Uncertainty analysis of temporal phase-stepping algorithms for interferometry Raul R. Cordero a, * , Jerome Molimard b , Amalia Martı ´nez c , Fernando Labbe d a Leibniz Universita ¨ t Hannover, Herrenha ¨ user Str. 2, D-30419 Hannover, Germany b Ecole Nationale Supe ´rieure des Mines, 158 Cours Fauriel, 42023 Saint Etienne, France c Centro de Investigaciones en O ´ ptica, Ap. 1-948, CP 37000, Leo ´ n, Gto., Mexico d Universidad Te ´cnica Federico Santa Marı ´a, Ave. Espana 1680, Valparaiso, Chile Received 11 October 2006; received in revised form 22 February 2007; accepted 22 February 2007 Abstract We have addressed the problem of the uncertainty evaluation of phase values rendered by two popular algorithms: the N-bucket and the (N + 1)-bucket, both used to exploit temporal phase-stepping techniques. These algorithms, are mainly affected by errors in the calibration of the piezoelectric transducers used to achieve the phase shift, external vibration and optical noise. We have characterized and compared the influences of these errors on the phase uncertainty. We applied a Monte Carlo-based technique of uncertainty propagation that allowed us to consider in the uncertainty evaluation the simultaneous contributions of different error sources. The uncertainty evaluation was per- formed for phase values in the range (0, 2p), with different values of N and assuming that the phase was calculated from fringe patterns generated by using either Moire ´ interferometry or electronic speckle-pattern interferometry. We found that the uncertainties associated with the phases rendered by both algorithms are similar and they can be significantly affected by the optical noise and the value of N. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.87.Bg; 42.30.Ms; 81.70.Fy Keywords: Laser interferometry; Phase shifting; Uncertainty analysis 1. Introduction The whole-field deformation induced on a specimen can be measured by using speckle-based techniques, such as shearography [1–3] and electronic speckle-pattern interfer- ometry (ESPI) [4–6] as well as Moire ´ interferometry [7–9]. These high-sensitivity optical techniques are based on the generation of fringe patterns formed by the interference of wavefronts coherently added. The determination of whole-field deformation field requires the measurement of the phase at each point of the generated fringe patterns. In order to measure the whole-field values of the phase in interferometric patterns, techniques of spatial and tem- poral phase-stepping [10] can be applied. The latter involves capturing several fringe patterns, generated at dif- ferent times by shifting the phases of the interfering beams in known amounts. This allows building up a set of simul- taneous equations that are solved for the phase by using algorithms [11–18]. There are two types of temporal phase stepping algo- rithms; those which only require the phase shifts to be equal, regardless of their exact value [14–16] and those which need a specific value of the phase shift. Two popular algorithms that require a specific value of the phase shift are the conventional N-bucket [17] and the self-calibrating (N + 1)-bucket [18]. The outcomes of both, the N-bucket and the (N + 1)- bucket algorithms, are mainly affected by the presence of harmonics in the signal; errors in the calibration of the pie- zoelectric transducers (PZT) used to achieve the phase shift; high-frequency environmental perturbations that 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.057 * Corresponding author. Tel.: +49 511 762 2679; fax: +49 511 762 4418. E-mail address: rrcorder@puc.cl (R.R. Cordero). www.elsevier.com/locate/optcom Optics Communications 275 (2007) 144–155