Optical vortices (nodal lines and phase singularities) are the generic singularities of scalar optics but are unstable in vector optics. We investigate experimentally and theoretically the unfolding of a uniformly polarized optical vortex beam on propagation through a birefringent crystal and characterize the output field in terms of polarization singularities (C lines and points of circular polarization; L surfaces and lines of linear polarization). The field is described both in the 2-dimensional transverse plane, and in three dimensions, where the third is abstract, representing an optical path length propagated through the crystal. Many phenomena of singular optics, such as topological charge conservation and singularity reconnections, occur naturally in the description

Flossmann, Florian

Schwarz, Ulrich T.

Maier, Max

Dennis, Mark R.

English

Schwarz, Uli

Dennis, M.

University of Regensburg Publication Server

PRL 95, 253901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending16 DECEMBER 2005Polarization Singularities from Unfolding an Optical Vortex through a Birefringent CrystalFlorian Flossmann, Ulrich T. Schwarz,* and Max MaierNaturwissenschaftliche Fakulta¨t II-Physik, Universita¨t Regensburg, Universita¨tsstraße 31, D-93053 Regensburg, GermanyMark R. DennisSchool of Mathematics, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom(Received 9 August 2005; published 12 December 2005)0031-9007=Optical vortices (nodal lines and phase singularities) are the generic singularities of scalar optics but areunstable in vector optics. We investigate experimentally and theoretically the unfolding of a uniformlypolarized optical vortex beam on propagation through a birefringent crystal and characterize the outputfield in terms of polarization singularities (C lines and points of circular polarization; L surfaces and linesof linear polarization). The field is described both in the 2-dimensional transverse plane, and in threedimensions, where the third is abstract, representing an optical path length propagated through the crystal.Many phenomena of singular optics, such as topological charge conservation and singularity reconnec-tions, occur naturally in the description.DOI: 10.1103/PhysRevLett.95.253901 PACS numbers: 42.25.Ja, 02.40.Xx, 03.65.Vf, 42.25.BsTopological defects are of significance in wave physicsbecause of their stability and persistence upon propagationin time or space [1,2]. This stability is due to their localstructure, which organizes the topology of the surroundingfield, and depends delicately on the symmetries of thesystem. In optical beams, different quantities often havedifferent symmetries: when beams carry both orbital andspin angular momentum (as is the case with the rotationalDoppler effect [3]), the total field intensity has cylindricalsymmetry, but the polarization does not.Here, we investigate the generic unfolding of the naturaldefects of scalar optics, namely, optical vortices (alsocalled phase singularities, nodal lines, or wave disloca-tions) [2], when a light beam propagates through a bire-fringent crystal. The uniform polarization of the incidentbeam is resolved into the two orthogonal linear polariza-tions according to the crystal’s birefringence, which propa-gate in different directions due to extraordinary refraction.The resulting interference of the two waves gives rise to acomplicated space-dependent polarization pattern, whichwe characterize in terms of the generic singularities ofpolarization optics, namely, L surfaces where the polariza-tion is linear and C lines where the polarization is circular[2,4,5]. These singularities intersect the transverse plane ofobservation along L lines and C points, respectively.Optical vortex lines are significant throughout optics, inpolarized components of the electromagnetic field. Theyare places where the complex scalar amplitude describingthe field is zero, and the phase change around the vortex isquantized in integer units (positive or negative) of 2(‘‘charge’’) [2]. Examples are vortices in laser beams,which can be tightly controlled by holograms, such as inthe creation of vortex knots and braids [6,7], as well asnaturally tangling in random speckle patterns [8]. Opticalvortices often studied in laser beams carry orbital angularmomentum (see, e.g., Refs. [2,3,9]) and are equivalentto quantized vortices in Bose-Einstein condensates [10],05=95(25)=253901(4)$23.00 25390superconductors, and superfluids [11]. Polarization singu-larities occur in fields where the state of (generally elliptic)transverse polarization varies with position. L lines andsurfaces separate regions of opposite handedness [2], and Clines and points are characterized by a disclination index1=2, analogous to defects in nematic liquid crystals[2,12]. These singularities occur naturally in polarizationspeckle fields [5,13], near field micro-optics [14], andskylight [15]. Polarization singularities in crystal opticshave been investigated previously [16,17], although appar-ently not with incident beams containing vortices.We find, when a monochromatic, uniformly polarizedlight beam with an embedded optical vortex propagatesthrough a birefringent crystal, that the emerging field’spolarization has a network of singularities with a simple,universal structure: a double helix of C lines, and an Lsurface topologically equivalent to Riemann’s periodicminimal surface. This structure is 3 dimensional; in addi-tion to transverse x; y structure of the emerging field, thereis an abstract third dimension given by optical path lengthpropagated through the crystal, determining the phase shiftbetween the two crystal eigenpolarizations. The physicalorigin of the effect is explained due to interference of theseeigenpolarizations, and the pattern is explored and verifiedexperimentally.The incident field is Einx; y   x; ydin, where the 2-dimensional complex vector din represents a uniform, pos-sibly elliptic state of transverse polarization, and the com-plex scalar  x; y is the amplitude. In the direction ofpropagation z, a transparent nonchiral crystal has twoorthogonal linear eigenpolarizations d1;d2 with refractiveindices n1; n2, distinct when away from the optic axes ofthe crystal (whether uniaxial or biaxial). Ein is resolvedinto the two crystal eigenpolarizations which are refractedappropriately, resulting, in the emerging beam, in a spatialshift sj (assumed to be in the x direction), includingbeam walk-off for the extraordinary beam(s). A relative1-1 © 2005 The American Physical SocietyPRL 95, 253901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending16 DECEMBER 2005phase shift , dependent on n2  n1 times the distance, isaccumulated during propagation through the crystal. Thesethree effects give an emerging polarization pattern of theformE x; y;  Xj1;2 x sj; ydin  djdjei1j=2 (1)up to an unimportant spatial translation. We are interestedin the case when  contains a singly quantized vortex, suchas a Laguerre-Gauss LG10 beam, represented in the focalplane by LG10x; y  x iy expx2  y2=w20	 (2)(w0 is the waist width). For an elliptically polarized inci-dent beam, din can be written d1 cos d2 sinei, withellipticity "  sin2 sin [see Fig. 1(a)]. From (1),   0results merely in a constant phase shift to the second term,so we need only to consider din linearly polarized, din d1 cosin  d2 sinin.In our experiment, a linearly polarized LG10 beam prop-agates through a uniaxial crystal, with the optic axis in thex; z plane [see Fig. 1(b)]. The topological agreementbetween the experiment and model (1) is sufficiently closethat the physical discussion can be illustrated using experi-mental results.Our experimental setup is represented in Fig. 1(c). ALaguerre-Gauss beam  LG10 with uniform linear polariza-tion din is synthesized from a He-Ne beam, using a polar-izer and spatial light modulator (SLM). Higher diffrac-tion orders from the SLM are removed by a telescopeand a pinhole. The KDP birefringent crystal is locatedFIG. 1 (color online). (a) Definition of ellipse parameters [cf.Eqs. (4) and (5)]:  is the angle of the major axis, tan is theratio of the y and x components of the field, and the ellipticity is"  sin2!; (b) orientation of a uniaxial crystal, and ordinary andextraordinary rays (angles are exaggerated); (c) experimentalsetup scheme (HeNe: laser; SLM: spatial light modulator;L1   L4: lenses; P: pinhole; KDP (potassium dihydrogen phos-phate): birefringent crystal; Pol1, Pol2: linear polarizer/analyzer;=4 plate; CCD: camera). Insets (d) and (e) show the totalintensity distribution before and after the birefringent crystal,respectively.25390in the waist plane, where  has the form (2) with w0 0:7 mm. Because of the large Rayleigh range (zR 2:4 m), phase front curvature, Gouy phase shift, and dif-fraction can be neglected over the change of optical pathlength relevant for the effects we describe.Our sample of uniaxial KDP has dimensions x yz  20 20 26 mm2 cut at 40 to the optic axis. In ourexperiment, the incoming beam is at normal incidence. Theordinary polarization d1 is 0; 1, the extraordinary polar-ization d2  1; 0. The relative phase shift is  2n2  n1l=vac: (3)Here n1  n0  1:507 for the ordinary and n2  n 40  1:490 for the extraordinary beam. We vary  ex-perimentally by rotating the crystal about the y axis, whichdoes not change d1;2; one  period corresponds to  0:056 [see Fig. 1(b)]. The optical path length changes dueto the change in n; the change in geometric path length lis negligible. We measure the outgoing Stokes parametersat 75 positions equally spaced through   0    4. Therelative transverse shift x  s2  s1  0:7 mm is deter-mined by the walk-off angle   0    1:52, where is the angle of normal velocity with respect to the opticaxis, and 0 is the angle of the ray velocity. For the smallrotation  the transverse shift x is effectively constant.The field emerging from the crystal can therefore be ap-proximated by Eq. (1) for a range of .For each x; y;, the four Stokes parameters S0    S3describing the polarization state are reconstructed fromseven intensity measurements by a CCD camera withdifferent combinations of a polarizer and a =4 plate[18]. The geometric quantities (angle  and ellipticity ")of the elliptic polarization state are determined by  12 argS1  iS2 12 argfI90  I0  iI45  I135g; (4)"  S3=S0  IRCP  ILCP=S0: (5)The subscript of I denotes the orientation of the analyzerand right- and left-handed circular polarization, RCP andLCP, determined using a =4 plate. We will describe thex; y; configuration of the emerging polarization field,which is further parametrized by the state din of initialpolarization. We begin by describing the 2-dimensionalfield for a fixed constant .We plot the intensity and polarization of a typical trans-verse plane of the emerging field, with phase shift 0 0:9 and linear input polarization in  40, in Fig. 2.Unlike the incident beam, the intensity vanishes nowhere,although there are minima near the zeros in the two shiftedcomponents. There are four isolated C points, and a closedloop L line, which encloses a region of right-handed ellip-tic polarization (the elliptic polarization in the rest of thefigure is left-handed). The C points are marked by coloredcircles which correspond to the 3-dimensional C lines,discussed later in Fig. 4. The original vortex is refracted1-2FIG. 3 (color). Two equivalent representations of the measuredpolarization angle , for the case plotted in Fig. 2. (a) Contourplot of ; black curves are isogyres; (b) polarization streamlines( is the curve tangent angle). The upper and lower C points arelemon-type, with index 1=2, the two on the left and right arestars, with index 1=2. The L line is shown in yellow, and blacksquares mark the positions of the doubly refracted originalvortex.FIG. 2 (color). Intensity and polarization in the x; y plane ofthe output field E, for an arbitrary choice   0  0:9. Theinput beam is linearly polarized at in  40. Polarizationellipses and axes are superimposed, plotted with major axesnormalized. The four C points of circular polarization aremarked by colored circles, and the L line loop of linear polar-ization is represented by a solid yellow line. Polarization is left-handed except within the L loop, where it is right-handed. Theblack squares are the positions of the original vortex after doublerefraction.PRL 95, 253901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending16 DECEMBER 2005differently in the two components, emerging at the twopoints (black squares) on the L line where the polarizationis purely parallel to d1 and d2. Since the central vortex isthe only zero in  LG10 , from (2), these are the only points onthe L line loop with these polarization angles.The singular nature of C points is emphasized inFig. 3(a), where isogyres (contours of constant ) areplotted. These isogyres end on the C points (where  isnot defined), and the  sense of  increase about the Cpoint determines its index, 1=2 [4,5]. This index’s half-integer nature is due to the -rotational symmetry of a(noncircular) ellipse, analogous to liquid crystal disclina-tion defects [12]. In Figs. 2 and 3, the upper and lower Cpoints (including the right-handed one) have index 1=2,the others 1=2. The geometry of the  pattern is plottedin Fig. 3(b), as a set of polarization streamlines. Here, thetangent to the curve through a point is the direction of .The two 1=2 C points have a ‘‘lemon’’-like geometry, the1=2 points have 3-fold ‘‘star’’ structure [2,4,5].The total index of the C points of E in Figs. 2 and 3agrees with the total topological charge of the incoming(scalar) field Ein   din, as expected in the unfolding of atopological singularity. This can be seen using the complexpolarization scalar ’  E E, which contains both polar-ization and phase information [5,19]. ’ has phase singu-larities at the C points of E, with (integer) charge equal tothe disclination index sign times the handedness sign (for right-handed, for left-handed). Evidently, ’ of thefield in Figs. 2 and 3 has 3 positive singularities (one right-handed lemon and two left-handed stars) and one negativesingularity [one left-handed lemon; lower red circle inFig. 3(b)]: the net topological charge is 2. For the in-25390coming field, ’in  Ein Ein   2din  din also has acharge of 2 (provided the incoming polarization is notcircular); topological charge is preserved as the beampropagates through the crystal.The effect on the fields of Figs. 2 and 3 of increasing by  (i.e.,   0  ) is equivalent to the action of a=2 plate: "! " (handedness reversed) and ! .Thus, in the 0   plane, C points have the same x; ycoordinates, but opposite handedness and disclination in-dex. The total vector field singularity index (handednesstimes index, summed over the C points) is unchanged.Properties of the full 3-dimensional structure can beinferred from these observations. C lines and L surfacesmeasured over a range of   0    4 for the input polar-izations in  40 (as in Figs. 2 and 3) and in  45 areplotted in Figs. 4(a) and 4(b). For typical fixed  (as inFigs. 2 and 3), there is a dominant polarization handedness.However, after half a  period, this dominant handednessmust change, leading to alternating handedness layers. Thepresence of the C point in the island of right-handedpolarization in Fig. 2, corresponding to the green line inFig. 4(a), implies that layers of the same handedness areconnected. The polarization at the points corresponding tothe refracted vortex in the two waves [i.e., y  0 and x sj in (1)] is linear for all . In x; y; space, volumes ofopposite handedness are separated by a single, connected Lsurface (see Fig. 4). This L surface is structurally stableand topologically equivalent to Riemann’s periodic mini-mal surface [20].Because of the conserved topological charge of 2 and theearlier observations, there are evidently two continuous Clines of opposite disclination index and handedness thread-ing through the periodic structure, as shown by the blueand green lines in Fig. 4(a). The =2 symmetry in implies that the C lines coil around each other in a doublehelix. The other two C points (red circles) of Fig. 2, withthe same handedness but opposite index, lie on the same C1-3FIG. 4 (color). Experimentally determined configuration ofpolarization singularities (L surface and C lines) in x; y;space for input polarizations (a) in  40 and (b) in  45. increases from 0 to 4, the L surface is plotted from 0 to 2.In (a) all C lines are disconnected; in (b) reconnections occur.PRL 95, 253901 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending16 DECEMBER 2005line which intersects the 0 plane twice. This C line isconfined to that handedness layer: this does not threadthrough the structure and does not increase monotonicallywith , but extends to 1 in the y direction. There areregions of  where this C line does not intersect thetransverse plane.The sense of twist of the C line double helix is deter-mined by which of d1;2 the incident din is closer to; itswitches when in  45. As in approaches this symmet-ric configuration, two pairs of C lines of the same handed-ness approach at intervals of  in , and reconnectionsoccur at in  45; such topological reactions occur stablyin three dimensions as a fourth parameter is varied [7,9,21].The symmetric configuration with reconnections is shownin Fig. 4(b). The topology of the C lines in the symmetricand nonsymmetric arrangements is reminiscent of thebraided vortices construction of Ref. [7]. That specificsuperposition shares some similarity to what occurs herenaturally; however, there are only two coiled C lines here,rather than three necessary to form a braid.We have described in detail the 2- and 3-dimensionaltopological structure of the polarization singularities of a25390scalar vortex beam propagated through a birefringent crys-tal. The physical origin of the structure has been described,and the features have been observed and illustrated experi-mentally. Many features of singular optics (e.g., conserva-tion of topological index, stability of structures, andreconnections) occur naturally in this simple physical situ-ation, showing the topological interplay between scalar andvector optics. The polarization singularity structure wouldbe more complicated through a more complicated crystal(with optical activity and dichroism), or with nonmono-chromatic light [16].1-4*Electronic address: ulrich.schwarz@physik.uni-r.de[1] J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).[2] J. F. Nye, Natural Focusing and Fine Structure of Light(Institute of Physics Publishing, Bristol, 1999).[3] J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, andM. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).[4] J. F. Nye, Proc. R. Soc. A 389, 279 (1983).[5] M. R. Dennis, Opt. Commun. 213, 201 (2002).[6] J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett,Nature (London) 432, 165 (2004).[7] M. R. Dennis, New J. Phys. 5, 134 (2003).[8] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 456, 2059(2000).[9] G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner,and E. M. Wright, Phys. Rev. Lett. 87, 023902 (2001).[10] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall,C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498(1999).[11] R. P. Feynman, in Progress in Low Temperature Physics,edited by C. J. Gorter (North-Holland, Amsterdam, 1955),Vol. 1, p. 17.[12] P. G. de Gennes and J. Prost, The Physics of LiquidCrystals (Oxford University Press, New York, 1993).[13] M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6,S281 (2004).[14] A. Nesci, R. Da¨ndliker, M. Salt, and H. P. Herzig, Opt.Commun. 205, 229 (2002).[15] M. V. Berry, M. R. Dennis, and R. L. Lee, New J. Phys.6, 162 (2004).[16] Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A6, S217 (2004); A. V. Volyar and T. A. Fadeyeva, Tech.Phys. Lett. 29, 111 (2003).[17] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 459, 1261(2003).[18] D. S. Kliger, J. W. Lewis, and C. E. Randall, PolarizedLight in Optics and Spectroscopy (Academic Press,Boston, 1990).[19] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 457, 141(2001).[20] B. Riemann, in Bernhard Riemann’s Gesammelte Mathe-matische Werke und Wissenschaftlicher Nachlass, editedby H. Weber (Teubner, Leipzig, 1892), p. 301.[21] J. F. Nye, J. Opt. A 6, S251 (2004).

Polarization singularities from unfolding an optical vortex through a birefringent crystal

Southampton (e-Prints Soton)

Flossmann, F

Schwarz, UT

Maier, M

Dennis, MR

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Commercial pressures on time-to-market often require the development of software in situations where deadlines are very tight and non-negotiable. This type of development can be termed ‘time-constrained software development.’ The need to compress development timescales influences both the software process and the way it is managed. Conventional approaches to modelling tend to treat the development process as being linear, sequential and static. Whereas, the processes used to achieve timescale compression in industry are iterative, concurrent and dynamic. That is, they replace the notion of ‘right-first-time’ with one of ‘right-on-time.’ In this paper we propose a new modelling technique, called Capacity-Based Scheduling (CBS), to control risk across a portfolio of time-constrained projects. We show how schedule constraints can be modelled in order to predict the consequences of alternative plans and control schedule risk across a portfolio of time-constrained projects

Powell, A.

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Polarization Singularities from Unfolding an Optical Vortex through a Birefringent Crystal

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Polarization singularities from unfolding an optical vortex through a birefringent crystal

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