Aberration Polynomials Research Articles

Ray-tracing and Aberration Formulae for a General Optical System

Abstract All aspects of the ray-tracing and of the calculation of the monochromatic and chromatic aberrations for a completely general optical system are treated. These include the specification of the system, the ray-tracing formulae for refraction, reflection and transfer, the introduction of pseudo-paraxial variables in the object and image spaces, the determination of the pupil domain and the best image plane, plus calculation of aberrations along rays. Hopkins' canonical coordinates are employed in the object and image space to facilitate the calculation of the aberration polynomial coefficients, for use in the evaluation of diffraction-based image criteria, such as the point spread function. All the formulae involved are always determinate and of good accuracy, no matter whether the object, image or either pupil, of the system is at finite distance or at infinity.

Design and evaluation of laser sources with high-quality wave fronts

The design and evaluation of laser sources that have high-quality wave fronts with minimal residual aberrations and irregularities are described. Two such sources have been developed-a collimated reference source and a point reference source. The collimated reference source provides a collimated wave front with a wave-front irregularity of <0.01 wave rms over a 6-mm clear aperture. Collimation is better than 0.067 wave peak to valley, with an angular ray departure of <1.75 x 10(-5) rad. The point reference source provides a diverging wave front with a wave-front irregularity of <0.01 wave rms over a 0.60 numerical aperture and 0.005 wave rms over a 0.35 numerical aperture. Source wavelengths are 670, 780, and 820 nm. These laser sources may be characterized in terms of their fourth-order wave-front polynomial coefficients by using the testing methods developed in this paper. Once characterized, the sources are used as a known input for testing the optical response of unknown systems. This enables complete characterization of an unknown system in terms of aberration polynomials or other criteria of interest when used in conjunction with a wave-front sensing interferometer. These sources are used to evaluate an equal-path, phase-modulated Mach-Zehnder interferometer. Applications include general-purpose laboratory sources as well as tools for evaluating the accuracy of optical systems.

Computing high order wave aberration coefficients from variations of best focus for small artificial pupils

Van den Brink (1962) measured the variation of best focus of vertical bar targets through a 1.4 mm aperture across the pupil of the eye. He expressed his results in contour maps of isodioptric lines across the pupil. The relative focus of a portion of the pupil is a function of the curvature of the wave front within the aperture. The curvature, in turn, is proportional to the second derivative of the wave aberration polynomial. Thus it is possible to make estimates of some of the coefficients of the wave aberration polynomial from Van den Brink's focusing data. This has been done for two states of accommodation. The results indicate that the wave aberration polynomial of the measured eye contains some significant terms of a power higher than hitherto measured.

Aberrations for grazing incidence telescopes

The wavefront aberration polynomial and transverse ray aberration expansions will be derived for grazing incidence two-mirror Wolter telescopes including all paraboloid-hyperboloid and paraboloid-ellipsoid combinations. The reference sphere is determined with the help of the principal surface of the telescope, and the aberration polynomials will be given as functions of the coordinates of the ray intersection with the reference sphere. Third, and some of the fifth- and seventh-order aberration terms will be analyzed. Also, the wellknown relationship between wavefront aberration polynomial and transverse ray aberration polynomials will be verified.

Fifth-order aberration theory of gradient-index optics—examples

Four examples are presented as verification of the validity and the usefulness of the equations derived earlier [ J. Opt. Soc. Am.73, 1149– 1161 ( 1983)]. In each example, the aberration coefficients derived by algebraic methods are compared with those obtained from ray-trace data to a seventh-order aberration polynomial including ninth-order spherical aberration and distortion. In all cases good agreement is obtained between the two methods.

Diffraction analysis of rotationally symmetric optical systems using computer-generated aberration polynomials

A ray trace scheme for the automatic generation of optical aberration polynomials to arbitrary orders pioneered by T. B. Andersen is successfully applied to the diffraction analysis of rotationally symmetric optical systems on a desk-top computer. The diffraction-based optical transfer functions at various field positions are computed using the relatively new Winograd Fourier transform algorithm. The coding includes aspherical and reflective surfaces.

The Calculation of the Wavefront Aberration Polynomial

This paper presents a method of generating the wavefront polynomial using a set of orthogonal polynomials that are particularly relevant to the ray fans used in optical design and evaluation. This method avoids the unsatisfactory aspects of conventional curve-fitting techniques that use collocation, osculation or least-squares to establish a polynomial in ascending powers of the radial pupil coordinate and the cosine of the pupil azimuth angle.

A Study on the Computation Accuracy of the Aberrational Diffraction Images

Some modifications are introduced in the numerical method described previously by Hopkins and Yzuel in order to improve its accuracy. The area elements in which the exit pupil is divided for the integration are equal in the methods proposed here. These methods differ on the radial variable used. The influence of the exact evaluation of the phase function derivatives on the final accuracy with respect to the approximate evaluation by finite differences is studied. The methods developed here are applied to several optical systems with different wave aberration polynomials. We study the minimum number of area elements needed on the integration to obtain the intensity on points of the image plane with a given accuracy. A discussion of the results is described in the graphs and the tables of the second part of this paper.

Catadioptric system with a gradient-index corrector plate

Sands showed the equivalence of the third-order aspheric surface contribution of the aberration polynomial to the third-order inhomogenous surface contribution. This fact is exemplified by the substitution of an axial gradient for the aspheric surface of the Schmidt corrector plate. It is shown that a gradient-index corrector plate system is limited in its off-axis performance by fifth-order oblique spherical aberration just as the conventional Schmidt system is.

Analytic merit function based on Buchdahl’s aberration coefficients

An equation has been developed that predicts the average size of the spot diagram over the field of view of a lens. The equation is based on an analytic integration of Buchdahl’s aberration polynomials, and is used to supply the figure of merit in an automatic correction computer program. The equation results in aberration balancing through the seventh order, and combines the procedures of optimization and analysis for designers who utilize the geometric spot size as an image quality criterion.

Proximate ray tracing and wave aberration coefficients*

Algebraic ray-trace equations for axially symmetric optical systems are expanded in terms of system parameters and paraxial variables. The transfer parameter used is optical path, and equations are given for second-, fourth-, sixth-, and eighth-order differences in optical path of a ray from the axial value. Selected rays are traced to a tilted reference sphere in the exit pupil, optical path differences of given order are equated to the wave aberration polynomial of corresponding order and with proper coordinates, and the resulting linear equations are solved for wave aberration coefficients.

A Fast and Simple Method of Determining the Coefficients of the Monochromatic Aberration Polynomial for Image Assessment and Automatic Optical Design

A rapid method of determining the coefficients of the monochromatic aberration polynomial of an optical system is described and tables of the universal coefficients for eleven polynomial types are given for ready use by an appropriate routine of a computer ray-trace programme. Results on the accuracy of aberrations represented by the polynomials so constructed are given for some optical systems. Applications of the method to image assessment and automatic optical design are discussed. Similar methods of calculating the integrated radius of gyration of the geometrical point-spread function and for determining the coefficients of a chromatic aberration polynomial are suggested.

Diffraction Theory of the Aberrations of a Slit Aperture

The aberrations of a slit aperture are classified in terms of Legendre polynomials instead of the usual power series. An expression is obtained for the Strehl criterion in the presence of all aberrations, provided they are small. Power-series expansions are developed for the point spread function for a single aberration. Finally, the Maréchal balancing theory is studied and shown to lead to the same aberration polynomials.

Use of Optical Aberration Coefficients in Optical Design

A general account is given of the way in which the aberration coefficients introduced by Buchdahl are made use of in the design of optical systems. Experience shows that for many applications the coefficients of third, fifth, and seventh orders provide a new and adequate technique for the analysis of the correction state of a system, the understanding of the raison d’etre of a particular type of system, and the differential correction of a rough design. The geometrical representation and classification of the individual terms of the aberration polynomials is discussed. The method is exemplified by excerpts from the design of a telephoto objective.