15 April 1998 Ž . Optics Communications 149 1998 213–216 Precision small angle measurements with a digital moire technique ´ Roberto Torroba a , A.A. Tagliaferri b, ) a ( ) Centro de InÕestigaciones Opticas CIOp and OPTIMO, UniÕersidad Nacional de La Plata, La Plata, Argentina b Instituto de Fısica, UniÕersidade Federal Fluminense and UniÕersidade Estadual do Rio de Janeiro, Rio de Janeiro, Brazil ´ Received 19 June 1997; revised 27 November 1997; accepted 8 January 1998 Abstract A method to measure the rotation angle of structures is presented. The method is based on the projection digital moire ´ Ž . technique, but instead of using a projection system we make use of the Talbot self-imaging phenomenon in order to produce the grating on the surface. q 1998 Elsevier Science B.V. Keywords: Digital moire; Talbot self-images; Metrology ´ 1. Introduction In the classical moire effect, fringe patterns are formed ´ wx when two gratings are superimposed 1 . Metrological wx applications of this effect include strain analysis 2 , vibra- wx wx tion analysis 3 and contour mapping 4 . Among the different alternatives we have the projection based moire ´ technique, which involves projecting an equidistant grating with collimated illumination to simplify the analysis. This projection is observed through another grating, placed in wx front of the recording system 5 . The recording step can be done in various ways, but electronic and digital process- ing are increasingly attractive, as they remove much of the cumbersome analysis and offer the potential of rapid fringe interpretation. These techniques were prompted by ad- vances in solid state detector arrays, processing technology and the development of powerful desk-top computers. The basic system consists of a CCD camera and equipment to store and compute analog or digitized images, where the results can be displayed on a monitor. Digitized moire ´ fringes were obtained in different ways, such as time ) Corresponding author. E-mail: alberto@if.uff.br averaging recording of vibrating surfaces, external video wx signals or electronic scanning and sampling techniques 6 . In any case, the difficulty of the analysis largely depends on the mathematics involved in describing the configura- tion of the moire fringes. In general, there is no unified ´ approach as to where and how the coordinate systems are set. In the present work we are concerned with the tilt measurement of structures by means of a digital moire ´ wx technique 7 . Rather than a mathematical description, we adopt an approach which is based on the use of a transfer curve, allowing a quick determination of the angle the structure rotated. The digital moire technique is of the ´ projection type, but instead of an imaging system for the projection, we make use of the Talbot or self-imaging wx phenomenon 8 to produce the grating image on the surface. The moire patterns are obtained by storing a first image ´ of the object under test in a frame grabber, and subtracting the actual incoming image from the previously stored. In this way we obtain the moire fringes in real time. In the ´ following sections, we will explain the practical technique to obtain the fringes and the transfer curve. In connection with this approach, the errors of the system are estimated. Experimental results will also be shown. 0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII S0030-4018 98 00021-2