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Optical phase shifts and diabolic topology in Möbius-type strips Indubala I Satija1 and Radha Balakrishnan2 (1) Department of Physics, George Mason University, Fairfax, VA 22030. (2)The Institute of Mathematical Sciences, Chennai 600 113, India 7 0 0 Wecomputetheopticalphaseshiftsbetweentheleftandtheright-circularlypolarizedlightafter 2 it traverses non-planarcyclic pathsdescribed by the boundary curvesof closed twisted strips. The evolution of the electric field along thecurved path of a light ray is described bythe Fermi-Walker n transport law which is mapped to a Schrödinger equation. The effective quantum Hamiltonian of a thesystemhaseigenvalues equalto0,±κ, whereκisthelocal curvatureofthepath. Theinflexion J pointsofthetwistedstripscorrespondtothevanishingofthecurvatureandmanifest themselvesas 7 thediabolic crossings of thequantumHamiltonian. Forthe Möbius loops, thecritical width where 1 the diabolic geometry resides also corresponds to the characteristic width where the optical phase shift isminimal. Inourdetailed studyofvarioustwisted strips, thisintriguingpropertysingles out ] r theM”obius geometry. e h PACSnumbers: 03.65.Vf,42.79.Ag t o . t a Thegeometricalphenomenonofanholonomyrelatesto is described by, m theinabilityofavariabletoreturntoitsoriginalvalueaf- nt - teracyclicevolution.[1]Anexampleofthis phenomenon x = (1+wcos )cost; d is the change in the direction of polarization of light in 2 n nt a coiled opticalfiber.[2, 3] This leads to an opticalphase y = (1+wcos )sint; o 2 shiftwhichcorrespondstoaphasechangebetweenaright c nt [ and a left-handed circularly polarized wave, when they z = wsin , travelalonganon-planarpath. [4]Theeffectisanoptical 2 1 manifestation of the Aharonov-Bohm phase, according (1) v to which two electron beams develop a phase shift pro- 3 with parameters α w α and 0 t 2π. Thus 9 portional to the magnetic flux they enclose.[5] For the α is the half-widt−h of≤the ≤strip and its≤leng≤th L = 2π. 3 polarized light the analog of magnetic flux is the solid The integer n is the number of half-twists on the strip 1 angle subtended on the sphere of directions kˆ, where kˆ (so that n = 1 refers to the well-known Mo¨bius strip). 0 represents the direction of propagation of light, which We are interested in the geometry and topology of the 7 changes as light passes through a twisted fiber. In pre- 0 boundary curve of the twisted strip. The parametric vious studies involving the propagation of light through / equationfor this spacecurveis foundby setting w= α t a helically wound optical fiber[2, 3, 4], the simple geom- ± a in Eq. (1). For odd n, the strip is a non-orientable m etry of a circular helix gives this solid angle to be equal surface with one boundary curve, so that the range of t to Ω = 2π(1 cos θ), where θ is the pitch angle of the - − is [0,4π]. For even n, the strip is orientable, with two d helix, or the angle between the axis of the helix and the boundary curves with similar geometries, and the range n local axis of the optical fiber. of t is [0,2π] for each of these curves. Topologically, for o In this paper, we compute the optical phase shift be- allwidths, the boundariesofthe odd-nstripswithn>1 c : tween the left and the right-circularly polarized light as are knotted curves (e. g., it is a trefoil knot for n = 3, v it passes through the optical fibers with a more complex a five-pointed star knot for n = 5, etc.), while those of i X geometry,namely,fibersshapedliketheboundary curves the even-n strips are not knotted. The boundary curve r ofclosedtwistedstrips. Althoughwestudyawholeclass ofthe Mo¨biusstripclearlydoesnotfallineitherofthese a of twisted strips, our main focus is on the Mo¨bius strip classes, since it is the only case in which the boundary whose unique geometry has been a source of constant curve of a non-orientable strip is not a knot. Figure 1 fascination. Here we explore the interplay between the shows the boundary curves of the Mo¨bius strip for vari- intrinsicgeometryofthestripsandthegeometricalphys- ous values of α. One of the characteristic feature of the icalphenomenon,namelythegeometricphaseshiftexpe- twisted strips are the inflexion points, distinguished by rienced by light as it passes through the boundary curve a local ”straightening” around that point. Various de- of the strips. Interestingly, the Mo¨bius geometry is sin- tailsregardingsuchpointsforn=twistedstripsandtheir gled out, since the optical phase shift is found to exhibit non-trivial dependence on the width-to-length ratio will acharacteristicminimumasthewidthofthestripiscon- be discussed later. An important point of this paper is tinuously varied. that for the Mo¨bius strip, the inflexion point manifests The class of twisted, closed strips under consideration itselfinaratheruniquewayinencodingthe polarization 2 properties of light as it traverses the Mo¨bius loop. We consider the propagation of circularly polarized light along an optical fiber which has the shape of the boundarycurveofatwistedstrip. Thedirectionofprop- Z agation of light is tangential to the boundary curve and hence completely encodes the geometry of the boundary 1 curve. As the light completes a full loop of the twisted 0.5 strip boundary, its direction of propagation,the tangent 0 indicatrix, completes a closed circuit on the surface of the sphere of directions defined by the propagation vec- -0.5 torkˆ. Byadiabaticallyvaryingthe directionofpropaga- -1 tion arounda closedcircuit on the sphere, the change in 2 1.5 the polarization of light is equal to the solid angle sub- 1 tended in k-space. Condition of adiabaticity is that the -1.5 -1 0 0.5 -0.5 -0.5 Y lengthoftheboundarycurvebelargeascomparedtothe 0 0.5 -1 wavelength of light. X 1 1.5 2-2-1.5 The boundary curve of a twisted strip, viewed as a space curve Γ :r(s) = (x,y,z), thus represents the path along which light propagates. Here s is the arc length FIG. 1: Boundary curvesof the closed Möbius strip (Eq. (1) measuredalongthecurve,withds=vdt,wherev = dr . with n = 1) for strip-widths α = 1.0,0.8,0.6 (outermost to |dt| Thegeometryofthe spacecurvecanbedescribedbythe innermost). Notethelocalstraighteningaroundtheinflexion right-handedorthonormaltriad(T,N,B)that represent point for thecritical curvewith α=0.8. unit tangent, normal and binormal at every point along the trajectory. The evolution of the triad on the curve can be described by Frenet-Serret equations[10], Here B is the ith-component of the binormal vector B. i dT dN dB A short calculation shows that the eigenvalues of this =κN; = κT+τB; = τN, (2) Hamiltonianare0, κ. Hence the quantumHamiltonian ds ds − ds − ± exhibits a 3-fold degeneracy at points where κ vanishes. whereB=T N. κandτ denote,respectively,thelocal These are just the inflexion points of the optical fiber, × curvature and the torsion. They are given by[10] and this is a general result that follows from the basic dT d2T evolution Eq. (4). For the example of a fiber shaped κ= T(s) ; τ = [T ]/κ2, (3) liketheboundarycurveofann-twistedstrip,therearen | | · ds × ds2 such inflexion points ( see below) that appear when the Intuitively, the curvature measures the deviation of the width ofthe striptakes onacertaincriticalvalue,which curvefromastraightline,whilethetorsionquantifiesthe depends on n. non-planarity of the curve. Thevanishing ofκ impliesthatthe quantumHamilto- In order to compute the change in the polarization of nianisdegenerateatthethecriticalpointandastandα light as it passes through a circuit in the shape of the vary on the surface of the strip, this degeneracy leads to boundary of a twisted ribbon, we follow the variation in adiabolicpoint. Fig. (2)illustratestheconicalgeometry the electric field vector with respect to t (or s ) as light nearthe criticalpoint. Furtherdetails aboutsuchpoints travels along the curve Γ. Using Maxwell’s equations, willbe discussedlaterinthis paper. Suchdiabolic cross- it can be shown that the evolution of the complex unit vector Eˆ associated with the complex three-component ings in t−α space are sensed by a circuit that does not pass through the degeneracy, but simply encloses it. In electricfieldonthisspacecurvefollowstheFermi-Walker particular,suchclosedloopsresultinwavefunctionsthat transport law[8, 9], acquire a Berry phase that depends on the geometry of dEˆ =κ(s)B Eˆ (4) tthioenp.aItnhcionnptraarsatmteoteBresrpraycephinasaen, wahdiicahbaitsicobctyacilnicedevboylua- ds × mathematicalmappingofanopticalsystemtoaquantum We first note that interestingly, Eq. (4) can be cast problem, and further, is associated with parametric cir- in the form of a Schrödinger equation with the following cuitsenclosingaconicalintersection,thegeometricphase Hamiltonian H(s), which is a three-dimensional, anti- thatweinvestigateinvolvescircuitsthatareinconfigura- symmetric Hermitian matrix, with pure imaginary ele- tion space,i.e.,boundarycurvesoftwistedstrips. Hence, ments. whileaspecialcircuitmaypassthroughadiaboliccross- ing,typicalcircuitsneverencloseone. Moreimportantly, 0 B B this geometric phase canbe directly measured in experi- 3 2 H =iκ BB3 −B0 −0B1  (5) msheanlltsd,ismcuakssinagtitthme oenred.interesting and applicable, as we 2 1 −  The geometric phase is obtained by solving Eq. (4)   3 Symbolicmanipulationfacilitatesthedeterminationof analytic expressions for the curvature κ(t) and torsion τ(t), by using Eq. (1) in Eq. (3). For the Mo¨bius case, t we get 7.5 6α[4cos t/2+α(5+8cos t)+α2(9cos t/2+cos 3t/2)] τ = , 7 κ2 6.5 where the curvature κ is given by 6 κ = 64 + 288αcos t/2 + α2(284 + 216cos t) + 5.5 α3(336cos t/2+64cos 3t/2)+α4(65+54cos t+6cos 2t) 5 Taylor expansion of κ and τ near α and t gives, 0.3 c c 0.2 -1 -0.5 0 -0.1 0 0.1 a κ ≈ 2v13/2[(α−αc)2+9b2vc2(t−tc)2]21 K 0.5 1-0.3 -0.2 c 3b(α α ) c τ − − , ≈ [(α α )2+9b2v2(t t )2] − c c − c FIG. 2: Conical topology of the eigenvalues ±κ(t,α) of the where b = 6 and v = dr/dt = 1/ (5). These equa- quantumHamiltonian(Eq. (5))nearthecriticalpoint(tc,αc) tionsshowt5hatτ chcang|essign|casthepparameterαpasses on thesurface of theMöbius strip. Here a=(α−αc)/αc through its critical value.[7] . Geometrically, the vanishing of κ results in the diver- gence in the torsion τ at the inflexion point. However, [9]. Writing Eˆ in terms of the complex unit vector M= as shown below, this singularity is integrable and the (N + iB)/√2, we have resulting integrated τ, the geometric phase is well defined. To show this, we calculate χ(t) near the inflexion Eˆ =α(s)M+β(s)M∗, (6) point. For arbitrary t0, the total twist in the time inter- val( t +t ,t +t )centeredaroundthecriticalpointt 0 c 0 c c witShub|αst|i2tu+ti|nβg|2t=his1.in the Eq. (4), it can be seen that gisr,aRltt−iccs−+ettq00uτavldtto=2A−rcRtttcac−+ntt[00 [(α−t0αbcv)c2(+α−v].c2αbAc2)(dst−twtce)2p]a.sTshthisroinutgeh- α(s)=eiχα(0) and β(s)=e−iχβ(0), where α ,itwillchangefrom (πbvtco(απ−α(ci)r)respectiveofthevalue c − of t ) giving rise to a jump of 2π. 0 S Figure 3 illustrates the evolution of the geometric χ= τ ds, (7) Z phase as the light propagates through the boundary 0 curves. For α < α , the phase factor χ(t) exhibits a c with τ as defined in Eq. (3). For a cyclic evolution, χ wiggle near the critical point which becomes saw-tooth canbe seento be equal to the phase increase( decrease) shapedat criticality. However,for α>α , the the phase c of the right (left) circularly polarized light.[9] Thus the change near the critical points is smooth. As discussed phase change determined solely by the geometry of the above, and seen in the figure, the polarization of light cyclic path Γ is fully characterized by its curvature and undergoes a full rotationof 2π after passing through the the torsion. In view of the transverse nature of the elec- critical point. tromagneticwaves,T Eˆ =0. This leadsto averydesir- Taylorexpansionof the unit normalnearthe inflexion able result that the dy·namical phase is zero and the net point shows that as we pass it at t = t , N rotates by c phase change is equal to χ, the geometric phase. π about a fixed direction. And for α <α , it rotates by c Oneofthecharacteristicsoftheboundarycurvesofthe π. The same is true also for B. Therefore,the number n-twisted strips is the existence of n inflexion points. At −of rotations of the pair N,B increases by 2π as we pass the inflexion point, the curvature vanishes and the tan- the inflection point. For an n-twisted strip, the number genttothecurveshowsacusp. FortheMo¨biusstrip(i.e., ofrotationsisequalton,foroddn. Forevenn,thereare n=1),thecriticalboundarycurvehasjustoneinflexion two boundary curves. The number of rotations for each point at t = 2π and α = 4/5. For n = 3, the boundary ofthemis(n/2). Thusthereexistsauniversalfunctional c curve is a trefoil, which is knotted, and there are three formforlocalgeometricalquantitiesτ andκandajump inflexion points. In general for the boundary curve of of 2π in the global variables near the inflexion point, for an n-twisted strip, n inflexion points occur at a critical the boundary curves of both orientable as well as non- width α = 1/(1 + n2/4), when cos(nt /2) = 1.[12] orientable strips. c c − As seen in the figure 1, the boundary curves encode the Figure 4showsthenetphaseaccumulatedbylightaf- widthofthestripandthecriticalcurve,namelythecurve terpropagatinginacyclicloopalongtheboundarycurve with aninflexion pointis distinguishedby a vanishing of ofann-twistedstrip,asits widthvaries,forn=1,2and curvature around that point. 3. In figure 5, we plot the phase modulo 2π. A striking 4 FIG. 3: (color on line) Geometric phase (in units of 2π) FIG. 5: (color on line) Same as Fig. 4, except that the acquired by polarized light as it passes through the closed geometric phase is plotted modulo 2π boundary curve of a Möbius strip. The top and the bot- . tom curves correspond respectively to strip widths just below (α = 0.79) and just above (α = 0.81) the critical width αc =0.8. origin of such a minimum rather mysterious. It is rather intriguing that the width for the existence ofthediaboliccrossingoftheeigenspectrumofthequan- tumHamiltoniancoincideswiththecriticalwidthresult- ing in the minimum phase shift. It would have been natural to speculate that the global phase shift is in- fluenced by a local diabolo geometry. However, such a minimumexistsonlyinthe Mo¨biusloop,whereasthedi- abolic crossings characterize the boundary curves of all n-twisted strips. Therefore, the fact that two different andsomewhatunique featuresoccuratthe sameparam- eteriseitheracoincidenceoritisconceivablethatthisis happeningbecausethe Mo¨biusgeometryandtopologyis unique inthe sense thatn=1 is the onlycasewhere the single boundary curve of the twisted strip is un-knotted and has a single inflexion point. The presence of multi- pleinflexionpointsinthen-twistedribbons(withn>1) seem to play a role in such a way as to wipe out this minimum. We conclude this paper by suggesting possible experi- mentsthatcouldverifyourtheoreticalresults. Webegin FIG. 4: (color on line) From Top to bottom : Geometric by noting thatin experiments thatdemonstrate the geo- phase(inunitsof2π)forthecyclicpathsalongtheboundary metrical phase phenomenon in optics, the usual nonpla- curvesofatriply-twistedstrip(n=3),adoublytwistedstrip narpaththathasbeenconsiderediseitheranopenhelix n=2andasingly-twisted,i.e.,Möbiusstrip,forvariousstrip [3],orahelixthatcanbeeffectivelyclosedbymakingthe widths. Herea=(α−αc)/αc fiber lie along planar paths (with τ =0) at the two ends of the helix[4]. Clearly, such a circuit has no inflexion pointswhereτ becomeslocallysingular,sinceκisacon- aspect is the characteristic minimum that we find in the stantforahelix. Toourknowledge,noexperimentshave phaseshiftinthecaseoftheMo¨biuscircuitcharacterized beenperformedonnonplanarfiberswithinflexionpoints. bythecriticalwidthofthestrip. Inotherwords,geomet- We hope that our theoretical results that show that the ric phase shows that the Mo¨bius geometry with n=1 is presence of inflexion points has a nontrivialeffect on the indeed special. The absence of such a minimum in the optical phase shift would motivate experimentalists to boundarycurvesofallother(integer)nvaluesmakesthe observe it. 5 Forexample,opticalfiberswhichhavetheshapeofthe curring at a critical width. Similar measurements can boundary curve of various n-twisted strips can be per- be done with optical fibers in the shape of the boundary haps be fabricated as follows. By twisting a sheet made curveofastriptwistedntimes,andrepeatedfordifferent ofsomepliablematerialsuchasplastic(oranyotherma- widths. terialappropriateforopticalexperiments)ofsomewidth Finally, our studies may be relevant in optical fibers αonce,andgluingtogetherthetwoshortedges,aclosed used in tuning of polarization, as such a phase can be Mo¨bius strip can be constructed. Instead of winding the seen in low birefringent optical fibers[14]. We hope opticalfiberonacylinderaswasdoneintheexperiments that our studies will also stimulate laboratory research of Tomita-Chiao[3] and Frins-Dultz [4], the fiber could involving lasers and condensates in this novel class of now be attached along the boundary curve of the above twisted geometries that we have studied. Mo¨bius strip, and a similar experimental setup could be used to measure the geometrical phase. The experiment canthenberepeatedwithvariousdifferentwidths ofthe RB thanksthe CouncilofScientific andIndustrialRe- Mo¨bius, to find the dependence ofthe geometricalphase search, India, for financial support under the Emeritus on the width, with a characteristic minimum phase oc- Scientist Scheme. [1] See, for instance, Geometrical Phases in Physics edited [8] M.BornandE.Wolf,PrinciplesofOptics6thed.(Perg- by A. Shapere and F. Wilczek (World Scientific, Singa- amon, Elmsford, NY, 1987); M. Kline and I. W. Kay, pore, 1989). Electromagnetic Theory and Geometrical Optics, Chap. [2] R.Y. Chiao and Y.S. Wu, Phys. Rev. Lett. 57, 933 V (IntersciencePublishers, New York,1965) (1986). [9] N.MukundaandR.Simon,Ann.Phys.228,205(1993). [3] A. Tomita and R.Y. Chiao, Phys. Rev. Lett. 57, 937 [10] D.J.Struik,Lectures on Classical Differential Geometry (1986). (Addison-Wesley,Reading, Mass., 1961). [4] E. M. Frins and W. Dultz, Opt. communications 136, [11] Radha Balakrishnan, A. R. Bishop, and R. Dandoloff, 354 (1997). Phys.Rev.Lett.64,2107(1990);Phys.Rev.B,47,3108 [5] Y.AharonovandD.Bohm,Phys.Rev.115,489(1959). (1993). [6] H.K. Moffatt and R. L. Ricca, Proc. R. Soc. Lond. A [12] E. Radescu and G. Vaman, Phys. Rev. E 64, 056609-1 439, 411 (1992). (2001). [7] Wewould liketopointout thatthefunctionalform ofκ [13] However,notethatthiscircuitcannotberegardedasthe andτ neartheinflexionpointisidenticaltothatobtained boundary curve of an n-twisted strip any more, since n in an earlier study[6] investigating the generic behavior is fractional. of a curve in the vicinity of an inflexion point, the only [14] P. Senthilkumaran,J. Opt.Soc. Am. B 22, 505 (2005). differencebeing that thevalueof b was unity there.