Phase unwrapping through demodulation by use of the regularized phase-tracking technique.

Most interferogram demodulation techniques give the detected phase wrapped owing to the arctangent function involved in the final step of the demodulation process. To obtain a continuous detected phase, an unwrapping process must be performed. Here we propose a phase-unwrapping technique based on a regularized phase-tracking (RPT) system. Phase unwrapping is achieved in two steps. First, we obtain two phase-shifted fringe patterns from the demodulated wrapped phase (the sine and the cosine), then demodulate them by using the RPT technique. In the RPT technique the unwrapping process is achieved simultaneously with the demodulation process so that the final goal of unwrapping is therefore achieved. The RPT method for unwrapping the phase is compared with the technique of least-squares integration of wrapped phase differences to outline the substantial noise robustness of the RPT technique.

Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system.

The Hartmann test is a well-known technique for testing large telescope mirrors. The Hartmann technique samples the wave front under analysis by use of a screen of uniformly spaced array of holes located at the pupil plane. The traditional technique used to gather quantitative data requires the measurement of the centroid of these holes as imaged near the paraxial focus. The deviation from its unaberrated uniform position is proportional to the slope of the wave-front asphericity. The centroid estimation is normally done manually with the aid of a microscope or a densitometer; however, newer automatic fringe-processing techniques that use the synchronous detection technique or the Fourier phase-estimation method may also be used. Here we propose a new technique based on a regularized phase-tracking (RPT) system to detect the transverse aberration in Hartmanngrams in a direct way. That is, it takes the dotted pattern of the Hartmanngram as input, and as output the RPT system gives the unwrapped transverse ray aberration in just one step. Our RPT is compared with the synchronous and the Fourier methods, which may be regarded as its closest competitors.

Phase unwrapping with a regularized phase-tracking system.

We develop a regularized phase-tracking (RPT) technique tounwrap phase maps. The phase maps that can be unwrapped with thisnew method may be bounded by arbitrarily shaped boundaries. Moreover, the RPT unwrapper has a higher noise robustness than previously reported phase-unwrapping schemes.

Axial tolerance in the position of aberration compensators placed in a converging beam.

To perform a null test of aspherical surfaces we used a computer-generated hologram or a lens or a mirror compensator to compensate the aspherical aberration. When compensating in a convergent light beam the axial position of this hologram or compensator is critical. A holographic compensator to be used in the convergent beam of light was designed and constructed. We have established some relations to determine the tolerance in the axial positioning of these compensators.

Path-independent phase unwrapping of subsampled phase maps.

A technique for unwrapping subsampled phase maps is presented. The subsampled phase map is obtained by standard phase-shifting methods that use subsampled interferograms. The technique then estimates the wrapped local curvature of the subsampled phase map. This local curvature is then low-pass filtered with a free-boundary low-pass filter to reduce phase noise. Finally the estimated local curvature of the wave front is integrated by the use of a least-squares technique to obtain the searched continuous wave front.

Wave-front recovery from two orthogonal sheared interferograms.

We present a new technique for using the information of two orthogonal lateral-shear interferograms to estimate an aspheric wave front. The wave-front estimation from sheared inteferometric data may be considered an ill-posed problem in the sense of Hadamard. We apply Thikonov regularization theory to estimate the wave front that has produced the lateral sheared interferograms as the minimizer of a positive definite-quadratic cost functional. The introduction of the regularization term permits one to find a well-defined and stable solution to the inverse shearing problem over the wave-front aperture as well as to reduce wave-front noise as desired.

Design of lenses to project the image of a pupil in optical testing interferometers.

When an optical surface or lens in an interferometer (Twyman-Green or Fizeau interferometer) is tested, the wave front at the pupil of the element being tested does not have the same shape as at the observation plane, because this shape changes along its propagation trajectory if the wave front is not flat or spherical. An imaging lens must then be used, as reported many times in the literature, to project the image of the pupil of the system being tested over the observation plane. This lens is especially necessary if the deviation of the wave front from sphericity is large, as in the case of testing paraboloidal or hyperboloidal surfaces. We show that the wave front at both positions does not need to have the same shape. The only condition is that the interferograms at both places be identical, which is a different condition. This leads to some considerations that should be taken into account in the optical design of such lenses.

First-order parameters for a general two-beam interferometer.

We formulate the first-order properties of a general two-beam interferometer. We show that it may be completely described by the light source position and the location, orientation, and size of the two images of this light source as seen from the observation plane, plus the image-plane position and the location, orientation, and size of the two images of this observation screen as seen from the light source position, plus the optical path difference between the two possible optical paths. The parameters are quite similar to those previously defined by Steel [W. H. Steel, Interferometry, 2nd ed. (Cambridge U. Press, New York, 1983)].

Effects on profile depth of changes in the thickness of the photoresist layers.

We propose a model to describe the behavior of the absorption profile as a function of the thickness changes of the photoresist layers (Shipley 1350-J), used to make high-quality holographic elements. We also show that the surface topology of the substrate is an important parameter that modulates the layer's thickness on the substrate. We obtain a depth modulation Δd in the exposure time (assuming a uniform irradiance), which is brought about by changes in the thickness of the photoresist layers at different points on the substrate; the thickness parameter is a function of the absorption coefficient of the photoresist.

Phase-locked-loop interferometry applied to aspheric testing with a computer-stored compensator.

A recently developed technique for continuous-phase determination of interferograms with a digital phase-locked loop (PLL) is applied to the null testing of aspheres. Although this PLL demodulating scheme is also a synchronous or direct interferometric technique, the separate unwrapping process is not explicitly required. The unwrapping and the phase-detection processes are achieved simultaneously within the PLL. The proposed method uses a computer-generated holographic compensator. The holographic compensator does not need to be printed out by any means; it is calculated and used from the computer. This computer-stored compensator is used as the reference signal to phase demodulate a sample interferogram obtained from the asphere being tested. Consequently the demodulated phase contains information about the wave-front departures from the ideal computer-stored aspheric interferogram. Wave-front differences of ~ 1 λ are handled easily by the proposed PLL scheme. The maximum recorded frequency in the template's interferogram as well as in the sampled interferogram are assumed to be below the Nyquist frequency.

Sub-Nyquist null aspheric testing using a computer-stored compensator.

A novel method to demodulate undersampled interferograms using a computer-stored undersampled compensator is presented. First, the sine and cosine of the computer-stored wave front is correlated with the interferogram that emerges from the asphere under test. Afterward, these two correlation images are used to find the phase map. The detected phase of the correlation fringes is the estimated phase difference between the software compensator and the frame-grabbed interferogram. The prior information required for this method is a good knowledge of the wave front being tested to a few wavelengths of error. Complying with this prior knowledge, the undersampled interferogram under analysis may be easily demodulated. Given that the proposed method is based on the correlation of the frame-grabbed interferogram and the computer-stored one, the method also withstands noise.

Analysis of light scattering from a cutting tool edge.

The scattering of light from cutting tools is studied. The contribution of cutting tool edge parameters (height and width) to scattering patterns and the influence of side surface roughness on scattering patterns are investigated. An angle-limited integrated scattering method is developed and analyzed for fast determination of edge parameters.

Aberrations of sphero-cylindrical ophthalmic lenses.

The authors have presented in two previous articles the graphic solutions resembling Tscherning ellipses, for spherical as well as for aspherical ophthalmic lenses free of astigmatism or power error. These solutions were exact, inasmuch as they were based on exact ray tracing, and not third-order theory as frequently done. In this paper sphero-cylindrical lenses are now analyzed, also using exact ray tracing. The functional dependence of the astigmatism and the power error for these lenses is described extensively.

Updated optical testing bibliography.

An updated bibliography on optical testing-some 1300 references-is now available on a DBASE diskette.

Analysis of the interferometric Ronchi test.

It is well known that the Ronchi test has two equivalent interpretations, Physical, as an interferometer, or geometrical, as if the fringes were just shadows from the fringes on the ruling. The second interpretation is nearly always used in practice because it is simpler. However, the disadvantage is that the irradiance profile of the fringes cannot be calculated with this theory. Here, the interferometric interpretation of the test will be used to obtain the irradiance profile and the sharpness of the fringes.

Design of Galilean-type telescope systems.

In this paper we present the design of three Galilean-type telescope systems with magnifications of 2.2x, 4x, and 5x. These systems are free of the large weight and length as well as the reduced field of view, which are frequent undesirable properties of Galilean telescopes. These designs have a moderate field of view and a short length, with reasonably good aberration correction, and may be used as binocular telescopes or magnifiers with a large working distance.

Tscherning ellipses and ray tracing in ophthalmic lenses.

In this paper the exact shape of the solutions to the equations for lenses free of oblique astigmatism, as well as those free from curvature of field or peripheral focus error, are presented. These solutions, as expected, resemble the Tscherning ellipses, but strongly deformed.

Tscherning ellipses and ray tracing in aspheric ophthalmic lenses.

The effect of conicoid asphericity in one of the surfaces of an ophthalmic lens is examined by means of exact ray tracing. Graphical solutions resembling the Tscherning ellipses are obtained for lenses free of oblique astigmatism as well as for lenses free of peripheral power error or curvature of field.