Radial Walsh functions form a closed set of orthogonal functions over a given finite interval, each function taking merely one constant value (either +1 or −1) in each of a finite number of subintervals into which the entire interval is divided. This set provides a remarkable set of orthogonal radial filters for the pupil of an imaging system. We report analytical expressions for members of the set and their Hankel transforms of order zero. Far-field diffraction characteristics, namely, the far-field amplitude distribution and the optical transfer functions, are presented for the first eight members of the set.

Walsh Functions in Lens Optimization: II. Evaluation of the Diffraction-based OTF for On-axis Imagery

Far-field diffraction properties of radial Walsh filters

The calculation of image assessment critiera, eg, the Strehl ratio, the point spread function, or the optical transfer function, involves the evaluation of an integral where the integrand is highly oscillatory over a large range of integration Prefaced with a brief description of the well-known numerical quadrature methods adopted for the purpose, this paper presents a new quadrature technique that obviates the need for knowledge of derivatives of the argument of the exponential integrand Some illustrative numerical results are presented

Numerical evaluation of a class of integrals for image assessment.

A simple and rapid numerical quadrature is developed for the evaluation of the diffraction-based optical transfer function for on-axis systems, using a Tchebycheff polynomial expansion of the pupil function. The integration of the autocorrelation integral of the pupil function is replaced by once and for all evaluations of the cross-correlation of respective polynomials. However, the expansion coefficients themselves of the Tchebycheff series are linear sums of the sampled pupil function and thus a series of coefficients can be generated that weight the pupil function at various points to give the resultant OTF. The coefficients for a tenth-order Tchebycheff expansion are included in the paper, and a set of tables of OTF values calculated with these coefficients and 64 2 64 Gaussian quadrature for a diffraction-limited system, and one with one wavelength of primary spherical aberration.

The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

This paper extends the analysis of optical imagery on the basis of Walsh functions to the determination of the diffraction-based optical transfer function (OTF) for on-axis systems. A semi-analytic and simplified expression for OTF is obtained which is directly amenable to subsequent operations which are often called for in optimization and tolerancing analyses. The computational speed for OTF evaluation is greatly increased where repetitive calculation of the OTF for different pupil plane parameters is required. Some computational results are presented and compared with the results obtained by other methods.

Walsh Functions in Lens Optimization: II. Evaluation of the Diffraction-based OTF for On-axis Imagery

Citations

Far-field diffraction properties of radial Walsh filters

Numerical evaluation of a class of integrals for image assessment.

References

The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

Walsh functions in lens optimization: I. FEE-based criterion