A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Simulated annealing: Practice versus theory

The coupled-mode theory (CMT) for optical waveguides is reviewed, with emphasis on the analysis of coupled optical waveguides. A brief account of the recent development of the CMT for coupled optical waveguides is given. Issues raised in the debates of the 1980’s on the merits and shortcomings of the conventional as well as the improved coupled-mode formulations are discussed. The conventional coupled-mode formulations are set up in a simple, intuitive way. The rigorous CMT is established on the basis of a linear superposition of the modes for individual waveguides. The cross-power terms appear logically as a result of modal nonorthogonality. The cross power is necessary for the self-consistency of the CMT for dissimilar waveguides. The nonorthogonal CMT, though more complicated, yields more-accurate results than the conventional orthogonal CMT for most practical applications. It also leads to the prediction of cross talk in directional couplers. The conventional orthogonal CMT is, however, reliably accurate for describing the power coupling between two weakly coupled, nearly identical waveguides. For dissimilar waveguides, a self-consistent orthogonal CMT can be derived by a redefinition of the coupling coefficients, and it predicts the coupling length and therefore the power exchange between the waveguides accurately if the two waveguides are far apart. Three typical coupler configurations—the uniform, the grating-assisted, and the tapered—are examined in detail. The accuracy, scope of validity, limitations, and extensions of the coupled-mode formulations are discussed in conjunction with each configuration. To verify the arguments in the discussions, comparisons with the exact analytical solutions and the rigorous numerical simulations are made.

Coupled-mode theory for optical waveguides: an overview

The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the diffraction theory of optical images

This paper describes the double exposure holographic formation of a contour map over a diffusely reflecting or transmitting of arbitrary shape and surface finish by using an immersion method with media of different indices of refraction between successive exposure. A wavelength λ effectively contributing to the fringe contour is given by λ0/Δn where λ0 is the wavelength of the laser and Δn is the index variation. Index variations were introduced by pressure difference in Freon to give λ=600 µ and 200 µ, and by the combination of two liquids to give 14 µ and 25 µ, thus extending the change of height corresponding to each fringe contour continuously up to one thousand times greater than with the optical Fizeau fringes. A correcting plate of a Schmidt camera in the grinding process was used as a test object and its figuring error were analyzed.

Holographic Generation of Contour Map of Diffusely Reflecting Surface by Using Immersion Method

Les franges de l'interferometrie holographique sont etudiees pour les mouvements types d'un objet, et la forme et la localisation des franges sont obtenues. Il resulte de ces etudes que le faisceau d'eclairage de l'objet est parallele a la direction d'observation et que la separation des franges d'egale inclinaison de celles d'egale epaisseur permet de mesurer le deplacement global et la deformation pure longitudinale et laterale.

Mesure de la Déformation d'un Objet par Interférométrie Holographique

A method for automatically measuring the three-dimensional (3-D) deformations of a diffuse surface is studied, using a real-time holographic interferometer and a television-computer system. The object surface is illuminated from three different directions. The scattered waves are recorded on a thermoplastic hologram. Real-time holographic interference patterns are observed through the hologram in a fixed direction by sequentially applying the three illuminating beams. The three fringe patterns are put into the computer using a television (TV) camera. The relative fringe order numbers are determined over the whole surface. Calculations for obtaining 3-D deformation distributions are performed, and the results are displayed on a cathode-ray tube (CRT) monitor.

Hybrid Holographic Interferometer For Measuring Three-Dimensional Deformations

We propose finding an optimal generalized color plane to carry out digital color image correlation. We seek the optimization of the correlation peak’s sharpness and, for the purpose of analysis, we make some assumptions regarding the statistical properties of images to be considered. The analysis leads to a simple solution involving a projection onto the Karhunen-Loeve vectors corresponding to the smallest eigenvalues of the interprimary colors covariance matrix. We use our results in the retrieval of the gastric mucosal surface structure from stereopairs taken through an endoscope. Study of the statistics of the endoscope images shows that the assumptions used earlier can be applied to this type of image. We present some results of stereopair matching with color image correlation. We conclude by proposing a pseudo-3-D surface reconstruction of the gastric surface by mapping the endoscopic image onto the retrieved elevation information.

Use of color image correlation in the retrieval of gastric surface topography by endoscopic stereopair matching

In the fringe-scanning methods such as the three- and four-bucket methods, the phase of a reference wave is generally changed at equal intervals from 0 to 2π. In practice, however, it is difficult or almost impossible for some interferometers, such as the dynamic zone plate interferometer, to realize an equal phase shift and to make a full scanning of 2π, which are necessary for discrete Fourier transform analysis. We show that even in such a case, phases in an interferogram can be obtained by using a Fourier analysis in a continuous space, and we analyze the accuracy of the phase determination in case of additive noise.

Jumpei Tsujiuchi

Papers

Holographic Generation of Contour Map of Diffusely Reflecting Surface by Using Immersion Method

Mesure de la Déformation d'un Objet par Interférométrie Holographique

Hybrid Holographic Interferometer For Measuring Three-Dimensional Deformations

Use of color image correlation in the retrieval of gastric surface topography by endoscopic stereopair matching

Accuracy of phase determination with unequal reference phase shift