Ocular accommodation and wavelength: The effect of longitudinal chromatic aberration on the stimulus–response curve

The longitudinal chromatic aberration (LCA) of the eye creates a chromatic blur on the retina that is an important cue for accommodation. Although this mechanism can work optimally in broadband illuminants such as daylight, it is not clear how the system responds to the narrowband illuminants used by many modern displays. Here, we measured pupil and accommodative responses as well as visual acuity under narrowband light-emitting diode (LED) illuminants of different peak wavelengths. Observers were able to accommodate under narrowband light and compensate for the LCA of the eye, with no difference in the variability of the steady-state accommodation response between narrowband and broadband illuminants. Intriguingly, our subjects compensated more fully for LCA at nearer distances. That is, the difference in accommodation to different wavelengths became larger when the object was placed nearer the observer, causing the slope of the accommodation response curve to become shallower for shorter wavelengths and steeper for longer ones. Within the accommodative range of observers, accommodative errors were small and visual acuity normal. When comparing between illuminants, when accommodation was accurate, visual acuity was worst for blue narrowband light. This cannot be due to the sparser spacing for S-cones, as our stimuli had equal luminance and thus activated LM-cones roughly equally. It is likely because ocular LCA changes more rapidly at shorter wavelength and so the finite spectral bandwidth of LEDs corresponds to a greater dioptric range at shorter wavelengths. This effect disappears for larger accommodative errors, due to the increased depth of focus of the eye.

The eye being calibrated (right eye in experiments 1 and 2, and left eye in experiment 3) was covered by an infrared filter, allowing to measure its refractive state while occluding the stimulus.A series of trial lenses from -4D to 7D in 1D steps were also placed in front of this eye, and refraction was measured binocularly for at least 30 seconds for each of the lenses.This method allows to obtain the defocus measured by the photorefractor for objective values of defocus introduced for the calibrated eye through the trial lenses, while also accounting for the changes in accommodation by concurrently measuring the refraction of the left eye that views the stimulus.An individual correction factor was obtained by plotting the average differences in refraction between the two eyes as a function of the trial lens used and fitting a linear regression through the linear portion of this function.The inverse of the resulting slope was then used to rescale all the refractive data obtained for this participant.An example of the results of the calibration procedure obtained for one subject are shown in Supp Fig 1.
In experiment 3, to account for the differences in refractive error between the two eyes, the average refractive difference with no trial lens (0D) was obtained for each participant and subtracted from their data.

Supp Fig 1.
Results of the calibration procedure for one participant.The left panel shows the median and 25 th and 75 th percentiles of the measured defocus in both eyes as a function of the power of the lens used in front of the left eye.The right eye was uncovered and accommodating on a fixed target, while the left eye was covered by an infrared filter and different lenses.The right panel shows the difference in defocus between both eyes and the fitted linear regression.The steep slope indicates that the photorefractor overestimates the defocus in the left eye of this participant, measuring 1.25 dioptres for each 1 dioptre of real defocus.The inverse of this slope can be used to rescale the refraction measurements and correct the overestimation.

Results.
Determining the accommodative range of observers.
Accommodative range varies widely among individuals, particularly in a sample of observers with differences in age and refractive error.Within the accommodative range the response is expected to be quasi-linear with respect to the demand, while beyond it -that is, for demands higher than the near point of accommodation or lower than the far point -the response becomes saturated as the power of the crystalline lens cannot longer increase or decrease, respectively.The slope of the accommodation response curve is usually assessed within this linear accommodative range; thus, it was important to determine the near and far point of accommodation for each individual observer in our sample.
To do this, we calculated for each participant the gradient of the accommodation response as a function of distance in dioptres for each illuminant.At distances where the gradient dropped by 50% or more when compared to the overall median gradient, the response was determined to be saturated, while the distances where the gradient was maintained were taken to be within the accommodative range or linear portion of the accommodation response curve.This process was done for individual illuminants, such that the response to each could saturate at different distances (due to the differences in accommodative demand caused by LCA).The results of this process were visually inspected and agreed well with the evaluation of the experimenter (see Figure 7 for an example of the results for one subject).In all the analyses presented in following sections, the saturated portion of the accommodation response was omitted (i.e., only accommodative and pupil responses within the linear portion of the accommodation response curve were included), except for section 2.3.6,where some analyses include responses beyond the accommodative range, as detailed there.
Effects of LCA on the accommodation response curve.Results of the linear mixed models of accommodation for individual participants.

Experiment
In this section, we present additional results of the linear mixed models used on the linear portion of the accommodation response curves, as described in the paper.The Supp Table 3. Individual slopes and intercepts estimated for the linear portion of the accommodation response curves of each participant to each illuminant used in experiments 1, 2 and 3.For each participant, the first row represents the estimated slopes for each illuminant, and the corresponding conditional standard deviations are shown between parentheses; and the second row in italics represents the estimated intercepts for each illuminant and the corresponding conditional standard deviations between parentheses.
Variability in the accommodation responses to narrowband and broadband illuminants.Accommodation response curves for individual participants.
In this section, we present the accommodation response curves of individual participants in experiments 1, 2 and 3.The plots in Supp Fig 2-4 present the accommodation response as a function of the distance in dioptres.
Supp Table 1.Linear mixed models of accommodation as a function of distance in dioptres and illuminant for each experiment: Accommodation ~ Distance * Illuminant + (1 + Distance * Illuminant | ID).The estimated coefficients and their 95% confidence intervals (95% CI) have been used to calculate the estimated slopes (in dioptres/dioptres) and intercepts (in dioptres)for each illuminant.The intercept specifies where the average person accommodates for a stimulus at infinity under the specified illuminant, and the slope specifies how that changes with dioptric distance.The random effects standard deviations (RE SD), t-ratios, degrees of freedom (df), and p-values are also shown.The t-tests compare within each experiment, the slope and intercept estimates of each illuminant with the estimates for the longest-wavelength illuminant.Supp Table 2. Post-hoc pairwise comparisons of the slope and intercept estimates betweenilluminants.The estimated differences (diff.),degrees of freedom (df), t-ratios and p-values are shown.The shaded p-values in italics represent significant results at the 0.05 level.The different illuminants have been named according to their colour for easier reading of the results.
tableshows the slopes and intercepts estimated for individual participants under each illuminant, which were calculated from the estimated random effects coefficients of the linear mixed models fitted to the data of each experiment.

Table 7 .
Linear mixed models' results of visual acuity for positive accommodative errors (top) and negative accommodative errors (bottom), as a function of error magnitude, pupil diameter, their interaction, and illuminant.The coefficient estimates, their 95% confidence intervals (CI 95%), t-ratios, degrees of freedom (df), p-values, and random effects standard deviations (RE SD) are shown.