Modeling and correction of fringe patterns in Doppler asymmetric spatial heterodyne interferometry.

Doppler asymmetric spatial heterodyne (DASH) interferometry is a novel concept for observing atmospheric winds. This paper discusses a numerical model for the simulation of fringe patterns and a methodology to correct fringe images for extracting Doppler information from ground-based DASH measurements. Based on the propagation of optical waves, the fringe pattern was modeled considering different angular deviations and optical aberrations. A dislocation between two gratings can introduce an additional spatial modulation associated with the diffraction order, which was seen in laboratory measurements. A phase correction is proposed to remove phase differences between different row interferograms, which is the premise for calculating the average interferogram to improve the signal-to-noise ratio. Laboratory tests, simulation results, and Doppler velocity measurements indicate that a matrix determined in the laboratory can be applied to correct interferograms obtained from ground-based DASH measurements.

Absolute characterization of high numerical aperture microscope objectives utilizing a dipole scatterer.

Measuring the aberrations of optical systems is an essential step in the fabrication of high precision optical components. Such a characterization is usually based on comparing the device under investigation with a calibrated reference object. However, when working at the cutting-edge of technology, it is increasingly difficult to provide an even better or well-known reference device. In this manuscript we present a method for the characterization of high numerical aperture microscope objectives, functioning without the need of calibrated reference optics. The technique constitutes a nanoparticle, acting as a dipole-like scatterer, that is placed in the focal volume of the microscope objective. The light that is scattered by the particle can be measured individually and serves as the reference wave in our system. Utilizing the well-characterized scattered light as nearly perfect reference wave is the main idea behind this manuscript.

Analysis and correction of distortions in a spatial heterodyne spectrometer system.

In this paper a method for correcting the radial distortion of interferograms generated by a spatial heterodyne spectrometer system is presented. Instead of utilizing calibration patterns, the distortion model parameters are estimated based on the distorted fringe features generated by projecting the straight interference stripes onto the detector. Comparisons between polynomial models and division models indicate that division models can deliver competitive performance on the reconstructed image with fewer parameters. Simulated interferograms based on ray-tracing are used to demonstrate the correction of errors in the spatial, phase, and spectral domain caused by optical distortion.

Effective wind and temperature retrieval from Doppler asymmetric spatial heterodyne spectrometer interferograms.

This paper presents a method for wind velocity and Doppler temperature retrieval from interferograms of a Doppler asymmetric spatial heterodyne spectrometer. This method is based on the analytic representation of the signal and the subsequent algorithms. It turns out to be more robust than the conventional Fourier transform method at low SNR. The influence of optical dispersion on the accuracy of the retrieved parameters is also characterized. The effective optical path difference is suggested for use in wind and temperature retrieval routines. Computer simulations are used to characterize the accuracy of the proposed method, in particular regarding the influence of optical dispersion.

Simultaneous measurement of phase transmission and birefringence of an object under test.

This paper presents a novel interferometric method for the simultaneous spatially resolved analysis of an object under test regarding the phase transmission function and the magnitude and orientation of the (uniaxial) birefringence. The measurement strategy is based on variations of the phase and polarization and processing the interference patterns so obtained. With this method, which is very similar to the classical phase-shifting interferometry, a complete analysis of birefringent properties of the object and its impact on the phase of the incoming light can be done in one measurement cycle. The theoretical description of the investigated methods and their experimental implementation are presented.

Incoherent averaging of phase singularities in speckle-shearing interferometry.

Interferometric speckle techniques are plagued by the omnipresence of phase singularities, impairing the phase unwrapping process. To reduce the number of phase singularities by physical means, an incoherent averaging of multiple speckle fields may be applied. It turns out, however, that the results may strongly deviate from the expected √N behavior. Using speckle-shearing interferometry as an example, we investigate the mechanism behind the reduction of phase singularities, both by calculations and by computer simulations. Key to an understanding of the reduction mechanism during incoherent averaging is the representation of the physical averaging process in terms of certain vector fields associated with each speckle field.

Simultaneous measurement of phase and local orientation of linearly polarized light: implementation and measurement results.

Optical components manipulating both polarization and phase of wave fields find many applications in today's optical systems. With modern lithography methods it is possible to fabricate optical elements with nanostructured surfaces from different materials capable of generating spatially varying, locally linearly polarized-light distributions, tailored to the application in question. Since such elements in general also affect the phase of the light field, the characterization of the function of such elements consists in measuring the phase and the polarization of the generated light, preferably at the same time. Here, we will present first results of an interferometric approach for a simultaneous and spatially resolved measurement of both phase and polarization, as long as the local polarization at any point is linear (e.g., for radially or azimuthally polarized light).

Interferometric homogeneity test using adaptive frequency comb illumination.

The homogeneity test of glass plates in a Fizeau interferometer requires the measurement of the glass sample in reflected as well as in transmitted light. For the measurement in transmitted light, the sample has to be inserted into the ray path of a Fizeau or Twyman-Green interferometer, which leads to a nested cavity setup. To separate the interference signals from the different cavities, we illuminate a Fizeau interferometer with an adaptive frequency comb. In this way, rigid glass plates can be measured, and linear variations in the homogeneity can also be detected. The adaptive frequency comb is provided by a variable Fabry-Perot filter under broadband illumination from a superluminescence diode. Compared to approaches using a two-beam interferometer as a filter for the broadband light source, the visibility of the fringe system is considerably higher.

Evaluation algorithms for multistep measurement of spatially varying linear polarization and phase.

Optical components manipulating both polarization and phase of wave fields find more and more applications in today's optical systems. In particular, the polarization orientation may vary across the aperture. New measurement techniques and evaluation algorithms are needed to simultaneously characterize the properties of such elements. In this Letter, a general measurement algorithm for locally linear polarization distributions is presented, extending the methods of phase shifting interferometry to the simultaneous determination of polarization and phase. A class of evaluation algorithms is derived, and some example algorithms are described and tested for their resilience against systematic and stochastic stepping errors.

Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters.

Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters is proposed. The first 45°-tilted cube beamsplitter splits object wave into two parallel copies: one copy is filtered by a pinhole in its Fourier plane to behave as reference wave, while the other one remains unchanged as object wave. The second cube beamsplitter combines the object and reference waves, and then split them together into two beams. Along with the two beams, two parallel phase-shifting interferograms are obtained in aid of polarization elements. Based on the proposed configuration, slightly-off-axis interferometry for microscopy is performed, which suppresses dc term by subtracting the two phase-shifting holograms from each other. The setup is highly stable due to its common-path configuration, and has been demonstrated to be suitable for measuring moving objects or dynamic processes.

Diffractive simultaneous bidirectional shearing interferometry using tailored spatially coherent light.

Measurements of wavefront deformations can be carried out with the help of lateral shearing interferometers. Here the focus is on a setup providing two shears along orthogonal directions simultaneously to generate the data needed for a reconstruction. We describe a diffractive solution using Ronchi phase gratings with a suppressed zeroth order for both the doubling of the wavefront under test and the bidirectional shearing unit. A series arrangement of the gratings offers an on-axis geometry, which minimizes the systematic errors of the test. For illumination, an extended incoherent monochromatic light source is used. High-contrast fringes can be obtained by tailoring the degree of coherence via a periodic intensity distribution.

Perfect imaging of hypersurfaces via transformation optics.

Conventional optical imaging systems suffer from the presence of many imperfections, such as spherical aberrations, astigmatism, or coma. If the imaging system is corrected for spherical aberrations and fulfills the Abbe sine condition, perfect imaging is guaranteed between two parallel planes but only in a small neighborhood of the optical axis. It is therefore worth asking for optical systems that would allow for perfect imaging between arbitrary smooth surfaces without restrictions in shape or extension. In this Letter, we describe the application of transformation optics to design refractive index distributions that allow perfect, aberration-free imaging for various imaging configurations in R(n). A special case is the imaging between two extended parallel lines in R(2), which leads to the well-known hyperbolic secant index distribution that is used for the fabrication of gradient index lenses.

Phase-shifting point-diffraction interferometry with common-path and in-line configuration for microscopy.

A new common-path and in-line point-diffraction interferometer for quantitative phase microscopy is proposed. The interferometer is constructed by introducing a grating pair into the point-diffraction interferometer, thus forming a common-path and in-line configuration for object and reference waves. Achromatic phase shifting is implemented by linearly moving one of the two gratings in its grating vector direction. The feasibility of the proposed configuration is demonstrated by theoretical analysis and experiments.

Phase-shift extraction for generalized phase-shifting interferometry.

A simple algorithm for blind extraction of phase shifts is proposed for generalized phase-shifting interferometry from only three interferograms. Based on the statistical property of the object wave, the algorithm calculates approximately the involved phase shifts as initial values. The extraction is further improved by an iterative method, considering the fact that the closer the phase shifts approach their real values, the more uniform the reconstructed reference wave will become. The feasibility of this algorithm is demonstrated by both simulation and experiment.

Interferometric quasi-absolute tests for aspherics using a radial shear position.

Increasing accuracy requirements in aspheric metrology make the development of absolute testing procedures for aspheric surfaces important. One strategy is transferring the standard practice three-position test for spheres to aspherics. The three-position test, however, involves a cat's eye position and therefore has certain drawbacks. We propose an absolute testing method for rotationally symmetric aspherics where the cat's eye position is replaced with a radially sheared position. Together with rotational movements of the specimen, the surface deviations can be obtained in an absolute manner. To demonstrate the validity of the procedure, we present a measurement result for a sphere and compare it with a result obtained by the standard three-position test.

Interferometric null test of a deep parabolic reflector generating a Hertzian dipole field.

We report on interferometric characterization of a deep parabolic mirror with a depth of more than five times its focal length. The interferometer is of Fizeau type; its core consists of the mirror itself, a spherical null element, and a reference flat. Because of the extreme solid angle produced by the paraboloid, the alignment of the setup appears to be very critical and needs auxiliary systems for control. Aberrations caused by misalignments are removed via fitting of suitable functionals provided by means of ray tracing simulations. It turns out that the usual misalignment approximations fail under these extreme conditions.

Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements.

Aspheric optical surfaces are often tested using diffractive optics as null elements. For precise measurements, the errors caused by the diffractive optical element must be calibrated. Recently, we reported first experimental results of a three position quasi-absolute test for rotationally invariant aspherics by using combined-diffractive optical elements (combo-DOEs). Here we investigate the effects of the DOE substrate errors on the proposed calibration procedure and present a set of criteria for designing an optimized combo-DOE. It is demonstrated that this optimized design enhances the overall consistency of the procedure. Furthermore, the rotationally varying part of the surface deviations is compared with the rotationally varying deviations obtained by an N-position averaging procedure and is found to be in good agreement.

Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference.

We have already reported a method for the quasi-absolute test of rotationally symmetric aspheres by means of combined diffractive optical elements (combo-DOEs). The combo-DOEs carry the information for the ideal shape of an aspheric surface under test as well as a spherical wave for the measurement at the cat's eye position. An experimental demonstration of the procedure is given. Measurements with two different designs of combo-DOEs have been conducted, and their relative advantages and disadvantages are discussed.

Calibration for cylindrical specimens in grazing-incidence interferometry via integration of difference measurements.

Cylindrical specimens may be tested advantageously by using grazing-incidence interferometry. A multiple positions test in combination with rotational averaging has recently been used to separate the surface deviations of the specimen from the interferometric aberrations. To reduce the measuring time and to check whether the results are reliable, a second procedure is now investigated, which uses the principle of the multiple positions test to determine quantities proportional to the difference quotients of the surface deviations. After numerical integration, the results can be compared with those obtained previously by rotational averaging. The measurement principle is described, and calibration results are presented.

Absolute calibration in grazing incidence interferometry via rotational averaging.

Interferometry in grazing incidence can be used to test cylindrical mantle surfaces. The absolute accuracy of the resulting surface profiles is limited by systematic wavefront aberrations caused in the interferometer, in particular due to an inversion of the test wavefront in an interferometer using diffractive beam splitters. For cylindrical specimens, a calibration method using four positions has therefore been investigated. This test is combined with another method of optical metrology: the rotational averaging procedure. The implementation for grazing incidence is described and measurement results for hollow cylinders are presented. The gain in accuracy is demonstrated.