Since the origin of the high interindividual variability of the chromatic difference in retinal image magnification (CDM) in the human eye is not well understood, optical parameters that might determine its magnitude were studied in 21 healthy subjects with ages ranging from 21 to 58 years. Two psychophysical procedures were used to quantify CDM. They produced highly correlated results. First, a red and a blue square, presented on a black screen, had to be matched in size by the subjects with their right eyes. Second, a filled red and blue square, flickering on top of each other at 2 Hz, had to be adjusted in perceived brightness and then in size to minimize the impression of flicker. CDM varied widely among subjects from 0.0% to 3.6%. Biometric ocular parameters were measured with low coherence interferometry and crystalline lens tilt and decentration with a custom-built Purkinjemeter. Correlations were studied between CDM and corneal power, anterior chamber depth, lens thickness, lens tilt and lens decentration, and vitreous chamber depths. Lens thickness was found significantly correlated with CDM and accounted for 64% of its variance. Vertical lens tilt and decentration were also significantly correlated. It was also found that CDM increased by 3.5% per year, and part of this change can be attributed to the age-related increase in lens thickness.
© 2014 Optical Society of America
Sharpness and contrast of the retinal image are affected by two types of optical aberrations: monochromatic and chromatic. Monochromatic aberrations result from imperfections in the refracting surfaces while chromatic aberrations result from dispersion of light in the ocular media  which makes the position of the focal plane and retinal image magnification dependent on wavelength .
Wavelength-dependent differences in optical power of the eye are referred to as longitudinal chromatic aberration (LCA). In the human eye, LCA amounts to about 2.5 D between the red and the blue ends of the visible spectrum, with more myopic refractions in the blue. Transverse chromatic aberration (TCA) has been less studied, but  described that it causes more loss in retinal image contrast than spherical aberration or coma. While LCA is similar among subjects because it is largely determined by the dispersion of water, almost all studies acknowledged that TCA is inherently variable not only among subjects [4–11] but sometimes also between the left and the right eyes [2,4–16].
One aspect of TCA is the chromatic difference in lateral image position, and the other is the chromatic difference in image magnification (CDM) [7,8,10,17]. The term TCA was used for slightly different phenomena in different studies. Conventionally, TCA occurs when light from an off-axis object enters in an oblique angle into the eye, as Ogboso and Bedell  measured it (the angle is then given from the achromatic axis of the eye). TCA will be larger the more oblique the incident angle becomes and should not depend on pupil size. This type of TCA also exists in the fovea as it is normally not coinciding with the achromatic axis of the eye. However, in the study by Zhang et al.  the measurements were performed for on-axis objects, with a small displaced aperture in the plane of the pupil. As an effect of the LCA, this displaced aperture resulted in a shift in transverse displacement of the rays on the retina depending on wavelength. This type of induced TCA was also used as a method to measure the LCA of the eye.
Three different psychophysical procedures have been developed to study CDM. Zhang et al.  estimated CDM from the physical tilt angle of the apparent frontoparallel plane (AFPP) under dichoptic viewing conditions when vision in one eye was limited to long-wavelength light (red) and the other was limited to short-wavelength light (blue). Due to the CDM, the image sizes were different on the retina. The subjects were asked to move rods on the AFPP until the same sizes on the retina were perceived. The adjusted lateral tilt of the AFPP was a measure of the magnitude of CDM. In the second method, CDM was measured as the perceived vertical misalignment of a red and a blue bar, the “two color vernier alignment method” . A shortcoming of this technique is that it can be used only near the fovea because vernier thresholds rapidly exceed CDM in the periphery of the visual field [10,18]. The third method is the combination of the two color vernier alignment method with an artificial pinhole aperture which is displaced at different lateral positions in front of the pupil . While all techniques described above measure CDM indirectly, a fourth technique would be more direct: matching the size of two targets that are presented at short and long wavelengths, respectively. However, until now, no study has been published that used this approach because size discrimination thresholds were assumed to be too high to resolve CDM . The authors of  perhaps based their conclusion in part on an article by Campbell et al. , who had looked at the ability of subjects to discriminate between sinusoidal gratings of different spatial frequencies. Campbell et al.  found that a difference in spatial frequency needs to exceed 3%–6% to become detectable. In contrast, objects with well-defined contours (such as squares) can probably be matched in size with much better resolution. An advantage of a size-matching procedure is that it can be done on a computer screen with software that allows adjusting the size of objects presented at different colors and sizes in fine steps. Therefore, we tested this approach. First, we measured the size-discrimination thresholds in our subjects for achromatic stimuli and found that they were sufficiently low to resolve the expected differences in size due to CDM. Second, the psychophysically determined CDM in each subject was correlated with optical and biometric variables in the eye, measured by low coherence interferometry and by a custom-built Purkinjemeter.
Twenty-one subjects, with an average age of years (range 21–58 years; 13 female and 8 male) and normal color vision, were recruited for the experiments. Nine subjects were myopic (average spherical equivalent : OD: D; OS: D) and three were hyperopic (OD: D; OS: D). Refractions of the left and right eyes were highly correlated among the subjects (refraction left * refraction right eyes , ; and the mean absolute difference between both eyes of the nine myopic subjects was D). Six of the myopic subjects wore spectacles, and three wore contact lenses, also during the experiments. The hyperopic subjects did not wear any corrections. To evaluate the impact of optical corrections on the measurements, CDM was correlated with the spherical refractive errors of the subjects. The study adhered to the tenets of the declaration of Helsinki and was approved by the local University Ethics Commission.
B. Psychophysical Measurements
A blue open square and a red open square with line width of 10 pixels were generated by software written in Microsoft Visual C++ 6.0. They were presented side by side on a black screen (experiment 1, Fig. 1), or they alternatingly covered each other at a frequency of 2 Hz (experiment 2, Fig. 2). Peak wavelengths of the squares, also used in a later simulation of CDM in ZEMAX, were at about 432 and 590 nm, respectively. According to the subjects’ reports, similar brightness was perceived for the red and the blue squares when the RGB coordinates were 0, 0, 255 (blue) and 200, 0, 0 (red). Both squares had the same side length of 34.9 mm, equivalent to 3.32 deg in the visual field at a viewing distance of 60 cm. A chin rest was used to minimize head movements. Subjects were tested with their right eyes; the left was covered. The test procedures were as follows.
1. Experiment 1
To verify that the subjects were able to discriminate the small differences in size between the red and blue squares, psychometric functions were measured in 11 of the subjects using a two-alternative forced choice (2-AFC) procedure. Two gray squares were presented on the monitor at the same places as the red and blue squares shown in Fig. 1. The sizes difference of the two squares was randomly varied by the software from 0% to 6.4% (diameter , , , , , , pixels), and subjects had to decide which of the two squares was larger. The procedure was repeated 10 times for each subject.
2. Experiment 2
Subjects adjusted the size of the blue square to match the size of the red square, using the left and right arrow keys of the keyboard. One step was equivalent to 0.294 mm (one pixel) on the LCD screen (EIZO FlexScan S1921, 19 in.), equivalent to 1.6 arcmin of visual angle at a distance of 60 cm. The size of the red square was fixed, but the initial size of the blue square was randomly varied by the software. Experiments were done with brighter red [RGB (255, 0, 0)] versus darker blue [RGB (0, 0, 200)] and with darker red [RGB (200, 0, 0)] versus brighter blue [RGB (0, 0, 255)]. The average of four repeated measurements provided the perceived size difference between the red and the blue squares for each subject, together with their standard deviations (SDs).
3. Experiment 3
A second paradigm was employed to measure CDM. A filled blue and a red square were alternatingly presented on top of each other (Fig. 2) with a flicker frequency of 2 Hz. The subjects had to adjust two variables. In the first step, they had to match the brightness of the blue and the red by minimizing perceived brightness differences. On average this condition was met when the blue square was set to RGB (0, 0, 255) and the red to (0, 0, 200). In the second step, they had to minimize the perceived size differences between both squares. As in experiment 1, the size of the red square was fixed at 34.9 mm (visual angle 3.2 deg). A black cross was presented in the center of the squares to lock fixation. As in experiment 1, the procedure was repeated four times, and averages and SDs were calculated.
4. Experiment 4
CDM was compared also in both eyes of five randomly selected subjects. As before, averages and SDs from four repetitions were determined.
C. Optical and Biometrical Measurements in the Eyes
All subjects were measured with a commercially available low-coherence interferometer, the Haag–Streit Lenstar LS 900 (HAAG–STREIT AG, Bern, Switzerland). The instrument provides central corneal thickness, corneal curvature (both flattest and steepest meridians along with the angle of the astigmatism, and the refractive powers in the two principal meridians), anterior chamber depth, lens thickness, and vitreous chamber depth. Pupils did not need to be dilated for the measurements.
D. Kappa Angle, Lens Tilt, and Decentration Measurement
A custom-built semi-automated device  was used to record Kappa angle, lens tilt, and lens decentration in 15 subjects who were selected based on availability from the full sample of 21 subjects. The positions of three Purkinje images (P1, P3, and P4) in the pupil were recorded for three different gaze positions. The software determined horizontal and vertical Kappa angle, horizontal and vertical lens tilt, and horizontal and vertical lens decentration. Pupils did not need to be dilated for the measurements.
Paired -tests were performed in Microsoft Excel (Asknet AG, Germany) to determine the significance of the differences between experiments. Linear regression analysis was used to determine correlations between psychophysical results and optical or biometrical measurements in the eyes. SPSS (version 16.0, SPSS, Chicago, Illinois) was used to build a multiple regression model for multivariable analysis. Using the “simultaneous” method, all the variables are set up together to determine which variable has the strongest impact on the dependent variable, that is, which provides the most accurate prediction of CDM. Since all variables such as lens thickness, lens tilt, lens decentration, and age may contribute to CDM, their relative impact on CDM is ranked. The four variables (: lens thickness, : lens tilt, : lens decentration, : age, : CDM) are assumed to satisfy the linear model: while , 2, 3, 4, where are values of an unobserved error term; and the unknown parameter constants. are called standard regression coefficients. The variable which had strongest impact on the dependent variable (CDM) was identified.
A. Experiment 1—Detection Thresholds for Size Differences of the Two Squares: Psychometric Function
Using a 2-AFC procedure, 11 subjects had to indicate which of two gray squares was larger (no chromatic cues). The percentage of correct responses was plotted as a function of size differences of the gray squares (Fig. 3). The psychometric function shows that size differences of 0.68% could be correctly discriminated in 75% of the cases.
B. Experiment 2—Perceived Differences in Magnification between the Blue and Red Squares
None of the subjects had reversed CDM (which would show up as a negative value in the percentage magnification difference shown in Fig. 4). All subjects judged the blue squares to be smaller than the red (Fig. 1), although the variability among subjects was large. Three perceived only a tiny magnification difference of 0.2%, while others saw differences of up to 3.2%. The average magnification difference between the red and blue squares were when the red square was presented at RGB (200, 0, 0) and the blue RGB (0, 0, 255), and when the red was presented at RGB (255, 0, 0) and the blue at RGB (0, 0, 200), suggesting that the perceived differences in size were not due to brightness differences. Figure 4 shows that the perceived magnification differences in the two conditions were highly correlated (; , ). There were also no statistically significant differences between the two data sets (paired -test, ), suggesting that we really measured the effects of CDM.
C. Experiment 3—Perceived Differences in CDM as Measured with the Second Paradigm
In this second test, three subjects perceived negative magnifications; that is, the blue filled square appeared larger to them than the red even though they were same size. CDM as measured in experiment 2 was plotted results from experiment 3 (Fig. 5). Two subjects perceived exactly the same size differences in both experiments, so that their data points are on top of each other. The results of experiments 2 and 3 were highly correlated (, , ).
D. Experiment 4—Correlations of CDM in Both Eyes
Five subjects were randomly selected to repeat experiment 2 also with their left eyes. CDM was found to be highly correlated between both eyes even though the number of subjects was small (, , ). That CDM is correlated in both eyes of a subject matches similar measurements by , who found that monochromatic aberrations display mirror symmetry in both eyes.
E. Measurements of the Optical Parameters in the Eyes of the Subjects with Low-coherence Interferometry
Biometrical and optical variables in the eyes of the 21 subjects, as well as their ages, are shown in Table 1. Dimensions of ocular structures, as well as corneal radius of curvature, were provided by the Lenstar LS-900. Effective corneal power was determined from the averages of corneal curvatures in the two principle meridians, using an effective refractive index of 1.332 [22,23].
F. Measurements of Kappa Angle, Lens Tilt, and Decentration in the Eyes of the Subjects
In 15 subjects, kappa, lens tilt angles, and lens decentration were measured with a custom-built Purkinjemeter . Results are shown in Table 2. The sign conventions used by the Purkinjemeter were as follows: In the horizontal direction, positive kappa indicates that the fovea is located in the temporal retina, relative to the intersection of the pupillary axis with the retina, and negative kappa indicates nasal retina location. A positive lens tilt angle indicates that the optical axis of the lens is tilted toward the nasal side, relative to the fixation axis, and negative value indicates temporal tilt of the lens optical axis. A positive value in horizontal lens decentration indicates nasal decentration, and a negative one indicates temporal decentration. In the vertical direction, a negative vertical kappa indicates that the fovea is above the intersection of the pupil axis with the retina, and a positive value indicates it is below the intersection of the pupil axis with the retina. A negative value indicates that the top of the lens is tilted toward the image space, and a positive one indicates tilt away from the image space. In lens decentration, a positive value indicates lens superior decentration, and a negative indicates lens inferior decentration.
In the horizontal direction, kappa ranged from to , and one subject had a negative horizontal kappa angle. Horizontal lens tilt angles varied from negative () to positive (). Horizontal lens decentration varied from to . In the vertical direction, three subjects had a negative kappa (, , and ) while the remaining ones had positive kappas. Vertical lens tilts ranged from to . The lenses of all subjects were decentered in the superior direction, relative to pupil center, by 0.29 mm on average.
G. Correlations of CDM with Optical and Biometrical Data of the Eyes
No significant correlations were found between CDM and corneal thickness, anterior chamber depth, corneal power, vitreous chamber depth, axial length, kappa angle (horizontal or vertical), and horizontal lens tilt and decentration (regression analyses not shown). However, the thickness of the crystalline lens was highly correlated to CDM (Fig. 6A; , , ). Significant correlations were also found between vertical lens tilt and CDM (Fig. 6B; , , ), and between vertical lens decentration and CDM (Fig. 6C; , , ). In conclusion, CDM increases with lens thickness and lens vertical decentration and decreases with lens vertical tilt.
We also analyzed the correlation of CDM versus age. On average, CDM increased by 3.5% per year of age (Fig. 6D: , , ).
H. SPSS Model: Multiple Regression Analysis
It was found that CDM increases with lens thickness and vertical decentration, and decreases with vertical lens tilt. To evaluate the relative importance of these variables, a multiple regression analysis was performed using SPSS. An enter model with one dependent variable, CDM, and three independent variables, lens thickness, lens vertical tilt, and lens vertical decentration, was analyzed. The highest impact variable was lens thickness (, ). The finding was significant (ANOVA: , ; ) and accounted for about 63.7% of the variance of CDM (adjusted ).
A. Comparisons of Our Measurements of CDM to the Literature Data
Our data are similar to the ones in the literature as long as we restrict the analysis to younger eyes. Hartridge  calculated that CDM can vary between 0.45% and 3.0%. Later, Thibos et al.  found that CDM varied from to in the fovea in six subjects (equivalent to about 0.6% to 2.8%), using the two-color vernier alignment task and an artificial pinhole pupil. However, the subjects in the study by Rynders et al.  were younger (19–36 years) than ours (22–58 years), and if we limit our analysis to subjects under 30 years of age, CDM was also less than 1 min of arc.
Also, schematic eye models were used to estimate foveal CDM. Zhang et al.  arrived at a theoretical value of less than 1%, but the age factor was not considered. CDM was also calculated from the chromatic difference in refraction  using the distance between the entrance pupil and the nodal point of the eye. The calculated CDM was only 0.37% between 400 and 700 nm and did not match the experimental data. It was proposed that this discrepancy can be attributed to the variability of the posterior nodal point positions in the different eyes . Zhang et al.  developed a formula to estimate CDM = delta-Rx*EN, where delta-Rx is the chromatic difference of focus (LCA measured in object space) and EN is the distance between the entrance pupil and the nodal point of the eye. According to the modeling, EN changes from 3.91 mm at 20 years to 4.35 mm at 60 years, an 11% increase. As delta-Rx does not seem to change with age, this is an estimate of increase in CDM with age. Our psychophysical data suggest that CDM is more than doubled between 20 years and 60 years.
B. Changes in Lens Thickness with Age
Changes in lens thickness with age have been studied, for instance, by Scheimpflug photography, magnetic resonance imaging (MRI), optical coherence tomography (OCT), or A-scan ultrasonography [25–31]. All authors report that the lens thickness increases with age, with an annual rate ranging from 19 to 26 μm. Our measurements with the Lenstar LS 900 in 51 subjects (including also the subjects of this study) were in line with these findings (average increase in lens thickness per year: 24 μm, regression plot not shown).
C. Optical Factors to Determine the Increase in CDM with Age
Vertebrate lenses consist of several onion-shaped layers with increasing refractive index from the periphery to the center . In some studies [29,33,34] the shape of the gradient refractive index (GRIN) profile was found to vary with age. These profiles are necessary to solve the lens paradox, the fact that lens power remains almost constant during aging despite the fact that lens thickness increases. MRI studies in the lens  show that the central plateau region maintains a constant refractive index with age while its size increases but that the peripheral refractive index gradient becomes steeper in older lenses. It is likely that, in addition to lens thickness, the change in gradient index profile may contribute to the increase in CDM. Until now, no accurate GRIN lens model has been published for the human eye, but it is known that there is considerable interindividual variability.
D. Lack of a Correlation of CDM with Horizontal Kappa
It was surprising that no correlation with the kappa angle was found, especially since variations in CDM are thought to be caused by TCA. The argument here is that since LCA is very similar between eyes, it would probably have given the same CDM for all subjects. A possible explanation could be that the younger subjects might have changed their accommodative state slightly when viewing the red and blue squares in experiment 2 and perhaps even experiment 3. Since the lens becomes thicker during accommodation it could have more LCA, resulting in more CDM. This possible explanation was further analyzed. During accommodation, lens thickness increases by about . This is a small effect. Even if the subjects would accommodate 10 D, the lens would have increased in thickness by only about 0.64 mm, 16.5% of the average total lens thickness in our subjects (3.88 mm). Therefore, a related effect on CDM should be small compared to the age-related increase in lens thickness from 3.21 to 5.11 mm. Another possible explanation why there is no significant correlation between CDM and horizontal kappa in experiment 2 is that the subjects scanned the red and blue squares continuously with their fovea, taking averages of their size with different eye positions.
E. Could the Measurements of CDM Be Confounded by the Power of the Spectacle Corrections?
It is well known that retinal image magnification is changed by spectacle correction. To evaluate the potential impact of this factor, CDM was correlated with the refractive errors of the subjects (Fig. 7A, all subjects; Fig. 7B, only the subjects wearing spectacles). There was no correlation detectable of CDM and refractive errors (, , ). To further analyze the potential impact of refractive errors, CDM was plotted against lens thickness only for the emmetropic subjects. The statistical significance of the correlation between CDM and lens thickness declined from to due to the smaller sample size, but the regression equations remained very similar (all subjects: CDM [%] = lens thickness*1.03–2.78, , ; only subjects with no refractive error: CDM [%] = lens thickness *0.92–2.28, , ). Therefore, the basic finding that CDM is correlated with lens thickness stands up even when refractive errors were corrected with spectacles or contact lenses.
F. Simulations of CDM with the Liou–Brennan Eye Model
We tested the performance of the Liou–Brennan eye model  using ZEMAX (ZEMAX Development Corporation, Bellevue, USA) but with a new crystalline lens as described by , who also simulated chromatic aberration. The optical parameters of the two eyes used in our simulation are shown in Table 1 (subject 1, 29 years old, and subject 2, 54 years old; denoted as filled diamonds and filled triangles, respectively, in Fig. 6A). The tested wavelengths were 432 and 590 nm. Both subjects had similar axial lengths. The major difference was lens thickness. In the image plane, subject 1 had also less CDM (0.38%) than subject 2 (0.58%), similar to data calculated by Bennett and Rabbetts  and Zhang et al. . The calculated differences in TCA were small compared to the ones that were actually measured in our psychophysical experiment (1.68% for the young subject and 2.52% for the older one). Possible explanations for the discrepancy between optical calculations and psychophysical measurements are the changes in the gradient index profile in the lens that occur with age.
G. Why Do We Not See Our CDM?
One of the puzzles in vision is that we do not see our CDM. At least three subjects in our study had a difference in image magnification in the red and the blue of about 3%, which should give rise to considerable color fringes at all edges that are tangential to the optical axis of the eye. A possible mechanism could be that the visual cortex scales image magnification in the cortical topographic maps according to wavelengths. For instance, if the blue image would be represented 3% larger than the red, color fringes would be removed in the neural representation. Interestingly, no electrophysiological recordings from cortical neurons are available with stimuli at different wavelengths to support or refute this hypothesis. Since CDM becomes more severe at older age, the wavelength-dependent scaling of the topographic maps would have to change with increase in age.
CDMs were psychophysically measured and ranged from 0.0% to 3.6% in our subjects. For the first time, we employed an object-size matching technique to measure CDM after we had found that subjects could discriminate 0.68% of a size difference in 75% of the cases. Size-discrimination thresholds were therefore low enough to resolve a significant part of the differences in CDM. Our major finding was that about 64% of the interindividual variance of CDM was explained by the variability in crystalline lens thickness.
This work was supported by OpAL, an Initial Training Network funded by the European Commission under the Seventh Framework Programme (PITN-GA-2010-264605).
1. D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A 22, 29–37 (2005). [CrossRef]
2. M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos, “Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes,” J. Opt. Soc. Am. A 12, 2348–2357 (1995). [CrossRef]
3. A. van Meeteren and C. J. Dunnewold, “Image quality of the human eye for eccentric entrance pupils,” Vis. Res. 23, 573–579 (1983). [CrossRef]
4. J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, “Measurement of the wave-front aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998). [CrossRef]
5. B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976). [CrossRef]
6. S. Marcos, S. A. Burns, E. Moreno-Barriusop, and R. Navarro, “A new approach to the study of ocular chromatic aberrations,” Vis. Res. 39, 4309–4323 (1999). [CrossRef]
7. S. Marcos, S. A. Burns, P. M. Prieto, R. Navarro, and B. Baraibar, “Investigating sources of variability of monochromatic and transverse chromatic aberrations across eyes,” Vis. Res. 41, 3861–3871 (2001). [CrossRef]
8. Y. U. Ogboso and H. E. Bedell, “Magnitude of lateral chromatic aberration across the retina of the human eye,” J. Opt. Soc. Am. A 4, 1666–1672 (1987). [CrossRef]
9. P. Simonet and M. C. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vis. Res. 30, 187–206 (1990). [CrossRef]
10. L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, and P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vis. Res. 30, 33–49 (1990). [CrossRef]
11. L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans,” Appl. Opt. 31, 3594–3600 (1992). [CrossRef]
12. R. E. Bedford and G. Wyszecki, “Axial chromatic aberration of the human eye,” J. Opt. Soc. Am. 47, 564–565 (1957). [CrossRef]
13. A. Ivanoff, Les aberrations de l’œil (Éditions de la Revue d’optique théorique et instrumentale, 1953).
14. H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14, 1684–1695 (1997). [CrossRef]
15. L. N. Thibos, “Calculation of the influence of lateral chromatic aberration on image quality across the visual field,” J. Opt. Soc. Am. A 4, 1673–1680 (1987). [CrossRef]
16. G. Wald and D. R. Griffin, “The change in refractive power of the human eye in dim and bright light,” J. Opt. Soc. Am. 37, 321–336 (1947). [CrossRef]
17. X. X. Zhang, L. N. Thibos, and A. Bradley, “Relation between the chromatic difference of refraction and the chromatic difference of magnification for the reduced eye,” Optom. Vis. Sci. 68, 456–458 (1991).
18. X. Zhang, A. Bradley, and L. N. Thibos, “Experimental determination of the chromatic difference of magnification of the human eye and the location of the anterior nodal point,” J. Opt. Soc. Am. A 10, 213–220 (1993). [CrossRef]
19. F. W. Campbell, J. Nachmias, and J. Jukes, “Spatial-frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
20. F. Schaeffel, “Binocular lens tilt and decentration measurements in healthy subjects with phakic eyes,” Investig. Ophthalmol. Vis. Sci. 49, 2216–2222 (2008). [CrossRef]
21. J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002). [CrossRef]
22. H. Littmann, “Grundlegende Betrachtungen zur Ophthalmometrie,” Albrecht v. Graefes Arch. Ophthalmol. 151, 249–274 (1951).
23. F. Schaeffel and H. C. Howland, “Mathematical model of emmetropization in the chicken,” J. Opt. Soc. Am. A 5, 2080–2086 (1988). [CrossRef]
24. H. Hartridge, “The visual perception of fine detail,” Philos. Trans. R. Soc. Lond. 232, 519–671 (1947).
25. K. Richdale, L. T. Sinnott, M. A. Bullimore, P. A. Wassenaar, P. Schmalbrock, C. Y. Kao, S. Patz, D. O. Mutti, A. Glasser, and K. Zadnik, “Quantification of age-related and per diopter accommodative changes of the lens and ciliary muscle in the emmetropic human eye,” Investig. Ophthalmol. Vis. Sci. 54, 1095–1105 (2013). [CrossRef]
26. J. J. Rozema, D. A. Atchison, S. Kasthurirangan, J. M. Pope, and M. J. Tassignon, “Methods to estimate the size and shape of the unaccommodated crystalline lens in vivo,” Investig. Ophthalmol. Vis. Sci. 53, 2533–2540 (2012).
27. K. Richdale, M. A. Bullimore, and K. Zadnik, “Lens thickness with age and accommodation by optical coherence tomography,” Ophthalm. Physiol. Opt. 28, 441–447 (2008). [CrossRef]
28. R. Navarro, F. Palos, and L. Gonzalez, “Adaptive model of the gradient index of the human lens. I. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24, 2175–2185 (2007). [CrossRef]
29. L. A. Jones, G. L. Mitchell, D. O. Mutti, J. R. Hayes, M. L. Moeschberger, and K. Zadnik, “Comparison of ocular component growth curves among refractive error groups in children,” Investig. Ophthalmol. Vis. Sci. 46, 2317–2327 (2005).
30. M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78, 411–416 (2001). [CrossRef]
31. S. A. Strenk, J. L. Semmlow, L. M. Strenk, P. Munoz, J. Gronlund-Jacob, and J. K. DeMarco, “Age-related changes in human ciliary muscle and lens: a magnetic resonance imaging study,” Investig. Ophthalmol. Vis. Sci. 40, 1162–1169 (1999).
32. S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Investig. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008). [CrossRef]
33. A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J.-M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. 58, 1781–1787 (2011). [CrossRef]
34. B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vis. Res. 42, 1683–1693 (2002). [CrossRef]
35. B. Jaeken, L. Lundstrom, and P. Artal, “Peripheral aberrations in the human eye for different wavelengths: off-axis chromatic aberration,” J. Opt. Soc. Am. A 28, 1871–1879 (2011). [CrossRef]
36. A. G. Bennett and R. B. Rabbetts, Clinical Visual Optics, 2nd ed. (Butterworths, 1984).