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Abstract
A gradual drop in visibility with obliquely incident light on retinal photoreceptors is namely described by the Stiles-Crawford effect of the first kind and characterized by a directionality parameter. Using a digital micromirror device in a uniaxial flicker system, here we report on variations of this effect with luminance levels, wavelengths within the visible and near-infrared spectrum and retinal regions ranging from the fovea to 7.5° parafoveal. Results show a consistent directionality in mesopic and photopic conditions. Higher directionality is measured for longer wavelengths, and a decrease with retinal eccentricity is observed. Results are discussed in relation to an absorption model for the visual pigments taking the outer-segment packing and thickness of the neural retina into account. Good correspondence is found without enforcing photoreceptor waveguiding.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The efficiency of light to trigger vision when obliquely incident onto the retina is described by the psychophysical Stiles-Crawford effect of the first kind (SCE-I) [1]. The SCE-I is characterized monocularly using a subjective visibility comparison of retinal fields produced by two Maxwellian sources of light projected onto the eye’s pupil: one near its center (reference) and another (test) at various locations across the pupil plane [1,2]. Two different methods have traditionally been used: (i) flickering of overlapping fields sequentially with incremental brightness [1,3–11] and (ii) simultaneous comparison of adjacent or concentric fields [12–15]. In either case, the intensity of the test (Itest) and/or the reference (Iref) field is adjusted until the brightness perception of the two is matched and an effective visibility, η, is determined as the intensity ratio at each pupil point of entry, . This visibility function is fitted to a Gaussian distribution in the pupil plane [2,12]:
where is the pupil position of maximum visibility and, is the directionality parameter that quantifies the drop-off rate with increased obliqueness at a given wavelength, λ. In photopic conditions, the directionality measured at the fovea is typically ∼0.05 mm−2 but it decreases for eyes with large axial lengths [10,13] and varies slightly with wavelength being typically smallest in the green spectral range for foveal vision [12,14,15]. The SCE-I is mainly a cone effect being largely absent in scotopic conditions where rod vision dominates [5,7]. The peak location of the sensitivity curve determines the pointing direction of cones [2,8] that often has a nasal bias which is more pronounced for myopic eyes [13]. The transition from Maxwellian (point) to Newtonian (normal) view is nontrivial and underestimates the role of the SCE-I in normal vision by up to an order of magnitude as we have recently shown using a mechanical flickering pupil with diameter in the range of 1.4 to 7.4 mm diameter [16] and followingly confirmed by others using a small flickering aperture on a spatial light modulator [17].Most SCE-I studies have been performed with foveal vision. Yet, the microscopic structure of the retina and the density of photoreceptors changes markedly towards the periphery. Westheimer [3] proposed to relate the variation in shape of the cones across the retina (i.e., rod-shaped at the fovea versus cone-shaped in the periphery) to an increase in directionality at eccentricities up to 3.75° in the temporal retina as determined with deep-red light (Wratten filter 29). Enoch and Hope [4] found similar results up to 10° using light-red light (Wratten filter 23A) but subsequently Bedell and Enoch [6], still using light-red light, reported a reduction in directionality to approximately foveal values at 35° eccentricity notwithstanding the very different shape of the parafoveal cones. In turn, scotopic vision shows a lack of directionality [5,7] despite of the fact that rods are similar in shape to foveal cones. Together, these findings show that the shape of individual photoreceptors does not provide a satisfactory explanation of the SCE-I neither in photopic nor scotopic conditions, but rather the density of visual pigments is a possible cause [16,18]. Also, it has been suggested that dynamical phototropism differences between rods and cones might be involved [19].
The SCE-I is accompanied by a related hue shift, viz., the Stiles-Crawford effect of the second kind (SCE-II), observed after brightness matching of the SCE-I [12]. Whilst the SCE-I has typically been explained by angular-dependent waveguide coupling [2,20], self-screening has detailed the SCE-II [21,22]. This incompatibility seems to suggest that waveguiding may play less of a role in vision than commonly assumed [16,18] although it may well be a relevant factor in high-resolution photoreceptor imaging [23].
The changes in SCE-I with retinal eccentricity is of relevance to understand whether it may play a role for emmetropization. Using light-red light, Choi et al. [13] found a gradual increase in directionality towards 15° eccentricity for emmetropes ( ± 0.75 D), moderate myopes (−3 to −5 D) and high myopes (<-5 D). Retinal thinning and elongation of outer segments have been reported for myopic eyes [24,25]. Thus, microscopic structural changes may plausible relate to changes in the SCE-I directionality and the density of visual pigments at different retinal locations.
Here, we report on SCE-I measurements from the fovea to the parafovea at up to 7.5° eccentricity using a uniaxial flicker system with a digital micromirror device (DMD) [10] at three different wavelengths within the visible spectrum and, to our knowledge, for the first time in the near-infrared. The dependence on retinal luminance is studied from scotopic to photopic conditions. All findings are discussed in relation to an absorption model for the visual pigments using a ray-optics model valid in the limit of weak absorption [16].
The paper is structured as follows: a thorough introduction to the volumetric absorption model is covered in section 2, followed by the experimental procedure performed to characterize the SCE-I at the fovea and parafovea with different brightness conditions and wavelengths, in section 3. Section 4 includes the directionality parameter results of the psychophysical SCE-I and its comparison to the modeled SCE-I plots. Discussion and conclusions of this study are found in sections 5 and 6, respectively.
2. Volumetric SCE-I modeling
The SCE-I is commonly attributed to waveguide properties of the photoreceptors resembling optical fibers without account for nonguided light [20]. However, discrepancies are inevitable due to low contrast of refractive indices within the cone-photoreceptors, the short length of inner segments (IS) and outer segments (OS), cellular irregularities, and the dense packing of photoreceptors that breaks the ideal conditions of isolated waveguides. Thus, the waveguide model cannot account for the highly diminished SCE-I in rods or for the appearance of the SCE-II. A geometrical-optics model of light absorbance based on the volumetric overlap between a beam of incident light and photoreceptor OS, was recently proposed by the authors [16]. This model accounts for the fragment of light that lies upon the OS, stimulating the visual sensation. The fraction of absorption for a single ray of light can be expressed in terms of Beer-Lambert’s law and is proportional to the effective visibility [16]:
where L is the total propagation distance across the OS with absorption coefficient , N is the molecular density, and is the molecular absorption cross section. To a first approximation Eq. (2) becomes linear, i.e., . Thus, the effective visibility is determined as the intersecting volume of the OS and the incident light beam.As Maxwellian view is typically used to characterize the SCE-I, here, a collimated plane-wave illumination is modeled as a cylindrical beam of parallel light rays, and its intersection volume with an array of cone-photoreceptor OS is calculated for different angles of incidence rotating about the entrance to the OS (rotating about the OS midpoint, as representative of the average location of incident light, results in almost identical results [16]). The diameter of the light beam is chosen to be inversely proportional to the density of cones of a certain kind, as neighboring cones of the same type will have a screening effect on that being analyzed. A foveal cone density of σ = 160,000/mm2 provides a hexagonal center-to-center spacing of 2.7 μm [26], to which a typical correspondence of 5% S-cones, 30% M-cones and 65% L-cones where broad absorption peaks centered at 420 nm, 534 nm and 564 nm respectively, are assumed [22]. Thus, lower parafoveal densities of cone OS are modeled with a wider beam of light. The length of the cone OS varies with retinal eccentricity with significant variations between reported values [20,27,28]. For the calculation of the OS and light intersection volume, we assume cylindrical OS length of 45 μm for the foveal cones and a 2 μm diameter [28]. In turn, for the parafoveal cones we assume an OS length of 35 μm with a slightly tapered conical reduction in diameter from 3 μm to 2 μm, and center-to-center spacing of 7.6 μm. A slight taper of the OS may partially account for the conical taper of the ellipsoid. For a schematic eye of axial length fAL = 22.2 mm, the angle of incidence of the light on the retina is given by , where r is the pupil entry location measured from the pupil center. The incident beam is angled between with respect to the axis of the OS, limited by a maximum pupil size of 8 mm. Figure 1 includes the normalized intersecting volume of the light beam with (a) a single OS and (b) a group of 3 identical OS in a hexagonal grouping for foveal cones. Figure 2 shows the same calculation for cylindrical and tapered conical OS of parafoveal cones with an assumed density of σ = 20,000/mm2.
Fig. 1 Normalized intersection volume between a light beam (blue cylinder) with a single and a group of 3 cylindrical cone OS (gray cylinder) of length 45 μm for foveal cones assuming 5% S-cones, 30% M-cones and 65% L-cones and a total density of σ = 160,000mm-2.
Fig. 2 Normalized intersection volume between a light beam (blue cylinder) with a single cylindrical and tapered cone OS (gray cylinder) of length 35 μm for parafoveal cones assuming 5% S-cones, 30% M-cones and 65% L-cones and a total density of σ = 20,000mm−2.
The dimensions of the OS here used are approximate values and can easily be varied to analyze different scenarios. A narrower OS will result in a slightly reduced drop-off with obliqueness whereas a wider OS will result in a slightly increased drop-off (differences <5%). In turn, a shorter OS will result in a slower drop-off and a longer OS in a faster drop-off with obliqueness (differences up to ~30%). Variations in the intersection volume with respect to the length and diameter of an individual cylindrical OS are represented in Fig. 3.
Fig. 3 Role of OS length (left) and diameter (right) in the calculation of the volumetric overlap between a single cylindrical beam of light and a cylindrical OS assuming green light with 30% M-cones and a foveal cone density of σ = 160,000 cones/mm2.
3. Experimental procedure
A uniaxial flicker system using a DMD has recently been implemented by the authors to characterize the SCE-I monocularly at the fovea [10]. Here, the system has been adapted for parafoveal measurements aided by a fixation LED (green) placed on a moveable stage and mounted in a plane conjugate to the retina. A schematic of the system is shown in Fig. 4. A quartz halogen lamp with a multimode fiber bundle (NA = 0.22) and a near-infrared (near-IR) laser, in the case of visible and IR light respectively, are used as a light source to illuminate the active area of the micromirror array. The DMD (Vialux V-7001 VIS) is programmed in LabVIEW to flicker between a 2.3 visual degrees circular reference, and a series of pupil scanning test patterns in monocular Maxwellian view at a flicker rate of 0.5 Hz. Both the reference and the test fields are the same size projected by the DMD which alternates between them in such a way that the subject sequentially sees the reference or the test field for 1 second each. This device and a 20° top-hat diffuser are placed in respective conjugate planes to the subject’s pupil, by 4-f systems with unit magnification. At each of these planes a 0.2 mm diameter bright spot is projected to provide a Maxwellian view. The very narrow beam makes the impact of ocular aberrations minimal as the light becomes a spherically expanding wave within the eye. This is also evidenced by the uniform appearance of the light within the retinal illumination test and reference patch. The presence of aberrations would cause variations of incidence angle and a possible spatial variation of perceived brightness. However, here, the translational motion of the illumination in the retinal plane is minimal when scanning the pupil by adjusting pupil centration and axial distance, and for subjects with refractive error, they wear their personal contact lenses, thereby ensuring that defocus and tip/tilt is minimal. The brightness of the test field is adjusted by the duty cycle of the micromirrors to an upper limit of 22.7 kHz, until the brightness is subjectively matched to that of the reference field. A Meadowlar liquid-crystal tunable color filter is used to set the wavelength at 450, 550 and 650 nm with corresponding bandwidths of 28, 22 and 30 nm respectively within the visible spectrum, whilst the near-IR laser provides a narrow bandwidth centered at 785 nm.
Fig. 4 Schematic of the uniaxial flicker system using a DMD to characterize the monocular SCE-I equipped with a LED fixation point. Section a) shows the reference field impinging on the fovea, while b) represents an off-axis pupil entry position of the test field for a parafoveal retina measurement. The DMD, diffuser and eye’s pupil are placed at conjugate planes by a unit magnification 4-f system to create a Maxwellian view.
Subjects’ right pupil were dilated with two drops of 1% tropicamide causing accommodation to be partially paralyzed. Centering of the pupil to the system was achieved in each case using an XYZ stage suited with a bite bar to reduce head movement. A single measurement of the SCE-I entailed scanning the test pattern horizontally across the pupil at 11 evenly spaced entrance positions and repeated 4 times to determine the average value and standard deviation. At each pupil entrance position, the subject was asked to adjust the duty cycle of the test pattern via a remote control until brightness was matched to that of the reference, corresponding to a luminance of 3.32 cd/m2. After testing the dependence of the SCE-I on luminance ranging from scotopic to photopic conditions, the chosen brightness for further measurements corresponds to the mesopic region where the directionality parameters plateaus. To measure the SCE-I at the fovea subjects looked straight on while for parafoveal measurements, analyzed at 2.5- and 5-degrees temporal retina and 2.5, 5 and 7.5 degrees nasal retina, the fixation LED was used to maintain a steady gaze while matching the reference and test pattern brightness subjectively. Centration was re-adjusted at each position of the LED until both sequentially imaged outermost test images at either extreme of the horizontal line across the pupil were similarly visible.
Measurements with green light were performed with the right eye of 6 emmetropes (up to ± 1D) and 2 myopes (marked with an asterisk) and 5 of those emmetropic subject were also measured with blue (450 nm) and red (650 nm) light. Finally, measurements with near-infrared light were performed for four of the emmetropic subjects, including the authors (AC and BV). Ocular data of all subjects can be found in Table 1. Myopic subjects wore their personal contact lenses for all measurements. The axial eye lengths were measured using an ultrasound biometer device (PalmScan A-2000 by Micro Medical Devices) and the refractive errors were determined using an auto-refractor (EyeNetra) with a nominal accuracy of 0.35 diopters. All procedures were in accordance with the Declaration of Helsinki for experiments involving human subjects and approved by the Human Research Ethics Committee at University College Dublin.
Table 1. Axial length and spherical refractive correction measured for the right eye of all subjects, including the authors AC and BV. The asterisk (*) denotes myopic subjects.
Results for the SCE-I measured at the foveal and parafoveal retina at the three different wavelengths are fit to the volumetric overlap absorption model by adjusting the total cone density at each retinal location, the diameter of light beam to comply with percentage of each cone type, as well as the likelihood of having adjacent cones of the same kind.
4. Results
Firstly, the dependence of the SCE-I with luminance of the reference field with a spectral bandwidth of 22 nm centered at 550 nm has been tested in the right eye of four emmetropic subjects. The power at the exit pupil of the system was measured with a power meter and converted to luminance in the logarithmic scale. Numerical values of the tested powers have been included in Table 2. Figure 5 graphs the results obtained from the psychophysical determination of the SCE-I for the different luminance values for foveal and 5° nasal parafoveal retina, with color coded scotopic, mesopic and photopic areas. At low brightness, the SCE-Iis not playing any significant role, as expected from previous studies [5,7], but directionality is seen to increase rapidly once cones begin to contribute to the visual sensation produced. A 63% increase is observed for a luminance rise of 1.294 cd/m2 (gray zone). After this point, the directionality appears to plateau, within the error, to an average value of ρ = 0.07 mm−2 at the fovea and ρ = 0.05 mm−2 at the parafovea. On average, a 30% reduction is observed at 5° nasal retina to that at the fovea. This differs from other parafoveal measurements done by others [3,4], which found that the directionality parameter at 10° nasal parafovea was up to 60% higher, on average, than that at the fovea. This may be a consequence of the different measurement technique as previously two-channel systems were used, or subject differences.
Table 2. Luminous power measured at the optical system's exit pupil and corresponding luminance values.
Fig. 5 Psychophysical dependence of the SCE-I with brightness variation measured with 550 nm light at the fovea and at 5 degrees nasal retina for 4 emmetropic subjects, covering scotopic, mesopic and photopic luminance conditions. Error bars correspond to ± 1 S.D.
Secondly, the determined dependence of the SCE-I directionality parameter, ρ(λ), at the fovea with wavelength is shown in Fig. 6(a) for the right eye of 5 emmetropes and 1 myopic subject. On average, the directionality parameter at the fovea is 0.044 ± 0.001 mm−2 and does not vary significantly when measured with blue or green light. However, on average, a 22% higher directionality is recorded for red and near-IR light. The common dip in directionality for green light reported in the literature [12] is only seen for 3 of the 6 subjects (subjects 2, 3 and 5). The green wavelength is between the absorption maxima of M- and L-cone opsins. Thus, both M- and L-cones contribute to the sensation produced corresponding to a larger grouping of cones with similar absorption. As seen in Fig. 1, grouping of OS results in a reduction of the directionality parameter. At wavelengths away from the absorption maxima of the cone opsins the absorption is less which may be modeled by either a lower in Eq. (2) or a shorter length of the OS when compared to the wavelength of highest absorption. This also results in an effective reduction of directionality. The directionality differences in green between subjects may plausible be due to different M- and L-cone densities although also the differences represented by error bars in Fig. 6(a) may account for some of the variations.
Fig. 6 Plot of (a) the measured directionality parameters at the fovea against the wavelengths for 6 subjects and (b) the directionality variation with retinal eccentricity along the horizontal cross-section of the pupil for 8 subjects. The asterisk (*) denotes myopic participants. Error bars correspond to ± 1 S.D.
For the parafovea, the right eye of all 8 subjects were measured using green light at retinal eccentricities ranging from 5° temporal to 7.5° nasal in steps of 2.5°. Results can be seen in Fig. 6(b). Overall, the SCE-I cone directionality parameter decreases with eccentricity, however, approximately a 10% increase is noted over the first step, at 2.5° from the fovea, which has been verified to occur in the nasal and temporal, as well as upper and lower retinal directions and is supportive of Westheimer’s findings [3]. Thereafter, 39% and 25% reductions were noted for consecutive 2.5° eccentricity increases, respectively, reaching up to a 52% decrement at 7.5° nasal compared to the foveal value. This drop-off occurs at smaller eccentricities than found by Bedell and Enoch [6], who found a lower directionality at 35°.
Similarly, the SCE-I directionality was measured both at 450 nm and 650 nm wavelengths. For blue light (Fig. 7(a)) sequential directionality reductions of 10, 16 and 28% per 2.5° retinal increase is observed, albeit with higher variability between subjects. In the case of red light (Fig. 7(b)), a distinct 24% increase is observed again at 2.5° nasal, temporal, upper and lower retina, followed by consecutive 23 and 24% reductions. A summary of all average directionality parameters and their respective percentages of reduction with respect to the ρ-value measured at the fovea for each wavelength is included in Table 3. In terms of the low directionality for rods, the large groupings of adjacent rhodopsin in the rod OS may well explain the low directionality [16] similar to the grouping effect for cones seen in Fig. 1.
Fig. 7 Directionality parameter against retinal eccentricity plots measured using a) 450 nm and b) 650 nm wavelength light for 5 emmetropic subjects including the authors. Error bars correspond to ± 1 S.D.
Table 3. SCE-I directionality parameter values measured at each wavelength and retinal eccentricity, and the corresponding percentage of reduction with respect to foveal values.
Third, based on the volumetric absorption model of photoreceptors described in section 2, the experimental data of the SCE-I for both authors (AC and BV), normalized and re-centered, have been compared to the theoretical plots of light absorbance in accordance to the angle of incidence on the retina. Similar plots have been obtained for the remaining subjects but, for simplicity, are not shown here. In each case, a single photoreceptor OS has been modeled as cylinder with 2 μm diameter and 45 μm length at the fovea, and tapered cylinder from 3 μm to 2 μm and 35 μm length at 5° nasal parafovea. Light beam diameters are obtained assuming a cone density of 160,000/mm2, 2.7 µm center-to-center spacing, and scaling according to the percentage of each cone type. Both a single cone and a group of 3 adjacent cones have been considered for best fit, taking into account the likelihood of a cone having neighboring cones of the same type. Comparison plots can be seen in Fig. 8.
Fig. 8 SCE-I characterization comparison between the normalized psychophysical experimental data and the theoretical plots based on the volumetric light absorption model of photoreceptors at a) the fovea with 45μm length cylindrical 2 μm outer segments, and b) parafoveal retina with 35 μm length tapered outer segments from 3 μm to 2 μm, for both authors AC and BV, and 450, 550 and 650 nm wavelengths. To ease comparison, curves at 550 and 650 nm have been vertically displaced 0.5 and 1 points, respectively. Error bars correspond to ± 1 S.D.
5. Discussion
The system used to characterize the psychophysical SCE-I provides directionality values at the fovea in good correspondence to those reported in the literature [8,12,14,15]. Its dependence with luminance at the pupil plane (Fig. 5) has shown that in very dim conditions this effect is virtually imperceptible, confirming the lack of visual directionality of rod photoreceptors in scotopic conditions [5,7] and explained by the large grouping of rhodopsin in adjacent rod OS [16]. The lower ρ-value measured at 5° nasal parafovea falls within prediction as the density of rods increases with eccentricity, and therefore the SCE-I is diminished. A drastic increase in directionality is seen in the mesopic region, when both cone- and rod-photoreceptors produce the visual sensation. A threshold is set at 3 cd/m2, after which the SCE-I maintains a constant contribution as luminance is increased up to the photopic region.
The wavelength is seen to not have a great influence on the directionality parameter in the lower half of the visible spectrum but its impact increases towards longer wavelengths, as reported by Lochocki et al. [14], and stabilizing into the near-IR region where the same L-cones drive the visual sensation produced. The cone-to-rod ratio gradually decays with distance from the fovea, causing the directionality parameter to diminish. The greatest appreciation of the effect is seen when measured using green light (Fig. 6(b)) and in terms of the volumetric absorption model this is explained by an increased contribution from both M- and L-cone opsins to the net absorption. In turn, when only a single cone type contributes to the sensation produced the directionality is expected to be higher due to a reduced screening by adjacent photoreceptors of a different cone class.
The geometrical-optics volumetric absorption model [16] has shown to fit well to this data. On one hand, as the density of cones lessens at higher eccentricities, the diameter of the modeled cylindrical incident light gets larger, as explained in section 2. This causes the modeled SCE-I curves to widen resulting in a lower directionality. On the other hand, photoreceptor outer segments located in the fovea tend to have a longer OS structure, which translates into a steeper curve provided by the model. Both these contributions are in good agreement with the experimental data seen in Fig. 6(b) and Fig. 7, where lower directionality is found at higher eccentricities and complies with results reported by Morris et al. [9]. However, consistent differences between the modeled SCE-I curve and the experimental data noted nearer the edge of the pupil in Fig. 8, could be caused by light propagating from the modeled OS into its neighboring cones of a different kind due to the overlap of the absorption spectra of each type of photopsin. Inclusion of one more term in the series expansion of Eq. (2) will reduce the predictions at the periphery of the modeled results further. For example, if measuring with 28 nm bandwidth red light centered at 650 nm, M-cones neighboring an L-cone can absorb a fraction of light according to the overlap of its absorption and incident light spectra. To best fit the model to experimental data, the addition of the percentages of each cone type exceeds 100%, which can also be explained by spectral overlaps between the photopsins absorbance and that of the incoming light. Finally, this effect may also account for the hue shift described by the SCE-II, as more oblique incidence will result in more light leakage reaching neighboring cones with a different spectral sensitivity. Thus, the SCE-II hue shift is absent when incident on axis of the photoreceptors but increases with obliqueness and towards both ends of the visible spectrum in agreement with measurements and expectations [12,22]. Taking the volumetric model one step further to account for partial absorption by other photoreceptors will be carried out in future studies.
6. Conclusions
Light capture efficiency with obliquely incident light on the retina is analyzed with a uniaxial SCE-I flicker system using a DMD to scan a 7 mm pupil with Maxwellian light and allow brightness matching via the duty cycle of the micromirrors. The SCE-I has been tested under luminance variations, confirming a minimum effect in scotopic vision, but maximized throughout the mesopic and scotopic luminance regions. A distinct variation with wavelength is reported, with similar SCE-I curves observed within the lower-half of the visible spectrum, and 22% higher directionality parameters for the upper-half and the near-IR region. The SCE-I has a highest impact at the fovea when measured with green light, where visibility is maximized, and a decrease with parafoveal eccentricity by up to 50% at 7.5° nasal. A similar variation is reported on average with blue light albeit higher variability between subjects. Slightly less of a drop in the directionality parameter with retinal eccentricity is seen with red light.
A volumetric absorption model [16] based on the geometrical overlap between incident light and the cone-photoreceptor outer-segments has proven to be adequate for the SCE-I visibility curves at foveal and parafoveal retinal regions, while accounting for cone-photoreceptor density and OS lengths. Ultimately, though, we expect that a full electromagnetic model will provide a more detailed understanding [18,29].
Funding
H2020 ITN MyFUN grant agreement No. 675137.
Acknowledgements
The authors would like to thank all the subjects who have participated in this study.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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