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Shaping optical beams with topological charge Anderson M. Amaral, Edilson L. Falcão-Filho, and Cid B. de Araújo* Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil *Corresponding author:
[email protected] Received February 20, 2013; revised April 2, 2013; accepted April 5, 2013; posted April 8, 2013 (Doc. ID 185695); published May 1, 2013 We show that by spatially arranging topological charges on a phase mask, it is possible to shape the spatial intensity profile of vortex beams in a controlled manner. As proof-of-principle experiments, we generated vortex beams with the spatial shape of straight lines, corners, and triangles. Potential applications for shaped beams include selective excitation of plasmonic modes, geometrically tunable Bose–Einstein condensates, and optical tweezers. © 2013 Optical Society of America OCIS codes: (260.6042) Singular optics; (050.4865) Optical vortices. http://dx.doi.org/10.1364/OL.38.001579
An optical vortex (OV) with a topological charge (TC) q is characterized by its azimuthal phase, which varies from 0 to 2πq for a full turn around its propagation axis. This characteristic is associated with remarkable features, [1] such as orbital angular momentum (OAM), a dark center (due to the phase singularity in the beam axis), and helix wave-fronts. There is a vast literature on this subject with interesting examples on laser trapping of atoms [2], transfer of OAM in optical tweezers [3], metrology [4], and plasmonics [5,6]. In some of these applications, not only is the phase structure of the optical field relevant, but its spatial intensity profile also plays an important role. Previous approaches to shape the spatial intensity profile of OAM beams include the use of the helical phase spatial filter (HPSF) [7] and the modification of phase pitch (MPP) [8] methods. However, although HPSF is a general method, it requires two spatial light modulators (SLMs) and a complex optical setup. On the other hand, the MPP method is limited to Lissajous patterns. In this work we demonstrate a simple method for shaping the spatial intensity profile of OAM beams based on spatially distributing the TC in a given geometry. Our method guarantees that the obtained spatial beam profiles have a chosen total TC as long as the TC distribution is contained inside the bright region of the beam. The experimental results are validated by numerical calculations. The phase structure of an OV to be applied at the SLM may be built on complex plane. Considering ϕ as the azimuthal angle that varies in the interval [0, 2π], and using u reiϕ , we represent an OV with TC equal to q as uq r q eiqϕ . The factor q in the phase determines the nontrivial topology of such beams and q is a topological invariant [9,10]. By suitably applying this phase pattern to an SLM, one can generate an OAM beam with a TC equal to q. In a typical experiment, light is incident normally on the SLM and generates a superposition of Laguerre–Gauss (LG) modes. The cylindrical symmetry of LG modes is inherited from the applied phase pattern: If the mask suffers perturbations, the beam shape may change [8,11]. The above considerations correspond to associate the TC to powers of u, and therefore we may construct a representation for arbitrary OAM beams by spatially distributing TCs, as in [12]. Denoting by pi and qi the (complex) position and value of the i-th TC, we obtain for the total 0146-9592/13/091579-03$15.00/0
phase ϕT generated by N TCs at the point r; ϕ on the SLM’s plane the expression " # N Y qi u − pi : (1) ϕT arg i1
Notice that for a sufficiently P large r, u − pi ≈ u, and ϕT may be approximated by i qi ϕ. So, the applied phase pattern is insensitive to small P TC displacements, and a beam with total TC Q i qi is obtained. The calculation of Q would be made by the winding number over the beam’s intensity maximum [10], and hence we have ! I N X 1 d ⃗x · ∇⃗ ϕT ≈ qi : (2) Q 2π beam i1 Equation (2) applies for beams where the bright region is far from the spatial distribution of TCs. Therefore, considering that a region containing TCs tends to have zero intensity, by spatially distributing the total charge Q in a given geometrical arrangement, one may obtain an OAM beam with the desired light intensity distribution. This is the key concept exploited throughout this Letter. The TCs’ arrangement can be discrete or continuous. The latter possibility corresponds to the limit N → ∞ with Q const. In practical terms, this case is obtained by distributing the total TC equally over the SLM pixels forming a given geometry. By a discrete charge distribution we mean a distribution of Q vortices with qi 1. Only the case where all the charges have the same signal (positive, in our convention) will be discussed here to avoid possible questions raised by vortex knots/ annihilation [12,13]. This will be discussed in a future longer paper. To generate the OAM beams we used a fiber-coupled diode laser (805 nm), whose output is collimated using a 25 cm focal length lens. This allows us to obtain an expanded beam that can be considered a plane wave. A beam splitter is used to direct light normally to the SLM (Hamamatsu—LCOS X10468-02). We inserted in the reflected beam a spatial filter made with 70 cm and 10 cm focal distance lenses. An iris at confocal region was used to select only the modulated beam. A CCD (Coherent—Lasercam HR) was used to analyze the beam profiles. To measure the total TC associated with the © 2013 Optical Society of America
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Fig. 1. Examples of obtained phase patterns (without overlapping with a plane wave carrier and apodization). Arrangement of TCs on a line: (a) discrete (qi 1) and (b) continuous (qi → 0) distributions for Q 10. There is only a subtle difference between the phase masks at the central region. (c) Corner shaped (Q 20) and (d) triangular shaped (Q 21) TCs arrangements.
optical beams we used the diffraction technique proposed in [14,15]. In this technique, the light diffraction by a triangular slit with 1 mm sides results in a triangular diffracted pattern. The number of bright spots over each triangle side is equal to Q 1, and the pattern orientation gives the Q signal. Figure 1 shows examples of obtained phase masks when TCs are distributed along a line, a corner (L-shape), and a triangle. Notice that discrete or continuous TCs distributions present only a subtle difference in the central region of the phase masks shown in Figs. 1(a)–1(b). Figure 2 shows the spatial intensity profile obtained using either discrete or continuous distributions of TCs on a line of length L parallel to the horizontal axis. Figures 2(a) and 2(b) show numerical results for discrete and continuous TC distributions, respectively, while Figs. 2(c) and 2(d) exhibit the corresponding experimental results. The region of maximum intensity is distorted when L is increased up to a length such that ring-like structures start to appear (L 4 mm, see Media 1). For the discrete TCs distribution, this is a result of the spatial separation of 5 qi 1 vortices, as can be seen for larger values of L. Figures 2(e) and 2(f) show the diffraction profile using the technique of [14,15] to determine the total TC. The six brilliant spots in each side of the triangle at Fig. 2 (Media 1) when L 0 mm and its orientation indicates that Q 5 (one of the sides of the triangular slit was perpendicular to the line of charges). This side corresponds to the bottom edge of the triangle on Figs. 2(e), 2(f), and 2 (Media 1). As would be expected from the patterns in Figs. 1(a) and 1(b), the respective beam profiles in Figs. 2(a)–2(d) are essentially equal. One remarkable result is that the dark region that contains the TC distribution is rotated in respect to the horizontal axis due to the Guoy phase shift [12,16,17]. This rotation increases with q and is reversed with a change in sign. By increasing L we observed that the diffraction pattern deforms and most of the triangle peaks
Fig. 2. Beam profiles for a linear distribution of TCs with L 3 mm. The total TC is Q 5. Numerical results: (a) discrete and (b) continuous charge distributions. Experimental results: (c) and (d) correspond to discrete and continuous TC distributions, respectively. (e), (f) Show the measurements of the total TC using a triangular slit. For experimental results at various L values, see (Media 1).
are blurred. However, the number of spots on the triangle’s edge remains equal to six, with some spots fading away for L 7 mm. This can be understood as a limitation of the measurement technique to highly deformed beams. Since the measurement is carried only at a finite region, for sufficiently large values of L the slit cannot diffract the whole beam. Therefore, it will measure only part of the beam’s TC. In order to demonstrate the validity of the present method to more complex geometrical structures, we constructed L-shaped (corner) and triangular OAM beams. Figure 3 shows the profile obtained considering (a)
Fig. 3. Experimental beam profile for L-shaped (corner) TC distributions with Q 10. (a) Discrete and (b) continuous TC distributions.
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proposed here, trapping potentials can be designed, such as to increase transition temperatures and/or shape the condensate. In optical tweezers, this method provides an alternative to shape noncircular routes for suspended particles [18]. In plasmonics, shaping of OAM beams may allow selective excitation of plasmonic modes, such as [6]. Also, even though we considered only linear TCs distributions, smooth 2D distributions can be designed. Fig. 4. (a) Intensity profile and (b) charge measurement of a triangular charges arrangement with Q 3. For more Q values, see (Media 2).
discrete and (b) continuous distribution of TCs. The value of Q 10 was chosen to minimize the amount of light over the TCs distribution and at the same time preserve the L-shape. When a smaller value of Q is selected, the vortices of a discrete TC distribution are well separated, and the destructive interference between them is less effective. On the other hand, if a very large value of Q is used, according to the argument leading to Eq. (1), the intense region radius can become so large that the profile tends to a deformed circle. To analyze the case of TCs distributed along the sides of a triangle we must consider the existence of two regions: (i) inside the triangle there are no charges and thus the winding number of any curve is zero; (ii) for curves encircling the triangle, the winding number is Q. We show in Fig. 4 the experimental results where we apodized the central region without charge. Figure 4(a) shows the beam triangular profile, and Fig. 4(b) confirms the TC of Q 3. Figure 4 (Media 2) shows that the pattern expands, rotates, and becomes more rounded for larger Q values. Rotation in this case is also due to Guoy phase. The examples discussed illustrate the potential of the present method to shape OAM beams with a large variety of geometries. The examples are building blocks for more complex arrangements. An immediate application of the concepts developed above is to precompensate on the SLM aberrations affecting vortex beams during propagation, as in [11]. Reference [2] shows that laser traps formed by LG beams have higher Bose–Einstein condensation temperatures due to modification of trapping potential by changing Q values. With the method
We acknowledge support of the Brazilian agencies CNPq (INCT—Fotônica) and FACEPE. A.M. Amaral would also like to thank professor A. M. J. Schakel for his lectures related to topological defects. We also acknowledge L. E. E. de Araujo for his help during the initial stage of this work. References 1. A. M. Yao and M. J. Padgett, Adv. Opt. Photon. 3, 161 (2011). 2. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, Phys. Rev. A 82, 023613 (2010). 3. D. G. Grier, Nature 424, 810 (2003). 4. J. Shamir, Opt. Eng. 51, 073605 (2012). 5. N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, Nano. Lett. 11, 2038 (2011). 6. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, Nano. Lett. 10, 529 (2010). 7. C.-S. Guo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, Opt. Commun. 259, 449 (2006). 8. J. E. Curtis and D. G. Grier, Opt. Lett. 28, 872 (2003). 9. M. Nakahara, Geometry, Topology and Physics (Taylor & Francis, 2003). 10. A. M. J. Schakel, Boulevard of Broken Symmetries (World Scientific, 2008). 11. A. Kumar, P. Vaity, and R. P. Singh, Opt. Express 19, 6182 (2011). 12. G. Indebetouw, J. Mod. Opt. 40, 73 (1993). 13. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, New J. Phys. 7, 55 (2005). 14. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010). 15. L. E. E. de Araujo and M. E. Anderson, Opt. Lett. 36, 787 (2011). 16. D. Rozas, C. T. Law, and J. G. A. Swartzlander, J. Opt.Soc. Am. B 14, 3054 (1997). 17. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, Opt. Express 17, 9818 (2009). 18. A. Jesacher, C. Maurer, S. Fuerhapter, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, Opt. Commun. 281, 2207 (2008).
