The paper considers a model of adiabatic waveguide modes in the zero approximation applied to the numerical solution of the problem of single-mode propagation of guided modes in a smoothly irregular integrated optical waveguide. Within the framework of the model, the solution to the Maxwell system of equations is reduced to a form that is expressed through the solution of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field. The multilayer structure of waveguides makes it possible to carry out one more stage of reducing the system of equations of the model to a homogeneous system of linear algebraic equations, the condition of nontrivial solvability of which specifies the dispersion equation. Auxiliary eigenvalue and eigenvector problems for describing adiabatic waveguide modes are solved. Example solutions for single-mode propagation of adiabatic waves are presented.
The paper considers the cross-sectional method and the small parameter method in application to the numerical solution of the problem of single-mode propagation of TE-modes in a smoothly irregular integrated optical waveguide invariant in the transverse direction. The cross-sectional method is formulated as a Helmholtz problem for a two-dimensional strip of smoothly irregular thickness. Using the Kantorovich method, the problem is reduced to a parametric problem for eigenvalues and eigenfunctions on a segment. The small parameter method is formulated as a Helmholtz problem on a two-dimensional plane, which in turn is also reduced to a parametric eigenvalue/eigenfunction problem on an axis. We implement various algorithms for the numerical solution of the resulting problems. Their coincidence high accuracy is shown with at the intersection of the definition domains.
The exactly soluble model of a train of zero-duration electromagnetic pulses interacting with a 1D atom with short-range interaction potential modelled by a δ-function is considered. The model is related to the up-to-date laser techniques providing the duration of pulses as short as a few attoseconds and the intensities higher than 1014 W/cm2.
The paper describes the classical and generalized Luneburg lens in the 3D and planar waveguide implementation. We demonstrate the relation between the focusing inhomogeneity of the effective refractive index of waveguide Luneburg lens and the irregularity of the waveguide layer thickness generating this inhomogeneity. For the dispersion relation of the irregular thin-film waveguide in the model of adiabatic waveguide modes we solve the problem of mathematical synthesis and computer-aided design of the thickness profile of waveguide layer for the Luneburg thin-film generalized waveguide lens with a given focal length.
The calculations are carried out in specially normalized coordinates convenient for computer calculations. The solution is compared with the one obtained using the cross-sections method.
We investigate the waveguide propagation of polarized monochromatic light in a smoothly irregular transition between two regular planar dielectric waveguides. The single-mode approximation of the cross-sections method is used. The smooth evolution of the electromagnetic field propagating mode is calculated. The calculation is performed using the regularized stable numerical method.
Basing on Dirac equation for interacting massless fermions, we propose a nonlinear model that describes a possible mechanism of ferromagnetism in graphene structures, resulting from electron-electron interaction and spontaneous breaking of spin symmetry of valence electrons. Qualitative predictions of the model are important for practical applications in spintronics. Localized kink-antikink patterns of valence electron spin density on the graphene surface are calculated, their interaction is described, and, finally, the formation of their quasi-bound metastable states (breathers) is investigated. The spectrum of breathers is calculated in both the analytical and the numerical form. Once created, the inverted population of the appropriate states may be used to generate quantum coherent nonlinear spin waves that can find practical applications in nanoelectronics and spintronics. Optical excitation and possible lasing transitions between the breather discrete states are discussed.
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