Optical encoders are commonly used for high accuracy position measurement, both linear and angular. In order to determine the position, the optical encoder generates two electrical signals that are combined using the arctangent algorithm. There are a number of situations, optical, mechanical and electronic, that affect these signals and produce an error in the position measurement. In this work, we analyze the error produced in optical encoders when the electrical signals vary from their nominal values. By using a linear expansion, simple expressions for the error estimation are obtained which can be used to improve the design of the optical encoders. In addition, an experimental verification of the theoretical results is performed.

A simple collimation technique based on measuring the period of one self-image produced by a diffraction grating is 11 proposed. Transversal displacement of the grating is not required, and then automatic single-frame processing can 12 be performed. The self-image is acquired with a CMOS camera, and the period is computed using the from the lens. The transmittance of the grating is tx 69 P n a n expiqnx being n integer numbers, a n are the Ix 3 ; z 2 ∝ I 0

We present a collimation technique based on a double grating system to locate with high accuracy an emitter in the focal plane of a lens. Talbot self-images are projected onto the second grating producing moiré interferences. By means of two photodetectors positioned just behind the second grating, it is possible to determine the optimal position of the light source for collimation by measuring the phase shift between the signals over the two photodetectors. We obtain mathematical expressions of the signal in terms of defocus. This allows us to perform an automated technique for collimation. In addition, a simple and accurate visual criterion for collimating a light source using a lens is proposed. Experimental results that corroborate the proposed technique are also presented.

Abstract. We propose an accurate technique for obtaining highly collimated beams, which also allows testing the collimation degree of a beam. It is based on comparing the period of two different self-images produced by a single diffraction grating. In this way, variations in the period of the diffraction grating do not affect to the measuring procedure. Self-images are acquired by two CMOS cameras and their periods are determined by fitting the variogram function of the self-images to a cosine function with polynomial envelopes. This way, loss of accuracy caused by imperfections of the measured self-images is avoided. As usual, collimation is obtained by displacing the collimation element with respect to the source along the optical axis. When the period of both self-images coincides, collimation is achieved. With this method neither a strict control of the period of the diffraction grating nor a transverse displacement, required in other techniques, are necessary. As an example, a LED considering paraxial approximation and point source illumination is collimated resulting a resolution in the divergence of the beam of δφ = ±1.57 µrad.

In this Letter, we analyze the near-field diffraction pattern 10 produced by chirped gratings. An intuitive analytical inter-11 pretation of the generated diffraction orders is proposed. 12Several interesting properties of the near-field diffraction 13 pattern can be determined, such as the period of the fringes 14 and its visibility. Diffraction orders present different 15 widths and also, some of them present focusing properties. 16The width, location, and depth of focus of the converging 17 diffraction orders are also determined. grating, an example is shown in Fig. 1. 61Now, let us divide the incident beam into a sum of narrow 62Gaussian beams, whose width is ω s and placed at different F1:1 Fig. 1. (a) Chirped binary diffraction grating with starting period F1:2 p 0 50 μm, final period p 1 10 μm, and length l 500 μm. F1:3For this case, the spatial frequency dependency is linear: