Directional sensitivity of the retina: A layered scattering model of outer-segment photoreceptor pigments.

Photoreceptor outer segments have been modeled as stacked arrays of discs or membrane infoldings containing visual pigments with light-induced dipole moments. Waveguiding has been excluded so fields diffract beyond the physical boundaries of each photoreceptor cell. Optical reciprocity is used to argue for identical radiative and light gathering properties of pigments to model vision. Two models have been introduced: one a macroscopic model that assumes a uniform pigment density across each layer and another microscopic model that includes the spatial location of each pigment molecule within each layer. Both models result in highly similar directionality at the pupil plane which proves to be insensitive to the exact details of the outer-segment packing being predominantly determined by the first and last contributing layers as set by the fraction of bleaching. The versatility of the microscopic model is demonstrated with an array of examples that includes the Stiles-Crawford effect, visibility of a focused beam of light and the role of defocus.


Introduction
Since the first observation of radiative mode patterns in light transmission by single photoreceptor cones [1] it has been widely accepted that photoreceptor cones and rods are biological waveguides with light concentration capabilities [2]. This may explain the strong directionality of cones in accepting light and the associated Stiles-Crawford effect of the first kind (SCE) [3,4] and plausibly also the appearance of mode-like radiation patterns seen in high-resolution retinal reflection images [5,6]. Nonetheless, it is striking that imaging across the layers of the in vivo retina is feasible and that different directionality parameters can be observed with optical coherence tomography when addressing the photoreceptor cells at different depths without effective angular filtering of backscattered light by waveguide modes [7]. Post mortem high-resolution optical microscopy of light transmission by photoreceptors depends on retinal tissues removed from the dark-adapted eye shortly after sacrificing the animal [1,2] and are therefore subject to severe perturbations from the natural state of the living eye. Normal visual function relies on less intense light that may not reach to the retinal pigment epithelium before being absorbed by visual pigments and thus the inclusion of only waveguided modes, even if present or effectively excited on the short scale of propagation, may oversimplify the light-photoreceptor interaction in vision.
High resolution in vivo retinal images show bright cones and rods presumably due to reflections caused by high-index mitochondria concentrated in the inner-to-outer segment junction, the outer-segment termination, and the retinal pigment epithelium layer. In turn, the visual pigment containing membranes and discs, that are responsible for the mechanisms of visual function, cannot be directly visualized with in vivo optical techniques. Perturbed reflective fluctuations caused by shedding of discs [8] or outer-segment gaps [9] have been observed with optical coherence tomography, and rapid bleaching-induced reflectivity changes have been revealed with flood-illumination ophthalmoscopy [10], but none of these have given direct evidence for the interaction of light with the visual pigments in the process of vision. Recent studies have attempted to bridge the gap by addressing retinal imaging while targeting single cones albeit still not permitting the actual 3-D interaction of light with the outer-segment pigments to be resolved or selectively probed [11].
Toraldo di Francia first pointed out that the photoreceptors have antenna-like properties with identical radiative and collective properties because of optical reciprocity [12]. This triggered the formulation of cylindrical waveguide models for the retinal receptors [3,4] and the development of retinal emulators based on waveguiding [13][14][15]. The biophotonic structure of the outer segment itself has been considered using finite-difference time-domain calculations in 2-D [16] and in 3-D confirming directionality [17][18][19]. The elevated refractive index of the outer segment is spatially modulated at the nanoscale by the densely packed pigment-containing membranes [20] which will perturb a simplistic waveguide model. Here, a microscopic approach is taken by considering the individual pigment molecules as induced point-dipole antennas [21,22] that coherently radiate light (or equally absorb incident light) in characteristic patterns defined by the coherent sum of them all across the multilayered outer segments of individual cones or rods. This links directly to the molecular absorption and thus gives insight into the electromagnetic role of the individual visual pigments in the process of vision. Rhodopsin has a static dipole moment [23] but for the interaction with light only the induced dipolar perturbation matters which is contained in the molecular polarizability and is directed along the incident electric field in the plane of the retina. Waveguiding is omitted in this study to allow the field to diffract beyond the physical boundaries of each photoreceptor cell into the matrix surrounding each cone and rod. Light incident from the rim of an 8 mm dilated eye pupil would strike the retina at an angle of ~10° and therefore requires a propagation length on the order of 12 µm to cross a 2 µm diameter outer segment. This makes it highly unlikely that multiple total internal reflections can efficiently drive waveguide modes and discriminate against uncoupled modes before the light is either absorbed or reaches the outer-segment termination. For incidence angles closer to the center of the pupil the assumption of multiple total internal reflections in the outer segment becomes even more compromised due to the short outer-segment length.
The paper is organized as follows. In section 2, two models (macroscopic and microscopic) are introduced that consider the combined radiative effect of an arrayed stack of circular apertures and an arrayed stack of membranes containing regular arrays of point dipoles driven by an externally-incident field, respectively. Numerical results are presented in Section 3 that include SCE directionality, scattering of a focused Gaussian beam by an ordered array of outer segments, with emphasis on the role of defocus and visibility, and the role of a conical outer segment on light confinement and acceptance angle. In Section 4 the results are discussed which is followed by the conclusions in Section 5. The variety of examples is used to demonstrate the strength of the analysis to give predictions that may stimulate new experiments aimed at furthering our understanding of the photoreceptor-light interaction for vision prior to neural mechanisms and for high-resolution retinal diagnostics.

Methods
The light acceptance angle of outer-segment pigments may be estimated by considering their collective radiative properties because of optical reciprocity. In the following, two different but related approaches are discussed for a schematic eye that excludes all intraocular scattering and absorption mechanisms apart from those contained in the photoreceptor outer segments themselves. First, a simplified arrayed aperture model which does not include pigment details, but only the parallel arrangement of outer-segment infoldings or discs, is introduced. Well-aligned rods and cones point toward a common pupil point [24] with the outer-segment infoldings or discs oriented perpendicular to the axis of their respective photoreceptor. Second, a dipolar antenna model that considers each visual pigment as an independent emitter or receiver of radiation is considered. The first model is macroscopic and based on optical diffraction from apertures to allow rapid calculations of paraxial scalar light propagation. The second model is microscopic and considers the full electromagnetic vectorial field propagation (and acceptance) of each visual pigment molecule due to the lightinduced component of its dipolar moment. This is similar to the use of elementary Huygens' wavelets when considering diffraction from apertures in optics. Thus, whereas the first model is only valid in the optical far field the latter allows calculations even in the vicinity of each molecule and disc membrane to distances within a fraction of the wavelength and can easily be extended to include the electromagnetic near field [21,22]. As will be shown both models give the same predictions at the pupil plane and thus the choice of model is dependent on the level of detail required but only the latter is suitable in the vicinity of the photoreceptors. Both models do not include absorption by the pigments per se but rather ad hoc as a damping from large depths of the outer segments. The term 'disc' will here refer to either rod outer-segment discs or cone membrane infoldings that are the outer-segment layers. The scattering properties of each element are assumed to be independent and thus multiple scattering is excluded and self-screening is only considered in the absorption along the photoreceptor axis. This is done for simplicity although it may lessen the accuracy of the predictions if near resonant conditions [21] or for very oblique incidence of light. The two models are shown schematically in Fig. 1. For convenience both models are shown in their normal representation, i.e., as diffraction from apertures and as scattering from single light-induced dipoles. The back illumination of the apertures is due to the fact that it is being used to model the reverse situation of scattering from stacked reflective discs that each will have the same far-field light distribution as the well-known case of aperture diffraction. Fig. 1. Photoreceptor outer-segment model: (a) schematic of a stacked-aperture array with N layers to represent the radiative and collective properties of visual pigments in the cone membrane infoldings or rhodopsin molecules of rod discs giving rise to a diffracted field, E diffr . (b) Stacked array of outer-segment dipoles driven by an incident field, E 0 , and the resulting Rayleigh-scattered field, E sc , from dipole (n,m) propagated a distance, R.

Arrayed aperture model of outer-segment pigment discs
A first approximation to the light-acceptance angle of outer-segment pigments can be made by considering the coherent radiation from a linear array of parallel equidistant circular apertures [see Fig. 1(a)]. This stems from the fact that the molecular density of visual pigment across each layer is high [25] and a uniform pigment density may therefore be assumed when calculating only the far-field radiation pattern. From optical reciprocity it thus follows that the radiative properties of a linear array of apertures will correspond to the collective properties of a linear array of discs with uniform pigment density.
When propagated to the pupil plane of a schematic eye with focal length f eye = 22.2 mm and refractive index n eye = 1.33, the diffracted paraxial far-field from a stacked array of N apertures can be calculated as the coherent addition of all of their individual contributions where r p is the radial coordinate in the pupil plane. In Eq. (1) n eye z f h n δ = + + − is the axial distance from aperture n to the pupil plane, the length of the inner segment to the first layer of the outer-segment is denoted h and will be taken as 30 µm, and δ is the spacing between adjacent apertures (discs) that will be taken as 20 nm. The outer-segment length is ( ) . Each aperture has a diameter d chosen as 2.0 µm. The dimensions are based on typical electron-microscopy data from the literature [25][26][27]. It should be stressed that Eq. (1) does not include any back reflections from the outer-segment cell wall as total-internal reflection is being excluded on account of the very localized dense pigment confined to the discs or membrane infoldings and ordinary reflections would be very weak given the small difference in the effective refractive index of the outer segment and the inter-photoreceptor spacing. If a 1.40 refractive index is assumed between the layers of the outer segment it would correspond to a 5% increase of the effective outer-segment length or a 5% decrease of the effective wavelength with a correspondingly small impact on the results. The refractive index of outer segments is determined by transverse-illumination interferometry averaging over many single layers of pigments and intra-spacing matrix [20,28] making it tremendously challenging to resolve dielectric variations on the scale of single membranes. For this reason the bilayer structure of the membranes is also excluded as the overarching aim is only to propagate the field to a distance where the effective electromagnetic relation between the light at the pupil and at the outer segment is determined predominantly by the average spacing and number of outer-segment layers. The wavenumber is 2 / eye k n π λ = where λ is the free-space wavelength. The incident field is propagated from aperture-to-aperture using 0, 1 0, exp( ) n n E E ikδ + = − to ensure that the fields are locked in-phase at the pupil. This is done since the approach is aimed at modeling the reverse situation of an incident wavefront of light entering the eye through the pupil before being distributed across the layers of the retina.
If only one disc is included (N = 1) Eq. (1) leads to the well-known Airy disc intensity distribution with an Airy spot diameter 2.44 / ( ) at the pupil. A single diffractive aperture model of the photoreceptor cones has previously been used to model the opticalscattering equivalent of the SCE [29] which was found to be more directional than its psychophysical counterpart [4,30].
Damping by pigment absorption in the photoreceptors may be introduced ad hoc in Eq. (1) by scaling the field amplitude E 0,n at each aperture in accordance with Beer-Lambert's absorption law where the fraction of light in the outer segment decreases exponentially with segment length. This is typically expressed by where σ is the molecular absorption cross section, N V is the density of pigments, and β is the absorption coefficient. For an outer-segment length L, the integrated absorption, 0 η , that triggers the visual system is therefore where 50% absorption is equivalent to 0.69 L β = . Equation (2) dampens the influence of deeply-located visual pigments unless the outer segment becomes strongly bleached for which 0 0 η → and likewise strong absorption is given by 0 For non-visual applications and retinal imaging with long-wavelength infrared light the illumination can penetrate the entire outer segment length ( 0 β → ) and thus, as far as light capture is concerned, all layers need to be included in the resulting directionality.

Induced dipolar-antenna model of stacked and ordered pigments
The scattering of light by individual induced dipoles may be considered as contributing to the elementary diffracted waves from the aperture model considered in Sec. 2.1. It adds insight, however, and permits calculations close to the pigments in any direction and at any point outside of the induced point dipoles themselves which are idealized representations of the nanometric space occupied by each pigment molecule. It is therefore not limited by paraxial optics or scalar waves but is a representation of the full electromagnetic propagation of a dipolar-driven electromagnetic field. The scattered field from an array of induced dipoles in the pigment molecules can be expressed as the sum of fields emitted by the individual dipoles. Each dipole field has a near, a middle, and a far-field component where only the last is retained at distances beyond a fraction of the wavelength [22]. Thus the total scattered far field from N stacked layers of uncoupled dipoles (in membrane infoldings or in discs) each containing M dipoles can be written as a sum of their dipolar spherical waves at any point in space has been assumed identical and isotropic for each pigment molecule, ε 0 is the permittivity of free space and the molecular radius is a. This is an idealization as the molecular polarizability is different in the plane of each membrane and outside of it but as the electric field of the incident light is predominantly in the plane of the retina only that part of the polarization will matter. The effective dielectric constant of the pigment is denoted ε d whereas the dielectric constant of the surrounding cellular matrix tissue is denoted ε t . Based on refractive indices of 1.40 for the dense outer segment [20] and 1.33 for the schematic eye, respectively, these will be taken as ε d = 1.96 and ε t = 1.77. For a molecule radius a = 2.5 nm [25] the polarizability is 38 6.04 10 The polarizability scales the strength of the scattering but it will not impact the distribution of the scattered light in any other way. The incident driving field is E 0 at location r m,n for dipole m,n. The incident field is only included as the driving mechanism for the scattering of light. From reciprocity of the dipolar antennas their directional scattering pattern is identical to their collective radiation pattern that when absorbing light triggers vision. In Eq. (4) all other retinal scattering mechanisms have been excluded, and thus as in Eq. (1) no surrounds of the cell have been included. This implies that dipoles from one photoreceptor can scatter freely towards or receive from other photoreceptors. Equation (4) excludes possible waveguiding that, if present, will only affect wave components propagating very close to the optical axis that may well not reach the cell wall within the length of the outer segment. In light of this assumption, the light-capture efficiency of the densely-packed visual pigments immersed in a uniform refractive index of the cellular matrix is expected to conform with Eq. (4) where the dense arrangement of induced dipoles is equivalent to narrow parallel layers of increased refractive index as modeled in Eq. (1).
Assuming linear polarization (x-polarization) of the incident light, the induced dipole moment of each pigment will be in the plane of its disc or membrane infolding. Thus, Eq. (4) for the total scattered field can be simplified to the following 2 2 , , 2 The assumption of linear polarization is important for a single dipole but not when adding the contribution from many dipoles where the combined effect is essentially the same as for unpolarized light. Equation (5) can be used to calculate the propagating scattered field and intensity at any location , m n ≠ r r both near and far from the array of the outer segment, and with only minor modifications can the middle and near-field terms be included [22].

Numerical analysis
Both models have been implemented in Matlab TM using a standard computer with a dual-core 2.53 GHz processor for numerical analysis to examine the capability of the models to produce directional light scattering (and acceptance) curves that resemble those of waveguides but without enforcing waveguiding. The arrayed aperture model is computational efficient and allows pupil intensity distributions to be found in less than 1 min. The impact of the contributing layers and the effective length of the outer segment were analyzed both without and with absorption for different fractions of bleaching. The spectral dependence of the directionality was examined as well as the influence of spacing variations in the layers.
The dipolar model was then used to examine propagating light distributions both near and far from the outer segments to show conformity between the macroscopic and microscopic models and to examine the possibility that mode-like scattering patterns might occur in the vicinity of the outer segment. Light scattering (and acceptance) patterns within the outer segments for oblique angles of incidence for both single and arrays of cone photoreceptors were examined and the light gathering efficiency by an array of outer segments for a focused and for a defocused beam of light were examined along with the possible role of a conical taper. The dipolar model is more computational demanding and single images were calculated in times of up to 6 hours.
The coherent scattering by the stacked dipolar pigments allows analysis of the possible leakage from the outer segment for different angles of incidence of the light. The angle of incidence on the retina, θ, for the schematic eye model is related to the pupil entrance point, Varying the angle of incidence of a plane wave at the retina thus allows an effective directionality to be determined from the amount of light contained within the outer segment. Individual outer segments may be organized into an array to represent a patch of retinal photoreceptors. A total of 19 hexagonally-packed outer segments spaced at a center-tocenter distance of 1.22d, where d is the diameter of each individual outer segment, has been analyzed. This patch can be used to evaluate possible radiative transfer between outer segments, to examine the impact of aberrations of a focused incident beam at the retina, and to analyze the role of scanning the beam across the outer segments to perform visual testing for different retinal locations or to do high-resolution retinal imaging from backscattered light originating inside of the outer segments. The incident beam is an x-polarized Gaussian beam propagating in the -z direction and expressed in cylindrical coordinates as a function of radial coordinate, p r , and axial coordinate, z [31]

Results
The methods described in Sec. 2 have been used to analyze light scattering properties of a simulated individual outer segment and a hexagonal array of outer segments to elucidate the impact of the parameters under different conditions.

Arrayed aperture model of outer-segment pigment discs
The results presented here are for a single outer segment of diameter d = 2.0 µm. Figure 2 shows the diffracted field intensity at the pupil obtained from Eq. where ρ is the characteristic directionality parameter and r p is the distance in the pupil plane from the entrance point of highest visibility. The curve becomes increasingly narrow and the photoreceptor more directional when the length of the outer segment is increased provided that all layers contribute equally which is representative of bleached conditions (or equivalently that all layers contribute such as in the case of long-wavelength infrared light). With low irradiance (mesoscopic to dim photopic range) a long outer segment would be expected to absorb the incident light in a limited number of contributing layers rather than the entire outer segment, i.e., effective N N < . Thus, the effective directionality parameter, ρ, is smaller for unbleached conditions than for bleached conditions where the entire outer segment is equally exposed to light. This agrees with psychophysical observations [32][33][34]. A narrow outer segment has reduced directionality which accounts for the fact that rods are less directional sensitive than cones [35]. For N = 1000: ρ = 0.044/mm 2 for d = 1.0 µm (for simplicity not shown here) and ρ = 0.0763/mm 2 for d = 2.0 µm. Psychophysical characterization of rods is done in the scotopic range and thus again effective N N < applies with a resulting reduced directionality. In turn, intense illumination in transmission of isolated rods from tissue produces high directionality as all outer-segment layers contribute [36].
The outer segment of parafoveal cones is shorter than for foveal cones suggesting a smaller directionality but this is counteracted by their larger diameter. Indeed, parafovea SCE measurements have found the cones to be more directional than at the fovea [37,38]. The quality of the fit to the SCE function is reduced for long outer segments where the deviation near the pupil rim increases. Note that the case of N = 2000 seen in Fig. 2 corresponds to a long outer-segment length of 40 µm that is totally bleached so that each layer contributes equally. This represents the optical SCE for which there is limited data available in the literature and that typically is captured across a 6 mm pupil [30,39] making it difficult to infer if the deviations seen in the simulations for 8 mm pupils occur in the experimental case.
The spectral variation of the predicted directionality can be obtained from Eq. (1) producing results like shown in Fig. 3 at three distinct wavelengths. As can be seen the predicted directionality decreases with increasing wavelength in agreement with experimental observations for the optical SCE [39]. Psychophysical measurements performed for a single pupil-entrance point at a time agrees only with this in parts of the visible spectrum [32,40]. This may be understood from the fact that different pigment densities are available for the S, M and L cones that weights the role of contributing layers differently across the spectrum when characterizing the SCE of the first kind. Thus in the middle of the visible spectrum, where the sensitivity of the eye is highest, less light is required and fewer layers contribute causing a reduction in the apparent directionality for green light [33]. and for even dimmer light the directionality approaches that of the single aperture case shown in Fig. 2. This agrees with experiments of the SCE and directionality following bleached conditions [32][33][34] and also with the fact that the optical equivalent of the SCE is more directional as it is performed in bleached conditions [4,30]. One may speculate if the change in directionality could be caused by a change in the refractive index rather than a change in effective length but this would require an approximate doubling of the optical path and thus a doubling of the refractive index which is not a realistic proposition. The curves plotted in Fig. 4 (from 0.000001%, i.e., 10 -6 % to 100%) cover a luminance range of 8 log units. Absorption has been omitted in the following dipolar analysis but the ad hoc approach expressed in Eq. (2) and Eq. (3) has been used to verify excellent correspondence between the aperture and dipolar antenna models for both bleached and unbleached cases. If light is very obliquely incident on the retina it may potentially traverse more than one outer segment. This is most likely to occur for bleached conditions whereas for dim light the absorption would effectively happen in a fraction of the outer segment corresponding to the case of effective N N < and a reduced risk of outer-segment light leakage.

Dipolar antenna model of stacked and ordered pigments
Each outer segment contains a large number of pigment molecules on the order of 10 6 or more with a molecular diameter of approximately 5 nm [25]. The molecules are distributed in an approximately square grid across each layer as evidenced by atomic force microscopy of rod discs [41]. Figure 5 shows the outcome for the field intensity in the pupil plane 2 2 , , sc x sc y I E E = + when adding the dipolar fields at the pupil with 740 molecules included in each disc. The results should be compared with those in Fig. 2 for the arrayed aperture model and as can be seen the estimated directionalities at the pupil are highly similar for the two models (differences are smaller than 0.8% for 1 disc and only 0.3% for 2000 discs). The effective directionality is dictated by the apparent aperture size rather than by the much wider directionality of the individual dipolar radiation patterns as a result of field interference. The dipolar model also allows calculation of the field in other planes and at other locations. Again it should be stressed that only the far-field dipole propagator was included in Eq. (4) as it is the only term that contributes to the field at the pupil [22]. Figure 6 shows results for the far-field part of the intensity pattern both within and outside of the outer segment when a different number of layers are included. Interestingly, the light appears partially focused by the arrayed dipolar antenna at about 2 µm above the outer segment giving the appearance of a narrowing of the photoreceptor. Inside and beyond the outer segment an annular mode-like pattern can be observed similar to that of a higher-order waveguide mode [1,4].
The dipolar model can be used to visualize light scattering when the light is incident on or off axis in order to judge the amount of possible leakage from the outer segment. Results for different angles of incidence in the x-z plane are shown in Fig. 7 for selected x-y and x-z cross sections. This shows axial confinement of the light for on-axis incidence but leakage for oblique incidence. Interestingly, for oblique incidence a fraction of the light remains within the outer segment as it scatters forward and gradually widens. The total power contained at its midpoint can be integrated to give a characteristic SCE curve that in this case matches with .
In bright light most of the outer segment contributes to the absorption and in consequence the directionality is highest. In dim light, absorption would be pronounced near the entrance to the outer segment and the directionality would be smaller in agreement with Fig. 4 and Fig. 5. As the entire outer segment in Fig. 7 is illuminated by an incident plane wave it will have lower directionality than in the real case with an array of outer segments that essentially screen for each other. Thus, in order to estimate individual photoreceptor directionality it is more appropriate to consider the radiative characteristics of the layered dipolar antenna array as demonstrated in Fig. 2 -Fig. 5. Note that the yz-sections in Fig. 7 show increased scattering with penetration depth in the outer segment which is a result of constructive interference of scattered light from each layer that becomes exponentially dampened once absorption is included by use of Eq. (2). The model allows assembling of the pigments into packed outer segments to simulate coherent light scattering of an incident plane wave by an array of photoreceptors. This is shown in Fig. 8. It can be seen that the oblique incidence again shifts the light towards one side in the outer segments (similar to that seen in Fig. 7 for an isolated photoreceptor) but some light is also present between the photoreceptors that partially spills into neighboring outer segments. This spilling is reduced when the outer segment is conical. A dark ring is formed around each outer segment when the light is axially incident. This is caused by destructive interference of the scattered light and it is also observed in experimentallyobtained high-resolution retinal images of photoreceptor cones [6]. It should be stressed that screening by neighboring outer segments has not been included in Fig. 8 but it would essentially restrict the incident light to the upper entrance of the outer segments. The situation would resemble that of a confined illumination to a single outer segment such as by a focused incident beam as explored in Fig. 9-Fig. 11.  It is important to examine what happens for a focused beam of light as this resembles the situation of viewing the ocular point-spread function as well as the cases of wavefront sensing and retinal imaging from backscattered light. For this purpose an incident Gaussian beam has been assumed which produces a spot size (radius) of w 0 = 1.00 µm that is focused on the entrance of the outer segment (0D) or slightly beyond (negative diopters) or before (positive diopters) for the central photoreceptor in the array. This is seen in Fig. 9 that shows an interesting asymmetry with respect to the defocus that can be explained by the different light scattering mechanisms in the forward and backward directions from the layers of the modeled outer segments. When focused at the upper entrance of the outer segment (0D) the beam spreads again towards the far end of the outer segment resulting in a slightly widening at the middle of the outer segment (as seen in the 0D image). When focused near the end of the outer segment (i.e. near −0.05D) the backscattered beam becomes slightly focused resulting in a reduced size of the spot in the middle of the outer segment (as seen in the −0.05D image). This refocusing effect for backscattered light is explained by the mirror-like reflections from each of the outer-segment layers as shown schematically below the images in Fig. 9. Larger defocus results in light at the neighboring outer segments and increased leakage of light outside of the outer segments. Fig. 9. Role of defocus seen in a cross section through the middle of the outer segments for an array of 19 hexagonally-packed outer segments each with N = 1000 layers each containing 12 dipoles. The array is illuminated by an incident Gaussian beam with beam waist (radius) w 0 = 1.00 µm striking axially onto the central photoreceptor and scattered by the array. The defocus in diopters refers to defocus of the focused incident beam with respect to the outer-segment entrance so that −0.05D is focused near the far end of the outer segment and positive values are focused before the outer segment is reached. The schematic drawings of a single outer segment show the case of focusing the incident light near the (a) upper entrance and (b) far-end exit. Blue arrows indicate the direction of back-scattered light from each layer causing a focusing effect when focused within or beyond the outer segment. Fig. 10. Cross section at the middle of the outer segments for an array of 19 hexagonallypacked outer segments each with N = 1000 layers that each contain 12 dipoles. A Gaussian beam of light with waist (radius) w 0 = 1.00 µm is axially incident and focused on the entrance facet at different horizontal locations of the outer-segment array and scattered. Left: the entire array is seen when illuminated by a plane wave, the following images towards the right show the scattering produced when the Gaussian beam is shifted increasingly towards the right. On the right-hand side: the beam is incident at the midpoint between two outer segments. For the centrally incident beam (0,0) cone 'b' carries only 5.6% of the power contained in cone 'a' and thus all 6 cones surrounding cone 'a' carry 33% of the power; when the beam is displaced by 25% of the outer-segment spacing cone 'b' carries 21% of the power with respect to the power in cone 'a'; and when displaced to the midpoint between two outer segments the ratio cone 'a' and 'b' carry an equal amount of power. In this case, the total power within the outer segments is 67% when compared to the on-axis case. θ. The dotted white circles show the 2.0 µm diameter of each outer segment.
The model allows analysis of the visual sensitivity to a focused beam of light entering on or off a given photoreceptor axis as shown in Fig. 10 by comparing the scattered power contained within all outer segments in its vicinity. This is similar to recent psychophysical experiments in which a modulated beam of light is being targeted to single parafovea cones [11,42]. A comparison of cones 'a' and 'b' shows that the total scattered power within the outer segments diminishes when the light is incident at an off-axis position in fair agreement with experimental values found outside of the central fovea [42]. The amount of light that is available to trigger visual pigments is dependent on the amount of bleaching. For 1000 contributing layers the predicted visibility (determined halfway down the active outersegment layers) is reduced to 67% when the focused beam is incident at the midpoint between two cones (Fig. 10 right) in comparison to a central incidence at (0,0). In comparison, for dim light with only 100 contributing layers the visibility for the off-axis case is only 45% when compared to the central incidence of the beam.
The possible role of a conical outer segment is shown in Fig. 11 in comparison to that of a cylindrical outer segment when the incident Gaussian beam is focused at the top and bottom of a conical outer segment, respectively. If the outer-segment cone angle is traced backwards to the eye pupil of the schematic eye it corresponds to a pupil diameter of 2.2 mm in fair correspondence to normal vision in bright-light conditions. The conical outer segment shows as expected an increased concentration of forward scattered light with depth. The cylindrical outer segment has a similar concentration effect for the scattered light suggesting that pigments at its far end contribute significantly only near to the photoreceptor axis. A conical outer aperture diminishes the role of the deeper layers towards the far-end apex of the outer segment and thus the directionality in the aperture model is predominantly determined by the large-diameter layers near the entrance to the outer segment, i.e., effective N N < . Finally, Fig. 12 shows a comparison of light scattering inside individual outer segments when illuminated on axis by an incident plane wave at three different wavelengths for both physical and non-physical disc gaps. A change in wavelength is similar to a change in disc spacing as it modifies the phase across the outer segment. In all cases, despite of the apparent differences in the intensity patterns within the outer segments, the corresponding directionality at the pupil remains identical and highly similar to the values shown in Fig. 3 as determined by the diameter and effective length of the outer segment at any given wavelength.

Discussion
The simplified aperture model for outer-segment layers in Sec. 2.1 implies that the pupilintensity distribution from a sum of amplitude Airy discs will make a good fit to the traditional Stiles-Crawford function as easily verified by plotting the two. The number of outer-segment layers included depends on the amount of bleaching and thus the method is tunable to both outer-segment diameter and effective length for the absorption of the incident light. For dim light the acceptance angle is large whereas for bright light the acceptance angle is small as determined from the layers in the outer segments needed to absorb the incident light. The summation will remove oscillations away from the central Airy spot given by for an 8 mm pupil. It should be stressed that a summation of apertures as expressed by Eq. (1) is needed to derive the best fitting function and the above is only suited within a limited range for dim light. Summation of only a few apertures, including the first and last disc (as determined by the fraction of bleaching), replicates closely the final result and the resulting directionality is highly insensitive to even random variations in disc spacing as has been confirmed numerically (not shown here). In bleached conditions the last disc is located at the outer segment termination and its reflectivity can be appreciated from in vivo OCT images [8,9] and post mortem tissue images [2]. Photoreceptor disarray can lower the apparent directionality but it is expected to have little influence for cones at or near the fovea centralis [43,44]. The Airy disc function is slightly more uniform near the pupil center than the Stiles-Crawford function and thus it bears similarities to the super-Gaussian function [45].
Oblique incidence of light increases the risk that light traverses more than one outer segment before being absorbed. For an outer-segment length L = 20 μm and outer-segment diameter d = 2.00 μm with photoreceptor intra-spacing of 1.22d this can only have an impact when the angle of incidence is θ > 7° corresponding to pupil entrance points r p > 2.7 mm with a reduced fall-off in the Stiles-Crawford effect. For dim light, where only a fraction of the outer-segment layers contribute, this is expected to be negligible.
The transient Stiles-Crawford effect, which is a recovery in visibility when light is shifted from one entrance point to its opposite counterpart in the pupil [46], may easily be understood from Fig. 7 which shows that for oblique incidence of light only pigments at one side of the outer segment contribute effectively. Thus a sudden change in the angle of incidence gives access to previously less-exposed pigments.
A possible role of defocus for accommodation can be inferred from Fig. 9 and Fig. 11 suggesting that the best focus is obtained when the amount of light within the outer segment is maximized. For a conical outer segment this occurs when the light is focused near its narrow far end. This may be at odds with the current understanding that light is focused at the inner segment entrance for best vision. The optical properties of the inner segment in itself, however, and possible cellular focusing effects [47] may support the image guidance [48] and light concentration towards the outer segment. This volumetric optimization of light overlap with the available pigments suggests that in dim light the best focus is possibly shifted forward towards the first layers of the outer segments by approximate −0.1 diopters. An even larger average shift of −0.8 diopter has been reported for night myopia [49] clearly also influenced by aberrations and subjective accommodative variations.
High-resolution retinal images show dark rings around the photoreceptors [6] that appear very similar to those seen in Fig. 8 and are caused by interference of scattered light. This brings the common waveguide model of photoreceptors into question. The longer penetration into the outer segment (bleached or weakly absorbed infrared light) the more directional it becomes and the waveguide character becomes more prominent with light leakage occurring mostly near the outer-segment entrance (see Fig. 7). Modes, inside the outer segments may be both forward and backward propagating but also unguided modes (leakage) needs to be evaluated. The dipolar scattering model circumvents this problem by allowing 3-D calculation of the electromagnetic field at any point (excluding the point dipoles) and thus gives complementary insight that is less readily available from a simplified axially-symmetric waveguide picture of the photoreceptors. For vision it is especially satisfying as it directly targets the visual pigments and their interaction with light.
Only monochromatic coherent light has been considered here as it directly impacts the wavefront at the retina in a very controllable manner [48,50,51]. When light enters the eye through different pupil points at a same time special consideration needs to be given to the degree of coherence although at the scale of single photoreceptors the results shown here are expected to remain valid also for incoherent light.

Conclusion
In this study, layered outer-segment models have been introduced that are based on the equivalence between the radiative and collective properties of individual visual pigments considered as optical antennas. As a result of this equivalence an alternative fitting function for the Stiles-Crawford equation is based on a scaled version of a central Airy disc representative for the individual contributions to the light at the pupil from each membrane layer or disc in the outer segments. The resulting function is dependent on the outer-segment diameter and effective length with respect to absorption as well as the wavelength of illumination.
Good agreement is found between a simplified arrayed aperture model and an induced dipolar antenna model that has almost identical radiative properties when propagated to the pupil and show good correspondence with experimental photoreceptor directionality parameters measured for unbleached and bleached conditions. For obliquely incident light the dipolar model shows that scattered light is confined to one side of the outer segment and light leakage to the intra-cone spacing is increased. For hexagonal arrangements of the outer segments the light scattering patterns appear highly similar to those seen in high-resolution reflection images of the cone photoreceptors. The method is demonstrated to have great flexibility to model important questions of light-photoreceptor interactions such the role of defocus and the role of a focused Gaussian beam of light incident on or between neighboring outer segments. The models do not include possible cellular waveguiding mechanisms as the attention is shifted to the nanometric refractive-index modulation of pigment containing discs in the outer segments that capture the incident light when triggering vision. Reflective cell wall surroundings may be included in further model refinements but for the short length scales of light propagation in the outer segments involved in vision it seems questionable that effective mode formation can occur so that light leakage as well as unguided modes must be included in more advanced waveguide models to break translational axial symmetry. In this study an isotropic molecular polarizability has been assumed without detailed consideration of the alterations produced by the physical neighborhood of other outer-segment components that are equally important from a waveguide modeling approach. Further model refinements and the inclusion of both rod and cones in the modeled array are expected to lead to further advancements.