Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides

We study the propagation of time-harmonic acoustic or transverse magnetic (TM) polarized electromagnetic waves in a periodic waveguide lying in the semi-strip $(0,\infty)\times(0,L)$. It is shown that there exists a Riesz basis of the space of solutions to the time-harmonic wave equation such that the translation operator shifting a function by one periodicity length to the left is represented by an infinite Jordan matrix which contains at most a finite number of Jordan blocks of size $> 1$. Moreover, the Dirichlet-, Neumann- and mixed traces of this Riesz basis on the left boundary also form a Riesz basis. Both the cases of frequencies in a band gap and frequencies in the spectrum and a variety of boundary conditions on the top and bottom are considered

Frozen light in periodic metamaterials

Wave propagation in spatially periodic media, such as photonic crystals, can be qualitatively different from any uniform substance. The differences are particularly pronounced when the electromagnetic wavelength is comparable to the primitive translation of the periodic structure. In such a case, the periodic medium cannot be assigned any meaningful refractive index. Still, such features as negative refraction and/or opposite phase and group velocities for certain directions of light propagation can be found in almost any photonic crystal. The only reservation is that unlike hypothetical uniform left-handed media, photonic crystals are essentially anisotropic at frequency range of interest. Consider now a plane wave incident on a semi-infinite photonic crystal. One can assume, for instance, that in the case of positive refraction, the normal components of the group and the phase velocities of the transmitted Bloch wave have the same sign, while in the case of negative refraction, those components have opposite signs. What happens if the normal component of the transmitted wave group velocity vanishes? Let us call it a "zero-refraction" case. At first sight, zero normal component of the transmitted wave group velocity implies total reflection of the incident wave. But we demonstrate that total reflection is not the only possibility. Instead, the transmitted wave can appear in the form of an abnormal grazing mode with huge amplitude and nearly tangential group velocity. This spectacular phenomenon is extremely sensitive to the frequency and direction of propagation of the incident plane wave. These features can be very attractive in numerous applications, such as higher harmonic generation and wave mixing, light amplification and lasing, highly efficient superprizms, etc

Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime

We study here the nonlinear Schrodinger Equation (NLS) as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell equations (NLM). The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current $\mathbf{J}$ producing an EM wave. The wave nonlinear evolution is analyzed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to some three small parameters $\alpha$, $\beta$ and $\varrho$. These parameters are introduced through the excitation current $\mathbf{J}$ to scale respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with $\beta ^{-1}$ scaling its spatial extension; (iii) its frequency bandwidth, with $\varrho ^{-1}$ scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current $\mathbf{J}$. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of $\alpha$, $\beta$ and $\varrho$, but it also produces new extended NLS (ENLS) equations providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compare with the classical NLS

Resonance enhancement of magnetic Faraday rotation

Magnetic Faraday rotation is widely used in optics. In natural transparent materials, this effect is very weak. One way to enhance it is to incorporate the magnetic material into a periodic layered structure displaying a high-Q resonance. One problem with such magneto-optical resonators is that a significant enhancement of Faraday rotation is inevitably accompanied by strong ellipticity of the transmitted light. More importantly, along with the Faraday rotation, the resonator also enhances linear birefringence and absorption associated with the magnetic material. The latter side effect can put severe limitations on the device performance. From this perspective, we carry out a comparative analysis of optical microcavity and a slow wave resonator. We show that slow wave resonator has a fundamental advantage when it comes to Faraday rotation enhancement in lossy magnetic materials