Mariela L. Alvarez

First-harmonic diffusion-based model applied to a polyvinyl-alcohol– acrylamide-based photopolymer

The photopolymerization diffusion models give accurate comprehension of the mechanism of hologram formation inside photopolymer materials. Although several models have been proposed, these models share the common assumption that there is an interplay between the processes of monomer polymerization and monomer diffusion. Nevertheless, most of the studies to check the validity of the theoretical models have been done by using photopolymers of the DuPont™ type, or photopolymer materials with values of the monomer diffusion time similar to those of the DuPont material. We check the applicability of a modified diffusion-based model to a polyvinyl alcohol–acrylamide photopolymer. This material has the property of longer diffusion times for the monomer to travel from the unexposed to the exposed zones than in the case of other polymeric materials. Some interesting effects are observed and theoretically treated by using the modified first-harmonic diffusion-based model we propose.

Higher-order analytical approximate solutions to the nonlinear pendulum by He's homotopy method

A modified Hes homotopy perturbation method is used to calculate the periodic solutions of a nonlinear pendulum. The method has been modified by truncating the infinite series corresponding to the first-order approximate solution and substituting a finite number of terms in the second-order linear differential equation. As can be seen, the modified homotopy perturbation method works very well for high values of the initial amplitude. Excellent agreement of the analytical approximate period with the exact period has been demonstrated not only for small but also for large amplitudes A (the relative error is less than 1% for A < 152°). Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. II: A More Accurate Approximate Solution

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the behaviour of a relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation A. Only one iteration leads to high accuracy of the solutions and for any value of A and the discrepancy between the approximate frequency and the exact one never exceeds 1.6%.

Generation of diffractive optical elements onto a photopolymer using a liquid crystal display

Liquid crystal displays (LCDs) are widely used as spatial light modulators (SLMs) in many applications (optical signal processing, holographic data storage, diffractive optics...). In particular, as an alternative microoptics recording scheme we have explored the possibility to use a LCD to display the diffractive optical element (DOE) to be recorded onto a photosensitive phase material, so as to enhance the flexibility of the recording architecture. In this application the LCD acts as an amplitude dynamic transparency. By means of an optical system we image the function addressed to the LCD onto the recording material. The element to be recorded onto the phase material can be easily changed simply by changing the function addressed to the LCD. Among the recording materials, photopolymers provide very attractive capabilities. They present a great flexibility in their composition, the recording layer can be manufactured in a wide range of possible thicknesses, and they are inexpensive. These properties make it an interesting material to generate the phase DOEs. Both the composition and the thickness need to be optimized for the application to DOEs. In this work we explore the results dealing with the calibration of the recording setup and the photopolymer material. We also analyse the performance of phase-only diffractive lenses generated onto the photopolymer. Promising results have been obtained, where the focalization of the diffractive lenses generated has been demonstrated.

Cubication of Conservative Nonlinear Oscillators.

A cubication procedure of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force, and this allows us to approximate the original nonlinear differential equation by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A, while in a Taylor expansion of the restoring force these coefficients are independent of A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain an approximate frequency–amplitude relation as a function of the complete elliptic integral of the first kind. Some conservative nonlinear oscillators are analysed to illustrate the usefulness and effectiveness of this scheme.

Real-time interferometric characterization of a polyvinyl alcohol based photopolymer at the zero spatial frequency limit

We characterize the optical modulation properties of a polyvinyl alcohol/acrylamide (PVA/AA) photopolymer at the lowest end of recorded spatial frequencies. To achieve this goal we have constructed a double beam interferometer in combination with the setup to expose the recording material. This is a novel approach since usually holographic recording materials are only characterized at high spatial frequencies. Some benefits are provided by the approach we propose: a direct calculation of the properties of the material is possible, and on the other hand additional information can be obtained since the results are not influenced by diffusion processes. Furthermore, this characterization is needed to optimize the PVA/AA photopolymers for another range of applications, such as recording of diffractive optical elements, where very low spatial frequencies are recorded. Different PVA/AA compositions and layer thicknesses have been analyzed. We have found that, depending on the layer characteristics, we can achieve high values of the phase-shift modulation depth and enhance the sensitivity of the material.

Nonlinear oscillator with discontinuity by generalized harmonic balance method

A generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(x). This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form. 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. We find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric, piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation we have shown that the relative error in the analytical approximate frequency is 0.24%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

Accurate control of a liquid-crystal display to produce a homogenized Fourier transform for holographic memories.

We show an accurate procedure to obtain a Fourier transform (FT) with no dc term using a commercial twisted-nematic liquid-crystal display. We focus on the application to holographic storage of binary data pages, where a drastic decrease of the dc term in the FT is highly desirable. Two different codification schemes are considered: binary pi radians phase modulation and hybrid ternary modulation. Any deviation in the values of the amplitude and phase shift generates the appearance of a strong dc term. Experimental results confirm that the calculated configurations provide a FT with no dc term, thus showing the effectiveness of the proposal.

Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

A modified generalized, rational harmonic balance method is used to construct approximate frequency-amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable. The procedure is used to solve the nonlinear differential equation approximately. The approximate frequency obtained using this procedure is more accurate than those obtained using other approximate methods and the discrepancy between the approximate frequency and the exact one is lower than 0.40%.

Exploring binary and ternary modulations on a PA-LCoS device for holographic data storage in a PVA/AA photopolymer

We focus on the novelty of three elements in holographic data storage systems (HDSS): the data pager, where we introduce a parallel-aligned liquid crystal on silicon (PA-LCoS) microdisplay; the recording material, where we consider the highly versatile PVA/AA photopolymer; and also in the architecture of the object arm, where a convergent correlator system is introduced. We show that PA-LCoS devices cannot implement pure hybrid-ternary modulated (HTM) data pages but a rather close approximation. Validation of the HDSS expressions for the convergent correlator and comparison with the widespread 4-f system is performed. Experimental results with PVA/AA material showing bit-error rates (BER) in the range of 10-3, further show its potential application for HDSS, and also demonstrate the validity of the testing platform and PA-LCoS calibration and optimization.