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Making Optical Vortices with Computer-Generated Holograms ͒ Alicia V

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Making Optical Vortices with Computer-Generated Holograms ͒ Alicia V Making optical vortices with computer-generated holograms ͒ Alicia V. Carpentier,a Humberto Michinel, and José R. Salgueiro Área de Óptica, Facultade de Ciencias de Ourense, Universidade de Vigo, As Lagoas s/n, Ourense, ES-32004 Spain David Olivieri Departamento de Linguaxes e Sistemas Informáticos, Universidade de Vigo, As Lagoas s/n, Ourense, ES-32004 Spain ͑Received 22 February 2007; accepted 16 June 2008͒ An optical vortex is a screw dislocation in a light field that carries quantized orbital angular momentum and, due to cancellations of the twisting along the propagation axis, experiences zero intensity at its center. When viewed in a perpendicular plane along the propagation axis, the vortex appears as a dark region in the center surrounded by a bright concentric ring of light. We give detailed instructions for generating optical vortices and optical vortex structures by computer-generated holograms and describe various methods for manipulating the resulting structures. © 2008 American Association of Physics Teachers. ͓DOI: 10.1119/1.2955792͔ I. INTRODUCTION optical tweezers that can trap neutral particulates,2 which re- ceive the angular momentum associated with the rotation of The phase of a light beam can be twisted like a corkscrew the phase.3 In nonlinear optics, waveguiding can be achieved around its axis of propagation. If the light wave is repre- inside the central hole.4,5 In astronomy the singularity can be ⌽ sented by complex numbers of the form Aei , where A is the used to block the light from a bright star to increase the amplitude of the field and ⌽ is the phase of the wave front, contrast of astronomical observations using optical vortex the twisting is described by a helical phase distribution ⌽ coronagraphs,6 which are useful for the search of extrasolar =m␪ proportional to the azimuthal angle of a cylindrical co- planets. Applications in quantum information employing ordinate system ͑r,␪,z͒, where z is the propagation direction quantized properties of the angular momentum of vortex of the wave train. light beams have also been proposed.7 Vortices are topologi- Because of this twisting, the light wave along the z-axis cal objects, which are important in many branches of physics cancels out because the value of the angular coordinate ␪ is such as fluid mechanics, superconductivity, Bose–Einstein not well defined at r=0, giving rise to a phase singularity1 condensates,8 and superfluidity.9 As we will show, the agree- where the amplitude vanishes. On a flat surface this light ment between theory and experiment is remarkable for linear beam will look like a bright ring with a dark hole in its center optics, and thus it provides a well characterized system for around which the phase rotates. Due to an analogous phe- the study of such phenomena. This visual verification be- nomena in fluids, this light beam is called an optical vortex tween theoretical and experimental results provides an added ͑see Fig. 1͒. way of gaining intuition about the underlying equations and To assure the continuity of the field at ␪=2␲, there exists theory. For this reason, we believe that it is desirable to teach an integer number m, which indicates the number of phase the basic laboratory techniques required to produce and ma- windings around the dark spot. This number is the topologi- nipulate optical vortices in the undergraduate physics cur- cal charge of the vortex, because it is conserved during riculum. propagation. The physical meaning of m corresponds to the The structure of the paper is as follows: In Sec. II we velocity of the phase rotation around the singularity, which explain how to fabricate computer-generated holograms,10,11 may be positive ͑for counterclockwise rotation͒ or negative which can be used to make optical vortices. References ͑for clockwise rotation͒. The definition of m is given by the 12–14 have described how such low-cost computer hologram z-component of the angular momentum Lz. For light beams experiments can be constructed. We build on this work and the value of Lz per photon is given by show how the generated holograms can be used to study several modern phenomena related with optical vortices. In -⌽ Sec. III we describe the necessary elements for experimenץ L =−iប =−iបm, ͑1͒ ␪ tally creating and manipulating optical beams with severalץ z vortices of topological charge m=1. In Sec. IV we describe where ប is Planck’s constant divided by 2␲. From Eq. ͑1͒ it using the same setup how to obtain structures with a central 15 is evident that the continuity condition at ␪=2␲ implies the vortex of charge m surrounded by single-charged vortices. 16 quantization of the angular momentum. The value of Lz in- Finally, we show how to precisely rotate the vortex lattices, dicates that there is a rotation of the momentum vector pជ which is of great utility for the optical manipulation of 17 around the dark hole as the beam propagates. This analogy trapped particulates. with the velocity fields of fluids suggests that the wave front dislocation observed in light beams should be called an op- II. DESIGN OF THE HOLOGRAPHIC MASK tical vortex. In recent years several important applications of optical Optical vortices can appear spontaneously or can be cre- vortices have been developed. For example, the absence of ated in one of several ways, such as by the manipulation of the gradient force in the central hole can be used to make the laser cavity,18 by employing mode converters,19 or by a 916 Am. J. Phys. 76 ͑10͒, October 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 916 Fig. 1. Two beams with the same charge propagating in parallel directions. The origin of the phase is arbitrary with respect to the horizontal axis. In this configuration the left side of the right beam and the right part of the left Fig. 3. Interference patterns of beams carrying topological charge m and beam have opposite phases. different reference waves ␺.In͑a͒ and ͑b͒ m=1 and m=−1, respectively, and ␺ is a tilted beam with planar phase. In ͑c͒ m=−1 and ␺ has a spherical phase. ͑d͒, ͑e͒, and ͑f͒ correspond to the same cases as the upper row for m=2 and m=−2, respectively. simple technique that uses computer-generated holograms.20,21 It has been shown in several classic papers how computer-generated holograms can be designed.11–14 We will show how to use computer-generated holograms to pro- hologram transmission function H. Thus, the irradiance in a duce optical vortices with optical equipment found in many plane far from the computer generated holograms is given by ͑ / ͒2 ␪ undergraduate laboratories. F͓␺ ͔ F͓ − r w ͉ ikxx im ͉2͔ ͑ ͒ I = GH = e e + e , 3 Our aim is to imprint a phase eim␪ in a Gaussian beam. Thus, we will make computer-generated holograms formed where kx is the x-component of the wave vector of the tilted ␺ ikx by the interference of a reference tilted plane wave 1 =e wave. The result can be calculated with standard mathemati- ͑where k is the spatial frequency indicating the tilting angle cal software and is plotted in Fig. 2͑g͒ for m=1. As can be ͒ ␺ im␪ ͑ ͒ ␺ of the wave and an object wave 2 =e , which carries the seen in Fig. 2 g , the Fourier transform of GH produces a singular phase. When this hologram is illuminated by a beam central Gaussian beam ͑diffraction order n=0͒ and two adja- containing the reference wave ͑in our experiment we will use cent beams corresponding to n=1 and n=−1 carrying the a Gaussian beam from a laser͒, the object is reconstructed phase singularities m=1 and m=−1, respectively. The inten- and the vortex appears in the output beam. sity pattern shown in Fig. 2͑g͒ is the same for the two vor- We generate a computer-generated holographic mask by tices. However, the phase distribution rotates in opposite painting an interference pattern proportional to the function, senses. For m=1 rotation is counterclockwise and for m= ͉␺ ␺ ͉2 ͉ ikx im␪͉2 ͓ ͑ ␪͔͒ ͑ ͒ −1 it is clockwise. The result can be explained by taking into H = 1 + 2 = e + e =2 1 + cos kx − m , 2 ␺ account that the Gaussian beam includes the plane wave 1 where ␪=tan−1͑y/x͒ is the polar coordinate. In Figs. used to register the hologram, and thus it can be used to ␺ ͑ ͒ 2͑a͒–2͑f͒ we show plots of H for m=1 to m=6. When one of reconstruct the object beam 2 the azimuthal phase . these holographic patterns is illuminated with a Gaussian To check the value of m for the beams in Fig. 2͑g͒ we 2/ 2 calculate numerically the interference of beams with m beam ␺ =e−r w of width w, the resulting far-field Fraun- G = Ϯ1 with a plane wave. The results are shown in Fig. 3, hofer diffraction pattern is proportional to the Fourier trans- where we show patterns corresponding to the interference of form F of the product ␺ H of the input function and the G a beam with m= Ϯ1 with a tilted plane wave ͓Figs. 3͑a͒ and 3͑b͒, respectively͔. The interference between a beam with m=1 and a spherical wave is shown in Fig. 3͑c͒. In Figs. 3͑d͒ and 3͑e͒, respectively, we show the pattern due to the inter- ference between beams with m= Ϯ2 with a plane wave and in Fig. 3͑f͒ a beam with m=2 with a spherical wave. III. FABRICATION OF THE HOLOGRAM The holographic masks that are used to produce optical vortices are fabricated in a two-step process.
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