We present the results of rigorous numerical calculations of the dependence of the reflection coefficient of a semi-infinite two-dimensional photonic crystal on the angle of incidence of the incoming plane wave. We show that,
contrary to some results published earlier, this coefficient is not strictly real even outside the crystal bandgaps.
We also propose a definition of the effective permittivity and permeability μ of a truncated photonic crystal
and specify the symmetry conditions to be satisfied by the truncation plane and the dominant crystal eigenmode
to assure continuity of ε and μ when the mode character changes from propagating to evanescent. The value of
the reflection coefficient obtained by treating the crystal as a homogeneous medium with ε and μ defined in the
proposed way is shown to be a good approximation to the rigorous value in a wide range of angles of incidence,
extending beyond that corresponding to propagating crystal modes.
The emission of a dipole in a finite-thickness photonic band gap structure is investigated. The dipole is located at a large value of the local density of states and its wavelength is taken at the edge of a full band gap. The resulting emission is highly enhanced and is confined in a small angular region. This is confirmed numerically for two different structures designed from two different tree-dimensional crystal: the woodpile and the simple cubic photonic crystals.
The diffraction by periodic structures using a representation of the field in some functional basis leads to a set of ordinary differential equations, which can be solved by numerical integration. When the basic functions are the exponential harmonics (Fourier decomposition) one arrives at the well-known classical differential method. In the case of simple lamellar profiles, the numerical integration can be substituted by eigenvalue-eigenvector technique, known in the field of diffraction by periodic systems under the name of rigorous coupled-wave analysis or method of Moharam and gaylord. When the basis functions are searched as the rigorous solutions of the diffraction problem inside the lamellar grooves, the theory is known under the name of modal method. A comparative analysis of the three methods is made to reveal the convergence rate for an arbitrary shaped grating using the staircase approximation. It is shown that in TM polarization this approximation leads to sharp peaks of the electric field near the edges. A higher number of fourir harmonics is then required to describe the field, compared with the case of a smooth profile, and a poor convergence is observed. The classical differential method, which does not use the staircase approximation does not suffer from this problem.
Conference Committee Involvement (3)
11th annual Sino-French “Photonics and Optoelectronics” PHOTONET International Research Network Workshop
11 April 2024 | Strasbourg, France
9th annual Sino-French “Photonics and Optoelectronics” PHOTONET International Research Network Workshop
7 April 2022 | Strasbourg, France
6th annual Sino-French “Photonics and Optoelectronics” PHOTONET International Research Network Workshop
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