1.

Introduction

Quantum wells (QW), quantum dots (QD), and other nanoscale devices created new possibilities for the development of microelectronics and microoptics. One of the promising approaches is based on photonic crystals, allowing the control of dispersion and propagation of light.^1,2 The best-known effects include transmission or rejection of light in a given wavelength range and waveguiding the light along linear and bent defects in a periodic photonic crystal structure. A surface plasmon is a transverse magnetic-polarized optical surface wave that propagates along a metal-dielectric interface. Surface plasmons exhibit a range of interesting and useful properties, such as energy asymptotes in the dispersion curves, resonances, field enhancement and localization, high surface and bulk sensitivities to absorbed molecules, and subwavelength confinement. Because of these attributes, surface plasmons found applications in a variety of areas such as spectroscopy, nanophotonics, imaging, biosensing, and circuitry.^3–10 Plasmon theory was extensively used to interpret the phenomena of light focusing at the nanoscale.^11–34 Technical applications of the optical nanodevices have also been widely discussed,^35–74 with several examples of natural nanostructures for light harvesting, transmission, and reflection described in plants and bacteria.^75,76

Specific filament-like biological nanostructures were described as necessary for the transparency of the vertebrate crystalline lens. The lens in the vertebrates is built of the so-called fiber cells; these cells are very narrow (3 to $5\mu \mathrm{m}$) and very long (millimeters), thus resembling fibers.⁷⁷ These cells have specialized beaded intermediate filaments (IFs) 10 to 15 nm in diameter, built of proteins filensin and phakinin. Such fibers were described as indispensable for the fiber cell transparency, with many alterations to these IFs of chemical or genetic origin resulting in the lens cataract formation.^78–86 Recently, a high level of transparency was also described in certain retinal cells. It was shown that light transmission in the vertebrate retina is restricted to specialized glial cells, which have many other physiological functions. Indeed, Franze et al.⁸⁷ in their pioneering studies demonstrated that the Müller glial cells function as optical fibers. They demonstrated that Müller cells (MCs) transmit visible-range photons from the retinal surface to the photoreceptor cells, located deep under the surface. In fact, the retina has inverted structure, thus the light projected onto it has to pass through several layers of randomly oriented cells with intrinsic scatterers before it reaches the light-detecting photoreceptor cells.^88,89 However, the guinea pig retina contains a regular pattern of MCs arranged mostly in parallel to each other, spanning the entire thickness of the retina ($\approx 120$ to $150\mu \mathrm{m}$). The MC main cylindrical process⁹⁰ that spans the retina resembles an optical fiber, because of its capacity to transmit light.⁸⁷ These cells typically have several complex side branches with functions not related to light transmission, with variable morphology.^90,91 Thus, the main processes of the MCs create a way for the light to go through the retina. Here, we suggest that these main processes may contain specialized molecular structures functioning as optical waveguides. Therefore, it is important to investigate the morphology of the MCs main processes and develop a theoretical model of light transmission by these cells.

Numerous materials form dielectric-conductive waveguides, including carbon-based materials with high carrier mobility and conductivity.⁹² Presently, we shall interpret the light transmission by MCs⁸⁷ based on the model of light transmission by a waveguide with conductive nanolayer coating. A dielectric waveguide with a metal nanolayer coating is adequately described by the plasmon theory.^1–10 This theory uses the quantum electronics approach to analyze the electromagnetic field (EMF) energy transmission by such a waveguide. A plasmon is understood as a conduction-band electron oscillation along the waveguide surface at the external EMF frequency.^{3–8,16–21,53–69} Such oscillations are induced by the EMF at the input end of the waveguide, and then the plasmon energy is transformed back into the EMF at the output end of the waveguide.^{3–8,16–21,53–69} Presently, we shall focus our attention on the role of the quantum confinement (QC) in the light transmission by MCs and nanotubes with a conductive coating, where the QC operates in the direction normal to the coating. We will analyze the EMF-induced transitions between the discrete states created in the conductive coatings, and the excitation transport from the input end of the waveguide to its output end. We shall demonstrate that the excitation transport should operate simultaneously with the state-to-state transitions, as the ground- and excited-state wavefunctions span the entire surface of the nanocoating. Now, we extend this approach to the analysis of the light transmission by the MCs, proposing a mechanism describing this phenomenon. This study has a direct impact on the medicine of human vision, and also on the development of nanomedical eye technology.

2.

Experimental Methods and Materials

The presently used histological material was collected in Russia in 1999 during the study of the bifoveal retina of the pied flycatcher (Ficedula hypoleuca). The eyeballs of the flycatcher chicks 27 days after hatching were fixed in 3% gluteraldehyde with 2% paraformaldehyde in 0.15 M cacodilate buffer and postfixed with 1% ${\mathrm{OsO}}_4$ in the same buffer. The eyeballs were oriented relative to the position of the pecten and embedded into Epon-812 epoxy resin. Ultrathin sections 60-nm thick were made on an LKB Bromma Ultratome ultramicrotome (L.K.B. Instruments Ltd., Northampton, United Kingdom) and examined on a JEM 100B (JEOL Ltd., Japan) electron microscope as described earlier.⁹³

3.

Experimental Results

The MCs in a Pied flycatcher eye extend from the vitreous body, where the light is entering the retina, through the entire retina to the photoreceptor cone cells, and thus the cell structure resembles the one previously described in the guinea pig eye.⁸⁷ The MC bodies are located in a specific layer inside the retina, sending the principal processes to both of its surfaces. Indeed, the electronic micrograms show that the MC endfeet (inverted cone-shaped parts of the MC covering the vitreous body, see Fig. 1, red arrows) form the inner surface lining called the inner limiting membrane (ILM), covering the entire inner surface of the retina. The basal processes of the MCs originating from the endfeet [Fig. 1(a), green arrows] protract away from the inner surface, normally to it and essentially parallel to each other [Fig. 1(a), green arrows], and spreading around branches of other cells. We suggest that similar to what was found in the guinea pig,⁸⁷ the light incident from the vitreous body is entering the endfeet [Fig. 1(a), red arrows], and next the excitation energy flows from the basal process to the cell body to the apical processes, where it is transferred to the cone photoreceptor outer segment, as shown in Fig. 1(c). The diameter of the MC main process in the Pied flycatcher may be $<1\mu \mathrm{m}$ [Fig. 1(a)]; therefore, these cells working as waveguides should be described as natural nano-optical structures.

Fig. 1

(a) Endfeet (red arrows) and basal processes (green arrows) of MCs in the Pied flycatcher retina. (b) High magnification insert from (a), showing a part of cytoplasmic structure (green arrows) that has parallel linear elements resembling IFs. This structure spans the cytoplasm from the narrow part of the basal endfoot to the apical end that wraps around a cone photoreceptor, in the direction of light transmission. Scale bar in (a) and (b)  is 500 nm. (c) Schematic presentation of the MCs (green arrows) with their endfeet (red arrows) and the cone photoreceptors (R, G, B). The light propagation direction coincides with the red arrows.

We also studied the cytoplasm of the MCs in the apical and basal processes, discovering parallel structures that span the cell cytoplasm all the way through from the inner membrane to the photoreceptors. A higher magnification insert of a part of the basal process of the MC⁸⁷ [see Fig. 1(b)] shows this structure within the cytoplasm of the cell process, repeating all of its curves. This structure resembles a bundle of parallel IFs with the outer diameter of 10 nm, with some smaller microparticles organized around the filaments. The IFs in the glial cytoplasm are most often associated with globular particles of chaperone proteins (usually termed crystallines)⁹⁴ that stabilize these long filaments, or with nucleoprotein particles.⁹⁵ These filaments span almost the entire 400 to $500\mu \mathrm{m}$ of the MC length, from the endfoot to the photoreceptor (the outer membrane), thus recalling the previous observations in other species.⁹⁶ The filamentary structure is absent in the MC endfeet. Unfortunately, neither the precise ultrastructure of this intermediate-filament bundles in the MCs nor the continuity of the filaments, which may be important for understanding the mechanism of their operation, could be resolved on the presently used equipment, and will be investigated additionally. Usually, the IFs have their external diameter varying within 8 to 13 nm, with each filament typically built of eight protofibrils.⁹⁷ Such specific IFs in the lens fiber cells are apparently responsible for the optical transparency of these cells.^78–86 In our opinion, the bundles of IFs in the MCs of the Pied flycatcher eye are the structures responsible for the transmission of luminous energy to the photoreceptor cells, deserving a close attention of quantum physics.

4.

Theoretical Models

Here, we will discuss the (1) quasiclassical and (2) QC approaches to the analysis of light transmission by the IFs in MCs. The two approaches may be summarized as follows:

4.1.

Quasiclassical Approach

We already mentioned numerous studies^1–74 that developed the theory of plasmonic quantum electronics and its applications to dielectric-conductive waveguides. Figure 2 shows this model schematically, where only the $E_{\parallel }$ component of the electric field may produce plasmons, exciting electronic oscillations polarized along the surface of the waveguide.

Fig. 2

Schematic presentation of the input section of the lightguide with a nanothick conductive wall. $\mathbf{P}$ is the Poynting vector, $\mathbf{E}$ is the electric field vector of the incident EMF, $\mathbf{H}$ is its magnetic field vector.

The efficiency of the energy transmission by a waveguide is significantly dependent on the profile of its input zone.^{3–18,64–74} Presently, however, we shall not discuss the plasmonic theory in detail, focusing our attention on the more exact quantum approach.

4.2.

Quantum Confinement

Analyzing tubes with nanometer-thick conductive walls, we must take into account the QC,^92,98,99 appropriate for the light transmission by tubes with the diameter smaller than the wavelength of light.

As we already mentioned, the parallel electric field component $E_{\parallel }$ contributes to the plasmonic excitations, whereas the normal component $E_{\perp }$ contributes to exciting the discrete states created by the QC in the tube nanowalls. Thus, we shall consider the interaction of these field components with the respective quasicontinuum and discrete electronic states, followed by the EMF emission at the other end of the waveguide. The theoretic analysis of the QC in the device shown in Fig. 1 is very complex. Therefore, we simplified the system by considering a cylinder with the internal diameter $r_0$, wall thickness $\rho$, and length $l$. The cylinder is linked to a cone, with the wall thickness $\rho \cos (\alpha )$, height $h$, and aperture $2\alpha$. Still, the problem admits numerical solutions only, with the results that we shall discuss next. We shall assume that the wavefunction amplitude is continuous at the contact surfaces between the cylinder and the two cones (see Fig. 3). Thus, we shall analyze the excitation at the input cone, the excitation transport via the cylinder, and the emission at the output cone, for the sake of symmetry.

Fig. 3

The model for qualitative analysis. The waveguide is composed of a tube and two cones, with conductive walls.

To evaluate these phenomena qualitatively, we need to determine the quantum states in the cylindrical part using cylindrical coordinates, and the quantum states in the conical part using conical coordinates. It is impossible to solve the Schrödinger equation analytically in a conical system, which we shall, therefore, analyze approximately. We shall also assume an increased excited-state population density in the output conical piece (the right side of the diagram) as compared to the input conical piece (the left side of the diagram). Thus, the energy transport in the model system reproduces the phenomena occurring in nature, where the excitation ends up on the chromophore molecules contained in the cone photoreceptor cells, with the higher excited-state density in the chromophores modeled by that in the output conical piece.

4.2.1.

Cylindrical coordinates

We solved the eigenstate problem for the conductive nanolayers, using the Schrödinger equation in the cylindrical coordinates (Appendix A). To this end, we used the boundary conditions equivalent to an axial potential box with infinite potential walls, an acceptable approximation for qualitative analysis. Generally, the Schrödinger equation in cylindrical coordinates reads

Eq. (1)

$$-\frac{{\hslash }^2}{2m}\mathrm{\Delta }\psi (r,\varphi ,z)=E\psi (r,\varphi ,z),$$

where

Eq. (2)

$$\psi (r,\varphi ,z)=\psi (r,\varphi )\psi (z),$$

and $m$ is the effective electron mass. A detailed analysis of this equation is presented in Appendices A and B. The emission intensity at the output end of our axisymmetric system may be written in the dipole approximation as follows:

Eq. (3)

$$\begin{gathered} I_{\mathrm{emiss}}\propto {|⟨{\psi }_g(r,\varphi ,z)|{\overrightarrow{r}}_e|{\psi }_{\mathrm{exc}}(r,\varphi ,z)⟩|}^2{P}_{\mathrm{exc}}(t) \\ ={|⟨{\psi }_g(r,\varphi ,z)|{\overrightarrow{r}}_e|{\psi }_{\mathrm{exc}}(r,\varphi ,z)⟩|}^2{e}^{-{\gamma }_z(E_{z,\mathrm{exc}})t}, \end{gathered}$$

where $⟨{\psi }_g(r,\varphi ,z)|{\overrightarrow{r}}_e|{\psi }_{\mathrm{exc}}(r,\varphi ,z)⟩$ is the matrix element of the optical dipole–dipole transition. We shall further use Eq. (3) in the approximate numerical analysis of the light transmission by the IFs.

4.2.2.

Qualitative analysis of the solutions at the input and output cones

Making use of the axial symmetry of the system, we shall use the cylindrical coordinates. Appendix C presents the secular equation (SE) and the respective boundary conditions. It is possible to solve the SE only numerically, although we may obtain some qualitative results even without solving it. Next, we present the numerical results for the light transmission by the system of Fig. 4. In the input cone (Fig. 3), the electric field component interacts with the internal conical surface, exciting both longitudinal and transverse states. A detailed analysis of the energy transmission coefficient for the device shown in Fig. 3 is presented in Appendix D, with the resulting coefficient given by (see Appendix D)

Eq. (4)

$$T=\frac{G_{\mathrm{em}}}{G_{\mathrm{abs}}}\approx \frac{{\mathrm{Z}}_{\mathrm{exc}}}{{\mathrm{Z}}_{\mathrm{abs}}}.$$

Fig. 4

The model for numerical analysis, showing a cross section of the waveguide with conductive walls. $R$ is the curvature radius referred to in Fig. 5 and 6.

Equation (4) gives the light transmission efficiency as determined by the QC mechanism. In Sec. 4.2.3, we will carry out the numerical analysis of the QC effects for the device shown in Fig. 4.

4.2.3.

Numerical analysis of the light transmission by the device of Fig. 4

All parameters used for numerical analysis are listed in Table 1 and shown in Fig. 4.

Table 1

Parameters used in the numerical analysis.

Parameter	Definition
$r_0$	The internal diameter of the cylindrical tube
$\rho$	Cylindrical tube wall thickness
$L_1$	Length of the device
$L_2$	Length of the cylindrical section
$L_3$	Length of each of the conical pieces
$R$	Curvature radius of the curved conical pieces
$R_d$	The larger radius of each of the conical pieces

Figure 4 shows the relevant geometrical parameters of the device. The internal diameter of the cylindrical tube is $r_0$, and its wall thickness is $\rho$, the same as that of the conical pieces. The length of the cylindrical tube is $L_2$, the length of the entire device is $L_1$, the length of the input and output pieces is $L_3=[(L_1-L_2)/2]$, and $R$ is the curvature radius of the curved conical pieces. We shall assume that $r_0\gg \rho$. The larger radius of the input and output cones is calculated as follows (Fig. 4):

Eq. (5)

$$R_d=r_0+R-\sqrt{R^2-L_3^2}=r_0+R[1-\sqrt{1-\frac{{(L_1-L_2)}^2}{4R^2}}].$$

The boundary ring shown in the zoom-in box in Fig. 4 is normal to both the internal and the external limiting surfaces.

Due to the axial symmetry of the system, we shall obtain the numerical solution of the SE in the cylindrical coordinates (see Appendix E). Since specific boundary conditions are created in the system considered (Fig. 4), the absorbed EMF energy is reemitted by the electronic excited states mainly in the output zone. This result is similar to the superemission observed in an active lasing medium. However, the similarity is only indirect, as it is impossible to create an inverse population distribution by an optical transition between the ground and the excited state. We shall address the spectral selectivity of the device in a follow-up publication.

4.3.

Numerical Calculations

Numerical analysis was carried out using homemade FORTRAN software and the method of finite differences.¹⁰⁰ We solved the Eqs. (26) and (30) numerically using ${\lambda }_{\mathrm{EMF}}=400\mathrm{nm}$ and several different sets of the parameter values: $\rho =10\mathrm{nm}$; $r_0=5\rho$, $10\rho$, $15\rho$, and $20\rho$; $L_1=n_1\rho$; $L_2=n_2\rho$; $L_3=\rho [(n_1-n_2)/2]$; $R=n_3\rho$ (see Fig. 4). We used the following values of the size multipliers: $n_1=1000$, $n_2=800$, and $n_3=\mathrm{100,200},\dots ,1000$. We present the results in terms of the absorption efficiency

Eq. (6)

$$\eta (n_3)=\frac{W}{W_0},$$

where $W_0$ is the EMF intensity incident within the device cross section

Eq. (7)

$$W_0=S_d{\int }_0^{\infty }{|E_{k_{\lambda }}({\omega }_{\lambda })|}^2\mathrm{d}{\omega }_{\lambda }.$$

We obtained the numerical solution of Eq. (76) using the finite difference method.¹⁰⁰ We used a homemade FORTRAN code for the numerical analysis of this equation combined with Eqs. (80) and (7), with Fig. 5 showing the calculated results.

Fig. 5

Calculated plots of the transmission efficiency $\eta$ on the curvature radius $R$ (see Fig. 4). The values of the variable model parameters are: $\rho =10\mathrm{nm}$, (1) $r_0=5\rho$, (2) $r_0=10\rho$, (3) $r_0=15\rho$, and (4) $r_0=20\rho$.

Figure 5 shows calculated plots of the transmission efficiency $\eta$ on the curvature radius $R$ (see Fig. 4). The values of the variable model parameters are: $\rho =10\mathrm{nm}$, (1) $r_0=5\rho$, (2) $r_0=10\rho$, (3) $r_0=15\rho$, and (4) $r_0=20\rho$.

The efficiency of the light transmission varies with the geometry, with the maximum values corresponding to the following parameter sets: (1) $R=2.23\mu \mathrm{m}$, with the maximum cone radius $R_{d,\max }=0.29\mu \mathrm{m}$; (2) $R=2.94\mu \mathrm{m}$, $R_{d,\max }=0.28\mu \mathrm{m}$; (3) $R=3.64\mu \mathrm{m}$, $R_{d,\max }=0.29\mu \mathrm{m}$; (4) $R=4.73\mu \mathrm{m}$, $R_{d,\max }=0.31\mu \mathrm{m}$.

4.6.

Numerical Calculations for the Device with $\rho =$ 0.5 nm

We carried out the model analysis for the following parameter values: $\rho =0.5\mathrm{nm}$; $r_0=5\rho$, $10\rho$, $15\rho$, and $20\rho$; $L_1=n_1\rho$; $L_2=n_2\rho$; $L_3=\rho [(n_1-n_2)/2]$; $R=n_3\rho$, where $n_1$, $n_2$, and $n_3$ are the size multipliers. Note that this time the size multipliers have larger numeric values, so that the total channel lengths are similar: $n_1=\mathrm{20,000}$, $n_2=\mathrm{19,960}$, and $n_3=100$, 200, 300, 400, 500, 600, 700, 700, 800, 900, and 1000. Figure 6 shows the numerical results. Here, all of the parameters $\rho$, $r_0$, $L_1$, $n_1$, $L_2$, $n_2$, $L_3$, $R$, and $n_3$ are the same as defined above.

Fig. 6

Calculated plots of the transmission efficiency $\eta$ on the curvature radius $R$ (see Fig. 4). The values of the variable model parameters are: $\rho =0.5\mathrm{nm}$, (1) $r_0=5\rho$, (2) $r_0=10\rho$, (3) $r_0=15\rho$, and (4) $r_0=20\rho$.

Once more, the EMF transmission efficiency has a maximum for each of the parameter sets, corresponding to the following values: (1) $R=95.7\mathrm{nm}$, $R_{d,\max }=3.0\mathrm{nm}$; (2) $R=125.6\mathrm{nm}$, $R_{d,\max }=5.4\mathrm{nm}$; (3) $R=155.5\mathrm{nm}$, $R_{d,\max }=7.8\mathrm{nm}$; and (4) $R=213.5\mathrm{nm}$, $R_{d,\max }=10.2\mathrm{nm}$. As it follows from the results, the optimal radius of the input and the output sections ($R_{d,\max }$) of the optical channels increases in comparison to the radius of the cylindrical section, while the difference $R_{d,\max }-r_0$ decreases, all with the increasing $r_0$. In effect, we obtain $(R_{d,\max }/r_0)-1\ll 1$, with the overall geometry very similar to that of a cylindrical tube. However, the most notable result is the high efficiency of the EMF transmission by such small diameter channels, as obtained in the calculations.

4.7.

Theoretical Description of the Absorption and Emission Spectra

Here, we shall calculate the shape of the spectral band in the absorption/resonance emission spectrum. The shape of the spectral band is described by (see Appendix F)

Eq. (9)

$$P_{\mathrm{GE}}(t)\approx D\times {\left[\frac{\gamma }{{({\omega }_{\mathrm{exc},g}-\omega )}^2+{\gamma }^2}\right]}^2,$$

where

Eq. (10)

$$D={|\frac{E_{0}e}{\hslash }⟨{\psi }_g(r,\varphi ,z)|{\overrightarrow{r}}_e|{\psi }_{\mathrm{exc}}(r,\varphi ,z)⟩|}^2.$$

Taking into account the algorithm developed above for the calculation of the transition matrix element $D$, and using also

Eq. (11)

$$\gamma \approx \frac{1}{{\tau }_{\mathrm{emiss}}}\propto {|⟨{\psi }_g(r,\varphi ,z)|{\overrightarrow{r}}_e|{\psi }_{\mathrm{exc}}(r,\varphi ,z)⟩|}^2,$$

we calculated the absorption band shape for the system of Fig. 4 using a home-made FORTRAN code with the following geometric parameters: $r_0=25\mathrm{nm}$, $\rho =4\mathrm{nm}$, $L_1=10\mu \mathrm{m}$, $L_2=9\mu \mathrm{m}$, and $R=0.5\mu \mathrm{m}$. The resulting spectrum is shown in Fig. 7.

Fig. 7

Calculated test absorption spectrum for device shown in Fig. 4, where the device parameters are: $r_0=25\mathrm{nm}$, $\rho =4\mathrm{nm}$, $L_1=10\mu \mathrm{m}$, $L_2=9\mu \mathrm{m}$, $R=0.5\mu \mathrm{m}$.

We see that the band maximum in Fig. 7 is located at $\mathrm{24,275}{\mathrm{cm}}^{-1}$, having the width of $6786{\mathrm{cm}}^{-1}$. A similar approach will be used in the follow-up papers to analyze the polarization selectivity¹⁰⁵ and the spectral selectivity of the IFs.

5.

Discussion

We analyzed in detail the role of the QC in the transmission of the electromagnetic energy by axisymmetric nanochannels with conductive walls, the channel diameter much smaller than the wavelength of the electromagnetic radiation. Note that the transmission of the EMF by such structures may be described by the well-developed and frequently applied plasmon–polariton theory.^1–37 The presently developed approach amounts to an extension of the plasmon–polariton theory in the QC limit.^31,73 Indeed, we describe the quantum states in a waveguide built of nanostructured conductive materials. We assume that the interaction of the EMF with the waveguide produces a transition into the quantum excited state that is delocalized over the entire device, in agreement with the standard plasmon–polariton theory approach, where the energy of the EMF is transported within the waveguide in the form of excited-state electrons.^{1–8,42–72} However, there are some terminological differences; here, the excited state may be described as an exciton, whereas the plasmon–polariton theory describes such excited states as plasmons or polaritons. We presume that these are different designations for the same physical phenomena, resulting from different qualitative descriptions of the behavior of the system. The difference between the quasiclassical and QC approaches was discussed above, under “the theoretical model.” Both approaches describe the efficiency of the EMF energy transmission from the input section to the output section. The energy loss in both cases is determined by the energy relaxation processes. In the first model, these are described as friction exerted by the lattice on the electrons, whereas the second model describes radiationless relaxation due to exciton–phonon interaction. In both cases, the relaxation may be characterized by the phenomenological parameter $\gamma$, identical for the two models in the zero-order approximation. Therefore, both models should produce identical energy transmission efficiencies in the zero-order approximation, with any deviations appearing in higher orders only. We did not address this issue at this time.

Presently, we analyzed a waveguide device with the same geometry of the input and the output sections. In principle, the energy transfer efficiency may depend on the geometry of each of these sections, thus allowing extra opportunities to optimize it. The optimum dimensions should also depend on the wavelength of the EMF, with a possibility of separate optimization for each of the color-vision photoreceptors. We shall explore these issues in a follow-up publication.

Our main goal was to develop an understanding of the mechanisms of optical transparency of the MCs and other specialized transparent biological cells; next, we shall discuss the possible chemical composition of the waveguide structures. Note that protein molecules are universal structural building blocks in animal cells. Apparently, appropriate proteins may form the conductive walls of the nanostructured waveguides, as proteins have high electronic conductivity even in the dry state, with values approaching those of typical molecular conductors. These high values of electrical conductivity result from the existence of conjugated bonds in the structure of proteins, enabling high electronic mobility.¹⁰² Note that the experimental data and a quantum-mechanical model for the light transmission by CNTs, enabled by their conjugated $\pi$-electron system, were reported by several authors.^106,107 Their results correlate quite well with the conclusions derived from our present models. Still, the exact composition and structure of the specialized IFs in the MCs remain unknown, setting goals for future studies.

Thus, as we already noted earlier,¹⁰⁸ we should never overlook the direct transmission mechanism of luminous energy by biological nanostructures, including cellular IFs, based on their quantum nano-optical properties, along with the classical mechanisms reported earlier.^83,109,110 Recently, Labin et al.¹¹¹ considered light transmission by MCs using a classical model of a light guide that uses the difference between the refraction index of these cells and that of the surrounding medium to confine the light and guide it toward the light-sensitive cones. Interestingly, they also arrive at the conclusion that the light-transmission coefficient of the MC depends on the wavelength, with red and green light guided to the cones more efficiently than blue light. We find this result counter-intuitive, as generally it is more difficult to focus or guide the light with the larger wavelength (red) as compared to that with the shorter wavelength (blue), with a thicker waveguide required for the red light.

All of the considerations presented here refer to conductive nanostructured IF bundles. Unfortunately, the structure and electric conductivity of the MC IFs remain unknown. However, the electric conductivity of different proteins was studied and reported earlier.^101,112 Such electric conductivity of the polypeptide systems ^101,112 was explained by their structure, including a common conjugated $\pi$-system involving the entire length of the polypeptide molecule. We already mentioned that the electric conductivity is needed for our models to be valid. It was also shown¹¹³ that self-assembled monolayers formed by conformationally constrained hexapeptides are electrically conductive and generate photocurrent. Thus, peptide-based self-assembled monolayers efficiently mediate electron transfer and photoinduced electron transfer on gold substrates. These results¹¹² may be described in terms of high electron mobility within the peptide monolayer in its electronic ground and excited states. We, therefore, infer that the IF polypeptides should be electrically conductive, enabling the presently proposed quantum mechanism as a complement and/or alternative to the classic mechanisms of the luminous energy transport within the retina.

6.

Conclusions

In this study, we report that MCs have long channels (waveguide structures), around 10 nm in diameter, spanning the larger part of the cell process, from the apex of the basal endfoot to the photoreceptor cells. Following the idea that such channels act as waveguides, we developed a QC model of the light transmission by the waveguides much thinner than the wavelength of the light quanta. We used our model for the qualitative analysis of the light transmission by such waveguides, with the calculations showing that such devices may transmit light with very high efficiencies from their input section to the output section, already possible in a device with a much simplified geometry. This mechanism was extended to the light transmission by the waveguides (specialized IFs) in the MCs, concluding that such waveguides are most probably built of proteins. Proteins have sufficient electric conductivity for the mechanism to be operational due to the presence of extended conjugated multiple bonds in their structure. The presently reported studies have a direct impact in the medicine of human vision and on the development of nanomedical vision technology.