The use of autorefractors using the image-size principle in determining on-axis and off-axis refraction. Part 1: Analysis of optical principles of autorefractors.

PURPOSE: To study the optical principles and properties of autorefractors that use the image-size principle in which the size of the reimaged retinal image determines refraction. METHODS: The retinal illumination and reimaging of the retinal image were described, as were variations in the basic system. Imaging was determined for systems in which the light source is either diverging or converging as it passes into the eye. Equations were determined to describe the dependence of refraction on the heights and angles of incoming and outgoing beams, and refraction error was determined when eye position was not correct. RESULTS: The fundamental refraction equation is DE=±(α+θ)/h1 where DE is refraction, h1 is the beam height entering the eye, and θ and α are the angles of the incoming and outgoing beams, respectively. The negative sign outside the brackets applies if the beam focuses before entering the eye, while the positive sign applies if the beam focuses after entering the eye. When light is diverging as it reaches the anterior eye, hyperopia produces greater retinal image sizes than myopia. The opposite is the case when light is converging as it reaches the anterior eye. The effect of incorrect ocular longitudinal position on the measured refraction was determined; this produced errors identical to those for vertex errors with ophthalmic lenses. CONCLUSION: For image-size principle autorefractors, simple equations describe the dependence of measured refraction on the height and angle of the instrument beam as it enters the eye and the angle of the light, reflected back from the retina, after it exits the eye. Further work will investigate the validity of such instruments for determining peripheral refraction.

The use of autorefractors using the image-size principle in determining on-axis and off-axis refraction. Part 2: Theoretical study of peripheral refraction with the Grand Seiko AutoRef/Keratometer WAM-5500.

PURPOSE: To determine, through simulations, the likely validity of Grand-Seiko autorefractors with annular targets in peripheral refraction. METHODS: Using a physical model eye, the distance inside the eye to which the Grand Seiko AutoRef/Keratometer WAM-5500 beam was converging and the effective size of its outer diameter at the cornea were determined. Grand-Seiko refraction was calculated from Rx  = (θ + α)/h1 , where θ is the angle of the ingoing radiation beam, h1 is the height of the beam at the anterior cornea and α is the angle of the beam emerging from the eye following reflection at the retina. Two eye models were used: a Navarro schematic eye and a Navarro schematic eye with a contact lens having a highly positive aspheric front surface. RESULTS: The instrument beam was determined to be converging towards the eye to a distance of 24.4 mm behind the corneal vertex, with a 2.46 mm effective size outer diameter of the beam at the anterior cornea. The Grand-Seiko refractions provided accurate estimates of peripheral refraction for the model eyes. The results were closer to Zernike refractions than to Zernike paraxial refraction. Spherical aberration influenced refraction by up to 0.5 D, and peripheral coma had limited influence. CONCLUSION: Grand-Seiko autorefractors in current use, and having a circular annulus with an ingoing effective outer diameter at the front of the eye of about 2.4 mm, are likely to give valid peripheral refractions.

A comparison of a traditional and wavefront autorefraction.

PURPOSE: To evaluate the agreement between the autorefraction function of the Canon RK-F2, an autorefractor/keratometer based on the ray deflection principle, and the Carl Zeiss Vision i.Profiler(Plus), an wavefront aberrometer, compared with each other and with a noncycloplegic subjective refraction. METHODS: Objective refraction results obtained using both instruments were compared with noncycloplegic subjective refractions for 174 eyes of 100 participants. Analysis of sphere, cylinder, and axis using spherical equivalent difference and a new measurement, cross-cylinder difference, was performed. The spherical equivalent refraction and cross-cylinder difference for the manifest refraction were compared using Bland-Altman limits of agreement and 95th percentile analysis. RESULTS: The 100 participants represent 52 women and 48 men with a mean (±SD) age of 51.7 (±13.8) years, an average (±SD) spherical power of -0.67 (±2.53) diopters (D), and an average (±SD) cylinder power of -0.94 (±0.87) D. The spherical equivalent difference is 0.03 D (Canon) and -0.11 D (Zeiss). The 95% limits of agreement for the spherical equivalent are -0.69 to 0.75 D (Canon) and -0.75 to 0.75 D (Zeiss). The mean cross-cylinder power difference is -0.08 D (Canon) and 0.02 D (Zeiss). The 95% limits of agreement for the cross-cylinder power difference are 0.63 to 0.50 D (Canon) and 0.49 to 0.75 D (Zeiss). The mean axis power difference is -0.04 D (Canon) and 0.05 D (Zeiss). The 95% limits of agreement for axis power difference are -0.71 to 0.63 D (Canon) and -0.78 to 0.78 D (Zeiss). The double-angle astigmatic plot center of distribution for the RK-F2 is 0.035 D at 70 degrees, and that for the i.Profiler(Plus) is 0.053 D at 32 degrees. CONCLUSIONS: Both instruments provided clinically useful spherical equivalent refractive data compared with a subjective refraction, whereas the Canon RK-F2 was slightly more accurate in determining the cylinder power compared with a subjective refraction.

Conditions under which two-element variable power lenses can be created. Part 1. Theoretical analysis.

The conditions under which a two-element variable power lens can be created are examined. Such a lens is defined as one in which the functional form of the optical effect created does not change as the elements translate with respect to one another--only the magnitude of the effect changes. It is found that only variable power optical effects that can be described by quadratic functions can be formed by laterally translating two-element variable power lenses. In the case of rotationally translating two-element variable power lenses, possible designs are found by mapping possible laterally translating designs from a Cartesian space to the polar coordinate space of the rotationally translating lens.

Conditions under which two-element variable power lenses can be created. Part 2. Application to specific designs.

It was found in Part 1 of this paper [J. Opt. Soc. Am. A 28, 2148 (2011)] that only variable power optical effects that can be described by quadratic functions can be formed by laterally translating two-element variable power lenses. In the case of rotationally translating two-element variable power lenses, possible designs are found by mapping possible laterally translating designs from a Cartesian space to the polar coordinate space of the rotationally translating lens. Several designs that have been manufactured or suggested theoretically are examined in Part 2 to see which ones are true variable power lenses.

Nested shell optical model of the lens of the human eye.

A nested shell model of the human lens is developed based on the known anatomical construction of the lens, on the known way in which the lens grows throughout its life, on the measured characteristics of the lens surfaces as a function of the age of the lens, on the measured changes in the shape of the lens during accommodation, and on measured material characteristics of the lens materials, such as density and index of refraction throughout. The observed changes in central surface curvature and thickness force the shell thicknesses to vary in a predictable way and in turn force the shell surface asphericity to take certain values. Thus, in addition to giving the shape of each shell, the model predicts the change expected in the asphericity of the lens surfaces as the lens ages and adds cortical cell layers. Two examples are given, one for a 25-year-old lens and one for a 40-year-old lens, to show how the cortical layers change their shapes throughout the cortex and over time as the lens ages. The performance of the model of this paper is compared to that of two other nested shell models, one where the layers have constant thickness and one where the lens posterior is fixed within the eye over time, to show the superior performance of this model with respect to maintaining a constant refractive error for the eye as the lens ages and grows.

Relative importance of sources of chromatic refractive error in the human eye.

The relative importance of the various optical elements of the human eye are analyzed to determine which contribute most to the chromatic variance in total refractive power of the eye. The concept of differential dispersion, defined as the change in the difference in index of refraction across a refractive surface with change in wavelength, is used to provide a theoretical tool for this analysis. The theoretical treatment shows that almost all the chromatic effect will be caused by the air-tear interface. Calculations of model eyes are made that support this view. Four model eyes are examined, an emmetropic eye, a hyperopic eye, a myopic eye, and an emmetropic eye accommodating 2.5 D.

Determining spherocylindrical correction using four different wavefront error analysis methods: comparison to manifest refraction.

PURPOSE: To evaluate methods that determine refractive correction from wavefront data that best matches the manifest refraction. METHODS: Wavefront data in the form of Zernike polynomial coefficients from several VISX US Investigational Device Exemption studies were used to calculate the spherocylindrical distance correction using four methods: full pupil second order, central curvature, reduced aperture, and 4-mm pupil. The data were analyzed in two groups, preoperative (3686 measurements, 882 eyes) and 6-month postoperative (2570 measurements, 797 eyes). Differences were formed between values found with each method and the manifest refractive values for the spherical equivalent (dSE), cylinder power (dCc), and axis power (dAc). The wavefront spherical equivalent values were corrected for chromatic shift, lane length, and vertex. RESULTS: Values derived using the reduced aperture and 4-mm pupil methods corresponded with manifest spherical equivalent values for both preoperative (x(-) = -0.03 diopters [D], x(-) =0.95 D) and postoperative (x(-) = -0.36 D, x =0.65 D) data. The full pupil second order method results corresponded best with manifest data for the astigmatic variables for both preoperative (x(-) dCc=0.15 D, σ dCc=0.31 D, x(-) dAc = 0.02 D, σ dAc =0.22 D) and postoperative data (x dCc=0.25 D, σ dCc=0.33 D, x dAc=0.01 D, σ dAc=0.21 D). However, all methods did well on these variables. CONCLUSIONS: The Zernike coefficient set as measured best calculates the astigmatic correction. The Zernike coefficient set resized to a 4-mm pupil is best in determining spherical equivalent.

Spherical aberration of a hydrogel contact lens when measured in a wet cell.

A method is given for calculating the spherical aberration induced when the power of a hydrogel contact lense is measured in a wet cell. Application of the method to measurement of contact lenses with varying power over the entire optical zone is included.

The range of local wavefront curvatures measurable with Shack-Hartmann wavefront sensors.

Shack-Hartmann wavefront sensor systems are studied to assess the range of local wavefront curvatures that may be measured for a given choice of lenslet size and focal length and illumination beam characteristics, with special emphasis on systems design for use in wavefront eye refractors. Effects considered include blur effects on detected spot size,spot differential movement due to local wavefront curvature and source size effects.Examples are given, using a full diffractive treatment of the spot intensity patterns, to illustrate that the maximum range values found using the method of this paper result in neighbouring spot intensity patterns that are at the limit for resolution.

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials.

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

Wavefront measurements of diffractive and refractive multifocal intraocular lenses in an artificial eye.

PURPOSE: To investigate whether wavefront measurements taken in eyes that have implanted multifocal intraocular lenses (IOLs) are reliable when creating laser refractive surgical treatments to remove residual refractive error from these eyes. METHODS: A specially designed fluid-filled model eye holding a multifocal IOL was refracted using a commercial wavefront eye refractor. The wavefront findings were then compared to the expected refractive error. The fluid-filled model was designed to match the optical characteristics of a human eye and to hold the IOL at the same distance from the cornea as in a human eye. Lenses tested were AMO Tecnis ZM900 series (diffractive design, +25.00 D, +4.00 D add), AMO ReZoom NXG1 series (refractive design, 25.00 D, +4.00 D add), and AMO 811E (diffractive with aspheric design 25.00 D, +4.00 add). The visible pupil diameter was 5.75 mm. RESULTS: Diffractive bifocal IOLs were found to create spot doubling in the Shack-Hartmann sensor image as expected from theoretical considerations. However, the residual sphere power and cylinder power reported were those expected. In addition, the higher order aberrations reported were reasonable in light of expected results. Refractive multifocal IOLs were found to create distorted Shack-Hartmann images as expected. The residual sphere power and cylinder power reported were not those expected nor were the higher order aberrations. In addition, the higher order values were sensitive to the position of the eye with respect to the refractor. CONCLUSIONS: Wavefront measurements from eyes with diffractive IOLs can be used, with caution, to plan laser refractive treatments. Wavefront measurements from eyes with refractive multifocal IOLs should not be used to plan post-implant laser refractive correction.

Analysis of wavefront-guided corrections to see if they fully correct ocular aberrations.

In a recent paper [J. Opt. Soc. Am. A22, 1473 (2005)], the authors concluded, as a result of analysis using an optical model of the eye, that wavefront-guided ablative correction of the refractive errors of the eye does not efficiently remove those errors. An analytical method is given that analyzes the effect of a wavefront-guided ablative treatment of the aberrations of the eye using only the measured wavefront error of the eye and a measured or assumed anterior corneal surface shape and shows that in fact a wavefront-guided ablative treatment does remove all clinically significant error of the eye if only the anterior surface of the cornea is physically altered as planned as a result of treatment. The use of a variation of the method to analyze the effect of treatment when other changes occur as a result of treatment is also given.

Generalized Coddington equations found via an operator method.

Generalized Coddington equations allow the optical properties of an arbitrarily oriented incoming astigmatic ray bundle to be found following refraction by an arbitrary surface. Generalized Coddington equations are developed using the abstract concept of vergence and refraction operators. After suitable incoming vergence and refraction operators have been formed, these operators are re-expressed in a common coordinate system via similarity transformations created from the series of space rotations necessary to align the coordinate systems. The transformed operators are then added together to produce the vergence operator of the refracted ray bundle. When properly applied, these generalized Coddington equations may be used with complex wavefronts and complex refracting surfaces if local surface curvature properties are known for both where the two intersect. The generalized Coddington equations are given in matrix form so that they may be easily implemented.

Variable power phase lenses.

A method is given to construct a phase lens capable of creating an optical aberration of variable power that is described by a single Zernike polynomial function whose meridional index is 1 or greater. The phase lens is created from two identical phase elements, each creating a single Zernike aberration, that can be rotated with respect to each other, thereby increasing the aberration effect from zero to twice the value of either. This is possible because these aberrations are vectorlike. Results are given from the testing of an example that was manufactured and designed to produce coma (Zernike term Z(3,1)).

A test eye for wavefront eye refractors.

PURPOSE: A multi-site study was conducted to test feasibility of a modified automatic refractor style test eye as a test device for wavefront refractors of various types and to determine whether a) they could be measured and b) when measurements could be made, to see if they were similar. This study did not attempt to assess which instrument most accurately measures the aberrations of the test eye or human eye. METHODS: Three automatic refractor style test eyes were modified for use as test devices for wavefront refractors. One had a simple spherical front surface, and two had additional aberrations added. The test eyes and holder were circulated to 11 test sites where attempts were made to measure them with eight different wave-front refractor systems. RESULTS: Eight (100%) of the eight wavefront refractor systems tested successfully measured the test eyes. The systems did not give similar results for the same test eye. In some cases, coma was reported where none was present. Differences in reported defocus values reflect different approaches for compensating for the dispersion of the eye. A corneal topography system could measure and recognize the aberrations of the test eyes as well as the wavefront refractor systems tested. Interferometry, on the other hand, did not prove to be a successful method to assess the surface of the test eyes. CONCLUSIONS: The test eye design may be used as a test device for wavefront refractor systems. This type of test eye can detect systematic differences between various wavefront refractors and can serve as a useful calibration and comparison tool.

Improving visual function diagnostic metrics with the use of higher-order aberration information from the eye.

PURPOSE: This paper reviews the currently used visual function diagnostic metrics, acuity, refractive error, and contrast sensitivity, and suggests ways to create new metrics using the information that has recently become available due to advances in measuring the higher-order aberrations of the eye. Particularly, emphasis is placed on finding metrics that address certain aspects of vision rather than on general metrics. METHODS: Two metrics based on the modulation transfer function are introduced, the Visual Quality Factor (VQF), which is a value based on the modulation transfer function between the spatial frequencies of 3 and 12 cycles per degree (c/deg) giving a measure of the overall degradation of visual quality due to aberrations, and the Subjective Sharpness Factor (SSF), which is a value based on the modulation transfer function between the spatial frequencies of 15 and 40 c/deg, giving a measure of the decrease in perceived image "sharpness" due to aberrations. Two metrics based on the point spread function are suggested, Point Spread Quality (PSQ), which is a measure of the "compactness" of the point spread, and Multiplicity Factor, which is a measure of the multiplicity of perceived images by measuring the number of discrete peaks in the point spread function. RESULTS: The VQF and SSF values for both monochromatic and polychromatic cases are analyzed with regard to the number of acuity letters lost using data from a published study. CONCLUSIONS: The SSF is found to be the best overall correlate with acuity performance if the degradation of the letter stimuli is not excessive.

System for the design, manufacture, and testing of custom lenses with known amounts of high-order aberrations.

Now that excimer laser systems can be programmed to correct complex aberrations of the eye on the basis of wave-front measurements, a method is needed to test the accuracy of the system from measurement through treatment. A closed-loop test method was developed to ensure that treatment plans generated by a wavefront measuring system were accurately transferred to and executed by the excimer laser. A surface was analytically defined, and a Shack-Hartmann-based wave-front system was used to formulate a treatment plan, which was downloaded to an excimer laser system. A plastic lens was ablated by the laser and then returned to the wave-front device, where it was measured and compared with the analytically defined wave-front surface. The two surfaces agreed up to 6th-order Zernike terms, validating the accuracy of the system.

Systematic underablation in laser in situ keratomileusis: ablation pattern identified by advanced topographical analysis.

Topographical analysis based on the differential geometry of surfaces-curvature topography-was developed and applied to a patient after laser in situ keratomileusis. The patient had a minimal residual refractive error and normal best corrected visual acuity but had multiple visual aberrations, including ghosting and glare, unless the pupils were maximally constricted. The corneal loci responsible for the aberrations were difficult or impossible to identify on axial topographies but were readily identified with curvature topography. The patient's ablations appeared to be miniature versions of the intended ablation profiles, with small areas of emmetropic central cornea surrounded by annuli of rapidly increasing keratometric power; that is, systematic underablation. This may explain why some patients have visual aberrations with pupil diameters smaller than the programmed optical zones.