We present a general theory of spin-to-orbital angular momentum (AM) conversion of light in focusing, scattering, and imaging optical systems. Our theory employs universal geometric transformations of non-paraxial optical fields in such systems and allows for direct calculation and comparison of the AM conversion efficiency in different physical settings. Observations of the AM conversions using local intensity distributions and far-field polarimetric measurements are discussed.
©2011 Optical Society of America
Together with energy and momentum, angular momentum (AM) is one of the most important dynamical characteristics of light [1,2]. For paraxial fields in free space, the eigenmodes of the AM operator are circularly-polarized vortex beams, where polarization helicity specifies the value of spin AM (SAM) per photon, whereas the vortex charge yields the orbital AM (OAM) per photon. Generation of the helicity-dependent vortices in optical systems signifies the spin-to-orbital AM conversion. This phenomenon is attracting noticeable attention in recent years and occurs in two basic situations: (A) upon interaction of paraxial light with anisotropic media possessing certain azimuthal symmetries [3–8] and (B) in essentially non-paraxial optical fields in locally-isotropic media [8–24]. (Optical fibers  can be regarded as an intermediate case.) In the case (B), nonparaxial AM states of light appear mostly upon (i) tight focusing [9–17], (ii) scattering by small particles or apertures [16–24], and (iii) in high numerical aperture (NA) imaging of small particles [16,17]. Importantly, the case (iii), involving a combination of focusing and scattering processes, represents a fundamental mechanism for translating the fine spin-orbit effects at micro- and nano-scales to the far-field.
The spin-to-orbital AM conversion in anisotropic paraxial systems is an extrinsic phenomenon produced by the azimuthally-dependent phase difference between ordinary and extraordinary modes, which is well-studied and reviewed [25,26]. On the other hand, the AM conversion in nonparaxial fields owes its origin to the intrinsic properties of light, geometric Berry phases, and fundamental separation of the SAM and OAM in the generic nonparaxial case [27–29]. The spin-to-orbital conversion in nonparaxial light has been considered for different systems using different ad hoc methods, such as Debye-Wolf theory for focusing or Mie theory for scattering. Furthermore, the spin-orbit interaction in a variety of similar imaging schemes is ascribed either to focusing [30,31], or to scattering , or to anisotropy [33,34]. Obviously, all these mechanisms co-exist and their unifying description and discrimination is necessary.
In the present paper we examine the spin-to-orbital AM conversion that appears in nonparaxial optical fields interacting with locally-isotropic media upon (i) focusing, (ii) scattering, and (iii) imaging. We develop a unifying theory of these effects based on universal, purely geometrical transformations of the fields and fundamental AM operators. Our approach highlights the common geometric origin of AM conversion due to different optical processes and allows us to compare the conversion efficiency in different physical settings depending on the aperture angles and properties of the incoming light.
2. Basic equations
We consider monochromatic wave electric fields characterized by their complex amplitudes : . Similar relations are implied when the wave magnetic field is involved. The Fourier or angular spectra of the fields are denoted by and . Throughout the paper we consider optical systems with axial symmetry about the -axis, and, for the AM description, it is convenient to use the basis of circular polarizations with respect to the optical axis. Given that are the basic vectors of the laboratory Cartesian frame and is the vector of complex amplitudes of a monochromatic electric field in this basis, the basic vectors and electric field components in the circular basis are:
The operators of the -component of the OAM and SAM of light in the Cartesian basis are : and , where is the azimuthal angle (either in coordinate or momentum space, depending on representation) and is the Levi-Civita symbol. In the circular basis, these operator become, correspondinglyEq. (3), for generic nonparaxial fields [29,35,36], but they do not affect the OAM and SAM expectation values which will be calculated for different optical systems below.
3. High-NA focusing
Let us consider the Debye–Wolf theory of focusing with a spherical lens [37,38], Fig. 1 . The incident field is paraxial, and one can neglect its -component: . Entering partial rays with the wave vector are marked by coordinates at the entrance pupil, which can be expressed via spherical angles defined with respect to the origin in the focal point: ,, and ( is the focal distance of the lens). After refraction, the partial rays converge at the focal point and have the nonparaxial k-vectors characterized by spherical coordinates in the momentum space: , , and (see Fig. 1) . The rays are distributed in the range , where is the aperture angle of the lens, and carry the electric fields . Thus,
The Debye–Wolf theory assumes that partial waves do not change their polarization state in the local basis attached to the ray, and the electric fields experience pure meridional rotations by the angle together with their k-vectors. This is an adiabatic approximation which neglects the polarization dependence of the refraction coefficients, cf . As a result, the focused field spectrum can be written using purely geometrical rotational transformation [16,40] , where , , is the matrix of rotation about the a-axis by the angle [ are the SO(3) generators of rotations, i.e., spin-1 matrices]. Explicitly, the transformation of the field in the circular basis takes the form [16,41]:38]. Remarkably, the same unitary transformation describes transition from the global circular basis (1) to the local helicity basis attached to the partial wave vector for a generic nonparaxial field . This reflects the fact that the refracted field does not change its polarization state in the helicity basis (labelled by subscript “H”):29], which could be generated, e.g., via focusing by an axicon lens with fixed . The focusing with a spherical lens differs only by a smooth -distribution of the produced plane-wave spectrum.
The three successive rotations, , , and , in the transformation indicate, respectively: (i) azimuthal rotation of the coordinate frame superimposing the -plane with the local meridional plane, (ii) refraction of the field on the angle therein, and (iii) the reverse azimuthal rotation compensating the first one. Because of the noncommutativity of the rotations, this transformation is accompanied by a generation of geometrical phases which appear in the form of vortices in the off-diagonal elements of Eq. (5). These elements describe effective transitions between different AM modes. For instance, if the incident wave is -circularly polarized, , i.e., has only the component, the refracted wave is , i.e., and . This indicates the generation of the component with the charge- vortex and the oppositely-polarized component with the charge- vortex . We emphasize once again that the actual helicities of partial waves remain unchanged, Eq. (6), and these components appear because of the observation of the redirected rays in the same laboratory frame. Nonetheless, the azimuthal phases signify real generation of the OAM in the laboratory frame, i.e., the spin-to-orbital AM conversion. Let the incident wave be a paraxial vortex beam with circular polarization and vortex charge , and let us denote the AM state of light with respect to the -axis by the OAM and SAM quantum numbers: . Then, the polarization transformation (5) can be symbolically written asEquation (7) exhibits the conservation of the total AM quantum number in each term: , where the last term corresponds to the longitudinal field which carries no SAM. Both the transverse field with the opposite polarization (b-term) and the longitudinal field (-term) contribute to the AM conversion. However, in the paraxial approximation, , one has and , so that the main contribution is due to the -component of the field .
First, we calculate the local (angle-resolved) OAM and SAM densities, and , without integration over the -distribution of the field. Using operators (3) in the circular basis, one can writeEq. (5) the refracted field is , and Eqs. (8) brings about39,42]) associated with the azimuthal distribution of partial rays with a fixed polar angle . Such distribution corresponds to the circular contour on the sphere of directions and the Berry phase gained after traversing this contour is . Equations (10) characterize a -dependent spin-to-orbital AM conversion which vanishes as in the paraxial limit . This conversion is described by the Berry-phase term, whose appearance is explained in  in terms of phase matching of the geometrical-optics rays and quantization of caustics.
To calculate the OAM and SAM expectation values, and (throughout the paper we imply values per photon), one has to integrate both numerators and denominators of Eqs. (8) over the spherical angles . Implying integration , the OAM and SAM expectation values are15,29])43,44]:
Evaluation of Eq. (13) for (i.e., the vortex-core distribution) brings aboutEqs. (15) yield . Then, for small aperture angle, , and . For maximal aperture , one has, , and the efficiency of the spin-to-orbital AM conversion reaches the value of . For higher values of , the efficiency of the conversion increases as the field becomes concentrated at higher angles (see Fig. 8 below).
Let us compare Eqs. (12) and (13) with other calculations of the OAM and SAM of the tightly focused circularly-polarized light [9,13,15,28,29]. First, our results differ from calculations  based on approach of  with a nonconserved total AM: . At the same time, the post-paraxial estimation is analogous to the multipole expansion of a strongly focused Gaussian beam . Equations (12) and (13) are similar to calculations in  and , but differ from the final result in , apparently due to an arithmetic inaccuracy therein. Finally, Eq. (12) is entirely analogous to the OAM and SAM of nonparaxial Bessel beams with well-defined helicity  modified by averaging over the polar angles .
So far, we used only the plane-wave spectrum of the focused field, . The actual real-space electric field near the focal point is determined by the interference of the partial plane waves and is given by the Debye integral similar to Fourier transform [37,38]:15,29].):Eq. (17) can be regarded as the total intensity of the electromagnetic field.
It is seen that the intensity distribution of the focused field depends on both the OAM (vortex) and SAM (helicity) quantum numbers, and . It is invariant with respect to the transformation but neither with respect to nor , which reflects symmetry typical of spin-orbit interaction. For a scalar wave, the intensity (17) would be . Nonparaxial vector terms proportional to describe spin-orbit coupling between the main mode of the order to the modes of the order and which appear owing to the azimuthal geometric phases in the matrix , Eq. (5). Dependence of the intensity distribution (17) on and is closely related to the spin-to-orbital AM conversion and the OAM expectation value , Eq. (12). Namely, generalizing the Bessel-beam results of  for the averaged polar angle , Eq. (13), the “mechanical” radius corresponding to the orbital angular momentum and underlying the focal intensity (17) can be estimated asFig. 2 . Similar polarization-dependent properties of the focal intensities have appeared in [10,11,14,15]. In particular, it follows from Eq. (17) that the intensity of the beams with antiparallel OAM and SAM, , does not vanish on the axis for . This was experimentally verified in [45,46]. In general, the size of the focal spot for a strongly focused beam has a fundamental lower bound that depends not only on the beam’s directional spread but also on its OAM and SAM, such that beams with antiparallel OAM and SAM can achieve tighter focal spots than those for which these AM are parallel . Note that asymmetric focusing with a truncated lens or distribution of the incident light brings about the - and -dependent transverse shifts of the focal-spot intensity centroid, which is proportional to , i.e., orbital and spin Hall effects [29,30,47].
4. Dipole scattering
Scattering of paraxial light by a nano-particle located at the origin essentially represents the spherical redirection of partial plane waves, Fig. 3 . We examine the simplest dipole approximation when the scattered spherical wave is generated by the dipole moment proportional to the incident field at the origin. Since higher-order paraxial beams (9) with have zero field in the center, , below we consider an incident plane wave with circular polarization, . The electric far field of the scattered wave is 16]:
Projection (20) is a nonunitary transformation, and it does not preserve the local helicity of the scattered field. Indeed, in the helicity basis attached to the sphere of partial -vectors, the scattered field is [cf. Equation (6)]:Eq. (20) demonstrates features that are quite similar to those of the focusing transformation (5). In particular, the off-diagonal geometric-phase elements of the matrix produces spin-to-orbital AM conversion which can be symbolically written similarly to Eq. (7):16–24]. While in the case of focusing the polarization is not changed in the helicity basis, the transformation (21) of the scattered field can be written as the following helicity transition:19,20], and the scattering transformation (20) becomes similar to the focusing one, Eq. (5).
Assuming an incident plane wave with circular polarization, , we determine the scattered field (20) and calculate the OAM and SAM angle-resolved densities using Eqs. (8):49], and similar results [but without factor] have appeared for the Rayleigh scattering . Equations (24) yield for the paraxial angles and the total conversion at (see Fig. 8 below). The integral OAM and SAM values are determined via Eqs. (11) with the spherical integration , which results in18,49] (see Fig. 8 below). Equations (24) and (25) demonstrate conservation of the local and integral total AM per photon, Eq. (14).
It is worth remarking that the converted part of the AM in Eq. (24) can be expressed asEq. (23). Here the quantity coincides with the degree of polarization of the scattered unpolarized light . Thus, the changes in the degree of polarization upon scattering are also connected with the above-considered geometric transformations of the wave field. In particular, the depolarization of multiply scattered polarized light and typical four-fold polarization patterns of the backscattered light are intimately related to the spin-to-orbital AM conversion [19,20]. In the weak-scattering approximation of small scattering angles ( in a single scattering), these depolarization effects can be explained via Berry-phase accumulation along the partial scattering paths [19,20,50,51]. This establishes a geometric-phase link between the AM conversions in focusing and scattering processes. For strong single scattering event (), the Berry-phase (adiabatic) approximation is not applicable because the geometric transformation (20) represents a projection rather than parallel-transport rotation (5) of the field.
5. Imaging of nanoparticles
Strongly focused or scattered fields are essentially nonparaxial, which is the main source of the spin-orbit phenomena. Accordingly, the AM conversion can be detected via various near-field methods: e.g., tracing motion of testing particles [11,12] or using near-field probes [14,21,24,47]. It should be noted that the use of testing particles is somewhat ambiguous as they can undergo orbital motion due to both orbital and spin energy flows, i.e., OAM and SAM [52,53]. At the same time, traditional paraxial-optics detectors are unable to measure adequately the spin-orbit phenomena in nonparaxial fields with strong longitudinal field component. It turns out, however, that a standard imaging system successfully resolves this dilemma by transfering the spin-orbit coupling into a paraxial far-field, where it can be easily detected. There were several experimental observations [16,31–34,54] of the spin-orbit interactions of light using imaging scheme: (i) focusing of the incident paraxial light with a high-NA lens; (ii) scattering by a small specimen; (iii) collection of the scattered light by another high-NA lens transforming it to the outgoing paraxial light, see Fig. 4 . In most cases the observed effects were ascribed either to focusing or to scattering process, although all the three elements of the imaging system contributed to the effect. The ‘lens-scatterer-lens’ system (Fig. 4) represents the basis for optical microscopy and it is important to describe the spin-orbit effects in such imaging system taking into account all its elements [16,17].
Since both the input and output fields in the imaging system are paraxial, the transformation of the field by the system can be described by an effective Jones matrix which gives the angle-resolved polarization state of the output field. Using successive applications of the geometric transformations of the first lens, Eq. (5), together with the Debye integral (16), dipole-scattering transformation (20), and the inverse transformation (5) of the collector lens, one can obtain the 3D transformation operator of the system , :
. (26)Here , the angles and mark the partial rays at the entrance and exit pupils, respectively (Fig. 4), is the position of the scattering particle near the common focus of the two lenses, and the phases are
First, let us consider the symmetric case when the scatterer is located precisely in the focus: . If the incident field is a homogeneous plane wave (which implies ), one can evaluate the integral (26) analytically. The resulting Jones operator is :Eq. (5), and it describes the spin-to-orbital AM conversion between two paraxial states of light:Eq. (23), one can see that, seemingly, the imaging scheme transfers the scattering-induced AM conversion which appears in the local helicity basis to the paraxial field and laboratory circular basis. AM conversion (29) strongly resembles paraxial spin-to-orbital convertors based on locally-anisotropic axially symmetric structures [3–7,25,26]. In our case, the -dependent helical phases arise from purely geometrical 3D transformations of the field inside the imaging system, which demonstrate effective birefringence of the radially (TM) and azimuthally (TE) polarized modes. Assuming -circularly polarized incident wave, , the output field is , i.e., and . Using Eqs. (8), we determine the local OAM and SAM densities at the exit pupil:Eqs. (11), implying integration over the exit pupil, , one can obtain the integral OAM and SAM values:
Since the output field is paraxial, its polarization properties can be characterized by space-variant Stokes parameters. The normalized Stokes vector can be calculated using the Pauli matrices :Eq. (30), coincides with the local degree of circular polarization, i.e., the third component of the Stokes vector: . Indeed, for paraxial fields it is determined by the transverse part of the operator : . The complete description of the polarization contains more information than the spin AM of light . In particular, the first two components of the Stokes vector (32) can evidence conversion to the OAM, Eq. (29). Indeed, calculating Eq. (32) for the incident circularly-polarized plane wave, we obtainFig. 5 . Parameters and demonstrate typical four-fold patters which signify generation of the oppositely polarized component with the optical vortex [16,19,20,33,50].
The cylindrical symmetry is broken in the imaging system if the specimen is transversely shifted in the focal plane. In this case, the system is not rotationally-invariant about the -axis, and the total AM is no longer conserved. This results in the AM-dependent orthogonal shift of the center of gravity of the output field, i.e., the Hall effect of light. Considering a small subwavelength displacement of the specimen in the plane, , , the effective Jones matrix can be evaluated analytically from Eq. (26) as a correction to Eq. (28) : ,Eq. (35), we obtain16]. Figure 6 displays the intensity distributions and shifts of the field centroid at different positions of the scatterer. Similar -dependent deformations of intensity in the imaging system with incident paraxial vortex beams, i.e., orbital Hall effect, were observed in . Another striking manifestation of the spin Hall effect is a strong separation of local SAM densities in the incident linearly polarized light .
Alongside the Stokes polarimetry, which reveals conversion to different polarization and OAM modes, the state of the output light can be fully described via intensities and phases of the two polarization components following from the Jones matrices (28) and (34). Distributions of the intensities and phases of the right- and left-hand circularly polarized field components for the on-axis and off-axis scatterer are shown in Fig. 7 . It is seen that the nonzero intensity and the vortices in the oppositely polarized (with respect to the incident light) component indicates the spin-to-orbital AM conversion (29). At the same time, the intensity redistribution and phase gradient in the main polarization component are responsible for the shifts (36), , for an off-axis scatterer.
We have examined the spin-to-orbital AM conversion in basic optical systems involving nonparaxial fields. The AM conversion originates from geometric transformations of the wave field: parallel-transport rotations in the case of focusing and spherical projections in the case of scattering. Despite the fact that problems of spin-to-orbital conversion were considered in recent years both in focusing and scattering systems, a number of controversies and the use of dissimilar approaches did not allow one to obtain solid quantitative results and compare the efficiency of the conversion in different systems. Our approach unifies various treatments of the AM in different nonparaxial optical systems and is based on the geometric field transformations and canonical AM operators. As a result, one can compare the efficiency of the spin-to-orbital conversion in high-NA lens focusing, Rayleigh (dipole) scattering, and in far-field “lens-scatterer-lens” imaging system. Figure 8 shows both angle-resolved OAM and SAM densities and integral OAM and SAM values (per photon) as dependent on the aperture angle. For comparison we also plot there the OAM and SAM of nonparaxial Bessel beams calculated in . One can see that for the plane incident wave () the conversion efficiency decreases in a sequence: “Bessel beam” – “scattering” – “focusing” – “imaging”. Curiously, the corresponding maximum efficiencies reached at the aperture angle are equal to 1, 1/2, 1/3, and 1/4. The total spin-to-orbital conversion for Bessel modes with was observed in a plasmonic experiment .
Although the OAM and SAM are important dynamical characteristics of light, it is difficult to measure them directly in optical systems. In particular, orbital motion of small testing particles [11,12] cannot be used as a measure of the OAM because both the spin and orbital energy flows can be responsible for it [52,53]. However, the close connection between the intensity distributions (in particular the radius of the focal spot) and the OAM values, Eq. (18) and Fig. 2, enables one to quantify the AM conversion via spin-dependent intensity profiles. Also, as we have shown, the angle-resolved polarimetry in the far-field imaging systems enables one to reconstruct the field distribution and unambiguously characterize its AM features  (Section 5).
Importantly, universal mechanisms of the spin-to-orbital AM conversion manifest themselves not only in optical but also in plasmonic [14,22–24,47] and semiconductor  systems. Therefore, the presented unified geometric and operator description of the AM conversions provides a useful link between spin-orbit phenomena in nonparaxial wave fields of various nature.
This work was supported by the European Commission (Marie Curie Action), Science Foundation Ireland (Grant No. 07/IN.1/I906), and the Australian Research Council.
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