Finished Theses

Institut für Theoretische Physik I

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Gruppenseminar der Theorie 1

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- Calculation of photonic band structures for 1D quasicrystals and 2D square crystals
- supervised by Michael Schmiedeberg
- finished 2018-05
- [Full text as PDF]

In the theoretical part of this thesis, the Master equation for dielectric material is deduced from Macroscopic Maxwell equations. Then the magnetic Field strength H are expanded as plane waves according to the Bloch theory, while the periodic permitivity e(r) is expanded as fourier series with respect to the reciprocal lattice vectors. By substituting them into the Master equation we obtained a eigenvalue matrix equation from which the photonic band structure of photonic crystal (w ∼ k) can be calculated. For the case of quasicrystal, since it can be seen as cross section of a higher dimensional periodic superspace, the Master equation can be modifed to calculate the photonic band structure, taking advantage of the fact that the permitivity being periodic with respect to the superspace, which makes the Bloch theory applicable. In the numberical calculation part, the photonic band structures of both two dimensional periodic square crystal and one dimensional Fibonacci-like quasicrystal are computed for various different ratios of permitivities. By comparing the results we found that for two dimensional square crystals, band gaps won't appear until the ratio of permitivity is large enough (w2 : w1 = 10), when there is a band gap visible, while the width of the gap will increase and another band gap will arise, as the ratio of permitivity gets larger. For one dimensional quasicrystal, a band gap can almost always been found if the ratio of permitivities are not one, unless when it gets too large (e.g. larger than 50). In the latter case, the band gap diappears and the bands get squeezed together at the position where the band gap exists for lower ratio of permitivity. We have noticed that by using different numbers of reciprocal lattice vectors in the calculation, the resolution of the band structure is affected, which is positively correlated to the number of Gs while the structure of the band remains the same, thus the position and width of the band gap is also unchanged.