Wigner-related phase spaces for signal processing and their optical implementation.

Phase spaces are different ways to represent signals. Owing to their properties, they are often used for signal compression and recognition with high discrimination abilities. We present several recently introduced Wigner-related sets of representations that have improved signal processing performance, and we introduce an optical implementation. This study deals with the generalized Wigner spaces, the fractional Fourier transform, and the x-p and the r-p representations. The optical implementations are demonstrated and discussed.

Perspective projections in the space-frequency plane and fractional Fourier transforms.

Perspective projections in the space-frequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented.

Comparison of fully three-dimensional optical, normally conducting, and superconducting interconnections.

Several approaches to three-dimensional integration of conventional electronic circuits have been pursued recently. To determine whether the advantages of optical interconnections are negated by these advances, we compare the limitations of fully three-dimensional systems interconnected with optical, normally conducting, repeatered normally conducting, and superconducting interconnections by showing how system-level parameters such as signal delay, bandwidth, and number of computing elements are related. In particular, we show that the duty ratio of pulses transmitted on terminated transmission lines is an important optimization parameter that can be used to trade off signal delay and bandwidth so as to optimize applicable measures of performance or cost, such as minimum message delay in parallel computation.

Space bandwidth-efficient realizations of linear systems.

One can obtain either exact realizations or useful approximations of linear systems or matrix-vector products that arise in many different applications by implementing them in the form of multistage or multichannel fractional Fourier-domain filters, resulting in space-bandwidth-efficient systems with acceptable decreases in accuracy. Varying the number and the configuration of filters enables one to trade off between accuracy and efficiency in a flexible manner. The proposed scheme constitutes a systematic way of exploiting the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.

Optical implementations of two-dimensional fractional fourier transforms and linear canonical transforms with arbitrary parameters.

We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.

Nonseparable two-dimensional fractional fourier transform.

Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example.

Toward an optimal foundation architecture for optoelectronic computing. Part I. Regularly interconnected device planes.

By systematically examining the tree of possibilities for optoelectronic computing architectures and offering arguments that allow one to prune suboptimal branches of this tree, I come to the conclusion that electronic circuit planes interconnected optically according to regular connection patterns represent an alternative that is reasonably close to the best possible, as defined by physical limitations. Thus I propose that this foundation architecture should provide a basis for future research and development in this area.

Toward an optimal foundation architecture for optoelectronic computing. Part II. Physical construction and application platforms.

Various issues pertaining to the physical construction of systems that are based on regularly interconnected device planes, such as heat removal and extensibility of the optical interconnections for larger systems, are discussed. Regularly interconnected device planes constitute a foundation architecture that is reasonably close to the best possible as defined by physical limitations. Three application platforms based on the foundation architecture described are offered.

Digital Fourier optics.

Analog Fourier optical processing systems can perform important classes of signal processing operations in parallel, but suffer from limited accuracy. Digital-optical equivalents of such systems could be built that share many features of the analog systems while allowing greater accuracy. We show that the digital equivalent of any system consisting of an arbitrary number of lenses, niters, spatial light modulators, and sections of free space can be constructed. There are many possible applications for such systems as well as many alternative technologies for constructing them; this paper stresses the potential of free-space interconnected active-device-plane-based optoelectronic architectures as a digital signal processing environment. Implementation of the active-device planes through hybridization of optoelectronic components with silicon electronics should allow the realization of systems whose performance exceeds that of purely electronic systems.

Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering.

We consider optical systems composed of an arbitrary number of lenses and filters, separated by arbitrary distances, under the standard approximations of Fourier optics. We show that every such system is equivalent to (i) consecutive filtering operations in several fractional Fourier domains and (ii) consecutive filtering operations alternately in the space and the frequency domains.

Fractional correlation.

Recently, optical interpretations of the fractional-Fourier-transform operator have been introduced. On the basis of this operator the fractional correlation operator is defined in two different ways that are both consistent with the definition of conventional correlation. Fractional correlation is not always a shift-invariant operation. This property leads to some new applications for fractional correlation as shift-variant image detection. A bulk-optics implementation of fractional correlation is suggested and demonstrated with computer simulations.

Fractional Fourier transform: simulations and experimental results.

Recently two optical interpretations of the fractional Fourier transform operator were introduced. We address implementation issues of the fractional-Fourier-transform operation. We show that the original bulk-optics configuration for performing the fractional-Fourier-transform operation [J. Opt. Soc. Am. A 10, 2181 (1993)] provides a scaled output using a fixed lens. For obtaining a non-scaled output, an asymmetrical setup is suggested and tested. For comparison, computer simulations were performed. A good agreement between computer simulations and experimental results was obtained.

Compact optical temporal processors.

Optical signal processing can be done with time-lens devices. A temporal processor based on chirp-z transformers is suggested. This configuration is more compact than a conventional 4-ƒ temporal processor. On the basis of implementation aspects of such a temporal processor, we did a performance analysis. This analysis leads to the conclusion that an ultrafast optical temporal processor can be implemented.

Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators.

The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as well as the possibility of exploiting the fractional Fourier transform as a mathematical tool in analyzing such systems.

Elements of a hybrid interconnection theory.

We present a textbooklike treatment of hybrid systems employing both optical and electrical interconnections. We investigate how these two different interconnection media can be used in conjunction to realize a system not possible with any alone. More specifically, we determine the optimal mix of optical and normally conducting interconnections maximizing a given figure-of-merit function. We find that optical interconnections have relatively little to offer if the optical paths are constrained to lie on a plane (such as in an integrated optics system). However, if optical paths are permitted to leave the plane, they may enable considerable increase in performance. In any event the prize in terms of performance is accompanied by a penalty in terms of system power and/or size.

Scaling of diffractive and refractive lenses for optical computing and interconnections.

We discuss both numerically and analytically how the space-bandwidth product and the information density of lenses scale as functions of their diameter and f-number over many orders of magnitude. This information may be useful for the design of optical computing and interconnection systems. For diffractive lenses, cost is defined as the number of resolution elements the lithographic production system must have; the relationship of this quantity to the space-bandwidth product and information density is also given.

Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform.

Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a function's Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

Chirp filtering in the fractional Fourier domain.

In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as additive noise, the FRT is used for filtering the line delta function in the appropriate fractional Fourier domain. Experimental filtering results for a Gaussian input function, which is modulated by an additive chirp noise, are shown. Excellent agreement between experiments and computer simulations is achieved.

Optical-coordinate transformation methods and optical-interconnection architectures.

The analogy between optical one-to-one point transformations and optical one-to-one interconnections is discussed. Methods for performing both operations are reviewed and compared. The multifacet and multistage architectures have the flexibility to implement any arbitrary one-to-one transformation or interconnection pattern. The former would be preferred for low-cost and low-resolution applications, whereas the latter would be preferred for high-cost and high-performance applications.