Proximity factor in image size magnification for optical instruments in general

A general expression is derived for the proximity factor in near image size magnification for an arbitrary instrument in front of an arbitrary eye. The proximity factor is a 2 x 2 matrix. The instrument and eye may be astigmatic and have decentred elements. The image on the retina may be blurred or not. The analysis is exact within the limitations of linear optics. The general results are specialized for the case of a stigmatic instrument and a stigmatic eye. The results are applied to the case of a thick, possibly bitoric, spectacle lens. The Appendix treats two numerical examples.     

Higher order aberrations,refractive error development and myopia control: a review

Evidence from animal and human studies suggests that ocular growth is influenced by visual experience. Reduced retinal image quality and imposed optical defocus result in predictable changes in axial eye growth. Higher order aberrations are optical imperfections of the eye that alter retinal image quality despite optimal correction of spherical defocus and astigmatism. Since higher order aberrations reduce retinal image quality and produce variations in optical vergence across the entrance pupil of the eye, they may provide optical signals that contribute to the regulation and modulation of eye growth and refractive error development. The magnitude and type of higher order aberrations vary with age, refractive error, and during near work and accommodation. Furthermore, distinctive changes in higher order aberrations occur with various myopia control treatments, including atropine, near addition spectacle lenses, orthokeratology and soft multifocal and dual-focus contact lenses. Several plausible mechanisms have been proposed by which higher order aberrations may influence axial eye growth, the development of refractive error, and the treatment effect of myopia control interventions. Future studies of higher order aberrations, particularly during childhood, accommodation, and treatment with myopia control interventions are required to further our understanding of their potential role in refractive error development and eye growth.     

Imposed retinal image size changes--do they provide a cue to the sign of lens-induced defocus in chick?

BACKGROUND: Young chicks can adjust their eye growth to compensate for both imposed hyperopia and myopia (using negative and positive spectacle lenses); the rate of eye elongation increases in the former and slows in the latter case. This emmetropizing behavior implies that the eye can distinguish the sign and magnitude of defocus, although the identity of the cue(s) involved is unknown. As the spectacle lenses used in these studies generally introduce significant retinal image size differences that are in opposite directions for negative and positive lenses (minification vs. magnification), we asked whether retinal image size might provide the required sign information. METHODS: This question was addressed by manipulating retinal image size while keeping lens power constant. We also investigated the effect of eliminating other potential cues, accommodation and chromatic aberration, under these conditions. Three negative "size" lenses of approximately -11 D optical power were used, with 2 of the lenses producing magnification rather than minification as typical of negative lenses (i.e. +1.9% and +6.9% compared to -2.9%). The lenses were fitted monocularly to 7-day-old chicks, which were subsequently measured at 9 and 11 days of age (refractive error and axial dimensions). The same lens-wearing schedule was applied to two other groups of chicks that had monocular ciliary nerve section surgery to prevent accommodation 2 days posthatching; one of these groups was reared under monochromatic yellow light instead of white light. RESULTS: Near-perfect refractive compensation was seen by the end of the treatment period with all three lenses, for all three treatment groups, and there was also little difference in the rate of compensation among the various groups. In all cases, the typical responses of axial (mainly vitreous chamber) elongation and myopia were observed. CONCLUSIONS: That manipulations to retinal image size, which either decrease or reverse the usual effects of negative lenses, did not disrupt compensation to the imposed hyperopic defocus, even in the absence of accommodation and chromatic aberration cues, argues against imposed retinal image size changes being the directional cue to defocus in experimental emmetropization.     

Modeling of lateral magnification changes due to changes in corneal shape or refraction

BACKGROUND AND PURPOSE: Especially after corneal surgery the lateral magnification of the eye providing the retinal image size of an object is a crucial factor influencing visual acuity and binocularity. The purpose of this study is to describe a paraxial computing scheme calculating lateral magnification changes (ratio of the image sizes before and after surgery) due to variation in corneal shape and spectacle refraction. CALCULATION STRATEGY: From the 4 x 4 refraction and translation matrices the system matrix representing the entire 'optical system eye' and the pupil matrix describing the sub-system from the spectacle correction to the aperture stop were defined for the state before and after surgery. As the chief ray is assumed to pass through the centre of the aperture stop, the 2 x 2 matrix of the lateral magnification ratio from preoperative to postoperative is described by the 2 x 2 sub-matrices of the respective pupil matrices. The cardinal meridians can be extracted by calculating the eigenvalues and eigenvectors. WORKING EXAMPLE: Vertex distance 14 mm, measured distance between corneal apex and aperture stop 3.6mm, keratometry 39 D+6D/0 degrees to 47D+3D/30 degrees and refraction 3.5D-5-5D/5 degrees to -4.0 D-3.5D/25 degrees preoperatively to postoperatively. The matrix of magnification ratio from preop to postop yields (0.8960 -0.0085;0.0074 0.9371) and the eigenvalues decomposition provided a 10.7% minified image at 170.1 degrees and a minified image of 6.1% at 78.7 degrees , which both are clinically relevant. CONCLUSION: We presented a straight-forward computer-based strategy for calculation of retinal image size changes using 4 x 4 matrix notation. With this model the meridional changes in lateral magnification from the preoperative to the postoperative stage or between follow-up stages can be estimated from keratometry, refraction, vertex distance and anterior chamber depth, which might be important for binocularity and vision tests in corneal surgery.     

Magnification, blur, and ray state at the retina for the general eye with and without a general optical instrument in front of it: 1. Distant objects.

PURPOSE: To derive general equations that characterize rays, magnification, and blur at the retina in the case of distant object points for a naked eye and for an eye looking through an arbitrary optical instrument. The eye and optical instrument may be astigmatic and noncoaxial. METHOD: The derivation is based on linear optics and makes use of the concept of the augmented ray transference of an optical system. Because the transference completely characterizes the linear optics, the analysis can claim completeness. RESULTS: Equations are presented for position and direction of rays at the retina from distant object points. They lead naturally to the definition of six properties that characterize blur, shape, size, orientation, and position of images of distant objects viewed by the naked eye and by the eye looking through an instrument. By way of example, the general equations are applied to the simple examples of a thin contact lens and a thin spectacle lens in particular. CONCLUSION: The analysis provides a framework, complete as far as linear optics is concerned, for the analysis of light arriving at the retina through any instrument from a distant point. In so doing, it unifies and generalizes concepts like blur and spectacle magnification, which, in the past, have been treated separately.     

Laboratory evaluation of blended lenticular aphakic lenses

Six types of blended lenticular aspheric lenses for the correction of aphakia were assessed with respect to mass, spectacle magnification, off-axis power errors, peripheral distortion, visual field size and cosmetic appearance. Relative to spherical surfaced lenses, these lenses showed considerably reduced mass, reduced peripheral distortion, increased visual field size and improved cosmetic appearance. All but one type showed considerably reduced off-axis power errors, but none of the lenses showed much reduction in spectacle magnification. In general, the manufacturers' claims made in favour of the blended lenticular lenses were supported.     

Blur due to pupil area when using progressive addition spectacles

The progression zone of a progressive addition spectacle lens cannot provide a sharp retinal image as it presents the pupil with a range of powers through which to view an object. The resulting blur is explored mathematically. The nature of this blur is extraordinary but its visual significance is very slight and is less than previously believed. Peripheral astigmatism, found in all such lenses, may lower their visual performance.     

Compensation of aniseikonia with toric intraocular lenses and spherocylindrical spectacles

BACKGROUND AND PURPOSE: Magnification disparity between the two eyes (aniseikonia) is one of the major unresolved problems in modern cataract surgery, potentially degrading binocular visual function or causing diplopia. The purpose of this study is to describe a paraxial computing scheme using 4x4 system matrices to simulate a corrected pseudophakic 'optical system eye' with a meridional magnification that matches the magnification of a given contralateral eye. METHODS: Based on the definition of a centred optical system in the paraxial Gaussian space containing astigmatic surfaces using 4x4 refraction and translation matrices, we derived a methodology for calculating the refractive power and axis of toric intraocular lenses and spherocylindrical spectacle corrections for (i) fully correcting the optical system eye and (ii) realizing an arbitrary meridional magnification by solving a linear equation system. RESULTS: The capabilities of this computing scheme are demonstrated with two examples. In example 1 we calculate a toric lens and a spherocylindrical spectacle correction for compensation of a corneal astigmatism to realize a predefined iso-meridional magnification. In example 2 we first determine the meridional magnification of the contralateral eye, which has been treated with cataract surgery and toric lens implantation, and then we compute the appropriate combination of a fully correcting toric lens and spherocylindrical spectacle refraction, which exactly matches the meridional magnification of the contralateral eye. CONCLUSION: We presented an en bloc matrix based strategy for the calculation of an optical system eye containing an astigmatic cornea, a toric lens implant and a spherocylindrical spectacle correction, where the toric lens and the spherocylindrical spectacle correction are determined to fully correct the system and to realize an arbitrary meridional magnification i.e. to eliminate aniseikonia.     

Schematic eye predictions of relative spectacle magnification in unilateral aphakia

Computer ray tracing provides a simple technique for investigating binocular relative spectacle magnification (BRSM) in the patient with unilateral aphakia. BRSM is defined as the retinal image size in the corrected aphakic eye relative to the retinal image size in the corrected phakic eye. The influence of preoperative ametropia and the mode of correction on BRSM is investigated, using Gullstrand's number 2 schematic eye as a model.     

Asphericity of the cornea and astigmatism

BACKGROUND: Regarding astigmatism of the cornea, curvature as well as asphericity depend on the meridional axis. Their functional dependence and relation to the quality of the retinal image are still unclear. METHODS: The astigmatic eye was modelled using a biconoid anterior corneal surface by means of a commercially available optical designer programme (Zemax EE, Zemax). The influence of asphericity and astigmatism on the quality of the retinal image was determined by means of ray tracing. Thirty eyes with astigmatism of up to 5.3 D underwent corneal topography (Keratograph C, Oculus) which allowed a numerical evaluation of the cylindric power as well as the asphericity in the main meridians. RESULTS: The quality of the retinal image of an eye with corneal astigmatism can be improved by a factor of 2.28 if the asphericity is optimised. Correction of the central astigmatism only (without considering asphericity) yields only a rather marginal improvement. The average difference of the asphericity in the main meridians is close to zero, however, the individual difference ranges from - 0.372 to + 0.444 which is definitely clinically relevant. CONCLUSIONS: Anisotropic asphericity of the cornea may significantly enhance or compensate central corneal astigmatism. Clinically manifest astigmatism is an individually variable combination of asphericity and curvature difference in the two main meridians and is dependent on the pupil size. Laser correction of corneal astigmatism must take meridional asphericity into account.     

Ophthalmoscopy and vitreoretinal surgery in patients with an ARRAY refractive multifocal intraocular lens implant

We developed a clinical strategy for dealing with situations in which ophthalmoscopic examination and vitreoretinal surgery are difficult in patients with an ARRAY refractive multifocal intraocular lens (IOL) implant. The ARRAY zonal-progressive IOL has a central 2.1-mm distance-vision zone for optically-unobstructed posterior pole observation. A concentric near-vision zone (+3.5-diopter add) surrounds this central zone. Optical ray-tracing is used to determine how a 2.1-mm pupil limitation restricts monocular and binocular retinal image size in head-mounted, slit-lamp, and operating microscope ophthalmoscopy. A 2.1-mm pupil decreases the retinal field of view of high magnification, narrow field lenses much more than that of wider-field, lower magnification lenses. This "worst-case" analysis suggests an ophthalmoscopic strategy, but is not strictly valid for the ARRAY lens because the near-vision zone surrounding its 2.1-mm central zone is not opaque. The near-vision zone contributes defocused information to the ophthalmoscopic image, diminishing its resolution and depth information. Wide-field, low magnification lenses are potentially less problematic than higher magnification lenses for examining and treating patients with an ARRAY IOL implant. This strategy is useful for panretinal photocoagulation or photodynamic therapy, but not for procedures requiring high magnification stereoscopic vision such as macular vitreoretinal surgery.     

Effect of myopia on visual acuity measured with laser interference fringes

The aim was to determine how visual acuity is affected by myopia when optical factors of the eye are controlled. Grating acuity was measured with interference fringes to avoid the effects of aberrations, and ocular biometry was used to compensate for differences in retinal image size among subjects. Distance spectacle refractions ranged from +2.25 to -14.75 D. The retinal magnification factor (RMF) in mm/deg was computed for each eye from the distance refraction, central corneal power and ultrasound biometry. A forced-choice orientation discrimination method was used to measure acuity for high-contrast 543 nm laser interference fringes in three retinal locations: the fovea, and at 4 deg and 10 deg eccentricity in the temporal retina. Acuity, expressed in c/deg and adjusted for spectacle magnification, was not significantly correlated with refraction at any of the three retinal locations. When acuity was converted to retinal spatial frequency units (c/mm) via the RMF, acuity decreased with increasing myopia at all three retinal locations (significantly at the fovea and at 10 deg eccentricity). Retinal acuity values in highly myopic subjects (>6 D) are consistent with retinal sampling distances that are larger than published values of human cone or ganglion cell spacing. The results imply that a highly myopic eye has retinal neurons that are more widely spaced than normal, but the increased axial length enlarges the retinal image enough to compensate for the retinal stretching. The data are consistent with a retinal stretching model that primarily affects the posterior pole.     

New matrix formulation of spectacle magnification using pupil magnification. I. High myopia corrected with ophthalmic lenses

In this paper we have applied the matrix method to calculating spectacle magnification (SM) and relative spectacle magnification (RSM) for myopic eyes corrected with ophthalmic lenses (OLs). We have been able to obtain very simple expressions of the magnifications and analogous values to the ones obtained using other geometric methods. Through the paraxial 2 × 2 object-image matrix for spherical systems, we can calculate the correcting lens pupil matrix ( M _L) and the corrected ametropic eye pupil matrix ( M _ca). These two matrices, that is M _L and M _ca, directly provide the lateral magnification and the lens-eye system's image focal parameters that enable us to easily calculate SM and RSM. The magnifications have been calculated for the case of myopic eyes corrected with OLs, which shows the validity of this method since its results are similar to the ones obtained using other methods. The analysis of other correcting systems such as contact intraocular lenses will be presented in two associated papers (to be published as parts II and III of this review).     

Off-axis monochromatic aberrations estimated from double pass measurements in the human eye.

Off-axis monochromatic aberrations in the human eye impose limits on peripheral vision. However, the magnitude of the aberrations off-axis, and in particular coma, has not been yet completely determined. We have developed a procedure to estimate third order aberrations in the periphery of the human eye. The technique is based on recording series of double pass retinal images with unequal entrance and exit pupil diameters (Artal, Iglesias, López-Gil & Green (1995b). J. Opt. Soc. Am. A, 12, 2358-2366.) which allows the odd asymmetries in the retinal image be assessed. The procedure that is described provides accurate estimates of the main off-axis aberrations: astigmatism, defocus and coma. We have measured these aberrations in four normal subjects. For a given eccentricity, the measured amount of coma and astigmatism are relatively similar among subjects, because the angular distance from the axis is the dominant factor in determining the magnitude of these aberrations. However, we found considerable variability in the values of peripheral defocus, probably due to a complicate combination of off-axis aberrations and fundus shape. The final off-axis optical performance of the eye for a given object location is determined by a particular mixture of defocus, astigmatism, coma and higher order aberrations.     

Peripheral power errors and astigmatism of eyes corrected with intraocular lenses

We compared the theoretical peripheral power errors and oblique astigmatism of eyes corrected with commercially available intraocular lenses (IOL's) with experimental data of normal phakic eyes. The peripheral power errors and the oblique astigmatism of the pseudophakic eye are larger than those of phakic eyes. The most sensitive component of the optical system of the pseudophakic eye to the peripheral power errors is the shape of the IOL, but only if the lens is away from the iris. The corneal and the retinal surfaces do not affect the peripheral power errors and oblique astigmatism significantly. A plano-convex lens with the flat surface facing the cornea gives the least peripheral power errors and oblique astigmatism and thus the value closest to the experimental data of phakic eyes. However, this design does not give the minimum spherical aberration. Therefore, eyes corrected with IOL's are expected to have poorer peripheral retinal image quality than normal phakic eyes.     

Laboratory evaluation of commercial aspheric aphakic lenses

Four types of commercial aspheric aphakic lenses were assessed with respect to mass, spectacle magnification, off-axis power errors, and peripheral distortion. The aspheric lenses have better distortion and off-axis imagery properties than spherical flat back lenses, but the improvement in distortion properties is small for those lenses whose design form specifies a flattish back surface. The claims concerning the distortion properties of these lenses are exaggerated. The shape (bending) of lenses has a large effect on spectacle magnification, peripheral distortion, and off-axis imagery.     

Ophthalmoscopic contact lenses for transpupillary thermotherapy

Ophthalmoscopic contact lenses for transpupillary thermotherapy (TTT) must provide effective visualization of retinal treatment sites and transmission of infrared diode laser radiation. Selection and proper use of retinal laser lenses requires knowledge of their lateral magnification, laser beam magnification factor, field of view and resolution. Optical performance is analyzed for Goldmann-type lenses and a series of inverted image lenses of differing magnification. Goldmann lenses have the highest resolution, but inverted image lenses of comparable magnification have 2.5 times or more their field of view. Inverted image lenses of similar magnification can differ in resolution. They require 2-4% more incident laser power to produce the same retinal irradiance as a Goldmann lens, but this difference is small in comparison to other clinical variables. Tilting an ophthalmoscopic contact lens up to 15 degrees causes little distortion in the circularity of the retinal spot formed by a laser beam or difference in retinal irradiance across the spot. Inverted image lenses produce higher anterior segment irradiances than Goldmann-type lenses, but anterior segment injuries are less likely in TTT than conventional visible light, short-pulse retinal photocoagulation because of the comparatively low irradiances used in TTT and the decreased absorption of diode laser infrared radiation in ocular media and melanin.     

Effect of spectacle and contact lenses on the effective corneal refractive zone

The effective corneal refractive zone is that portion of the cornea traversed by the light that enters the pupil of the eye from object points at a specified angle from the line of sight. It is of relevance in corneal surgery and for understanding the effect of corneal opacities and lesions on vision. Gaussian optics is used in this paper to obtain explicit equations for the geometry of the effective corneal refractive zone for a simplified eye, when spectacle and contact lenses are worn. The theory shows that lenses of positive power increase the diameter of the effective corneal refractive zone and lenses of negative power decrease the diameter. For axial object points the diameter of the effective corneal refractive zone increases by about 0.015 mm per dioptre increase in the power of the spectacle or contact lens. For object points at 30 degrees from the longitudinal axis, the increase is about twice as much in the case of contact lenses and more than four times as much in the case of spectacle lenses.     

Ophthalmoscopic contact lenses for transpupillary thermotherapy

Ophthalmoscopic contact lenses for transpupillary thermotherapy (TTT) must provide effective visualization of retinal treatment sites and transmission of infrared diode laser radiation. Selection and proper use of retinal laser lenses requires knowledge of their lateral magnification, laser beam magnification factor, field of view and resolution. Optical performance is analyzed for Goldmann-type lenses and a series of inverted image lenses of differing magnification. Goldmann lenses have the highest resolution, but inverted image lenses of comparable magnification have 2.5 times or more their field of view. Inverted image lenses of similar magnification can differ in resolution. They require 2-4% more incident laser power to produce the same retinal irradiance as a Goldmann lens, but this difference is small in comparison to other clinical variables. Tilting an ophthalmoscopic contact lens up to 15° causes little distortion in the circularity of the retinal spot formed by a laser beam or difference in retinal irradiance across the spot. Inverted image lenses produce higher anterior segment irradiances than Goldmann-type lenses, but anterior segment injuries are less likely in TTT than conventional visible light, short-pulse retinal photocoagulation because of the comparatively low irradiances used in TTT and the decreased absorption of diode laser infrared radiation in ocular media and melanin.