Color Sorting by Retinal Waveguides

Light is being detected by the two distinct types of photorecep-tors in the human retina: cones and rods. Before light arrives at the photore-ceptors, it must traverse the whole retina, along its array of higher-index Müller cells serving as natural waveguides. Here we analyze this optical process of light propagation through Müller cells by two independent optical methods: numerical beam propagation and analytical modal analysis. We show that the structure and refractive index profile of the Müller cells create a unique spatio-spectral distribution of light. This distribution corresponds to the positions and spectral sensitivities of both cones and rods to improve their light absorption.Müller cells are living optical fibers in the vertebrate retina," Proc. Natl. Müller cells separate between wavelengths to improve day vision with minimal effect upon night vision, " Nature Communications 5, 4319 (2014).Efficient colour splitters for high-pixel-density image sensors,


Introduction
Light absorption by the photoreceptors of the retina is the first step of the visual process.This absorption of light by the cones and rods is determined by the properties of the visual pigments as well as the spectral, temporal and spatial characteristics of the incident light.There is a ~100 fold difference in sensitivity between rods and cones, which forms the basic mechanism of two parallel visual systems: the photopic system, active under well-lit conditions and supportive of color vision (cones), and the scotopic system which is active at low light levels and enables night vision (rods) [1].The entire human retinal surface, except for the fovea (the important central part of the retina), is an ensemble of the two photoreceptor arrays, where most of the cells are the high-sensitivity rods, Fig. 1(a).Another prominent feature of the retina (of all vertebrates) is a seemingly 'inverted' structure with respect to the light path, where the photoreceptors are located at the bottom of the mostly transparent retina, behind five neuronal layers, Fig. 1(b).These layers and their inner structures are hard to observe because phase objects are not visible near the focus of imaging systems.
However, experimental studies using confocal microscopy [2] have supplied indications that light is being channeled by natural waveguides (glial Müller cells) inside the retina, from the inner limiting membrane, where light is first incident on the retina, down to the photoreceptors, where it is being absorbed.The resulting enhancement of visual acuity was shown by a computational and numerical analysis [3].A recent study showed that the retinal waveguides separate between wavelengths to improve day vision without hampering night vision [4].However, the properties of this mechanism were not evaluated nor explained.Here we address this optical process by two independent analytical-computational methods.
A Müller cell has a wide inner part, facing the pupil (Fig. 1), and narrowing down to couple in its outer part to a single cone [4,5].The wide, funnel part comes in front of ~20 rods around this cone (the number of nearby rods increases away from the fovea).Müller glial cells have an important metabolic role for the photoreceptors besides their recently discovered optical part.The influence of Müller cells on human vision can only be identified by examining the light passing the cell array into the cones and rods at the bottom of the retina.
To address this process we analyzed a single optical unit, which is composed of a Müller cell coupled to a single cone and ~20 surrounding rods, Fig. 1(a),(b).We examined the light interaction with a Müller cell, using previous phase microscopy measurements of the refractive index profile of the cell, Fig. 1(c),(d) [2].The methods applied were an optical modal analysis [6] and a numerical FFT split-step beam propagation method [7,8].

Light propagation analysis
We digitized images [9] of human glial cells in order to define their width along the cell length.The images were scaled with 0.13 µm/pixel, to form a 1000×256×256 data-cube which maintains the two characteristic features of the cell: the entrance diameter of the upper funnel cup is ~12 µm, and along the light guide ~2.4 µm.For each realization, perturbations were added randomly, up to 5% of the width, as well as for its center line, in order to simulate the uneven boundaries and undulations of the cells.According to the measured refractive index profile, obtained by phase microscopy measurements of human retinas, the cell was divided into four refractive indices domains, with a smooth transition between them, Fig. 1(c).In this simplified model, the soma was ignored being beside the cell and of lower refractive index [2], as well as the short gap after the cell [10].To account for the scattering and distortion of the various retinal fiber layers, we added random transverse perturbations of the refractive indices on a scale of 1 µm and ~10% of the local refractive index difference between the cell and its surrounding.An initial light distribution entering the cell was taken as a diffraction pattern from the eye's pupil, which is broadened by aberrations, to create an average Gaussian distribution of ~40 µm width [11].Next, the field was propagated down the medium, plane by plane, at 0.13 µm steps.The split-step beam propagation method (BPM) of the global third order [7,8] solves numerically the Helmholtz equation for generic refractive index structures and n is the refractive index of the medium.By writing the Laplacian as x y ⊥≡ and z is the propagation direction), and using the fact that the refractive index variations are small, / 1 n n δ << , the equation can be rewritten as Here n and 0 n are the refractive indices of the cell and its surrounding respectively in a single z layer, and ϕ is the electromagnetic field amplitude.We define A and B operators as Here A and B are 256×256 matrices, updated by the field propagation and the refractive index data cube.Eq. ( 2) can be rewritten accordingly as The first order numerical solution of Eq. ( 4) is known as the FFT-BPM where h is an infinitesimal propagation step in the z direction (of 0.13 µm length), F is the two dimensional Fourier transform, and operator A f is the A operator representation in Fourier space.As a first step of the analysis, we examined the light confinement during its propagation along the cell for 25 distinct visible wavelengths (400-700nm).A characteristic light intensity obtained by the simulation along the cell is shown in Fig. 2(a), for blue (450nm), green (530nm) and red (650nm) wavelengths.The final intensity is calculated by squaring the field matrices and then by using a circular mask of Müller cell's radius, in order to obtain only the intensity within the cell's bottom area of ~4.5 µm 2 .Finally, the pixels inside the circle mask are summed to obtain the transmitted signal.
The results show that a substantial portion of the energy entering the Müller cell leaks out during propagation, due to the significant reduction of the cell's width in its first 30-40 µm, Fig. 2(a).In addition, the electro-magnetic field is less confined within the cell for the blue wavelength band than for the green band, Fig. 2(b),(c).A complete analysis of Müller cell's light transmission for the entire visible spectrum demonstrates an enhanced confinement of light around 550nm, Fig. 2(c).These results correspond very well with the spectral sensitivity of the retinal cones, attached to the Müller waveguiding cells: Nearly 90% of all cones possess maximum sensitivity in the green (530nm) and yellow (560nm) [12] and only ~10% in the blue.Notably, the light leaking out of the waveguide cell, ~85% of the incident energy for blue light, is spectrally compatible with the high sensitivity of the surrounding rods to short wavelengths (490nm).Due to Müller cells light guiding, up to 40% of the incident light is concentrated into the cones at 550 nm, Fig. 2(c).Without the wave guiding functionality of Müller cells, light would have been distributed uniformly over twenty photoreceptors at the bottom of a single Müller cell, and only 5% of the energy would have reached the cone [4].

Chromatic distribution analysis
A color (RGB) representation of light allows to gain insight into the spatial color distribution at the bottom of Müller cells, just before light is incident on cones and rods.The 25 distinct (monochromatic) spatial distributions (Fig. 3), were combined to red, green and blue representations by the CIE human color matching functions [13], enabling a transformation from each wavelength into its corresponding RGB human weights.Composition of the RGB images into a true color distribution generates a clearer spectral representation of the latter.We see that all colors spread after the funnel, but the green-yellow light is recaptured into the center (the cone's area), while violet-blue light leaks outside the Müller cells, where the surrounding rods are located, Fig. 3(c).These results provide a computational description for the spectral transmission of Müller waveguides to the photoreceptors spectral absorption.Two physical parameters should be addressed in order to assess the stability of this phenomenon.The first is the cell's width, which as a biological parameter possesses typical statistical variations [9].The second parameter is the angle between the incident light and the cell's main axis.For that purpose we have calculated the modal distribution in a Müller circular waveguide with an average 1.2 µm inner radius.In this simplified analysis, the upper-cup funnel part of the cell was not taken into account, thus relating only to the light transmitted into the cone photoreceptor.Since the refractive indices of the cell and vicinity fulfill ( ) / 1 cell surrounding cell n n n − < <, the waveguide can be considered as weakly guiding and its modes are given by the characteristic equation [6] 1 J ha K qa h q J ha K qa where J l is the Bessel function of the first kind and K l is the modified Hankel function of the second kind.Here l is an integer, k = 2π / λ, a is the fiber's radius, and h and q follow the relation . For a given index l there could be a few distinct solutions for the equation, which are represented by a second non-zero integer m, which together give the complete set of linear polarized modes, LP lm .In our case of the Müller cell, the waveguide is multimodal and supports a group of six modes: {LP 01 , LP 11 , LP 21 , LP 02 , LP 31 , LP 12 }.We calculated the confined power inside the cell's core for the six modes, Fig. 4(a).Next, the total power of all normalized modes was summed for various cells' widths, Fig. 4(b).The results show that for the mean Muller cell width (~1.2 micron), wavelengths with λ >600 nm possess core power smaller than 30%.This is calculated with the assumption that all modes have equal input power.In order to take into account also the cell's funnel-like structure (considering also its upper cup) and refractive index variations, we applied the BPM numerical simulation, using the same global parameters of wavelength and inner-cell's width as was used for the modal analysis.For the mean Müller cell's width the BPM solution also shows the drop of core power for wavelengths with λ <500 nm, thus providing a light transmission maximum at ~560 nm, as can be seen in the blue curve of Fig. 4(c).Notice how similar is this modal calculations to that of the myoid and cone spectral transmission [14,15].
When light enters the pupil away from its center, it reaches the retina as a tilted wavefront, rather than perpendicularly.At night time, the pupil dilates up to 8 mm, and with an average eye length of 23 mm, the maximum incidence angle with respect to the retina is ~10 0 .Therefore, we calculated also the average transmission of light in the waveguide cells up to 10 0 incidence slant and found the same peak transmission, as the red curve in Fig. 4(c).The relative intensity is lower, as can be expected when the leakage increases.The high transmission drift, as a function of the cell's width, as in Fig. 4(d), is consistent with the modal analysis calculation, Fig. 4(b).For the mean width of the human Müller cell (1.2 µm), the peak po-tential light transmission is at ~560nm.These results do not depend on the glial cells being even, straight, or smooth [4].

Conclusions
In this study we investigated, by analytical and computational methods, the spatial and spectral light distribution as being transmitted by Müller waveguides to the cone and rod photoreceptor cells in the human retina.We found that the specific refractive index properties and structure of the retinal Müller cells lead to a spectrally dependent light guiding, compatible with the natural spectral sensitivity of the cones, located in their distal end.At the same time, light of short wavelengths is not captured in the straight guide, and ~85% of the total power in these wavelengths leaks out to the surrounding rods, most sensitive to short wavelengths.Thus, Müller cell waveguiding provide a considerably higher photon dose for the photopic system of cones, while increasing the optical signal-to-noise ratio for the scotopic system of the rods.Subjective measurements such as the Stiles-Crawford effect (SCE) can also provide a link between the Müller cell optical phenomena and the resulting visual process.The SCE, which shows higher central sensitivity in cones, is missing in the rods [16,17] and might be attributed to the fact that light reaching the rods arrives after diffraction and scattering, resulting in loss of light directionality, unlike the cones, which receive mostly guided light by Müller cells.It should be noticed that this natural special optical arrangement can lead to biomimetic applications, such as novel micro color splitters for high density and color image sensors [18], or for photovoltaic cells [19].

Fig. 1 :
Fig. 1: (a) Schematic representation of the photoreceptors layer.The high sensitivity rods (represented in orange) surround the lower sensitivity cones, with three spectral peak sensitivities: blue (430nm), green (530nm) and red-yellow (560nm).(b) Side view of the retina, where the Müller cells concentrate and channel light from the top layer to the photoreceptors, down from a ~12 µm diameter into a cone of 2.4 µm diameter.(c) Refractive index profile of a human Müller cell and vicinity neurons, and (d) the corresponding structure.The refractive index in the cell is higher than its surrounding by ~1%-3%.

Fig 2 .
Fig 2. (a) Light intensity distribution propagating along a Müller cell for blue (450 nm), green (530 nm) and red (650 nm) wavelengths (notice different scales).(b) Light intensity inside Müller cells (normalized to the input power), as a function of propagation distance along the cell (~130 µm) for the same three colors.Most of the energy leaks out to the surrounding rods during propagation.The waveguide confinement of green light (delivered to the cone) is higher than the corresponding red and blue.(c) Normalized light intensity as function of propagation distance inside the cell for 25 wavelengths.

Fig 3 .
Fig 3. RGB representation of Müller cell's light transmission.(a) The spatial distributions at the bottom of a Müller cell for 25 discrete wavelengths (three shown).(b) CIE standard observer color matching functions (xyz) used for a transformation from wavelengths to RGB.(c) True color spatial distribution at the bottom of a Müller cell, reconstructed by composing the red, green and blue distributions.High concentration of medium wavelengths into the center is evident.

Fig. 4 :
Fig. 4: Spectral transmission by a human Müller cell: modal analysis (left) and BPM (right).(a) Mode power confined within the cell's core (mean radius 1.2 µm, no funnel), normalized by the total power for the mode.(b) Normalized total core power (summed over all modes) for a characteristic distribution of cells widths (1-1.4 µm), only for the cylindrical constant width part of the cell.(c) Transmission within the whole cell obtained by BPM for normal incidence (blue curve) and averaged tilted field (red curve) for the whole cell.(d) Normalized total intensity for the characteristic cell's radius distribution obtained by the BPM for normal-incidence field (average of 20 realizations each).