Color sorting by retinal waveguides Amichai M. Labin and Erez N. Ribak* Department of Physics, Technion - Israel Institute of Technology, Haifa 32000, Israel *
[email protected]
Abstract: Light is being detected by the two distinct types of photoreceptors in the human retina: cones and rods. Before light arrives at the photoreceptors, 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. ©2014 Optical Society of America OCIS codes: (330.0330) Vision, color, and visual optics; (060.0060) Fiber optics and optical communications.
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#220144 - $15.00 USD (C) 2014 OSA
Received 30 Jul 2014; revised 31 Oct 2014; accepted 28 Nov 2014; published 22 Dec 2014 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.032208 | OPTICS EXPRESS 32208
1. 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), 1(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), 1(d) [2]. The methods applied were an optical modal analysis [6] and a numerical FFT split-step beam propagation method [7, 8]. 2. 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.
#220144 - $15.00 USD (C) 2014 OSA
Received 30 Jul 2014; revised 31 Oct 2014; accepted 28 Nov 2014; published 22 Dec 2014 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.032208 | OPTICS EXPRESS 32209
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%.
The split-step beam propagation method (BPM) of the global third order [7,8] solves numerically the Helmholtz equation for generic refractive index structures
∇ 2 E (r ) + k 2 n 2 E ( r ) = 0 .
(1)
Here r = ( x, y, z ) , E (r ) is the electromagnetic field, k = 2π / λ and n is the refractive index of the medium. By writing the Laplacian as ∇ 2 = ∇ ⊥2 + ∂ 2 / ∂z 2 ( ⊥≡ ( x, y ) and z is the propagation direction), and using the fact that the refractive index variations are small, δ n / n
