Tne salient features of "classical" speckle patterns are reviewed. The relaxation of assumptions leading to the classical model yields statistical properties that differ from those associated with conventional speckle. Attention is turned to a summary of those new statistical aspects of speckle that have been discovered or explored since the last SPIE conference on speckle, held in August of 1990.

The statistical theory of speckle is presented in detail including discussions of the transmission functions for a surface roughness type of diffuser and the associated characteristic function. The correlation function for the output amplitude is calculated for a variety of optical systems. These second-order moments are written for two states of position and temporal-frequency variables so that one can analyze both the spatial and the wavelength dependence of speckle. We parallel the usual cases which are treated in Fourier optics summarizing the following configurations: Space-Variant, Fourier Transform, Far-Zone, and Imaging.

A rigorous electromagnetic theory was developed to compute the speckle of a 2D random rough surface with perfect conductivity, when the incident wavelength is of the same order of magnitude as the lateral size of the asperities (resonance domain). The failure of Beckmann's theory in this domain is shown. On the other hand, a new approximate theory leads to accurate results when the mean depth of asperities is small with respect to the wavelength. A comparison of numerical results with some predictions of speckle theory is made. In particular, the limit between partially developed speckle and developed speckle is studied both numerically and theoretically.

The technique of clipping is applied to speckle intensity correlation. With clipping the speckle intensity is converted to a binary signal by comparing the speckle intensity to a fixed threshold. The magnitude of the space-time correlation coefficient is determined from measurements of the correlation function of the clipped speckle. The results of experiments performed using a 2-D clipping detector are presented.

The Cauchy/Schwarz inequality is applied to image reconstruction and power spectrum analysis in the case where the image or autocorrelation in question is assumed to be non-negative. The inequality is a restriction on the Fourier transform and is shown to be a useful diagnostic in phaseless multidimensional image reconstruction and power spectrum analysis. Since all power spectra are non-negative, the inequality holds for all autocorrelations.

The theory of image plane speckle is very well confirmed by experimental data, except the case in which a considerably small-N speckle (N<0.04) is produced by a deep phase screen. N is an effective number of scatterers that contribute to the complex amplitude of the scattered field. Here a new theoretical approach is proposed. It is assumed that for such a value of N there is a close relationship between the local value of parameters that characterize a phase screen at an arbitrary point and the value of intensity at the conjugate point of the image plane. Therefore the intensity fluctuation is considered similar to the phase-contrast effect and not the speckle phenomenon. A relation that expresses the intensity fluctuation in terms of first and second spatial derivatives of phase is calculated. This relation takes into account both the finite bandwidth of the imaging system and the displacement of the observation plane with respect to the image plane (defocusing). A formula for contrast versus N and defocusing is derived. Results obtained for a perfectly focused image of a gaussian screen are compared with the theory of Jakeman and McWhirter and with the experimental results of Levine and Dainty. 1 This current research has shown that for a moderately rough screen (rms phase ≈2¶ rad) and N<0.012 the approach presented here is better confirmed by the experiment than the theory presented by Jakeman and McWhirter It is found that the key parameter of the presented theory is a product of N and phase variance. The above conclusion can also be obtained from Jakeman and McWhirter's theory when approximations for N<<1 are performed.

The study deals with an analysis of dichromatic speckle patterns (SP) formed in the far-field diffraction region by optically heterogeneous materials under illumination with light of two near-by wavelengths. The existence of the radial-fibrillar appearance of SP is explained; it is shown that the occurrence of the radial-fibrillar "structure" of SP cor-responds to the measurable correlation of dichromatic SP as specified by the spectral spatial cross-correlation function of the intensity fluctuations. A speckle pattern cor-relation measuring method for the determination of az (standard deviation of optical path fluctuations due to the passage of coherent light through a layer of heterogeneous material) is described, and the relations between the parameter az and the structural parameters for two simple types of heterogeneous materials are discussed. Experimental results obtained with a series of samples of polyethylene filled with CaCO3 are in a comparatively good agreement with theoretical values.

A model for the transmission of light through a thin diffuser is presented which accounts for the angle of incidence of an input plane wave. This model is then used to make several illustrative calculations of the spectral intensity of the light scattered by the diffuser for arbitrary correlation lengths. In these calculations, it is assumed that the diffuser roughness is represented by a jointly normal random process. The spectral intensity is evaluated analytically for a triangular correlation function and arbitrary diffuser roughness. Experimental curves are also presented of the radiation patterns of ground glass for a wide range of illumination angles with the output angle 0 being swept over the range ± 90°.

A first communication "Electromagnetic study of speckle" has shown that the problem of the diffraction by a random rough surface in Optics can be solved rigorously using the laws of electromagnetism. In this communication, we show that these laws may be used to deduce some parameters of the rough surfaces fro the speckle pattern, in some conditions, even though this problem appears at present time to be extremely difficult in the general case.

The speckle phases are studied statistically in both image and diffraction fields. The general analysis for these studies is given by introducing an optical "system function" and the results are applied to the analysis of the speckle phases in both fields. The speckle phase distributions are evaluated by a phase angle defined as the extent of the equi-probability density ellipse stretched from the origin in the complex plane of the speckle amplitude.

A surface diffuser is said to be strong, if scattered radiation does not contain a specular component. Making use of the expression which relates the magnitude of this component to the density function of surface roughness the condition of strong scattering is formulated in terms of surface statistics. It is shown that rms roughness, being the functional of the roughness density, can be minimized under the condition that specular component does not appear, however a standard method of calculus of variations can not be applied. In the present study two sequences of the continuous density functions are proposed. Each density fulfills the strong scattering condition and variances of subsequent densities decrease. In both cases the limit density is discrete and characterizes a binary diffuser which introduces two phase shifts ¶/2 and -¶/2 (each one with the probability equal to 0.5). Rms phase of such a diffuser equals ¶/2 rad. Reasoning is presented which justifies that any other density function either yields rms phase greater than ¶/2 or causes the specular component to appear. It is shown that there are two types of the continuous density functions which closely approximate the limit function. Both types are bimodal and symmetric. An example of the continuous density of phase is given which is positive in the (-φo,φo) interval, where φ =2.4 rad and whose rms phase equals 1.7 rad (i.e. it is greater than limit value ¶/2 by 8ξ)). The present study shows that strong scattering can be performed by quite smooth surfaces and consequently a speckle pattern of high contrast can be observed in this case.

Fundamentals of speckle metrology are reviewed, with special emphasis on principles governing speckle formation and dependence of this phenomenon on objects' motion and/or deformation. Processes relating to recording and reconstruction of specklegrams are discussed and are followed by presentation of methods for quantitative interpretation of specklegrams. Solution of the resulting equations, based on the method of projection matrices and the fringe vector theory, is outlined. Applications of speckle metrology techniques to measurement of displacements and strains are presented. Presentation of these applications is illustrated by representative examples from the fields of engineering and medicine, including measurement of displacements and vibrations of objects with diffusely scattering surfaces in real time.

Statistical Optics are used to derive a set of equations that govern the spatial movement of laser speckles. Experiments were designed to verify the validity of these equations.

A novel ellipsometric method is introduced for the analysis of speckle patterns. The method is based on the illumination of the scattering object by a plane-polarized light beam with its polarization plane rotating. The speckle pattern produced by the scattered light is then investigated for its polarization characteristics. An approximate theory is outlined and compared with experimental measurements demonstrating the usefulness of the method as a powerful surface analytic tool.

We present results of an investigation into the visualization and measurement of retinal blood-flow distribution by means of single-exposure laser-speckle fundus photography. The technique relies on the speckle effect produced when laser light is scattered at a diffusing structure and on the fact that the speckle will be averaged out to some extent when the structure concerned is moving and/or decorrelating. We discuss two alternative techniques used to process the specklegrams obtained from the retina. The first technique uses an analog optical spatial filtering procedure to enhance the resulting variations in speckle contrast. Although first results have shown the basic usefulness of this technique it suffers from fundamental disadvantages. In the second technique we digitize the specklegram and use digital image processing techniques (1) to convert to contrast variations in the fundus photography into colour variations and (2) to obtain the blood flow of the vessels with a reasonable low statistical error.

The work described in this paper represents an effort to demonstrate the validity of the hybrid approach to the analysis of structural deformation. Mathematical modelling was incorporated into the process of reducing data from a digital correlation analysis of a speckled surface. The experimental data was collected both by directly imaging the surface onto a digitizing vidicon system and by transmitting the image through a coherent optical fiber bundle. Considerable savings of time and resources can be realized through the hybrid approach and the strengths of the theoretical and experimental procedures complement each other beautifully. The final results agreed within a few percent of each other and with values obtained by another independent method (high frequency moire), demonstrating the accuracy of the procedure.

The boiling phenomenon of the subjective laser speckle produced by a moving diffuser is applied for lens testing by two methods, in one of which the temporal auto-correlation of the fluctuating signal detected at a certain point in the speckle field is used for measuring the modulation transfer function (MTF) of the lens through which the speckle is produced. Using an automated MTF measuring system developed on this principle, some experimental results are obtained for an ordinary camera lens as well as a gradient-index (GRIN) rod lens. In the other method, the spatial cross-correlation of two speckle patterns produced before and after the diffuser translation is used for visualizing the non-uniformity of imaging capability of a GRIN rod-lens-array.

An improved version of the laser speckle strain gauge is presented in which speckle displacement caused by deformation of a laser illuminated object is detected in real time using new photodetectors in place of linear image sensors and microcomputer in the conventional version. The detector, called a spatial filtering detector with electronic scanning facility, produces a voltage directly proportional to speckle displacement. Therefore, this strain gauge, which utilizes differential speckle displacement, delivers a voltage that is proportional to surface strain. It has a strain sensitivity of a few mV/microstrain. We applied this gauge to strain measurements of high-polymer films in various directions under loading at frequencies up to several tens of Hz and were able to evaluate their Poisson's ratios.

We have developed a displacement meter which simultaneously measures the (distance and direction of) movement of an object in three dimensions. A piece of retroreflective "Scotchlite" adhesive tape, with a grid printed on it, is attached to the object (target) and illuminated by a single laser beam. The displacements in the X-Y plane perpendicular to the beam are determined by measuring the movement of the grid images appearing at the X and Y detectors. (The reflected beam is split, and transmitted to X, Y and Z detectors). The displacement along the Z axis parallel to the beam is determined -- using a Michelson interferometer -- from the movement of interference fringes which occur when the light reflected from the target is superimposed on a direct "reference" beam from the laser. A new high speed image sensor is used for X, Y and Z detectors to measure the movement of the grid image and the interference pattern -- as a result, the meter has a frequency response from DC to 20 kHz. The sensor is a high resolution spatial filter (array of photocells connected or scanned in such a manner as to be sensitive to a particular light pattern on the array). Using a spatial filter eliminates the need for correlation or other complex signal processing. The accuracy is of the order of 20 μm for X and Y axes and 1.5 μm for the Z axis. The optical system (lenses, mirrors and sensors) is compact, so the meter is portable.

Second order statistics have been derived for the speckle in diagnostic ultrasound that arises from diffuse (incoherent) scattering in the presence of distributed and organized specular (coherent) scattering. They serve as the basis for a three-dimensional feature space in which tissue textures can be classified. The covariance matrix of the measurements in this space is a generalization of the speckle spot number or sampling concept that arises in the study of signal or lesion detectability.

Tissue signatures are obtained from the first and second order statistics of ultrasonic B-scan texture. Laboratory measurements and early clinical results show that the image may be segmented to discriminate between different normal tissues and to detect abnormal conditions based on a three-dimensional feature space. These features describe the intrinsic backscatter properties of the tissues imaged and may be used as the basis of an automatic tissue characterization algorithm.

An image correlation theory of white light speckle is presented which describes the light intensity as a continuous function for both the reference and deformed images. The reference image is divided into small regions corresponding to small subimages of the object surface. The subimage forms the basic element of the process of image correlation. The subimage of the undeformed intensity pattern is mapped to the deformed surface in order to compare its location to a corresponding location on the deformed intensity surface. This mapping function mathematically translates, rotates and deforms small subimages until the best fit is obtained with the recorded data. The following is thus obtained: (1) translation, (2) rotation, and (3) deformation.

Speckle appearing in synthetic aperture radar (SAR) images is generated by coherent interference of radar echoes from target scatters. Basically speckle noise has the nature of a multiplicative noise. In this paper procedures for defining and verifying a statistical noise model are developed and two multiplicative noise smoothing algorithms are presented. These two algorithms are computationally efficient, and have the potential of achieving real or near real-time processing. Several SEASAT SAR and SIR-B images are used for illustration.

Equations for laser radar heterodyne efficiency are derived assuming the heterodyne detection is degraded by phase-front distortions produced by target speckle and atmospheric turbulence. These equations are numerically evaluated for representative target detection scenarios to illustrate the effects of laser speckle on coherent laser radar detection.

After a homomorphic transformation has been used to transform speckle noise into additive signal-independent noise, classical techniques are used to evaluate the number of discerni-ble grey levels and the information capacity of images degraded by fully developed speckle noise. The probability distributions developed earlier by us are used to evaluate the required probabilities of error. Expressions are derived for square and for circular apertures. The information capacity is found to have a maximum of about 0.2 bits per speckle. The spatial frequency resolution required to obtain a given signal to noise ratio by spatial integration is evaluated.

In many cases the speckle maxima required for the our realization of the Shift-and-Add method are not well defined. This is due mainly to Poisson noise, inherent in the detection process, which obliterates the shape of faint speckles. The problem is aggravated for extended objects with local peaks, such as binary stars. As a remedy, we use a filter that smoothes out each speckle and at the same time defines its location. The best filter should be very close to the mean speckle itself: a matched filter. The initial guess for this filter is a bell function, slightly wider than the expected mean speckle. This initial guess is used to locate filtered speckle maxima which are then used to produce a better mean speckle estimate by shift-and add. The procedure is iterated until the mean speckle converges. We find that the iterative speckle estimate is not the optimum matched filter. The most suitable filter must suppress the variable background created by coalescing speckles in a large speckle cloud as well as smooth the single-photon event noise. Thus we combine the mean speckle with a band-pass filter into a matched filter. Local speckle maxima are thus enhanced, whereas single photons are discriminated against by using a comparison low-pass filtered frame, since they do not contain much power. The combined process, speckle identification and weighted-shift-and-add, can be carried out in the image plane or in the Fourier plane. We have experimented in both domains.

The synthetic aperture radar (SAR) aboard Seasat in 1978 demonstrated a unique sensitivity to oceanic and geologic features imaged over a 100-kilometer swath with 25-meter resolution. The ability of the remote sensor to resolve the fine details of large environmental systems has resulted in the orbiting of a similar system, the shuttle imaging radar (SIR-B), aboard the space shuttle Challenger during October 1984. Coherent speckle noise observed for these Doppler imaging-radar systems is caused by random correlations of the illuminating radar chirp with the surface reflectance down range. Radar speckle is similar to optical speckle in that respect, but it is also influenced by along-track sampling statistics. The Ray-leigh statistics of coherent scattering and the Poisson statistics of radar pulse detection are employed to model the speckle observed in spatially random data samples. Speckle degradation of a radar scene may obstruct interpretations of scene detail, but it can also be useful in determining the spatial response of the remote sensor and scene correlator. Randomly speckled scenes that are otherwise featureless provide a white-noise input to the Doppler imaging process. Several such scenes have been processed with fast Fourier trans-form (FFT) methods to estimate the point-spread function and its Fourier-domain equivalent, the wavenumber response function. These measurements of spatial resolution are used to compare the Seasat SAR and the Challenger SIR-B remote sensors. In addition, two ground-scene correlators are compared in terms of the point-spread estimate of spatial resolution for common input data from the SIR-B sensor.

The CLEAN technique was introduced in radio astronomy in the last decade to reduce sidelobe-induced artifacts. It was designed to operate on noncoherent radiation fields. This paper extends CLEAN to the coherent radiation field as found in microwave radar, acoustic sonar and ultrasonic scanning. In addition to sidelobe-artifact suppression, the method reduces speckle-induced artifacts. CLEAN coherently decomposes the radiation field of a complex target into signals from individual scattering centers. This signal decomposition permits the elimination of the constructive (sidelobe) and destructive (speckle) interference, resulting in increased target dynamic range and reduction of the mottled effect in extended targets. The experimental high resolution microwave imagery obtained from a Valley Forge Research Center imaging radar (20 m long, 3 cm wavelength) demonstrates the strength of the technique.

A non-linear speckle filter based on geometric concepts is defined and an example of its effectiveness on synthetic aperture radar imagery is shown. A comparison with look-averaging is made using artificial imagery with synthetic speckle.

Recent progress in optical inverse scattering is reviewed with emphasis on the detection of a phase grating hidden by diffusers using optical correlation techniques.

In a recent paper we investigated the problem of reconstructing the magnitude of a 2-D complex signal f from samples of the Fourier transform of f lying in a small region off-set from the origin. The primary application of interest was synthetic aperture radar. We showed that high quality speckle reconstructions are possible so long as the phase of f is highly random. In this paper we explore the possibility of Fourier-offset reconstruction from just the phase of the Fourier transform. We provide and compare a large number of computer simulated image reconstructions from phase plus magnitude, phase only (constant magnitude), magnitude only (zero phase), and from magnitude plus quantized phase. A number of conclusions are drawn regarding Fourier-offset phase-only reconstruction and several topics are suggested for further research.

The atmosphere of the earth restricts the resolution of conventional astrophotography to about 1". We will discuss various high-resolution speckle methods which can overcome image degradation caused by the atmosphere and telescope aberrations. All methods yield diffraction-limited resolution, for example 0.03" for a 3.6m telescope. We will show various astronomical applications of speckle interferometry, observations of asteroids, Pluto/Charon, double stars and the gravitational-lens triple quasar. Speckle interferometry can yield a true diffraction-limited image if there is a point source (reference star) in the isoplanatic neighbourhood of the object (holographic speckle interferometry). We show an application of holographic speckle interferometry to the central object R136a of the 30 Doradus nebula in the Large Magellanic Cloud. A true diffraction-limited image of R136a has been reconstructed. The image shows that R136a consists of 8 stars. Speckle masking is a triple correlation method which yields diffraction-limited images of general astronomical objects. A point source near the object is not required. We will describe an application of speckle masking to the central object in the giant H II region NGC 3603. The reconstructed image shows that this central object is a star cluster of 4 stars with magnitudes 11m, 12m, 13m and 13m. The signal-to-noise ratio of the speckle masking image is about the same as the signal-to-noise ratio of the speckle interferometry autocorrelation. Speckle spectroscopy is a speckle method which yields diffraction-limited objective prism spectra. It is also possible to apply speckle masking to multiple-mirror interferometers or long-baseline interferometers on the ground or in space. True images of very high angular resolution can be obtained by these techniques. We will show computer simulations which illustrate the dependence of the signal-to-noise ratio on photon noise.

The infrared emission of astronomical sources is buried in a large background due to the thermal emission of the telescope. The real time deformations of an adaptive flexible mirror are likely to produce a spurious modulation of this background, adding noise to the signal. We estimate here the amount of noise introduced by such a mirror.

We have applied a variant of the Shift-and-Add algorithm originally developed by Lynds, Worden & Harvey to astronomical speckle interferometric data. A set of impulses corresponding in location and magnitude to the local maxima in each specklegram is generated and used to obtain an average speckle by means of a Wiener-type filter deconvolution procedure. This technique yields diffraction limited images which appear to be self calibrating for seeing effects. Realistic point spread functions have been obtained for a number of telescopes at different wavelengths and results are also presented for the resolved red supergiant Alpha Orionis. The limiting signal-to-noise of the technique as indicated by the results presented here suggests a dynamic range of ~6 stellar magnitudes with no evidence of residual seeing effects.

Recent partially-published and previously unpublished extensions of the shift-and-add principle are summarised. An up-to-date bibliography of shift-and-add is included. A new technique, potentially capable of imaging any object viewed through severely distorting media, is presented and illustrated with computer generated data (corresponding to one-dimensional speckle images). The technique, which is called zero-and-add, can be extended to two dimensions by appealing to the theory of image reconstruction from projections.

Roll deconvolution is a speckle method that can improve the resolution of the 2.4m Hubble Space Telescope (HST) at short UV wavelengths. A diffraction-limited image can digitally be reconstructed from two degraded images recorded at two different roll angles of the HST. The reconstruction is performed by complex inverse filtering of the two degraded images. In digital simulations we have investigated how the signal-to-noise ratio of the reconstruction depends on photon noise (104 to 50 photon counts per pixel), on the structure and size of the point spread function (caused by aberrations) and other parameters.

In November 1980 we implemented a one-dimensional infrared speckle interferometry system for the University of Hawaii (U H) 2.2-m telescope at Mauna Kea. Currently there are functioning systems on the 2.2-m telescope, the 3-m NASA Infrared Telescope Facility (IRTF), and the 3.8-m United Kingdom Infrared Telescope (UKIRT). With several detectors, these have the capability of operating over a range of wavelengths from 1 to 12 μm; this part of the infra-red is well suited for observing objects with real or apparent temperatures 250 < T < 3000 K. We now routinely measure diffraction-limited visibility functions for a variety of astronomical sources. These include volcanic eruptions on I0, circumstellar dust shells around evolved stars, young stars in giant molecular clouds, and pre-main-sequence stars. Spatial structure as small as 100 milliarcsec has been measured with the largest telescopes. The speckle interferometer has also been used together with a cooled grating spectrometer to achieve high spectral resolution and spatial resolution simultaneously. We have the ability to display real-time power spectra for evaluation of signal-to-noise ratios (SNR) and save all of the raw data for later processing. This latter feature has recently been used successfully to retrieve phases and construct one-dimensional pictures. The current version of the interferometer features an IBM PC that can be moved to various telescope facilities. It has been used on the Kitt Peak 4-m telescope as well as the telescopes on Mauna Kea.

The Weighted Shift-and-Add algorithm (WSA) is an image reconstruction technique whereby diffraction limited images of astronomical objects are obtained from speckle interferometric data. This paper attempts to put the understanding of WSA on a firm mathematical basis by a statistical analysis of the algorithm. The approach follows that of the Hunt, Fright, and Bates study carried out for the Simple Shift-and-Add algorithm. The expected WSA profile is found to be linearly dependent on the square modulus complex coherence function of a speckle pattern. The last section of this paper contains a discussion of how the statistical analysis compares to the results obtained with our new Weighted Shift-and-Add procedures using cross-correlation (WSA/XC) and deconvolution (WSA/WD).

A new technique known as Differential Speckle Interferometry has been applied to data obtained using the fully-phased six-mirror aperture of the Multiple Mirror Telescope. By observing stellar objects at two distinct wavebands simultaneously, differences in the object resulting from the different wavelengths can be derived. Observations were made of the supergiant star Alpha Orionis in order to investigate the Hydrogen-alpha emission from the surrounding envelope. The data reduction process consists of a frame-by-frame weighted deconvolution procedure. This process involves an inherent Wiener-type filtering which must be removed in order to preserve high spatial frequency information. Results for Alpha Orionis and for the unresolved source Gamma Orionis are presented.

Coherent arrays of telescopes, either ground or space based, are likely to become in the next decade a new class of powerful and wide-spread astronomical instruments. It has often been argued that at optical wavelengths these arrays are only able to measure the modulus of the source's spatial spectrum, and therefore are poorly suited for imaging. In this paper we show that phase reconstruction techniques developed for speckle interferometry can be applied to two telescopes Michelson interferometry. We make the comparison between image plane and pupil plane interferometry and compute a merit factor: for modulus determination, pupil plane is always preferable, while image plane is better for phase retrieval.

A cross spectrum analysis technique, which makes it possible to compute an unbiased estimate of the speckle pattern Wiener spectrum, has been developed at the Nice University Astrophysics Department. The technique is based upon the use of a dual channel experiment, each of the channels giving a record of the image intensity with its specific noise. A cross spectrum analysis between the two channels gives an estimate of the Wiener spectrum free of background noise bias. Using a one-dimensional interferometer, the technique has given results on several subjects such as the study of atmospheric MTF including anisotropies due to the geometry of the telescope aperture, the time-space properties of speckles, the angular structure of Betelgeuse and the convective motions of solar photospheric microstructures. The cross spectrum analysis technique has been applied also to the speckle spectroscopy technique and to the differential speckle interferometry technique. A theoretical study of the bichromatic cross spectrum has been elaborated and the first experimental results are shown, they are in good agreement with the theory. Expressions are given for the signal-to-noise ratio on the amplitude and the uncertainty on the phase of the cross-spectrum of two speckle images in the presence of photon shot noise. They are applied to the very promising differential speckle interferometry technique which already permitted us to measure angular micro-displacements, due to atmospheric dis-persion, into the submilliarcsecond range.

A new holospeckle interferornetric technique is developed for visualizing two dimensional velocity field of solids and fluids. The method requires only single exposure and results in sinc function fringes representing velocity contours.

If a rough surface is illuminated by a coherent light wave of wavelength λ1 it is not possible to determine the surface profile from the phases of the speckle field formed by the scattered light. If the rough surface is illuminated, however, by an additional coherent wave of wavelength λ2, the phase differences between the two speckle fields do contain information about the macroscopic surface profile even if subject to a statistical error. We present the pertinent statistical properties of dichromatic speckle fields and show (1) that the macroscopic surface profile may be determined from the phase differences if the effective wavelength Δ = λ1λ2/|λ1-λ2| sufficiently larger than the standard deviation of the microscopic profile of the illuminated surface and (2), that the statistical error is reasonably small if the phase measurements are obtained from speckles with sufficient intensity.

Computer generated optical fringe patterns provide valuable information to the experimentalist. They can be used to compare the results of an experiment with a theoretical solution, aid in the planning of experiments or used to perform conceptual experiments. This paper discusses several straightforward fringe plotting algorithms that have been developed by the authors for the purpose of generating fringe patterns on a computer. The emphasis of the paper is on the generation of fringe patterns using high-speed line printers or dot matrix printers which allow the formation of pseudo-gray scale images. Examples of particular interest to speckle researchers, including displacement patterns and Young's fringes, are presented to illustrate the algorithms developed.

An analysis of the fringe visibility of a speckle interferometer has been given as a function of the source position. The interferometer consists of a double exposure speckle recording which, for an infinite aperture, produces in monochromatic light a set of unit visibility rectilinear fringes localized at infinity. In practice, the plane conjugate to that of the point source of light is taken to be the localization as well as the observation plane of the fringes. A longitudinal displacement of the light source gives rise to a reduction in the fringe visibility which has been monitored by positioning a line detector in the observation plane. The sensitivity of such a position sensing device depends upon the size and location in reference to the fringe profile and it has been estimated to be of the order of 0.1 micrometer. This study has been applied to the design and construction of a remote electro-optic displacement sensor. The range and the sensitivity of the device are mutually complementary and one could only be increased at the loss of the other or vice-versa. Furthermore, a study of the linearity of the sensor has been made as a function of the other sensor parameters.

Speckle patterns have high-frequency phase data, which make finding the absolute phase of a single speckle pattern difficult. However, the phase of the difference between two correlated speckle patterns can be determined by applying phase-shifting techniques to speckle interferometry, which will quantitatively determine the phase of double-exposure speckle measurements. The technique uses computer control to take data and calculate phase without an intermediate recording step. The randomness of the speckle causes noisy data points that are removed by data processing routines. A study of the phase errors attributable to decorrelation of the speckle patterns shows a 33° rms error for 10 waves of tilt. One application of this technique is finding the phase of deformations, where up to 10 waves of wavefront deformation can easily be measured. Results of deformations caused by tilt of a metal plate and a disbond in a honeycomb structure brazed to an aluminum plate are shown.

A holospeckle technique for 3D displacement using sandwich plates is developed whereby the resulting holospeckle has a variable sensitivity in both the hologram and specklegram modes.

The interference fringe intensity fluctuations generated by speckle are considered. It is shown that the ensemble average variance of interference pattern intensity consists of the fourth-order correlation terms. A method of determination of these terms is proposed. The fluctuations generated by Gaussian speckle are discussed.