In applied science, resolution has always been, and still is, an important issue. Since it is not unambiguously defined, it is interpreted in many ways. In this paper, which reviews the concept of optical resolution, a number of these interpretations are discussed. A discussion of resolution has to be preceded by a discussion of what is actually understood by an ‘‘optical image.’’ In a remarkable paper, Ronchi1 distinguished ethereal images, calculated images, and detected images. The term ethereal image was introduced only to represent the physical nature of the imaging phenomenon. As is customary in science in general, attempts have been made to give a mathematical representation of this phenomenon, both geometrically and algebraically. According to Ronchi, the images that have thus been calculated are mere mathematical constructions and should therefore be called calculated images. In the past, many approaches to the concept of resolution concerned these calculated images. This resulted in the so-called classical resolution criteria, such as Rayleigh’s criterion and the associated reciprocal bandwidth of the image. These criteria provide resolution limits that are determined solely by the calculated shape of the point-spread function associated with the imaging aperture and the wavelength of the light. From now on, they will be called classical resolution limits. Calculated images are by their very nature exactly describable by a mathematical model and thus noise free. Such images do not occur in practice. Therefore Ronchi stated that the resolution of detected images is much more important than the classical resolution, since it provides practical information about the imaging system employed. Hence one should consider primarily the resolution of detected images instead of that of calculated images. This means a necessary introduction of some new quantities of interest, such as the energy of the source and the sensitivity properties of the detector. Since Ronchi’s paper, further research on resolution— concerning detected images instead of calculated ones— has shown that in the end, resolution is limited by systematic and random errors resulting in an inadequacy of the description of the observations by the mathematical model chosen. This important conclusion was independently drawn by many researchers who were approaching the concept of resolution from different points of view, which will be discussed in the subsequent sections.

Resolution: a survey

The concept of a quasihomogeneous source is introduced. Unlike a source that is strictly homogeneous in its statistical properties, a quasihomogeneous source may be finite. Many physical sources, both primary and secondary ones, are adequately approximated by this model. Coherence and radiometric properties of light generated by such sources (assumed, for simplicity, to be planar) are discussed and an important reciprocity relation is shown to exist between light in the far zone and in the source plane. This relation implies that the degree of coherence in the far zone is given by the classic form of the van Cittert-Zernike theorem, even though the source may have a high degree of spatial coherence over arbitrarily large areas. The reciprocity relation also provides a generalization of a recently derived result that expresses the angular dependence of the radiant intensity in terms of the degree of spatial coherence of light in the source plane. The dependence of all the basic radiometric quantities on the distribution of the optical intensity across the source and on the degree of spatial coherence of the light emerging from the source is discussed and is illustrated, for some typical sources, by computed curves.

Coherence and radiometry with quasihomogeneous planar sources

The second-order coherent properties of Gaussian-Schell beams propagating through turbulent media as described by the Kolmogorov turbulence model are studied and some theoretical results are presented in this paper.

Propagation of a Gaussian-Schell beam through turbulent media

The theoretical relationship between the coherence length and phase-front randomness is analysed using a Gaussian-Schell source and beam model for light propagating through a turbulent atmosphere. Computer results are given which show that partially coherent sources are less influenced by atmospheric turbulence than are coherent sources.

Coherence length of a Gaussian-Schell beam and atmospheric turbulence

Bibliography on higher-order statistics

Superresolution by Holography

Merits of symmetric scanning for detection of coherence function in incoherent imaging system

A new method for tomographic reconstruction of a two‐dimensional (2‐D) velocity field that consists of a set of curved streams with corresponding constant curvatures along each stream with statistically independent fluctuations is proposed. In this method, five ultrasonic beams, which are arranged in proper positions and directions, are used as the line detectors and the fifth‐order correlation analysis is applied to the detected data. In this case, the information about each stream is transformed into a corresponding peak in the fifth‐order correlation space. Hence, the velocity field can be reconstructed by finding flow parameters from the measured correlation data in connection with the geometries of the measuring system. The usefulness of this method is confirmed by computer simulations as well as experiments that are carried out by using a 4.5‐MHz ultrasonic wave system.

Flow velocity field tomography using multiple ultrasonic beam detectors and high‐order correlation analysis

Superresolution in incoherent imaging system

A new method to detect the stress conditions inside metals such as the limiting stress value of elastic deformation is proposed. It is based on the nonlinear dependence of ultrasound velocity on stress. First, the nonlinear dependence of sound velocity on stress is observed in an Al alloy. Then equipment which consists of a wave generator for electromagnetic monopole stress perturbing pumping and a 50 MHz ultrasonic sound velocity change detecting unit is presented. In this system the stress perturbation is applied at the desired measuring region in the metal along with the probe wave and changes in the wave are observed. Finally, it is applied for the estimation of the elastic deformation limit of Al alloy and the usefulness of the method is confirmed.

Takuso Sato

Papers

Superresolution by Holography

Merits of symmetric scanning for detection of coherence function in incoherent imaging system

Flow velocity field tomography using multiple ultrasonic beam detectors and high‐order correlation analysis

Superresolution in incoherent imaging system

Estimation of the stress state inside metals using stress perturbing waves and probe waves