This paper deals with the non-linear oscillation of a simple pendulum and presents an approach for solving the non-linear differential equation that governs its movement by using the harmonic balance method. With this technique it is possible to easily obtain analytical approximate formulas for the period of the pendulum. As we shall see, these formulas show excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation.The most significant feature of the treatment presented is its simplicity because for the level of approximation considered in this paper the required work can be done "by hand". KEY WORDS: Simple pendulum, large-angle period, harmonic balance method 3

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a simple pendulum, and an approximate expression for its period is obtained. Only one iteration leads to high accuracy of the solutions and the relative error for the approximate period is less than 2% for amplitudes as high as 130°.Another important point is that this method provides an analytical expression for the angular displacement as a function of time as the sum of an infinite number of harmonics, although for practical purposes it is sufficient to consider only a finite number of harmonics. We believe that the present study may be a suitable and fruitful exercise for teaching and better understanding perturbation techniques in advanced undergraduate courses on classical mechanics.

Parallel aligned liquid crystal (PA-LC) devices are widely used in many optics and photonics applications to control the amplitude, phase and/or state of polarization (SOP) of light beams. Simplified models yet with a good predictive capability are extremely useful in the optimal application of these devices. In this paper we propose and demonstrate the validity of a novel model enabling to calculate the voltage dependent retardance provided by parallel-aligned liquid crystal (PA-LC) devices for a very wide range of incidence angles and any wavelength in the visible. We derive the theoretical expressions, and both experimental and theoretical retardance results are obtained showing a very good agreement. The proposed model is robust and well adapted to a reverse-engineering approach for the calibration of its parameters, whose values are obtained without ambiguities. The model is based on only three physically related magnitudes: two off-state parameters per wavelength and one global voltage dependent parameter, the tilt angle of the LC molecules. To our knowledge it represents the most simplified model available for PA-LC devices yet showing predictive capability. Not only eases the design of experiments dealing with unconventional polarization states or complex amplitude modulation, but it also serves to analyze the physics and dynamics of PA-LC cells since we have estimation for their voltage dependent tilt angle within the device.

We present an analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display. An ideal display would produce continuous phase-only modulation, reaching a maximum phase-modulation depth of 2pi. We introduce the concept of modulation diffraction efficiency that accounts for the effect of nonlinearities only in the phase modulation of the display. We review a diffractive model with which to evaluate this modulation efficiency, including modulation defects such as nonlinear phase modulation, coupled amplitude modulation, phase quantization, and a limited modulation depth. We apply this diffractive model to Fresnel lenses and show that these modulation defects produce a lens multiplex effect. Finally we demonstrate that the application of a minimum Euclidean projection principle leads to high modulation diffraction efficiency even if the phase-modulation depth is much less than 2pi. We demonstrate that the modulation efficiency can exceed 90% for a modulation depth of 1.4pi and can exceed 40% (the equivalent for a binary phase element) for a modulation depth of only 0.7pi. Experimental results from use of a twisted nematic liquid-crystal display are presented to confirm these conclusions.

Abstract:In this paper we experimentally analyze the performance of a twisted nematic liquid crystal on silicon (LCoS) display as a function of the angle of incidence of the incoming beam. These are reflective displays that can be configured to produce amplitude or phase modulation by properly aligning external polarization elements. But we demonstrate that the incident angle plays an important role in the selection of the polarization configuration. We performed a Mueller matrix polarimetric analysis of the display that demonstrates that the recently reported depolarization effect observed in this type of displays is also dependant on the incident angle.