People studied cones and rods by applying the optical model with three sections, which includes myoid, ellipsoid and outer segment [21] . In this paper, we first propose to study cones and rods by applying the optical model with five sections, which includes nucleus, outer fiber, myoid, ellipsoid and outer segment (Figure 1(a)).

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 [22] . The inner and outer segment diameters of human foveal cones are about 2 μm and 1 μm in size respectively and the diameter of cones differs in going from the fovea to the periphery [23] , while those of rods are both about 2 μm [24] . To our knowledge, there are not the refractive indexes of human cones and rods available. But the values of refractive indexes of the same kind of cells in vertebrate species are close. In this paper, we take the average values of refractive index from different vertebrates species in Sidman’s paper as human’s refractive indexes, where the refractive indexes of myoid, ellipsoid, outer segment and surrounding medium are about 1.36, 1.39, 1.385 and 1.3476 for cones (and 1.36, 1.40, 1.41 and 1.3476 for rods) respectively [25] .

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 Figure 1(b), the outer segments and myoids of foveal LWS and MWS cones are single-mode optical waveguides, which lead human cones to see objects very stable and with high spatial resolution and low noise in daylight. The outer segments of foveal SWS cones are single-mode optical waveguides as well, however the myoids of foveal SWS cones, whose normalized frequency is 2.635 which is slightly larger than 2.405, work at multimode. Although the myoids of foveal SWS cones work at multimode, the main power is still contained in the fundamental mode instead of in high order modes.

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 [15] , and their structure is similar to the structure of foveal cones, which work at single-mode and are with tapered structure. In optical communication systems, 10Gbit/s and lower speed photodetectors work at multimode and are with straight structure, and their structure is similar to the structure of rods, which work at multimode and are with straight 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 [12] [13] . A photoreceptor’s inner segment is hexagon [26] , whose Laplacian MFD can be calculated from the MFD definition of arbitrary two dimensional optical waveguides. In this paper, we take the outer segment, ellipsoid and myoid as circular fibers for simplicity. In circular fibers, our Laplacian MFD definition for two dimensional waveguides will be simplified to the reference definition of MFD, which was proposed by Petermann [16] [17] .

In this part, we mainly talk about MFDs of foveal cones, because the most part of rods work in multimode. Figure 2 shows the Laplacian MFD of outer segments of human foveal cones. As shown in Figure 2(b) and Figure 2(c), at cones’ real geometrical diameter (1 μm) and relative refractive index difference delta = 2.66% (calculated using the refractive index given by Sidman), the Laplacian MFDs of foveal MWS and LWS cones’ outer segments are 1.34 and 1.42 μm respectively, which are very close to the minimum values. It is worth to note that a small MFD is benefit to reduce the crosstalk between neighboring cone outer segments, where optical signals convert to electrical signals, therefore a small MFD is benefit to improve the spatial resolution. As shown in Figure 2(a), the Laplacian MFD = 1.14 μm (at d_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 [5] [6] . Our calculation results of MFD of cones’ inner segments based on fiber parameters are close to their experimental results. The MFDs of myoids of foveal SWS, MWS and LWS cones are 2.05, 2.35, and 2.46 μm respectively. The diameter of outer fiber of foveal cones is about 1.6 μm [27] . As shown in Figures 3(a)-(c), the MFDs at the geometrical diameter of 1.6 μm and 2 μm are nearly equal in foveal cones (which are 1.916, 2.303 and 2.455 μm respectively for outer fibers of SWS, MWS and LWS cones), and this lead the coupling loss between myoids and outer fibers to be very small (0.035, 0.020 and 0.022 dB respectively for outer fibers of SWS, MWS and LWS cones). We can found that there are strong coupling effects between inner segments of adjacent cones and outer fibers of adjacent cones, (see Ref. [7] ), and these strong coupling are very important to unify the two important color theory: opponent color theory and tricolor theory.

Figure 4 shows the Laplacian MFDs of foveal cone myoid and outer segment change with the incident wavelength. The MFDs increase as the wavelength increases. This may be one of reasons that lead red objects appear larger than blue or green objects with same physical sizes. It is also interesting that the MFD of outer segment at long wavelength end is nearly equal to that of myoid at short wavelength end.

The ellipsoid of inner segment of a cone is with tapered shape as shown in Figure 1(a). We propose that the ellipsoid of a cone may act as a spot size converter between myoid and outer segment to reduce coupling loss. The coupling loss between myoid and outer segment at each cone’s most sensitive wavelength is calculated by Formula (4) and Formula (5) respectively in Figure 5 and we can see that the values are quite close. If the light had been launched from myoid to outer segment directly (i.e. without ellipsoid), the coupling losses between outer segment and myoid of foveal SWS, MWS and LWS cones would have been 1.39, 1.06, and 0.96 dB respectively (calculated with Formula (4)), which are quite large and will reduce the sensitivity and induce leaked light power significantly. As shown in Figures 5(a)-(c), if the foveal cones are with the ellipsoid in real situation, the minimum values of coupling losses between myoid and outer segment of foveal SWS, MWS and LWS cones are 0.21, 0.33 and 0.38 dB respectively (calculated with Formula (4)).

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 Figure 6 shows, nearly at all incident wavelengths, the coupling loss between foveal cone myoid and outer segment (calculated with formula 4) is at the lowest point when the refractive index is at the golden ration point value 1.3785.

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 Figure 7 that the values of coupling loss calculated by Formulas (4) and (5) at the same wavelength are quite close when the MFDs are close to each other at adjacent end faces.

The coupling loss curves are quite flat in the right side region between the refractive index at golden ratio point and 1.39 in Figure 5. In this flat region, even if the ellipsoid’s refractive index changes for some reasons (e.g., the amount of liquid in ellipsoid changes under some conditions etc.), the ellipsoid’s function to reduce coupling loss still works well and is very stable. It is hard to measure the refractive index of photoreceptors in living condition very precisely, and it is possible that the real refractive index of foveal cone’s ellipsoid in living condition may be smaller than that of Sidman’s result 1.39 in some cases [28] . It is possible that the real refractive index of foveal cone’s ellipsoid in living condition may be in the right side region between the refractive index at golden ratio point and 1.39, and even it is possible for the real refractive index to be at the golden ratio point to make the lowest coupling loss between myoid and outer segment, and this can save energy.

Figure 8 shows the function of foveal cones’ ellipsoid as a spot size converter. Spot size converters, which can reduce coupling losses induced by MFD mismatch between two waveguides, have been widely used in single-mode fibers, laser diodes, photodetectors, modulators etc [29] -[33] . The MFD of a myoid is larger than that of the outer segment, and the MFD of ellipsoid reduces gradually from the interface between myoid and ellipsoid to the interface between ellipsoid and the outer segment. The ellipsoid of cones converts the large MFD to the small one in order to match the different MFDs of myoid and the outer segment. Therefore, we proposed that the ellipsoid of a cone acts as a spot size converter to reduce the coupling loss.