Analytical solution of Maxwell’s equations in lossy and optically active crystals (original) (raw)

Photonic Crystals Transmission Phenomena

Light travelling through a slab made of a photonic material is expected to decay exponentially if its frequency lies within a bandgap. Numerical calculations as well as experiments show that sometimes a more complex behaviour occurs: The field inside the crystal may decay bi-exponentially or show saturation i.e. a small part of the incoming field is transmitted without attenuation. These propagation properties can be explained by a generic model which is solved analytically. The model medium consists of a separable dielectric function which is the sum of two one dimensional square well functions or trigonometric functions in two perpendicular directions.

Transport properties of nonlinear photonic crystals

Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics, 2005

Keen interest has been shown in exploiting the transport properties of nonlinear photonic crystals for modulation, switching and routing applications at telecommunication frequencies. This is due to the rich anisotropy evidenced by a highly-textured dispersion surface, and its dependence on the permittivity profile of the structure. We consider a two-dimensional photonic crystal, presume an instantaneous change in profile due to an optical Kerr effect, and show how the beams refraction angle depends on its intensity. For computational simplicity, we employ a self-consistent approach and the linear eigenvalue equation, which we solve using a finite element method. Previous studies have extracted directions of optical power flow from equifrequency contour gradients. Such gradients, more formally the group velocity, lack local definition and are only meaningful in the context of a spatial average over the unit cell. As a result, the relationship between the local character of the optical transport and the induced permittivity profile is obscured. By contrast, we explicitly consider the spatial dependence of the Poyntings vector, from which we also extract mean transport directions with much greater computational efficiency. We thereby demonstrate the interrelationship between optical transport and nonlinear response at the nanoscale. A consequent analysis of refraction in the context of the superprism effect reveals new aspects to the optical transport in such nonlinear systems.

The generalized solutions of Maxwell equations for the uniaxial crystal

The exact analytical solutions of Maxwell's equations are constructed by means of the method of generalized functions in vector form for unbounded uniaxial crystals. The fundamental solutions of a system of Maxwell's equations for uniaxial crystals are obtained. In particular, the solutions for elementary electric dipoles are deduced and the radiation patterns for Hertzian radiator dipole in such media are presented. Validity of the solutions have been checked up on balance of energy by integration of energy flow on sphere.

Accurate model of electromagnetic wave propagation in unidimensional photonic crystals with defects

Journal of …, 2007

Complete vectorial studies of electromagnetic wave propagation in complex media are difficult and computationally intensive. A mathematical continuous model can simplify some of such difficulties. Using this technique, we have modeled the propagation of electromagnetic plane waves through a semiinfinite photonic crystal with defects. The model is exact in the sense that the only approximations are the numerical accurate solution of ordinary differential equations and the simulation of defects by a high frequency sinusoidal continuous variation of the main crystal period.

Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals

Physical Review E, 2006

The effective medium theory is a useful approach for investigating the electromagnetic wave propagation in periodic multilayer slabs. It allows accurate computation of transmission and reflection spectra as well as of phase and group velocities. In this paper we derive an exact analytical expression for the energy velocity of a one-dimensional finite photonic crystal based on the effective medium approach. It accounts for the multiple reflections within the structure which results in the characteristic oscillations of the transmission spectrum. Our analytical expression holds for an arbitrary refractive index contrast and goes beyond the limits of the standard homogenization method. In order to validate our approach, results obtained by using the all-frequency effective energy velocity have been compared to those obtained using the transfer matrix method.

Spectral properties of conservative, dispersive, and absorptive photonic crystals

GAMM-Mitteilungen, 2018

This article reviews both recent progress on the mathematics of dispersive and absorptive photonic crystals and well-established results on conservative photonic crystals. The focus is on properties of the photonic band structures and we also provide results that are of importance for the understanding of lossy metal-dielectric photonic crystals.

Light propagation in an imperfect photonic crystal

Physical Review B, 2006

The envelope function approach for the electric and magnetic fields of the light wave in the photonic crystals has been used to investigate the propagation of light in disordered photonic crystals and its reflection or refraction at the boundary. We showed that small long-range distortion of the crystal lattice can explain peculiarities of the transmittance spectrum of the polystyrene colloid film we have grown.