Model of polarization selectivity of the intermediate filament optical channels

PhotonicsandNanostructures–FundamentalsandApplications16(2015)24–33

Availableonlineatwww.sciencedirect.com

ScienceDirect

Model

of

polarization

selectivity

of

the

intermediate

filament

optical

channels

Igor

Khmelinskii

,

Lidia

Zueva

,

Michail

Inyushin

,

Vladimir

Makarov

a_{UniversidadedoAlgarve,FCT,DQBandCIQA,Faro,8005-139,Portugal}

b_{SechenovInstituteofEvolutionaryPhysiologyandBiochemistry,RussianAcademyofSciences,St.Petersburg,Russia}

c_{UniversidadCentraldelCaribe,SchoolofMedicine,Bayamón,00960-6032PR,USA}

d_{UniversityofPuertoRico,RioPiedrasCampus,POBox23343,SanJuan,00931-3343PR,USA}

Received25May2015;receivedinrevisedform28July2015;accepted5August2015 Availableonline14August2015

Abstract

RecentlywehaveanalyzedlighttransmissionandspectralselectivitybyopticalchannelsinMüllercellsandothertransparent cells,proposingamodeloftheirstructure,formedbyspecializedintermediatefilaments[1,2].Ourmodelrepresentseachoptical channelbyanaxiallysymmetrictubewithconductive walls.Presently,weanalyze theplanarpolarizationselectivityin long nanostructures,usingthepreviouslydevelopedapproachextendedtostructuresoftheellipticcross-section.Wefindthattheoutput lightpolarizationdegreedependsonthea/bratio,withaandbthesemiaxesoftheellipse.ExperimentaltestsusedaCrnano-strip devicetoevaluatethetransmittedlightpolarization.Themodeladaptedtotheexperimentalgeometryprovidedanaccuratefitof theexperimentalresults.

PublishedbyElsevierB.V.

Contents

1. Introduction...25

2. Thetheoreticalmodelandmethods...26

2.1. Thequantumconfinementmodel...26

2.2. Theperturbationtheoryapproach ... 27

2.3. Thenumericalanalysis...28

3. Experimentsonathin-filmwaveguide ... 28

3.1. Experimentalmethodsandtechniques...28

3.2. Experimentalresultsandanalysis ... 29

4. Discussion...30

5. Conclusions...31

Acknowledgments...31

AppendixA...31

References...32

∗_{Correspondingauthor.Tel.:+17875292010;fax:+17877657717.}

E-mailaddresses:[email protected],[email protected](V.Makarov).

http://dx.doi.org/10.1016/j.photonics.2015.08.001 1569-4410/PublishedbyElsevierB.V.

1. Introduction

The celestial distribution of the angle of the sky-light polarization, being the same under all possible skyconditions(clear, fog,clouds,etc), isusedfor the orientationbypolarization-sensitiveanimals,including manyvertebrates[3].Thereisconsiderablebehavioral andphysiologicalevidenceforpolarization-based nav-igationinvertebrates,includingfish,reptilesandbirds, butnotinmammals,whileit isknownthatmammals, includinghumans,canstillperceivethepolarizationof light tosomeextent[4,3].Thesearchfor thephysical mechanismsofthepolarizationsensitivityhastakentwo differentpaths:someresearchersarelookingforoptical polarizingfilters infront of the photoreceptors; while otherssuggestthatphotoreceptorcellsthemselvesmay havedifferentintrinsicsensitivitytodifferentlypolarized light[3].

Photoreceptors would be intrinsically sensitive to polarization,iftheyhadsomedichroicabsorbanceatthe molecularlevel. Indeed, it wasshown for the inverte-braterabdomericphotoreceptorsthattheirchromophore ispreferentiallyalignedalongtheaxisofthe microvil-lus andimmobilized, allowing for robust polarization sensitivity [3]. On the other hand, the first spectro-scopic measurements found rhodopsin dipoles in the vertebratephotoreceptorsfreetorotatewithinthe pho-toreceptormembranewithoutanypreferredorientation to the incident light, thus rejecting the possibility for theirpolarizationsensitivity[5,6].Theseearlierresults have been criticized later. Namely, a preferred ori-entation of rhodopsin was discovered in some fish species (anchovy conephotoreceptor outersegments), where it is contained in transversely-oriented lamel-larmembranes[7–9].Additionally,rhodopsinmobility was significantly restricted in some species [10], explained by its possible oligomerisation [11]. These data suggest that the photoreceptors may be intrinsi-callysensitivetopolarization,atleastinsomevertebrate species.

Ontheotherhand,thesameanchovyandsomeother fishhavespecialized guanine crystalssurrounding the outersegmentsthatmayworkaspolarizedlight reflec-tors [12,13]. The suggested polarization sensitivity in birds is due to specialized oil droplets present in the opticpathofonlyoneoftheconephotoreceptorsinthe specialized cone pair [14–17].All of theseadditional elementspresent inthe opticalpathmayworkas spe-cializedfilters.Herewesuggestanovelpossibilitythat the specialized opticalchannels inside the transparent cellsmayworkasadditionalpolarizationfiltersinfront ofthephotoreceptors.

Wehaveearlierproposedthatbundlesof nanoscale filaments(witheachfilament10–12nmindiameter)in the transparent cells of the optical tract may directly participateinthetransmissionoflightenergy,and devel-oped aphysical model of the light energy transfer in longcarbon-basedconductivenanostructures[1,2].We developed a quantum mechanism (QM) of the elec-tromagnetic field(EMF) transmissionby awaveguide [1,2],acapillarywithconductivewalls,withthe diam-etersignificantlysmallerthantheEMFwavelength.We suggestedthat the intermediatefilamentsfoundinthe transparentcellsmaybetheidealmatchtothe nanotube-basedmodel,becauseoftheirdiameter(10–12nm)and because their axial structure resembles that of nano-tubes,withalow-densitycoreandhigh-densitywalls, accordingtotheX-raydiffractiondata[18].Thus,our modelsprovidethetheoreticalbackgroundforthe exper-imentalresultsobtainedearlierbydifferentauthorsthat implicatethespecialized intermediatefilamentsinthe celltransparency [1,2].Note that genetic deletions or mutations,orchemicalmodificationsofthese interme-diatefilamentsmayleadtotransparencyloss [19–23], underliningthe importance of their structure for their light-guidingproperties.

Interestingly,theretinalMüllercells(MC)andtheir intermediatefilamentsshouldbeincludedintothe opti-cal path before photoreceptors invertebrates, as they werefoundtotransferlighttotheconesintheirinverted retina [24]. We found [1,2] that the QM reproduces the high efficiency of the EMF transmission by the nanoscaletubes,providedtheirshapeisoptimized.We alsoproposed thatsuch mechanism mayexplainlight transparencyoftheMC,withouttheexactknowledgeof thewaveguidechemicalstructure[1,2].Generically,we model each of the waveguides/channels inthe bundle byan axisymmetric tube withconductive walls. Note that extended π-conjugated carbon systems are elec-tricsuperconductors,typicalexamplesbeingsingle-wall carbonnanotubesandgraphene[25–32].

The optical selectivity in different nanoscale sys-temshasbeenexploredquiteintensivelybefore[32–44]. RecentlyweappliedanapproachproposedbyMakarov etal.[1]toexplorethe spectralselectivity in axisym-metricnanoscalewaveguides[2].Wereportedthatthe transmissionspectra of the model waveguideshave a well-definedspectralband,itswidthdependentonthe waveguidediameterandwallthickness.Thus,we con-cludedthattheMCwaveguidescomposedofbundlesof specializedintermediatefilamentsmaytransmitvisible lightwithinadeterminedspectralrange,dependenton thegeometricalparametersoftheindividualfilaments. Presently,weextendthemodelingapproachesdeveloped

earlier[1,2]toexploretheplanarpolarization selectiv-ityintheopticalwaveguides(includingthespecialized intermediate filaments in Müller cells, etc), allowing themtofunctionintheopticaltractaspolarizationfilters infront of the photoreceptors. Generally,polarization selectivityarises inless-than-axially-symmetric wave-guides.Inparticular,hereweanalyzelightpolarization inthewaveguideswithellipticalcross-sectionin func-tionofthea/bratio,aandbbeingthesemiaxesofthe ellipse,by using boththe perturbation theory andthe numericalanalysisofthecompletemodel.

2. Thetheoreticalmodelandmethods

Recently,wepresentedthebasictheoretical descrip-tion of the light (Electromagnetic field, EMF) transmissionbynanochannels[1].Sincethediameterof theintermediatefilamentsisabout10–20nm,theEMF cannotbe transmitted by such channels in the classi-cal description of the Maxwell theory. Therefore, we proposedthat theEMFtransmissionbysuchchannels shouldbedescribedbytheQuantumMechanism(QM) basedontheQuantumConfinement(QC)ofthe excita-tionsinlongstructuresofsmalltransversesize[1].Inthe firstapproximation,wedescribedthesechannelsaslong hollowcylinderswithconductivewalls.Awell-known exampleofsuchmolecularsystemsthatwealready con-sidered is given by the single-wall carbon nanotubes (SWCNT),whichareelectricallyconductiveduetoan extendedconjugated␲-electronicsystemoftheircarbon backbone[1].Thus,theexternalEMFinteractswiththe intermediatefilaments,inducingformationoftheexcited electronicstates,delocalizedovertheentirelengthofthe filament.Duetothegeometry oftheintermediate fila-ments,theexcitedstatesmainlyemitinthetwoendzones ofthefilament,withthephotonsimmediatelyreabsorbed by the photosensor cell [1]. Alternatively, the excita-tionmaybetransferreddirectlyfromthefilamenttothe sensorcellchromophores,withoutgenerationofa pho-ton.Presently,weusethemainideasofthepreviously developedtheoreticdescription[1].

In the present study, we extend the earlier devel-opedmodelingapproachesdescribinglighttransmission byintermediatefilaments,present,forexample,inthe Müllercells.Thissectionanalyzesthepolarization selec-tivity of the optical channels in the Müller cells and other biological systems using the earlier developed models andmethods, including amodel for the EMF transmission by optical channels [1,2]. Note that the channeldiameterandwallthicknessaremuchsmaller than the wavelength of light, withthe wall thickness andthediameteroftheopticalchannelbeingthemodel

parameters.Thismodelemploysquantumconfinement (QC) to describe the EMF transmission by the opti-cal channels [1,2].Presently, we use the same model to analyze the polarization selectivity of the optical channelsinfunctionoftheirgeometricparameters.We analyze a waveguide with a coaxial elliptical cross-sectionusingboththeperturbationtheoryandthedirect numericalanalysis.Notethatbothofthethesetheoretical approacheswereinitiallyproposedearlier[1,2]. 2.1. Thequantumconfinementmodel

Presently, weextendtheearlierdeveloped theoreti-calmodels[1,2]tothewaveguideswithanelliptictube symmetry,analyzingthelighttransmissionbyacoaxial waveguidewithanellipticcross-section(seeFig.A1.1), wheretheellipseisdescribedby

x2 a2 ext + y2 b2 ext = x2 a2 int + y2 b2 int =1 (1)

Here,aextandbext arethe semiaxesof theexternal ellipse, aint and bint are the semiaxes of the internal ellipse, with bext

aext =

bint

aint, L isthe length of the device.

The Laplaceoperatorintheellipticcylinderreference system(ξ,η,z;ECRS)mayberepresentedasshownin the Appendix,while the Schrödinger equation for the electronintheECRSisgivenby:

− 2 2meΔψ (ξ,η,z) =Eψ (ξ,η,z) (2) or (a) −  2 2me 1 f2_Sh2_{(ξ) +Sin}2_(η)  _∂2 ∂ξ2 + ∂2 ∂η2  ψ (ξ,η) =Eξ,ηψ (ξ,η) (b) −  2 2me ∂2_{ψ (z)} ∂z2 =Ezψ (z) (3)

where, x=aCh (ξ) Cos (η) x=aSh (ξ) Sin (η) z=z (4) f =√a2_−b2 a≥b (5)

above,ξisapositiverealvalue,η∈[0,2π],and ψ (ξ,η,z) =ψ (ξ,η) ψ (z)

E=E␰,η+Ez

(6) Weanalyzedtheproblemforthefollowingboundary conditions: U (ξ,η) = ⎧ ⎪ ⎨ ⎪ ⎩

∞; on the external surface ∞; on the internal surface 0; between the two surfaces U (z) =

0; 0<z<L ∞; z≤0; z≥L

(7)

Thesolutionofthez-dependentpartisgivenby: En,z= π 2_2_n2 2meL2 n=1,2,... (8) ψn(z) =  2 Lsin πn Lz (9) Thus, we obtain the completewavefunction in the form: ψ (ξ,η,z) =ψ (ξ,η) ψ (z) =  2 Lsin πn Lz ψ (ξ,η) (10)

The ψ (ξ,η) functionsmaybe determined approxi-matelyforsmalleccentricities,i.e.,fora/b−1«1,using theperturbationtheory.Alternatively,theproblemmay be solved numerically for any a/b value. Next, we describebothapproaches.

2.2. Theperturbationtheoryapproach

TheLaplaceoperatorintheellipticalsystemmaybe presentedasasumoftwoterms[51](seetheAppendix):

=CRS+Pert (11) where, (a) CRS= ∂ 2 ∂ρ2 + 1 ρ2 ∂2 ∂ϕ2 + ∂2 ∂z2 (b) Pert=

1−1−2cos4(η)ε2+ε4cos2(η) sin2(η)

1−ε2_+ε4_cos2_{(η) sin}2_(η) 1 ρ ∂ ∂ρ (12)

Notethat the first term coincides withthe Laplace operatorinthecylindricalreferencesystem(CRS),while the second term adds the contributions uniqueto the ellipticcylinderreferencesystem(ECRS).Weusedboth termsintheperturbation-theoryanalysis.Thefollowing relationsrepresenttheperturbationoperatorintermsof theCRSvariables: ρ=a2_−b2_Ch2_{(ξ) cos}2_{(η) +Sh}2_{(ξ) sin}2_(η) a≥b (13) aint≤a≤aext bint≤b≤bext ϕ=ArctgTh (ξ) Tg (η) (14)

Thus,wemayobtaintheexpressionsforξ,ηofthe ECRSintermsofρ,ϕoftheCRS.

Thesolutionsof the Schrödingerequation withthe Laplaceoperator(12a)havebeenobtainedearlier,and wereusedhere[1].Usingthezero-ordersetofthe elec-tronicstatesintheaxisymmetricsystem,theirenergies, andthe previously proposedmethods tocalculate the lighttransmissionefficiency[1],wecalculated numer-icallythepolarizationfactorofthetransmittedlightin functionofa/b.Wemadethesecalculationsinthe first-orderperturbationtheory,withtheperturbedstatesand theperturbationmatrixelementsgivenby:

|nΛmPert= |nΛm +  n _{Λ m} VnΛm,n _Λ m E(0) n m=E(0)n Λ m _{n Λ m}  VnΛm,n _Λ m =−  2 2me  nΛm |ΔPert| n Λ m  ˆ V =− 2 2me Pert (15)

Thesecalculationswereperformedusingtheearlier developedmethods, withappropriate modifications to the respective homemadeFORTRAN code [1,2].The simulations used two different parameter sets of the optical channel: 3.0 or 4.0nm wall thickness, 5nm internalradius,10.0␮mchannellength,and34,700or

1 a/b ratio 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Polar iza ti on D egree 0.00 0.02 0.04 0.06 0.08 0.10 2

Fig.1.Perturbation-theorycalculationsfortheelliptic-tubemodel:(1) 3.0nmwallthickness,5nminternalradius,10.0␮mchannellength, 34,700cm−1 energyoftheincidentEMFradiation;(2)4.0nmwall thickness,5nminternalradius,10.0␮mchannellength,19,500cm−1 energyoftheincidentEMFradiation.

19,500cm−1energyofthetransmittedlight.Theresults werepresentedintermsoftherelationship:

P =Wa−Wb Wa+Wb

(16) where,Waisthecalculatedoutputenergywiththe polar-izationparalleltothelargeraxisandWbisthecalculated outputenergywiththepolarizationperpendiculartothe largeraxis,bothobtainedfortheinputlightwithP=0. Thea/bratiovariedfrom1.0to1.5,withFig.1showing theresultsobtainedforthetwomodelsystems.

Fig.1 shows that thepolarization degree increases from0toabout0.08,beinglargerforthehigher-energy radiation, although the calculated difference between thetwoparametersetsis<0.01.Weshalldiscussthese resultsindetailbelow.

2.3. Thenumericalanalysis

TheEq.(3a)wassolvednumericallyusingthe finite-differencemethodsasdescribedearlier[1].Fig.2shows the polarization degree calculated for the same two parametersets,infunctionofthea/bratio.

Fig.2showsthatthepolarizationdegreeachieves val-uesclosetounityforbothof theparametersets atthe a/bratio of 5.5. Wecompared the results obtained in thetwoapproachesfora/b<1.5.Thevaluescalculated bytheperturbationtheoryaretypicallyslightlysmaller thanthoseobtainedbythenumericalanalysis,withthe maximumrelativedifferencesbelow1%.Therefore,the perturbationtheoryadequatelydescribesthedegreeof polarizationata/b<1.5. 1 a/b ratio 1 2 3 4 5 6 Polar ization D egree 0.0 0.2 0.4 0.6 0.8 1.0 2

Fig.2.Numericalanalysisoftheelliptic-tubemodel:(1)3.0nmwall thickness,5nminternalradius,10.0␮mchannellength,34,700cm−1 energyoftheincidentEMFradiation;(2)4.0nmwallthickness,5nm internalradius,10.0␮mchannellength,19500cm−1 energyofthe incidentEMFradiation.

3. Experimentsonathin-filmwaveguide

Wediscussed above the theoreticalmodels analyz-ingthepolarizationselectivityoftheopticalwaveguides. Themodelwasprimarilyaddressingtheintermediate fil-aments,existinginthestructureofthelight-transmitting cells,suchastheMüllercells.Presently,weareunable toperformexperimentsonbiologicalobjects;therefore, wemeasuredtheangleofpolarizationexperimentallyon athin-filmmodelwaveguide,aCrmetaltrackdeposited on AlNsubstratewith0.1×1.0cm2 _{sizeandvariable} thickness, in the nanometer range. Two fiber-optical light-guides wereconnectedtothe trackatbothends, normal tothe tracksurface.The effective diameter of the light-guideswas1.0mm,equaltothetrackwidth. ThetransmissionspectraoftheCrtrackswererecorded withapolarizerinthebeampath.

3.1. Experimentalmethodsandtechniques

The experimental setup described earlier [2] was partiallymodifiedforthepresentmeasurements. Com-mercialAlNsubstrates12.5mmindiameterand1.5mm thick(ValleyDesignCorp.)wereusedtodeposit rectan-gularCrtracks0.1×1.0cm2insizewiththepreset thick-nessin thenm range.CommercialCr targets (Sigma-Aldrich)were usedtoproduce nano-tracksona com-mercialsputtering/thermo-evaporationBenchtopTurbo depositionsystem(DentonVacuum).Acopperfoilmask wasusedtodeposittheCrtracksoftherequired geome-try. The trackthicknesswascontrolledby XRD,with the XPert MRD system (PANalytic) calibrated using standardnanofilmsofthesamematerial.Theestimated

3 Energy (cm-1, 103) 12 14 16 18 20 22 24 26 28 30 Tr an sm issi on E fficien cy 0.0 0.2 0.4 0.6 0.8 1.0 2 1

Fig.3. Transmissionspectraofthe0.1×1.0cm2_{×6.4nmCrsample:}

(1)9mmdistancebetweentheinputandoutputlightguides;(2)7mm distance;(3)5mmdistance.

absoluteuncertaintyoftheCrtrackthicknesswas7%; therelativeuncertaintiesweremuchsmaller,determined bytheshutteropeningtimesofthedepositionsystem.

ThetransmissionspectrawererecordedonaHitachi U-3900H UV–visibleSpectrophotometer.Thespectral peakmaximaandwidthswerelocatedusingthe Peak-Fitsoftware(Sigmaplot). TheAlNsubstrateswiththe depositedCrtracksweremountedintothesampleholder. Two multimode fiber-optic light-guides with 1.0mm diameterwerealsomountedonthesamesampleholder, normaltothetrack.Thelight-guidesweretransparentin therangeof800–445nm.

Thus,oneendofeachofthelight-guideswasindirect contacttotheCrtrack,whiletheotheronewasmounted into a home-madefiber-optic adaptor, connecting the experimentalassembly tothe spectrophotometer,with alinearpolarizerinstalledintotheinputbeampath.The blankscanmadewiththetwosampleendsofthe light-guidesconnecteddirectlytoeachotherwassubtracted fromtheexperimentalspectra.

3.2. Experimentalresultsandanalysis

We tested two different Cr tracks: (a) 0.1×1.0cm2×6.4nm;and(b)0.1×1.0cm2×8.3nm. Fig.3showsthespectraofthesample(a)without polar-izer atdifferentdistancesbetweentheendsof thetwo light-guidesconnectedtothesample.

We saw that the transmitted light intensity is only weakly dependent on the distance between the ends. We therefore concluded that the entire surface of the Crtrackemitslightalmosthomogeneouslyinfunction ofthedistancefromtheinputlight-guide.

1 Energy (cm-1, 103) 10 12 14 16 18 20 22 24 26 28 30 Tr an sm issi on E fficien cy 0.0 0.2 0.4 0.6 0.8 1.0 2 3 4 5 6 7 a b

Fig.4.Transmissionspectraofthe(a)0.1×1.0cm2_{×6.4nmand(b)}

0.1×1.0cm2_{×8.3nmCrtracksontheAlNsubstraterecordedwith}

thepolarizeratdifferentangles:(1)0o_;(2)15o_;(3)30o_;(4)45o_;(5)

60o_;(6)75o_;(7)90o_.

Fig.4showsthetransmissionspectraofthesamples (a)and(b),recordedwiththepolarizerinthebeampath, atdifferentangles,withthezeroangle,correspondingto theelectricfieldvectorEparalleltothetrackaxis.

ComparingthespectraoftheFig.4,weconcludethat the polarization efficienciesof the Cr tracksare quite high,withthethinnersample(a)havingaslightlyhigher polarizationefficiency.

Thus,theCrsampletracksareefficientlight polariz-ers.Recallingthatthetransmissionofalinearpolarizer variesascos2(α),αbeingtheanglebetweentheelectric oflinearly-polarizedlightandthepolarizationdirection, weplottedtheintegratedtransmissionspectrainfunction ofcos2(α),withtheresultsshowninFig.5.

The data of Fig. 5 were fitted by linear functions (y=ax+b) withtheparameters:a=0.886,b=0.117;

b Cos2(α) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Norm alize d S pect ru m In tegral 0.0 0.2 0.4 0.6 0.8 1.0 1.2 a

Fig. 5.The integrated transmission spectra vs. the theoretical cos2(α) factor, describing the intensity in function of the polar-izerangle,forthetwosamples:(a)0.1×1.0cm2×6.4nmand(b) 0.1×1.0cm2_×8.3nm.

b Polarizer Angle 0 20 40 60 80 100 Norm alized Integra ted S p ectrum 0.0 0.2 0.4 0.6 0.8 1.0 1.2 a

Fig. 6.The integrated transmission spectra vs. the polarizer angle for both samples (a) 0.1×1.0 cm2×6.4nm and (b) 0.1×1.0cm2×8.3nm.Theexperimentalpointsarecomparedtothe modelcalculations(lines).

and a=0.817, b=0.195, respectively, and very good qualityfits(R2>0.9997),showingthattheresultsfollow thecos2(α)behaviorexpectedforthelinearlypolarized light.

Now,we modifiedthe earlierdevelopedtheoretical model [1,2] in order to accommodate waveguides of rectangularcross-section,usingthesamephysicalideas, withthe bandwidthinthe absorptionspectrum deter-minedbytheinteractionsbetweenthediscretequantum statesandthequasi-continuumconductionzone,andby thedensityofstatesinthelatter.Thehome-made FOR-TRANcode[1,2]wassuitablymodifiedtoaccommodate rectangularwaveguides.Fig.6showsthatmodel calcu-lationsreproducetheexperimentalresultsquitewell.

Weconcludethis section bystating that nanosized waveguideshaveintrinsicpolarization selectivity, pro-vided they are not cylindrically symmetrical, as seen bothinmodelcalculationsandintheexperimentalstudy ofthe rectangularCrtracks.Webelievethatthe same mechanismofthe polarizationselectivity mayoperate intheopticalchannelsoflivecellsbuildofbundlesof theellipticintermediatefilaments.

4. Discussion

We investigated the polarization selectivity of the nanosized optical waveguides theorically, and tested it experimentally on amodel stripnanostructure. The polarization selectivity for the transmitted light arises whenwe considerawaveguidewithan elliptic cross-section,being absentfor awaveguide of acylindrical cross-section.Notethatthetransversaldimensionsofthe waveguidearemuchsmallerthanthewavelengthofthe light,transmittedintheformofanexcitedstate(exciton) fromoneendofthewaveguidetotheother.Presentlywe

consideredamodel basedonquantumconfinementin nanostructuredconductive materials[1,2].Thismodel was used toexplore theEMF energy transmissionby cylindrical channels and more complex systems with axialsymmetry;thesameapproachwaspresentlyusedto analyzethespectralselectivityoflongoptical nanochan-nels. Note that the light transmission by nanowires, nanotubesandsimilarsystemshas beenpartially ana-lyzedusing theplasmon–polarontheory [45–47],with the transmissionspectrumalso dependenton thewire radius.Thetransmissionspectraofthenanowireswere significantly wider[45–47] thanthoseobtained previ-ously[1,2]forthethin-walledtubes,whilecomparable to thoseobtained for the thick-walled tubes[2].This result is dependent on the γ; in the classical case of the plasmon–polaron theory, this parameter describes the friction coefficient describing the electron motion in such systems; in the QM description, this parame-terdescribestherelaxationrateoftheexcitedelectronic state.Theparameterγincreaseswiththewallthickness ofthenanotubes[1,2],resulting,asalreadymentioned, inthespectralwidthsoftheplasmonexcitationsbeing comparabletothespectraobtainedforthethick-walled tubes,ascalculatedusingtheQMmechanism[2].

The polarization selectivity of the optical channels made of filaments was analyzed in the present study usingthequantumconfinementapproachfornanotubes with elliptic cross-section. We found that such chan-nelsproducepolarizationdegreesinexcessof95%for an ellipsewitha/b = 5.5, functioningas anefficient polarizer.

Ourtheoreticalmodelwasprimarilyaddressing bio-logicalsystemstransparenttolight,includingbundlesof intermediatefilamentsinMüllercells.Weproposed[1,2] thatintermediatefilamentsarebuiltofelectrically con-ductiveproteins[48,49].Indeed,thestructureofprotein molecules was found to facilitate long-range electron transferinsolutionsandevenindrysolidstate, trans-mittingelectriccurrentsmuchhigherthanthoserecorded forsaturatedorganicsubstanceswithcomparablelayer thickness. In fact, proteins demonstrate the electrical conductivity comparabletothat ofnanowiresmade of carbonnanotubes [48].Taking intoaccount the infor-mationontheelectricconductivityofproteinsandthe experimentaldataobtainedinthepresentstudy,we con-clude thattheintermediatefilamentswithnon-circular cross-section should have some polarization selectiv-ity as regards light/excitation transmission. Thus, the currentlyproposedquantumconfinementmodelshould appropriatelydescribetheintermediatefilaments, wire-like biologicalstructures built of proteins andpresent inmanytypesofcells.Webelievethattheintermediate

filamentsfoundintransparentcellsshouldbedescribed quite well by the nanotube-basedmodel, due to their small diameter (usually, 10–12nm) andbecause their cross-section closely resembles that of athick-walled nanotube,withthe lower-densitycoreandthe higher-density walls, according tothe X-ray diffraction data [18].Notethat SWCNTsare indeedanefficientEMF polarizer,infullagreementwiththepresentlydiscussed theoreticideas[53].

Whilethe sensitivityof the vertebrateeyeis well-documented behaviorally, the physical basis of its polarizationsensitivityisstillunderinvestigation.There isapossibilityofthe opticalpolarizingfiltersexisting in front of the photoreceptors, or else the photore-ceptors themselves may havean intrinsic polarization sensitivity [3]. Currently, we have reasons to believe that naturemayuse bothpossibilitiesindifferent ver-tebrate species. Here we suggest a novel mechanism of the polarization-dependent filtering by specialized nanofilaments existinginthe retina, namelythe inter-mediate filamentsin the Müller cells, which transmit lighttotheconephotoreceptorsinthevertebrateretina [24,50].Themechanismdiscussedinthepresentstudy is radically different from those proposed previously in the literature; however, it is a definite possibility. We believethat this mechanismmay also be atwork inothertransparentbiologicalsystemswithspecialized filaments.

5. Conclusions

WereportthatthemodeldeviceshowninFig.A1.1 hasthepolarizationselectivityasregardsthe transmis-sionoflight.Thepolarizationdegreeofthetransmitted light strongly depends on the a/b ratio, transfor-ming strongly ellipticnanofibers into efficientoptical polarization filters. We believe that similar structures existinnature,includingintermediatefiberbundlesin Müller cells, conducting light from the internal sur-face of the retina to the sensory photoreceptor cells (cones).

Presently,wediscussedthepolarizationselectivityof thespecializedintermediatefilamentsintheMüllercells. Asweproposed,apparentlyallofthebiologicalobjects thataretransparenttovisiblelighthavebundlesof inter-mediate filaments that may directly transmitthe light energythroughthecellavoidinglightabsorptionby var-iouschromophorespresentinothercellularstructures. Weinferthatsomeoftheselight-transmitting interme-diatefilamentsmaybepolarization-selective,provided they have an elliptic cross-section. To illustrate the developedtheoreticalmodels,wetestedthepolarization

selectivity of nanometer-thick macroscopically-sized rectangularCrstrips.Themodelsmodifiedto accommo-date waveguidesof rectangular cross-section describe theexperimentaldatawithgoodaccuracy.Webelieve thattheproposedmechanismofthepolarization selectiv-itymaybeatworkintransparentlivecellsthattransmit lightoverbundlesofintermediatefilaments.

Acknowledgments

V. M. is grateful for the NASA EPSCoR grant PR NASA EPSCoR (NASA Cooperative Agreement NNX13AB22A), and M. I. is grateful for the NIH grantG12MD007583thatpartlysupportedthepresent research.

AppendixA.

Presently,weanalyzedthelighttransmissionbya co-axialcylindrical systemwiththe ellipticcross-section (Fig.A1.1).Thegeometryisdescribedbytheequations:

x2 a2 ext + y2 b2 ext = x2 a2 int + y2 b2 int =1 (A1)

Here,aext,bextarethesemiaxesoftheexternal,and aint,bintoftheinternalellipse(Fig.A1.1),b_aext_ext =b_aint_int,L isthelengthofthedevice.

TheLaplaceoperatorintheECRSreferential(ξ,η,z) maybepresentedasfollows:

= 1 f2_Sh2 (ξ) +sin2(η) _∂2 ∂ξ2 + ∂2 ∂η2  + ∂2 ∂z2 (A2)

Thelatterrelationshipmayberewrittenusingthe vari-ablesoftheCRSreferential(ρ,ϕ,z),usingthefollowing relationscouplingthetworeferencesystems:

ρ=f 

Ch2_{(ξ) cos}2_{(η) +Sh}2_{(ξ) sin}2_{(η) (A3)} ϕ=Arctg (Th (ξ) Tg (η))

f =a2_−b2 ε= 1

Thus,theLaplaceoperatorintheECRSexpressedin termsofthederivativescalculatedintheCRSisgiven by[51]: =  1−ε2+ε4cos2(η) sin2(η) 1−ε2_cos2_(η)_1−ε2_sin2 (η)  ∂2 ∂ρ2+ 1 ρ2 ∂2 ∂ϕ2  + ∂2 ∂z2 + 

1−ε2cos2(η)cos2(η) +1−ε21−ε2cos2(η)1−cos2(η) 1−ε2_+ε4_cos2_{(η) sin}2_(η) 1 ρ ∂ ∂ρ =_∂ρ∂2₂+1− 

1−2cos4(η)ε2+ε4cos2(η) sin2(η) 1−ε2_+ε4_cos2_{(η) sin}2_(η) 1 ρ ∂ ∂ρ+ 1 ρ2 ∂2 ∂ϕ2 + ∂2 ∂z2 (A4)

Thus,thelatteroperatorisasumoftwoterms:

=CRS+Pert (A5) where, CRS= ∂ 2 ∂ρ2 + 1 ρ2 ∂2 ∂ϕ2 + ∂2 ∂z2 Pert=

1−1−2cos4_(η)_ε2_+ε4_cos2_{(η) sin}2_(η)

1−ε2_+ε4_cos2_{(η) sin}2_(η)

1

ρ∂ρ∂

(A6)

Theoperator(A2)wasusedinthenumerical analy-sisoftheproblemusingfinitedifferencemethods[52], whilethe operator(A5) was used inthe perturbation-theoryapproach.

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