Optical angular momentum derivation and evolution from vectorial field superposition

Optical intrinsic angular momentum can be regarded as derivation from spatial superposition of optical vector fields embodied by spinning or and spiraling the electric field vector. We employ vectorial formulation derivation to comprehensively study all angular momentum contents of optical vector fields in arbitrary superposition states, including the longitudinal and transverse, spin and orbital (SAM and OAM) components. As for the orthogonal superposition fields, there inherently exists spin-orbit shift from longitudinal SAM to OAM, and the whole local spin flow manifests local multiple fold helical trajectories. Especially, both the spin-orbit shift and transverse SAM could become considerable in the non-paraxial condition. Our studies here provide an explicit insight into the derivation and evolution, intrinsic correlations and salient features of various types of angular momentum components.

Recently, extensive attention and academic interest have been increasingly attracted by photon's angular mo mentu m, including spin and orbital angular mo mentum (SAM and OAM). The well-known longitudinal SAM and OAM with the same direction as the optical axis are associated with polarization states and spiral phase distribution of light beam, respectively, which is the salient intrinsic feature of light [1]. Photon's longitudinal OAM, as a new degree of freedo m since pioneered by Allen and co-workers [2], has open a door to a large research fields with nu merous applications in optical co mmunication [3], quantum optics and informat ion [4], optical tweezers and micro mechanics [5], and even astronomy [6]. Apart fro m the longitudinal angular mo mentu m, the classical transverse OAM as an extrinsic properity of structured beam is associated with beam trajectory dependent of transverse coordinates of the beam centroid, similar to mechanical AM of classics particles [7]. Furthermore, the new-discovered transverse SAM has attracted a rapidly growing interest [8]. It arises as a result of the prominent longitudinal electric field, and exh ibits unique features in sharp contrast to the usual longitudinal SAM, such as the effects of spin-mo mentum locking and lateral forces [9].
It is well known that vectorial fields are characterized by spatially inhomogeneous state of polarizat ion, and man ifest the full vectorial nature of electro magnetic wave [10]. They can be potentially applied to particle acceleration, microscopy, and sensing because of its unique properties [11]. Vectorial fields are supplied with higher-order solutions of vector Helmholt z equation [12]. In weakly guid ing condition, fiber-guided modes carrying integer longitudinal SAM and OAM can be regarded as an attribution to a superposition with a phase shift of 2  between two odd and even vector modes as eigenmodes of optical fiber waveguides [13]. Ho wever, when spatially arbit rarily superposing two vectorial fields, the resultant fields may exhibit d istinct features on polarization state and spatial phase distribution that correspondingly spins and spirals the electric field vector. As a result, the longitudinal SAM and OAM as mean values are not confined to integers [14,15]. Non-integer OAM also can be generated by using non-integer spiral phase plates [17], or with differential operators [18], astig matic elements [19], and etc [20]. Especially, in high-contrast-index waveguides or in the case of nonparaxial propagation, there exists a nonnegligible angular mo mentum shift fro m SAM to OAM. It is analogous to the spin-to-vortex conversion of a paraxial beam in uniaxial anisotropic crystal or the polarization dependence of both SAM and OAM in nonparaxial case due to Berry-phase shift [16]. No-integer OAM and spin-orbit shift provide a discrete mu lti-d imensional state space for photons, which may find applications in quantum informat ion processing, encryption and quantum digital spiral imag ing [21]. Furthermore, in this case of nonparaxial fields, the transverse SAM increases sharply. The dominant ability of transverse spin is to achieve spin-controlled unidirectional propagation of light in nanofiber, surface plas mon-polaritons, and photonic-crystal waveguide [22], connected with the quantum spin Hall effect of light [23].
In this Letter, beyond paraxial appro ximation, we employ fully vectorial formulat ion derivation to comprehensively study the derivation and evolution of all optical angular mo metu m co mponents fro m spatially arbitrarily superposing vectorial fields. We visualize the superposition to get arbitrary no-integer longitudinal SAM and OAM, and present the inherent spin-orbit shift and transverse SAM as a result of large longitudinal electric fields. We also show the whole spin flow as a co mbination of longitudinal and transverse SAM in the nonparaxial orthogonal superposition fields, and discuss the handedness of spin flow and the property of spin-momentum locking of transverse SAM.
Firstly, we derive the optical angular mo mentu m components of general optical fields in fully vectorial formulat ion. Based on the canonical mo mentum exp ression [24], the longitudinal OAM density can be deduced in the cylindrically symmetric coordinate systems, (1) where  is the angular frequency of light, 0  stands for permittiv ity in vaccu m. In general, r  ,   , and z  are the radial, azimuthal, and longitudinal field co mponents of vector fields, respectively. It represents the intrinsic vortex-dependent OAM that depends unpon spatially varying phase distribution. As for the transverse OAM density, it can be given by . (2) which is origin-dependent and belongs to the extrinsic OAM. The SAM density with three components that is along the radial, azimuthal and longitudinal direct ion, respectively, can be written as The longitudinal SAM component z S is well-known commonplace, of wh ich the direction is determined by the polarization degrees of freedom. Ho wever, the unusual transverse SAM r S and  S are independent on the polarization of beam, but are determined by the longitudinal electirc field. Seeing Eqs. (1) and (3), the longitudinal OAM has an intrinsic longitudinal-SAM-dependence, associated with the origin of spin-orbital shift. Additionally, the t ime averaged energy density per unit length can be given as where  is permittivity of waveguide material. When superposing two azimuthal-dependent vectorial fields with a phase shift of 2  and a normalized energy allocation, one can get the arbitrary resultant vectorial field that propagates along the +z direct ion. It can be formu lated by

 
, it corresponds to the azimuthal polarized mode TE01.
Insetting three vectorial electric field co mponents of Eq. (5) into Eqs. (1) and (3), beyond paraxial appro ximation, we can express the longitudinal OAM density and the classical longitudinal SAM density The transverse SAM density can be written by, . The direction of transverse SAM only depends upon the propagagtion direction of light inherently regulating the phase correlation of rad ial and longitudinal electric field co mponents. It gives rise to the phenomenon of spin-mo mentum locking belonging to the intrinsic property of transverse SAM, which makes sense that it enables spin-controlled unid irect ional propagation of light [22].
The mean SAM and OAM in the unit of 1  can be written as the ratio of the integral angular mo mentum to the averaged energy in the Minkowski expression form [25], as follows, for OAM, where the i n is the refract ive index of optical med iu m, subscript i corresponds to different waveguide layer, and  represents the amount of spin-orbit shift in orthogonal superposition state, and given by   It is just dependent on the superposition states and the winding order m . In any superposition states, it is conserved for the total longitudinal angular momentum. Analogously, we give the expressions in terms of the rat io of the integral transverse OAM and SAM to the averaged energy, as follows, cos sin  . These superposition states do not carry SAM and OAM. Beyond them, any other states on the semi-spherical surface would spin and spiral the spacial electric field vector. It gives rise to spatial-variant asymmetry of polarization ellipticity, analogous to the combination of hybrid states of polarization in Refs. (26). It thereby induces intriguing arbitrary non-integer values of mean SAM and OAM. The colour map on semi-spherical surface represents different mean OAM values. The yellow semicircle divide the semi-spherical surface into two components. The polarization orientation of the first quadrant is characterized by left handedness, and that of the second quadrant is right handedness. In the center of two quadrants, i.e.     as orthogonal superposition states, the polarization states display uniform ellipse distribution with an azimuthal symmetry. Th is nonparaxial superposed vectorial field carrying non-integer OAM can be found in high-contrast-index waveguides, for instance, hollow ring-core fiber [27], where exists considerable spin-orbit shift. It degrades into the fully circular polarization carry ing integer SAM and OAM in the weakly guid ing fibers due to the nearly identical rad ial and azimuthal field co mponents. In reverse, when 0   or 2  , and 1   , the resultant vectorial fields can describe fiber-guided vector modes and can be combined by two purely circu larly polarized OAM modes [28]. Note that when 2  varies, it changes the azimuthal angle of the overall superposition fields, but does not affect the investigation on SAM and OAM. It is worth pointing out that there is a noticeable difference between the resultant fields expressed by Eq. (5) and the fields described in h igher-order Poincaré sphere [28]. in the nonparaxial case. For nonparaxial fields, TM01 and TE01 modes are split, whereas even and odd HE/ EH modes are degenerated associated with the effective refractive index. Considering the combination of longitudinal and transverse SAM density, we plot the spin flow trajectories of uniform vector field TM01 that is closed loop as shown in Fig. 2(a). Note that the TE01 vectorial field does not have transverse SAM, because of no longitudinal electric field, as shown in Fig. 2(b). We further present the spin flo w of orthogonal superposed fields   give the spacial phase distribution of these fields that reflect optical OAM states. These spin trajectories as a co mb ination of longitudinal and transverse SAM are characterized by helixes. It arises as a result of phase offset of transverse SAM lobes along the propagation around the optical axis based on the defination of SAM fro m Eqs. (3) and (9) with the reverse direct ion as the propagation direction, disaligned with the linear mo mentu m k , as shown in Figs. 2(d) and 2(f). Significantly, in all cases above, the direction of transverse SAM remains the same, and is locked to the linear mo mentu m k , which is the intrinsic property of transverse SAM [9].
In conclusion, the resultant field by arbitrarily superposing vectorial fields has been comprehensively investigated in terms of various optical anguler mo mentum in this Letter. For a nonparaxial superposed field, there are remarkably inherent spin-orbit shift and transverse SAM due to the large longitudinal electric field. We can obtain arbitrary non-integer OAM and fractional SAM by arbitrarily superposing the vectorial fields. The whole spin flow of TM01 and higher-order superposing fields man ifests mu lple-fold helical trajectories, of which the fold nu mber is the same as the azimuthal order. This helical optical spin can enrich optical fo rce and torque as new degree of freedo m that enables to expliot the next generation of photonic traps. Our revelation and presentation in this Letter provide a systematic physical insight into optial angular mo mentum derived fro m vectorial field superposition, including the longitudinal and transverse, SAM and OAM components, spin-orbit shift and evolution, and the property of transverse SAM. It may facilitate the development of optical vectorial fields and optical angular mo mentu m in fundamental studies and various applications, such as optical commun ications, optical manipulation, and quantum application, etc.