Abstract
In-plane light propagation in two-dimensional (2D) photonic crystals (PCs) has been investigated by using the finite element method (FEM) in frequency domain. Conventionally, the band structures of 2D PCs were calculated by either the plane-wave expansion method (PWEM) or the finite difference time domain method. Here, we solve the eigenvalue equations for the band structures of the 2D PCs using the adaptive FEM in real space. We have carefully examined the convergence of this approach for the desired accuracy and efficiency. The calculated results show some discrepancies when compared to the results calculated by the PWEM. This may be due to the accuracy of the PWEM limited by the discontinuous nature of the dielectric functions. After acquiring the whole information of the dispersion relations within the irreducible Brillouin zone of the 2D PCs, the in-plane photon density of states for both the transverse electric (TE) and transverse magnetic (TM) modes can be calculated, accurately. For the case, the width of the complete band gap predicted by the FEM is much smaller, only about 65 % of that calculated by the PWEM. Therefore, the discrepancy in the prediction of complete band gaps between these two methods can be quite large, although the difference in band structure calculations is only a few percent. These results are relevant to the spontaneous emission by an atom, or to dipole radiation in two-dimensional periodic structures.
Highlights
INTRODUCTIONMany numerical methods have been developed and applied to the analysis and investigation of photonic crystals (PCs) including the plane-wave expansion method (PWEM) [1,2,3,4,5,6,7], the finitedifference frequency-domain/finite-difference time-domain (FDTD) method [8,9,10], the multiplescattering method [11, 12], and the finite-element frequency-domain/finite-element time-domain method [13,14,15,16,17,18,19,20], and others
The discrepancy is getting larger as the frequency is going to higher regime. This may be due to the accuracy of the plane-wave expansion method (PWEM) limited by the discontinuous nature of the dielectric functions [7]
As the contrast of the dielectric constant is high, the step-like dielectric function is usually approximated by limited number of Fourier basis in the PWEM
Summary
Many numerical methods have been developed and applied to the analysis and investigation of photonic crystals (PCs) including the plane-wave expansion method (PWEM) [1,2,3,4,5,6,7], the finitedifference frequency-domain/finite-difference time-domain (FDTD) method [8,9,10], the multiplescattering method [11, 12], and the finite-element frequency-domain/finite-element time-domain method [13,14,15,16,17,18,19,20], and others. The finite-element method (FEM) has proved to be a flexible and efficient numerical tool with which to design various types of microwave components with inhomogeneous and complex structures [22]. The finite-element method is employed to discretize the dielectric function profile of the PCs and the in-plane band structures are calculated by solving eigenvalue equations with proper periodic boundary conditions following the Bloch theorem [23]. Based on the finite-element analysis of the in-plane band structures of the 2D PCs in the irreducible Brillouin zone, the in-plane photon density of states (PDOS) of the 2D PCs for the TE and TM modes can be calculated accurately
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
