Jorge Francés

An accurate closed-form approximate solution for the quintic Duffing oscillator equation

An accurate closed-form solution for the quintic Duffing equation is obtained using a cubication method. In this method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude. The replacement of the original nonlinear equation by an approximate cubic Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn, respectively. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is lower than 0.37%.

Surface relief model for photopolymers without cover plating

Relief surface changes provide interesting possibilities for storing diffractive optical elements on photopolymers and are an important source of information to characterize and understand the material behaviour. In this paper we present a 3-dimensional model based on direct measurements of parameters to predict the relief structures generated on the material. This model is successfully applied to different photopolymers with different values of monomer diffusion. The importance of monomer diffusion in depth is also discussed.

Retardance and flicker modeling and characterization of electro-optic linear retarders by averaged Stokes polarimetry

A polarimetric method for the measurement of linear retardance in the presence of phase fluctuations is presented. This can be applied to electro-optic devices behaving as variable linear retarders. The method is based on an extended Mueller matrix model for the linear retarder containing the time-averaged effects of the instabilities. As a result, an averaged Stokes polarimetry technique is proposed to characterize both the retardance and its flicker magnitude. Predictive capability of the approach is experimentally demonstrated, validating the model and the calibration technique. The approach is applied to liquid crystal on silicon displays (LCoS) using a commercial Stokes polarimeter. Both the magnitude of the average retardance and the amplitude of its fluctuation are obtained for each gray level value addressed, thus enabling a complete phase characterization of the LCoS.

Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices

Recently we proposed a novel polarimetric method, based on Stokes polarimetry, enabling the characterization of the linear retardance and its flicker amplitude in electro-optic devices behaving as variable linear retarders. In this work we apply extensively the technique to parallel-aligned liquid crystal on silicon devices (PA-LCoS) under the most typical working conditions. As a previous step we provide some experimental analysis to delimitate the robustness of the technique dealing with its repeatability and its reproducibility. Then we analyze the dependencies of retardance and flicker for different digital sequence formats and for a wide variety of working geometries.

Educational Software for Interference and Optical Diffraction Analysis in Fresnel and Fraunhofer Regions Based on MATLAB GUIs and the FDTD Method

Interference and diffraction of light are elementary topics in optics. The aim of the work presented here is to develop an accurate and cheap optical-system simulation software that provides a virtual laboratory for studying the effects of propagation in both time and space for the near- and far-field regions. In laboratory sessions, this software can let optical engineering undergraduates simulate many optical systems based on thin slits. The numerical method used is the finite-difference time-domain (FDTD) method that has been successfully applied in many engineering fields. Using this numerical method, the irradiance distribution can be successfully evaluated in different planes far from the simulation grid without degrading performance. In addition, an easy-to-use MATLAB GUI handles all the parameters of the FDTD simulation and computes theoretical values of irradiance for both the Fresnel and Fraunhofer regions. Therefore, by using this software, the student is able to analyze the behavior of the Fresnel and Fraunhofer expressions as a function of the distance. This distance is defined as the space between the slits plane and the plane that contains the virtual screen on which the irradiance pattern is represented.

Relief diffracted elements recorded on absorbent photopolymers

Relief surface changes provide interesting possibilities for storing diffractive optical elements on photopolymers and are an important source of information for characterizing and understanding the material behavior. In this paper we use a 3-dimensional model, based on direct parameter measurements, for predicting the relief structures generated on without-coverplate photopolymers. We have analyzed different spatial frequency and recording intensity distributions such as binary and blazed periodic patterns. This model was successfully applied to different photopolymers with different values of monomer diffusion.

Higher accurate approximate solutions for the simple pendulum in terms of elementary functions

A closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained. To do this, the exact expression for the maximum tension of the string of the pendulum is first considered and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation. We obtain the parameters for the approximate by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. We believe that this letter may be a suitable and fruitful exercise for teaching and better understanding nonlinear oscillations of a simple pendulum in undergraduate courses on classical mechanics.

Electrical dependencies of optical modulation capabilities in digitally addressed parallel aligned liquid crystal on silicon devices

Abstract. Parallel aligned liquid crystal on silicon (PA-LCoS) displays have found wide acceptance in applications requiring phase-only modulation. Among LCoS devices, and PA-LCoS as a specific case, digital addressing has become a very common technology. In principle, modern digital technology provides some benefits with respect to analog addressing such as reduced interpixel cross-talk, lower power consumption and supply voltage, gray level scale repeatability, high programmability, and noise robustness. However, there are also some degradating issues, such as flicker, which may be enhanced. We analyze the characteristics of the digital pulse width modulated voltage signals in relation to their effect on the optical modulation capabilities of LCoS displays. We apply calibration techniques developed in our laboratory, basically the classical linear polarimeter extended to take into account the existence of flicker. Various digital sequence formats are discussed, focusing the analysis on the variations in the magnitude of the applied voltages across the LC layer. From this analysis, we obtain how to amplify the retardance dynamic range and how to enhance linearity in the device without enhancing flicker and without diminishing the number of available quantization levels. Electrical configurations intended for phase-only and intensity modulation regimes, useful in diffractive optics, are given.

Diffractive lenses recorded in absorbent photopolymers.

Photopolymers can be appealing materials for diffractive optical elements fabrication. In this paper, we present the recording of diffractive lenses in PVA/AA (Polyvinyl alcohol acrylamide) based photopolymers using a liquid crystal device as a master. In addition, we study the viability of using a diffusion model to simulate the lens formation in the material and to study the influence of the different parameters that govern the diffractive formation in photopolymers. Once we control the influence of each parameter, we can fit an optimum recording schedule to record each different diffractive optical element with the optimum focalization power.

Volume Holograms in Photopolymers: Comparison between Analytical and Rigorous Theories

There is no doubt that the concept of volume holography has led to an incredibly great amount of scientific research and technological applications. One of these applications is the use of volume holograms as optical memories, and in particular, the use of a photosensitive medium like a photopolymeric material to record information in all its volume. In this work we analyze the applicability of Kogelnik’s Coupled Wave theory to the study of volume holograms recorded in photopolymers. Some of the theoretical models in the literature describing the mechanism of hologram formation in photopolymer materials use Kogelnik’s theory to analyze the gratings recorded in photopolymeric materials. If Kogelnik’s theory cannot be applied is necessary to use a more general Coupled Wave theory (CW) or the Rigorous Coupled Wave theory (RCW). The RCW does not incorporate any approximation and thus, since it is rigorous, permits judging the accurateness of the approximations included in Kogelnik’s and CW theories. In this article, a comparison between the predictions of the three theories for phase transmission diffraction gratings is carried out. We have demonstrated the agreement in the prediction of CW and RCW and the validity of Kogelnik’s theory only for gratings with spatial frequencies higher than 500 lines/mm for the usual values of the refractive index modulations obtained in photopolymers.