Lakshminarayan Hazra

Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets

The authors use a genetic algorithm in the structural design of cemented doublets in accordance with a prespecified set of Gaussian characteristics and primary aberration targets. Actual glasses are taken as variables. Suitable modifications in the algorithm are sought with which to explore the solution space in search of the global as well as useful local optima. Some illustrative examples of structural design are given.

Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming

An approach for ab initio synthesis of the thin lens structure of linearly compensated zoom lenses is reported. This method uses evolutionary programming that explores the available configuration space formed by powers of the individual components, the intercomponent separations, and the relative movement parameters of the moving components. Useful thin lens structures of optically and linearly compensated zoom lens systems are obtained by suitable formulation of the merit function of optimization. This paper reports our investigations on three-component zoom lens structures. Illustrative numerical results are presented.

Structural design of multicomponent lens systems

The thin lens design of multicomponent systems with a prespecified set of primary aberration residuals and Gaussian characteristics usually starts with the choice of a lens type followed by a selection of the glasses to be used for the elements of its different components. An alternative scheme is reported in this paper which aims at utilizing these degrees of freedom in a more effective way. The detailed structure of the individual components and the glasses to be used in the lens elements comprising them are determined after the feasibility of attaining the given aberration targets with viable components has been ascertained on a preliminary layout of components obtained from a consideration of Gaussian requirements and Petzval curvature of the desired system.

Far-field diffraction properties of radial Walsh filters

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.

Toraldo filters with concentric unequal annuli of fixed phase by evolutionary programming

The resolving power of an optical imaging system is limited by residual aberrations and diffraction effects. The Rayleigh-Abbe diffraction limit of resolution corresponds to radius of the central lobe of the point spread function of an aberration free diffraction limited system. An attempt to circumvent this limitation was proposed by Toraldo di Francia, who showed that suitable pupil plane filtering can overcome this resolution limit, albeit over a restricted field. This paper reports results of our investigations on the use of evolutionary programming to obtain globally or quasi-globally optimum solutions in synthesis of lossless Toraldo filters consisting of concentric unequal area zones of fixed phase.

Structural design of cemented triplets by genetic algorithm

We present an implementation of a genetic algorithm in the structural design of cemented triplets according to a prespecified set of Gaussian characteristics and primary aberration targets. The approach is directed toward obtaining a suite of promising solutions allowing some floating of the primary aberration targets. This method obviates the need for any heuristic preselection of glasses for the three elements of the triplet. Searches for optimal solutions are conducted in the total configu- ration space of the degrees of freedom, consisting of continuous vari- ables like shape factor and power distributions, and discrete variables like available glass types. The role of genetic diversity in the evolving population on convergence of the optimization runs has been investi- gated, and, accordingly suitable modifications are incorporated in the basic genetic algorithm to eliminate undue stagnation and premature convergence problems. Some illustrative examples are given.

Imaging by zone plates: axial stigmatism at a particular order

A Fresnel zone plate (FZP) produces simultaneously multiple images, both real and virtual. In general, none of the images is stigmatic beyond the paraxial domain. Axially stigmatic imaging by FZP’s calls for unique values of zone radii corresponding to each combination of the order number of the image and the object distance. In a recent paper [ J. Opt. Soc. Am. A10, 69 ( 1993)] we presented a set of nomographs for ready estimation of zone-plate parameters. The treatment dealt with the special case of a positive first-order image produced by a FZP. We now extend the study to incorporate imaging of any specific order. It is shown that, with a simple redefinition of the variables, the nomographs presented in the earlier paper retain their practical usefulness.

Structural design of doublet lenses with prespecified aberration targets

The authors present a systematic approach for the structural design of doublet lenses in accordance with a prespecified set of Gauss- ian parameters and primary aberration targets. This approach obviates the need for heuristic selection of glasses for the constituent lens ele- ments. An identical approach is implemented for the structural design of both cemented and broken contact doublets. The optimization procedure is the well-known least-squares method. For facilitating convergence a second-derivative damping technique is utilized. Constraints are taken care of by suitably defined continuously differentiable penalty functions that are treated as pseudoaberrations in optimization. A strategy for dy- namic weightage of the components of the defect function is devised for overcoming stagnation at the local optimum in the immediate neighbor- hood of the starting point. The strategy is also effective in constraining the glasses of the constituent elements of the doublet within the permis- sible glass domain. Some illustrative examples are given.

Design of the individual components of a multicomponent lens system: use of a singlet

It has been recently shown that the complex problem of thin lens structural design of a multicomponent lens system with a prespecified set of primary aberration residuals and Gaussian characteristics can be reduced to the problem of design of individual components with specified Gaussian parameters and central aberration targets. This paper reports a systematic approach to study the feasibility of using singlets for the individual components. The method provides an analytical procedure to select the optimum glass in a semiautomatic manner. It also attempts to ascertain the practical viability of a singlet solution by considering the possible effects of the structure of the singlet on higher-order aberrations. A set of graphs for rapid study of the feasibility of a singlet solution is also presented.

Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern

Pupil plane filtering by radial Walsh filters is a convenient technique for tailoring the axial intensity distribution near the focal plane of a rotationally symmetric imaging system. Radial Walsh filters, derived from radial Walsh functions, form a set of orthogonal phase filters that take on values either 0 or π phase, corresponding to +1 or -1 values of the radial Walsh functions over prespecified annular regions of the circular filter. Order of these filters is given by the number of zero-crossings, or equivalently phase transitions within the domain over which the set is defined. In general, radial Walsh filters are binary phase zone plates, each of them demonstrating distinct focusing characteristics. The set of radial Walsh filters can be classified into distinct groups, where the members of each group possess self-similar structures. Self-similarity can also be observed in the corresponding axial intensity distributions. These observations provide valuable clues in tackling the inverse problem of synthesis of phase filter in accordance with prespecified axial intensity distributions. This paper reports our observations on self-similarity in radial Walsh filters of various orders and corresponding axial intensity distributions.