1. INTRODUCTION

The impact of pupil size on retinal image quality (IQ) is modulated by the combined effects of diffraction (because optical bandwidth is proportional to pupil diameter) and blur generated by optical aberrations. In the absence of higher-order aberrations, defocus causes the diameter of the eye’s point spread function (PSF) to scale directly with pupil diameter for monochromatic and polychromatic light [1]. Further deterioration of the PSF due to higher-order monochromatic aberrations also increases with pupil size [2,3], an effect that is amplified in eyes with elevated monochromatic aberrations [4]. As pupil size increases, therefore, retinal contrast increases due to increases in the diffraction bandwidth, but contrast decreases because of blurring of the retinal image by aberrations. This tradeoff produces optimum photopic visual acuity and IQ with pupil diameters between 2 and 3 mm [5–9].

When considering visually weighted IQ [10], an additional factor, quantal fluctuations, influences the tradeoff between diffraction and aberration blur when retinal illuminance declines. At all light levels below high photopic levels [below 1000 Trolands (td)], reduced retinal illuminance reduces neural contrast sensitivity (CS) [11], primarily due to the impact of quantal fluctuations on signal to noise ratios (SNRs) [11–14]. The impact of photon noise on contrast detection is known as the de Vries–Rose or square root law [15]. Within the de Vries–Rose range, increasing pupil size will have the beneficial effect of increasing the SNR for detecting contrast by increasing retinal illuminance [7]. Thus, quantum fluctuations shift the tradeoff between factors that increase and decrease visual quality with pupil dilation toward larger pupils, with the optimum pupil diameter increasing from $<3\mathrm{mm}$ at photopic light levels to about 6 mm at mesopic light levels [6–9].

Given these multiple factors influencing the optimum pupil size over a range of ambient illumination levels, a simple question emerges. Will the increased impact of photon noise outweigh the beneficial effects of increased retinal image contrast provided by small pupils [16]? This question has gained increased relevance recently due to surgical treatments for presbyopia that implant small (1.6 mm diameter) artificial pupils into the cornea [17], which has raised concerns that these small pupils may be visually detrimental at low light levels [18].

To provide a framework for exploring the questions asked above, this report builds upon traditional optical models that capture the combined optical effects of diffraction and aberrations [19,20], both of which vary with pupil size [21,22]. These models are made visually relevant by adding the neural effects of human vision, e.g., by computing the neurally weighted IQ [10,23–25]. Typically, visually weighted optical modeling employs some measure of neural contrast threshold, but it does not explicitly include the effects of retinal illuminance. This deficiency is addressed here by making the neural contrast sensitivity function (CSF) depend explicitly on retinal illuminance [13,14]. By this method, we are able to examine the impact of pupil size on visually weighted metrics of IQ as light levels are changed.

The tradeoff between the effects of diffraction, aberrations, and photon noise can be formulated two ways that give seemingly different, yet mutually consistent, impressions of the importance of pupil size. One formulation describes the tradeoff when stimulus luminance is held constant as pupil size changes, which is the natural situation for most practical applications. Our analysis will show that optimum pupil size increases with decreasing stimulus luminance in this case. The second formulation describes the tradeoff when retinal illuminance is held constant as pupil size changes, an approach that provides greater insight into the underlying mechanisms. Our analysis will show that the optimum pupil is independent of retinal illuminance under these conditions.

2. METHODS

A. Computational Methods

1. Optical Model

We quantified the impact of pupil size and light level on IQ using an optical model written in MATLAB R2014b (www.mathworks.com) that uses wave optics (WO) theory to compute the optical PSF and optical transfer function (OTF) from the amplitude and phase of the pupil function [10]. To reveal the WO effects of diffraction + interference, results were compared with similar calculations, using geometrical optics (GO) that includes only aberration effects. The range of pupil sizes investigated (1–7 mm) covers most of the physiological range [26]. The phase function (i.e., wavefront error) included Zernike polynomials from the second order up to the sixth order, representative of presbyopic eyes [9]. Individual coefficients of the model are derived from the young adult population mean coefficient absolute values [27], which were scaled up by the reported age effects on higher-order aberrations [28]. Polychromatic computations employed 22 visible wavelengths in increments of 0.1 D of longitudinal chromatic aberration (LCA), where LCA was set to fit the average adult eye as defined by the Indiana Chromatic Eye model [29,30]. The polychromatic OTF (${\mathrm{OTF}}_{\text{poly}}$) and modulation transfer function (${\mathrm{MTF}}_{\text{poly}}$) for an equal-energy white-light spectrum were computed as the Fourier transform of the polychromatic PSF, computed as the luminance-weighted sum of 22 monochromatic PSFs [31]. Transverse chromatic aberration (TCA) was set to zero.

To model different pupil sizes, Zernike coefficients in Table 1 were rescaled by Schwiegerling’s method [21]. To simulate the effect of target distance for a nonaccommodating distance corrected eye, we ran through-focus analysis [32] by including defocus levels from $-3\mathrm{D}$ to $+1\mathrm{D}$ (increment 0.1 D) into the phase function. Zero defocus is defined as the target vergence that produces peak IQ for the reference source wavelength (589 nm) in an aberration-free model. Negative defocus means the eye is underpowered (i.e., the target is closer than the presbyopic eye’s far-point).

Table 1. Third- through Sixth-Order Zernike Aberration Coefficients (ANSI Standard Convention) for a 7 mm Pupil Used in Our Model^a

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2. IQ Metric Calculations

Quality of a polychromatic PSF was quantified in the spatial frequency (SF) domain for the optimally focused eye by three metrics. The first is the visual Strehl ratio determined from the polychromatic OTF (VSOTF) in a manner analogous to the monochromatic definition [20],

 

Fig. 1. Neural CSFs for different retinal illuminances (0.9, 9, 90, 500, 900, and 9000 td). Note the 900 and 9000 td superimpose.

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Fig. 2. VSOTF is plotted as a function of pupil diameter for geometrical (red filled symbols) and WO (black open symbols) at fixed retinal illuminances of 900 td (A) and 0.9 td (B) for four optical scenarios: diffraction-limited optics (circles), monochromatic aberrations only (squares), chromatic aberrations only (triangles), and both monochromatic and chromatic aberrations (diamonds). Values of VSOTF are for the optimally focused eye normalized to the monochromatic diffraction-limited 3 mm pupil case.

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Fig. 3. VSOTFs are plotted as a function of pupil diameter in mm for three fixed luminance levels: 2000 (black open circles), 10 (black open squares) and $1\mathrm{cd}/{\mathrm{m}}^2$ (black open triangles). For comparison, we also plot VSOTF at a fixed retinal illuminance level of 0.9 td (red thick line), and the SROTF is shown in blue filled squares.

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Fig. 4. Sample through-focus plots of polychromatic area MTF (A) and (B) and VSOTF (C) and (D) are plotted for our presbyopic eye model for a series of pupil diameters (1 to 7 mm) for high photopic [$1000\mathrm{cd}/{\mathrm{m}}^2$, (A) and (C)] and low mesopic [$0.1\mathrm{cd}/{\mathrm{m}}^2$, (B) and (D)] luminances. Absolute metric values are normalized to the best polychromatic IQ at 7068 td ($1000\mathrm{cd}/{\mathrm{m}}^2$) for a 3 mm pupil that generated denominators of 22.97 for area MTF, 900.51 for VSOTF.

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Fig. 5. Impact of pupil size on VSOTF IQ is plotted for three defocus levels (0, $-1$, and $-2$ diopters, black, blue, and red, respectively) and three luminance levels [1000 (A), 20 (B), and 0.1 (C) $\mathrm{cd}/{\mathrm{m}}^2$].

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Fig. 6. Polychromatic VSOTF is plotted as a function of retinal illuminance in Trolands for three defocus levels: 0 D (A); $-1\mathrm{D}$ (B); and $-2\mathrm{D}$ (C). Retinal illuminance was manipulated by using five stimulus luminances (1000, 100, 10, 1, $0.1\mathrm{cd}/{\mathrm{m}}^2$), producing five sets of data for each defocus level. Also, retinal illuminance was varied by varying pupil size (1, 1.2, 1.6, 2, 3, 4, 5, 6, and 7 mm), which generated the sets of nine data points connected by lines. The red solid lines with a slope of $+0.5$ and black dashed line with a slope of 0 represent the photon noise slope prediction and Weber’s law prediction, respectively. The right-most data point in each series is for the 7 mm pupils, and the left most data points for 1 mm pupils. A VSOTF equal to 1.0 is the maximum anticipated value obtained for well-focused retinal images of high retinal illuminance (7068 td) formed through an optimum (3 mm) pupil. The solid line with $\text{slope}=0.5$ was fit by eye to the upper left-most symbols in each data set. This reference line serves as an outer envelope for the data, while simultaneously indicating the anticipated behavior of VSOTF in the de Vries–Rose domain, where photon noise causes CS to be proportional to the square root of retinal illuminance. The intersection of these two reference lines occurs at retinal $\text{illuminance}=500$ td, which mirrors the transition illuminance (between 90 and 900 td) from Weber’s law to de Vries–Rose law reported in the literature [11,13].

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Fig. 7. NVC computed at 5 cpd (A) and CS at 5 cpd (B) and (C) are plotted as a function of retinal illuminance for focused (black squares) and $-2\mathrm{D}$ defocused (red triangles) grating stimuli, at three luminance levels (0.2, 20, $2000\mathrm{cd}/{\mathrm{m}}^2$) for a series of pupil diameters (see text for the detailed pupil sizes), illustrated as sets of data points connected by lines. Two lines with slopes of $+0.5$ and 0 are photon noise and Weber’s law predictions, respectively. The range of retinal illuminance levels within each group in (A) sequentially reflects 10 varying pupil sizes (1, 1.2, 1.6, 2, 2.6, 3, 4, 5, 6, and 7 mm), and similarly eight pupil sizes (1, 1.3, 1.6, 2, 2.5, 3, 4, and 6 mm) for the first subject in (B) and nine pupil sizes (1, 1.3, 1.6, 2, 2.5, 3, 4, 6, and 7 mm) for the second subject in (C). The model simulation in (A) has a $+0.6\mathrm{\mu m}$ spherical aberration of a 7 mm pupil, which is close to the SA of our two presbyopic subjects [9].

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Fig. 8. Effect of retinal illuminance on retinal IQ metric VSOTF for three optical models. Dashed reference line has slope 0.5 (pupil $\text{diameter}=3\mathrm{mm}$).

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In Eq. (1) ${\mathrm{NCSF}}_L$ is the radially symmetric, neural CSF for retinal illuminance $L$ in Trolands, ${\mathrm{OTF}}_{\text{Poly}}$ is the polychromatic OTF for the aberrated eye computed for some particular pupil size, ${\mathrm{NCSF}}_{7KTd}$ is the neural CSF for 7000 td. The VSOTF metric is well correlated with foveal visual acuity for high-contrast targets [24,33–35], and it emphasizes high SFs because the elemental area ${df}_x{df}_y$ in Eq. (1) is an area equal to the square of the elemental step size in SF.

The second metric of IQ is area MTF, defined as the area under the radially averaged ${\mathrm{MTF}}_{\text{poly}}$ and above neural threshold (NT) (i.e., the inverse of the neural CSF) [10],

The third metric is SROTF, the conventional Strehl ratio computed from the polychromatic OTF,

This metric was included because it employs the same optical components as the VSOTF but lacks weighting by neural CS. As with the VSOTF, this metric integrates across the diffraction limit set by the pupil diameter. Differences between VSOTF and SROTF as luminance and pupil size vary are therefore attributable to changes in neural CS caused by quantal fluctuations. Further details of these IQ metrics are described elsewhere [10,33].

To facilitate a direct comparison with psychophysical experiments using single-frequency gratings (rather than a continuous spectrum) with variable orientation, we modified Eq. (1) to produce a normalized measure of visual contrast (NVC) for individual SF $f$ defined as

3. Neural CSFs

The neural threshold curve published in Campbell and Green’s 1965 study [36] obtained with a retinal illuminance of 500 td is widely used in visual IQ metric calculations [10,23–25]. However, as light levels are lowered or raised from 500 td, neural contrast thresholds will increase or decrease because of quantal fluctuations, as shown psychophysically by Van Nes and Bouman [11] and computationally by Banks et al. [14] and Rovamo et al. [13] for sinusoidal gratings. Accordingly, photon noise effects were incorporated in our model by making ${\mathrm{NT}}_L$ dependent upon retinal illuminance. Neural contrast threshold functions were extracted from a modified whole-eye model of CS that is light-level sensitive [13]. The first step was to remove the optical component of the model by multiplying the model’s contrast thresholds by the model’s optical MTF. We then scaled the neural CS predicted at 500 td to match the experimentally observed neural CSF reported by Campbell and Green [36]. The neural CSFs, shown in Fig. 1, confirm that the model obeys Weber’s law at the highest light levels and lowest SFs. The model also exhibits square root law behavior (curves shift vertically in proportion to square root of illuminance) at all lower light levels and at all but the lowest SFs, in accordance with human vision [11,14].

B. Psychophysical CS Testing

Psychophysical grating CS varies in direct proportion to optical image contrast [36] and with the square root of retinal illuminance when photon noise determines sensitivity [11,14], and therefore provides a direct approach for assessing the combined effects of both optical and neural factors on human vision. In real-world situations, small pupils will affect both optical quality (reduced aberrations, increased diffraction) and NTs (increased photon noise effects due to reduced retinal illuminance). We examined these two effects of pupil size changes concurrently by maintaining a fixed stimulus luminance rather than the more typical approach of isolating the optical effects of changing pupil size by maintaining retinal illuminance fixed [37,38] or isolating the effect of neural changes by keeping pupil size fixed [11].

We ran psychophysical CS tests with five cycles per degree (cpd) gratings while controlling pupil size and luminance levels. This medium SF (5 cpd) stimulus was chosen to show the combined effects of photon noise, defocus, and aberrations. CS was not tested with higher SFs, due to their invisibility at low light levels [39] and/or with significant defocus. The freeware (FrACT) CS test [40] running on an iMac (Apple, Cupertino, California) displayed sinusoidal grating stimuli onto a rear projection screen with a DELL digital light projector. Grating contrasts at 5 cpd were attenuated by a factor of 0.1 by simply defocusing the projector, which reduces all displayed contrasts at 5 cpd by the same factor. All photometric calibrations were made with a Photo Research 0.25 deg spot photometer focused to sample a 1 mm diameter area of the grating stimuli. The optical attenuation was selected to display 5 cpd contrasts as low as 0.10%, and thus measure human contrast thresholds below 1% with the 8 bit D/A projector (actual contrast thresholds were $0.1\times$FrACT reported thresholds). The display luminance was linearized and SF calibrated using the tools provided by the FrACT program [40]. Contrast thresholds were determined using three repeated 18 trial four-alternative forced choice QUEST staircases [41].

CS was measured at 0.2, 20, and $2000\mathrm{cd}/{\mathrm{m}}^2$ to represent typically encountered environmental light levels. Pupil size was controlled with eight artificial pupils (1, 1.3, 1.6, 2, 2.5, 3, 4, and 6 mm) projected into the pupil plane by an optical system containing two telescopes, each with unit magnification. Two defocus levels (0 D and $-2\mathrm{D}$) were tested in a random order by adding trial lenses to the distance correction, which was imaged into the pupil plane through the same telescope. These two defocus levels are experienced by distance-corrected presbyopes viewing distant and 50 cm stimuli. Defocus was added to the best distance correction obtained subjectively with white light and natural pupils at high photopic light levels. Before starting the experiment, subjects dark-adapted for at least 25 min, and light levels and artificial pupils were placed in an ascending order, to ensure that the retina was progressively light-adapting during the experiment.

We tested two subjects with no ocular or systematic disease, and lacking accommodation (age $>50$ years [42]). We also used an open field autorefractor (Grand Seiko, Japan) [43] to confirm that each subject had zero measurable accommodation (no observable change in refractive error as subjects were instructed to focus on a near target). Levels of ocular spherical aberration ($C_4^0$ in μm), measured with the COAS aberrometer [33], were found to be $+0.5$ and $+0.6\mathrm{\mu m}$ for 7 mm pupil in the eyes of subject #1 and #2, respectively [9]. Long-lasting pupil dilation effects needed for this study were achieved with an initial drop of 2.5% phenylephrine combined with 1% tropicamide. During the experiment, pupil size was checked, and if it approached 7 mm, a single drop of 1% tropicamide was added. Experimental protocols were approved by the Indiana University institutional review board, adhered to the tenets of the Declaration of Helsinki, and required informed consent from each subject.

3. RESULTS

A. Optimum Pupil As a Function of Retinal Illuminance

The tradeoff between the effects of diffraction and aberrations as pupil size varies was examined without including the impact of pupil size on photon noise by evaluating visual IQ at fixed levels of retinal illuminance. The role of aberrations was assessed by comparing models that were aberration-free, or included monochromatic aberrations, chromatic aberrations, or both monochromatic and polychromatic aberrations. To isolate the role of aberrations and diffraction, the WO calculations were compared with a GO ray-tracing model (Fig. 2). In these computational experiments, neural CS did not vary with pupil size because retinal illuminance was held constant at either 900 [Fig. 2(A)] or 0.9 td [Fig. 2(B)].

The impact of diffraction and interference on retinal IQ is indicated graphically in Fig. 2 by the gap between VSOTF values for the aberration-free configuration as computed by GO (red filled circles) and WO (black open circles). For the WO solution, increasing pupil diameter from 1 to 2 mm dramatically increases VSOTF as the diffraction effects diminish, but further increases in pupil diameter provide more modest increases because the increased bandwidth associated with larger pupils benefits mainly SFs beyond the visual limit. For the aberration-free model, the diffraction effect is effectively eliminated for the largest pupil size as the WO solution asymptotically approaches the GO solution.

When typical levels of monochromatic aberrations are present, VSOTF declines approximately linearly with pupil diameter for the GO solution (red filled squares), and therefore the smallest pupil is also the optimum pupil. However, the WO solution (black open squares) indicates the presence of an optimum pupil diameter of about 3 mm resulting from a tradeoff: smaller pupils produce a diffraction penalty, whereas larger pupils produce an aberration penalty. Interestingly, VSOTF values for the WO solution exceed those for the GO solution when the pupil is large. This result raises the possibility that, in some circumstances, diffraction plus interference may partially protect the eye against image degradation by aberrations.

The effect of chromatic aberration in an otherwise aberration-free eye (triangles) is similar to that described above for monochromatic aberrations, again leading to an optimum pupil diameter of about 3 mm. For larger pupils, IQ is slightly greater when only chromatic aberrations are present than when only monochromatic aberrations are present, and worse when both are included in the model, which is consistent with experimental evidence [2]. With both types of aberration present, the optimal pupil diameter is reduced from 3 to approximately 2 mm.

All the observations and conclusions drawn above apply for high [Fig. 2(A)] and low [Fig. 2(B)] levels of retinal illumination, despite the fact that the ordinate scales in Figs. 2(A) and 2(B) are very different. We may conclude, therefore, that quantum fluctuations dramatically reduce visual IQ for a fixed pupil size as illuminance declines, in approximate proportion to the square root of the illuminance ratio (i.e., $\surd 1000$ in Fig. 2). This result is to be expected mathematically because neural CS scales with the square root of illuminance (Fig. 1) over most of the SF spectrum in the Rovamo model. We therefore conclude that, provided retinal illuminance is held constant as pupil size varies, optimum diameter is essentially independent of retinal illuminance, and for our presbyopic model remains between 2 and 3 mm. Mechanistically, this means the tradeoff between WO phenomena (diffraction and interference) and GO phenomena (optical aberrations) that optimizes medium-sized pupils is largely independent of quantum fluctuations even though neural CS changes dramatically with light levels.

B. Optimum Pupil As a Function of Stimulus Luminance

The results described above pertain to the laboratory situation in which an experimenter adjusts target luminance as pupil size varies in order to maintain constant retinal illuminance. Outside the laboratory, where luminance is typically fixed as pupil size varies, retinal illuminance grows in proportion to the square of pupil diameter. Consequently, visual IQ at each pupil size is associated with a different neural contrast threshold function for light levels where the square root law determines neural CSF. The consequences of this change in paradigm are shown in Fig. 3, which compares VSOTF for constant luminance ($1\mathrm{cd}/{\mathrm{m}}^2$) with constant illuminance (0.9 td). Retinal illuminance for these two conditions is the same for a 1.5 mm pupil diameter, and therefore VSOTF is the same, and the two curves cross. As pupil size increases, the increased retinal illuminance for the fixed luminance condition leads to elevated neural CS, and therefore increased VSOTF despite greater attenuation of image contrast due to aberrations. The converse is true for very small pupils, where the lowered neural CSF associated with the reduction in retinal illuminance adds to the detrimental impact of diffraction, creating a greater decline in VSOTF for the constant luminance condition. The net result is a doubling of optimum pupil diameter from 2 to 4 mm in this example.

A comparison of the visual quality metric VSOTF with the purely optical metric SROTF is also drawn in Fig. 3 to illustrate the detrimental effects of quantal fluctuations. At the highest luminance ($2000\mathrm{cd}/{\mathrm{m}}^2$) Weber’s law will apply to all pupil diameters, and thus the neural CSF will not vary with pupil diameter. Therefore, the differences between VSOTF and SROTF reflect the effect of neural weighting unencumbered by changing quantal fluctuations. For small pupils, neural weighting protects against loss of IQ due to diffraction, whereas for large pupils, neural weighting protects against loss of IQ due to aberrations. The SROTF metric is more susceptible to diffraction with small pupils and aberrations with large pupils because it includes invisible, very high SFs.

As stimulus luminance decreases from 2000 to $10\mathrm{cd}/{\mathrm{m}}^2$, VSOTF declines due to reduced neural CS caused primarily by quantal fluctuations. VSOTF declines more with small pupils than with large pupils due to the lowered retinal illuminances associated with small pupils. For example, VSOTF drops 0.82 log units when luminance drops from 2000 to $10\mathrm{cd}/{\mathrm{m}}^2$ for a 1 mm pupil, but the drop is only 0.45 log units for a 3 mm pupil, and a $<0.1$ log unit drop for the 7 mm pupils. With additional reductions in stimulus luminance from 10 to $1\mathrm{cd}/{\mathrm{m}}^2$, VSOTF is reduced by approximately the same value (0.5 log units) at all pupil sizes, which is predicted if the entire neural CSF is photon noise limited for all pupil diameters at both 10 and $1\mathrm{cd}/{\mathrm{m}}^2$.

C. Effect of Defocus on Optimum Pupil Size at High and Low Luminance

Results described above were for optimal focus, which was determined computationally by computing IQ for a range of defocus values as illustrated for two metrics (VSOTF and area MTF) in Fig. 4. These through-focus curves are instructive for showing how pupil optimization and depth-of-focus (DoF) are affected by quantum fluctuations. Both metrics include the human CS/threshold data as well as the OTF or MTF, but differ in their emphasis on low and high SFs (see Section 2, Methods). Weber’s law applies for the high luminance condition [$1000\mathrm{cd}/{\mathrm{m}}^2$, Figs. 4(A) and 4(C)], whereas quantal fluctuations decrease neural CSF as pupil size declines for the low luminance condition [$0.1\mathrm{cd}/{\mathrm{m}}^2$, Fig. 4(B) and 4(D)].

In the absence of blur due to higher-order aberrations (HOAs), we would expect peak IQ for a polychromatic point source of equal-energy white light to occur for slightly negative target vergence due to the 555 nm peak in the V-lambda curve (our eye model is emmetropic at 589 nm). We see this in the through-focus plots for small pupils (e.g., 1 mm diameter, VSOTF peaks at $-0.12\mathrm{D}$). Also, because of the significant Zernike spherical aberration included in the eye model, the embedded negative $r^2$ term introduces a small hyperopic shift in optimum focus that grows as pupil size increases. This behavior is revealed in all four panels of Fig. 4 by a rightward shift of the peak in each curve as pupil size increases, as noted previously [32]. These through-focus plots also confirm that the increased DoF provided by small pupils exceeds any possible expansion of DoF caused by elevated spherical aberration (SA) levels that accompany larger pupils [44]. As pupil size is reduced, the curves become flatter, which means nearly maximum IQ is preserved over a larger range of defocus. However, absolute IQ also declines as pupils get smaller, an effect that is especially dramatic for the VSOTF metric and dim light [Fig. 4(D)].

Reducing stimulus luminance 4 orders of magnitude (from 1000 to $0.1\mathrm{cd}/{\mathrm{m}}^2$), the peak value of area MTF drops by about a factor of 2 regardless of pupil size [note the different ordinate scales in Figs. 4(A) and 4(B)]. By comparison, VSOTF drops by a factor of 25 for large pupils and nearly a factor of 60 for the smallest pupil. This latter value approaches the $100\times$ reduction expected from the de Vries–Rose square root law, emphasizing that VSOTF, but not area MTF, reveals the impact of increased photon noise effects on CS at low luminance. This result is expected because the VSOTF is computed from the product of the OTF and the CSF, and therefore any increase in contrast thresholds due to photon noise are proportionally reflected in the VSOTF. This is not the case for the area MTF metric, which is based upon the linear difference between the MTF and contrast thresholds. Therefore, if photon noise increased thresholds from, say, 0.01 to 0.1 for a SF that was imaged with an MTF of 1.0 (i.e., a low SF), the contribution to VSOTF will be reduced by a factor of 10, but the contribution to area MTF for this SF would change by $100\%*[(1-0.01)-(1-0.1)]/(1-0.01)$, or only 9%. Also, because the VSOTF is defined as the two-dimensional integral of the product of the OTF and neural CSF, it will be dominated by high SFs even though the integrand is relatively low. The opposite is true for the area MTF, which computes the one-dimensional integral of the difference between radially averaged MTF and the neural contrast threshold function. This integral gives equal weight to all SFs, and the integrand is larger for low SFs, so the area MTF metric is dominated by low SFs.

Selected data from Fig. 4 are replotted in Fig. 5 to show how IQ varies with pupil size for constant levels of defocus. For zero defocus, optimum pupil size increases as stimulus luminance decreases. At mesopic luminance, the increased blur due to aberrations present in a 4–5 mm pupil is compensated partially by the increased neural CS associated with the higher retinal illuminance, resulting in an optimal pupil diameter between 4 and 5 mm. However, the opposite is true when image blur is amplified by adding $-1\mathrm{D}$ of defocus. In this case, reducing optical blur by making the pupil smaller improves IQ even though neural CS is lower for the small pupils. When defocus is raised to $-2\mathrm{D}$, the smallest pupil always produces higher visual quality in the retinal image regardless of stimulus luminance.

Although defocused IQ continues to improve as the pupil gets smaller, the gain at low luminance is less than that observed at high luminances. For example, for 1 mm pupils, VSOTF for $-2\mathrm{D}$ of defocus is 103.8 (0.2698/0.0026) times, 26.9 (0.069/0.00257) times, and 4.1 (0.0049/0.0012) times higher than for a 7 mm pupil at 1000, 20, and $0.1\mathrm{cd}/{\mathrm{m}}^2$, respectively. All three curves in Fig. 5 converge with a 1 mm pupil, indicating the impact of defocus on IQ is negligible with a small pupil at any luminance. The fact that small pupils retain their superiority at low luminance in the defocused eye implies that the gain from increased DoF with small pupils exceeds the loss due to the combined effects of photon noise and diffraction.

An important methodological conclusion to be drawn from the preceding narrative is that quantum fluctuations play an important role in determining visually weighted IQ. Except at high light levels, where Weber’s law operates, the impact of quantal fluctuations is determined by retinal illuminance level, which is in turn determined by stimulus luminance and pupil size. To reveal the impact of retinal illuminance on IQ, therefore, we replot the VSOTF data for a 5 log unit range of stimulus luminance as a function of retinal illuminance in Fig. 6. Groups of data connected by lines represent a single luminance levels and a range of pupil sizes. Due to the de Vries–Rose law [14], we expect to see the signature of quantal fluctuations in the form of a square root law for IQ, just as there is for CS [11,14]. This anticipated result is clearest in the VSOTF values for the well-focused eye [Fig. 6(A)], where several data points (filled symbols) from each set lie on the dashed reference line of slope $+0.5$. This constellation of special points form a type of “main sequence,” to borrow a term from astrophysics [45] and the oculomotor literature [46] that, in the present context, exhibits quantum-limited IQ as measured by VSOTF. Each stimulus luminance except $1000\mathrm{cd}/{\mathrm{m}}^2$ contributes to this main sequence, which we interpret as evidence that photon noise is the major factor influencing visual quality over the range of retinal illuminances we typically encounter. For smaller pupils, the data begin to depart slightly from the reference line, which we attribute to the effects of diffraction. For larger pupils in each data set, VSOTF again falls, which we attribute to the additional effect of optical blur due to aberrations. In this case, the increased blurring effect of aberrations for large pupils can overcome the benefits associated with improved SNR, resulting in a decrease in VSOTF in spite of increased retinal illuminance.

For the two highest levels of stimulus luminance (100, $1000\mathrm{cd}/{\mathrm{m}}^2$), peak $\mathrm{VSOTF}=1$, indicating that increasing retinal illuminance beyond the transition value of about 500 td provides no benefit. This is the signature of Weber’s law, as revealed by the IQ metric VSOTF. In this Weber domain, increasing pupil size beyond the optimum produces a cost (due to aberrations) without a benefit (due to increased retinal illuminance), and therefore image contrast is reduced because of aberrations, but there is no compensating benefit of improved photon SNR.

With $-1\mathrm{D}$ of defocus [Fig. 6(B)], VSOTF values are always below the main sequence line established for the well-focused eye [Fig. 6(A)], emphasizing that other factors besides quantal fluctuations play the major role in determining visual quality, which in this case is the added defocus. This interpretation is consistent with the observation that adding defocus shifts the balance point, resulting in a much smaller optimum pupil size [9]. The shift in the balance point is even more striking when defocus is increased to $-2\mathrm{D}$ [Fig. 6(C)]. For example, at $1000\mathrm{cd}/{\mathrm{m}}^2$ (+ symbols in Fig. 6), VSOTF drops as pupil is increased from 3 to 7 mm by a factor of $2.6\times$, $4.4\times$, and $4.6\times$ for focused, and $-1\mathrm{D}$ and $-2\mathrm{D}$ defocused presbyopic model. At lower luminance, the drop in VSOTF when expanding pupil diameter from 3 to 7 mm is retained in the presence of defocus, e.g., for the focused, $-1\mathrm{D}$ and $-2\mathrm{D}$ conditions, VSOTF drops by $1.3\times$, $2.2\times$, and $3.0\times$ at $1\mathrm{cd}/{\mathrm{m}}^2$, and $1.1\times$, $1.7\times$, and $2.0\times$ at $0.1\mathrm{cd}/{\mathrm{m}}^2$, respectively.

D. Psychophysical Tests of Model Results

We tested the conclusions drawn from our computational model by measuring human CS for a medium SF ($5\mathrm{cyc}/\mathrm{deg}$) at three luminance levels, two levels of defocus (0, $-2\mathrm{D}$) and for a series of pupil sizes. The results for two subjects are shown in Fig. 7 as a function of retinal illuminance using the same format as Fig. 6. As predicted, for the well-focused eye (black squares), our computationally determined normalized visual contrast [NVC, Eq. (4) in Section 2, Methods] at $5\mathrm{cyc}/\mathrm{deg}$ showed a main sequence of points that lie on a line of slope 0.5 indicative of quantum-limited IQ [Fig. 7(A)]. The transition from the de Vries–Rose domain to the Weber domain for NVC occurs at a lower retinal illuminance ($200\mathrm{cd}/{\mathrm{m}}^2$), as shown experimentally at this SF [11] and as predicted by a photon noise-limited neural model of contrast detection [14]. Deviations from this main sequence behavior are most noticeable with pupils $>5\mathrm{mm}$, where aberrations lower visually weighted image contrast in spite of increased retinal illuminance. In the presence of $-2\mathrm{D}$ of defocus (red symbols), the NVC data do not fit the main sequence observed with the focused eye, but instead pupil dilation $>1.5\mathrm{mm}$ leads to a significant drop in image contrast (mirroring the VSOTF data shown in Fig. 6). However, unlike the VSOTF data, the single SF defocused NVC data reveal distinct oscillations as pupil size increases, which reflect the familiar contrast-reversals (i.e., “spurious resolution”) in the OTF for eyes with positive spherical aberration and negative defocus [47–49].

A similar pattern of results was observed in the psychophysical CS data [Figs. 7(B) and 7(C)] for both subjects. For example, as pupil size and stimulus luminance change, both subjects exhibit CS that mirrors the main characteristics reported for the model: focused CS is dominated by quantal fluctuations below $200\mathrm{cd}/{\mathrm{m}}^2$, but with significantly lower CS than predicted by the de Vries–Rose law with large pupils; for luminance above $200\mathrm{cd}/{\mathrm{m}}^2$, Weber’s law dominates. Therefore, without any gain from increased retinal illuminance, the full effect of aberrations is revealed. In the presence of $-2\mathrm{D}$ of defocus, CS is lowered for all pupil diameters $>1.5\mathrm{mm}$, and again we see the familiar oscillations reported previously [48]. The quantitative differences between the model NVC and experimental CS data may reflect individual differences in the aberration structure and neural CSFs. The fact that the model’s predictions based on population averages from the literature are in reasonable agreement with the experimental performance of our individual subjects suggests the main features of the model’s predictions do not depend strongly on the specific aberration structure or neural sensitivity of individual eyes.

4. DISCUSSION

The pupil of the human eye varies in size from a minimum of about 2 mm experienced at very high luminance up to a maximum approaching 8–9 mm as environmental lighting drops to mesopic levels [50]. These pupil size changes are insufficient to act as an effective gain control mechanism (e.g., as luminance changes by ${10}^6\times$, pupil area changes by only $4\times$), but instead they retain a near optimum pupil for imaging high SFs in well-focused eyes [6,7,9]. To gain a clear understanding of the underlying mechanisms that affect visual quality as luminances and pupil sizes change, we developed an optically based model of the human eye that included the retinal illuminance dependency of the neural CSF. Unlike a purely optical model, which is independent of light level, neural CS [11,13] is dominated by two well-established rules of human vision. At high photopic levels ($>1000$ td), Weber’s law holds true (CS is unaffected by retinal illuminance), but at lower levels of retinal illuminance, CS is limited by quantal noise fluctuations known as the de Vries–Rose or square root law. This innovation in our modeling allowed us to examine the impact of pupil size at different lighting levels because stimulus retinal illuminance is the product of luminance and pupil area.

The impact of quantal fluctuations on neural CS and their interaction with the OTF of the eye can be appreciated if we consider a simple experiment in which a target with mean luminance $L_0$ and spatial contrast $C=\mathrm{\Delta }L/L_0$ is viewed through a neutral density filter with attenuation factor $F$. The effective target luminance is therefore ${\mathrm{FL}}_0$ but contrast, as a ratio, is unaffected by the filter. When performance for target detection is limited by quantal fluctuations, the detectability $d^{\prime }$ of the target is $d^{\prime }=C\sqrt{{\mathrm{FL}}_0}/\sqrt[4]{1+C}$ [51]. The denominator of this expression accounts for the superior performance of an ideal observer for detecting contrast decrements ($C<0$) compared to contrast increments ($C>0$). For present purposes, we ignore this asymmetry associated with the sign of contrast by using the approximate formula $d^{\prime }=C\sqrt{{\mathrm{FL}}_0}$. Thus, the contrast required to achieve a fixed level of performance is $C=d^{\prime }\sqrt{{\mathrm{FL}}_0}$. For simplicity, assume threshold performance in a detection experiment corresponds to $d^{\prime }=1$, in which case CS is $S=1/C=\sqrt{{\mathrm{FL}}_0}$. When written in terms of retinal illuminance $I_0=L_0*A$, where $A=\text{pupil}$ area, CS is $S=1/C=\sqrt{{\mathrm{FI}}_0/A}$.

Optical aberrations and diffraction will attenuate retinal image contrast by an amount determined by the MTF and the SF spectrum of the target. For a point source of light, the SF spectrum is flat and therefore the SF spectrum of the retinal image will equal the eye’s MTF. If the observer’s CS at any given SF $f$ is designated $S(f)$, then neural sensitivity expressed in terms of the attenuated retinal image will be $N(f)=S(f)/\mathrm{MTF}(f)$ [36]. For example, suppose a grating with SF $f$ is at threshold for detection when $C=0.5$, then $\mathrm{CS}=S=1/C=2$. If the eye’s MTF at that SF is, say, 0.25, then retinal contrast for the same $\text{grating}=0.5*0.25=0.125$, so “neural contrast sensitivity” $N$ at this frequency is $N=1/0.125=8=S/\mathrm{MTF}$. Rearranging the equation, we conclude that for a point source of light, $S(f)=N(f)*\mathrm{MTF}(f)$. In other words, stimulus CS is the product of neural CS and the eye’s OTF [36].

The IQ metric VSOTF is obtained by integrating $S(f)$ over SF. Thus, by the arguments given above, VSOTF should be proportional to $\sqrt{{\mathrm{FI}}_0/A}$. Consequently, when pupil area A is held fixed, VSOTF should vary in proportion to $\sqrt{{\mathrm{FI}}_0}$, the square root of retinal illuminance. As shown in Fig. 8, this prediction was verified by numerical calculations for three optical models: $\text{Case}1=\text{physical}$ optics calculation of the PSF of an aberrated optical model (normal LCA and the monochromatic Zernike aberrations of the presbyopic eye); $\text{Case}2=\text{physical}$ optics calculations of the PSF for an eye with no aberrations (neither monochromatic nor LCA); $\text{Case}3=\mathrm{GO}$ calculation of the PSF for the same aberrated eye as in Case 1. In all three cases, the computed value of VSOTF varied as the square root of retinal illuminance (dashed reference line has slope 0.5) as predicted by the preceding arguments. Thus, the prediction of square root behavior is robust enough to survive major variations in the optical model of an eye.

Although the calculations reported in Fig. 8 were carried out for a particular pupil size (3 mm diameter), the square root behavior of VSOTF should apply for any pupil size. Thus, a second prediction is that the relationship between pupil size and logVSOTF for a fixed level of retinal illuminance will have the same functional form regardless of retinal illuminance. This prediction was verified by the computed results shown in Fig. 2, where the curves for all three cases described above simply slide up and down the logVSOTF axis without changing shape or relative position as $F$ is varied to change stimulus luminance over the full domain of the de Vries–Rose law (0.9–900 td). There are some subtle differences in the shapes of the 900 and 0.9 td data in Figure 2, which we attribute to the reduced spatial bandwidth of the neural CSF at lower retinal illuminances (Fig. 1), since lower SFs are less affected by optical degradation that are high SFs. Because the plots in Fig. 2 share the same shape, they reveal that the optimum pupil will be independent of retinal illuminance in an experiment that holds retinal illuminance constant as pupil area $A$ varies.

The above calculations make a third prediction for an experiment configured to hold target luminance $L_0$ constant as pupil area changes (this was our psychophysical experimental method). In that case, optimum pupil area should vary with $F$ because retinal illuminance is changing as pupil area changes. The data in Figs. 3 and 5 reveal that this increase in optimum pupil diameter occurs as long as the de Vries–Rose law makes neural CS vary with luminance, but is absent at higher photopic luminance where Weber’s law applies.

Psychophysical studies of CS typically are designed to study neural properties of human vision, and to accomplish this, optics must remain constant, which is achieved by employing a fixed artificial pupil (e.g., 2 mm pupil diameter [11]). Alternatively, to isolate the impact of optics, retinal illuminance must be held constant (e.g., 2150 td [5]). However, in real life, both pupil size and retinal illuminance vary together as environmental luminance levels alter. Our CS measurements (Fig. 7) were designed to capture the combined effects of stimulus luminance and pupil size on visual quality, and they confirmed that our light-level sensitive model provided an accurate representation of human vision in which neural and optical components both vary with pupil size.

Being able to include the impact of pupil size and environmental lighting levels into a model of visual quality has direct relevance to the issue facing the clinical community exploring use of small pupils as a treatment for presbyopia. For example, will the increase in optical DoF generated by reducing pupil size (Fig. 4) successfully overwhelm the increased photon noise effects associated with the accompanying reduced retinal illuminance? When pupil size shrinks to 1 mm, photon noise problems combine with diffraction to significantly worsen focused image qualities at all but the very highest luminance. But, in presence of $-1\mathrm{D}$ and $-2\mathrm{D}$ defocus (Figs. 5 and 6), the 1–2 mm pupils always yield the best overall image quality (VSOTF) at all retinal illuminance levels. Significant gains in defocused IQ could be obtained while minimizing the negative impact on focused vision by limiting the pupil miosis to avoid very small ($<2\mathrm{mm}$) pupils.

As environmental light levels change, both luminance and pupil size vary [26], and it has been suggested that active pupil adjustment retains pupil size at or near that required to optimize visual acuity [6] and IQ [9]. We examine this hypothesis directly in Fig. 9, where we compare VSOTF for focused images achieved with the typical pupil size of this presbyopic population [26] to that achieved with two different fixed pupil diameters (1.6–7 mm) and with optimum pupils. As expected, VSOTF values obtained with a large 7 mm pupil match those achieved with the actual pupil (7 mm) at $1\mathrm{cd}/{\mathrm{m}}^2$, but they fall significantly behind the natural pupil (0.3 log units) at $1000\mathrm{cd}/{\mathrm{m}}^2$. The converse is true for the small 1.6 mm diameter pupil, which produces VSOTF values matching those of the natural pupil at $1000\mathrm{cd}/{\mathrm{m}}^2$ (4 mm), but it produces a VSOTF that is 0.3 log units lower than that obtainable with the natural pupil at $0.1\mathrm{cd}/{\mathrm{m}}^2$. Clearly, the natural pupil is superior to either a fixed large (7 mm) or fixed small (1.6 mm) pupil.

 

Fig. 9. Effect of luminance on retinal IQ metric VSOTF for 7 mm (blue filled circles), 1.6 mm (black open triangles), optimum pupils (blue open circles) and natural pupils (red filled squares) at six luminance levels (0.1, 1, 10, 20, 100, and $1000\mathrm{cd}/{\mathrm{m}}^2$). Natural pupil diameters were 7 mm, 7 mm, 6 mm, 6 mm, 5 mm, and 4 mm for 0.1, 1, 10, 20, 100, and $1000\mathrm{cd}/{\mathrm{m}}^2$, respectively [26].

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Significantly, we find that the VSOTF IQ with natural pupils approaches that of the optimum pupils at all light levels (0.1 log unit lower VSOTF values), which is consistent with the hypothesis that invokes image optimization as the driving force behind changing pupil size [6]. The slightly suboptimal natural pupil sizes are always larger than our predicted optimal pupil diameters, which are 4.0, 4.0, 3.6, 3.52, 3.16, and 2.54 mm for light levels of 0.1, 1, 10, 20, 100, and $1000\mathrm{cd}/{\mathrm{m}}^2$, respectively [9]. The model-predicted smaller optimum pupil sizes could reflect excessive weighting of high SFs by the VSOTF metric. Although sensitive to high SFs, human vision also relies heavily on low SF content in the environment [52,53], and image contrast at lower SFs is affected less by defocus [36], but below about 100 td, low SF CS is photon noise limited (Fig. 1). Therefore, photon noise will have a larger effect than optical blur at the lower SFs, and for low luminances, optimized pupil size will tend to be larger. Also, in young eyes with lower levels of HOAs than present in our presbyopic model, larger pupils will be more beneficial at lower light levels [6]. The converse is also true. When aberrations are higher, optimum pupil sizes will be smaller, which raises the possibility that individual differences in pupil size may reflect individual differences in aberrations.

In summary, by revealing the competing forces at work when ambient light levels change, we achieve a clearer understanding of why more light improves vision. As stimulus luminance increases, reflexive constriction of the pupil reduces blur from aberrations, which compensates partially for increased diffraction. Although pupil constriction reduces SNR of quantum fluctuations, the net effect is still a gain in IQ. For example, if increasing luminance tenfold from 10 to 100 reduces pupil size from 6 to 5 mm, then retinal illuminance increases by $10*{(5/6)}^2=7$ times. At lower light levels, this increase in retinal illuminance will increase retinal IQ by about $\surd 7=2.6$, as shown in Fig. 9. From a clinical perspective, our modeling results provide convincing evidence that small pupils should provide an effective presbyopic correction option that will improve near IQ at both photopic and mesopic luminance with a relatively small loss of best-focused IQ that can be regulated by careful choice of small pupil size. Careful assessment of night vision will provide important insight into the safety and practicality of implementing small pupil corrections for presbyopia.