## Abstract

The generation of light beams carrying orbital angular momentum (OAM) has been greatly advanced with the emergence of the recently reported integrated optical vortex emitters. Generally, optical vortices emitted by these devices possess cylindrically symmetric states of polarization and spiral phase fronts, and they can be defined as cylindrical vector vortices (CVVs). Using the radiation of angularly arranged dipoles to model the CVVs, these beams as hybrid modes of two circularly polarized scalar vortices are theoretically demonstrated to own well-defined total angular momentum. Moreover, the effect of spin–orbit interactions of angular momentum is identified in the CVVs when the size of the emitting structure varies. This effect results in the diminishing spin component of angular momentum and purer OAM states at large structure radii.

© 2014 Optical Society of America

Recently, several novel integrated photonic devices have been proposed for the generation of optical vortex beams that carry photonic orbital angular momentum (OAM) [1–3], aiming at revolutionizing the research and application of OAM optics by enabling compact, robust, and complex photonic integrated circuits that incorporate arrays of such devices. The optical vortex beams emitted by these integrated devices are vectorial in nature, often with cylindrical symmetry in their state of polarization (SOP), hence they are generalized as cylindrical vector vortices (CVVs).

In scalar optical vortices with a homogeneous SOP, such as linearly polarized Lagurre–Gaussian (LG) beams, it is understood that the OAM per photon is well-defined as $\ell \hslash $ in the paraxial limit [4]. The classical manifestation of photonic OAM is in the form of a helical phase front resulting from the azimuthal varying phase term $\mathrm{exp}(j\ell \phi )$, where $\ell $ is the topological charge and $\phi $ is the azimuthal angle. On the other hand, in CVVs, as it is the rotating vector field components that have the OAM-indicating phase dependence of $\mathrm{exp}(j\ell \phi )$ [5], the phase evolution around the optical axis is accompanied by the SOP evolution. It has been suggested in [6,7] that the vector elegant LG modes, with a similar spatial dependence of SOP and phase structure, possess the well-defined OAM states. However, the CVVs generated by these devices, as the hybrid modes of circularly polarized Bessel-like vortices [5], still imply an ambiguity in the total angular momentum (TAM), OAM, and spin angular momentum (SAM) values. For many potential applications involving the use of OAM, either exploiting momentum transfer via light–matter interaction or using the OAM states to carry information, it is fundamentally important to clarify such ambiguity by quantifying the angular momentum (AM) and its constituents carried by the CVVs.

In this Letter, we thoroughly investigate the TAM and its spin and orbital components carried by such CVVs using a classical approach. The TAM per photon of the emitted CVVs is found as well-defined by the topological charge as $\ell \hslash $ within the paraxial limit. On the other hand, the spin component per photon, which can be regarded as the contribution from the scalar vortex constituents carrying spin eigenstates of photon ($\pm \hslash $, respectively), is generally nonzero for all orders of topological charge. Furthermore, the spin-to-orbit conversion of AM in CVVs is revealed when the size of the emitter varies.

In general, the OAM-carrying helical phase front of the CVVs emitted by the previously mentioned devices results from the local manipulation of phase or polarization of the input light by a number ($q$) of scattering/diffracting elements arranged in a circular fashion, characterized by a phase increment of $2\pi \ell /q$ between adjacent elements. Furthermore, these structures are fed by input optical modes with cylindrical symmetry. Therefore, all aforementioned CVV emitters can be represented by the composition of a common structure as shown in Fig. 1, i.e., a group of angular-distributed dipoles. The dipole elements in such groups have the SOPs arranged in a cylindrically symmetrical fashion (either radially or azimuthally), and the initial phase states are defined by the topological charge as $\{{\varphi}_{m}=2\pi \ell m/q\}$, where $m=\mathrm{1,2},\dots ,q$. Hence, the CVV emitters can all be analyzed using the theoretical model developed in [5]. All dipoles $\{{\mathbf{P}}_{m}^{A}\}$ are assumed with a uniform moment of ${\mathrm{P}}_{A}$ and a time dependence of $\mathrm{exp}(-j\omega t)$, and they are located evenly along the circumference with $\rho =1$ on the emitter plane ($\zeta =0$), where ($\rho $, $\phi $, $\zeta $) is the dimensionless cylindrical coordinates, with $\rho =r/R$ and $\zeta =z/R$ as the polar radius and vertical distance normalized to the radius of the dipole group, $R$.

The case of azimuthally polarized angular dipoles (APADs) is considered first. Under the paraxial limit and in the far-field zone of dipole radiation ($\zeta \gg \lambda /R$), the cylindrical components of the emitted beam are [5]

The Jones vector of the transverse field of the APAD-emitted CVV can be written from Eqs. (1a) and (1b) as

The $\zeta $ components of the SAM and OAM per photon per unit length for the paraxial beams in Eq. (2) can be calculated by, for example, Eq. (8) in [9]. It should be noted that Eqs. (1a)–(1c) and (2) have been modified into a more precise form ($\propto 1/{({\rho}^{2}+{\zeta}^{2})}^{1/2}$), compared with those in [5] ($\propto 1/\zeta $). The transverse components thus vanish sufficiently fast to result in a convergent integral across the $x\u2013y$ plane for the energy density. After some straightforward algebra, the SAM and OAM components have the form

The AM carried by the APAD/RPAD-emitted CVVs should also be investigated without invoking the paraxial approximation, such as in the case of small emitting structures ($R\ll \lambda $). However, it is well-known that the identification of spin and orbital parts of optical AM beyond the paraxial limit is not straightforward and meets fundamental difficulties [12–14]. In 2002, Barnett demonstrated that it was the AM stored in a unit length, often taken as the AM “flux,” leading to the theoretical difficulty and introduced the correct form of AM flux with the AM continuity equation [15]. The $z$ component of the cycle-averaged AM flux through a plane oriented in the $z$ direction can be separated into physically meaningful spin and orbital parts, as [15]

Here, a numerical approach is used. First, instead of from the analytical expressions as in Eqs. (1a)–(1c), the electric and magnetic fields are obtained by numerically calculating the far-field interference of the $q$-element APADs, and then Eqs. (6a) and (6b) are integrated numerically. The calculated ratio of the AM fluxes to the energy flux as the function of topological charge is illustrated in Fig. 2, where $q=36$, $R=3.9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, $\lambda =1.55\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and the ratio is presented as multiplied by $\omega $. Returning to the aforementioned $\ell =1$ case in which ${S}_{\zeta}^{A}$ was found nonzero in the paraxial limit, the numerically calculated nonparaxial fluxes are $({M}_{S,\zeta \zeta}^{A}/F)\xb7\omega \approx 0.00396$ and $({M}_{L,\zeta \zeta}^{A}/F)\xb7\omega \approx 0.996$. The SAM components (indicated by + marks in Fig. 2) for other $\ell $ values generally have nonzero values and only equal zero when $\ell =0$. Although the TAM per photon of the APAD-emitted CVV is perfectly defined as $\ell \hslash $ (x marks), its OAM component generally is not (○ marks). A similar conclusion can also be drawn for RPAD-emitted CVVs.

Additionally, one may notice that in Eq. (5), apart from the topological charge, ${S}_{\zeta}^{A}$ also varies with the normalized propagation constant $\nu =2\pi R/\lambda $, or with the radius of the angular-dipole structure for a certain wavelength. When $\nu \gg \ell $, which requires $R/\lambda \gg 1$, ${S}_{\zeta}^{A}$ tends to zero and ${L}_{\zeta}^{A}$ approaches $\ell \hslash $. On the other hand, if the radius satisfies $R/\lambda \ll 1$, the APAD/RPAD-emitted CVVs have relatively significant longitudinal components, and the paraxial approximation no longer holds. Neither does Eq. (5). Again by means of the numerical model for the APAD/RPAD far-field radiation, the SAM and OAM variations as a function of the normalized radius ${R}_{n}=R/\lambda $ are shown in Fig. 3, where $\lambda =1.55\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$.

In fact, for ${R}_{n}\ll 1$, the angular-dipole structure is equivalent to the extensively studied case of circularly polarized dipole radiation [15,16]. In [15], it is indicated that the flux ratio of AM to energy through a sphere, which is centered on the dipole of complex moment ${\mathrm{P}}_{i}\text{\hspace{0.17em}}\mathrm{exp}(-j\omega t)$ $(i=x,y)$, is ${\sigma}_{z}/\omega $, where ${\sigma}_{z}=j({\mathrm{P}}_{x}{\mathrm{P}}_{y}^{*}-{\mathrm{P}}_{y}{\mathrm{P}}_{x}^{*})$. Therefore, it is analogous in the $\ell =\pm 1$ case that when ${R}_{n}\ll 1$, the two dipoles of each dipole pair, (${\mathbf{P}}_{m}^{A}$, ${\mathbf{P}}_{m+q/4}^{A}$) or (${\mathbf{P}}_{m}^{R}$, ${\mathbf{P}}_{m+q/4}^{R}$), have the phase difference of $\ell ({\varphi}_{m+q/4}-{\varphi}_{m})=\pm (\pi /2)$ and orthogonal polarizations (${\widehat{\phi}}_{m}\xb7{\widehat{\phi}}_{m+q/4}=0$, ${\widehat{\rho}}_{m}\xb7{\widehat{\rho}}_{m+q/4}=0$), forming a beam with an AM flux ratio of $\pm 1/\omega $. Consequently, all $q/2$ pairs of dipoles emit a beam of ${M}_{\zeta \zeta}^{A}/F=\pm 1/\omega $ or ${M}_{\zeta \zeta}^{R}/F=\pm 1/\omega $, constructively. Although it has not been suggested in [15] that this AM flux can be separated into spin and orbital components, as shown in Fig. 3(a) and 3(c), half of the AM flux is attributed to spin and the other half to OAM. This agrees well with the analytical prediction in [16]. For $\ell =\pm 2$, the dipole elements on the opposite sides of the circumference tend to cancel each other when ${R}_{n}\ll 1$, and the AM flux only starts to emerge when ${R}_{n}\approx 1$, as shown in Figs. 3(b) and 3(d). In general, for odd orders of $\ell $, the AM flux evolves from ${M}_{S,\zeta \zeta}/F={M}_{L,\zeta \zeta}/F=\pm (1/2)/\omega $ when ${R}_{n}\ll 1$, while from ${M}_{S,\zeta \zeta}/F={M}_{L,\zeta \zeta}/F=0$ for even orders of $\ell $. It should also be noted that as the radius increases, there exists a critical radius for the APAD structure, where the spin component becomes unitary [1 for $\ell >0$ and $-1$ for $\ell <0$, see Fig. 3(a) and 3(b)] and the emitted beam is globally circular-polarized over the transverse plane. The reason of this difference between APADs and RPADs lies in the fact that APAD elements are linearly polarized along the azimuthal direction, in which the spinning of circular polarization is also defined. However, RPAD elements are polarized in the orthogonal direction of the polarization space, and it is not periodic in this dimension, which consequently allows no complete spinning.

On the other hand, when ${R}_{n}\gg 1$, the distance between the two dipoles of each pair $\sqrt{2}R\gg \lambda $ and the correlation between them vanishes (note the rapid variation in SAM and OAM when ${R}_{n}\approx 1$ in Fig. 3), resulting in the emergence of a spatially helical phase front, as shown in Fig. 4. This diminishes the spin component and defines the OAM as $\ell \hslash $ per photon. In general, the increase in $R$ leads to spin-to-orbit AM conversion, by radially “decentralizing” the linear polarization components of the spinning light field. In other words, if the linear components of a spinning light source are detached in a geometrically significant distance ($R\gg \lambda $) from each other, the intrinsic phase delay between them is reconstructed into the geometric phase in the polarization state space, or Pancharatnam phase [17], which has been shown to account for the OAM carried by cylindrical vector beams [18].

In conclusion, we investigated the AM carried by a general class of cylindrical vector vortices (CVVs), typically generated by optical vortex emitters with cylindrically symmetric emission structures. By calculating the AM components of the angular-dipole-emitted CVVs, it can be concluded that these optical vortex emitters, with practical dimensions ($R>\lambda $), are shown to emit well-defined TAM of $\ell \hslash $ per photon. It has also been shown that the OAM–SAM ratio is dependent on the emitter size, indicating a dynamic conversion between the two mechanically equivalent counterparts. It is further revealed that as emitter dimension increases, the intrinsic SAM carried by the cylindrically rotating SOP of the dipole collection is gradually converted into OAM, resulting in a practically pure and well-defined OAM emitter. These results clarify the common misconception of the OAM states carried by CVVs and provide useful instructions for optical vortex generation techniques.

S. Yu and X. Cai are grateful to Prof. Sir Michael Berry for the useful discussions. The project is partially funded by the Chinese Ministry of Science and Technology under project 2014CB340000. J. Zhu’s research at the University of Bristol is funded by the China Scholarship Council (201306100054).

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