Summary The maximum visibility of the interferences obtainable from two points in a wave field is defined as their degree of coherence γ. By a simple statistical method general formulae are found for deducing γ from illumination data. For any extended lightsource γ is found equal to the amplitude in a certain diffraction image. It does not change by the use of a condensing lens, but depends only on the aperture of the illuminating cone. These properties are applied to the microscopic observation of objects in transmitted light.

The concept of degree of coherence and its application to optical problems

A macroscopic theory of interference and diffraction of light in stationary fields produced by finite sources which emit light within a finite spectral range is formulated. It is shown that a generalized Huygens principle may be obtained for such fields, which involves only observable quantities. The generalized Huygens principle expresses the intensity at a typical point of the field in terms of an integral taken twice independently over an arbitrary surface, the integral involving the intensity distribution over the surface and the values of a certain correlation factor, which is found to be the ‘degree of coherence’ previously introduced by Zemike. Next it is shown that under fairly general conditions, this correlation factor is essentially the normalized integral over the source of the Fourier (frequency) transform of the spectral intensity function of the source, and that it may be determined from simple interference experiments. Further, it is shown that in regions where geometrical optics is a valid approximation, the coherence factor itself then obeys a simple geometrical law of propagation. Several results on partially coherent fields, established previously by Van Cittert, Zernike, Hopkins and Rogers, follow as special cases from these theorems. The results have a bearing on many optical problems and can also be applied in in vestigations concerned with other types of radiation.

A macroscopic theory of interference and diffraction of light from finite sources. I, Fields with a narrow spectral range

The radiometric theory of spatial coherence is presented with special attention to the validity of the approximations on which it is based. A new definition of the transverse coherence area is introduced and shown to be in general agreement with earlier definitions. In free-space propagation the product of the transverse coherence area and the intensity is shown to be constant along rectilinear rays, and, for radiation from uniform Lambert sources, a well-known paraxial formula for the transverse coherence area is extended to the extraparaxial domain. A decrease of the spatial coherence in free-space propagation takes place in regions with an increase of the intensity. For imaging systems this occurs in a finite part of image space whenever a real image of a diffusely radiating, extended object is formed at a finite distance.

Radiometric theory of spatial coherence in free-space propagation

Complex amplitude and polarization distribution of an optical beam plays a dominant role in shaping the focused structure of the beam. It is therefore possible to engineer the focal spot using the pupil function manipulation. Helical phase structure arising due to phase singularity in the wave front plays an important role in shaping the focal spot. Tight focusing of an optical beam produces intensity distribution in the focal volume different from the well-known results based on scalar theory, and polarization distribution shows space variant characteristics. In the present paper, roles of the amplitude-, phase-, and polarization distribution on the tightly focused structure of the optical beams are reviewed. Impact of the helical phase structure in the pupil function engineering and subsequently on the focused structure is discussed with special reference to the authors' investigations at IIT Delhi. Certain applications in which tight focusing is desired are briefly discussed. Some miscellaneous investigations are also mentioned.

https://web.iitd.ac.in/~psenthil/Conferences/InvertJSTP1-2009.pdf

Tight Focusing of Optical Beams; A Review - Part 1

It is shown that a 'phase-coherence factor' may be defined in a manner which leads, without recourse to explicit statistical analysis, to the theorems established by van Cittert (I934) and Zernike (I938) for analogous factors. An invalid approximation made in their calculations of the phase-coherence factor for a plane illuminated directly by a source is corrected. The new treatment is applied to the theory of Young's experiment, the stellar interferometer, and illumination in the microscope. The phase-coherence factor defined here enables a general theory of the formation of optical images to be formulated. Further, it is shown that the diameter of the area of coherence on a plane illuminated by a source of angular radius α is given by d = 0.16λ/N sin α, where N is the refractive index of the intervening medium.

The concept of partial coherence in optics

The analogy between the van Cittert-Zernike theorem and diffraction in focal regions is discussed. For wavefields that can be approximated by a generalized quasihomogeneous wave model, the spatial coherence is described by a generalized van Cittert-Zernike theorem even close to the source. The main difference between the classical and the generalized theorem is that the source intensity in the former is replaced by the source radiance in the direction of the observation point in the latter. The spatial coherence is related to the source radiance by a modified Debye integral, which implies that the wave energy is propagated according to the laws of classical radiometry. No assumption of source incoherence is involved, and the results apply to radiation from both primary sources and secondary sources like rough scattering objects or illuminated apertures in optical systems.

J J Stamnes

Papers

Van Cittert-Zernike theorem for quasihomogeneous wavefields and the modified Debye integral