We describe a method that we believe will for the first time allow diffraction-limited imaging through ground-level turbulence with large apertures and at large distances (e.g., 1mm resolution at 1km and at a wavelength of 1μm). The key lies in collecting image data in the spatial frequency domain via the method of Fourier telescopy and in taking suitable time averages of the Fourier telescopy signal magnitude and phase. The method requires active illumination of the target with laser light, and the time averages required will likely be of many seconds duration, if not minutes. The scheme will thus not be suitable for time-varying scenes.

Building on the work of Goodman and Lawrence [1], we have extended digital holographic imaging to gigapixel scales with 2-D aperture synthesis. Sub-pixel registration algorithms were required to mosaic together thousands of arrays of data, and phase-error correction algorithms were required to correct for system instabilities.

In this talk we present a series of illustrative topics in Fourier Optics that are proving valuable in the design of EDOF camera systems. They are at the level of final examination problems that have been made solvable by a student or professoi having studied from one of Joseph W. Goodman's books---our tribute for his 75fr year. As time permits, four illustrative topics are l) Electromagnetic waves and Fourier optics;2) The perfect lens; 3) Connection between phase delay and radially varying focal length in an asphere and 4) tailored EDOF designs.

Goodman's popular linear systems formulation of scalar diffraction theory includes a paraxial (small angle) approximation that severely limits the conditions under which this elegant Fourier treatment can be applied. In this paper a generalized linear systems formulation of non-paraxial scalar diffraction theory will be discussed. Diffracted radiance (not intensity or irradiance) is shown to be shift-invariant with respect to changes in incident angle only when modeled as a function of the direction cosines of the propagation vectors of the usual angular spectrum of plane waves. This revelation greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate diffraction calculations. Non-paraxial diffraction grating behavior (including the Woods anomaly phenomenon) and wide-angle surface scattering effects for moderately rough surfaces at large incident and scattered angles are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this extension to Goodman's Fourier optics analysis. The resulting generalized surface scatter theory has been shown to be valid for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckman-Kirchhoff theory. This has enabled the development of a complete linear systems formulation of image quality, including not only diffraction effects and geometrical aberrations from residual optical design errors, but surface scatter effects from residual optical fabrication errors as well. Surface scatter effects can thus be balanced against optical design errors, allowing the derivation of optical fabrication tolerances during the design phase of a project.

The white-light compensated rotational shear interferometer (coherence interferometer) was developed in an effort to study the spatial frequency content of passively illuminated white-light scenes in real-time and to image sources of astronomical interest at high spatial frequencies through atmospheric turbulence. This work was inspired by Professor Goodman's studies of the image formation properties of coherent (laser) illuminated transparencies. We discovered that real-time image processing is possible using white-light interferometry. The concept of a quasimonoplanatic approximation is introduced as a parallel to the quasimonochromatic approximation needed to describe the theory of Fourier transform spectrometers. This paper describes the coherence interferometer and reviews its image formation properties under the conditions of quasimonoplanacity and describes its development and its applications to physical optics, optical processing and astrophysics including the search for exoplanets.

We review two techniques of unconventional holography, coherence holography and photon-correlation holography, which we recently proposed and experimentally demonstrated. We will emphasize the importance of noticing mathematical analogies in optics and physical phenomena, which give insights into the methodology for developing new techniques.

Since VanderLugt's famous 1964 paper showing that complex valued filters for optical matched filters could be made, an extremely large number of papers have been written - some by me and some by Professor Goodman, and many more by others. Yet optical Fourier transform filtering is almost never used for anything but university research. The reasons are fairly well known but seldom stated. We have found one new approach to pattern recognition that solves most of the problems and introduces a totally new and very useful thing that can be done with Fourier filtering.

In this presentation we discuss a new type of optical microscope that is capable of obtaining spectral and spatial information from biological tissue samples. The current system uses multiplexed volume holograms to probe multiple depths of a tissue sample without the need for scanning. This greatly simplifies the instrument and should allow it to be adapted for laproscopic applications. The technique can be combined with fluorescent dye markers to identify cancerous tissue.

We review the history of scalar diffraction theory from the classical approach of Kirchkoff, Rayleigh and Sommerfeld, to the modern ideas introduced by Duffieux that eventually led to the use of distribution theory by Arsac , and finally to the derivation of the expressions using quantum mechanics and relativity. The latter exploits the invariant properties of the energy-momentum four-vector of light.

Electromagnetic waves carry energy, linear momentum, and angular momentum. When light (or other electromagnetic radiation) interacts with material media, both energy and momentum are usually exchanged. The force and torque experienced by material bodies in their interactions with the electromagnetic field are such that the energy as well as the linear and angular momenta of the overall system (i.e., the system of field plus matter) are conserved. Radiation forces are now used routinely to trap and manipulate small objects such as glass or plastic micro-beads and biological cells, to drive micro- and nanomachines, and to contemplate interstellar travel with the aid of solar sails. We discuss the properties of the electromagnetic field that enable such wide-ranging applications.

Optical transforms are represented in computers by their discrete versions. In particular, Fourier optics is represented through Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT). Being discrete representation of the optical Fourier transform, these transforms feature a number of peculiarities that cast a new light on such fundamental properties of the Fourier Transform as sampling theorem and the uncertainty principle. In this paper, we formulate the Discrete Sampling Theorem and the discrete uncertainty principle, demonstrate that discrete signals can be both bandlimited in DFT or DCT domains and have strictly limited support in signal domain and present examples of such "bandlimited/ space-limited" signals that remain to be so for whatever large of their samples.

Speckle was re-discovered after the invention of the laser in 1960. The unpublished 1963 Stanford Electronics Laboratory report by J W Goodman was the first comprehensive derivation of the first and second order statistics of the speckle intensity. This short paper describes how the senior author came to know Professor Goodman through their mutual, and lasting, interest in laser speckle. New results in speckle continue to be discovered, and we briefly describe one of these, the elimination of phase vortices using cascade adaptive optics systems.

This paper leverages many successes of the Fourier theorem in modeling the measurable experimental data in the fields of optical physics while underscoring some of the misinterpretations by focusing on the light-matter interaction processes, which give rise to the measurable data. We are proposing that we need to introduce Interaction Process Mapping Epistemology (IPM-E) to complement the currently successful Measurable Data Modeling Epistemology (MDM-E). We show that IPM-E helps us recognize that EM waves cannot produce interference fringes by themselves as they do not interact with each other. Accordingly we have been missing the NIW-principle (Non-Interaction of Waves). Application of the NIW-principle to Fourier theorem, as applied to optical physics reveals and helps us understand many optical phenomena much better than so far we have understood. We discuss Fourier optics, Fourier transform spectrometry, coherence theory, spectrometry theory and laser mode locking theory and summarize for each case, the deeper understanding that we have been missing by neglecting the NIW-principle.

Optical singularities are localized regions in a light field where one or more of the field parameters, such as phase or polarization, become singular with associated zero intensity. Focused to a small spot, the electromagnetic field around the singularity has interesting characteristics, in particular when it interacts with matter. The light scattered by a material object within the strongly varying optical field around the singularity is extremely sensitive to changes and can be exploited for metrology with high sensitivity and the study of physical processes on a nanometer scale.

We explore the use of coherent optical processors for visualizing phase-space representations, as well as for designing complex amplitude masks, which reduce the impact of focus errors on the modulation transfer function (MTF).

Fundamental Fourier optics is applied to metallic near-field superlens, whose transfer function is computed with the transfer matrix, the Surface Plasmon Polariton (SPP) resonance and the SPP waveguide theory. However, when the object nano-structure consists of feature nano-slits and nano-holes etc, which are as the basic object elements to scatter the light, especially when the objects are metal, the electrical dipoles are induced at the nano-slits and nano-holes by the illuminating light, the space invariance condition can be not respected within the dimension of the nano-meter scale objects, so that the point spread function becomes approximate and the superlens is usually characterized by the image of a two nano-slit pattern. The superlens is designed and optimized based on the transfer function. Improvement in the transfer function can improve significantly the image quality. The real image of the near-field superlens can be computed with numerical simulation using the FDTD method.

In this paper we present technical evolution at Physical Optics Corporation (POC), from Fourier Optics, inspired by Professor Joseph Goodman's classic book: Introduction to Fourier Optics, to recent directions at POC, related to socalled "Integrative Engineering."

The theory of partial coherence has a long and storied history in classical statistical optics. The vast majority of this work addresses fields that are statistically stationary in time, hence their complex envelopes only have phase-insensitive correlations. The quantum optics of squeezed-state generation, however, depends on nonlinear interactions producing baseband field operators with phase-insensitive and phase-sensitive correlations. Utilizing quantum light to enhance imaging has been a topic of considerable current interest, much of it involving biphotons, i.e., streams of entangled-photon pairs. Biphotons have been employed for quantum versions of optical coherence tomography, ghost imaging, holography, and lithography. However, their seemingly quantum features have been mimicked with classical-state light, questioning wherein lies the classical-quantum boundary. We have shown, for the case of Gaussian-state light, that this boundary is intimately connected to the theory of phase-sensitive partial coherence. Here we present that theory, contrasting it with the familiar case of phase-insensitive partial coherence, and use it to elucidate the classical-quantum boundary of ghost imaging. We show, both theoretically and experimentally, that classical phase-sensitive light produces ghost images most closely mimicking those obtained with biphotons, and we derive the spatial resolution, image contrast, and signal-to-noise ratio of a standoff-sensing ghost imager, taking into account target-induced speckle.

Digital holography, Fourier optics and speckle are combined to enable a new direction in drug discovery. Optical coherence imaging (OCI) is a coherence-gated imaging approach that captures dynamic speckle from inside living tissue. The speckle temporal fluctuations arise from internal motions in the biological tissue, and the changes in these motions caused by applying drugs can be captured and quantified using tissue dynamics spectroscopy (TDS). A phenotypic profile of many reference drugs provides a training set that would help classify new compounds that may be candidates as new anti-cancer drugs.

In the mid-1990s when digital photography began to enter the consumer market, Professor Joseph Goodman and I set out to explore how computation would impact the imaging system design. The field of study has since grown to be known as computational photography. In this paper I'll describe some of its recent advances and challenges, and discuss what the future holds.