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

The influence of primary astigmatism on the performance of optimum apodizers has been investigated theoretically. The analytical method used in.this study is quite general in the sense that it can be extened to make similar studies regarding the influence of other aberrations, such as spherical aberration and coma, on the performance of any of a general class of optimum apodizers. The aberration function used in this study is that given by the ZERNIKE-NIJBOER theory of wave aberrations. Analytical expressions have been obtained for various quantities, for example the intensity distribution, the fractional encircled energy distribution, the Strehl ratio etc., in order to make a thorough study of the diffraction pattern. Numerical results have been computed for two chosen apodizers and for certain amounts of primary astigmatism. These results have been illustrated graphically and discussed. Isophote diagrams for a few cases have also been presented.

Influence of Primary Astigmatism on the Performance of Optimum Apodizers

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

This paper presents the report of investigations on a priori apodization of the pupil of a telescope operating in a turbulent medium. The criterion of apodization has been taken to be the maximization of the factor of encircled energy on a pre-specified area on the far-field plane. It has been observed that the optimum pupil function designed on the basis of a perfectly space-coherent wave being incident on the pupil has got to be modified by a small but significant amount to obtain the optimum function when a partially space-coherent wave is incident on the same. This leads to the possibility of using turbulence-corrector pupils in conjunction with the optimum pupil function designed for the coherent case to achieve an overall apodization. Numerical results are presented.

On Atmospheric Turbulence and the Problem of Optimization of the Telescopic Pupil

Imaging characteristics of a partially coherent, apodized optical system in the presence of defocusing are presented. To examine them, three types of apertures, whose amplitude transmittances are s...

Performance of an Apodized Aperture in a Defocused Optical System under Partially Coherent Illumination

The performance characteristics of optical systems having apodized apertures and employing partially coherent illumination have been investigated. The pupil function has been taken to be of the form f(x y) = 1 m g (x 2 + y 2), where g is the amplitude shading parameter of the filter. The generalized cross-transfer functions for the system have been calculated for different values of the coherence parameter which is the ratio of the numerical aperture of the condenser to that of the objective. The modulations of periodic targets commonly employed, namely, the sine wave, square wave and triangular wave objects, have also been investigated. It has been shown that for the partially coherent and the incoherent illuminations ( S 1) the apodization improves the low frequency response at a cost of the high frequency response of the system. However, for coherent and near coherent illuminations (

Performance of Apodized Apertures under Partially Coherent Illumination

The effects of condenser transmission characteristics upon the performance of partially coherent systems have been investigated. It has been shown that the transmittance of the condenser aperture affects the coherence conditions prevailing upon the object plane and modifies the image forming properties of the system. Two different types of irradiance distribution functions have been assumed, in the first type the irradiance is maximum at the centre and tapers off towards the edges of the aperture, the reverse being true for the second type. Their influence upon the spatial frequency response and the edge response of the system has been investigated. The effect of the condenser size and transmission upon the two-point resolution of the system has also been examined.

R.K. Bhogra

Papers

Performance of Apodized Apertures under Partially Coherent Illumination

Influence of Condenser Transmission Characteristics upon the Performance of Microscopic Systems