A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

Absolute planarity measurements by interferometry are classically made using three flats, compared two by two in the course of four or more tests. Data reduction is performed with various analytical methods. Here we present instead a data processing algorithm that converges to solution numerically by iteration. Examples are presented both on synthetic interferograms and on experimental data. High accuracy and versatility of the approach are demonstrated.

Iterative algorithm for three flat test

Fritz’s method [Opt. Eng.23, 379 (1984)] of
using Zernike polynomials to assess the absolute planarity of test
plates is revisited. A refinement is described that takes into
account the data decorrelation that appears in experiments. An
uncertainty balance is defined by propagation of error contributions
through the steps of the method. The resultant measuring procedure
is demonstrated on a data set from experiments, and a nanometer level
of uncertainty is achieved.

Absolute measurement of planarity with Fritz's method: uncertainty evaluation.

Measuring flats in the horizontal posture with interferometers is analyzed in detail, taking into account the sag produced by gravity. A mathematical expression of the bending is provided for a plate supported at three unevenly spaced locations along the edge. It is shown that the azimuthal terms of the deformation can be recovered from a three-flat measuring procedure, while the pure radial terms can only be estimated. The effectiveness of the iterative algorithm for data processing is also demonstrated. Experimental comparison on a set of three flats in horizontal and upright posture is provided.

Three-flat test with plates in horizontal posture

To determine the absolute flatness deviations of optical surfaces, a novel method using two optical plates to achieve the absolute flatness test is presented. Absolute deviations of three surfaces, the rear surface of plate I and the front and rear surfaces of plate II, are obtained by four measurements. Wavefront error due to the inhomogeneity of plate II is measured beforehand and is then subtracted from the test results. Vertical profiles of the three surfaces are compared with the measurement results obtained by Zygo's three-flat application. Good agreement validates our method. The advantage of our method is that only one transmission flat is needed during the absolute test, which is especially useful for large-aperture interferometer calibration.

Method for absolute flatness measurement of optical surfaces

A large-aperture high-accuracy phase-shifting digital flat interferometer that is a combination of optics, mechanics, electricity, and algorithm is described. The aperture size is 250 mm and a liquid surface is used as an absolute flat to eliminate system error. The accuracy is better than ?/50 (?50.6328 µm, peak-to-valley value). The tested aperture can be enlarged to 500 mm. This interferometer has been used as an optical flat standard instrument for China. The optical interferometer, phase shifter, and calibration of precision are described.

Large‐aperture high‐accuracy phase‐shifting digital flat interferometer

A 2-D fast Fourier transform (FFT) algorithm for analyzing interferograms and the fringe shift is described. The phase errors caused by nonlinear response of a detector and by a random noise are analyzed theoretically. From the analysis, it is concluded that (1) the phase error due to the nonlinear response of a detector can be canceled by the proper filter window in the transform plane, and (2) the 2-D transform permits better separation of the desired information components from unwanted components than a 1-D transform. The relationship of 2-D FFT algorithm accuracy with factors such as the quantization of grey levels, spatial carrier frequency, spatial scanning direction, pixel array, form of the wavefront to be tested, etc., are discussed by analyzing a simulated ideal interferogram. An example of analyzing an actual interferogram and measuring the displacement of a piezoelectric transducer (PZT) device is given. In principle, the 2-D FFT algorithm can attain to an accuracy of (lambda) /100 approximately (lambda) /200 under optimum parameter conditions.

Two-dimensional Fourier transform algorithm analyzing the interferogram and the fringe shift

This paper describes a large-aperture high-accuracy phase-shifting digital flat interferometer which is a combination of optic, mechanics, electricity, and algorithm. The aperture size is 250 mm and liquid surface is used as an absolute flat to eliminate systemic error. The accuracy is better than (lambda) /50 ((lambda) is Hz-Ne laser wavelength) (peak-to-valley value). The tested aperture can be enlarged to 500 mm. This interferometer will be used as our country optical flat standard instrument. This paper describes the optical interferometer, phase shifter, and calibration of precision.

Daojiong Chen

Papers

Large‐aperture high‐accuracy phase‐shifting digital flat interferometer

Two-dimensional Fourier transform algorithm analyzing the interferogram and the fringe shift

Large-aperture high-accuracy phase-shifting digital flat interferometer