Mariela Lázara Álvarez López

A novel rational harmonic balance approach for periodic solutions of conservative nonlinear oscillators

An analytical approximate procedure for a class of conservative single degree-of-freedom nonlinear oscillators with odd non-linearity is proposed. This technique is based on the generalized harmonic balance method in which analytical approximate solutions have rational forms. Unlike the classical harmonic balance techniques, in this new procedure the approximate solution and the restoring force are expanded in Fourier series prior to substituting them in the nonlinear differential equation. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. Four nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving a class of conservative nonlinear oscillatory systems.

SF-FDTD analysis of a predictive physical model for parallel aligned liquid crystal devices

Recently we demonstrated a novel and simplified model enabling to calculate the voltage dependent retardance provided by parallel aligned liquid crystal devices (PA-LCoS) for a very wide range of incidence angles and any wavelength in the visible. To our knowledge it represents the most simplified approach still showing predictive capability. Deeper insight into the physics behind the simplified model is necessary to understand if the parameters in the model are physically meaningful. Since the PA-LCoS is a black-box where we do not have information about the physical parameters of the device, we cannot perform this kind of analysis using the experimental retardance measurements. In this work we develop realistic simulations for the non-linear tilt of the liquid crystal director across the thickness of the liquid crystal layer in the PA devices. We consider these profiles to have a sine-like shape, which is a good approximation for typical ranges of applied voltage in commercial PA-LCoS microdisplays. For these simulations we develop a rigorous method based on the split-field finite difference time domain (SF-FDTD) technique which provides realistic retardance values. These values are used as the experimental measurements to which the simplified model is fitted. From this analysis we learn that the simplified model is very robust, providing unambiguous solutions when fitting its parameters. We also learn that two of the parameters in the model are physically meaningful, proving a useful reverse-engineering approach, with predictive capability, to probe into internal characteristics of the PA-LCoS device.

Interpolatory fixed-point algorithm for an efficient computation of TE and TM modes in arbitrary 1D structures at oblique incidence

We develop the Interpolatory Fixed-Point Algorithm (IFPA) to compute efficiently the TE and TM reflectance and transmittance coefficients for arbitrary 1D structures at oblique incidence. For this purpose, we demonstrate that the semi-analytical solutions of the Helmholtz equation provided by the fixed-point method have a polynomial dependence on variables that are related to the essential electromagnetic parameters -incidence angle and wavelength-, which allows a drastic simplification of the required calculations taking the advantage of interpolation for a few parameter values. The first step to develop the IFPA consists of stating the Helmholtz equation and boundary conditions for TE and TM plane incident waves on a 1D finite slab with an arbitrary permittivity profile surrounded by two homogeneous media. The Helmholtz equation and boundary conditions are then transformed into a second-order initial value problem which is written in terms of transfer matrices. By applying the fixed-point method, the coefficients of such transfer matrices are obtained as polynomials on several variables that can be characterized by a reduced set of interpolating parameters. We apply the IFPA to specific examples of 1D diffraction gratings, optical rugate filters and quasi-periodic structures, for which precise solutions for the TE and TM modes are efficiently obtained by computing less than 20 interpolating parameters.We develop the Interpolatory Fixed-Point Algorithm (IFPA) to compute efficiently the TE and TM reflectance and transmittance coefficients for arbitrary 1D structures at oblique incidence. For this purpose, we demonstrate that the semi-analytical solutions of the Helmholtz equation provided by the fixed-point method have a polynomial dependence on variables that are related to the essential electromagnetic parameters -incidence angle and wavelength-, which allows a drastic simplification of the required calculations taking the advantage of interpolation for a few parameter values. The first step to develop the IFPA consists of stating the Helmholtz equation and boundary conditions for TE and TM plane incident waves on a 1D finite slab with an arbitrary permittivity profile surrounded by two homogeneous media. The Helmholtz equation and boundary conditions are then transformed into a second-order initial value problem which is written in terms of transfer matrices. By applying the fixed-point method, the coefficients of such transfer matrices are obtained as polynomials on several variables that can be characterized by a reduced set of interpolating parameters. We apply the IFPA to specific examples of 1D diffraction gratings, optical rugate filters and quasi-periodic structures, for which precise solutions for the TE and TM modes are efficiently obtained by computing less than 20 interpolating parameters.