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The characteristics of optical waveguide of human photoreceptors play important roles in vision. The mode-field-diameter (MFD) is a very important parameter of a single-mode waveguide, and it is related to many important optical characteristics of a single-mode waveguide. Here we show that MFDs of outer segments of human foveal cones are close to the minimum values at their geometric diameter for outer segments of foveal cones. Small MFD of outer segment is important for eyes to have high spatial resolution and low interaction between neighboring cones. We propose that the ellipsoids of foveal cones act as spot size converters to reduce the coupling losses between myoids and outer segments.

The characteristics of optical waveguide of human photoreceptors play important roles in vision. Enoch first discovered that the outer segments of vertebrate photoreceptors are optical waveguides [

The mode-field-diameter (MFD) is a very important parameter of single-mode waveguides, and it is related to many important optical characteristics (e.g. coupling loss, angular misalignment loss, transverse offset loss, Rayleigh scattering loss, macro-bending loss, cross talk between two waveguides, effective area, the power percentage in the core and waveguide dispersion etc.) of single-mode waveguides [

In this paper, we first discuss the normalized frequency V of foveal cones and rods so that we can find out whether each part of photoreceptors works at single-mode or multimode; then we study the MFD of photoreceptors; finally we study the role of ellipsoids in foveal cones as spot size converters to reduce the coupling loss and we find the coupling loss is close to minimum at the golden ratio point.

The relative refractive index difference of a waveguide is defined as:

where

where

Anhui Liang et al proposed the Laplacian mode-field width for arbitrary noncircular optical waveguide [

where

For two optical waveguides working at fundamental mode, the coupling loss at the interface between them is given by [

where

When the cores of two optical fibers are aligned well, the coupling loss at the interface between them can be calculated approximately by the following formula [

where _{ }and

People studied cones and rods by applying the optical model with three sections, which includes myoid, ellipsoid and outer segment [

In human eyes, there are rods and cones. The cones can contain different visual pigments and according to their absorption characteristics are called SWS, MWS and LWS cones (short, middle and long wave sensitive cones). The absorption maxima of SWS, MWS, LWS cones and rods are 437 nm, 533 nm, 564 nm, and 498 nm respectively [

When the normalized frequency V of an optical waveguide is smaller than 2.405, which is the cut off normalized frequency of LP11 mode, it will work at single-mode. There is no mode interference in single-mode waveguides, therefore the light intensity in single-mode waveguides is more stable than that in multimode waveguides and the light, whose mode field distribution is like bell shape within the center at the fiber axis, is insensitive to environmental perturbation as well. As shown in

The myoids of rods are single-mode waveguides, but the outer segments of rods are multimode optical waveguides. Because the light is launched from the myoids as the fundamental mode, the main power at the outer segments of rods is still contained in the fundamental mode. Because the rods and cones are responsible for scotopic and photopic vision respectively, one advantage for the outer segments of rods to have multimode optical waveguides structure is to improve sensitivity by gathering all of power in fundamental mode and high order modes.

The single-mode waveguide structure of foveal cones is good for their functions of high spatial resolution and high detection speed. Because the diameter of foveal cones’ outer segments (about 1 μm) is smaller than that of rods (about 2 μm), the smaller diameter d_{o} is helpful to reduce cones’ capacitance and increase cones’ response speed to light. In optical communication systems, 40 Gbit/s high speed photodetectors work at single-mode and are with tapered structure [

For arbitrary two dimensional single-mode waveguides (e.g. laser diodes, photo detectors etc.), there are several MFD definitions, but the Laplacian MFD, which was proposed by the coauthor of this paper and Fan, is the most important MFD definition [

In this part, we mainly talk about MFDs of foveal cones, because the most part of rods work in multimode. _{o} = 1 μm) of SWS cones is close to the smallest

Laplacian MFD and its real geometrical diameter 1 μm. The Laplacian MFDs of foveal cones are sensitive to the relative refractive index differences.

Figures 3(a)-(c) show that Laplacian MFDs of photoreceptor’s myoid are insensitive to the geometrical diameter variation around their real geometrical diameter (about 2 μm). Vohnsen et al. used the Gaussian MFD of 2.2 μm of cones’ inner segments in their model based on the Stiles-Crawford effect measurement and the Gaussian approximation [

The ellipsoid of inner segment of a cone is with tapered shape as shown in

With the help of ellipsoids of foveal SWS, MWS and LWS cones, the coupling losses are reduced by 1.18, 0.73 and 0.58 dB respectively (at the optimum ellipsoid refractive index). This coupling loss reduction can help eyes to improve sensitivity and to reduce energy consumption; meanwhile it will help eyes to improve spatial resolution and to reduce noise because of very low leaked power to neighboring cones. The refractive index of ellipsoid is 1.39 according to Sidman’s result, and our results show that the coupling losses of foveal SWS,

MWS and LWS cones will be 0.30, 0.46 and 0.52 dB respectively at this refractive index (calculated with formula 4). With the helps of ellipsoid, the coupling losses of foveal SWS, MWS and LWS cones are reduced by 1.09, 0.60 and 0.44 dB respectively. At the minimum coupling loss points, the corresponding refractive indexes of ellipsoid of foveal SWS, MWS and LWS cones are 1.3774, 1.3780 and 1.3782 respectively, and they are quite close to the refractive index 1.3785 at the golden ratio point between myoid’s refractive index (n_{m} = 1.36) and ellipsoid’s refractive index (n_{e} = 1.39) (especially in MWS and LWS cones). As

We have calculated the coupling loss between foveal cone myoid and outer segment using formula 4 and 5 respectively, when the refractive index of ellipsoid is 1.3785. We can see from

The coupling loss curves are quite flat in the right side region between the refractive index at golden ratio point and 1.39 in

We first propose the five-segment optical model of human photoreceptors. We find MFDs of outer segments of

human foveal cones are close to the minimum values at their geometric diameter for outer segments of foveal cones. Small MFD of outer segment is important for eyes to have high spatial resolution and low interaction between outer segments of neighboring cones. We propose that the ellipsoids of foveal cones act as spot size converters to reduce the coupling losses between myoids and outer segments.

This work is supported by The National Natural Science Funds (81470661), Jiangsu Province Creative Talent and Startup Talent Program (0900613001), Jiangsu Province Creative Team Program, and NUPT Research Foundation for Advanced Talents (NY212001).