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

A comparative analysis of algorithms for fast computation of Zernike moments

Fast computation of Legendre and Zernike moments

In this paper we review a special set of orthonormal functions, namely Zernike polynomials which are widely used in representing the aberrations of optical systems. We give the recurrence relations, relationship to other special functions, as well as scaling and other properties of these important polynomials. Mathematica code for certain operations are given in the Appendix.

Zernike polynomials: a guide

A novel approach to the fast computation of Zernike moments

(1976). On the Mathematical Properties of the Zernike Polynomials. Optica Acta: International Journal of Optics: Vol. 23, No. 8, pp. 679-680.

On the Mathematical Properties of the Zernike Polynomials

Using a series expansion in Zernike polynomials to express the pupil function of an optical system, a means for computing the Optical Transfer Function has been found which avoids explicit numerica...

A New 'Analytic' Method for Computing the Optical Transfer Function

A simple condition is shown to be necessary and sufficient to suppress edge-ringing in optical imaging. This criterion is valid in the presence of aberrations and apodization, and for all cases of (spatially stationary) partially coherent illumination. The edge-ringing tendencies of an optical system may be assessed through the use of two new performance functions which form a Fourier-transform pair. One of these performance functions is related to the ‘transmission cross-coefficient’ which appears in the theory of partially coherent imaging. In the coherent limit this performance function reduces to the amplitude transfer function (pupil function), and in the incoherent limit it reduces to the Optical Transfer Function (OTF). The use of this performance function for the general assessment of partially coherent imaging systems is suggested.

Eric C. Kintner

Papers

On the Mathematical Properties of the Zernike Polynomials

A New 'Analytic' Method for Computing the Optical Transfer Function

Edge-ringing in Partially Coherent Imaging

A Recurrence Relation for Calculating the Zernike Polynomials

On the Theory of Light Beam Propagation in Non-linear Media