Absolute Flatness Testing Research Articles

Absolute flatness testing of skip-flat interferometry by matrix analysis in polar coordinates.

A new method utilizing matrix analysis in polar coordinates has been presented for absolute testing of skip-flat interferometry. The retrieval of the absolute profile mainly includes three steps: (1) transform the wavefront maps of the two cavity measurements into data in polar coordinates; (2) retrieve the profile of the reflective flat in polar coordinates by matrix analysis; and (3) transform the profile of the reflective flat back into data in Cartesian coordinates and retrieve the profile of the sample. Simulation of synthetic surface data has been provided, showing the capability of the approach to achieve an accuracy of the order of 0.01 nm RMS. The absolute profile can be retrieved by a set of closed mathematical formulas without polynomial fitting of wavefront maps or the iterative evaluation of an error function, making the new method more efficient for absolute testing.

Establishing a flatness standard

A further-developed three-plate rotation method [Appl. Opt. 31, 3767 (1992)] for the absolute testing of flats is used as a measuring method for establishing flatness standards. A special phase-stepping Fizeau interferometer was built that permits flats of diameters up to 200 mm to be tested. First results show a mean error between 0.002 and 0.003 λ in the determination of the absolute flatness deviations (λ = 632.8 nm). These deviations are obtained at 500-600 points on each plate. A number of experimental conditions connected with the thermal and mechanical stability of the plates must be fulfilled.

Absolute testing of flats by using even and odd functions

We describe a modified three-flat method. In a Cartesian coordinate system, a flat can be expressed as the sum of even-odd, odd-even, even-even, and odd-odd functions. The even-odd and the odd-even functions of each flat are obtained first, and then the even-even function is calculated. All three functions are exact. The odd-odd function is difficult to obtain. In theory, this function can be solved by rotating the flat 90°, 45°, 22.5°, etc. The components of the Fourier series of this odd-odd function are derived and extracted from each rotation of the flat. A flat is approximated by the sum of the first three functions and the known components of the odd-odd function. In the experiments, the flats are oriented in six configurations by rotating the flats 180°, 90°, and 45° with respect to one another, and six measurements are performed. The exact profiles along every 45° diameter are obtained, and the profile in the area between two adjacent diameters of these diameters is also obtained with some approximation. The theoretical derivation, experiment results, and error analysis are presented.

Absolute flatness testing by an extended rotation method using two angles of rotation

The rotation method for the absolute testing of three flats is extended by adding a second rotation of one of the flats. This means that altogether five interferograms of pairs of flats (positional combinations) are evaluated: three basic combinations and two rotational combinations. The effect of random measuring errors is minimized by fully applying least-squares methods. Here the addition of a second rotation leads to a substantial increase of accuracy of the results and enables the lateral resolution to be further enhanced.

Absolute flatness testing by the rotation method with optimal measuring-error compensation

The rotation method for the absolute testing of three flats by the evaluation of four interference patterns of pairs of these flats is developed further. Least-squares methods for determining and minimizing the effect of random measuring errors are fully applied. This application makes an optimal resolution in depth and an enhanced lateral resolution possible. The computational effort mainly consists of a repeated solution of a linear equation system with 3N unknowns if N diameters of each flat are to be evaluated. The rms error of determining a flatness deviation is calculated as a function of the rms measuring error, the desired lateral resolution, and the position on the surface. The algorithm is extended to the case of using square-grid detector arrays by a special interpolation method.