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Abstract
One and half decades ago, Müller glia cells of the retina became subjects of extended research as optical waveguides. It was demonstrated that outside the fovea, these cells are capable of providing light transmission through the thicker parts of the retina from the vitreous surface to the photoreceptor cells. We combined optical modeling of the eye’s optical system with numerical methods that describe light guiding within Müller cells to analyze efficiency of light capture and guidance at different peripheral positions. We show that higher order guided modes play an important role, especially in the case of higher incidence angles and extended geometry of the electromagnetic field distributions predicted by the eye’s optical model. We analyze the mode structure excited at different retinal peripheral positions and show that actual construction of these cells optimizes light guiding. Our results refine previously published modeling results regarding Müller cells as waveguides and provide extension to the whole area of the retina.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
It is well known that vertebrates, among them also humans, possess inverted retinas, where the photoreceptor cells are on the opposite side of the retina to the incoming light: in large part of the active retina light has to penetrate through cell layers of 100–200 µm thickness before being absorbed by the receptors [1,2,3].
The inverted retina is apparently an “evolutional mistake”, but has many advantages [4,5], such as volume saving when arranging retinal cells in smaller vertebrate eyes [6], pushing light absorbing pigmented RPE cells to the outer zones [6,7] through invagination during the development of the eye and allowing simultaneous penetration of sigma carrying axons through the wrapped neural tissue [8]. Due to the retina inversion light must penetrate through a layer of neurons and glia cells before reaching the light sensing structures of the specialized neurons, hence some absorption, refraction and reflection may occur before light is captured. The resolution and quality of the image may be seriously influenced and the threshold of the detectable amount of light may be increased [1,2,4,9].
The hypothesis that the inner layers of the retina may behave as anti-aliasing filter, similar to many current digital cameras, was not confirmed by measurements demonstrating aliasing artifacts during vertebrate vision at resolution values approaching the physical limit provided by the closely packed receptor cell structure [2].
Other studies find solutions for the problematic of intermediate layers in the information processing mechanisms at a higher level – e.g. in the brain. Many experiments demonstrate that continuous perception occurs only for excitations containing movements, only scenes containing temporal changes are processed [10,11]. For this purpose the eye globe continuously performs micro-movements called saccades [11].
The excitations, scenes that do not change temporally at the receptor level quickly become invisible. This function has the main task to filter out the stationary always present entities from the information stream from outside, such as blood vessels, and different functional retinal cells. These can be made visible when moving light sources are used to project shadows from e.g. vessels onto the receptor cells [10,12,13].
One important aspect of the image deterioration effect of the intermediate cells is in the morphology of the retina: in the case of the vertebrates, the retina is split into two different functional parts, the foveola and the perifovea (between these two we find the fovea and parafovea) (Fig. 1(a)). The perifovea is relatively thick (>200 µm) and contains all glial and neural structures that may disturb the light propagation path. On the other hand the foveola is a deepening in the surface of the retina, relatively thin (<100 µm) and does not contain neural structures and vessels, only thin amount of tissue on the inner side of the photoreceptive layer. Hence sharp vision is achieved in the foveola, where light scattering is minimized [5].
Fig. 1. (a) OCT registration of a subject’s retina (b–e) and two-photon registration of the photoreceptor layer of the same human sample: (b) Outer segments, rods and some cones; (c) Level of photoreceptor–bipolar–(horizontal) synapses (triades); (d) Posterior part of bipolar cells; (e) Large ganglion cells with nuclei. The shown distances from the axes correspond to those given in Fig. 4 as important zone limits from the point of view of our analysis. The two-photon registrations show what kind of retinal layers and different cell structures appear in the way of light in reaching the photoreceptors.
A relatively new finding [1,2,3] proposes a mechanism for enhancement of arrival of light to the photoreceptors outside the fovea, through the intermediate neural and glial layers, consisting on the light guiding function of Müller glia cells. These are multifunctional but primarily glia-cells that appeared in an early stage during the evolution of the vertebrate retina [4]. As a primary function, due to their rigid mechanical properties they form the skeleton of the whole retina, providing mechanical support and protection for the neural cells. Moreover they isolate particular neurons electrically, take part in their metabolism and in the annihilation of debris after cell decease.
Newest research demonstrated their regulating role in the information transfer of the neurons – they control the concentration of K+ ions and store materials necessary for cell regeneration [14], take part in the maintenance of homeostasis and in the generation and evolution of different neurons. The ramifications and other complicated structures of these glia are in connection with these multiple functions [2,3,14]. However the morphological structure of their central cell-pillar shows highly similar parameters throughout the retina, hence the idea to act as waveguides through the retinal layers to the photoreceptors was introduced [2,3,15,16,17,18]. Two main Müller cell types can be identified in the retina: one is mainly appearing around the foveola, it is Z shaped combined with a conical structure [19], the others appear mainly in the perifovea, with straight main axes, parallel to each other, with a nearly cylindrical shape, excepting the conical termination on the inner side of the retina.
A research group observed high contrast of the Müller cells under the microscope in guinea pig retinal samples when compared to the surrounding cells and tissue [3]. This indicates high refractive index difference between the glia and other surrounding tissue. This observation together with the uniform arrangement and parallelism of neighboring glia resembled the fiber-optic transmitter arrays used for low distortion image forwarding in modern imaging equipment.
Although the cylindrical part of the Müller cell’s phylum is of nearly constant diameter, the end part joining with the vitreal border between retina and vitreous forms a funnel like structure. It was found that the surfaces of these funnels at the vitreal border nearly completely cover the inner surface of the retina, hence they may collect almost all light reaching the retina from the vitreous, and direct it toward the cylindrical cell body [3,5,14–18].
In this paper we concentrate on the understanding of light guiding parameters of the straight Müller cells, lying in the areas of the retina outside the fovea. We provide an analysis from an optical point of view of the light guiding capabilities of these structures, to better understand their role in maintaining spatial information encoded in the image generated by the optical system of the eye on the retina. This information would be completely lost without light guiding through the relatively thick layers of different cells in the periphery. However, although blurred, the spatial distribution is maintained in the image finally formed through visual perception. Light guiding through scattering layers by Müller cells is one important step in this. We intend to reveal characteristic parameters of this step, depending mainly on the distance from the eye’s axis. Numerical modeling of the light reaching the retina and propagation through Müller cells was used to reveal the possible outcomes and limitations introduced by this step in the process of visual perception.
Our principal task is to confer the morphological uniformity of glial cells throughout the retina with the different illumination conditions leading to different light guiding outcomes. Distribution of light on the inner surface of the retina changes much with the position and incidence angle on the surface. We used a geometrical optical model of the eye’s optical system to calculate realistic light distribution in different positions on the inner surface of the retina, inside and outside the fovea. For modeling we used the Zemax software, which contains many useful functions, including physical optical approach, which we used for the calculations presented in this paper. However accurate modeling of light propagation through biological cells of sizes comparable with the wavelength is not possible with it. Numerical calculation of the wave guiding through the Müller cells was performed using beam propagation method [20] to assess the amount of light reaching the outer end of these cells under different input conditions. We cross-checked the results with a modal structure of the guided light using RP Fiber Calculator software. Systematic analysis based on this numerical modeling of light guiding within these cells led to the conclusion that light collection efficiency is greatly enhanced by the morphological structure and uniformity of these cells, and the presence of different guided modes is of high importance, especially in the red range of the visible spectrum.
Finally we analyzed the effect of the variation of structural and material parameters on the light guiding ability of Müller cells and introduced a stability/instability indicator, which demonstrates that actual morphological and material properties of these are giving the highest stability in light guiding against fluctuations of these parameters.
2. Light distribution on the inner surface of the retina
2.1 Model of the eye’s optical system
Labin et al [1,16,17] reported about theoretical investigation of light guiding in Müller cells at multiple wavelengths within the visual spectrum. They found that some aberrations, especially chromatic aberrations of the optical system anteceding the retina may be corrected by the light collecting behavior of the Müller cell funnels. The lights at different wavelengths are collected by the Müller cell end cups and led to the photoreceptors by the cylindrical cell body. The cone photoreceptors play a major role here, since in many cases they are placed immediately below the extensions of the Müller cell [21]. These simulations also show that light incident under different angles will be transmitted in different ways by these cells. Light incoming under bigger incidence angles will leave the central pillar of the cell, where a kind of leakage occurs. Their hypothesis is that light that is not effectively guided by the Müller cells reaches the photoreceptors grouped around the central cone to which the guided light arrives. Hence some wavelengths are directed to the cones placed at the output of the cylindrical cell bodies and others (mainly in the blue region) arrive to the rods, which surround these cones. This assumption is also maintained in [1,17], where the authors extended their numerical simulation with the wavelength dependence of the linearly polarized modes of the cell-waveguide. They reported calculation of the light transmittance of these cells averaged in the 0–10° incidence angle range. The incident light used for simulation was of Gaussian distribution, with 40 µm waist diameter, used as a maximum generated by the optical system of the eye [16,17]. We provide a more detailed analysis here, in which we demonstrate how the image spot size is varying within the 0–10° incidence angle, where the spot size remains below the cell’s maximal diameter, and further for higher angles, where the spot becomes bigger than a single cell. This size mismatch seriously influences the coupling into the Müller cells, transfer efficiency to the receptors and resolution of the captured image details.
Our simulations use optical models of the eye for estimating light distribution on the retina. The models were custom enhanced based on measurement results [22,23], based on the Liou–Brennan [24,25] and Gullstrand eye models [26,27] – respectively. Figure 2 shows the Zemax eye model scheme, developed starting from the original Gullstrand model, that divides the crystalline lens into three zones. The models were improved upon measurements of optical and mechanical parameters of real human eyes under clinical conditions. We emphasize here the use of irregular surface in the model for the first surface of the cornea, where astigmatism and curvature parameters were introduced from measured results. More then 20 eyes were measured and their aberration values were averaged. We also averaged eye and vitreous lengths and used these values to better fit reality, as described in [22].
Fig. 2. Zemax model of the eye generated from the Gullstrand model, showing beams passing the pupil and arriving to different locations on the retina. Model parameters were improved based on clinical measurements.
For the PSF calculations we used the eye model constructed based on the Liou–Brennan eye model. This model contains the inhomogeneous distribution of the refractive index in the crystalline lens, in contrary to the Gullstrand model that assesses more precisely the interception angle and position of incoming rays on the retina. Hence the combination of the models provides realistic and precise field distribution estimation also in the points on the retina far from the eye’s axis.
The angles between the rays falling onto the inner surface of the retina at an arbitrary point can be defined relative to the local retina normal, the direction of the radius connecting the geometrical center of the approximated eye and the analyzed point. Based on detailed measurement results presented about the human eye [27–29], the geometrical calculations show that the incidence angles on the inner retinal surface relative to the surface normal may be far bigger than 10° estimated in previous analyses [16–17]. The highest angle appears at the edges of the retina, called ora serrata.
The distance between the ora serrata and the equator of the eyeball varies slightly throughout the contour of the equator, average nasal, temporal, inferior and superior values are given measured along the four sections: 5.81 mm, 6 mm, 4.79 mm and 5.07 mm, respectively [28]. We take an average value of 5.41 mm, to estimate the maximum incidence angle θ according to Fig. 3. The sphere matching the inner surface of the retina is of 11.3 mm radius (as given by the curvature of the retina in the eye model) [27] and the distance between the sphere’s center and the pupil is 9.53 mm. Average anterior chamber depth (3.17 mm) and matching sphere radius (11.3 mm) were subtracted from the average eye length (24 mm) [26,27].
Fig. 3. Scheme of the ray path reaching the extremity of the retina at the ora serrata, at an angle of 50.77°.
The incoming rays arriving at a given location on the retina may pass through different points in the pupil’s aperture. The coupling efficiency in the light guiding cells is defined by the angle relative to the cell’s axis and the incoming beam’s axis. The light guiding cell can be photoreceptor or Müller cell, depending on the location on the retina. In the peripheral zones mainly Müller cells conduct the light to the receptors in green and red ranges of the spectral sensitivity spectrum [16,17].
The photoreceptor cells being directly exposed e.g. in the fovea may be oriented towards the pupil to enhance coupling efficiency for the light arriving from that direction [29]. However there is no systematic data about such orientation of the Müller cells in locations where they could be responsible for light transmission. Micro-photos (e.g. Figure 2(c), 3(a) of [18]) show that they may be also tilted, but the relation between this tilt and the direction of incoming light is not yet clarified. The most simple approach would be to assume that the axes of these cells are oriented along the retina normal at the given location. In this case incidence angle for the cell is the angle between the axis of the incoming light and the retina normal. Hence we can assume that the light entering the Müller cells may have a maximum incidence angle θ with the cell’s axis when the cell is near the end of the retina, at the ora serrata, and it is oriented perpendicularly to the retina, corresponding to the calculated 50.77°. In fact, if we still would have cells oriented away from the retina normal, we can calculate the coupling and light guiding efficiency also for this modified incidence angle, and apply the values to estimate the amount of possibly transmitted light.
2.2 Incident field at different retinal locations
Our systematical calculations show that different portions of the retina “see” different portions from the total field of view. Size and orientation of the image spots corresponding to different object points are determined by the calculated point spread functions (PSFs) by determining e.g. 80% of the encircled energy. Results show that the size of the spots may become very large in retinal zones lying apart from the eye axis, when compared with the average size of photoreceptor cells and even of the Müller glia cells. The role of the Müller glia cells becomes very important in the formation of the perceived image in the retina areas outside the parafovea. The light collecting and concentrating function of the end funnel directly determines sampling of the image generated by the optical system of the eye and the final optical resolution of it.
The PSF corresponding to an object point is formed by the diffraction of light emanating from that point through the cornea, pupil and lens, where the pupil limits the light cone arriving to the retina – see Fig. 2. The incidence angle corresponding to a spot determined by a calculated PSF can be approximated by the angle enclosed by the axis of the cone of light forming the spot – this is the line connecting the center of the pupil with the center of the spot – with the line normal to the retina in the center of the PSF. The retina normal is approximated with the normal of the sphere matching in big part the retina’s inner surface. This is the incidence angle used during modeling light distribution on the retina. Our simulations with the eye model show the direct relation between the size – in fact the intensity distribution – of the spot and the incidence angle defined as above.
In the fovea the incidence angle is nearly perpendicular to the surface of the retina, the spot size is of 3-4 µm diameter, depending on the diameter of the pupil, as shown in Supplementary Figs. S1 and S2. The cone receptors are averagely of 1.5 µm size in this zone [30], providing a perfect sampling of the image according to the Nyquist criterion [31,32]. Here the Müller cells do not play a significant role in light guiding, since they are present only at the sides with a specific Z form [19], which are probably even not involved in light collection and conduction.
We identified two crucial distances from the axis on the retina where changes in image formation happen. This is not the symmetry axis of the eye, but the axis connecting the pupil center with the center of the fovea. One of these locations is the edge of the parafovea – Fig. 1(a) – outside which the Müller cells play a role in light guiding and image sampling. Approximately outside this limit, the Z shaped Müller cells are replaced by cells with straight pillars, better suited for light conduction. The other zone border is where the spot size corresponding to one object point becomes comparable with the maximal size of the Müller cell’s inner termination. Our simulation shows that this is at about 10° incidence angle relative to the defined axis. Figure 4 shows these locations and the geometry used to calculate the corresponding parameters, these are also appearing in Fig. 1(a).
Fig. 4. Visualization of the 3.44° and 10° incidence angles for the rays passing though the center of the pupil marked in red relative to the surface normal of the retina (approximated by the radius of the matching sphere of 11.3 mm length).
The distance of the parafovea’s edge from the optical axis is approximately 1.25 mm [30]. Hence from the triangle with the 11.3 mm long radius as hypotenuse the opening angle α1 is 6.33°. The average distance between the parafovea edge and pupil center becomes:
Incidence angle θ1 results from the trigonometry in the corresponding triangle:
In the zone between 3.44° and 10° incidence the spot size remains below 12 µm in diameter, the outer diameter of the Müller cell’s funnel, at least at smaller than 4 mm pupil diameter, hence the light from a single object point can be collected by a single Müller cell. The average distance of these points from the center of the pupil is
We intended primarily to determine from the optical model of the eye how the PSF and spot size changes in the different zones of the retina, and how these are related to shapes and sizes of the Müller cells, which are identified as primary units responsible for image formation, determining also achievable optical resolution in peripheral zones. Comparison of the cell size and structure within different zones with the spot sizes and incidence angles delivered by the eye’s optical system assessing the local mechanism of sensing and image formation.
Table 1 presents the spot diameters that contain 80% of the total entering light as a function of incidence angle and pupil diameter. Table 2 gives the variation of the Strehl ratio with the same parameters. It is obvious that increasing incidence angles and pupil sizes lead to increasing spot sizes and aberrations, which is indicated by the strongly decreasing Strehl ratio.
Table 1. Variation of the PSF spot radius in microns (80% encircled energy) with incidence angle and pupil diameter
Table 2. Variation of the Strehl ratio with incidence angle and pupil diameter
Figure 5 shows the spot distribution at 10° incidence. The smaller spot at 1.5 mm pupil diameter is distributed symmetrically, however at higher pupil openings there is a considerable smearing. The average spot is of higher diameter than the Müller cell’s funnel, marked with the white semicircle.
Fig. 5. Incident spot variation at 10° incidence on the back surface of the retina (a) at 1,5 mm pupil diameter, (b) at 4 mm pupil diameter, (c) at 6 mm pupil diameter The spot size is around 4–15 µm in diameter, side of the shown area is 25,6 µm.
Above 10° angle of incidence, on average more than 3.4 mm off axis, the Müller cell’s end funnels are smaller than the light spot arriving from a single object point, hence sampling of the image is performed by these funnels directly.
Many model programs (RP Fiber Calculator, BPM Matlab) only accept Gaussian-like light distributions at the entrance surface (in this case that of the Müller cells), therefore approximating the input beam as Gaussian is a necessary restriction in our model. This poses a problem, since the beam distribution becomes more extended spatially at locations far from the axis, and only parts of the total diffraction pattern coming from the pupil can be collected by individual Müller cells. These diffraction patterns, especially at incidence angles exceeding 14° cannot be treated as Gaussian according to the literature [1,16,17]. Although the spots for 10° and 14° incidence angles are apparently smeared at higher pupil openings (Figs. 5(b) and 5(c) and Suppl. Figs. S4(b) and S4(c)) the power is still concentrated in the central area, and a Gaussian fitting provides less than 9% RMS error between the intensity profiles. For this reason, below 14° incidence angle we treated the spots as Gaussian distributions. A method that better approximates the incident beam might utilize a Gaussian with larger beam radius, truncated in a way that results in a better fit to the calculated PSF shape.
For a rough analysis we divided the retina into a central area, where the PSF is smaller than the input diameter of the Müller cell and a peripheral one, where the spots cover more Müller cells. In the fovea and it’s immediate neighborhood the role of Müller cells in light guiding is negligible, the image processing and resolution is done directly by the photoreceptors on a first level [29,31]. In the zones, where the size of the spot indicated by the PSF becomes comparable to the area of the Müller cell’s funnel, we have a one by one relation between the optical resolution provided by the optical system of the eye and the Müller cell. Our Zemax simulation shows that such PSF sizes belong to light cones falling on the retina under incidence angles between around 10°and 45°. The pupil size highly influences the spot size and hence the limits of this zone.
It is interesting to see how imaging is performed at these extreme periphery values. We calculated the image formed by the optics of the eye with the object scene centered around the point corresponding to the 40° incidence angle. The input object sample is a picture of 13.3*10 mm size in the object plane, as shown in Fig. 6(a). The image of this object formed by the cornea and lens through the inner parts of the eye on the inner surface of the retina is smeared and not sharp, as shown in Fig. 6(b). This image will be sampled by the Müller cells with 10–15 µm funnel diameter and this only helps to recover parts of the contours. Figure 6(c) shows the image after sampling with Müller cells of 12 µm size, assuming that the cells are densely packed, as shown in Fig. 1 of Ref. [8]. The image blurred by the aberrations of the optics of the eye becomes more pixilated after sampling by the cells. Importantly, the cells are smaller here than the sizes of the image spots, calculated from the PSFs. Probably detection of movement is the main purpose of this imaging at very peripheral locations.
Fig. 6. Simulated image portion created by the eye’ optical system at the surface of the retina, around 40° incidence angle. (a) Input picture centered at the corresponding object point, of 13.3*10 mm size. (b) Image generated from the object by the optical system of the eye, with size of 2.22*1.67 mm; the simulation takes into account only geometric aberrations. Size of the funnels are pictured with three tiny white rings in the upper side of the image. (c) Image from (b) sampled by Müller cells of 12 µm average size, densely packed.
3. Müller cells as waveguides
3.1 Numerical simulation of light propagation through Müller cells
We also analyzed light throughput of the Müller cell with BPM method using freely accessible software [20]. Numerical simulation of the beam propagation through the cell reveals the dependence of the throughput on the wavelength and field distribution of the PSF at the cell’s collector surface. The simulation helps to approximate the amount of light transmitted by the cells at different retinal positions by applying the field distribution calculated by the Zemax software as input. During image formation the total incoming light at a particular cell entrance is a superposition of such PSF’s – see Supplementary Figs. S4 and S5. In the zone above 10° incidence angle on the retina the PSF corresponding to one object point is distributed over more than one Müller cell, hence one image point may reach more than one photoreceptor. It is also interesting to calculate whether the light from such a PSF can be efficiently collected by the funnel like end of the cell, and how the light is concentrated by its main pillar to the photoreceptors. If all light reaching a cell’s funnel is conducted, then the possible optical resolution is determined by the ratio of the funnel diameter and the spot size. Light collection and guiding is only efficient when the light incident on the cell is able to generate waveguide modes conducted by the cell’s pillar.
The software in its present form does not accept the exact field distribution calculated by Zemax; as an alternative we approximated the calculated PSF with truncated Gaussian functions that resembled the average size of the spot, leading to approximate transmission results, especially in the case of the extended and non-symmetrically distributed PSF’s, as in Supplementary Figs. S3, S4 and S5.
The numerical aperture of the eye’s optical system is small; with maximum pupil opening of 8 mm diameter the NA is about 0.23 [27]. The numerical aperture of the Müller cells can be estimated from the refractive indices and anatomical measurements, and varies between 0.12 and 0.3, maximum being in the thinnest pillar. Hence the acceptance angle of the cell waveguide is bigger in the thinner parts, than the light cone generated by the eye optics. The smaller acceptance angle in the thinner beginning parts of the funnel causes coupling losses, but they are negligible, since these parts are thin, having only a couple of wavelengths thickness along the propagation direction.
The software used for numerical simulation allows modeling of the effect of refractive index and cell diameter changes along the cell’s main pillar. Other changes in the geometry or material composition of the cells can be taken into account in future simulations, when new measurement results would indicate it.
The size and refractive index variation in funnels were modeled according to the description found in [1,16,17]. The funnel zone was divided into five slices, of nearly equal length, the refractive index was increased from 1.345 to 1.38 and the diameter decreased from 12 µm to 2.4 µm, along the propagation direction from the retina–vitreous interface towards the photoreceptors. The used model with the parameters is shown in Fig. 7.
Fig. 7. Model of the Müller cells with a funnel termination at the vitreous, used in the numerical simulation of light throughput with BPM Matlab. All sizes are in microns.
Light guiding characteristics of the cells with funnels are shown in Figs. 8,9 and 11, for straight and oblique incidence. In section 2 we have shown that PSF distributions and spot sizes are directly related to their position on the retina relative to the axis of the eye, and this position was associated with the corresponding incidence angles, defined as the angle between light propagation direction and retina normal. Incidence angle, as defined by the angle of the incoming light and cell axis may be different, if the cell is not aligned parallel to the retina normal. Therefore it is worth to calculate throughput of the cells for spot size and incidence angle pairs different from those resulted from Zemax modeling. For example, in [29] a tilt of the photoreceptors was described, which provided perpendicular incidence at different retinal locations. There is no experimental evidence yet of the tilt of Müller cells respective to the retinal normal towards the pupil in humans, but it is not excluded, hence the numerical results of perpendicular coupling of different spot distributions into these cells may be useful and realistic.
Fig. 8. Intensity distribution along the Müller cell with funnel structure at 500 nm wavelength, with incidence parallel to the cell’s axis, spot size of 12 µm, comparable with the diameter of the cell’s funnel. It is visualized, that the funnel concentrates the light around the cylindrical axis despite of the low refractive index difference of the funnel end part and the surrounding tissue.
Fig. 9. Intensity distribution along the Müller cell (a) with funnel structure and (b) without funnel structure – considering only the central pillar, at 500 nm wavelength, with incidence parallel to the cell’s axis, spot size of 100 µm, comparable with the diameter of the cell’s funnel. It is visualized, that the funnel concentrates the light around the cylindrical axis despite of the low refractive index difference of the funnel end part and the surrounding tissue. Although without a funnel the amount of light in the core is impressive, the total amount of transmitted light is bigger with funnel. Color bars indicate intensity, and they are common for Figs. (a) and (b).
Figure 8 shows the distribution of the field inside the Müller cell for approximately 12 µm input field diameter (truncated Gaussian with 6 µm waist radius). The light guiding efficiency is high, since most of the light is concentrated on the axis.
When comparing Fig. 8 with Fig. 9(a) one can see the transmission efficiency difference of the Müller cells with different incident spot sizes. If the spot size is bigger (∼20–100 µm) than the funnel’s highest diameter, the collection efficiency is smaller. For spot diameters smaller or comparable to the funnel size (∼12 µm), the light collection is highly efficient. It is clear that the light in the bigger spot outside the funnel’s edge will not enter the cell, hence at the input we took the large Gaussian spot with 100 µm diameter and truncated it with the funnel’s diameter, generating a realistic input and compared it with the Gaussian input of 12 µm. The difference in the transmission efficiency is mainly because different transversal modes are excited: the excitation efficiency and propagation leakage of these modes are also different.
Clearly the effect of light concentration by the funnel like beginning of the cell at the inner retina border can be demonstrated, as shown in Fig. 9. In Fig. 9(b) a hypothetical comparison to the real situation is given, with the cell main pillar as a light guide, without a funnel termination. The coupling efficiency for such large spots, e.g. 100 µm in diameter is 0.11% for cells with the funnel, and 0.0125% for the hypothetical cells without the funnel. In Fig. 9(a) the intensity distribution, represented by colors in the entrance part – left side of the pictures – indicates more efficient coupling into the cell when compared to Fig. 9(b).
The amount of light reaching the terminal of the cell (the integration of intensities over the cross section area of the cell) is bigger in the case with funnel (integrating through the cross section at the right end in Fig. 9(a)) than without (integrating through the cross section at the right end in Fig. 9(b)). This is despite of the apparent color difference, which indicates only the higher light concentration around the core in Fig. 9(b). Hence our numerical calculation demonstrates an almost tenfold increase in light collection efficiency for large spots using light concentrating function of the funnel like end structures.
Light throughput difference for different input spot sizes is visualized in Fig. 10. The figure reveals that transmission efficiency is not only affected by coupling efficiency difference, but also by the leakage of the excited transversal modes during propagation.
Fig. 10. Throughput efficiency of Müller cells calculated using the numerical model for fiber transmission, (a) spot size of 100 µm, (b) spot size of 12 µm diameter.
When calculating the throughput for inclined incidence, it is straightforward to apply the assumption that these cells are parallel to the retina normal, and apply the corresponding spot size – incidence angle pairs revealed by calculation of the incoming light distribution with the Zemax eye model. One example is shown in Fig. 11(a), with 15 µm spot radius and 14° incidence angle. Here the PSF spot size corresponds to that calculated by the Zemax model for 14° incidence angle and 4 mm pupil diameter.
Fig. 11. (a) Calculated intensity distribution along the Müller cell with a funnel structure at 500 nm wavelength, at 15 µm incident spot diameter, comparable with the size of the funnel, at incidence angle of 14° relative to the cell’s axis. (b) Light intensity distribution at the end of the cell, at the output towards the photoreceptor layer.
The modeling result shows that light coupling into the main pillar is efficient, and the light is guided until the end of the cell, although part of it is outside the pillar (the waveguide core) and the output intensity distribution is asymmetric, as shown in Fig. 11(b). The incidence angle is marked by the white arrow in Fig. 11(a) – it is obvious that the Müller cell effectively transforms the beam axis to become parallel to the cell’s axis and facilitates the light reaching the targeted receptive cells on the other end.
3.2 Multi-modal light guiding of Müller cells
Inspection of the light throughput at incidence angles around 10°–15° revealed that the role of modes is crucial in the visible wavelength range. Waveguides with diameter of 2.4 µm and 0.035 refractive index difference between core and cladding divide the visual spectrum roughly into thee zones: between 380 nm and 419 nm there are 5 guided modes, between 419 and 565 nm there are four, and above 577 nm only two modes. Cut-off wavelengths are at 419.08 nm, 565.12 and 576.7 nm respectively. Our modal analysis shows that increase of incidence angles and incident beam diameters increases the role of the modes. Above 10° incidence angle the total amount of light is divided almost equally between the modes, hence the throughput is highly determined by their presence. Although the total coupling efficiency decreases considerably with increasing incidence angles, the role of the excited higher order modes becomes crucial. We calculated the amount of light carried by the different available modes for the incidence angle and incident spot size pairs determined by the Zemax model of the eye’s optical system.
The largest size of the spot was of 12 µm diameter, corresponding to the typical maximum distance between the Müller cells. In this calculation we only considered the thinnest part of the waveguide, the thinnest pillar of the Müller cell, of 2.4 µm diameter. Numerical modeling of light guiding of the whole structure – funnel and thin waveguide together – presented in the previous section revealed that the light in the spot comparable to the funnel size is collected to an amount of 80–90% into and around the main wave guiding pillar. Within these pillars the light propagates through the available modes until the termination of the cell, after which it is absorbed in the photoreceptor layer.
The numerical simulations show that when the spot size is around or below the diameter of the funnel, (Figs. 8 and 11) light guiding happens in the central pillar of the Müller cell of minimum diameter along the light path. Hence the analysis of the modes guided by this waveguide-like cell part is straightforward and realistic when one wants to assess the amount of light arriving to the photoreceptors at different wavelengths. We calculated the modes excited in the main pillar of the Müller cell at the incidence angles and spot sizes calculated in the Zemax model of the optical throughput of the eye. Based on the numerical results – e.g. Figures 8 and 11 – for spot sizes smaller than 12 µm the role of modes in the guiding trough the narrowest cell pillar is of major importance.
We analyzed the mode structure in this central pillar at different incidence angles and corresponding spot sizes at different wavelengths. Figures 12(a), (b), (c) show the distribution of the transmitted radiation among the modes.
Fig. 12. Distribution of the light power efficiently coupled into the cell among the modes at different incidence angles (a) at 410 nm, (b) at 560 nm, (c) at 575 nm wavelength.
The mode distribution at each incidence angle was calculated with the spot size calculated with the Zemax optical model for 4 mm pupil diameter. For the mechanical and refractive index parameters of the Müller cells taken from [1] the wavelength range of the visible spectrum can be divided into three regions. Between 380 nm and 419 nm there are five supported modes: LP01, LP11, LP02, LP21, LP31. Between 419 nm and 565.12 nm there are four modes – LP01, LP11, LP02, LP21 – and above 576.79 nm there are only two: LP01 and LP11. It is important that simulations, presented in Figs. 12(a), (b), (c) revealed that the distribution of coupled light over more modes increases relative light guiding ability at higher incidence angles. At 577 nm and 14° incidence only two modes can be excited, here the efficiency is 6.4% for LP11 (summation of negative and positive modes) and 2.35% for LP01 mode, with 4 µm spot diameter (smallest). At 560 nm there are four modes excited at the same incidence angle and spot size: 6.02% LP11, 2.04% LP01, 5.26% LP21 and 0.61% LP02. The possibility to excite the LP02 and LP21 modes as well increased the overall excitation efficiency by 5.18%, by more than half of the value at 577 nm, from 8.75% to 13.93%. With higher spot sizes (higher pupil aperture) the light collection efficiency is less – 2–5% –, but the increase in light guiding capacity due to the additional modes can be of 60–70%, at only 15 nm wavelength difference. Hence the number of possible guided modes and the exact position of the mode cut-off wavelength are of high importance.
4. Analysis of morphological effects on the cell’s wave guiding ability
We have shown in the previous section how the angle of incidence determines the coupling of the light into the Müller cells and that the efficiency of the throughput to the photoreceptors is affected by the number of the guided modes.
In the case of biological entities, it is an important question whether the fluctuations of the mechanical and optical parameters of the cells originating from the biological nature influence the light guiding properties of them, especially the number and intensity of the transmitted modes. The mechanical sizes (diameter, length) of the cell may vary within the same retina from location to location, and may be quite different in different eyes. The refractive indices of the tissues also fluctuate; hence it is useful to analyze the tolerance of the light guiding property of the cells, especially the number of supported modes against these variations. We call this tolerance as stability and introduced a stability/instability indicator to quantitatively characterize this property.
The value of this indicator is the square root of the sum of the squares of the products combining mode number and mode cut-off wavelength changes caused by cell radius and refractive index variations. The formula is inspired by the method used to calculate mean squared deviation generally used in statistics and signal analysis [33].
This parameter depends only on the physical properties of the cells and is not influenced by the field distribution arriving to their input. Hence this indicator is adequate to compare the different structures from the point of view of their light guiding efficiency. One limitation is that only those parameters are appearing in the indicator, where mode cut-off occurs. However, as shown before the mode cut-off will be of crucial importance for the light guiding cells lying in the retina segments far from the center.
The formula of our suggested stability/instability indicator:
SIRn1 – The value of the indicator for a fiber with main pillar radius R and refractive index n1
N – the number of modes in a fiber of radius R and refractive index n1
ΔNΔR – change in the number of modes caused by a radius change ΔR
ΔλΔR = ∂λ/∂R*ΔR – change of the cut-off wavelength of one mode when the radius changes by ΔR
ΔNΔn1 – change of the number of modes when the refractive index changes by Δn1
ΔλΔn1 = ∂λ/∂n1*Δn1 – change of the cut-off wavelength of one mode when the refractive index changes by Δn1
The formula of the stability indicator is composed by the changes in the number of the guided modes and their cut-off wavelengths caused by the variation of the cell pillar radius R and refractive index n1. The changes in the cut-off wavelength are expressed by the terms N*ΔλΔR and N*ΔλΔn1, which are in fact approximations. The number of modes as a multiplier takes into account the cumulated effect of each possible cut-off change (as many possible cut-offs as many modes). Generally higher values of this indicator indicate bigger sensitivity to radius and refractive index variation.
We analyzed the variation of the stability indicator based on diagrams produced by numerical evaluation of the light guiding possibilities of the cells. Diagrams based on mode calculations performed with the software RP Fiber Calculator were generated to visually represent the wavelength and cell radius dependence of the light guiding ability of the Müller cells. These diagrams emphasize the effect of the morphological change of the radius and less that of the refractive indices, mainly because of the lack of reliable biological data.
Figure 13 shows the values of the cell radii where cut-offs for the different linearly polarized guided modes appear as a function of wavelength. These represent linear relations and show that Müller cells with an average radius of 1.2 µm guide four modes in the blue region, three modes in the green and yellow region and two modes in the red and near infrared portions of the spectrum. If the cell radii fluctuate (in the case of biological entities it is implicit), then the mode cut-off wavelengths will also shift. This diagram is calculated assuming all possible guided modes determined by the geometry and relative refractive index of the narrowest cell pillar. We have shown in the previous section that as we depart from the axis connecting the pupil and foveola centers the incidence becomes oblique and the amount of the higher order modes in the total throughput becomes more and more dominant. Hence the cut-off wavelength will have a bigger importance in cells on the lateral sides of the retina.
Fig. 13. Wavelengths of the different modes as a function of the fiber radius R. Color indicates the order of the modes: yellow are first order modes (LPx1), green are the second order (LPx2), blue the third order (LPx3), pink fourth order (LPx4), purple fifth order (LPx5), etc. Some are emphasized in the figure, since they are the possible modes of the central pillar of the Müller cells. LP01 is the fundamental mode, it is visualized at R = 0 value, indicating that it does not have cut-off wavelength. Refractive indices used for the calculation are those given in the literature [17] for Müller cells: n1 = 1.38 and n2 = 1.345.
First we fix the possible variations of the cell radius and the refractive index. These could be also arbitrary values, but considering optical and biological assumptions we restrict the values of these for the following sets:
In principle we adapted the variation of these parameters from references 16, 17 and 18. The refractive index variation is mainly based on Fig. 1(c) of [17]. The radius variation of 0.1 µm indicates small structural variations, errors, usually being smaller than the wavelengths of the conducted light. The indicator contains only the width variation of the main cell pillar. The variation range is realistic based on experimental results, when comparing e.g. to Figs. 1(b), 2(c), 3(a) of [18].
We indicated some specific points on Fig. 14 for which we calculated the values of the indicator based on our formula. During the calculation we normalized the values of the products and the results of the quadratic summations. Main results are shown in Figs. 15 and 16.
- a) The absolute values of the indicators have little importance, their relative values are mainly of higher interest. This is because the absolute values depend on the particular values of the chosen radius and refractive index variations for which the investigation took place.
- b) We use normalized values of the indicators, hence the values of the indicators depend also on the chosen point representing a parameter configuration (e.g. in the case of the point no. 1 there is a value of 100, because we got the highest value of the indicator for that particular configuration).
- c) The relative high value of the indicator indicates higher instability, whereas small values indicate stable behavior of the cells in their light guiding and mode supporting function.
- d) The variation of the refractive index and the radius are independent, hence they can strengthen their mutual effect on the cut-off wavelengths, or can compensate each other mutually. Most interesting is when they both contribute with a different sign to the wavelength shift; hence cancel the effect of each other.
- e) The indicator handles the number of possible supported modes as a factor that increases instability, mainly in the shorter wavelength range.
Fig. 14. Value map of the stability indicator as a function of fiber radii and wavelength. Based on the relative difference between values we defined zones, within the map that bear specific characteristics, marked by capital letters from A to G. Coloring does not precisely corresponds to the stability indicator values, the borders of the zones serve rather intuition. The different zones contain configurations with similar values of the stability indicator, we marked connected zones according to this. At configurations far from the actual possible Müller cells these zones are not detailed. We only emphasize fine changes of the indicator for the zones directly related to Müller cells, using transient coloring. Lack of colors in bigger zone areas – e.g. below the E zone – indicates that there are no cutoff wavelengths. Encircled numbers represent sample configurations; the corresponding indicator values are given for each configuration with black numbers.
Fig. 15. Stability zones as depending on the wavelength for the possible variations of the Müller cell radius.
Fig. 16. Wavelength dependence of the effective mode diameter for the various guided modes by the Müller cell, (a) at hypothetical 1 µm cell pillar radius and (b) at the real 1.2 µm cell radius. Relation between the area of the modes and area of the waveguide cross-section is important from the point of view of leakage losses. Color code for the mode types: black: LP0x, red: LP1x, yellow: LP2x, green: LP3x, blue: LP4x, pink: LP5x, where x is the mode order.
Grouping the stability indicator based on their normalized values may lead to a classification of the cell’s behavior within the different portions of the diagram. The main indicator is the average value of the Müller cell’s radius, which places the cell into different zones for different wavelengths. However, comparison between zones for values other than the characteristic cell radii and refractive indices may lead to some conclusions regarding the possible consequences when the cells have different parameters than actual. We discuss these alternative parameter sets based on the diagram of Fig. 14 and show that the actual natural average is “optimal”.
For example the highest instability in this context appears for small wavelengths and radius values higher than the Müller cells – in the G zone – where many guided modes will be possible, and hence small variations in the refractive index and radius will result in fluctuation of many mode cut-off wavelengths. This will appear as loss in the light guiding efficiency of the cells, hence it is not optimal for efficient light throughput. The contrary is valid for the E zone, where the variations in cell radius and refractive index cause only one mode jump. This zone is on the limit of the single mode guidance. At short wavelengths, the cut-off wavelength does not change much, at longer wavelengths this increases together with the relative value of the stability indicator. Despite of this in this mode, light guiding would be of highest stability. But the optical system of the eye does not allow small spot sizes on the input of these cells, hence coupling in these small cross section area cells is inefficient. We show later that in this small diameter case the ratio of the mode areas relative to the pillar’s cross section area is bigger than optimal, and considerably more light leakage occurs, than it would in the case of bigger pillar diameters.
Their actual parameters (average radius and refractive index) place the Müller cells into zones A and B. These zones are shown as continuous because of the similar values of the indicator that slightly increase with the wavelength. After zone E the smallest relative indicator values are found in the A zone. The small values are consequence of the small contribution of all factors in the formula 5. These values are not the smallest possible but none of them are high, when compared to the parameters appearing in other zones. For example, in point 2 the cut-off wavelength changes are small, but the number of guided modes is high, multiplying the small wavelength change values. In point 19 the number of modes increases the indicator value, however the local mode-number fluctuation is small. In neighboring point 15 the mode number fluctuation is bigger. In points 28 and 31 the number of modes is smaller than e.g. in point 16, the mode number fluctuation is of the same value, but the wavelength fluctuation values are bigger. As a result, in zone A the components of the stability indicator are of small and moderate values hence the relative indicator values remain small. In zone B the indicator values are somewhat bigger due to the higher wavelength and wavelength-fluctuation values. E.g. in point 22 the indicator has a relatively high value of 12.71, despite of the smaller number of guided modes: 3. As a conclusion the analysis of the stability indicator shows the light guidance of the Müller cells from the point of view of guided modes is optimal in the actual A and B zones.
Based on the diagram of Fig. 15 we show how the stability zones can be used to evaluate the cell’s behavior with eventually varying parameters. For example, increase of the cell radius to 1.4 µm causes narrowing of the spectrum in the A and B zones. The average sensitivity peak of the “blue” cone photoreceptors would fall outside the A zone, and higher wavelengths would also be less effectively transmitted. On the contrary, when the cell radius is decreased to e.g. 1 µm, the stability zones A and B would cover a larger spectrum portion.
However, any decrease of the radius leads to decrease of the useful transmitting cross section, hence the effective mode area would increase compared to the cross section area, which would lead to increased losses. The mode area and cell cross-section area relations are shown in Figs. 16(a) and (b), for the actual 1.2 µm average cell diameter and the hypothetic 1 µm diameter.
The above analysis shows that the actual size and structure of the Müller cells is close to ideal regarding the stability and light guiding efficiency. Within the A and B zones the variation of the stability indicator along the vertical axis is minimal, the values e.g. in points 16 and 17, as well as in 21 and 22 are nearly identical. This indicates that in the biggest part of the visible wavelength range the cells show high stability in light guiding against radius variation, at least in the 1–1.4 µm range. This supports the hypothesis that Müller cells are optimized light collectors and guides in the sensitivity range of the photoreceptor cells.
5. Conclusion
In this analysis we have shown that Müller cells are adequate to collect and guide light from the inner surface of the peripheral retina to the photoreceptor cells. Light distribution on the inner surface of the retina has been calculated using Zemax eye model based on biological data collected from many human subjects. Numerical modeling of the light guiding within Müller cells including the funnel-like termination shows that in locations outside the fovea the calculated spot sizes and incidence angles allow efficient coupling into the Müller cells as fibers. The numerical results also show that light guiding property of the Müller cells is determined by the size and refractive index of the narrowest pillar. Importantly this cell process supports few mode light guiding, and the cut-off wavelengths of the higher order modes influence the wavelength range of the transmission. Variation in anatomic properties leads to shifts in number of guided modes at a given wavelength; hence the transmission efficiency is seriously affected. The stability indicator introduced in the previous section and the corresponding diagram helps greatly to have a visual representation of the light guiding efficiency of an actual cell or cell group.
As Müller cells play a crucial role in light transmission through the scattering retina outside the fovea, their anatomical modification may lead to increase or decrease of the transmission efficiency and thus in light sensing in different wavelength ranges. Light sensing efficiency decrease at short or long wavelengths may be symptoms of a disease, which is in fact caused by size or refractive index variations in Müller cell’s processes. Examination of these anatomic parameters can help to identify the reasons of this type of eye malfunction.
Funding
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (NBP 24 (ED-17-1-2017-0011), NKFIH-873-4/2020, VKSZ-12-1-2013-80); EU Commission QuanTELCO (862721).
Acknowledgments
OCT picture of the retina in Fig. 1(a) courtesy of the Department of Ophthalmology, Semmelweis University. Two-photon registrations of retina details in Fig. 1(b)–(e) courtesy of Tamás Tompa, Femtonics Ltd. Special thanks to Dr. Gábor Vass for his critical remarks and to Prof. Dr. Özséb Horányi for his inspiring hints.
This project was funded by “Medical technological research and development on the efficient cure of cataract” (VKSZ-12-1-2013-80); Subprogram no 24. of the National Bionics Program in Hungary (NBP24); NKFIH-873-4/2020 Quantum Information National Laboratory project of Hungarian National Research, Development and Innovation Office.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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