Hiroshi Miyashiro

Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running

Abstract A common path lateral-shearing interferometer with a minimum number of optical components has been developed. Because the interferometer is little affected by mechanical vibrations and air turbulence, it can be mounted on an ultraprecision lathe and can be used to measure the shapes of workpieces. A plane parallel glass plate is used to shear the wavefront under test in the interferometer. To analyze the interference fringes obtained by the interferometer precisely, a fringe-scanning method using a slight tilt of the glass plate is used. Zone plates that are computer-generated holograms are used to measure spherical and aspherical surfaces with the interferometer. A spherical and a parabolic concave mirror were measured with the interferometer. The spherical mirror was also measured by a Fizeau interferometer to compare the error with that measured by the lateral-shearing interferometer. The experimental results agreed well with those measured by the lateral-shearing interferometer.

High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials

Interferograms obtained with ordinary interferometers, such as the Fizeau interferometer or the Twyman-Green interferometer, show the contour maps of a wave front under test. On the other hand, lateral shearing interferograms show the difference between a wave front under test and a sheared wave front, that is, the inclination of the wave front. Therefore the shape of the wave front under test is reconstructed by means of analyzing the difference. To reconstruct the wave front, many methods have been proposed. An integration method is usually used to reconstruct the wave front under test rapidly. However, this method has two disadvantages: The analysis accuracy of the method is low, and part of the wave front cannot be measured. To overcome these two problems, a new, to our knowledge, integration method, improved by use of polynomials, is proposed. The validity of the proposed method is evaluated by computer simulations. In the simulations the analysis accuracy achieved by the proposed method is compared with the analysis accuracy of the ordinary integration method and that of the method proposed by Rimmer and Wyant. The results of the simulations show that the analysis accuracy of the newly proposed method is better than that of the integration method and that of the Rimmer-Wyant method.

An instrument for manufacturing zone-plates by using a lathe

Abstract A laser writing system is developed in order to manufacture precise and inexpensive zone-plates. The zone-plates are a key component of an interferometer that measures shape error of spherical or aspherical mirrors on an ultrapecision lathe. The laser writer is mounted on the sliding table of the lathe. A glass plate coated with photoresist is attached to a chuck of the lathe and rotated. The zone-plate grating is written on the glass plate by a focused laser beam. Zone-plate writing is sped up by reducing the necessary number of computer instructions, causing no loss in the accuracy of the zone-plate. The accuracy of the zone-plate obtained by this inexpensive method is 0.1 μm and the same as that achieved by an electron beam writer.

Shape measurement of workpiece surface with zone-plate interferometer during machine running

Abstract A zone-plate interferometer is in the family of common-path interferometers that have the following feature: interference fringes are hardly influenced by air turbulence and machine vibrations. A new zone-plate interferometer with a three-dimensional optical system is developed in order to measure the shape error of spherical and aspherical mirrors manufactured with an ultra-precision cutting machine. Shape measurements of the mirrors are performed when they are turning at a speed of 900 rpm. The position of a zone-plate is arranged quickly and easily by using a Moire technique. The results obtained with the zone-plate interferometer agree well with those obtained with a Fizeau interferometer.

Zone-plate inteferometer to measure the positioning error of a cutting tool

Abstract A simple type of zone-plate interferometer has been developed to measure precisely the positioning error of a cutting tool. Interference fringes obtained by the interferometer are little affected by air turbulence in the optical paths and by machine vibrations. The shape of the mirrors surface being tested is spherical and is manufactured with an ultraprecision lathe. A zone plate is set at the midposition between the vertex of the spherical mirror being tested and the center of curvature of the mirror. The error in the shape of the mirror and the positioning error of the tool can be determined by analyzing the interference fringes. Two spherical concave mirrors were measured. One mirror was manufactured with a tool that had same positioning error. The positioning error was observed as distortion of the interference fringes. The images obtained by the zone-plane interferometer agree well with the images obtained by a Fizeau interferometer and a computer simulation based on the experimental results. The other mirror was manufactured with the tool after the positioning error had been eliminated, based upon the results of the above experiments. The interference fringes of the mirror show no distortion, and the error in the shape of the mirror is small.

Method to obtain a clear fringe pattern with a zone-plate interferometer

When a zone-plate interferometer is used, a bright spot appears at the center of the image plane. The spot makes it difficult to analyze the interference fringes. A simple technique that is based on the principle of fringe-intensity reversal is proposed to analyze the fringes efficiently. A zone plate with a phase fraction of π/2 or 3π/2 is used in this technique to diminish the bright spot. Unlike the masking technique, no part of the data on the fringes is lost. The fringes can, therefore, be analyzed completely. The technique is described in detail, and the results of an experiment in which the shape error of a concave mirror was measured with the proposed zone plate is presented. The experimental results agree well with the results obtained with the Fizeau interferometer.

Improvement of precision in the analysis of a lateral shearing interferogram using integration

In the method for obtaining an error in a shape under test using the integration from a lateral shearing interferogram, it is assumed that the lateral shear of the shape is so small that the interferogram pattern is considered to be representative of the wavefront slope. To obtain the shape, the slope is analyzed by integration. When the lateral shear of the wavefront is not small, an accurate shape cannot be obtained. Furthermore, the analyzed area of the wavefront is limited by the amount of shear, and the whole area cannot be obtained by this method. A method for reconstructing the whole accurate shape from the wavefront analyzed using the integration process is presented in this paper. In a computer simulation to investigate the efficacy of the method, the reconstructed shape agreed with the original shape error.

Analysis of zone-plate interference fringe pattern

The zone-plate interference fringe pattern is analyzed by FFT. The spectrum obtained by FFT contains information on the shape error. To select the information on the shape error from the spectrum of the fringe pattern is difficult, because the resolution of the spectrum is low. An FFT with over-sampling is applied to increase the spectrum resolution. A method is proposed to select the necessary information from the spectrum of the fringe pattern. Experimental results obtained by the proposed method agree well with those obtained by Fizeau interferometer.

Application of a fringe scanning method to zone-plate interferometry

Fringe scanning for analyzing shape error of a test surface is performed by fringe shift of interference images with different phases. The method of proposed fringe shift is carried out by a zone-plate, which has fan-shaped sections of four different initial phases of 0, (pi) /4, (pi) /2, and 3(pi) /4. Experimental results agree with the theory.