© 2008 OSA / FiO/LS/META/OF&T © 2008 OSA / Solar 2008 a891_1.pdf FThO2.pdf FThO2.pdf
Measuring orbital angular momentum of a photon using the diffraction reciprocal lattice of a triangular slit W. C. Soares1, I. Vidal1, D. P. Caetano1, E. J. S. Fonseca1, S. Chávez-Cerda2, and J. M. Hickmann1 1
Optics and Materials Group – OPTMA, Instituto de Física, Universidade Federal de Alagoas CAIXA POSTAL 2051, Maceió, AL, 57061-970, Brazil 2 Instituto Nacional de Astrofisica, Optica y Electronica, Apdo Postal 51/216 Puebla, México
[email protected]
Abstract: We demonstrate that the reciprocal lattice in the Fraunhofer diffraction pattern of single photons with orbital angular momentum passing through an equilateral triangular slit provides quantitative information of the total amount of orbital angular momentum. © 2008 Optical Society of America OCIS codes: (270.0270) Quantum Optics; (260.1960) Diffraction theory; (260.1180) Crystal optics
High-order Laguerre-Gauss (LGpm) beams are one of the most studied beams possessing Orbital Angular Momentum (OAM). In 1992, it was demonstrated that such light beams carry a well-defined OAM of mÑ per photon [1], where m is an integer number. In this work, we investigate diffraction patterns of photons with OAM by an equilateral triangular slit. Using an analogy with solid-state physics, the far field diffraction pattern is treated as the reciprocal lattice of a direct lattice formed by the sides of the triangle. Given the orbital angular momentum conservation, only selected parts of the reciprocal lattice are unveiled, allowing us to fully determine the OAM eigenstate of the photon. The theoretical approach starts with the Fraunhofer integral, Ed (k^) ∂ ! τ(r^)Ei (r^)exp(i k^◊r^)dr^, where τ(r^) is the function describing the slit and Ei (r^) is the incident field. In the analogous integral of solid state physics [2], τ(r^) corresponds to the function describing the unitary cell of the direct lattice. We use this analogy to analyze the diffraction pattern features considering that the incident beam carries OAM and the equilateral triangular slit is a unit cell of an equilateral triangular lattice defined by the vectors a1 and a2, as showed in Fig. 1(a).. The reciprocal lattice is generated by the vectors b1and b2 , also showed in Fig. 1(a). The reciprocal lattice, by definition, is generated by the set of vectors k^= v1 b1 + v2 b2 satisfying exp(i k^◊r^) =1 for all vectors r^ in the Bravais lattice We can show that the integer components (v1, v2) must be v1+v2 = m, with m being the topological charge of the incident single photon field (see Fig. 1(b)). In this way, the diffraction pattern must reveal the corresponding part of the reciprocal lattice associated with the charge m. Figure 1 (c, d) shows 6 x 6 mm matrices corresponding to the experimental results. The number N of maxima along any side of the triangular structure gives the amount of OAM of the incident photon, according to m = (N -1). From figure 1 (b) and (c), N = 4 corresponds to m = 3, in agreement with the OAM eigenstate of the photons diffracted. This rule applies for any OAM eigenstates from the diffraction patterns, by counting the number of maxima [3]. It is also possible to determine the sign of m. For m = - 3 the orientation of the diffraction pattern changes as showed in Fig. 1 (d). We also can understand this result by noting that m < 0 implies in v1 and v2 < 0 therefore, the orientation of reciprocal lattice vectors should change as well.
Fig. 1. The direct and reciprocal vectors associated with the triangular slit (a); the portion of the reciprocal lattice to m = 3 (b) ;experimental results of measurements of far field diffraction pattern at the single photon count regime for m = 3 (c) and m = -3 (d)..
In conclusion, we have demonstrated that the reciprocal lattice associated with a triangular slit can be probed by light’s OAM. We have also showed a direct way to measure an OAM eigenstate corresponding to a single photon field. The equilateral triangular shape of the slit seems more appropriated to determine the OAM of light, but we can probe others 2D Bravais lattices at the single photon regime by using a slit corresponding to their respective unit cell. References [1] Allen, L., M. W. Beijersbergen, et al., "Orbital Angular-Momentum of Light and the Transformation of Laguerre-Gaussian Laser Modes." Physical Review A 45, 8185-8189 (1992). [2] C. Kittel, Introduction to Solid State Physics, (8th Ed., Wiley 2005), Chap. 2. [3] D. P. Caetano, E. J. S. Fonseca, S. Chávez-Cerda, W. C. Soares, and J. M. Hickmann, “Generating optical lattices using light's orbital angular momentum,” submitted (2008).
