Van Cittert Zernike Research Articles

Measurement of two-dimensional optical system MTF by computation of second order speckle statistics

An interferometric approach to the calculation of the two-dimensional MTF of an optical system is proposed. The technique is based on the computation of the second-order spatio-temporal statistics of a fluctuating speckle pattern. The theorem of Van Cittert-Zernike is invoked to relate the speckle, due to the illumination of a perfect diffuser by the point spread function of an optical system, to the two-dimensional MTF of the system. The computed MTF is displayed in the form of a contour map and can also be represented in the conventional form of a one-dimensional vertical cut. Preliminary measurements have yielded qualitatively useful results and clearly illustrate the suitability of two-dimensional maps for the detection of transfer function anisotropies.

An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals

The dependence of the far-field coherence and radiant intensity of light scattered from a liquid crystal on the correlation length for the fluctuations of molecules in the crystal are analyzed theoretically. The results are useful for understanding data obtained in scattering experiments that study the hydrodynamic properties of liquid crystals excited into internal flow by a dc electric field or by application of sound vibrations. It is found that if the molecular correlation length is small relative to the dimensions of the illuminated volume, the scattered field behaves according to the van Cittert–Zernike theorem, and the motion of the molecules is better studied by measurement of the radiant intensity. If instead the molecular correlation length is much longer than the dimensions of the illuminated volume, the van Cittert–Zernike theorem does not apply, and the spatial correlation properties of the motion of the molecules can be studied by measurement of the far-field coherence.

Partially Coherent Wave Propagation and Generalized Van Cittert-Zernike Theorem

The exact theory of partially coherent wave propagetion from a non-singular curved boundary surface is developed by solving the boundary value problem for the wave equation satisfied by the mutual coherence function. Various cases are considered to obtain rigorous or approximate formulas for the mutual coherence functions some of which have already been derived by the Rayleigh or Fresnel diffraction formula. A generalized van Cittert-Zernike theorem for the light emitted from a source with the curved surface is also obtained.

Evanescent waves and the van Cittert Zernike theorem in cylindrical geometry

The cylindrical angular spectrum of the wavefield is introduced. In this representation the field consists of homogeneous as well as evanescent waves. The representation is applied to propagation problems and an analogue of van Cittert Zernike theorem is obtained in cylindrical geometry.

Van Cittert–Zernike Theorem and Lambert’s Law

Van Cittert–Zernike Theorem and Lambert’s Law

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.

Conditions for Microdensitometer Linearity*

A simple microdensitometer system is analyzed using the principles and analytical techniques of the theory of partial coherence. A specification of the physical conditions under which the instrument is linear is obtained, for incoherent illumination. The illuminating mutual intensity is then generalized, by the Van Cittert–Zernike theorem, and conditions on the partial coherence of the preslit illumination necessary for effective incoherence are determined. The new conditions determine a mode of linear operation for the microdensitometer for which an optical transfer function may be unambiguously defined.

Spatial Coherence in Periodic Systems*

The van Cittert–Zernike theorem states that quasimonochromatic light from a spatially incoherent source, propagating in free space, will increase in spatial coherence. Wolf has shown that quasimonochromatic light, which is initially only partially space coherent or even incoherent, may become coherent upon propagating through a periodic system of apertures. In this paper we employ the mode theory of optical resonators and transmission systems to obtain the results of the van Cittert–Zernike theorem and to study the rapidity with which coherence builds up in periodic systems of apertures and confocal lenses. A perturbation calculation for modes in circular geometries with small Fresnel numbers is included.

Generalized Source and the van Cittert–Zernike Theorem: A Study of the Spatial Coherence Required for Interferometry

If two wavetrains are to exhibit interference at some point in space they must be at least partially coherent, one with the other, at this point. This partial coherence between the two interfering wavetrains can, in most interferometers, easily be related to the mutual coherence between separated points in the single wavetrain from which the two interfering wavetrains are derived. The mutual coherence in this wave-train depends upon the distribution of the source which produces it in a manner given by the van Cittert-Zernike theorem, which in most practical cases can be economically stated in terms of source intensity as a function of direction from the point under consideration. The mutual intensity is the three-dimensional Fourier transform of this generalized source. Knowing this, we can, within the limits set by fact that the generalized source is a distribution spread upon the unit sphere, design to order source distributions to produce the mutual coherence necessary for different kinds of interferometry. Examples are given; the case of an infinitely thin annular source is worked out in detail.

Microwave Measurements of Partially Coherent Fields

This paper describes a new technique for producing an extended incoherent microwave source and its use in generating calibrated partially coherent fields. The source consists of a planar array of commercial fluorescent lamps for which the thermal radiation corresponds to electron temperatures of over 10,000°K. The technique is significant in that, heretofore, laboratory measurements of radiation from extended microwave sources have been restricted principally to coherent fields, even though many radio propagation studies involve partially coherent fields. Radiometric measurements were made of the coherence of the extended microwave source fields and found to agree well with the van Cittert–Zernike theorem. Diffraction patterns of circular apertures illuminated by partially coherent fields were also measured and showed similar good agreement with theory.