October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
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Observer-dependent sign inversions of polarization singularities Isaac Freund Physics Department, and Jack and Pearl Resnick Advanced Technology Institute, Bar-Ilan University, Ramat-Gan ISL52900, Israel ([email protected]) Received July 28, 2014; revised September 14, 2014; accepted September 15, 2014; posted September 16, 2014 (Doc. ID 219897); published October 9, 2014 We describe observer-dependent sign inversions of the topological charges of vector field polarization singularities: C points (points of circular polarization), L points (points of linear polarization), and two virtually unknown singularities we call γ C and αL points. In all cases, the sign of the charge seen by an observer can change as she changes the direction from which she views the singularity. Analytic formulas are given for all C and all L point sign inversions. © 2014 Optical Society of America OCIS codes: (260.0260) Physical optics; (260.6042) Singular optics; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.39.005873
The polarization singularities of vector optical fields include lines of circular polarization, C lines that meander throughout the field, and lines of linear polarization, L lines [1–16]. In the nonparaxial, three-dimensional (3D) fields of interest here, also L lines meander throughout the field, whereas in paraxial fields these lines are seen as lines in a plane where L surfaces intersect the plane [1,4,6]. In 3D fields, where a C line or an L line pierces a plane, here the observation plane P, a point of circular or linear polarization, a C point or an L point, appears. C and L lines are observer-independent, invariant features of the optical field. A remarkable property of these lines is that the polarization ellipses that surround every C or L point on the line rotate about the point [1–16]. Perhaps no less remarkable is the fact that as the observer changes the direction from which she views the singularity (her line of sight, LOS) she finds that the sense of these rotations are seen to change. For one LOS, the observer may see the surrounding ellipses rotate around the singularity in the counter-clockwise direction, whereas for another LOS that differs infinitesmally from the first, she sees clockwise rotations. These inversions are a fundamental, intrinsic property of the 3D structure of light. Sign inversions of C and L points were first found by Nye and Hajnal [4] for a particular LOS (discussed in detail later); a quantitative theory for this LOS has been given by Berry [7]. Here we discuss the general case, and present analytical formulas for all directions of observation. We also introduce two classes of polarization singularities, γ C points on C lines, and αL points on L lines, and describe their sign inversions. C points are singular because the major axis α and minor axis β of the polarization ellipse are undefined when the ellipse degenerates into a circle, the C circle; L points are singular because the minor axis β and the ellipse normal γ are undefined for a linear polarization figure. Axis α, however, remains well defined and is along the direction of linear polarization. The projections onto P of the corresponding axes of the surrounding polarization ellipses whose centers lie in P rotate around C and L points with signed net winding, i.e., rotation, angle ΘC π and associated signed topological index, winding 0146-9592/14/205873-04$15.00/0
number, or “charge,” I C ΘC ∕2π 1∕2 for C points, ΘL 2π, and I L ΘL ∕2π 1 for L points. When the polarization ellipse is viewed down one of its principal axes, α, β, or γ, the projections onto P of the corresponding axes of the surrounding polarization ellipses whose centers lie in P are also seen to rotate around the point with index I O 1 [17–19]. Thus, all ordinary points are seen as singularities for three directions of observation; we call these directional singularities “O points” [20,21]. When the LOS is along α, β, or γ, we describe the O point as an αO , βO , or γ O point. If the central point is a C point, rather than an ordinary point, we emphasize this by writing γ C . Similarly, if the central point is an L point, we emphasize this by writing αL . Figure 1 reviews basic properties of the various singularities. In what follows, we specify the observer’s LOS by means of a unit vector n (the normal to the observation plane P) that points from the singular point to the observer.
Fig. 1. Singularity streamline patterns. Streamlines formed by the projected axes of the surrounding ellipses are shown by thin lines, separatixes, here Λ lines, that separate regions of opposite curvature are shown by thick lines. Throughout, positive/ negative half index singularities are shown by single white/ black circles, integer index singularities by nested circles. These patterns are characterized by index I and the number of Λ lines, N Λ . (a)–(a″) C point patterns [2]. (a) “ Lemon,” I C 1∕2, N Λ 1. (a′) “Monstar,” I C 1∕2, N Λ 3. (a″) “Star,” I C −1∕2, N Λ 3. (b)–(b″) L, O, γ C and αL point patterns [12,21]. here I I L;O;γC ;αL . (b) “ Spiral,” I 1, N Λ 0. (b′) “Sink” or “node,” I 1, N Λ 4. (b″) “Saddle,” I −1, N Λ 4. © 2014 Optical Society of America
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In Fig. 2, we present a C point index sphere that shows the changes in I C of a given C point as the point is viewed from different directions; index spheres for axes α and β are always identical. In the representation of Fig. 2(b), the C circle corresponds to the equator. As can be seen from this figure, on any path on the sphere that connects two antipodal points, index I C inverts an even number of times. This occurs because for antipodal points the observer views the same plane from opposing directions: say, from the “top” for a point in the northern hemisphere, and from the “bottom” for its antipodal partner in the southern hemisphere. On a line of constant longitude, for example, there are two inversions: one inversion [on the tilted great circle in Fig. 2(b)] occurs when n is perpendicular to the tangent to the C line at the C point, and therefore, P is tangential to the line; the other inversion [on the equator in Fig. 2(b)] occurs when n is perpendicular to axis γ and therefore lies in the plane of the C circle. The degeneracy produced by the intersection of these two great circles is an essential one that is not lifted by perturbation. All C points generate qualitatively similar index spheres; a number of typical examples are shown in Fig. 2(c) using a Cartesian representation. In all cases, only monstars can transform into stars, and vice versa, so that preceding an inversion lemons always transform into monstars. One can understand why sign inversion occurs when P is tangential to the C line by considering what an observer sees as she translates P relative to the line. Before P intercepts the line, she sees no topological charge since there are no C points. Charge conservation requires that the net charge remain zero also when P first contacts the line, i.e., is tangent to the line. Now, the charge
Fig. 2. Dependence of I C on the direction of observation. (a) Spherical coordinates θ, φ. (b) Index sphere, the north pole (N) is along the z axis in (a). Light areas correspond to I C 1∕2, dark areas to I C −1∕2. (c) Cartesian representations of I C θ; φ for three typical examples. The interested reader may enjoy sketching possible vector triplets n; γ; ω , Eqs. (1), for different points in these plots.
seen in the tangent plane is a property of the plane, and is therefore always seen to be zero whenever P is tangential to the line, including the case in which P rotates around a C point on the line. In generic fields, a nonzero charge dropping to zero under a smooth, continuous rotational transformation heralds sign inversion, as is observed when P rotates through the tangent to the C line. Sign inversion also occurs when n is perpendicular to γ, because the sign of I C depends on the handedness of the light (right or left) as seen by the observer: if γ points into the observer’s half-space, she sees right-handed polarization, whereas if γ points away from her halfspace she sees left-handed polarization. When n is perpendicular to γ, the handedness seen by the observer is undefined; in generic fields this ambiguity heralds inversion of the handedness, and therefore inversion of the sign of I C , as n rotates through a plane normal to γ. This definition of handedness for 3D fields in which there are partial waves traveling in many different direction is consistent with that used in paraxial fields; it appears to differ from a definition proposed by Nye [1]. We have developed a simple formula that describes I C at all points on the index sphere: I C 1∕2signn · ωsignn · γ;
(1a)
ω ∇RE · E × ∇IE · E;
(1b)
γ IE × E:
(1c)
Equations (1) are to be evaluated at the C point. The vorticity ω is a continuous vector field introduced by Berry and Dennis [5]; on the C line itself ω lies along the tangent to the line. All data presented here (Figs. 1–4) were obtained using simulations that are exact solutions of Maxwell’s equations, and local expansions of these equations [17–21]. The white lines (inversion curves) in Figs. 2(b) and 2(c) were calculated from Eqs. (1), and the color-coded regions show the numerically measured signs of I C . We have verified Eqs. (1) for 106 independent realizations of C points on C lines in random optical fields by measuring the index on both sides of both types of inversion curves in Figs. 2(b) and 2(c). These results also confirm that the only inversion points are those described by Eqs. (1). The index theorem [22] permits a change in the net charge of a bounded region only if charges enter or leave the region, i.e., cross the boundary. In accord with this requirement, an observer that examines P as it rotates through an inversion point sees a chemical-like reaction involving one-or-more additional singularities entering and/or leaving the field of view. The generic sign inversion reactions that are seen for C points are illustrated in Fig. 3. L point sign inversions are similar to C point sign inversions: sign inversion occurs when the observation plane is tangent to the parent L line, and when it contains the direction of linear polarization. Proceeding as for I C , we obtain for I L ,
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Fig. 3. Generic sign inversion reactions of C points. (a)– (a″) Sign inversion of an initial positive (I C 1∕2) C point as the observation plane P rotates through the tangent to the parent C line. The example shown here is for the case in which the observer studies the rotation of axis α of the ellipses that surround the central C point. The orientation of these axes projected onto P is shown gray-scale coded 0 to π, black to white. In (a), as P approaches the tangent, in addition to the positive C point at the origin, the observer sees a negative C point (the “intruder”) enter the field of view. In (a′) she sees the negative intruder collide with, displace, and usurp the position of the initial positive C point, and in (a″) she sees the initial positive C point flee the scene. The observer sees the same sequence of events when studying axis β of the surrounding ellipses. (b′)–(b″) Sign inversion of the negative C point in (a″) as P continues to rotate, passing though γ. Panel (b) shows the negative C point as P approaches γ. In (b′) P contains γ, and the observer sees the central negative C point emit a negative αO point, whose index is I αO −1. Conserving charge, the negative C point changes sign and becomes a positive, I C 1∕2, C point. In (b″) the observer sees the emitted negative αO point float off into the surrounding sea of O points. Studying axis β of the surrounding ellipses, instead of axis α, at the positions of P corresponding to (b′)–(b″), the observer sees the following: at (b) she again sees a negative (I C −1∕2) C point at the origin, together with a positive βO point with index I βO 1 that floats in from the surrounding sea of O points; at (b′) the observer sees this βO point collide with and absorb the initial negative C point, and at (b″) she sees the end product, a positive, I C 1∕2, C point. In general, as P rotates through γ, α∕β transition combinations are seen to occur with approximately equal probabilities: emission/emission, emission/absorption, absorption/emission, and absorption/absorption.
I L signn · τsignn ·b z;
(2a)
τ ∇IE x × ∇IE y :
(2b)
Equations (2) are to be evaluated at the L point. At the point, τ is along the tangent to the L line, zˆ is along the direction of linear polarization, and E x and E y are the optical field components along an arbitrary pair of orthogonal xy-axes in the plane normal to zˆ. Expanding the components of the optical field around an L point as E x P xx iQxx x P xy iQxy y P xz iQxz z, Ey P yx iQyx x P yy iQyy y P yz iQyz z, Ez K z P zx iQzx x P zy iQzy y P zz iQzz z, we obtain for the components of τ, τx Qxy Qyz − Qxz Qyy , τy Qxz Qyx − Qxx Qyz , τz Qxx Qyy − Qxy Qyx , which follows Eq. (2b). The index spheres and inversion curves for L points are qualitatively similar to those shown for C points in
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Fig. 2. Using our simulations of random optical fields [17–21], we have verified numerically Eqs. (2) for the L point inversion curves of 106 independent L points. The sign inversion “chemical” reactions of L points seen when P passes through the tangent to the L line follow the schema in Figs. 3(a)–3(a″), except that for L points, the winding numbers are 1. L point sign inversion when P contains the direction of linear polarization, however, differs from Figs. 3(b)–3(b″). In the case of L points, the observer sees a sequence that superficially resembles Figs. 3(a)–3(a″), but with the important difference that the “usurper” is not another L point, but rather a βO or a γ O point from the surrounding sea of O points, depending upon whether the observer studies the rotations of axes β or γ of the surrounding polarization ellipses. In all cases, only sinks can transform into saddles, and vice versa, so that preceding sign inversion, spirals transform into sinks. During the directional sign inversions found by Nye and Hajnal [4], the observer continually rotates the direction of her LOS: as she traverses a C line, she maintains the observation plane P coincident with the C circle, whereas on an L line, she maintains P perpendicular to the direction of linear polarization. For C points, sign inversion of the winding angle of axes α and β of the surrounding ellipses occurs when the C circle is tangent to the C line; for L points, sign inversion of axes β and γ occurs when the direction of linear polarization is perpendicular to the L line. For both C and L points, NH inversions occur when P is tangent to the parent line, and are thus seen to be special cases of Eqs. (1) for C points and Eqs. (2) for L points. Berry has given explicit formulas for NH inversion points with their special lines of sight [7]. In Eqs. (1) and (2), we generalize his results to arbitrary lines of sight, and add the important class of sign inversions that occur when n is perpendicular to γ for C points, and n is perpendicular to the direction of linear polarization for L points. These latter inversions cannot occur under NH constraints. As our keen-eyed observer tracks an NH inversion, she notes that in addition to the expected rotation of axes α and β of the surrounding ellipses around C points, axis γ of the surrounding ellipses also rotate around the point with integer winding numbers 1 and streamline patterns shown in Figs. 1(b)–1(b″). She similarly finds that in addition to the expected rotation of axes β and γ of the surrounding ellipses around L points, axis α of the surrounding ellipses also rotate around these points with integer winding numbers 1 and streamline patterns shown in Figs. 1(b)–1(b″). Tracking these unexpected rotations as she moves along the lines, she encounters points on the lines at which the signs of their winding numbers invert—our observer has found two new classes of directional singularity sign inversions; she christens these γ C point inversions for C lines, αL point inversions for L lines. Similar to L points, for both γ C and αL inversions, only sinks can transform into saddles, and vice versa, so that preceding sign inversion, γ C and αL spirals transform into sinks. The observer finds that γ C and αL inversion points do not coincide with NH inversion points, so that C lines and
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Fig. 4. γ C inversions on a C line. The C line is shown by the thick curve, regions of the line that contain positive γ C points are colored gray and labeled γ C , regions of the line containing negative γ C points are colored black and labeled γ C− . The tracks of positive γ O points are shown by thin gray lines labeled γ O , tracks of negative γ O points are shown by thin black lines laC beled γ O − . Two γ inversion points, labeled A and B, respectively, are present. The directions shown for the arrowheads O on the γ O and γ − tracks are for the case in which the C line is traversed from bottom to top. If the observer traverses the line from top to bottom, she sees the same tracks, but with the directions of all arrowheads reversed.
L lines contain two distinct, often interleaved sets of directional inversion points. Closely studying these new inversions, the observer finds that as she moves along a C line, a γ O point emerges from the surrounding sea of O points and approaches the line. Similarly, the observer finds that as she moves along an L line, an αO point emerges from the surrounding sea of O points and approaches that line. Plotting the positions of these O points, here their tracks, the observer finds what appear to be lines of γ O points that intercept, and exchange charge with the C line at a γ C inversion point, and lines of αO points that intercept, and exchange charge with the L line at an αL inversion point. Typical examples of these phenomena for γ C inversions on C lines are shown in Fig. 4; analogous events are seen for αL inversions on L lines. In summary: Observer dependent sign inversions of polarization singularities have been described, and general analytical formulas have been presented for sign inversion of C points on C lines, Eqs. (1), and L points on L lines, Eqs. (2); these equations are extensions to the general case of earlier results by Berry [7] for the special case of Nye and Hajnal inversions [4]. Because of the spe-
cial conditions on the orientation of the C circle, or the direction of linear polarization, relative to the parent line, NH inversions correspond to accidental coincidences that occur at apparently random, sparse locations on the line. In contrast, as Eqs. (1) and (2) show, in the general case, every point on every C line, and every point on every L line, undergoes sign inversion in a doubly infinite set of planes. For both C and L points, one set consists of all planes that contain the tangent to the line. For C points, the second set consists of all planes that contain axis γ, whereas for L points, this second set consists of all planes that contain the direction of linear polarization. Two directional singularities, γ C points on C lines and αL points on L lines, have been introduced, and their sign inversions described. The sea of O points that surround C and L lines [20,21] has been shown to play essential roles in the sign inversions of all currently known optical polarization singularities. References 1. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999). 2. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977). 3. J. F. Nye, Proc. R. Soc. A 389, 279 (1983). 4. J. F. Nye and J. V. Hajnal, Proc. R. Soc. A 409, 21 (1987). 5. M. V. Berry and M. R. Dennis, Proc. R. Soc. A 457, 141 (2001). 6. M. R. Dennis, Opt. Commun. 213, 201 (2002). 7. M. V. Berry, J. Opt. A 6, 675 (2004). 8. R. W. Schoonover and T. D. Visser, Opt. Express 14, 5733 (2006). 9. S. Zhang, B. Hu, Y. Lockerman, P. Sebbah, and A. Z. Genack, J. Opt. Soc. Am. A 24, A33 (2007). 10. R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, Phys. Rev. Lett. 100, 103901 (2008). 11. I. Freund, Opt. Commun. 283, 1 (2010). 12. I. Freund, Opt. Commun. 283, 16 (2010). 13. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, Appl. Opt. 51, 2925 (2012). 14. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, Opt. Express 21, 8815 (2013). 15. M. S. Soskin and V. I. Vasil’ev, J. Opt. 15, 044022 (2013). 16. F. S. Roux, Opt. Lett. 38, 3895 (2013). 17. I. Freund, Opt. Commun. 242, 65 (2004). 18. I. Freund, Opt. Commun. 249, 7 (2005). 19. I. Freund, Opt. Commun. 256, 220 (2005). 20. I. Freund, Opt. Lett. 37, 2223 (2012). 21. I. Freund, Opt. Commun. 285, 4745 (2012). 22. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994), Chap. 6.
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Observer-dependent sign inversions of polarization singularities Isaac Freund Physics Department, and Jack and Pearl Resnick Advanced Technology Institute, Bar-Ilan University, Ramat-Gan ISL52900, Israel ([email protected]) Received July 28, 2014; revised September 14, 2014; accepted September 15, 2014; posted September 16, 2014 (Doc. ID 219897); published October 9, 2014 We describe observer-dependent sign inversions of the topological charges of vector field polarization singularities: C points (points of circular polarization), L points (points of linear polarization), and two virtually unknown singularities we call γ C and αL points. In all cases, the sign of the charge seen by an observer can change as she changes the direction from which she views the singularity. Analytic formulas are given for all C and all L point sign inversions. © 2014 Optical Society of America OCIS codes: (260.0260) Physical optics; (260.6042) Singular optics; (260.5430) Polarization. http://dx.doi.org/10.1364/OL.39.005873
The polarization singularities of vector optical fields include lines of circular polarization, C lines that meander throughout the field, and lines of linear polarization, L lines [1–16]. In the nonparaxial, three-dimensional (3D) fields of interest here, also L lines meander throughout the field, whereas in paraxial fields these lines are seen as lines in a plane where L surfaces intersect the plane [1,4,6]. In 3D fields, where a C line or an L line pierces a plane, here the observation plane P, a point of circular or linear polarization, a C point or an L point, appears. C and L lines are observer-independent, invariant features of the optical field. A remarkable property of these lines is that the polarization ellipses that surround every C or L point on the line rotate about the point [1–16]. Perhaps no less remarkable is the fact that as the observer changes the direction from which she views the singularity (her line of sight, LOS) she finds that the sense of these rotations are seen to change. For one LOS, the observer may see the surrounding ellipses rotate around the singularity in the counter-clockwise direction, whereas for another LOS that differs infinitesmally from the first, she sees clockwise rotations. These inversions are a fundamental, intrinsic property of the 3D structure of light. Sign inversions of C and L points were first found by Nye and Hajnal [4] for a particular LOS (discussed in detail later); a quantitative theory for this LOS has been given by Berry [7]. Here we discuss the general case, and present analytical formulas for all directions of observation. We also introduce two classes of polarization singularities, γ C points on C lines, and αL points on L lines, and describe their sign inversions. C points are singular because the major axis α and minor axis β of the polarization ellipse are undefined when the ellipse degenerates into a circle, the C circle; L points are singular because the minor axis β and the ellipse normal γ are undefined for a linear polarization figure. Axis α, however, remains well defined and is along the direction of linear polarization. The projections onto P of the corresponding axes of the surrounding polarization ellipses whose centers lie in P rotate around C and L points with signed net winding, i.e., rotation, angle ΘC π and associated signed topological index, winding 0146-9592/14/205873-04$15.00/0
number, or “charge,” I C ΘC ∕2π 1∕2 for C points, ΘL 2π, and I L ΘL ∕2π 1 for L points. When the polarization ellipse is viewed down one of its principal axes, α, β, or γ, the projections onto P of the corresponding axes of the surrounding polarization ellipses whose centers lie in P are also seen to rotate around the point with index I O 1 [17–19]. Thus, all ordinary points are seen as singularities for three directions of observation; we call these directional singularities “O points” [20,21]. When the LOS is along α, β, or γ, we describe the O point as an αO , βO , or γ O point. If the central point is a C point, rather than an ordinary point, we emphasize this by writing γ C . Similarly, if the central point is an L point, we emphasize this by writing αL . Figure 1 reviews basic properties of the various singularities. In what follows, we specify the observer’s LOS by means of a unit vector n (the normal to the observation plane P) that points from the singular point to the observer.
Fig. 1. Singularity streamline patterns. Streamlines formed by the projected axes of the surrounding ellipses are shown by thin lines, separatixes, here Λ lines, that separate regions of opposite curvature are shown by thick lines. Throughout, positive/ negative half index singularities are shown by single white/ black circles, integer index singularities by nested circles. These patterns are characterized by index I and the number of Λ lines, N Λ . (a)–(a″) C point patterns [2]. (a) “ Lemon,” I C 1∕2, N Λ 1. (a′) “Monstar,” I C 1∕2, N Λ 3. (a″) “Star,” I C −1∕2, N Λ 3. (b)–(b″) L, O, γ C and αL point patterns [12,21]. here I I L;O;γC ;αL . (b) “ Spiral,” I 1, N Λ 0. (b′) “Sink” or “node,” I 1, N Λ 4. (b″) “Saddle,” I −1, N Λ 4. © 2014 Optical Society of America
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In Fig. 2, we present a C point index sphere that shows the changes in I C of a given C point as the point is viewed from different directions; index spheres for axes α and β are always identical. In the representation of Fig. 2(b), the C circle corresponds to the equator. As can be seen from this figure, on any path on the sphere that connects two antipodal points, index I C inverts an even number of times. This occurs because for antipodal points the observer views the same plane from opposing directions: say, from the “top” for a point in the northern hemisphere, and from the “bottom” for its antipodal partner in the southern hemisphere. On a line of constant longitude, for example, there are two inversions: one inversion [on the tilted great circle in Fig. 2(b)] occurs when n is perpendicular to the tangent to the C line at the C point, and therefore, P is tangential to the line; the other inversion [on the equator in Fig. 2(b)] occurs when n is perpendicular to axis γ and therefore lies in the plane of the C circle. The degeneracy produced by the intersection of these two great circles is an essential one that is not lifted by perturbation. All C points generate qualitatively similar index spheres; a number of typical examples are shown in Fig. 2(c) using a Cartesian representation. In all cases, only monstars can transform into stars, and vice versa, so that preceding an inversion lemons always transform into monstars. One can understand why sign inversion occurs when P is tangential to the C line by considering what an observer sees as she translates P relative to the line. Before P intercepts the line, she sees no topological charge since there are no C points. Charge conservation requires that the net charge remain zero also when P first contacts the line, i.e., is tangent to the line. Now, the charge
Fig. 2. Dependence of I C on the direction of observation. (a) Spherical coordinates θ, φ. (b) Index sphere, the north pole (N) is along the z axis in (a). Light areas correspond to I C 1∕2, dark areas to I C −1∕2. (c) Cartesian representations of I C θ; φ for three typical examples. The interested reader may enjoy sketching possible vector triplets n; γ; ω , Eqs. (1), for different points in these plots.
seen in the tangent plane is a property of the plane, and is therefore always seen to be zero whenever P is tangential to the line, including the case in which P rotates around a C point on the line. In generic fields, a nonzero charge dropping to zero under a smooth, continuous rotational transformation heralds sign inversion, as is observed when P rotates through the tangent to the C line. Sign inversion also occurs when n is perpendicular to γ, because the sign of I C depends on the handedness of the light (right or left) as seen by the observer: if γ points into the observer’s half-space, she sees right-handed polarization, whereas if γ points away from her halfspace she sees left-handed polarization. When n is perpendicular to γ, the handedness seen by the observer is undefined; in generic fields this ambiguity heralds inversion of the handedness, and therefore inversion of the sign of I C , as n rotates through a plane normal to γ. This definition of handedness for 3D fields in which there are partial waves traveling in many different direction is consistent with that used in paraxial fields; it appears to differ from a definition proposed by Nye [1]. We have developed a simple formula that describes I C at all points on the index sphere: I C 1∕2signn · ωsignn · γ;
(1a)
ω ∇RE · E × ∇IE · E;
(1b)
γ IE × E:
(1c)
Equations (1) are to be evaluated at the C point. The vorticity ω is a continuous vector field introduced by Berry and Dennis [5]; on the C line itself ω lies along the tangent to the line. All data presented here (Figs. 1–4) were obtained using simulations that are exact solutions of Maxwell’s equations, and local expansions of these equations [17–21]. The white lines (inversion curves) in Figs. 2(b) and 2(c) were calculated from Eqs. (1), and the color-coded regions show the numerically measured signs of I C . We have verified Eqs. (1) for 106 independent realizations of C points on C lines in random optical fields by measuring the index on both sides of both types of inversion curves in Figs. 2(b) and 2(c). These results also confirm that the only inversion points are those described by Eqs. (1). The index theorem [22] permits a change in the net charge of a bounded region only if charges enter or leave the region, i.e., cross the boundary. In accord with this requirement, an observer that examines P as it rotates through an inversion point sees a chemical-like reaction involving one-or-more additional singularities entering and/or leaving the field of view. The generic sign inversion reactions that are seen for C points are illustrated in Fig. 3. L point sign inversions are similar to C point sign inversions: sign inversion occurs when the observation plane is tangent to the parent L line, and when it contains the direction of linear polarization. Proceeding as for I C , we obtain for I L ,
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Fig. 3. Generic sign inversion reactions of C points. (a)– (a″) Sign inversion of an initial positive (I C 1∕2) C point as the observation plane P rotates through the tangent to the parent C line. The example shown here is for the case in which the observer studies the rotation of axis α of the ellipses that surround the central C point. The orientation of these axes projected onto P is shown gray-scale coded 0 to π, black to white. In (a), as P approaches the tangent, in addition to the positive C point at the origin, the observer sees a negative C point (the “intruder”) enter the field of view. In (a′) she sees the negative intruder collide with, displace, and usurp the position of the initial positive C point, and in (a″) she sees the initial positive C point flee the scene. The observer sees the same sequence of events when studying axis β of the surrounding ellipses. (b′)–(b″) Sign inversion of the negative C point in (a″) as P continues to rotate, passing though γ. Panel (b) shows the negative C point as P approaches γ. In (b′) P contains γ, and the observer sees the central negative C point emit a negative αO point, whose index is I αO −1. Conserving charge, the negative C point changes sign and becomes a positive, I C 1∕2, C point. In (b″) the observer sees the emitted negative αO point float off into the surrounding sea of O points. Studying axis β of the surrounding ellipses, instead of axis α, at the positions of P corresponding to (b′)–(b″), the observer sees the following: at (b) she again sees a negative (I C −1∕2) C point at the origin, together with a positive βO point with index I βO 1 that floats in from the surrounding sea of O points; at (b′) the observer sees this βO point collide with and absorb the initial negative C point, and at (b″) she sees the end product, a positive, I C 1∕2, C point. In general, as P rotates through γ, α∕β transition combinations are seen to occur with approximately equal probabilities: emission/emission, emission/absorption, absorption/emission, and absorption/absorption.
I L signn · τsignn ·b z;
(2a)
τ ∇IE x × ∇IE y :
(2b)
Equations (2) are to be evaluated at the L point. At the point, τ is along the tangent to the L line, zˆ is along the direction of linear polarization, and E x and E y are the optical field components along an arbitrary pair of orthogonal xy-axes in the plane normal to zˆ. Expanding the components of the optical field around an L point as E x P xx iQxx x P xy iQxy y P xz iQxz z, Ey P yx iQyx x P yy iQyy y P yz iQyz z, Ez K z P zx iQzx x P zy iQzy y P zz iQzz z, we obtain for the components of τ, τx Qxy Qyz − Qxz Qyy , τy Qxz Qyx − Qxx Qyz , τz Qxx Qyy − Qxy Qyx , which follows Eq. (2b). The index spheres and inversion curves for L points are qualitatively similar to those shown for C points in
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Fig. 2. Using our simulations of random optical fields [17–21], we have verified numerically Eqs. (2) for the L point inversion curves of 106 independent L points. The sign inversion “chemical” reactions of L points seen when P passes through the tangent to the L line follow the schema in Figs. 3(a)–3(a″), except that for L points, the winding numbers are 1. L point sign inversion when P contains the direction of linear polarization, however, differs from Figs. 3(b)–3(b″). In the case of L points, the observer sees a sequence that superficially resembles Figs. 3(a)–3(a″), but with the important difference that the “usurper” is not another L point, but rather a βO or a γ O point from the surrounding sea of O points, depending upon whether the observer studies the rotations of axes β or γ of the surrounding polarization ellipses. In all cases, only sinks can transform into saddles, and vice versa, so that preceding sign inversion, spirals transform into sinks. During the directional sign inversions found by Nye and Hajnal [4], the observer continually rotates the direction of her LOS: as she traverses a C line, she maintains the observation plane P coincident with the C circle, whereas on an L line, she maintains P perpendicular to the direction of linear polarization. For C points, sign inversion of the winding angle of axes α and β of the surrounding ellipses occurs when the C circle is tangent to the C line; for L points, sign inversion of axes β and γ occurs when the direction of linear polarization is perpendicular to the L line. For both C and L points, NH inversions occur when P is tangent to the parent line, and are thus seen to be special cases of Eqs. (1) for C points and Eqs. (2) for L points. Berry has given explicit formulas for NH inversion points with their special lines of sight [7]. In Eqs. (1) and (2), we generalize his results to arbitrary lines of sight, and add the important class of sign inversions that occur when n is perpendicular to γ for C points, and n is perpendicular to the direction of linear polarization for L points. These latter inversions cannot occur under NH constraints. As our keen-eyed observer tracks an NH inversion, she notes that in addition to the expected rotation of axes α and β of the surrounding ellipses around C points, axis γ of the surrounding ellipses also rotate around the point with integer winding numbers 1 and streamline patterns shown in Figs. 1(b)–1(b″). She similarly finds that in addition to the expected rotation of axes β and γ of the surrounding ellipses around L points, axis α of the surrounding ellipses also rotate around these points with integer winding numbers 1 and streamline patterns shown in Figs. 1(b)–1(b″). Tracking these unexpected rotations as she moves along the lines, she encounters points on the lines at which the signs of their winding numbers invert—our observer has found two new classes of directional singularity sign inversions; she christens these γ C point inversions for C lines, αL point inversions for L lines. Similar to L points, for both γ C and αL inversions, only sinks can transform into saddles, and vice versa, so that preceding sign inversion, γ C and αL spirals transform into sinks. The observer finds that γ C and αL inversion points do not coincide with NH inversion points, so that C lines and
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Fig. 4. γ C inversions on a C line. The C line is shown by the thick curve, regions of the line that contain positive γ C points are colored gray and labeled γ C , regions of the line containing negative γ C points are colored black and labeled γ C− . The tracks of positive γ O points are shown by thin gray lines labeled γ O , tracks of negative γ O points are shown by thin black lines laC beled γ O − . Two γ inversion points, labeled A and B, respectively, are present. The directions shown for the arrowheads O on the γ O and γ − tracks are for the case in which the C line is traversed from bottom to top. If the observer traverses the line from top to bottom, she sees the same tracks, but with the directions of all arrowheads reversed.
L lines contain two distinct, often interleaved sets of directional inversion points. Closely studying these new inversions, the observer finds that as she moves along a C line, a γ O point emerges from the surrounding sea of O points and approaches the line. Similarly, the observer finds that as she moves along an L line, an αO point emerges from the surrounding sea of O points and approaches that line. Plotting the positions of these O points, here their tracks, the observer finds what appear to be lines of γ O points that intercept, and exchange charge with the C line at a γ C inversion point, and lines of αO points that intercept, and exchange charge with the L line at an αL inversion point. Typical examples of these phenomena for γ C inversions on C lines are shown in Fig. 4; analogous events are seen for αL inversions on L lines. In summary: Observer dependent sign inversions of polarization singularities have been described, and general analytical formulas have been presented for sign inversion of C points on C lines, Eqs. (1), and L points on L lines, Eqs. (2); these equations are extensions to the general case of earlier results by Berry [7] for the special case of Nye and Hajnal inversions [4]. Because of the spe-
cial conditions on the orientation of the C circle, or the direction of linear polarization, relative to the parent line, NH inversions correspond to accidental coincidences that occur at apparently random, sparse locations on the line. In contrast, as Eqs. (1) and (2) show, in the general case, every point on every C line, and every point on every L line, undergoes sign inversion in a doubly infinite set of planes. For both C and L points, one set consists of all planes that contain the tangent to the line. For C points, the second set consists of all planes that contain axis γ, whereas for L points, this second set consists of all planes that contain the direction of linear polarization. Two directional singularities, γ C points on C lines and αL points on L lines, have been introduced, and their sign inversions described. The sea of O points that surround C and L lines [20,21] has been shown to play essential roles in the sign inversions of all currently known optical polarization singularities. References 1. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999). 2. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977). 3. J. F. Nye, Proc. R. Soc. A 389, 279 (1983). 4. J. F. Nye and J. V. Hajnal, Proc. R. Soc. A 409, 21 (1987). 5. M. V. Berry and M. R. Dennis, Proc. R. Soc. A 457, 141 (2001). 6. M. R. Dennis, Opt. Commun. 213, 201 (2002). 7. M. V. Berry, J. Opt. A 6, 675 (2004). 8. R. W. Schoonover and T. D. Visser, Opt. Express 14, 5733 (2006). 9. S. Zhang, B. Hu, Y. Lockerman, P. Sebbah, and A. Z. Genack, J. Opt. Soc. Am. A 24, A33 (2007). 10. R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, Phys. Rev. Lett. 100, 103901 (2008). 11. I. Freund, Opt. Commun. 283, 1 (2010). 12. I. Freund, Opt. Commun. 283, 16 (2010). 13. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, Appl. Opt. 51, 2925 (2012). 14. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, Opt. Express 21, 8815 (2013). 15. M. S. Soskin and V. I. Vasil’ev, J. Opt. 15, 044022 (2013). 16. F. S. Roux, Opt. Lett. 38, 3895 (2013). 17. I. Freund, Opt. Commun. 242, 65 (2004). 18. I. Freund, Opt. Commun. 249, 7 (2005). 19. I. Freund, Opt. Commun. 256, 220 (2005). 20. I. Freund, Opt. Lett. 37, 2223 (2012). 21. I. Freund, Opt. Commun. 285, 4745 (2012). 22. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994), Chap. 6.
