The Generalised Lau Effect

Metrological errors in optical encoders

Abstract: 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.

Apparatus for producing patterned illumination including at least one array of light sources and at least one array of microlenses

Abstract: An apparatus for producing structured light comprises a first optical arrangement which comprises a microlens array comprising a multitude of transmissive or reflective microlenses which are regularly arranged at a lens pitch P and an illumination unit for illuminating the microlens array. The illumination unit comprises an array of light sources for emitting light of a wavelength L each and having an aperture each, wherein the apertures are located in a common emission plane which is located at a distance D from the microlens array. For the lens pitch P, the distance D and the wavelength L, the following equation applies P 2=2LD/N, wherein N is an integer with N≧1. High-contrast high-intensity light patterns can be produced.

Collimation method using a double grating system.

Abstract: 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 moire 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.

Double grating systems with one steel tape grating

Abstract: Steel tape gratings are used in different metrology applications. As the period of these gratings was large (around 100μm,), its analytical study has been performed, up to date, using a geometrical approach. Nowadays, steel tape gratings can be manufactured with lower periods, around 20–40 μm, and diffractive effects must be taken into account. Also, due to the roughness of the surface, statistical techniques need to be considered to analyze their behavior. In this work, an analysis of the pseudo-imaging formation in a double grating system including one steel tape grating is performed. In particular Moire and Lau configurations are analyzed. We have found that roughness significantly affects to Moire configuration. However, its effect is negligible in Lau configuration. Generalized grating imaging configuration is also studied in depth. It is shown that roughness does not affect to the contrast of pseudoimages, but it modifies their depth of focus.

Invariant grating pseudoimaging using polychromatic light and a finite extension source

Abstract: The Talbot effect is a well studied phenomenon by which grating pseudoimages appear at certain periodic distances when monochromatic light is used Recently, numerical simulations have shown a new phenomenon; when a polychromatic light beam is used in a double grating system, the intensity of the pseudoimages presents a transverse-profile that remains unaffected over a wide range of propagation distances This effect can be used to increase the tolerances of gratings based optical devices, such as displacement measurement systems, interferometers, and spectrometers The pseudoimages formation with a polychromatic and finite extension light source is analytically and experimentally demonstrated Relatively simple analytical expressions for the intensity and the contrast allow us to predict when pseudoimages present a constant contrast and when they disappear Furthermore, we experimentally obtain the pseudoimages using the proposed configuration, corroborating the theoretical predictions

Beugungserscheinungen an Doppelrastern

Abstract: Stellt man 2 gleichartige Raster hintereinander, so wird in bestimmten Abstanden ein Raster durch das andere Raster abgebildet. Diese Abstande ergeben sich aus der Beugungstheorie. Sie werden berechnet und mit dem Experiment verglichen. Es ergibt sich dabei eine ganze Schar von Abstanden, die sich nach einem einfachen Gesetz aneinanderreihen. Bei der Berechnung der Gangunterschiede an den einzelnen Spalten bei den verschiedenen Abstanden ergaben sich auffallende, symmetrisch angelegte Folgen von Zahlen, die in einer Tabelle wiedergegeben sind.

The Lau effect (a diffraction experiment with incoherent illumination)

Abstract: In 1948 Ernst Lau described a diffraction experiment, whereby the object was illuminated by an extended white light source. The object consisted of two coarse gratings, positioned behind each other at a distance of a few centimeters. Colored fringes appear at infinity. We present an explanation, based on diffraction theory. We also note similarities to Talbot's effect called “self-imaging”, or sometimes “Fourier-imaging”. A specific application of the Talbot effect, the Talbot interferometer, is equivalent to the Lau effect in terms of the theory of partial coherence. Based on this equivalence we suggest two applications of the Lau effect, a simple interferometer and a Fourier spectrometer.

Lau effect and coherence theory

Abstract: We show that the Lau effect can be explained in a simple manner by means of the coherence theory. Such an explanation leads to predict some new features of the effect that are confirmed by the experiment.

Incoherent Superposition of Multiple Self-imaging Lau Effect and Moiré Fringe Explanation

Abstract: Characteristics of the diffraction field of two spatially separated linear diffraction gratings under incoherent illumination are studied. The analytical model is based on the spatial superposition of many mutually incoherent Talbot (self-imaging) effects. It permits a simple explanation of the basic parameters of the ‘incoherent’ diffraction images: axial localization, spatial period and lateral displacement. The moire fringe formation in space, the imaging of a grating by a second grating, and the Lau effect are considered; the results are compared with the former studies in the literature.

An explanation of the Lau effect based on coherence theory

Abstract: The Lau effect is investigated as a result of the propagation of partially coherent light. It is found that the fringe structure is dependent on the relationship between the grating separation and the periodicity of the complex degree of spatial coherence in the plane of the second grating. This provides a simple and elegant means of interpreting and extending the Lau effect.