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Channel and memory capacities are limited and expensive resources in optical communication. In this work we propose several approaches to reduce the required capacity by using the a priori knowledge about the optical system used to capture the spatial information. The a priori knowledge is related to the triangularlike shape of the optical transfer function and to its spectral symmetry. In our work we will demonstrate a reduction in memory capacity required for the same image quality or a resolution enhancement for the same physical memory capacity. We will also present a nearly all-optical implementation of these techniques.

The propagation of stationary wave fields that exhibit simultaneously lateral and longitudinal periodicity is investigated. As a model, we use a Fabry-Perot resonator with periodically structured mirrors under monochromatic plane wave illumination. The resonator leads to a longitudinal periodicity, the grating mirrors to a lateral periodicity. The angular spectrum of the transmitted wave field is given as the product of two terms, one related to the lateral, the other to the longitudinal properties. Its modal structure can vary significantly depending on the ratio of the lateral and longitudinal periods and the reflectivity of the resonator's mirrors. For example, it is possible to generate bandgap behavior despite the fact that the periods may be significantly larger than the wavelength. The results of this investigation apply to the design of phase-coupled array resonators and multiplexers.

Holography is reformulated by using the framework of phase-space optics. The Leith-Upatnieks off-axis reference hologram is compared with precursors, namely, single-sideband holography. The phase-space representation of complex amplitudes focuses on similarities between different holographic recording schemes and is particularly useful for investigating the degree of freedom and the space-bandwidth product of optical signals and systems. This allows one to include computer-generated holography and recent developments in digital holography in the discussion.

We propose to use a differential operator for representing the influence of phase-only filters on the defocused modulation transfer function of the clear pupil aperture. We present a phase-only filter that implements optically Taylor's theorem in phase space. We show numerical simulations of the modulation transfer functions and the images that can be obtained by using the proposed filter.

Self-imaging means image formation without the help of a lens or any other device between object and image. There are three versions of self-imaging: the classical Talbot effect (1836), the fractional Talbot effect, and the Montgomery effect (1967). Talbot required the object to be periodic; Montgomery realized that quasiperiodic suffices. Classical means that the distance from object to image is an integer multiple of the Talbot distance Z(T) = 2p2/lambda, where p is the grating period. Fractional implies a distance that is a simple fraction of Z(T): say, Z(T)/2, Z(T)/4, 3Z(T)/2.... We explore the most general case of the fractional Montgomery effect.

We present a theoretical analysis of the temporal behavior of double-diffraction setups. It applies, in particular, to Talbot and Montgomery interferometers, whose operation is based on the self-imaging effect. The use of both types of interferometer as temporal filters for optical and terahertz applications was recently suggested. We show that double-diffraction setups can be modeled as communications channels with dispersive behavior caused by diffraction. We develop mathematical expressions for the phase delay, the group velocity, and the group-velocity dispersion for both quasi-monochromatic and polychromatic case. Based on these results, the temporal impulse response of a double-diffraction setup is derived. Finally, a general description of its practical implementation are presented.

In the previous two sections of "Flatland optics" [J. Opt. Soc. Am. A 17, 1755 (2000); 18, 1056 (2001)] we described the basic principles of two-dimensional (2D) optics and showed that a wavelength lambda in three-dimensional (3D) space (x, y, z) may appear in Flatland (x, z) as a wave with another wavelength Lambda=lambda/cos alpha. The tilt angle alpha can be modified by a 3D-Spaceland individual, who then is able to influence the 2D optics in a way that must appear to be magical to 2D-Flatland individuals-in the spirit of E. A. Abbott's science fiction story of 1884 [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)]. Here we show how the light from a white source can be perceived in Flatland as perfectly monochromatic, so diffraction with white light will be free of color blurring and the contrast of interference fringes can be 100%. The basic considerations for perfectly achromatic diffraction are presented, along with experimental illustration of Talbot self-imaging performed with broadband illumination.

In "Flatland optics: fundamentals" [J. Opt. Soc. Am. A 17, 1755 (2000)] we described the basic principles of two-dimensional (2D) optics and showed that a wavelength lambda in three-dimensional (3D) space (x,y,z) may appear in Flatland (x,z) as a wave with another wavelength, lambda = lambda/cosalpha. The tilt angle alpha can be modified by a 3D (Spaceland) individual who then is able to influence the 2D optics in a way that must appear to be magical to 2D Flatland individuals-in the spirit of E. A. Abbott's science fiction story [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)] of 1884. We now want to establish the reality or objectivity of the 2D wavelength lambda by some basic experiments similar to those that demonstrated roughly 200 years ago the wave nature of light. Specifically, we describe how to measure the 2D wavelength lambda by mean of five different arrangements that involve Young's biprism configuration, Talbot's self-imaging effect, measuring the focal length of a Fresnel zone plate, and letting light be diffracted by a double slit and by a grating. We also performed experiments with most of these arrangements. The results reveal that the theoretical wavelength, as predicted by our Flatland optics theory, does indeed coincide with the wavelength lambda as measured by Flatland experiments. Finally, we present an alternative way to understand Flatland optics in the spatial frequency domains of Flatland and Spaceland.

A unified mathematical formulation for designing and analyzing even the most general optical processor is presented. It exploits the Wigner distribution function to characterize the illumination, the input, the inherent filter, and the output results. To characterize the propagation of the light through the optical processor setup, we exploit the Wigner matrix formalism, which is appealing because it allows simple geometric analysis. The Wigner distribution function was extended to include illumination of arbitrary coherence so that processors using either coherent light or partially coherent light can be designed and analyzed with the same Wigner formalism. The basic principles, design, and analysis of the imaging and Fourier-transform operations and use of the Wigner formalism to evaluate the performance and tolerances of optical processors are presented.

Superresolution permits fine details to be observed that cannot be resolved by standard optical instruments. We demonstrate that superresolution is nothing more than an adaptation of degrees of freedom converted from the spatial domain to some other domains and vice versa. The tool used for the required conversion and adaptation is the Wigner chart. In addition, we show that the Wigner chart may be naturally applied for analyzing and understanding complex optical setups used for obtaining superresolution.

"Flatland" is the title of a 120-year-old science fiction story. It describes the life of creatures living in a two-dimensional (2D) Flatland. A superior creature living in the three-dimensional (3D) spaceland, as we do, can easily inspect, for example, the inside of a Flatland house, as well as the content of a flat man's stomach without leaving any trace. Furthermore, the 3D person has supernatural powers that enable him to change the laws of physics in Flatland. We present here the concept of a 2D Flatland optics with one transversal coordinate x and one longitudinal coordinate z. The other transversal coordinate y allows total inspection of Flatland optics, and the freedom to change the wavelength, without using something like nonlinear optics or a Doppler shift. Monochromatic 3D light can be converted reversibly into polychromatic 2D light. A large variety of 2D systems and 2D effects will be presented here and in follow-up contributions. An epilogue faces the question, how "real" is Flatland optics?

New optical configurations for performing general coordinate transformation operations of shear, rotation, and their combination are presented. These configurations consist of refractive spherical and cylindrical lenses that are readily available. Typically, high-resolution imagery can be obtained, depending on the size of the input object, the illumination wavelength, and the f-number of the lenses. Basic and more general configurations are presented, along with experimental results clearly showing image shearing, rotation, and a combination of these with high-quality output imagery.

Optical correlation, or matched filtering, can now be applied more widely than before, because the light is now allowed to be totally incoherent, spatially and spectrally. Two such correlators were demonstrated recently. Their state of chromatic correction can be called achromatic, since the scaling error has two zero crossings within the visible range of wavelengths. We present a new apochromatic correlator, in which the scaling error has three zero crossings. The maximum error and the rms error are reduced by a factor of 5. Our apochromatic correlator is composed of two highly dispersive heavy flint lenses that are in contact with two diffractive lenses and two chromatic corrected refractive lenses. The uncommon combination of flint dispersion and diffractive dispersion enabled us to achieve apochromatic correction of the scaling factor of the correlator.

An optical correlator that can operate with totally incoherent light is presented. Such a correlator can be designed to compensate completely for the inherent chromatic aberrations by resorting to elements with specialized, possibly impractical, dispersion characteristics. Nevertheless, a practical configuration that exploits available achromatic lenses and Fresnel zone plates was designed, built, and tested experimentally. The results reveal that detectable correlation peaks can be obtained with totally incoherent white light. The designs, experimental procedures, and results are presented.

Many photographic systems are located on moving platforms. Because of mechanical vibrations that exist on those platforms, the photographed image is distorted. To avoid those distortions requires that very complicated mechanical attenuators be used. We suggest an optoelectronic image-processing algorithm to overcome the distortions caused by such vibrations. The algorithm for different types of vibration is reformulated in an exact manner, as opposed to the reformulations of previously known calculations. Furthermore, new types of vibration are explored. The algorithm is fully computerized and thus avoids the need for attenuating devices.

We report on a simple, spatially incoherent, wavelength-independent imaging system that, in contrast to the conventional case, exhibits a dispersion-compensated point-spread function. Our hybrid (diffractive-refractive) three-lens imaging configuration thus acts as an all-incoherent dispersion-compensated optical irradiance correlator. So the optical arrangement is well adapted to processing color information (both spatially and temporally incoherent) under natural illumination.

Spatial coherence plays a major role in characterizing quasi-monochromatic, partially coherent, optical signals. Here a fairly simple system for synthesizing special cases of spatial coherence is proposed. The special cases, called hybrid, include the case of total coherence in the x direction and, simultaneously, total incoherence in the y direction. The optical setup is based on a quasi-monochromatic, spatially coherent light source, such as a laser, and a simple moving optical element.

The mutual intensity function plays a major role in characterizing quasi-monochromatic, partially coherent optical signals. We propose to use the mutual intensity as a carrier of information to avoid speckle noise in coherent illumination systems and to permit the use of complex functions that are prohibited spatially incoherent sources. To do this we require methods for encoding the information as a coherence function. An optical system for synthesizing a beam with a given mutual intensity function is proposed. The optical system permits the synthesis of any desired mutual intensity function. The illumination is supplied by a quasi-monochromatic, spatially incoherent source. Experimental results demonstrate the performance of this system for several cases.

The mutual-intensity function plays a major role in characterizing quasi-monochromatic, partially coherent optical signals. We demonstrate an optical system for displaying the mutual intensity of a one-dimensional input beam. The experimental system is based on the fact that the mutual intensity of a signal can be expressed as the ensemble averaging of a cross-correlation operation between two related optical signals. The setup consists of a Sagnac interferometer followed by an optoelectronic joint transform correlator. Experimental results demonstrate the capabilities of the mutual-intensity analyzer.

A system for superresolution of temporally restricted objects allows one to transmit ultrawideband optical signals (objects) through a narrow aperture. Such a system was proposed, improved, and implemented for one- and two-dimensional objects. Implementing frequency-plane enhancement in this system provides the ability of spatial filtering of ultrawideband objects. A method for spatial filtering in a system for superresolution of temporally restricted objects by use of a generalized Dammann grating is suggested. An experimental demonstration of high-frequency enhancement (high-pass filtering) of one-dimensional objects is presented.

The expanding use of optical communication by means of optical fibers and the situation of drastically increasing amounts of data to be transmitted urge the exploration of novel systems permitting the transmission of large amounts of spatial information by fiber with smaller spatial resolution. An optical encoding and decoding system is suggested for transmitting one- or two-dimensional images by means of a single-mode fiber. The superresolving system is based on wavelength multiplexing of the input spatial information, which is achieved with diffractive optical elements. Preliminary experimental results demonstrate the capabilities of the suggested method for the one- and two-dimensional cases.

We combine two concepts: the joint transform correlator and the fractional Fourier transform. This combination is almost as convenient experimentally as the classical joint transform correlator. As a processor it is as versatile as the standard fractional Fourier correlator.

We show how the focus longitudinal error z can be deduced from the observed image. This deduction requires some digital image processing. Once the focus error z is known, one may compensate for it either by shifting the lens or by deblurring the image digitally. We have studied systems with a circular or with a rectangular aperture. In the former case, the defocus causes some dark rings in the frequency spectrum of the image. With a rectangular aperture, the defocus creates a set of straight, orthogonal dark lines. These dark zones provide a quantitative diagnosis of the focus error. Some experimental evidence supports our concept.

Objects that vary slowly over time may be superresolved by use of moving gratings. A system of this kind was proposed three decades ago. However, such a system creates some distortion of the spectral response of the resolved objects, and it achieves superresolution in only one dimension. We propose an enhanced method based on Dammann gratings instead of regular gratings for achieving two-dimensional superresolution. The modified approach achieves results with undistorted output and relatively high light efficiency, and it is effective for both coherent and incoherent illumination. Preliminary results demonstrate the feasibility of the new approach.

The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is pi-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.

The auto triple correlation has several fundamental advantages over the ordinary autocorrelation of second order. We present an optoelectronic processor for the computation of the auto triple correlation.

Objects that temporally vary slowly may be superresolved by use of moving gratings. A system of this kind had been proposed three decades ago. However, it provides some distortion of the spectral response of the resolved object. In this project, an enhanced method based on Dammann gratings instead of regular gratings is suggested. The modified approach achieves results with an undistorted output and relatively high light efficiency, and it is effective for both coherent and incoherent light. Experimental results are provided for demonstrating the ability of the new approach.

A novel compact holographic crossbar architecture based on planar optics is presented. It consists of a pair of identical planar holographic elements, a two-dimensional array (N x N) transmission mask, a one-dimensional array (N) of input light sources, and a one-dimensional array (N) of detectors. Each planar element contains two cylindrical holographic lenses, both of which are recorded on a single glass substrate. The design of the overall compact configuration is presented along with experimental results.

Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon-Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon-Wigner display has been demonstrated experimentally.

We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are combined into a fractional Hilbert transform, which is characterized by two parameters. Computer simulations are presented.

The fractional Fourier transform is redefined for working with incoherent light. As a real transformation, the incoherent fractional Fourier transform overcomes coherent system disadvantages such as the speckle effect and the need for incoherent-coherent conversion. It also might have some applications for digital image and signal processing owing to its decreased computing complexity. An incoherent optical implementation of the new transform based on the shearing interferometer is suggested. Laboratory experimental results are given.

The fractional Fourier transform has been used in optics so far for wave propagation and for signal processing. Now we show that this new transform can also be helpful for lens design, especially for specifying a lens cascade.

Several aspects of graphic codes of computer-generated holograms are discussed. This is in contrast to the algorithmic studies that have dominated research in the field of computer holography in recent years. We study the graphic cells of binary cell-oriented holograms under the aspects of specificfabrication and performance problems such as the pen-width problem and zero-order scattering. Asresult, we present experiments with new or modified cell structures that avoid some of these problems.

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.

We measure mutual intensity in quasi-monochromatic light. Our setup is simple, without a lens. A rotating grating and a one-dimensional receiver are the only pieces of hardware. The action of our system can be understood as that of a mock Lau interferometer.

We describe a new procedure for generating complex amplitude distributions along the optical axis of an optical processor by the use of binary masks in the form of binary angular sectors. This type of binary spatial filter acts as a gray-level apodizer with rotational symmetry. Experimental verifications are included.

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.

The family of fractional Fourier transforms permits presentation of a temporal signal not only as a function of time or as a pure frequency function but also as a mixed time and frequency function with a continuous degree of emphasis on time or on frequency features. We show how it is possible to implement the fractional Fourier transform on time signals by using optoelectronic modulators and optical fibers with suitable dispersion. We also show how a fractional-Fourier-transform-based photonic signal-processing system could be composed.

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.

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.

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.

Image formation of three-dimensional objects suffers from out-of-focus noise if the light is coherent. Nevertheless, it is possible to generate two noiseless image intensities (I(A)and I(B)) at two depth locations by means of a single computer hologram. The phases related to I(A) and I(B) provide design freedom. To accomplish this goal, we use a ping-pong algorithm that bounces back and fourth between the two planes. Our ping-pong algorithm is the fourth member of a family of algorithms. The first member is known as the Gerchberg-Saxton algorithm. Basic considerations and experimental results are presented. Details of the algorithm are explained in Appendix A.

A function that is its own Fourier transform is coined a self-Fourier function. The complete set of self-Fourier functions has been recently defined by Caola [J. Phys. A: Math. Nucl. Gen. 24, L1143-L1144 (1991)]. We study the behavior of a self-Fourier function as a self-imaging function. The space-bandwidth product of these functions is studied. An illustration of a self-Fourier function that was a Wigner distribution function is given. Some experimental results are also presented.

We present an optical setup that converts planar binary curves into two-dimensional amplitude distributions, which are proportional, along one axis, to the Bessel function of order n, whereas along the other axis the order n increases. This Bessel displayer can be used for parallel Bessel transformation of a signal. Experimental verifications are included.

A novel optical free-space mesh-connected bus interconnect network architecture is proposed. A mesh-connected bus [IEEE Trans. Comput. C-30, 264-273 (1981)] is known to have the capability of interconnecting, with a three-stage switching, N nodes with a power distribution loss proportional to √N and is therefore advantageous for networking a large number, say over 1000, of communicating ports. Based on conventional space-invariant optical components in a compact and efficient geometry, the proposed optical mesh-connected bus system concept can be used to build either free-space optical interconnect links for parallel processing applications or central switching systems for local or global lightwave communication networks. The proposed architecture lends itself to networking under both the wavelength-division multiple access and other multiple-access environments. In this paper, based on the wavelength-division multiple-access environment, various optical system implementation and performance issues are discused and parameters are analyzed. It was found that by use of a reasonably compact three-dimensional free-space volume, more than 100,000 dispersion-limited communication nodes at a uniform channel spacing of 0.75 nm can be linked with a moderate power distribution loss of 28 dB. Some preliminary optical wavelength-division multiple-access mesh-connected bus experiments based on a 27 × 27 panchromatic optical source array were performed to confirm the operational principle of the proposed concept.

The problems of spatial diffraction and optical dispersion in the time domain was extended by analogy to optical ultrafast pulse filtering. An analog setup in the 4-f spatial-filter configuration is proposed, and several applications such as the temporal signal convolver or correlator and the joint transform processor are suggested, followed by ways to generate the temporal filter. The implementation aspects and an example of the speed superiority of this approach are also discussed.

The incoherent optical correlator was extended to perform rotation-invariant pattern recognition by using a set of two real-positive filters presenting a single-order circular harmonic function.

The job of an array illuminator is to provide an array of optical gates or smart pixels with photon power or with synchronous clock signals. So far it has been common to take the power from one big laser and distribute it to perhaps a million gates. An obvious alternative is to assign one private small source to each gate. We favor an in-between approach: a few medium-size sources share the job of providing photons. This hybrid approach has several advantages, such as better homogeneity, less coherent noise, and a distributed risk of source failure. We propose several setups and present some experimental results. Our concept calls for an array of incoherent point sources. We simulate such an array experimentally with a single source, which is virtually expanded into a source array by grating diffraction. Ordinarily these virtual sources are mutually coherent, which is undesirable for our aims. We destroy the mutual coherence by moving the grating during the photographic recording of the output array.

We demonstrate a one-dimensional optical perfect-shuffle interconnect system for temporal signals. A temporal imaging configuration with a segmented time lens is used.