Pixel-based absolute topography test for three flats

Surface shape determination with a stitching Michelson interferometer and accuracy evaluation

Stitching methods are increasingly used for determining the surface shape of large and high precision optical elements used in synchrotron beamlines. They consist in reconstructing the surface topography from multiple measurements on overlapping parts of the optics aperture by various algorithms. This paper is an attempt to investigate how true and accurate such a reconstruction can be. Error sources are identified and evaluated throughout the acquisition and processing steps. The analysis is based on the example SOLEIL Michelson interferometer for nano-topography, a dedicated measurement bench for stitching interferometry. We propose a method for determining the error made on the estimate of the interferometric reference surface from the stitching dataset. This determination is made before and independently of the stitching procedure itself.

Iterative algorithm for absolute planarity calibration in three-flat test

An iterative algorithm is presented to calibrate the shapes of the reference surfaces in three-flat test. Three measurements in three-flat test determine the even part of the flat surfaces. The odd part can be obtained by adding one or more rotation measurements and using analytical recovery algorithm. Here we present an iterative algorithm to obtain the odd part instead of special analytical recovery algorithm. The iterative algorithm is purely numerical, and only involved in data rotation operation. A simulation experiment proves the validity and high accuracy of the proposed algorithm method.

Absolute surface metrology by rotational averaging in oblique incidence interferometry

A modified method for measuring the absolute figure of a large optical flat surface in synchrotron radiation by a small aperture interferometer is presented. The method consists of two procedures: the first step is oblique incidence measurement; the second is multiple rotating measurements. This simple method is described in terms of functions that are symmetric or antisymmetric with respect to reflections at the vertical axis. Absolute deviations of a large flat surface could be obtained when mirror antisymmetric errors are removed by N-position rotational averaging. Formulas are derived for measuring the absolute surface errors of a rectangle flat, and experiments on high-accuracy rectangle flats are performed to verify the method. Finally, uncertainty analysis is carried out in detail.

Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data

A method to provide absolute planarity measurements through an interferometric oblique incidence setup and an iterative algorithm is presented. With only three measurements, the calibration of absolute planarity is achieved in a fast and effective manner. Demonstration with synthetic data is provided, and the possible application to very long flat mirrors is pointed out.

Long-term deformation at room temperature observed in fused silica

Cases of long-term deformation of fused silica flats are reported. The phenomenon is detected at the scale of the nanometer, and exhibits a time constant of the order of 9 years. The observed deformation appears related to gravity and constraints, but a change of physical properties locally resulting in non-homothetic behavior is also hypothesized.

Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors

Interferometers accurately measure the difference between two wavefronts, one from a reference surface and the other from an unknown surface. If the reference surface is near perfect or is accurately known from some other test, then the shape of the unknown surface can be determined. We investigate the case where neither the reference surface nor the surface under test is well known. By making multiple shear measurements where both surfaces are translated and/or rotated, we obtain sufficient information to reconstruct the figure of both surfaces with a maximum likelihood reconstruction method. The method is demonstrated for the measurement of a 1.6 m flat mirror to 2 nm rms, using a smaller reference mirror that had significant figure error.

Method for absolute flatness measurement of optical surfaces

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.

Absolute testing of the reference surface of a Fizeau interferometer through even/odd decompositions

Absolute testing of spherical surfaces is a technological necessity because of increased accuracy requirements. In a Fizeau setup, the main part of the interferometer deviations thereby comes from the reference surface. We demonstrate the validity of an absolute testing procedure for the reference surface that has been proposed earlier. The procedure relies on the decomposition of the surface deviations into odd and even parts and could be used in partially coherent illumination. The odd deviations are obtained from a basic and a 180 degree-rotated position of an auxiliary sphere, and the even deviations can be measured with the help of a cat's eye position in double pass using an opaque half screen in the interferometer aperture.

Iterative algorithm for three flat test

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.

Three-flat test solutions based on simple mirror symmetry

In interferometric surface and wavefront metrology, three-flat tests are the archetypes of measurement procedures to separate errors in the interferometer reference wavefront from errors due to the test part surface, so-called absolute tests. What is believed to be a new class of solutions of the three-flat problem for circular flats is described in terms of functions that are symmetric or antisymmetric with respect to reflections at a single line passing through the center of the flat surfaces. The new solutions are simpler and easier to calculate than the known solutions based on twofold mirror symmetry or rotation symmetry. Strategies for effective azimuthal averaging and a method for determining the averaging error are also discussed.

Estimating the root mean square of a wave front and its uncertainty

The root mean square (rms) of the surface departure or wave-front deformation is an important value to extract from an optical test. The rms may be a tolerance that an optical fabricator is trying to meet, or it may be a parameter used by an optical designer to evaluate optical performance. Because the calculation of a rms involves a squaring operation, the rms of the measured data map is higher on average than the rms of the true surface or wave-front deformation, even if the noise is zero on average. The bias becomes significant as the scale of the noise becomes comparable to the true surface or wave-front deformation, as can be the case in the testing of ultraprecision optics. We describe and demonstrate a simple data analysis method to arrive at an unbiased estimate of the rms and a means to determine the measurement uncertainty.

Absolute interferometric testing based on reconstruction of rotational shear

A method, believed to be new, for the absolute interferometric testing of flat or spherical surfaces is presented. It is based on the classic three-flat test, combined with additional measurements of one test piece in different rotational positions. Full-surface absolute maps for each test piece are determined with a data-processing technique based on the rotationally sheared maps of the rotated surface. An optimized numerical reconstruction algorithm employing linear filtering and superposition of the different rotational shear spectra in the angular frequency domain is used to reconstruct the rotationally sheared data. The technique does not require any assumptions about the surfaces under test; has low error propagation, even in the case of high spatial resolution; and is computationally efficient.

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

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.