Paraxial Ray Tracing Research Articles

Theoretical analysis of accommodation amplitude and ametropia correction by varying refractive index in Phaco-Ersatz.

Phaco-Ersatz is a surgical procedure for restoring accommodation in which the presbyopic crystalline lens is replaced by a flexible polymer gel in the lens capsule. We assessed the feasibility of simultaneously correcting ametropia while restoring accommodation using Phaco-Ersatz by utilizing polymer gel of the appropriate refractive index. Computation results using paraxial equations and ray tracing on two model eyes indicate that although this approach might be feasible for the hypermetrope, its usefulness for correcting myopia is limited, principally due to significant reductions in the resultant amplitude of accommodation.

The art of back-of-the-envelope paraxial raytracing

Paraxial raytracing is a valuable tool for making quick back-of-the-envelope calculations in optical system design. Its popularity has been highly diminished as a consequence of the growing sophistication of computer-aided raytracing and the availability of powerful computers. The availability of raytracing computer programs does not harm in itself the usefulness of graphical raytracing, but it makes people forget, or never learn the basics of graphical raytracing. In this paper, some of the basic concepts of graphical raytracing are presented, along with some examples that illustrate how it can be applied to multielement systems. The mastering of the methods presented in this paper constitute some of the fundamentals required in the use of any optical software. Such knowledge is absolutely necessary for the conception of novel optical systems.

General scheme for paraxial ray tracing and aberration calculation in inhomogeneous media

A set of general formulae for paraxial ray tracing in inhomogeneous media are derived, which can be used together with the aberration formulae by Sands for the first- and third-order analyses of inhomogeneous lenses of all kinds of gradient function. The choice of an optimum step size for paraxial ray tracing and aberration integration is discussed, and it is shown that results with good accuracy can be obtained with a relatively large step size and very economic computing effort, even for complex gradient functions such as the tapered hyperbolic secant.

General scheme for paraxial ray tracing and aberration calculation in inhomogeneous media

Abstract A set of general formulae for paraxial ray tracing in inhomogeneous media are derived, which can be used together with the aberration formulae by Sands for the first- and third-order analyses of inhomogeneous lenses of all kinds of gradient function. The choice of an optimum step size for paraxial ray tracing and aberration integration is discussed, and it is shown that results with good accuracy can be obtained with a relatively large step size and very economic computing effort, even for complex gradient functions such as the tapered hyperbolic secant.

Refractive index of the crystalline lens in young and aged eyes

Background: When the ageing crystalline lens is modelled on the basis of a constant equivalent lens, the changes in ocular dimensions would lead to an increase in power of the order of two dioptres. A comparable increase in myopia is usually not evident with increasing age and this inconsistency has been referred to as the lens paradox. It has been proposed that this paradox can be resolved if the refractive index is modelled as a gradient refractive index. The purpose of this paper was to study differences in the equivalent, gradient and surface refractive index of the crystalline lens between a young and old age group.Methods: Biometric data was collected for 96 subjects: 48 young adults with an age range 19 to 31 years (mean 22.10 ± 2.93 years) and 48 old adults with an age range 49 to 61 years (mean 53.88 ± 3.88 years). The equivalent refractive index was determined for each subject by paraxial ray tracing and a merit function based on refractive error and Purkinje image height. The refractive index gradient was determined by modelling the crystalline lens as a bi‐elliptical iso‐indicial structure in a three‐surface Gullstrand‐Emsley schematic eye and a merit function based on the surface power, the gradient refractive index power and the equivalent power of the lens. The central refractive index of the lens was assumed to be 1.406.Results: The differences between the groups included a decrease in the mean equivalent refractive index from 1.427 ± 0.007 to 1.418 ± 0.006, an increase in surface refractive index from 1.386 ± 0.007 to 1.394 ± 0.006 with a concurrent change in the gradient refractive index profile. The refractive index changes maintained a constant mean lens power in each group.Conclusions: The so‐called ‘lens paradox’ whereby an increase in the power of the crystalline lens does not lead to an increase in myopia is resolved by a decrease in the equivalent refractive index of the lens or when modelled as a gradient refractive index structure, by an increase in the surface refractive index and an associated change in gradient for an assumed central refractive index of 1.406.

Field restriction and vignetting in contact lenses with opaque peripheries

Background: Some previous researchers have shown empirically that visual fields can be restricted by contact lenses with opaque and semi‐opaque peripheral zones. However, there has been no formal statement of the optical theory behind such restriction.Methods: Calculations are made of fields of view with opaque periphery contact lenses, based on paraxial theory and meridional finite ray tracing in the Gullstrand‐Emsley schematic eye.Results: This analysis shows that paraxial theory is adequate for predicting ‘half fields of full illumination’ for ocular pupil diameters smaller than six millimetres, although paraxial estimates of ‘half total field’ can substantially underestimate those obtained by finite ray tracing for a wide range of ocular pupil sizes. Therefore finite ray tracing may be a more appropriate method of calculating half total field. ‘Total field’ will be smaller with smaller ocular pupils and smaller contact lens apertures. A clinical method is suggested for assessing total field with such contact lenses in situ. The analysis applied to opaque periphery contact lenses is extended to the semi‐opaque periphery contact lenses used for cosmetic alteration of eye appearance.Conclusion: Semi‐opaque periphery contact lenses will decrease retinal illuminance most at regions outside those tested by previous empirical static perimetry studies.

Methods to obtain quantitative parametric descriptions of the optical surfaces of the human crystalline lens from Scheimpflug slit-lamp images. I. Image processing methods.

Of the methods developed (e.g., phakometry, magnetic resonance imaging, etc.) for noninvasive measurement of the geometry of the anterior segment of the human eye, Scheimpflug photography offers the best resolution and the highest precision. The primary obstacle encountered with this or any other image-based method has been in obtaining quantitative measurements directly from the images. Image enhancement (gray-scale gradient analysis) and pattern recognition methods (Hough transform and recursive least-squares algorithms) are developed so that parametric representations of lens surfaces and zone boundaries can be obtained directly from the images. Methods to correct for nonlinear Scheimpflug camera reproduction ratios and provide error estimates for geometrical parameters are also developed and will be presented separately. Combined, these techniques yield representations of lens geometry having sufficient precision, to which paraxial ray tracing can be applied to determine lens optical properties by using well-posed optical models with one unknown.

Paraxial ray tracing in inhomogeneous optical media

Abbas A. RangwalaUniversity of MumbaiDepartment of PhysicsMumbai 400 098, IndiaAshok M. GhodgaonkarRamdulal TewariUniversity of MumbaiWestern Regional Instrumentation CenterMumbai 400 098, IndiaAbstract. Starting from the variational form of Fermat’s principle, weidentify the optical Lagrangian and obtain the standard form of the rayequations in inhomogeneous optical media. Specializing to cylindricallysymmetric media, we define the paraxial ray approximation through theexact first integral of the ray equation. This enables us not only to relatethe constants of integration to the boundary conditions at the entry pointof the ray into the medium, but more importantly furnishes a systematicway of introducing corrections to the paraxial approximation. We alsoexhibit power-law refractive index profiles, which result in periodic raytrajectories.

Improved computing scheme for measuring eye alignment with Purkinje images I and IV

SummaryThis study introduces an improved computing scheme for determining eye rotation from Purkinje images I and IV. The original computing scheme systematically underestimated eye rotation. Paraxial raytracing calculations revealed that this error resulted from failure to account for the fact that Purkinje images I and IV fall al different distances behind the cornea. The error could be overcome with a correction factor derived from paraxial raytracing calculations. A series of experiments were carried out to test the validity of this correction factor; involving exact raytracing calculations as well as measurements on physical model eyes and human eyes. The influence on the correction factor of ocular surface asphericity, accommodation, age and ocular component variations were examined. The new method was also compared to Hirschberg's technique, which makes use of Purkinje image I alone, as a means of screening for strabismus.

Improved computing scheme for measuring eye alignment with Purkinje images I and IV

This study introduces an improved computing scheme for determining eye rotation from Purkinje images I and IV. The original computing scheme systematically underestimated eye rotation. Paraxial raytracing calculations revealed that this error resulted from failure to account for the fact that Purkinje images I and IV fall at different distances behind the cornea. The error could be overcome with a correction factor derived from paraxial raytracing calculations. A series of experiments were carried out to test the validity of this correlation factor, involving exact raytracing calculations as well as measurements on physical model eyes and human eyes. The influence on the correction factor of ocular surface asphericity, accommodation, age and ocular component variations were examined. The new method was also compared to Hirschberg's technique, which makes use of Purkinje image I alone, as a means of screening for strabismus.

Prediction of intraocular lens power using the lens haptic plane concept

Purpose: To test algorithms for the preoperative estimation of the lens haptic plane (LHP) and to assess these in terms of predictability of postoperative refraction.Setting: SA Erik's Eye Hospital, Stockholm, Sweden.Methods: Preoperative axial length, anterior chamber depth (ACID), and cataractous lens thickness were measured in consecutive cases scheduled for phacoemulsification and posterior chamber intraocular lens (IOL) implantation. The algorithms tested used ACID and cataractous lens thickness to estimate LHP. The exact geometry of the IOL was used to calculate postoperative ACID from LHP. Thick lens theory and paraxial ray tracing were used to calculate predicted postoperative refraction. The calculated value was compared with actual refraction 3 to 6 months postoperatively.Results: Mean absolute average error in predicted refraction was 0.38 diopters (D), with 78% of eyes within 0.50 D and 97% within 1.00 D for the best algorithm.Conclusions: The predictability in postoperative refraction found by applying the LHP concept and paraxial ray tracing was excellent. However, the small sample, with its unusually slight variation in axial lengths, did not allow statistically significant differences between different postoperative refraction prediction methods to be demonstrated.

Calculation and optimization of Narcissus using paraxial ray tracing

The Narcissus effect occurs in IR optical systems that employ a cooled quantum detector and can degrade the image quality very severely. Therefore, it is essential that it is controlled in the design stage. A detailed description is given of how Narcissus can be calculated, by the use of a routine based on paraxial ray tracing, and reduced automatically and simultaneously with the system aberrations, by inclusion of its results in the merit function of a damped least-squares optimization program.

The Lens Haptic Plane (LHP) a Fixed Reference for IOL Implant Power Calculation

OBJECTIVE: To establish an unambiguous method to assign A-constants and similar constants required by the various formulas in use for the calculation of intraocular lens (IOL) implant power, without recourse to clinical investigations. METHOD: Retrospective analysis of clinical data involving widely different IOL designs and several surgeons was performed to find the position of the lens haptic plane (LHP). The clinical data had been generated previously for entirely different purposes. Inclusion criteria were visual acuity of 0.5 (20/40) or better and availability of a postoperatively (3-12 months) measured ACD (distance from anterior cornea to anterior IOL). Ray-tracing calculations were performed to find the ACD compatible with the optical data (preoperative L and K, and IOL power). The exact design (front and back radii, and central thickness) of the IOL was used in these calculations. The calculated ACD was verified against the postoperatively measured ACD. Lens haptic plane was calculated from ACD using the geometry of each particular design. RESULTS: Data from 688 cases, 8 different clinics and 4 IOL models were analysed. On average the calculated ACD coincided with the postoperatively measured ACD when applying paraxial ray-tracing and thick lens theory (Gaussian optics). An average eye with K = 43.63 D, L = 23.24 mm and LHP = 4.02 mm was established. For any new IOL design, the power to make this average eye emmetropic can be determined by applying paraxial ray-tracing. Subsequently the A-constant or any other formula constant can be determined. LEMMA: There is no need to utilize constants inherent to various formulas. The calculation model used to find the constants could just as well be used directly to find the exact implant power, provided that the position of the LHP is known and that the manufacturer supplies the necessary design information. Because the LHP concept is independent of IOL model, the advantage would be that the surgeon could concentrate on refining his technique to estimate LHP preoperatively, rather than personalizing constants for every new IOL model.

Limited‐view diffraction tomography in a nonuniform background

The main problems in geophysical diffraction tomography are (1) complicated media and (2) rather limited acquisition geometries. Existing algorithms solve the limited‐view problem in an iterative manner, but are valid only for line sources and 2-D homogeneous background models. In this paper, we derive an iterative algorithm based on asymptotic wave theory that can compensate for a limited acquisition geometry. The method is valid for a 2-D nonuniform background model and point‐source illumination (i.e., a 2.5-D geometry). Paraxial ray tracing is employed to model the arbitrary background wave response, and the general structure of the algorithm has a strong resemblance to the iterative ART‐algorithm used in straight ray tomography. Our method is shown to be stable in the presence of moderate white noise and gives reasonable results, both geometrically and quantitatively, when applied to synthetic crosshole data involving a nonhomogeneous background model and limited view.

An automated ray method for diffraction modelling in complex media

SUMMARY We present the computational concept and first results of an automated 2-D ray-tracing algorithm which combines the standard ray method with the method of edge waves and paraxial ray tracing. Reliable ray synthetic seismograms are obtained for subsurface structures of high complexity. Both diffracted and multiple diffracted arrivals are automatically computed, complementing all types of primary arrivals (reflected, multiple reflected, converted waves, etc~} where geometric shadow zones are caused by edges (inhomogeneities) in the subsurface model. The method of computation can be summarized as follows: (1) during standard ray tracing, properties of central and paraxial rays are computed for a set of neighbouring rays. (2) Diffraction points (edges) are identified by comparing the amplitude and traveltime differences of neighbouring rays with the corresponding values of their paraxial approximation. ~3) Detected edges are used as source points for diffracted rays. ( 4) Repetition of (1)-(3) for diffracted rays allows computation of multiple diffractions ('diffracted diffractions'). (5) The amplitude decay of diffracted arrivals is computed according to the theory of edge waves. Its critical variables are expressed in terms of second-order paraxial traveltimes. The method is demonstrated for a simple and complex synthetic model and a real data complex

Analytic design of computer-generated holograms focusing in nonplanar curves

An approximate method for the analytical determination of transmittances which are able to focus an incident plane wave into nonplanar focal curves is proposed. First, the solution for the projection of the demanded focal curve onto the main focal plane is determined by using the stationary phase method. Next, the regions of the hologram focusing into the particular points of the projected curve are determined and, as a third step, the value of the focal lenght is made variable for each, accordingly to the assumed final form of the focal curve. This is completed by a discussion of the limitations of the proposed method, based on the paraxial ray tracing of the obtained elements, as well as by a presentation of the experimental results supporting the suggested solution.

Visual optics and retinal cone topography in the common marmoset (Callithrix jacchus).

The common marmoset (Callithrix jacchus) is a small, diurnal, New World monkey amenable to vision research. In this paper we describe the visual optics and cone photoreceptor topography of the normal adult marmoset. Paraxial optical ray-tracing shows that the marmoset eye is well represented as a scaled-down version of the human eye. The density of foveal and perifoveal cone photoreceptors in the marmoset is as high, and in peripheral retina higher, than those reported in humans and macaques. The foveal acuity predicted by the Nyquist limits set by the cone mosaic (30 c/deg) is in agreement with behavioral measures of visual acuity. Foveal depth of focus is remarkably small (< 0.2 D) for an eye of this size (axial length about 11 mm). Estimates of the amplitude of accommodation using infrared photorefraction indicate that the marmoset is capable of more than 20 D of accommodation.

Generalized Zone Plates Focusing Light into Arbitrary Line Segments

Abstract A new computer-generated generalized zone plate is proposed, which is able to concentrate the incident beam into a line segment of various length and arbitrary inclination to the optical axis, as well as unrestricted distribution of the longitudinal intensity. The approach is based on the energy conservation principle and equations of the paraxial ray tracing, both considered in the cartesian coordinate system. Hyberbolic, elliptic, linear, and conical zone plates are shown to be limiting cases of the proposed element. Other specific cases, e.g. a new element focusing into a segment of the optical axis, are also derived.

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

SUMMARY Both paraxial ray tracing and two-point ray tracing are powerful tools for solving wave propagation problems. When a slowness model is mildly perturbed from a reference model, one can use perturbation theory for the determination of the ray positions and the traveltimes. An extension of Fermat’s theorem is presented, which states that the traveltime is stationary with respect to the perturbations in the ray position provided that the endpoints of the ray are perturbed along the wavefront of the unperturbed ray. It is shown that when the ray perturbation satisfies this condition the second-order traveltime perturbation can be computed from the first-order ray perturbation. A perturbation analysis of the equation of kinematic ray tracing leads to a simple second-order differential equation for the ray deflection expressed in ray coordinates. This constitutes a perturbation method based on a Lagrangian formulation, and leads to a first-order expression for the ray deflection and a second-order expression for the traveltime perturbation. This is of relevance to non-linear traveltime tomography because it leads to an efficient method for evaluating the lowest order ray deflection and the non-linear effect this has on the traveltimes. The theory is applicable both to two-point ray tracing and to the determination of paraxial rays. The derivations in this paper are completely self-contained. All expressions, including the transformation to ray coordinates, are derived from first principles. In this way one obtains insight in the approximations that are actually made. A scale analysis leads to dimensionless numbers that give an indication whether the theory is applicable to a specific problem. For the special case of a layered reference medium the final equations are particularly simple. Plane discontinuities in the reference model and the slowness perturbation are incorporated in the theory. The final expressions for the ray deflection and the traveltime perturbation can be implemented numerically in a simple way. It is indicated how applications to very large-scale problems can be achieved. Several examples, including the propagation of waves through a quasi-random model of the earth’s mantle illustrate the theory.

Paraxial ray tracing in gradient-index (GRIN) media: problems of convergence

With the use of the recursive paraxial ray-trace procedure described by Moore [ J. Opt. Soc. Am.61, 886 ( 1971)] to examine the properties of the crystalline lens of the eye, several cases of false or nonconvergence of the series were discovered. Two cases are presented here.