Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems

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 nonparaxial 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.

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][28][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 [32], 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.

Basic equations
We consider monochromatic wave electric fields ( ) 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: so that ( ) , , . Thus, transition from the Cartesian to circular basis is realized via the unitary transformation Throughout the paper we mostly analyze the fields in the circular basis, and the subscripts C are omitted.
, where φ is the azimuthal angle (either in coordinate or momentum space, depending on representation) and ijl ε is the Levi-Civita symbol. In the circular basis, these operator become, correspondingly Note that the vortex fields , with the eigenvalue 0 σ = . There are some fundamental mathematical difficulties in using canonical operators L and Ŝ , Eq. (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.

High-NA focusing
Let us consider the Debye-Wolf theory of focusing with a spherical lens [37,38], Fig. 1. The incident field ( ) 0 E r is paraxial, and one can neglect its z -component: Entering partial rays with the wave vector 0 , where c θ is the aperture angle of the lens, and carry the electric fields ( )  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 θ together with their k-vectors. This is an adiabatic approximation which neglects the polarization dependence of the refraction coefficients, cf. [39]. As a result, the focused field spectrum ( ) E k ɶ can be written using purely geometrical rotational transformation [16,40] ( ) , and cosθ is the apodization factor that ensures the conservation of the energy flow [38]. Remarkably, the same unitary transformation ( ) , U θ φ describes transition from the global circular basis (1) to the local helicity basis attached to the partial wave vector for a generic nonparaxial field [29]. This reflects the fact that the refracted field does not change its polarization state in the helicity basis (labelled by subscript "H"): (6) It follows from here that the high-NA focusing of a paraxial circularly-polarized field generates a nonparaxial pure helicity state of light. Recently, some of us considered nonparaxial vector Bessel beams with well-defined helicity [29], which could be generated, e.g., via focusing by an axicon lens with fixed 0 θ θ = . The focusing with a spherical lens differs only by a smooth θ -distribution of the produced plane-wave spectrum.
The three successive rotations,  vortex e 2iσφ . 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 σ and vortex charge ℓ , and let us denote the AM state of light with respect to the z -axis by the OAM and SAM quantum numbers: ,σ ℓ . Then, the polarization transformation (5) can be symbolically written as Equation (7) exhibits the conservation of the total AM quantum number in each term: const corresponds to the longitudinal field z E ɶ which carries no SAM. Both the transverse field with the opposite polarization (b-term) and the longitudinal field ( 2ab -term) contribute to the AM conversion. However, in the paraxial approximation, 1 θ ≪ , one has 2 / 4 b θ ≃ and 2 / 2 ab θ ≃ , so that the main contribution is due to the z -component of the field [11].
First, we calculate the local (angle-resolved) OAM and SAM densities, z l and z s , without integration over the ( ) , θ φ -distribution of the field. Using operators (3) in the circular basis, For the field σ E ℓ ɶ , this results in (cf. [29,15]) where the averaged polar angle represents a measure of the directional spread of the field and is defined as [43,44]: Evaluation of Eq. (13) for the efficiency of the spin-to-orbital AM conversion reaches the value of 1 / 3 . For higher values of ℓ , the efficiency of the conversion increases as the field becomes concentrated at higher angles θ (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 [9] based on approach of [27] with a nonconserved total AM: z At the same time, the post-paraxial estimation 2 c / 4 z L σ θ ≈ is analogous to the multipole expansion of a strongly focused Gaussian beam [13]. Equations (12) and (13) are similar to calculations in [15] and [28], but differ from the final result in [15], 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 [29] modified by averaging over the polar angles θ .
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]: Here , , x y z = r ( / f k r ≪ and the common phase kf is subtracted). Calculating the Debye integral (16) and then the intensity, ℓ , for the incident paraxial beam (9) with the focusing transformation (5), one can derive (cf. [15,29]): Here we denoted sin k  (17) is given in Fig. 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, 0 σ < ℓ , does not vanish on the axis ρ = 0 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 [44]. Note that asymmetric focusing with a truncated lens or distribution of the incident light brings about the ℓ -and σ -dependent transverse shifts of the focal-spot intensity centroid, which is proportional to z L , i.e., orbital and spin Hall effects [29,30,47].

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 0 E at the origin. Since higher-order paraxial beams (9) with 0 ≠ ℓ have zero field in the center, where / r = r r ⌢ is the unit radial vector with spherical coordinates ( ) , θ φ of the observation point. Akin to focusing, the real-space spherical angles ( ) , θ φ serve as the coordinates in momentum space for the scattered far field. In this manner, the r ⌢ -vector plays the role of the / k k -vector for scattered waves and transformation (19) is reminiscent of the vector transformation ( ) Here ( ) is the projector onto the orthogonal plane,  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 k -vectors, the scattered field is [cf. Eq. (6) (21) Other than that, 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 Π produces spin-to-orbital AM conversion which can be symbolically written similarly to Eq. (7): The AM conversion upon scattering of light on various objects was analyzed in several papers [16][17][18][19][20][21][22][23][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: In the linear approximation in θ (for small scattering angles 1 θ ≪ ), the helicity (23) is conserved [19,20], and the scattering transformation (20) becomes similar to the focusing one, Eq. (5).
Assuming an incident plane wave with circular polarization, ( ) 0 σ σ ∝ E 0 e , we determine the scattered field (20) and calculate the OAM and SAM angle-resolved densities using Eqs. (8): Identical results were obtained for the field radiated by a rotating dipole [49], and similar results [but without ( ) Thus, the spin-to-orbital AM conversion has the efficiency 1/ 2 [18,49] (see Fig. 8 below). Equations (24) and (25)  Here the quantity ( ) P θ coincides with the degree of polarization of the scattered unpolarized light [48]. 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 ( 1 θ ≪ 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 (~1 θ ), 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.

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 nearfield 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][32][33][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 [16],   First, let us consider the symmetric case when the scatterer is located precisely in the focus: r s = 0 . If the incident field is a homogeneous plane wave 0 E (which implies 0 = ℓ ), one can evaluate the integral (26) analytically. The resulting Jones operator is [16]: where the aperture-dependent coefficient is ( ) ( ) ( ) Note that the spin density z s , Eq.
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 zaxis, 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 ( )