aa r X i v : . [ phy s i c s . op ti c s ] O c t Multi-twist optical M¨obius strips
Isaac Freund
Physics Department, Bar-Ilan University, Ramat-Gan ISL52900, Israel
Circularly polarized Gauss-Laguerre GL and GL laser beams that cross at their waists at a smallangle are shown to generate a quasi-paraxial field that contains an axial line of circular polarization,a C line, surrounded by polarization ellipses whose major and minor axes generate multi-twistM¨obius strips with twist numbers that increase with distance from the C point. These M¨obiusstrips are interpreted in terms of Berry’s phase for parallel transport of the ellipse axes around theC point. OCIS codes:
We show here how to create quasi-paraxial, ellipticallypolarized light fields in which the major and minor axesof the polarization ellipses generate M¨obius strips withradially increasing numbers of half-twists; to our knowl-edge, these polarization structures are unique.The polarization ellipses in elliptically polarized parax-ial fields lie in parallel planes (here the xy -plane) ori-ented normal to the propagation direction ( z -axis). Thegeneric point polarization singularity in any plane of sucha field is a C point, an isolated point of circular polariza-tion embedded in a field of polarization ellipses [1 − π ) I C = ± /
2. As thebeam propagates in space the C point traces out a line,a C line [1 − I C = ± / non -paraxialfield? The (possibly surprising) answer is that the majorand minor axes of these ellipses generate M¨obius strips[6]. The canonical M¨obius strip has a single half-twist,and can be either right-handed (RH) or left-handed (LH).Such strips have recently been shown [6] to occur natu-rally in random fields. Illustrated in Fig. 1 is an exampleof a left-handed M¨obius strip. In random fields left- andright-handed strips are equally probable.M¨obius strips with a larger, odd number of half-twistsare possible, and strips with three half-twists are alsofound in random fields [6]; in such fields ∼ / paraxial laser beam containingan axial C line can be generated by a coaxial, coherentsuperposition of a circularly polarized RH (LH) Gauss-Laguerre GL mode and a circularly polarized LH (RH) GL mode. The LH (RH) GL mode contains a centraloptical vortex (phase singularity) at which the amplitudevanishes [7], so at that point the polarization in the com-bined beam is RH (LH) circular. At other points inthe beam RH and LH components combine to produceelliptical polarization.Within the paraxial approximation, at the waists ofthe individual beams, and therefore at the waist of thecombined beam, there exist only the two transverse fieldcomponents E x and E y . However, if the two beams aremade to intersect in the xz -plane at small angles ± θ rel-ative to the z -axis, a third field component E z ∼ E x sin θ develops. At its waist, the resulting field, describedquantitatively below, has the following unique property: (a) (b) FIG. 1: Computed M¨obius strip surrounding a C point in anon-paraxial, elliptically polarized optical field. (a) Threedimensional view of the M¨obius strip. Here the major axes ofthe ellipses on a circle centered on the C point rotate through180 o around the circle circumference to generate a one-half-twist M¨obius strip. Throughout, for clarity, only half-axesare shown. (b) The strip in (a) seen from above. In this viewellipse centers are shown by filled gray circles and semi-axes bystraight black lines; axis endpoints that lie above (below) thecircle of ellipse centers are shown by filled white (black) cir-cles. The ellipse axes can be seen to rotate counterclockwiseabout the circle circumference, forming a half-turn segmentof a left-handed circular screw. Surrounding the central C point are circular rings of ra-dius r on which the major and minor axes of the polar-ization ellipses generate multi-twist M¨obius strips thatcontain an odd number of half-twists; these strips havethe unique property that the number of half-twists in-creases with radial distance r from the C point !In Fig. 2 we show an example of such a M¨obius strip.In principle, r , and therefore the number of half-twists,increases without limit; in practice, of course, the Gaus-sian envelope of the beam reduces the intensity to im-measurably small levels far from the beam center. FIG. 2: Computed multi-twist optical M¨obius strip generatedby the major axes of the polarization ellipses on a circle sur-rounding a C point. An observer walking (gray arrow) alongthis circle sees the ellipse axes rotate in the clockwise direc-tion (black arrow), generating a M¨obius strip containing 13half-twists that forms a circular, right-handed screw (helicalworm gear) containing 6 . We turn now to a quantitative description of thesemulti-twist optical M¨obius strips. For the sake of defi-niteness, we take the GL mode to be to be RH and thevortex containing GL mode to be left handed. We as-sume that both beams have the same wavelength λ , andthe same waist parameter w ≫ λ , and that they inter-sect maximally at their waists which are centered on theorigin. Although there are a number of different exper-imental approaches to generating such beams, the onemost likely to be used involves liquid crystal modulators.These are usually addressed as a Cartesian grid of pixels,and so in what follows we use Cartesian coordinates.We define a Gaussian envelope function for a GL l beam, l = 0 ,
1, by G l = (cid:0) w /W l (cid:1) B l exp (cid:16) − (cid:0) ρ l (cid:1) / (cid:0) W l (cid:1) (cid:17) × exp (cid:16) − ik (cid:0) ρ l (cid:1) / (2 R l ) (cid:17) exp (cid:0) − ikZ l (cid:1) , (1)where (cid:0) ρ l (cid:1) = (cid:0) X l (cid:1) + y , R l = Z l + Z /Z l , k =2 π/λ, Z = k w / , W l = w q (cid:0) Z l / Z (cid:1) , B l = (cid:0) iZ l / Z (cid:1) / q (cid:0) Z l / Z (cid:1) . Writing θ l for the an-gle that the GL l beam makes with the z -axis, X l = x cos θ l + z sin θ l , Z l = − x sin θ l + z cos θ l . (2) The field components of the combined beam E = E x ˆx + E y ˆy + E z ˆz , with ˆx , ˆy , ˆz unit vectors along the correspond-ing coordinate axes, are E x = (cid:0) E (cid:1) x + (cid:0) E (cid:1) x , E y = (cid:0) E (cid:1) y + (cid:0) E (cid:1) y ,E z = (cid:0) E (cid:1) x sin θ + (cid:0) E (cid:1) x sin θ , (3)where (cid:0) E (cid:1) x = G , (cid:0) E (cid:1) y = iG , (4a) (cid:0) E (cid:1) x = √ G B (cid:0) X + iσy (cid:1) /W , (4b) (cid:0) E (cid:1) y = − i √ G B (cid:0) X + iσy (cid:1) /W . (4c)In Eqs. (4b) and (4c) σ = +1 ( σ = −
1) for a positive(negative) vortex. For all figures presented here, w =100 λ , θ = − θ = 5 o , and σ = +1.The major axis α and minor axis β of the ellipses sur-rounding the C point in Figs. 1 and 2 (and Figs. 3and 4 below), are obtained from Berry’s formulas [5]: α = Re (cid:16) E ∗ √ E · E (cid:17) , β = Im (cid:16) E ∗ √ E · E (cid:17) .The exact r dependence of the number of half-twistsand handedness of the M¨obius strips requires an analysesthat is too complicated to describe here. Instead, insightinto what happens can be obtained by considering theangle α xy = arctan ( α y , α x ) shown in Fig. 3, where α x and α y are the x, y -components of α . Because α is aline, not a vector, it is plotted modulo π . On the smallcircle at the center of the figure α xy winds around thecentral C point, which in such plots appears as a vortex,with winding number I C = − / I C = − / α xy decreases by π . FIG. 3: Rotation angle α xy , plotted 0 to π black to white, ofthe projection of the major axes α onto the xy -plane. The corresponding M¨obius strip, shown in Fig. 1, canbe interpreted in terms of Berry’s phase [8]: as one movesalong the circle surrounding the C point, α undergoesparallel transport, rotating through π during one com-plete circuit; this leads to winding number I C in the xy -projection, and the one-half-twist M¨obius strip in 3D.In addition to the central C point, Fig. 3 contains anumber of π -fringes that are analogous to the 2 π -fringesin the forked-fringe method for measuring vortices [9 , α passes through a π -fringe it rotates through π , so that the total number of half-twists equals the totalnumber of fringes traversed during a circuit about the Cpoint. The large circle in Fig. 3 that passes through 13 π -fringes corresponds to the M¨obius strip in Fig. 2, whichhas 13 half-twists. Similar considerations apply to axis β , and in Fig. 4 we show a M¨obius strip generated bythis axis which has 29 half-twists (14 . FIG. 4: M¨obius strip containing 29 half-twists generated byminor axis β . Here radius r = 42 λ is twice that of Fig. 2. The forgoing leads to a simple, heuristic expression fortwist number τ , the number of full twists, as a functionof the radial distance r ≪ w from the C point. For θ = − θ = θ , far from the C point the fringe spacingalong the here horizontal x -axis, is s = λ/ (2 sin θ ), sothat the number of fringes traversed by a line of length2 r , the circle diameter, is 2 r/s . On a circle centered onthe C point the circumference passes through each fringetwice, and upon adding in the extra π -fringe induced bythe C point we obtain τ calc ≃ int(4 r sin ( θ ) /λ ) + 1 / . (5)In Fig. 1, r = 2 λ , τ calc = 0 .
5, in Fig. 2, r = 20 λ , τ calc = 6 .
5, and in Fig. 4, r = 42 λ , τ calc = 14 . − ineach case in agreement with Eq. (5). We find Eq. (5) tobe in general agreement with our computer simulationsfor r < w /
2; for larger r , s begins to decrease signifi-cantly with increasing r due to wavefront curvature andthe Gouy phase shift, and Eq. (5) underestimates thetwist number.To be reported on elsewhere are the twist number andhandedness of the M¨obius strips as a function of r andother system parameters, the n -foil knots generated bythe axis endpoints of n -twist M¨obius strips, the strips(twisted ribbons) with an even number of half-twists thatappear on circles that do not enclose a C point, the multi-twist optical M¨obius strips that are generated by othercombinations of laser modes, and by other beam config-urations, and the z dependence of these structures.In summary, we have shown how to combine two circu-larly polarized laser beams, one of which contains a vor-tex, to create a quasi-paraxial field containing a centralC point surrounded by polarization ellipses whose ma-jor and minor axes generate multi-twist optical M¨obiusstrips with twist numbers that increase radially from theC point. These M¨obius strips are structurally stable,changing unimportantly when, for example, 3% noise is added to the simulation. Coherent nanoprobe techniques[11 −
18] capable of determining the field structure on sub-wavelength scales should permit experimental measure-ments of these highly unusual objects. As for possibleapplications, we note, inter alia, that the M¨obius stripscould be embedded in polarization sensitive photoresiststo create devices with unique optical properties.email address: [email protected] (I. Freund).
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