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The Dipole Vortex Optics Communications 231 (2004) 115–128 www.elsevier.com/locate/optcom The dipole vortex Henk F. Arnoldus *, John T. Foley Department of Physics and Astronomy, Mississippi State University, P.O. Drawer 5167, Mississippi State, MS 39762-5167, USA Received 3 October 2003; accepted 4 December 2003 Abstract We show that the field lines of the Poynting vector of the radiation field of an electric dipole are vortices if the radiation carries angular momentum. When such a dipole is located near the surface of a perfect conductor, it induces a current density on the surface, and it is shown that the field line pattern of this current density consists of infinite spirals. We have identified a Master Spiral to which all field line spirals converge asymptotically. It is also shown that the field lines of the Poynting vector of the radiation field near the surface contain a vortex. Ó 2003 Elsevier B.V. All rights reserved. PACS: 03.50.De; 42.25.Bs Keywords: Dipole radiation; Optical vortex; Mirror dipole; Singular circles; Master spiral 1. Introduction reflected waves. The earliest example is the dif- fraction of a plane wave by a half-infinite screen, A singular point in an optical radiation field is a where vortices appear at the illuminated side of the point where the amplitude of the field vanishes, and screen [2]. More recently, it was found that vortices hence the phase in that point is undefined. For a occur in the diffracted field of a plane wave by a slit long time, such phase singularities were considered in a screen [3,4], and in interference between three more of a curiosity, until Nye and Berry [1] showed plane waves [5]. Another example is the field that singular points appear in a radiation field quite structure in the focal plane of a focusing lens, naturally, as phase dislocations in traveling waves. where the energy flow exhibits vortex lines [6,7]. It A particularly interesting phenomenon is the op- seems that optical vortices which appear in the field tical vortex, the center of which is a singular point. of a Laguerre–Gaussian (laser) beam are the most Such vortices appear, for instance, in interference widely studied [8–11]. patterns between incident waves and diffracted or In an optical vortex, the field lines of the Po- ynting vector, representing the energy flow, circu- late around the singular point (two dimensions) or * Corresponding author. Tel.: +1-662-325-2919; fax: +1-662- 325-8898. around a vortex axis (three dimensions). There E-mail addresses: [email protected] (H.F. Arnoldus), seems to be an intimate relation between the [email protected] (J.T. Foley). angular momentum of the light [12–15], and the 0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.12.043 116 H.F. Arnoldus, J.T. Foley / Optics Communications 231 (2004) 115–128 existence of vortices in the radiation field. Al- terms of the complex amplitudes of the electric and though this relation is not clear in general, it is well magnetic fields as established that for the Laguerre–Gaussian beam 1 à vortices are only observed for modes that carry SðrÞ¼ Re½EðrÞBðrÞ ; ð6Þ 2l0 angular momentum [16,17]. Conversely, if a radi- where we have dropped terms that oscillate at ation field carries angular momentum, one would twice the optical frequency. For time harmonic expect that there is a possibility that vortices are fields, the Poynting vector is time independent, so present. In this paper we shall show that for the we write SðrÞ instead of Sðr; tÞ. We then substitute field of the simplest and most important radiating the right-hand sides of Eqs. (3) and (5), and work system, the electric dipole, the appearance of a out the cross product. This yields vortex and the presence of angular momentum in ck6 the field go hand-in-hand. SðrÞ¼ ½d Á dà ðd Á ^rÞðdà Á ^rÞ^r 32p2e q2 0 2 1 2. Dipole radiation À 1 þ Im½ðd Á ^rÞdà : ð7Þ q q2 We consider an electric dipole moment dðtÞ, The significance of the Poynting vector is that with complex amplitude d and oscillating har- n^ Á SðrÞdA,withn^ a unit vector perpendicular to monically with angular frequency x: the surface element dA represents the energy per dðtÞ¼Re½deÀixt: ð1Þ unit of time flowing through dA into the direction of n^. We now consider a sphere with radius R, and The electric field, emitted by the dipole, is written centered at the origin. We then have n^ ¼ ^r, which as gives Im½ðd Á ^rÞdÃÁ^r ¼ 0, showing that the second Eðr; tÞ¼Re½EðrÞeÀixt; ð2Þ term in braces in Eq. (7) does not contribute to the and when the dipole is located at the origin of radial power flow. Since q ¼ kR on the sphere and 2 coordinates, the complex amplitude EðrÞ is given dA ¼ R sin hdhd/ in spherical coordinates, we see by [18] that ^r Á SðrÞdA is independent of R. We then obtain for the emitted power by the dipole k3 1 1 EðrÞ¼ d ðd Á^rÞ^r þ À i ½3ðd Á^rÞ^r À d eiq; 3 dUem k à 4pe0q q q ¼ ^r Á SðrÞdA ¼ x d Á d; ð8Þ dt 12pe ð3Þ 0 a well-known result [19], which is usually derived with k ¼ x=c. The dimensionless radial distance by only considering the fields for R large (far field). between the dipole and the field point r is q ¼ kr, and ^r is the radial unit vector. The corresponding magnetic field is 4. Spherical unit vectors Àixt Bðr; tÞ¼Re½BðrÞe ; ð4Þ The Poynting vector SðrÞ, Eq. (7), has a radial part, proportional to ^r, and a part that depends on with complex amplitude the orientation of the complex amplitude d of the 3 1 k i iq dipole. We shall now assume that d is proportional BðrÞ¼ 1 þ ð^r  dÞe : ð5Þ to a spherical unit vector c 4pe0q q iw d ¼ d0e es; s ¼1; 0; 1; ð9Þ 3. Poynting vector and emitted power with d0 > 0 and w an overall phase. When the di- pole is an atom, the emitted radiation is fluores- The Poynting vector is defined as Sðr; tÞ¼ cence and the dipole moment has the form as in Eðr; tÞBðr; tÞ=l0 in terms of the time dependent Eq. (9). For s ¼ 0 we have a Dm ¼ 0 transition and fields. With Eqs. (2) and (4) we can express this in s ¼1 corresponds to a Dm ¼ 1 transition. The H.F. Arnoldus, J.T. Foley / Optics Communications 231 (2004) 115–128 117 helicity s refers to a preferred direction, taken to At any field point, SðrÞ is proportional to ^r, with be the z-axis, and for a quantum field this is the the constant of proportionality positive. There- quantization axis. For s ¼ 0 we have e0 ¼ ez,and fore, the field lines are straight lines, starting at the with Eq. (1) this gives for the dipole moment origin and radially outward. dðtÞ¼d0ez cosðxt À wÞ, representing a linear di- More interesting is the case of a rotating dipole pole directed along the z-axis. moment in the xy-plane. The field lines of the For s ¼1 the unit vectors are defined as vector field SðrÞ are curves rðuÞ, with u a dummy parameter, for which at each point of the curve, 1ffiffiffi eÆ1 ¼ p ðex À ieyÞ; ð10Þ 2 SðrÞis the tangent vector of the curve. Therefore, the field lines are the solutions of and the corresponding dipole moment is dr d0ffiffiffi ¼ SðrÞ: ð17Þ dðtÞ¼p ½ex cosðxt À wÞey sinðxt À wÞ: du 2 In spherical coordinates this is equivalent to the 11 ð Þ following set of three coupled differential equations For s ¼ 1 the dipole moment rotates counter- for the coordinates r, h and / as a function of u: clockwise (from þx to þy) in the xy-plane, and for dr s ¼1 the rotation is clockwise. ¼ SðrÞ^r; ð18Þ The spherical unit vectors are normalized as du à à 2 es Á es ¼ 1, and therefore we have d Á d ¼ d0. For the emitted power, Eq. (8), we shall write P0, which is dh r ¼ SðrÞeh; ð19Þ k3d2 du P ¼ x 0 : ð12Þ 0 12pe 0 d/ r sin h ¼ SðrÞe : ð20Þ Furthermore, it follows by inspection that du / à À2Im½ðes Á ^rÞes ¼se/ sin h ð13Þ A great simplification in the computation of field in spherical coordinates, and therefore the Poyn- lines arises if one realizes that the vector fields SðrÞ ting vector becomes and f ðrÞSðrÞ, with f ðrÞ an arbitrary positive function of r, have the same field lines. For in- 3k2P 0 ^ à ^ ^ stance, if one would take f ðrÞ¼1=jSðrÞj, then the SðrÞ¼ 2 ½1 ðes Á rÞðes Á rÞr 8pq parameter u equals the arc length of the field line. s 1 For the problem at hand, we take f ðrÞ as þ 1 þ sin he/ : ð14Þ q q2 3k3P 1 À1 The radial part can be simplified as f ðrÞ¼ 0 1 À sin2 h : ð21Þ 8pq2 2 sin2 h; s ¼ 0; 1 ðe Á ^rÞðeà Á ^rÞ¼ s s 1 À 1 sin2 h; s ¼1: 2 The set of Eqs. (18)–(20) for the field lines then ð15Þ becomes dq 5. Field lines of the Poynting vector ¼ 1; ð22Þ du For a dipole with a dipole moment along the z- dh axis, the Poynting vector is given by Eq.
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