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Optical Vortices and the Flow of Their Angular Momentum in a Multimode Fiber

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Optical Vortices and the Flow of Their Angular Momentum in a Multimode Fiber Ô³çèêà íàï³âïðîâ³äíèê³â, êâàíòîâà òà îïòîåëåêòðîí³êà. 1998. Ò. 1, ¹ 1. Ñ. 82-89. Semiconductor Physics, Quantum Electronics & Optoelectronics. 1998. V. 1, N 1. P. 82-89. ÓÄÊ 535.2, PACS 4265.Sf, 4250.Vk Optical vortices and the flow of their angular momentum in a multimode fiber A. N. Alexeyev, T. A. Fadeyeva, A. V. Volyar Physical Department, Simferopol State University, Yaltinskaya 4, 333036 Simferopol, Ukraine M. S. Soskin Institute of Physics, NAS Ukraine, 46 prospekt Nauki, Kyiv, 252028, Ukraine Abstract. The problem of propagation of optical vortices in multimode fibers is considered. The struc- tural changes experienced by the wave and ray surfaces in their transformation from the free space to a fiber medium are determined. The continuity equation is obtained for the flow of the vortex angular momentum in an unhomogeneous medium. Keywords: optical vortex, multimode fiber, angular momentum, continuity equation. Paper received 23.06.98; revised manuscript received 14.08.98; accepted for publication 28.10.98. I. Introduction Studies of the properties of single optical vortices in optical fibers have been started quite recently due to devel- Defects of the radiation wavefront in a multimode fiber are opment of low mode fiber applications, techniques of selec- usually associated with a distortion of the field structure. tive excitation [9] and radiation field isolation from a fiber Thus, it can be concluded that such effects hinder the use of [10, 11]. There is certain difference between optical vorti- multimode fibers in optical communication lines and sen- ces in a fiber and in the free space. First of all, it applies to sors. However, a thorough study of the nature of wave defi- σ the relation between the light wave polarization z and its ciency makes us take a new look on the problem of data σ topological charge l. The helicity z characterizes the direc- transmission through a multimode optical fiber, where waves tion of vector rotation of the electric (magnetic) field of a with wavefront defects play the main role. σ σ light beam ( z = +1 is right and z = –1 is left circular polari- Defects of a scalar wave field structure were explicitly zation). studied in [1] by J. Nay and M. Berry. They classify such For a paraxial Gaussian beam in the free space, it is pos- defects by dividing them into purely screw, purely edge and sible to change the values and signs of the topological charge mixed screw-edge dislocations of the wavefront. This clas- σ l and helicity z independently [11]. sification is based on the fact that the real and imaginary Amazing properties of optical vortices in the free space parts of the field strength should be simultaneously equal to were presented in [12, 13]. It was shown that an optical vor- zero tex transmits the angular momentum which can be calcu- Re[e(x, y, z)] = 0, Im[e(x, y, z)] = 0. (1) lated as The problem of light beams in the free space deals mainly =×1 MrP2∫ , (2) with the issues of generation of wavefront dislocations in- dS side laser resonators [2], on phase optical holograms [3, 4] cS or on an astigmatic mode converter [5]. Sometimes a light where r is the radius-vector, P is the Pointing vector, S is the field with a purely screw dislocation is called an optical area of beam’s cross section. When passing through the mode vortex [2]. converter, the angular momentum M is able to change its Wavefront dislocations in a multimode fiber field were value and sign [14]. first described in [6], and a correlation was found to exist Moreover, the remarkable experiments recently reported between the average number of dislocations and the number in [15] have shown that an optical vortex can trap and screw σ of fiber eigenmodes. Purely screw dislocations are experi- microscopic particles. Change of the polarization z of a mentally observed in the form of «forks» in the interference Gaussian beam may change the state of these particles [15]. pattern. Changes of external conditions result in a move- Contrary to optical vortices in the free space, guided vorti- ment, birth, and death of random dislocations in the optical ces in an optical fiber are rigidly defined by the pair of num- σ fiber [7, 8]. bers: the topological charge l and helicity z. Values of l and 82 © 1998 ²íñòèòóò ô³çèêè íàï³âïðîâ³äíèê³â ÍÀÍ Óêðà¿íè K. N. Alexeyev et al.: Optical vortices and angular momentum... σ z cannot be changed independently from one another. In the polarization correction. From equations (4) and (6) it is addition, the requirement for an optical vortex to be stable possible to obtain σ ≠ is determined by the selection rule l + z 0 [18]. 3 Thus, a question arises about the possibility to use fields ()2∆2 δβ~=∇⋅∫()ρρθθ~~{}⋅∇∫ ⋅~2, (7) in optical fibers for practical purposes. In an optical fiber, a ρ 00eett / e 20θθ∞∞tttfd d set of spatially distributed optical vortices may exist simul- V θ taneously. Furthemore, by means of an optical fiber a light where ∞ is the cross section area. vortex can be placed in locations where it would be impos- The solution of equation (6) can be obtained in the form: sible to use conventional optical devices. ± () The objective of this paper is to study the properties of ee=±$exp{}ϕβ exp{}~ , (8) il Fl R i z optical vortices and their angular momenta in the field of a ± where e$ is the unit vector of the right (+) or left (–) circu- multimode optical fiber. ϕ In the second section we, consider eigenfields of an axi- lar polarization, is the azimuth coordinate, l is the azimuth index (l = 0, 1, 2, ...). The radial function F (R) is obtained ally symmetric low-mode fiber presented in a circularly po- l larized basis. The angular momentum flow, the continuity from equation [16] equation for the angular momentum flow, and correlation 2 1 2 +−+−~22()=0 with that in the free space are analyzed in the third, fourth d2 d l 2 , (9) UVfFR and fifth sections of this paper. dR R dR R where ~ is the waveguide mode parameter in the fiber core, 2. Guided vortices in an optical fiber determinedU from the boundary conditions. Consider the peculiarities of the propagation of circularly Taking into account in (8) particular solutions of equa- polarized waves in a locally isotropic axially symmetrical tions (7), its can be shown that there are three groups of medium of a multimode optical fiber with the refractive in- eigenmodes [17]: dex: 1) Circularly polarized homogeneous optical CV vortices 2 2 for l = 1, 2, 3... (“CV” is for circular vortex [9]) n (R) = n (1 – 2∆f(R)), (3) co where n is the refractive index along the fiber axis for R = =±$±() {}ϕ co eet exp FRl il = 0, R = ρ/ρ , ρ is the radial coordinate, ρ is the core radius, 0 0 2∆ =±+−(){} ( )ϕ 22 exp 1 − eizlGR il ∆= co cl V ×{}β nn2, n is the refractive index of clad, f(R) is ε exp 1 2 cl ±0 () he=−$ µexp{} ± ϕ iz nco tcolin0 FR il (10) the function of the refractive index profile. ε2∆ =±+0 −(){} ( )ϕ The stationary vector wave equation for the electric field µ exp 1 hnzco0 GR l il strength in an inhomogeneous medium can be written in fol- V lowing form: 2) Circularly polarized unhomogeneous optical CV vorti- {}{}∇+2222 −β =−∇ ⋅∇ 2 eettln , (4) ces (l > 1) tttkn n where k is the wave number in vacuum, β is the propagation constant. =±$m () {}ϕ eet exp For weak guiding fibers with a low energy loss n is a real FRl il value and n ≈ n , then the profile parameter ∆ can be writ- 2∆ co cl +() ( ) =±−exp{}1ϕ ten as eizlGR il ∆ ≈ V ×{}β (n – n )/n . (5) ε exp 2 co cl co =−m 0 () {} ± ϕ iz he$ µexp (11) In this case, we can neglect the term in the right side of tcolin0 FR il equation (4) [16] and rewrite the wave equation in the form ε2∆ =±−0 +(){} ( )ϕ µ exp 1 hnzco0 GR l il V {}∇+2222 −~β =0, (6) ~et t kn 3) Linearly polarized azimuth-symmetrical fields (l = 1) where ~β is the propagation constant in scalar approxima- =+()ϕϕ() ~ ex$cos y$ sin 1 tion, e is the electric field in this approximation, t FR t ∆ 2+ 2 2 = () ∂∂ 1 ∇≡2 + eiz GR × ()β 2 2 V exp 3 t ∂∂. ε iz TM : =−0()ϕϕ − () (12) xy 0m µxy$ sin $ cos 1 hntco 0 FR If the field distortion e in a weak guiding fiber in the limit ∆ → 0 in equation (4) is assumed to be small, then =0 ≈ β ~β δβ δβ hz e ~e , and the propagation constant = + , where is ÔÊÎ, 1(1), 1998 83 SQO, 1(1), 1998 K. N. Alexeyev et al.: Optical vortices and angular momentum... =−()ϕϕ() The transversal components of the TM0m and TE0m ex$sin y$ cos 1 t FR modess (l = 1) have azimuth-symmetrical distribution of the =0 ez electric et and magnetic ht fields (see (12), (13)). ε At the fiber axis, these fields turn into zero. In accord- =+0()ϕϕ()×()β µxy$ cos $ sin 1exp 4 tco 0 iz ance with the polarization singularities [19] we conclude TE : hn FR (13) 0 ε2∆ that these fields have purely screw declinations of polariza- =0 +() µ 1 tion. hinzco 0 GR V It should be noted that the electric (magnetic) field of a 2π = ρ∆2 Gaussian beam in a void has also a longitudinal e (or h ) where λ is the waveguide fiber parameter, z z Vnco component.
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