Analysis of the π phase–shifts obtained in the Fourier spectra of phase gratings and grids by using two–window grating interferometry
N.–I. Toto–Arellanoª, G. Rodríguez–Zuritab, A. Martínez Garcíaª, J.F. Vázquez–Castillob, and J.A. Rayas Álvarezª
ª Centro de Investigaciones en Óptica, A.C., Loma del Bosque #115 Col. Lomas del Campestre, Apartado Postal 1–948, 37150 León, Gto. México. E–mail: ivantotoarellano@cio.mx
b Facultad de Ciencias Físico–Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado postal 1152, 72001 Puebla, Pue., México.
Recibido el 22 de octubre de 2009 ]]> Aceptado el 16 de junio de 2010
Abstract
As it is very well known, the Fourier coefficients of only–phase gratings are Bessel functions of the first kind of integer order. Because the values of these real–valued functions oscillate around zero, they can adopt negative values, thereby producing phase shifts of π between some diffraction orders. To better understand the practical implications of this effect, in this work the phase shifts in the Fourier spectra of several phase gratings and grids are numerically found in order to compare them with experimental phase shifts. These experimental shifts induce changes in the modulation of the interference fringes formed by grating (or grid) interferometers when placing two windows at object plane of the system. Measurements of the changes in the interferograms are presented.
Keywords: Diffraction; phase grating; phase shift; interferometry; fringe analysis; Bessel function.
Resumen
Es conocido, que los coeficientes de Fourier de rejillas de fase son funciones de Bessel de la primera clase de orden entero. Ya que los valores de estas funciones reales oscilan alrededor de cero, entonces pueden adoptar valores negativos, de tal manera que pueden producir cambios de fase de π entre algunos órdenes de difracción. Para entender mejor las implicaciones prácticas de este efecto, en este trabajo se encuentran numéricamente los corrimientos de fase en el espectro de Fourier de varias rejillas y mallas de fase y son comparadas con los corrimientos de fase experimentales. Estos corrimientos de fase inducen cambios en la modulación de franjas de los patrones de interferencia formados por interferómetros de rejilla (o malla) al colocar dos ventanas en el plano del objeto del sistema. Se presentan mediciones de los cambios surgidos en los interferogramas.
Descriptores: Difracción; rejillas de fase; corrimiento de fase; interferometría; análisis de franjas; funciones de Bessel.
]]> PACS: 42.87.Bg; 42.79.Ci; 42.79.Dj; 42.15.Eq; 42.25.Hz; 07.05.Pj
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Acknowledgements
Authors thank M.A. Ruiz Berganza for his contribution in proofreading the manuscript. Enlightening comments and references from anonymous referees are also acknowledged. One of the authors (NITA) occupies a postdoctoral position at CIO and expresses sincere appreciation to Luisa, Miguel, Patty and Damian for the support provided and to CONACyT, for grant 102137/43055. Partial support from PROY: 48286–F is also acknowledged.
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