Interferometric grazing incidence test of rough steep convex spherical and aspherical surfaces: first simulations and experimental proof of principle.

In the past, grazing incidence interferometry has been applied for rough plane, cylindrical, acylindrical, and general rod-like surfaces using diffractive beam splitters. Here, we demonstrate that also rough convex steep rotational symmetric spherical or aspherical surfaces can be measured along one meridian in a single step using diffractive beam splitters and phase-shifting techniques. The extension to the whole surface can be attained by successive meridional measurements of the surface under test by azimuthal adjustments. The principle of the method is given and for a spherical ball lens as an extremely curved surface simulated, and experimental data are presented. The features of the interferogram are discussed, and the experimental evaluation of a single meridian including the unwrapped phase data is shown.

Sharp and rectified imaging of plane test objects in diffractive grazing incidence interferometers.

Fine-ground plane surfaces can be interferometrically tested in grazing incidence with laser illumination in the visible. Since the roughness of the surfaces is in the micrometer-range, the incidence angle of the probing wave front is very close to 90°, causing a strong anamorphic distortion of the length of the test sample on an imaging detector. In contrast to this the width of the sample remains undistorted in the image plane. Here, the case of a diffractive grazing incidence solution will be discussed. In particular, the rectification of the anamorphic distortion combined with sharp imaging of the whole length of the test sample will be described together with solutions for matching the illuminating and imaging beam cross section to the lateral extension of the sample.

Multi-pass Shack-Hartmann planeness test: monitoring thermal stress.

The multi-pass solution for surface measurements with the help of a Shack-Hartmann sensor (SHS) on the basis of a Fizeau cavity enables fast access to surface deviation data due to the high speed of the SHS and easy referencing of the measured data through difference measurements. The multi-pass solution described in a previous publication [J. Schwider, Opt. Express 16, 362 (2008)], provides highly sensitive measurements of small displacements caused by thermal non-equilibrium states of the test set up. Here, we want to demonstrate how a pulsed thermal load changes the surface geometry. In addition the temporal response for different plate materials is monitored through a fast wave front measurement with very high sensitivity. The thermal load close to a delta-function in time will be applied from the back-side of a plane plate by heating a small Peltier element with a heat impulse of known order of magnitude. The development of the surface deviation on the time axis can be monitored by storing a set of successive deviation pictures.

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.

Fizeau-type multi-pass Shack-Hartmann-Test.

The stability of the Shack-Hartmann sensor against mechanical disturbances from the environment can advantageously be exploited in highly sensitive wave front tests of surfaces. Here, a Fizeau-type multiple beam test is investigated. The enhancement of the phase sensitivity in a Fizeau-resonator formed by two high reflecting mirror surfaces enables tests in reflected light which includes also opaque surfaces. The multiplication of the sensitivity with the number of passes through the Fizeau resonator provides a big margin against the rather limited repeatability of wave front measurements with such a wave front sensor. The method has been tested for planeness Fizeau measurements. It could also be exploited in spherical Fizeau tests. But in the latter case the two spherical surfaces forming the resonator should have radii which differ only by e.g. 1mm.

Nonnull testing of rotationally symmetric aspheres: a systematic error assessment.

An optical setup for the testing of rotationally symmetric aspheres without a null optic is proposed. The optical setup is able to transfer the strongly curved wave fronts that stem from the reflection of a spherical testing wave front at a rotationally symmetric asphere. By simulation it is proved that the algorithms of the Shack-Hartmann sensor that is used can cope with the steep wave-front slopes (approximately 110lambda/mm) in the detection plane. The systematic errors of the testing configuration are analyzed and separated. For all types of error, functionals are derived whose significance is proved by simulation. The maximum residual errors in the simulations are fewer than lambda/500 (peak to valley).

Dispersion error in white-light linnik interferometers and its implications for evaluation procedures.

White-light interferometry is a standard optical tool with which to measure profiles of discontinuous structures such as diffractive optical elements. But there is one outstanding technological problem: The interferometers have to be symmetric; i.e., the geometrical path lengths in glass have to be the same for both interferometer arms. If these paths in glass are not equal within the field of view, a dispersion error will occur that is rather complicated to compensate for. The error appears in the measured profile in the form of steps of lambda/2 in height. A simulation of interferograms disturbed by dispersion deviations is presented, and an algorithm is introduced that eliminates the steps without changing the actual phase information or averaging neighboring pixels. The results are shown with simulated and real data.

Wave-front reconstruction with a shack-hartmann sensor with an iterative spline fitting method.

One limitation of the conventional Shack-Hartmann sensor is that the spots of each microlens have to remain in their respective subapertures. We present an algorithm that assigns the spots to their reference points unequivocally even if they are situated far outside their subaperture. For this assignment a spline function is extrapolated in successive steps of the iterative algorithm. The proposed method works in a single-shot technique and does not need any aid from mechanical devices. The reconstruction of a simulated steep aspherical wave front (approximately 100 lambda/mm slope) is described as well as experimental results of the measurement of a spherical wave front with a huge peak-to-valley value (approximately 400 lambda). The performance of the method is compared with the unwrapping method, which has been published before.

Testing of rod objects by grazing-incidence interferometry: experiment.

A grazing-incidence interferometer for the testing of technical surfaces for macroscopic surface deviations is described. Computer-generated holograms serve as beam splitters and references for the workpieces tested. The sensitivity of the interferometer depends on the period of the computer-generated holograms. The method is demonstrated at a rod object of convex profile. Using phase-stepping techniques, the grazing-incidence interferometer provides fast measurements of the entire mantle surface of the test sample with submicrometer precision.

Some considerations of reduction of reference phase error in phase-stepping interferometry.

Positioning errors and miscalibrations of the phase-stepping device in a phase-stepping interferometer lead to systematic errors proportional to twice the measured phase distribution. We discuss the historical development of various error-compensating phase-shift algorithms from a unified mathematical point of view. Furthermore, we demonstrate experimentally that systematic errors can also be removed a posteriori. A Twyman-Green-type microlens test interferometer was used for the experiments.

Ultraviolet interferometry with apochromatic reflection optics.

To improve the resolution and the sensitivity of optical metrology, we have constructed an interferometer for vacuum-ultraviolet wavelengths. To examine the influence of the wavelength, especially with regard to the period of the object's structure, we chose an apochromatic design. With reflective optics, wavelengths from 157 to 900 nm can be employed for interferometric measurements. The benefits and also the technological problems that accompany the use of vacuum-ultraviolet wavelengths are discussed. The design of this interferometer and measuring results with different wavelengths are presented.

Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack-Hartmann sensor.

A comparison of absolute sphericity measurements with a ShackHartmann sensor and a TwymanGreen interferometer is presented. The absolute deviations of a test sphere from its ideal shape were calculated in both cases from the measured wave aberrations of three different positions. Very good qualitative and quantitative agreement of the results was achieved. The difference of the root-mean-square values of the two methods was 1/1000 of a wavelength.

Confocal microscopy with a refractive microlens-pinhole array.

The stable setup of a confocal arrangement consisting of a combination of a refractive microlens and a pinhole array is presented. The focal plane of the microlenses lies at the rear surface of the substrate in the pinhole plane. By using a microscope objective one can image the stop array onto the object at a reduced size. Surface profiles of refractive and diffractive optical elements were measured with the help of this confocal microscope.

Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm.

An algorithm for expanding the dynamic range of Shack--Hartmann sensors is proposed. The distribution of the spot dislocations is treated with a modified unwrapping algorithm that is widely used in interferometry. The algorithm unwraps the spot dislocations and assigns the spots to their original subapertures, leading to a huge expansion of the dynamic range. For the proposed algorithm there remains a limitation on the maximum wave-front curvature instead of on the maximum wave-front slope. Examples are given that show spot fields that were wrapped four times; the measured wave front had a peak-to-valley value of 116 lambda .

Misalignment effects of the Shack-Hartmann sensor.

The Shack-Hartmann sensor uses a microlens array and a CCD camera for wave-front measurements. To obtain wave-front measurements with high accuracy, an accurate relative alignment of both is essential. The different states of misalignment of the Shack-Hartmann sensor are divided into groups and are treated theoretically and experimentally. Their effect on the accuracy of wave-front measurements is evaluated. In addition, a practical method for proper alignment of the Shack-Hartmann sensor is proposed.

Lateral shearing interferometer based on two Ronchi phase gratings in series.

Shearing interferometers are very popular and have a growing range of applications, especially in the field of optical testing. We describe a new kind of a lateral shearing interferometer. The lateral shear is produced by two Ronchi phase gratings in series. The phase can be shifted by a lateral movement of one grating relative to the other, and the amount of shear can easily be adjusted by variation of the distance of the gratings. The simplicity of the device is an important advantage, especially in the near IR.

Axicon-type test interferometer for cylindrical surfaces: systematic error assessment.

The principle and the alignment aberration functions are described for an axicon-type test interferometer for measuring cylindrical mantle surfaces. Additionally, we show that the derived systematic alignment functions fulfill for reasonably small misalignments the requirements for measurements in the range of approximately 1/100 of a fringe. We verify this with optical path-length calculations, using ray tracing.

White-light Fizeau interferometer.

A white-light Fizeau interferometer is described. Commonly, white-light fringes can be produced only by using a virtual wedge instrument such as a Michelson interferometer. By use of a series arrangement of a Fabry-Perot interferometer in front of a two-beam Fizeau interferometer, white-light fringes can be produced. For white-light fringes to be obtained, the thickness of the air gap between the Fizeau plates has to be adjusted to the same thickness as the air gap between the Fabry-Perot plates (or in more general terms to a rational multiple of this value). The contrast of the two-beam type of Fizeau fringes depends on the reflectivity of the Fabry-Perot plates.

Design and fabrication of computer-generated beam-shaping holograms.

For the design of computer-generated holograms reconstructing certain intensity patterns with phase freedom, we use an object-oriented approach. The given intensity pattern is decomposed into elementary objects for which appropriate phase-only hologram functions can be constructed. The total hologram function is found by the subsequent superposition of its constituents, with a relative amplitude and phase weighting for each of them. Thus, the degrees of freedom are dramatically reduced compared with those of sampling approaches. The design algorithm allows us to compensate on the one hand for the intensity and phase distribution of the impinging laser beam and on the other hand for the shape of the hologram aperture. We report on the computer-aided design of such holograms, as well as their fabrication through the use of laser lithography and reactive ion etching. Optical reconstructions are shown.

Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption.

Computer-generated phase-only holograms can be used for laser beam shaping, i.e., for focusing a given aperture with intensity and phase distributions into a pregiven intensity pattern in their focal planes. A numerical approach based on iterative finite-element mesh adaption permits the design of appropriate phase functions for the task of focusing into two-dimensional reconstruction patterns. Both the hologram aperture and the reconstruction pattern are covered by mesh mappings. An iterative procedure delivers meshes with intensities equally distributed over the constituting elements. This design algorithm adds new elementary focuser functions to what we call object-oriented hologram design. Some design examples are discussed.