Imaging by a phase grating used in displacement measurement

Precision displacement measurement with a phase grating based on grating imaging is proposed. In this imaging system, a conventional amplitude grating as a scale is replaced by a phase grating. That generates a sinusoidal pattern, which works as an optical scale, with higher amplitude than the conventional imaging system with an amplitude grating. In this paper, analysis using an optical transfer function is described, so that optimal image contrast can be achieved. The experimental results with the image contrast of about 60 percent demonstrate good agreement with theoretical value. The results indicate the proposed method yields very good performance on displacement measurement. Introduction Grating imaging, used in displacement measurement, is well known [1,2]. Analyses of the grating imaging based on the optical transfer function (OTF) theory and experimental demonstration have been reported when using an amplitude grating as a pupil [3,4]. However, in amplitude grating imaging, an image contrast and received light power on an image plane, which determine accuracy and resolution in displacement measurement, were not satisfactory. In order to improve the contrast and the power, we propose using a phase grating as the pupil instead of the amplitude grating. In this paper, analysis by the OTF theory [3,4] is discussed using a phase grating with a concavo-convex shape with a 50 percent duty ratio as the pupil. To confirm theoretical results, we measured the image contrast from the intensity distribution on the image plane. Furthermore, the displacement measurement when the pupil phase grating moved was performed under high contrast imaging conditions. Theory The optical system of grating imaging consists of spatial incoherent light and three planes; an object plane, a pupil plane and an image plane. The optical distribution on the object plane is imaged on the image plane by the pupil. The contrast of the image, which influences measurement accuracy and resolution, depends on the wavelength, the shapes of the pupil and the distances between three planes. To study the image contrast, OTF analysis when using a phase grating as the pupil was performed. Figure 1 is a schematic diagram of the grating imaging system. An object grating with pitch p1 is placed on the X1-axis, and illuminated by spatial incoherent light. A pupil grating with a concavo-convex shape with 50 percent duty ratio is located at a distance Z1 from the object grating. The pitch of the pupil grating is set to p. The phase-shift amount of the pupil grating is controlled by the depth of the concavo-convex shape. The image plane is placed on the X2-axis located at a distance Z2 from the pupil grating. Fig.1 Schematic diagram of the optical system The pupil function P(x) of the pupil grating is defined as