Many approaches for producing optical vortices have been developed both for fundamental interests of science and for engineering applications. In particular, the approach with direct excitation of several emitters has a potential to control the topological charges with a control of the source conditions without any modifications of structures of the system. In this paper, we investigate the propagation properties of the optical vortices emitted from a collectively polarized electric dipole array as a simple model of the several emitters. Using an analytical approach based on the Jacobi-Anger expansion, we derive a relationship between the topological charge of the optical vortices and the source conditions of the emitter, and clarify and report our new finding; there exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of the dipole fields.
© 2015 Optical Society of America
Optical vortices, which carry orbital angular momentum of light, are characterized by a spatially distributed phase profile of eilϕ where ϕ is an azimuthal angle on the plane orthogonal to the propagation direction of light, and l is the topological charge of optical vortices [1–3]. Optical vortices has been actively studied over a wide range of fields such as optical manipulation , optical sensing , laser fabrication  and classical and quantum information processing [7,8]. In order to generate and control the optical vortices, the mode conversion of electromagnetic field has been mainly proposed in various structural devices. The spatial light modulators were used to generate a Laguerre-Gaussian beam with an arbitrary topological charge by holographic patterns [9, 10]. The pinhole plates with various arrangements were designed to create arrays of optical vortices by illuminating with a plane wave [11, 12]. The spin-to-orbital angular momentum conversion were performed using artificial optical devices such as Q-plate  and a plasmonic metasurface . The compact optical vortex emitters were fabricated to convert the optical modes from whispering-gallery-modes to optical vortices [15, 16]. On the other hand, the direct excitation of several emitters for producing optical vortices were recently studied in a ring molecules  and an antenna array . This method has a potential to produce optical vortices by controlling the source conditions of the emitters without any modification of the system structure.
In this paper, we analytically investigate the propagation properties of the optical vortices generated by a collectively polarized electric dipole array, which becomes a simple model for the direct excitation of several emitters. We derive an analytical solution of the electromagnetic field decomposed by vortex fields via several approximations, and clarify the following properties: (1) there is a certain relationship between the topological charge of the optical vortices and the source conditions of the emitters, and (2) There exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of light between the collective electric dipole moments and the intrinsic radiation patterns of each emitter without any split of the singular point in the magnetic field.
2.1. Far-field electromagnetic field decomposed by Jacobi-Anger expansion
In our system, N infinitesimal electric dipoles are placed on the x–y plane with the radius of a such that the center of the circle is located at the origin of the coordinate. The coordinate of the j-th dipole is given by (a cosφj, a sinφj, 0) where the azimuthal angle . The electric dipole moment of all of the electric dipoles is represented by a collective polarization vector p = p(sxex + syey)eiqφj where the real value p shows the amplitude of the electric dipole moment, the complex parameters of sx, sy represent the polarization of each dipole with the normalized amplitude and eiqφj is the phase factor of each electric dipole with an integer q which decides the topological property of the initial phase distribution of the system. The source condition of the system is characterized by the three parameters; the number of dipoles in the array N, the polarization of each dipole (sx, sy) and the initial topological factor q. Figure 1(a) depicts an illustration of our system with the source condition of (sx, sy) = (1, 0), N = 3 and q = 1. An analytical solution of the electromagnetic field generated by the electric dipole array is given by the superposition of each electric dipole field ,
In order to verify the possibility for the generation of the optical vortices in our system, we calculate the phase distribution of the electromagnetic field in the case where the electric dipoles are collectively polarized to x direction (sx = 1, sy = 0). Figures 1(b)–1(g) shows the phase distribution of the x component of the electric and the y component of the magnetic field with the three different couple of the parameters (N, q) = (3, 1), (5, 2) and (7, 3) at z = 100λ. The horizontal and the vertical axes are scaled by the wavelength of the electromagnetic field emitted from each dipole λ. The magnetic field has a singular point at the center of the xy plane with a topological charge q. On the other hand, the singular points of the electric field appears with the split in the case of the higher order of q. The distance between the singular points is as long as the wavelength of the electromagnetic field. Although the exact solutions of Eqs. (1) and (2) give us a certain distribution of electromagnetic filed, however, it is not clear for the relationship between the propagation properties of the optical vortices and the source conditions.
In order to discuss the singular points in the electric fields split in the far-field with the distance as long as the wavelength of λ, we perform two approximations: the far-field approximation in which the higher terms of λ/z ∼ a/z are neglected and the near-axis approximation in which the higher terms of λ/z ∼ x/z ∼ y/z are neglected. These two approximations are suitable for analyzing the internal phase structure of the optical vortices in the far-field regime. At first, we perform the approximation to the function of the phase factor in a cylindrical coordinate (ρ, ϕ, z) by omitting the higher orders than the second order of x/z, y/z, a/z, λ/z. Moreover, we perform the Jacobi-Anger expansion to expand the electromagnetic field by the superposition of the vortex field which propagates along the z axis.Appendix A), the far-field and the near-axis solution of the electromagnetic field is obtained as the following; Eqs. (12)–(14) is one of our main results. In the far-field and the near-axis regime, we obtain a certain relationship between the source conditions and the topological charge of the optical vortices. Figure 2 show the value of Qmin with respect to the system parameters (N, q) [Fig. 2(a)], and several phase profiles calculated from the exact solutions in Eqs. (1) and (2) fixing the collective polarization to (sx = 1, sy = 0) [Figs. 2(b)–2(d)]. Qmin obviously corresponds to the topological charge calculated from the exact solution, and becomes a good index for characterizing the dipole array as an optical vortex emitter corresponds to the topological charge calculated from the exact solution.
2.2. Intrinsic split of singular points in electric field with higher order topological charge
In the case where |Qmin| ≥ 2, the transverse components of electric fields are represented by the sum of the two terms. This implies that the singular points of the transverse components are not located at the center of xy plane. For simplicity, we consider only the case of Qmin ≥ 2 in the following discussion. The transverse components of the electric field are given byAppendix B). (17)–(19) are completely characterized the intrinsic split of the singular points in the electric field of Ex with respect to the source condition of the emitter. If the electric dipoles are horizontally polarized (β = 0 and γ = 0), the singular points in the electric field of Ex are vertically split with the distance of the singular points simply expressed by Fig. 3.
The split of the higher-order optical vortex has been discussed in the holograhic method [20, 21]. In general, the higher-order vortex is easily split into several vortices with the topological charge one under the practical fluctuation and incompleteness. We emphasize that if the perfect system removing any practical problems is assumed for generation of the optical vortices, nevertheless, there exists intrinsic split of the singular points in the electric field in the dipole array due to the spin-orbit interaction caused by the radiation patterns of each electric dipole.
3. Numerical simulation
To verify the consistency of our analytical approach with several approximations, we numerically calculate a topological charge around one of the singular points ζ1(β, γ) from the exact solution of Eqs. (1) and (2). The topological charge that we numerically calculated is defined asFig. 4(a)]. The topological charges for each Qmin reach to one when the z coordinate is over several ten times of the wavelength. This implies that there is a split vortex with a topological charge one in the coordinate of the singular points given by Eqs. (17)–(19) around the far-field regime, and the singular points propagate in parallel to the z axis. Figure 4(b) shows the numerical topological charges under Qmin = 2. At all of the points without the case where β = γ = π/2, the singular point with the topological charge one appears. This implies that the singular points split into two, and one of them propagates along z axis with the coordinate of ζ1(β, γ). On the other hand, there is an optical vortex with the topological charge two which has no splitting at the center of the xy plane in the case of β = γ = π/2 because the coordinate of singular points degenerate into ζ1(β, γ) = (0, 0). All numerical result calculated from the exact solutions is supported by our analytical approach with the far-field and the near-axis approximations.
We mainly focused on the phase distribution of the electromagnetic field to characterize the structures of the optical vortices generated by the electric dipole array. On the other hand, the energy distribution is also important to discuss the function of the system. Figure 5 shows the energy density in the zx plane calculated from the exact solutions for each source condition Qmin with the polarization condition of (sx, sy) = (1, 0). The larger is the area where the energy becomes weak around the z axis, the larger is the topological charge of the optical vortex. It stems from the terms of amplitude including the Bessel function with the order of its topological charge.
An electric dipole is the most fundamental element for emitting electromagnetic field to the far-field with a wide range of wavelengths. For the implementation of our optical vortex generator, it is necessary to fix the position of each dipole on a circle and to control each phase. It is possible to easily implement our system in radiowave or microwave regions because the phase of the electric dipoles can be directly controlled with conventional systems. In addition, the intrinsic split of the singular points in the electric field can be experimentally detectable due to the long wavelengths. Our system has a possibility to switch the topological charges only by controlling the phase distributions of the electric dipoles q without the number of the electric dipoles N.
In this paper, we discuss the propagation properties of the optical vortices emitted from a collectively polarized dipole array. Under the far-field and the near-axis approximations, we clarify the following two properties: (1) there is a certain relationship between the source conditions and the topological charge of the optical vortices, (2) There exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of light between the collective electric dipole moments and the intrinsic radiation patterns of each emitter without any split of the singular point in the magnetic field. These analytical results give us a certain perspective for controlling the topological charge of optical vortices.
Appendix A: Derivation of the far-field solution
In order to derive the far-field solution by decomposing the optical vortices which propagates on the z axis, we start from the electric and the magnetic field indicated by the superposition of the field emitted from each dipoleEqs. (4) and (5) respectively. We obtain the selection rules for the topological charges given by the Dirac’s delta via the direct calculation of the sum of Λj and Πj.
Appendix B: Derivation of the singular points in Ex
In the case Qmin ≥ 2, the Ex given in Eq. (15) includes the non-trivial vortices. In order to find the coordinate of the singular points of the non-trivial vortices, we derive the product form of eiϕ1eiϕ2 from the final terms of Eq. (15) where ϕ1 and ϕ2 are defined as new azimuthal angles in cylindrical coordinates achieved by translating the z axis into the radial directions respectively. At first, we define , and expand to the Cartesian coordinate to find the transformation factor ζx and ζy.
This work was supported by Program for Leading Graduate Schools: ”Interactive Materials Science Cadet Program”.
References and links
1. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). [CrossRef] [PubMed]
2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]
3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]
6. T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010). [CrossRef] [PubMed]
7. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012). [CrossRef]
10. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order laguerre-gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008). [CrossRef]
11. K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987). [CrossRef]
12. Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013). [CrossRef] [PubMed]
14. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014). [CrossRef]
15. A. A. Savchenkov, A. B. Matsko, I. Grudinin, E. A. Savchenkova, D. Strekalov, and L. Maleki, “Optical vortices with large orbital momentum: generation and interference,” Opt. Express 14, 2888–2897 (2006). [CrossRef] [PubMed]
16. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012). [CrossRef] [PubMed]
17. M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014). [CrossRef]
18. X. Ma, M. Pu, X. Li, C. Huang, W. Pan, B. Zhao, J. Cui, and X. Luo, “Optical phased array radiating optical vortex with manipulated topological charges,” Opt. Express 23, 4873–4879 (2015). [CrossRef] [PubMed]
19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
20. I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993). [CrossRef]