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We present a formalism for optical waveguiding in two-dimensional periodic structures by means of Bragg reflection. The theory provides analytic expressions for the field modal profiles, dispersion, and attenuation or gain in the waveguide. Line defects in two-dimensional photonic crystal slabs are used as the principal example.

The model used for analyzing the gain characteristics of Er-doped waveguide amplifier (EDWA) is based on the rate-propagation equations with intensity profiles of pump and signal light (IPPSL). This simulation indicates that the IPPSL have significant effect on the gain of signal light. The results show that a gain change 2.3 dB of signal light at 1534 nm is obtained for a 4 cm long channel waveguide amplifier with different intensity profiles pumped by 80 mW at 980 nm. Experiment also shows that a gain change about 2.31 dB at 1534 nm for the channel waveguide of 4 cm length is obtained for different IPPSL. Thus, the effect of IPPSL should be considered to enhance the pump efficiency in the design of EDWA.

We consider an eigenvalue problem for a certain type of quasi-linear second-order differential equation on the interval (0, ∞). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a self-focusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.

We propose a new method for calculating dispersion spectra of shear waves in the two-dimensional free phononic plates made of solid matrix with periodically distributed inclusions and in the waveguides composed of a phononic layer between two periodic substrates. The method proceeds from the propagator M which involves exact integration in the depth coordinate. Because the components of M can be very large, the dispersion equation for a free plate is recast in terms of the resolvent of propagator R = (αI - M)^-1 (α is a constant) which is numerically stable. The resolvent is the central object of the method. Another key tool, which comes into play in the case of a waveguide, is a projector P expressed as a contour integral of the resolvent of the substrate. The projector allows to extract the "physical" modes decreasing into the depth of the substrates without solving the wave equation. The resulting dispersion equation for a waveguide defined via the projectors for the substrates and the resolvent for the enclosed layer is numerically stable. We provide several options for the calculation of the resolvent and projector. Besides, special attention is given to derivation of the dispersion equations for the uncoupled symmetric and antisymmetric dispersion branches in the case of mirror-symmetric structures.

We consider the nonlinear propagation of an optical beam in a two-dimensional waveguide matrix, which is described by the discrete nonlinear Schrödinger equation with a self-focusing nonlinearity. Our study focuses on the stability domain of the discrete vortex solitons with its topological invariant S equal to 1. Such domain is identified with the propagation constant k and the coupling constant C. Our simulations show that there is a critical value C_cr for any fixed value of k. At C ≤ C_cr, the stable domain is continuous. However, at C > C_cr, the stable domain becomes discontinuous and fuzzy. The distribution of the continuous stable regions is identified and plotted. Moreover, we give the relation of the total power P to C and k, which enable us to figure out the continuous stability area through the (P, C) plane.

We present a formalism for optical waveguiding in two-dimensional periodic structures by means of Bragg reflection. The theory provides analytic expressions for the field modal profiles, dispersion, and attenuation or gain in the waveguide. Line defects in two-dimensional photonic crystal slabs are used as the principal example.