Daniel Malacara

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.

Focusing Errors in a Collimating Lens or Mirror: Use of a Moiré Technique

The Fresnel diffraction pattern of a grating periodically gives an exact reproduction of the grating. By superposing a second grating on such an image, we form a moiré pattern that depends upon the degree of parallelism of the incident beam. We present here the possibility of using this technique for studying the focusing errors in a collimating lens or a mirror.

IV - Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images

This chapter discusses the direct spatial reconstruction of optical phase from phase-modulated images (PMI). The spatial variations of the phase play a key role in the optical recording of wavefronts, optical testing, metrology, and optical information processing. The direct spatial reconstruction of optical phase (DSROP) and its applications in various interferometric and moire methods can generate a large number of new methods in which optics and computer image processing can find optimal interconnections. The chapter aims to improve some DSROP methods such as the smoothing method (with the averaging window) and the band-pass filtering method (with the differential Gaussian filter) in the space domain. Some new methods are also proposed, including (1) DSROP in the space domain for patterns with chirped radially symmetric (Fresnel and Gabor zone plates) carriers, (2) DSROP for phase-modulated images, and (3) some DSROP methods for space- and time-PMI.

Bibliography of Various Optical Testing Methods

A bibliography of various methods of optical testing has been compiled, with the papers grouped by subject, e.g., Fizeau interferometer. Only interferometer papers-or those on similar devices-that have direct relevance to shop testing are included. This reasonably complete compendium should be of use to workers in optical fabrication and testing.

Some applications of the Murty interferometer: A review

Several applications of the Murty interferometer in the testing of optical systems and components are reviewed in this article. The interferometer has applications in the testing of lenses, spherical mirrors, paraboloidal mirrors and parallel plates. The interferometer can also be used for the determination of homogeneity of optical materials, measurement of refractive indices of glasses and liquids, measurement of radii of curvature and location of cardinal points, checking the laser beam collimation, determination of the power of ophthalmic lenses, measurement of refractive indices of simple lenses, measurement of birefringence of optical materials, thermal expansion coefficient of metallic bars and optical distortion in transparencies or glass windows.

Geometrical parameters in the Hartmann test of aspherical mirrors.

The Hartmann test has been used with great success to determine figuring errors in large aspherical concave surfaces for telescope mirrors. Here, a mathematical model is presented that allows us to compute the optimum geometrical parameters for this test. It is assumed that the light source is placed near the center of curvature.

Complex linear filters for phase shifting with very low detuning sensitivity

A new way of synthesizing phase stepping formulae based on linear complex filters is described. These complex linear filters remain in quadrature even for a large amount of detuning (linear phase miscalibration). The linear complex filter is obtained as the minimizer of a quadratic cost functional weighting of several desirable filter properties such as range of quadrature.

Mathematical Interpretation of Radial Shearing Interferometers

The procedure for computing a radial shearing interferometric pattern is given. The interferometric pattern is analyzed to obtain the wavefront shape. Restricting the discussion to wavefronts having rotational symmetry, we give two different methods of finding the wavefront. One approach is to scan along a diameter of the interferometric pattern and the other is to examine the shape of the fringes. The relative sensitivity of a radial shearing interferometer with respect to that of a Twyman-Green interferometer is also analyzed.

Noisy fringe pattern demodulation by an iterative phase locked loop

Abstract The phase locked loop (PLL) technique applied to demodulate two-dimensional carrier-frequency fringe patterns has been developed recently. Here we present an extension to the basic PLL scheme to demodulate noisy fringe patterns. This modified technique estimates the phase in the fringe pattern iteratively; that is, the first wavefront estimation is done using a flat reference phase and the second iteration takes the demodulated phase found in the first iteration as the new reference. The third demodulating iteration uses the second phase estimation as the reference and so on, until further changes in the detected wavefront fall below a predefined threshold. During the iterative process the bandwidth of the iterative PLL system is gradually decreased to improve the signal-to-noise ratio of the detected phase as well as to resolve noise-generated phase inconsistencies.

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.