G. Toraldo di Francia

Degrees of Freedom of an Image

The maximum number of degrees of freedom of an image, which is usually derived by a rough application of the sampling theorem, has a precise mathematical meaning. This is established by applying the theory of prolate spheroidal functions, as discussed by Slepian and Pollack and other authors. Both coherent and incoherent illumination are considered. The number of degrees of freedom of the image can be substantially different in both cases. A dramatic example is represented by the thin-ring pupil, which is virtually unidimensional in the former case and two dimensional in the latter case.

Resolving Power and Information

The degrees of freedom of an image formed by any real instrument are only a finite number, while those those of the object are an infinite number. Several different objects may correspond to the same image.It is shown that in the case of coherent illumination a large class of objects corresponding to a given image can be found very easily. Two-point resolution is impossible unless the observer has a priori an infinite amount of information about the object.

ON THE THEORY OF SOME CERENKOVIAN EFFECTS

The field generated by a charged article in uniform straight motion is expanded into a set of evanescent waves. The expansion is valid in any half-space with no points in common with the path of the particle. The evanescent waves may impinge on the surface of an optical diffraction grating and be diffracted. Some of the diffracted waves turn out to be ordinary plane waves, which carry energy away from the grating. It is possible in this way to explain the Smith and Purcell effect and to derive some quantitative conclusions.

Geodesic Lenses for Guided Optical Waves

Two-dimensional lenses are essential elements in integrated optics. As an alternative to the techniques already suggested and tested to obtain such optical systems, a different method is reported that consists of extending to thin film optics the principle of configuration lenses already studied for application to microwave antennas. The working principle of such lenses is that the propagation occurs along a curved surface in a two-dimensional Riemann space. The rays are the geodesics of the surface. Two optical geodesic lenses having planar input and output have been constructed and tested. They can be easily inserted in planar film circuits. The advantage of these optical systems is that the substrate can be prepared in advance with the accuracy of glass optics. Then the dielectric film can be deposited with the same method used for planar components.

Individuals, Kinds and Names in Physics

It has been recognised for a long time that language plays a crucial role in science. The neopositivist philosophers were right when they put special emphasis on the analysis of the language of science. But is this analysis sufficient to clarify all the problems of the philosphy of science? Can science be reduced to a pure linguistic game — as Wittgenstein would put it — or is there something else? Surely science talks of something. Accordingly, the scientist must be aware of the relation existing between the language he uses and the things he is talking about. But here ends the ‘game’, if ever there was one, and formidable problems arise.

Flat-Roof Resonators

A microwave or laser resonator with flat-roof mirrors is discussed from a theoretical point of view. When the angle of the roof is extremely small, on the order of 10−3 rad for microwaves or 10−3–10−6 rad for optics, this resonator has some features in common with the confocal resonator.

Electromagnetic cross-section of a small circular disc with unidirectional conductivity

SummaryThe scattering of an electromagnetic field by a small circular disc is investigated with the assumption that the disc can conduct current in only one direction. The analysis follows closely that given byBethe in a similar case. Although Bethe’s theory is open to criticism and leads to some wrong results, the present discussion appears to the author to be perfectly correct. Both the scattering cross-section and the angular momentum cross-section are evaluated. This is sufficient for explaining why the experimental measurement of the torque exerted upon the screen is in poor quantitative agreement with the theory based on the geometric cross-section.RiassuntoViene studiata la diffrazione di un’onda elettromagnetica da parte di un disco sottile, di piccolo diametro, perfettamente conduttore in una direzione e isolante nella direzione ad essa ortogonale. Si applica un metodo già seguito daBethe nel caso di un foro circolare in uno schermo perfettamente conduttore. Viene calcolata la sezione efficace di assorbimento di momento angolare. Viene dimostrato un teorema generate che mette in relazione questa sezione efficace con la sezione di scattering.

Identity Questions from Quantum Theory

To what extent does quantum mechanics give rise to violations of Leibniz’ theory of identity? This is a highly controversial problem. With different arguments, the question has been positively answered by Margenau, van Fraassen, Mittelstaedt, da Costa, Krause and other scholars. A negative answer, instead, has been defended by Aerts, Piron, Costantini, Garibaldi.

The Inverse Scattering Problem in the Born Approximation and the Number of Degrees of Freedom

We formulate, in the framework of the Born approximation, the problem of reconstructing the refractive index distribution of a three-dimensional object. We discuss the lack of uniqueness and stability of this problem and we propose a method for determining the number of degrees of freedom. The analysis, concerning the latter point, is illustrated with the aid of numerical computations. A reconstruction procedure is also briefly discussed.

Many-element lasers

SummaryTheoretical and experimental results are reported, obtained with a solid-state laser consisting of many active elements in series. A statistical treatment is developed to derive the oscillation conditions. Asymptotic expressions are given of the threshold gain per unit length holding either for a very large number of elements or for inner surface reflectivities close to 1. It turns out that lossless reflections at the inner surfaces can increase the cavityQ with respect to a one-element laser of the same active length. Preliminary experiments with five and tenelement lasers demonstrate the mode selection property of the manyelement laser and show the tendency of the system to a «quasicontinuous» emission. The possibility of obtaining some optical correction of the defects inherent in the solid material has also been demonstrated.RiassuntoSi descrivono i risultati sia teorici che sperimentali, ottenuti con un laser allo stato solido composto da più elementi attivi in serie. Per ricavare le condizioni di oscillazione è stato usato un trattamento statistico. Si danno delle espressioni asintotiche per il guadagno per unità di lunghezza alla soglia, che valgono o quando il numero degli elementi è molto grande, o quando la riflettivita delle superficie interne tende a 1. Risulta che riflessioni senza perdite alle superficie interne possono aumentare ilQ della cavita rispetto a un laser composto da un solo elemento avente la stessa lunghezza di materiale attivo. Risultati sperimentali preliminari su laser composti da cinque o dieci elementi dimostrano che laser a più elementi possono essere utilmente impiegati per la selezione dei modi. Si mostra inoltre che tali sistemi tendono all’emissione «quasi continua». Un altro vantaggio dei laser a più elementi attivi, rispetto a quelli composti da un unico elemento, è la possibilità di correggere in maniera semplice i difetti ottici del materiale solido.