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This topic provides a number of bandstructure calculation examples for different lattice types. (For layered structures with no patterning, one can also use the technique described in Slab mode analysis of a OLED layer structure to obtain the dispersion relation.)



This example assumes that you are familiar with the concepts of bandstructure, Brillouin zones and Bloch's theorem.


We intend to study structures that are periodic in at least one dimension. For simplicity, consider a structure periodic in the x direction, with period a. Bloch's theorem tells us that the modes of a periodic structure can be written as \( E_{k}(x)=\exp (i k x) u_{k}(x) \), where u(x) is a periodic function of x, with period a. This means that u(x+a)=u(x). Another way to write Bloch's theorem is then \( E_{k+a}(x)=\exp (i k a) u_{k}(x) \). This defines Bloch boundary conditions in one dimension and this condition can be generalized to two or three dimensions in a straightforward way. Bloch boundary conditions are used in FDTD to calculate the properties of an infinite periodic structure by simulating only one unit cell. The photonic bandstructure is calculated by determining the angular frequencies, wn(k), as a function of wave vector k for all the Bloch modes in a given frequency range. Also, the spatial field information for a particular Bloch mode, \( E_{n, k}(\omega, \vec{r}) \), in two and three dimensions can also be determined.

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