The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the diffraction theory of optical images

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

Bibliographic Review on “Studies on Partial Coherence”

The detection and resolution of an object, and an estimation of its parameters, such as position and radius, have been treated by signal detection theory. The object was viewed through an optical system that had a circular diaphragm and which suffered from spherical aberration and defect of focus. It was observed that the detectability, resolution, and accuracy in the determination of the parameters decreased with an increase in the aberration coefficient, though the decreases were smaller for partially coherent light than for coherent light.

The detection and resolution of incoherent objects by an aberrated optical system in the presence of background noise

Far-field intensity distribution has been plotted for a circular aperture suffering from spherical aberration and illuminated by quasi-monochromatic partially space-coherent light. Gaussian and besinc forms of correlation have been assumed. It is shown that higher values of coefficient of aberration can be tolerated as the value of number of correlation intervals contained in the aperture increases.

B N Gupta

Papers

Effect of spherical aberration on the Fraunhofer diffraction of an optical system illuminated by partially coherent light