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Ciattoni et al. Vol. 20, No. 1 / January 2003 / J. Opt. Soc. Am. A 163 Circularly polarized beams and vortex generation in uniaxial media Alessandro Ciattoni Dipartimento di Fisica, Universita` Roma Tre, I-00146 Rome, Italy, and Istituto Nazionale di Fisica della Materia Unita` di Roma 3, Rome, Italy Gabriella Cincotti Dipartimento di Ingegneria Elettronica, Universita` Roma Tre, I-00146 Rome, Italy, and Istituto Nazionale di Fisica della Materia Unita` di Roma 3, Rome, Italy Claudio Palma Dipartimento di Fisica, Universita` Roma Tre, I-00146 Rome, Italy, and Istituto Nazionale di Fisica della Materia Unita` di Roma 3, Rome, Italy Received June 19, 2002; revised manuscript received August 20, 2002; accepted August 22, 2002 We deduce the expressions for the two circularly polarized components of a paraxial beam propagating along the optical axis of a uniaxial crystal. We ?nd that each of them is the sum of two contributions, the ?rst being a free ?eld and the second describing the interaction with the opposite component. Moreover, we expand both components as a superposition of vortices of any order, thus obtaining a complete physical picture of the interaction dynamics. Consequently, we argue that a left-hand circularly polarized incoming beam, endowed with a circular symmetric pro?le, gives rise, inside the crystal, to a right-hand circularly polarized vortex of order 2. The ef?ciency of this vortex generation is investigated by means of a power exchange analysis. The Gaussian case is fully discussed, showing the relevant features of the vortex generation. © 2003 Optical Society of America OCIS codes: 260.1180, 260.1960. 1. INTRODUCTION Polarization dynamics of the optical ?eld is an interesting topic in optics, since it exhibits a variety of attracting phenomena,1 from both a theoretical and an applied point of view. In this perspective, optical propagation inside anisotropic media plays a prominent role because of the capability of these media affecting the polarization state of the radiation.2 As far as the plane-wave case is concerned, the effect of a crystal on the polarization is well understood and is exploited in a large number of optical devices. Going into the paraxial regime, the effect of the crystal becomes more involved because a coupling between polarization evolution and diffraction emerges: This implies that, generally, a beam in a crystal is nonuniformly polarized.3,4 Although the description of paraxial beams is more complex than that of plane waves, it is much more interesting because of the wide freedom in choosing the ?eld impinging on the crystal, whose only constraint is the paraxial requirement that the waist be much larger than the wavelength. This entails the possibility of employing anisotropic media to generate beams with peculiar polarization patterns with an appropriate choice of the incoming ?eld and, obviously, of the crystal. The study of circularly polarized radiation has attracted a good deal of attention mainly because of its remarkable property of carrying angular momentum. Since a circularly polarized beam can be thought of as the superposition of two orthogonal linearly polarized ?elds that are in phase quadrature, one may expect that the effect of a crystal on such a beam is nontrivial. In particular, the most interesting case is that of a circularly polarized beam propagating along a direction that is a rotational symmetry axis of a crystal because two cylindrical symmetries are simultaneously present; in fact, at every point, circularly polarized radiation is characterized by an electric ?eld vector rotating around the direction of propagation, and, consequently, it exhibits cylindrical symmetry in the polarization state, no direction being preferred. Following this symmetry argument, we are led to consider a uniaxial crystal and to choose its optical axis as the direction of propagation, since it is also a rotational symmetry axis. Among the various examples of paraxial ?elds with rotational symmetry properties, the vortices play a signi?cant role, because of the angular momentum that they carry.5 In recent years, the study and the generation of optical vortices have attracted a considerable research interest.6–9 The most famous example of physically realizable ?elds characterized by vortices are perhaps the higher-order Laguerre–Gaussian beams that can be produced directly from a laser.10–13 More common is the generation of vortices by converting the lowest-order Hermite–Gaussian beam into a Laguerre–Gaussian beam by means of spiral phase plates14,15 or computer- 1084-7529/2003/010163-09$15.00 © 2003 Optical Society of America 164 J. Opt. Soc. Am. A / Vol. 20, No. 1 / January 2003 generated holograms.16,17 Higher-order Hermite– Gaussian beams can be converted into higher-order Laguerre–Gaussian beams by using mode converters based on cylindrical lenses.18 In the present paper, we deduce the expressions for the circular components of a paraxial beam propagating along the optical axis of a uniaxial medium. To achieve this goal, we appropriately superimpose the Cartesian components of a paraxial beam whose expression is given in Ref. 19. The obtained expressions are more symmetric and more readable than the corresponding Cartesian ones; in fact, each circular component decomposes into the sum of a free ?eld and a second ?eld describing the interaction with the other component. To get a deeper physical understanding of the circular component dynamics, we express them as a superposition of vortices of any order. We show that there is a nontrivial coupling among the various vortices belonging to the opposite circular components. More precisely, we ?nd that the vortex of order n of the boundary distribution of the left-hand circular component affects only the vortex of order n ?2 of the propagating right-hand circular component. Analogously, the vortex of order n of the boundary distribution of the righthand component affects only the vortex of order n ?2 of the propagating left-hand component. This unusual kind of mixing is interpreted as a consequence of the coupling between polarization evolution and diffraction that occurs inside an anisotropic crystal in the paraxial regime. Moreover, we discuss the possibility offered by this scheme to generate a vortex of order 2. In fact, from the above-mentioned analysis, it emerges that an incoming left-hand circularly polarized and circularly symmetric beam (which is a zero-order vortex) gives rise to a righthand circularly polarized vortex of order 2. This is an interesting effect from an applied point of view, since it represents a simple method to actually generate a vortex beam. To prove that the vortex generation process is ef?cient, we evaluate the powers carried by the two components and we ?nd that the total initial power undergoes, asymptotically, an equipartition between the two components. The case of a Gaussian beam is analytically investigated, showing in a concrete case the way in which the vortex is generated. 2. CIRCULARLY POLARIZED COMPONENTS OF A PARAXIAL BEAM Let us start our analysis by considering a monochromatic paraxial ?eld E(r, t) ?ReSE(r)exp(阨$;t)T propagating along the optical axis of a uniaxial nonabsorbing crystal.19 If we choose a reference frame whose z axis coincides with the optical axis of the crystal, the transverse part of the electric ?eld is given by E ?Exeˆ x ?Eyeˆ y ?exp(ik0noz) ?(Ao ?Ae), where k0 ?$;/c, r ?xeˆ x ?yeˆ y , no and ne are the ordinary and extraordinary refractive indices, respectively, and Ao and Ae are the slowly varying ordinary and extraordinary envelopes, given by u i j AoQr , zR ? d2k exp i k • r ? i k2 z 2k0no k l 1 k 2 y ?k2 阫xky 阫 x k y k 2 x • E˜Q kR, Ciattoni et al. u i j AeQr , zR ? d2k exp i k • r ? i n o k2 2 k 0 n 2 e z k l 1 k 2 x ?k2 kxky kxky k 2 y • E˜Q kR, (1) where k ?kxeˆ x ?kyeˆ y and u1 E˜ Q kR ?内容过长，仅展示头部和尾部部分文字预览，全文请查看图片预览。 ). 20. A. Ciattoni, G. Cincotti, and C. Palma, ‘‘Propagation of cylindrically symmetric ?elds in uniaxial crystal,’’ J. Opt. Soc. Am. A 19, 792–796 (2002). 21. A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, ‘‘Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,’’ J. Opt. Soc. Am. A 19, 1894– 1900 (2002). 22. G. Cincotti, A. Ciattoni, and C. Palma, ‘‘Hermite–Gauss beams in uniaxially anisotropic crystal,’’ IEEE J. Quantum Electron. 12, 1517–1524 (2001). [文章尾部最后500字内容到此结束,中间部分内容请查看底下的图片预览]请点击下方选择您需要的文档下载。

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