Metrol. Meas. Syst., Vol. XVII (2010), No. 2, pp. 173-194 ____________________________________________________________________________________________________________________________________________________________________________________ Article history: received on Mar. 26, 2010; accepted on May 27, 2010; available online on Jun. 16, 2010, DOI: 10.2478/v10178-010-0016-6. METROLOGY AND MEASUREMENT SYSTEMS Index 330930, ISSN 0860-8229 www.metrology.pg.gda.pl NEW CARRÉ EQUATION Pedro Americo Almeida Magalhaes Junior, Perrin Smith Neto, Cristina Almeida Magalhães Pontificia Universidade Catolica de Minas Gerais, Av. Dom Jose Gaspar 500, CEP 30535-610, Belo Horizonte, Minas Gerais, Brazil ( paamj@oi.com.br, +55 31 3221 6813, psmith@pucminas.br, crisamagalhaes@hotmail.com) Abstract The present work offers new equations for phase evaluation in measurements. Several phase-shifting equations with an arbitrary but constant phase-shift between captured intensity signs are proposed. The equations are similarly derived as the so called Carré equation. The idea is to develop a generalization of the Carré equation that is not restricted to four images. Errors and random noise in the images cannot be eliminated, but the uncertainty due to their effects can be reduced by increasing the number of observations. An experimental analysis of the errors of the technique was made, as well as a detailed analysis of errors of the measurement. The advantages of the proposed equation are its precision in the measures taken, speed of processing and the immunity to noise in signs and images. Keywords: metrology, Moiré techniques, fringe analysis, phase measurement, phase shifting technique, Carré equation. © 2010 Polish Academy of Sciences. All rights reserved 1. Introduction Phase shifting is an important technique in experimental mechanics [1-3]. Conventional phase shifting equations require phase shift amounts to be known; however, errors in phase shifts are common for the phase shift modulators in real applications, and such errors can further cause substantial errors in the determination of phase distributions. There are many potential error sources which may affect the accuracy of the practical measurement, e.g. the phase shifting errors, detector nonlinearities, quantization errors, source stability, vibrations and air turbulence, and so on [4]. Currently, the phase shifting technique is the most widely used technique for evaluation of interference fields in many areas of science and engineering. Its principle is based on the evaluation of the phase values from several phase modulated measurements of the intensity of the interference field. It is necessary to carry out at least three phase-shifted intensity measurements to determine the phase unambiguously and very accurately, at every point of the detector plane. The phase shifting technique offers fully automatic calculation of the phase difference between two coherent wave fields that interfere in the process. There are various phase shifting equations for phase calculation that differ on the number of phase steps, on phase shift values between captured intensity frames, and on their sensitivity to the influencing factors during practical measurements [4]. The general principle of most interferometric measurements is as follows. Two light beams (reference and object) interfere after an interaction of the object beam with the measured object, i.e. the beam is transmitted or reflected by the object. The distribution of the intensity of the interference field is then detected, e.g. using a photographic film, CCD camera. The phase difference between the reference and the object beam can be determined using the mentioned phase calculation technique. The phase shifting technique is based on an