circles) for a 2D PC with a triangular lattice of air cylinders etched into silicon $({\varepsilon }_d=11.90)$ at a filling ratio of $67\%$ as used in Ref. [21]. The total PDOS is contributed from both the radiative (red dotted line) and evanescent (blue dashed line) modes. The PBG calculated by the FEM for the off-plane radiative waves ranges from $0.395(2\pi c/a)$ to $0.399(2\pi c/a)$ while that for the in-plane case ranges from $0.382(2\pi c/a)$ to $0.400(2\pi c/a)$ [13].

Figure 3

3D PDOS for (a) TE modes and (b) TM modes of a triangular array with air cylinders etched into a dielectric $({\varepsilon }_d=12.96)$ at a filling ratio of $75\%$ [19] and for (c) TE modes and (d) TM modes of a square array with air cylinders etched into a dielectric $({\varepsilon }_d=16.00)$ at a filling ratio of $72.38\%$ [20]. The blue dashed (red dotted) lines correspond to the PDOS of evanescent (radiative) waves. The PBGs calculated by the FEM for the radiative waves are (a) $0.310\text{–}0.495(2\pi c/a)$ and $0.827\text{–}0.834(2\pi c/a)$, (b) $0.403\text{–}0.434(2\pi c/a)$, (c) $0.410\text{–}0.487(2\pi c/a)$, and (d) $0.218\text{–}0.259(2\pi c/a)$ and $0.389\text{–}0.416(2\pi c/a)$.

Figure 4

3D total PDOS (black solid lines) of (a) a triangular array and (b) a square array normalized to that of the vacuum. The blue dashed (red dotted) lines correspond to the normalized PDOS of evanescent (radiative) waves. The PBGs calculated by the FEM for the radiative waves in (a) and (b) range from $0.403(2\pi c/a)$ to $0.434(2\pi c/a)$ and from $0.410(2\pi c/a)$ to $0.416(2\pi c/a)$, respectively. In comparison, those calculated by the PWEM in Refs. [19] and [20] range from $0.423(2\pi c/a)$ to $0.437(2\pi c/a)$ and $0.4045(2\pi c/a)$ to $0.4197(2\pi c/a)$