E. W. Marchand

Radiometry with sources of any state of coherence

The basic laws of radiometry are generalized to fields generated by a two-dimensional stationary source of any state of coherence. Important in this analysis is the concept of the generalized radiance function, introduced by Walther in 1968. The concepts of generalized radiant emittance and of generalized radiant intensity are introduced and it is shown how all these quantities may be expressed in terms of coherence functions of the source. Both the generalized radiance of Walther and the generalized radiant emittance may take on negative values, indicating that these quantities have, in general, a less-direct physical meaning than have the corresponding quantities of traditional radiometry (which presumably represents the incoherent limit of the present theory). The generalized radiant intensity is, however, always found to be non-negative and, just as in the incoherent limit, represents the angular distribution of the energy flux in the far zone.

Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields*

In the first part of this paper, the concept of the angular correlation function of a stationary optical field is introduced. This function characterizes the correlation that exists between the complex amplitudes of any two plane waves in the angular spectrum description of the statistical ensemble that represents the field. Relations between this function and the more commonly known correlation functions are derived. In particular, it is shown that the angular correlation function is essentially the four-dimensional spatial Fourier transform of the cross-spectral density function of the source. The angular correlation function is shown to characterize completely the second-order coherence properties of the far field. An expression for the intensity distribution in the far zone of a field generated by a source of any state of coherence is deduced. Some generalizations of the far-zone form of the Van Cittert–Zernike theorem are also obtained.

Derivation of the Point Spread Function from the Line Spread Function

In dealing with optical imaging systems, it is more feasible experimentally to measure line spread functions than point spread functions. When the intensity distribution is known to possess rotational symmetry, the point spread function can be obtained mathematically from the corresponding line spread function by solving an integral equation. A direct solution of this equation is given which represents a procedure that is simpler for practical use than the usual one involving Fourier transforms.

Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UK−UR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.

Consistent Formulation of Kirchhoff’s Diffraction Theory*

Kirchhoff’s diffraction theory, which is often criticized because of an apparent internal inconsistency, is shown to be a rigorous solution to a certain boundary-value problem that has a clear physical meaning. This new interpretation of Kirchhoff’s theory is a direct consequence of the Rubinowicz theory of the boundary diffraction wave.We argue that these true boundary conditions of Kirchhoff’s theory are physically reasonable for diffraction at an aperture in a black screen whose linear dimensions are large compared with the wavelength. The boundary values which Kirchhoff’s solution takes in the plane of the aperture and in the near zone on the axis for the case of a normally incident plane wave diffracted at a circular aperture are compared with previously published results of experiments with microwaves, and reasonable agreement is found. (Strict agreement cannot be expected since the screens used in the microwave experiments were not black.) Moreover, Kirchhoff’s solution is found to be in closer agreement with the experimental results than the “manifestly consistent” Rayleigh–Sommerfeld theory.We also suggest an extension of the Kirchhoff theory, which might provide a physically reasonable approximation to the solution of the problems of diffraction at an aperture in a black screen whose linear dimensions are of the order of magnitude of or smaller than the wavelength of the light.

Ray Tracing in Gradient-Index Media

A gradient-index element is one in which the refractive index varies appreciably in a layer just below the surface of the glass. The present paper deals with practical methods of tracing rays through such a medium. One method, due to L. Montagnino, lends itself to rapid computer ray tracing. A second method, through restricted to the case of spherical symmetry in the index function, lends itself both to computer tracing and to analytical investigations.

From Line to Point Spread Function: The General Case

Several methods are known for the mathematical conversion from line spread functions to point spread functions in the rotation-symmetric case. The present paper solves this problem without any assumptions about the symmetry of the functions.

Boundary Diffraction Wave in the Domain of the Rayleigh–Kirchhoff Diffraction Theory*

Maggi and Rubinowicz formulated a boundary wave theory of diffraction at an aperture, which may be regarded as developments of early ideas of Young about the nature of diffraction. These investigations were based on the Kirchhoff diffraction theory. In the present paper it is shown that a theory of the boundary diffraction wave may also be formulated by taking as starting point the representation of the field in terms of the Rayleigh diffraction integrals, which are free from a mathematical inconsistency inherent in the Kirchhoff theory. It is shown that the Rayleigh integrals (with physically approximate but mathematically consistent boundary conditions of the Kirchhoff type) may be transformed into the sum of two terms. One term represents the effect of a disturbance which originates in each point of the boundary of the aperture (boundary wave); the other represents the combined effect of disturbances originating in certain special points situated within the aperture. In the special cases when the field incident upon the aperture is plane or spherical, the second term represents precisely the disturbances predicted by geometrical optics, in strict analogy with the classical results of Maggi and Rubinowicz. In these special cases the boundary wave of the present theory differs from the boundary wave of the earlier theories only in the form of an inclination factor.

Ray Tracing in Cylindrical Gradient-Index Media

There are two families of gradient-index functions of cylindrical type for which the differential equations of the rays can be completely integrated without any approximations. One of these has focusing properties within the medium and so can be considered for use in GRIN rods. Both examples can prove useful in the design of conventional types of optical systems involving gradient-index elements.

Diffraction at Small Apertures in Black Screens

In an earlier paper dealing with a consistent formulation of Kirchhoff’s theory, an extension of the theory was proposed which seems to be applicable to treatment of diffraction at very small apertures in black screens. The extension consists of the addition to Kirchhoff’s solution of the effect of waves arising from multiple diffraction at the edge of the aperture. In the present paper, calculations based on this modified theory are presented. Corrections to Kirchhoff’s theory are obtained for the case of a plane wave incident normally on a small circular aperture in a black screen. Appreciable departures from Kirchhoff’s theory are found only when the diameter of the aperture is small compared with the wavelength of the incident wave or when the angles of diffraction are very large. It is also shown that in the asymptotic limit ka→∞ (k is the wave number, a the radius of the aperture) our results are consistent with those obtained on the basis of Keller’s geometrical theory of diffraction.