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.

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.

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).

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.

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.

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.

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.

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 .

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.

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 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.

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.

Holographic perfect shuffle permutation element for a miniaturized switching network

A holographic perfect shuffle element with 80 channels for a miniaturized switching network is reported. An array of vertical-cavity, surface-emitting lasers is used as a transmitter. The whole permutation is carried out totally in glass. The 80 channels are permuted within a rectangle with a volume of 3 mm × 4 mm × 2 mm. Four planes of stacked volume holograms recorded in dichromated gelatin form this perfect shuffle element with an angular spectrum between 7° and 35°. Changes in the wavelength of the diode lasers to Δλ = ±10 nm can be compensated with this setup. The overall efficiency per channel lies between 40% and 60%. When Fresnel reflections and absorption are taken into account, a transmission per hologram between 78% and 90% is achieved.

Dispersive interferometric profilometer

The usual automated interferometric profilometers suffer from phase-unwrapping problems. We discuss a one-dimensional method for the absolute determination of the path difference in interferometers to obtain unique surface profiles with high accuracy.

Binary blazed reflection gratings

A reflection grating with a binary surface profile is presented that has high diffraction efficiency. The measured intensity for the + 1st diffracted order was 77%. The binary grating is composed of a minilattice with feature sizes comparable with the wavelength of the incident light. The overall structure is designed in such a way that it imitates a conventional blazed grating. The grating also has interesting polarization properties. The main part of the TE-polarized light is diffracted into the 1st diffracted order, and most of the TM-polarized light remains in the 0th diffracted order. The measurements of the grating are compared with rigorous diffraction theory and found to be in reasonable agreement.

Possibilities and limitations of space-variant holographic optical elements for switching networks and general interconnects

Space-variant optical elements are necessary for realizing optical interconnects of arbitrary design. For such applications, planar holographic optical elements offer the highest degree of flexibility and ease of production; however their diffraction-based operation gives rise to chromatic aberrations. Here estimates for the number of independent space-variant interconnects, their spatial tolerances, and their wavelength stability are considered for close-cascade, and related, geometries.

Measuring phase dispersion of dielectric multilayer stacks

Superposition-fringes interferometry is a valuable tool to detect the phase dispersion of highly reflecting semitransparent layers. Phase dispersion shows as a fringe deviation of spectrally dispersed fringe patterns. If the fringes are spectrally dispersed, the zero optical path difference fringe should be straight and run parallel to the edge of the spectrum with white-light illumination. This is not the case with dielectric films. These phase deviations give rise to measurement errors in interferometry in general. Here the equation for the superposition fringes is given together with a few measuring examples. Automated evaluation and calibration posibilities are discussed and limitations of the measuring method are outlined.

Diffractive beam splitter for laser Doppler velocimetry

A miniaturized sensor head for the optical measurement of velocities of fluids based on laser Doppler velocimetry is demonstrated. Holographic optical elements mounted on a glass substrate are used for beam splitting and deflection. Volume holograms in dichromated gelatin exhibit good optical efficiency (75% transmission of a cascade of two holographic optical elements). With diffractive devices one can achieve achromatic behavior that makes the sensor insensitive to wavelength drifts or mode hopping of a semiconductor laser.

Phase shifting interferometry: reference phase error reduction

The accuracy of phase shifting interferometry is limited by reference phase errors caused by nonlinearities in the phase shifter movements, calibration errors of this device, phase drifts during the measuring interval, and low frequency mechanical vibrations. Since the error depends on the phase to be measured in a sinusoidal fashion one can greatly reduce its influence on the measuring results. For this purpose a function also containing the characteristic error function, is fitted to the measured data. In a second step the error function is subtracted from the measured phase values. In most cases an error reduction by 1 order of magnitude seems achievable as has been demonstrated by computer simulations.

Accuracy of phase shifting interferometry

The accuracy of phase shifting interferometers is impaired by mechanical drifts and vibrations, intensity variations, nonlinearities of the photoelectric detection device, and, most seriously, by inaccuracies of the reference phase shifter. The phase shifting procedure enables the detection of most of the errors listed above by a special Lissajous display technique described here. Furthermore, it is possible to correct phase shifter inaccuracies by using an iterative process relying solely on the interference pattern itself and the Fourier sums used in phase shifting interferometry.

Array illuminator based on phase contrast

An array illuminator converts a uniformly wide beam losslessly into an array of bright spots. These spots provide the necessary illumination for microcomponents such as optical logic gates or bistable elements. Such elements may serve as devices in a 2-D discrete parallel processor. We propose an array illuminator with a phase grating on its front end. The phase grating is illuminated uniformly and then converted into an amplitude image by means of a phase contrast setup. The bright spots of the amplitude image are to be used for illuminating the array of microdevices of a digital optical computer.

Continuous lateral shearing interferometer

A simple lateral shearing interferometer based on the use of gratings is discussed, which facilitates the display of the spatial complex degree of coherence (CDC). A first version consisting of two identical gratings inclined relative to each other was used to demonstrate the working principle. For this purpose partially coherent light was used which was generated by imaging different object distributions on a rotating scatterer (e.g., double-slit or linear grid patterns). Furthermore, an achromatic lateral shearing interferometer based on the use of three gratings in a series arrangement is discussed and a theoretical proof for the achromatism of this arrangement is given.

Single sideband Ronchi test

A new lateral shear interferometer based on the Ronchi grating is proposed. The normal Ronchi test is impaired by multiple beam interference (Talbot effect). Furthermore, the fringe pattern is somewhat restricted by fringe number and orientation. The new shear interferometer suppresses the Talbot effect by spatial filtering and by using a second grating enables arbitrary fringe orientation and number.

Superposition fringe shear interferometer

A new white light shear interferometer based on the use of superposition fringes is described (superposition fringe shear interferometer, SFSI). The SFSI enables one to test for plane waves; chromatic as well as all other aberrations can be measured. Test examples and the secondary spectrum of a microscope objective are given. By spectroscopically dispersing a slit section of the shear interferogram the chromatic aberrations can be displayed. The mean phase difference between the two interfering waves can be adjusted by tilting one of the two interferometer etalons. The whole setup is mechanically stable. Shear interferograms can be obtained by inserting an interference filter in the ray path.

Superposition fringes as a measuring tool in optical testing

Superposition fringes are suitable for making highly accurate measurements of the thicknesses of air layers. A combination of a multiple beam interferometer (Fabry-Perot) and a two-beam interferometer is discussed. In this case the contrast degradation is small. Two possible applications are discussed to some extent: the first deals with the adjustment of air gaps between lenses or other optical elements, and the second is a special spherical Fizeau interferometer. This interferometer makes possible surface testing of spheres with the help of interference colors or rings as is usual with the proof glass method, with radius differences of several centimeters. The latter application seems especially promising.

Establishing an optical flatness standard

Methods proposed by the authors to establish a flatness standard without using a liquid mirror are proved in practice and extended. The extension is performed by a development of methods for the determination and compensation of random and systematic measuring errors by means of condition equations which must be satisfied by the measured sums of deviations from absolute planeness. Linear errors of these sums of deviations which can lead to ambiguities and errors of planeness deviations can be discovered and completely eliminated. Also nonlinear errors, for example, as a result of temperature differences or of mechanical stress, can be recognized without repeating the interference photography procedure. The deviations from absolute planeness of three fused silica plates were determined along seven diameters (angular distance 2pi/14) with an accuracy of lambda/500 (mean square error). This was performed by evaluating two sets of four different interference photographs, each with contour plane distances of lambda/50 (from fringe to fringe).

Precise measurement of planeness

Interference methods are reviewed-particularly those developed at the German Academy of Sciences in Berlin-with which the deviations of an optically flat surface from the ideal plane can be measured with a high degree of exactness. One aid to achieve this is the relative methods which measure the differences in planeness between two surfaces. These are then used in the absolute methods which determine the absolute planeness of a surface. This absolute determination can be effected in connection with a liquid surface, or (as done by the authors) only by suitable evaluation of relative measurements between unknown plates in various positional combinations. Experimentally, one uses two- or multiple-beam interference fringes of equal thickness(1) or of equal inclination. The fringes are observed visually, scanned, or photographed, and in part several wavelengths or curves of equal density (Aquidensiten) are employed. The survey also brings the following new methods: a relative method, where, with the aid of fringes of superposition, the fringe separation is subdivided equidistantly thus achieving an increase of measuring precision, and an absolute method which determines the deviations of a surface from ideal planeness along arbitrary central sections, without a liquid surface, from four relative interference photographs.