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This paper describes two superresolution coherent imaging techniques, which both use a diffraction grating to direct high spatial frequency information that would otherwise be lost through the imaging system pupil. The techniques employ digital holography to measure the optical field in the image plane and rely on capturing multiple holograms with the illumination condition altered between exposures. In one case, we used linear signal processing to separate aliased spectral regions, while in the second case we directly measured spectral regions without aliasing. In both cases, we stitched together higher-bandwidth synthetic aperture spectra and used them to reconstruct superresolution images. Our experimental results validated the approaches, demonstrating a resolution gain factor of approximately 2.5.

Previously,1 we introduced a new method of optical analog-to-digital conversion based on oversampling and error diffusion coding that offers the potential to extend resolution and conversion rates beyond that currently possible with other electronic or optical converters. In this new approach, the analog input signal is first optically sampled at a rate much greater than the Nyquist frequency and is then quantized by a modulator that incorporates one-bit quantizers and linear filters in a negative feedback architecture for the purpose of reducing in-band quantization noise. The output of the modulator is subsequently processed by a digital decimation filter, which removes the out-of-band noise and generates a high resolution digital approximation to the analog input signal at the signal's Nyquist rate. In this paper, we present experimental results from the first demonstration of an optical first-order noninterferometric modulator and discuss both spatial and temporal extensions of this optical error diffusion technique to multidimensional applications.

Handwritten-signature verification is treated as a two-class synthetic discriminant function (SDF) problem. Images of valid and casually forged signatures are collected and binarized, using an electronic digitizing camera. Performance of this approach with a small number of valid signatures in the training set is examined, and substantial improvement is demonstrated when forgeries are included in the set. In particular, the equal-error rate for the SDF classifier with forgeries included is shown to average approximately 4% across nine different subjects. The effects of image preprocessing on false acceptance and true rejection rates are examined. The use of alternatives to forged signatures in the training matrix is explored. Finally, SDF performance is shown to deteriorate when the tested forgeries are produced with some a priori knowledge of the target signature.

A method for designing and recording a holographic optical element that is used as a waveguide focusing grating coupler is presented. It is based on recording the holographic coupler with two predistorted wave fronts, derived from interim holograms, whose readout and recording geometries are different. The corrected holographic coupler has almost aberration-free performance even for couplers with very small f/numbers or with large wavelength shift between recording and readout.

A method for designing a double grating for coupling a polychromatic wave out of a waveguide with negligible angular dispersion is presented. It is based on calculating analytically two different grating functions, such that the chromatic dispersion of the first grating function is compensated by the second grating function. It is shown that the usable wavelength range can be increased by more than 2 orders of magnitude relative to conventional grating couplers for typical angular tolerance.

Some thirteen papers in this 10 Mar. 1990 issue of Applied Optics consider the technology and use of optical interconnects. This introduction also briefly discusses where the field is going.

Almost all coherent pattern recognition architectures are based on optical correlation of the input with a designed filter. However, the filter can be implemented via many different media, and each medium will impose different realizability constraints on the filter. That is, different media will have different regions of physical realizability. In the past, there has not been much work addressing the problem of designing an optimal filter given an arbitrary region of realizability. This paper presents the theory for just such an optimal filter design. A fast algorithm is presented to implement the theory. The algorithm is demonstrated with two examples.

Many spatial light modulators and computer-generated hologram techniques can very efficiently implement binary phase-only filters. At present, almost all binary phase-only filters are designed by first designing a matched spatial filter and then binarizing it. There is no theoretical basis to expect that such a filter will be optimal. This paper first develops a theory for optimal binary phase-only filters and then presents a numerical algorithm which designs the optimal binary phase-only matched filter for a given image. The filter is optimal in the matched filter sense of maximizing SNR at the output origin. Characteristics of the optimal filter compared to conventional binary phase-only filters are discussed.

The geometrical design characteristics of multiple-image holograms are evaluated. A figure of merit expressing these characteristics as a function of the hologram diameter and the distance between the hologram and the image plane is developed. This value is then used to compare two designs which are capable of forming several hundred interconnections. The results indicate that these connections can be formed between points on the substrate separated by 2-3 cm provided that the holograms are separated from the substrate plane by 0.5-1 cm. Each hologram design is experimentally demonstrated in bleached photographic emulsions.

For a multilayer structure illuminated by a laser beam, absorption of optical energy in the absorptive layers and the diffusion of the resultant heat throughout the structure are studied. Analytical and numerical procedures for this study are described, and, as a specific example, the profiles of temperature distribution during recording on a magnetooptical disk are presented. The technique is also expected to be of value for studies of thermal marking and laser annealing.

Using residue arithmetic it is possible to perform additions, subtractions, multiplications, and polynomial evaluation without the necessity for carry operations. Calculations can, therefore, be performed in a fully parallel manner. Several different optical methods for performing residue arithmetic operations are described. A possible combination of such methods to form a matrix vector multiplier is considered. The potential advantages of optics in performing these kinds of operations are discussed.

Peter Kellman and Joseph W. Goodman Stanford University, Department of Electrical Engineering, Stanford, California 94305. Received 20 May 1977. Recent interest in Mellin power spectra, complex filtering, and correlation is due to the scale invariance of the Mellin transform. Mellin correlation has been cited for application to 2-D target detection and 1-D signal processing where parameters of scale and Doppler are unknown. In such applications, Fourier domain matched filtering requires either sequential hypothesis testing or multichannel processing. Loss of correlation in the Fourier implementation due to scale difference between received signal and reference is exchanged, however, for an equally severe loss in the Mellin correlation due to time or positional displacement. A coherent optical technique for Mellin transforms is described here in which signals may be processed in real time to provide scale invariant correlations. The parameterization on time uncertainty is provided for automatically by utilizing an acoustooptic spatial light modulator. Methods for performing general linear integral transforms using parallel processing in both spatial dimensions allow real time operation. Several of these methods have been outlined recently; one configuration, well suited for implementing Mellin spectrum analysis, is reviewed here. The general linear superposition integral may be written as

In the absence of noise, a linear spatially invariant blur is completely removed by the application of a transfer function proportional to the inverse of the blur transfer function. However, most blur transfer functions of interest either contain zeros or approach zero; and, consequently, the inverse should cover an infinite dynamic range. In these cases, the inverse can be approximated by a holographic spatial filter that employs a finite range of amplitude transmittance. The effect of finite range on the restored impulse response or point spread function of the combined blurring and deblurring operations are studied. A computer simulation is compared with experimental results for the case of severe defocusing through a circular aperture. (auth)

Nonlinear filtering operations can be performed in coherent optical systems with the help of the halftone screen process. Theoretical and practical considerations regarding this type of system are presented. The use of these methods to achieve logarithmic filtering is emphasized. Applications to separation of multiplicative signals and noise, speckle noise reduction, and processing of radiographic images are considered. Experimental results are presented.

In discrete-cosine-transform-based (DCT-based) compressions such as JPEG it is a common practice to use the same quantization matrix for both encoding and decoding. However, this need not be the case, and the flexibility of designing different matrices for encoding and decoding allows us to perform image restoration in the DCT domain. This is especially useful when we have severe limitations on the computational power, for instance, with in-camera image manipulation for programmable digital cameras. We provide an algorithm that compensates partially for a defocus error in image acquisition, and experimental results show that the restored image is closer to the in-focus image than is the defocused image.

We propose and evaluate the feasibility of a multiprocessor interconnect based on free-space optics for reducing network latency. The proposed interconnect supports high-throughput network links and can exploit locality of memory references through reconfiguration. Our analysis shows that the proposed interconnect can, in principle, support hundreds of nodes in the static mode and tens of nodes in dynamic mode.

Heat removal, rather than finite interconnect density, is the major mechanism that limits how densely we can pack three-dimensional computing systems of increasing numbers of elements. Thus highly interconnected approaches can be employed without a further increase in system size. The use of optical interconnections for implementing the longer connections of such systems is advantageous. In fact, if the optical communication energy is sufficiently low and large-bit repetition rates are employed, conductors are useful for only the shortest connections and can be dispensed with altogether with little disadvantage. This justifies consideration of an optical digital computer.

The relationships between the synthetic discriminant function and the filters constructed from two eigenvector decompositions are developed and examined.

A method for recording a substrate-mode holographic interconnect system has been developed. The system is composed of two identical holographic optical elements (HOEs) that were recorded on the same plate. The first one collimates a coherent point source into a plane wave that is trapped inside the plate by total internal reflection. The second HOE focuses the collimated wave into a detector. Because the possible recording wavelengths for efficient holograms are usually different from the readout wavelengths, the holographic elements must be recorded with complex wavefronts in order to assure high diffraction efficiencies and low aberrations. The complex wavefronts are derived from simple spherical holograms whose readout geometries differ from those used during recording. The method is illustrated with HOEs recorded at 488 nm and read out at 633 nm. Nearly diffraction-limited imaging and high efficiencies were achieved.

We present a simple technique for observing and studying the onset of microstructure in silver halide gratings, which can also be used to improve processing chemistry.

The design and experimental results of a new 2 x 2 optical routing switch based on liquid crystal is reported. The switch uses nematic liquid crystal for polarization beamsplitting and ferroelectric liquid crystal for polarization rotation. The polarization insensitive, broadcast capable, full duplex switch has a signal-to crosstalk ratio of ~20 dB, a switching speed below 250 micros and a measured insertion loss of 1.4 dB. We show that the basic 2 x 2 switch can be extended to form a N x N rearrangeable permutation network.

Methods for using neural networks for computation are considered. The success of such networks in finding good solutions to complex problems is found to be dependent on the number representation schemes used. Redundant schemes are found to offer advantages in terms of convergence. Neural networks are applied to the combinatorial optimization problem known as the Hitchcock problem and signal processing problems, such as matrix inversion and Fourier transformation. The concept of programming complexity is introduced. It is shown that for some computational problems, the programming complexity may be so great as to limit the utility of neural networks, while for others the investment of computation in programming the network is justified. Simulations of neural networks using a digital computer are presented.

The number of holograms that can be linearly superimposed on one photographic plate has two fundamental limits; one is the finite extent of the characteristic amplitude transmittance vs energy-of-exposure (Ta − E) film curve, and the other is holographic reciprocity law failure [ K. M. Johnson L. Hesselink J. W. Goodman , Appl. Opt.23, 218 ( 1984)], the chronological decrease in brightness of reconstructed images in a multiple-exposure hologram. In this paper a multistep hologram, called the multiple multiple-exposure hologram, is presented which increases the number of holograms that can be linearly superimposed on one photographic plate and reconstructs images of equal brightness. The brightness of the reconstructed images in the final display hologram is a function of the reference-to-object beam ratio K in the intermediate and final steps of the multiple multiple-exposure hologram. By choosing different values for K in each step of this technique, a cross section can be displayed with the same diffraction efficiency in the initial multiple-exposure holograms and the final multiple multiple-exposure hologram. The trade-off between the maximum number of holograms that can be linearly superimposed on one photographic plate and the signal-to-noise ratio in the final display hologram is also discussed.

A coherent optical system for performing continuous Fourier transforms can be modified to perform discrete Fourier transforms. Such a system is capable of diagonalizing circulant matrices presented at its input. The diagonal elements of the new matrix are the eigenvalues of the original matrix. A suitable modification allows the eigenvalues of many different circulant matrices to be found simultaneously. Such a technique can be used for the initial portion of a coherent optical matrix inversion system, which can find the inverses of circulant matrices. The method can also be applied to the problem of inverting Toeplitz matrices in a hybrid digital and optical system.

An imaging system is proposed as an alternative to metallized connections between integrated circuits. Power requirements for metallized interconnects and electrooptic links are compared. A holographic optical element is considered as the imaging device. Several experimental systems have been constructed which have visible LEDs as the transmitters and PIN photodiodes as the receivers. Signals are evaluated at different source–detector separations. Multiple exposure holograms are used as a means of optical fan out allowing one source to simultaneously address several receiver locations. Limitations of this technique are also discussed.

A ground-based astronomical telescope with a segmented primary mirror will suffer image-degrading wavefront aberrations from at least two sources: (1) atmospheric turbulence and (2) segment misalignment or figure errors of the mirror itself. This paper describes the derivation of a mirror control feedback matrix that assumes the presence of both types of aberration and is optimum in the sense that it minimizes the mean-squared residual wavefront error. Assumptions of the statistical nature of the wavefront measurement errors, atmospheric phase aberrations, and segment misalignment errors are made in the process of derivation. Examples of the degree of correlation are presented for three different types of wavefront measurement data and compared to results of simple corrections.

An iterative method for finding the eigenvectors and eigenvalues of a matrix via incoherent optical matrix-vector multiplication and simple electronic feedback is described.

Previous authors have reported a falloff in diffraction efficiency of reconstructed images in multiple-exposure holograms. We call this phenomenon holographic reciprocity law failure. We propose an explanation for this phenomenon based on latent image formation in photographic materials and discuss a method for superimposing holograms that reconstruct images of equal brightness.

The limitations inherent in a recently proposed analog method for solving simultaneous linear equations are examined, and methods for overcoming some of these limitations are discussed. In its original form the method requires that all eigenvalues of the matrix of coefficients lie in a unit circle centered on (1,0) in the complex plane. Proper scaling of the matrix and the data vector extends this region to the entire right half of the complex plane (neglecting the effects of noise). A modification of the algorithm is described that allows the region to be further extended to the entire complex plane. If the product of the gains in the forward and feedback branches is not unity, the solution produced by the algorithm is shown to be in error. Finally, the effects of noise, which is inevitably significant in any analog realization of the algorithm, are examined. Noise is found to produce a limiting mean square error of the solution, thus preventing perfect convergence to the ideal solution vector. A procedure for determination of when to stop the iteration is proposed.

A procedure for predicting the error rates of a computer-generated holographic memory with general transform domain quantization error is developed. The kinoform, ROACH, detour phase binary hologram, and generalized detour phase binary hologram are compared on the basis of error rates and encoding efficiency. Error rates predicted by the analysis are compared with those obtained by computer simulation.

A ground-based adaptive optics imaging telescope system attempts to improve image quality by measuring and correcting for atmospherically induced wavefront aberrations. The necessary control computations during each cycle will take a finite amount of time, which adds to the residual error variance since the atmosphere continues to change during that time. Thus an optical processor may be well-suited for this task. This paper investigates this possibility by studying the accuracy requirements in a general optical processor that will make it competitive with, or superior to, a conventional digital computer for adaptive optics use.

In this paper, we consider a two-wave mixing system using a photorefractive medium and compute the diffraction efficiency as a function of the writing beam ratios. In certain conditions, the diffraction efficiency is shown to be exactly proportional to the inverse of the object beam intensity. Experimental results, obtained using a Bi12GeO20 crystal, are compared to the theoretical expression. In particular, we investigate the dependence of the range of beam ratios over which inversion occurs on the experimental parameters of total writing beam intensity and applied electric field. Results of performing inversion of gray-scale objects using a degenerate four-wave mixing setup are shown.

It has been demonstrated previously that a thin phase hologram, recorded in a weak reference condition, is capable of inverting a complex field. Using a computer simulation of the properties of a thin phase hologram, we find the operating conditions and dynamic range for wave-front inversion. The conclusions of the simulation are used for designing an experiment to invert a circulant matrix and the results of the experiment well support the analysis.

A coupled wave model for two multiplexed reflection holograms recorded in a volume material is presented. Both simultaneously and sequentially formed gratings are evaluated. The effects of a strong intermodulation grating, the thickness change with emulsion processing, and the construction geometry on the desired diffraction orders are investigated over a range of reconstruction angles. The model results are compared with experimental data from holograms formed in bleached photographic emulsions.

In this paper we present a new method for estimating the bandwidth of multimode optical fibers based on the frequency correlation function of the speckle patterns generated by the interference of fiber modes. This technique, which does not require a pulse or signal generator, can be utilized to estimate the bandwidth of a multimode fiber using a relatively short length of fiber. By applying this method to a test fiber we obtained a bandwidth of approximately 36 MHz km which is in relatively good agreement with the approximately 44-MHz x km bandwidth measured by a conventional pulsed technique.

In the past, most optical data processing systems have been restricted to performing linear space-invariant operations. However, a wide class of interesting data processing operations require linear space-variant filtering. Three methods for performing linear space-variant processing of 1-D inputs are described. Experimental results obtained with all three systems are presented, and their relative advantages and disadvantages are discussed.

Long-range holographic experiments have been performed over distances as large as 12 km. Holograms and images of retroreflector and Scotchlite objects were obtained. The reference wave was supplied by a retroreflector placed next to the object, and the source of illumination was a Q-switched ruby laser. Resolutions obtained varied with atmospheric seeing conditions, ranging from 3 see of arc to 0.5 see of arc. The primary practical problem encountered was detector saturation, a consequence of the limited dynamic range of photographic film. The use of a low contrast developer only partially alleviates this problem.

The ease with which an optical system can achieve an extremely large time-bandwidth product makes optical filters and cross-correlation function generators attractive for signal detection and parameter estimation when signal-to-noise ratios are very low. The principles of a new optical filter that overcomes the sometimes severe positioning requirements of conventional optical filters are described. The relationship between the signal amplitude and matched-filter impulse response amplitude is discussed, and a method for reducing a noise component that is common to all optical filters is given. A matched filter was constructed and used to detect a radar-like pulse when the signal-to-noise ratio was -40 dB. A digital computer may be used with an optical system to filter time signals with an accuracy that potentially is as good as the accuracy of an all-digital system, but with a significant reduction in computation. The theory of an optical digital filter is presented.

A hologram may be regarded as a linear temporal filter operating on the time-varying object fields incident during exposure. Such a point of view allows the effects of object vibrations and other simple motions to be predicted in a straightforward fashion. In addition, this filtering property suggests that holograms might be used to provide doppler maps of moving objects. Finally, the temporal-filtering viewpoint offers insight into the effect of the wavefront reconstruction process on images of noncoherent objects.