Spin–orbit coupling and conservation of angular momentum flux in non-paraxial imaging of forbidden radiation

Rigorous, closed form expressions are derived for non-paraxial imaging of sources of multipole radiation from which conservation of angular momentum (AM) flux is established. Coupling of spin and orbital optical AM flux is also quantitatively investigated, highlighting the importance of spin–orbit interactions in high numerical aperture imaging.


Introduction
Radiation from atoms or molecules arises through electron transitions; however, quantum selection rules dictate that many such transitions are forbidden under an electric dipole approximation. So-called forbidden radiation can nevertheless still be emitted via electric dipole forbidden transitions due to higher order multipole corrections to the transition matrix, such as the magnetic dipole or electric quadrupole coupling terms. Despite the inherent lower transition probabilities, sources of forbidden radiation now play a greater role in much of today's science. coincide. More complex optical setups can in turn be modelled as a combination of one or more lens pairs [20], such that fundamentally only imaging through a 4 f system need be considered. Critically, introduction of an aperture stop in the common focal plane yields an afocal and telecentric system from both object and image spaces, such that vectorial ray tracing can be used to compute the image field [21].
Consider then the action of the imaging system on a plane wave, originating in the object space, travelling in a direction s(θ 1 , φ 1 ) = (sin θ 1 cos φ 1 , sin θ 1 sin φ 1 , cos θ 1 ) and with associated generalized (3 × 1) Jones vector E(θ 1 , φ 1 ) (see figure 1). Each lens in the system rotates the polarization vector of the ray about an axis perpendicular to the meridional plane in which it propagates, as can be treated using a generalized Jones matrix formalism [21]. The geometric field distribution on the Gaussian reference sphere of the focusing lens, assuming a transmission geometry, for a single ray is hence where a(θ) is an apodization factor (= √ cos θ for an aplanatic lens) required to conserve energy at each lens and R µ (α) denotes a rotation of α about the µ coordinate axis [21]. Note (θ 2 , φ 2 ) define the direction of propagation of the ray in image space.
Devaney and Wolf have however given the angular spectrum of multipole radiation of order (l, m) as [22] Here, P m l (cos θ ) are the associated Legendre polynomials. Rewriting equation (2) in terms of Cartesian field components allows direct substitution into equation (1), yielding The final image field E(ρ 2 ) at a position ρ 2 relative to the geometric image point for a monochromatic field of wavelength λ, can then be found using the Debye-Wolf [23] integral, originally formulated by Ignatowsky [24], are the cylindrical polar coordinates of ρ 2 , while f 2 and α 2 are the focal length and semi-angle of convergence of the focusing lens respectively and k = 2π/λ. Using the identity [25] 2π 0 e inβ exp[ia cos(β − γ )]dβ = 2π i n e inγ J n (a), where J n (a) is the nth order Bessel function of the first kind, the azimuthal integration can be performed analytically, yielding The extra term z mp cos θ 1 has been introduced to the exponent to allow lateral displacements of the multipole source. Transverse shifts to a position ρ mp can further be incorporated, assuming shift invariance of the imaging system such that E ν,off-axis Using the curl equations it can similarly be shown that the associated magnetic field distributions  Figure 2 shows the calculated field distributions (amplitude and phase) in the image plane of a 4 f imaging system (100×) in which the collector lens has a NA of 0.95, for assorted 6 electric and magnetic multipoles. Azimuthally periodic phase distributions are clearly evident, suggesting the presence of OAM. It is furthermore interesting to note that due to the form of the K ±ν lm and K 0ν lm integrals, zero total energy density is always seen on axis when imaging multipole radiation with |m| > 1.

Angular momentum (AM) flux
Following Barnett's definition [15], the total AM flux density (3 × 3) tensor is defined here as M = ρ × T, such that M i j = mn imn ρ m T n j , where T is Maxwell's electromagnetic stress tensor and imn is the Levi-Civita symbol. Restricting to time harmonic fields of angular frequency ω and performing cycle averages, the elements of Maxwell's stress tensor are given by where δ i j is the Kronecker delta, such that the flux of the z-component of the OAM and SAM in the z-direction is respectively In general, these fluxes are a function of position ρ 2 , however this dependance has been dropped for clarity. The integrated flux M zz through a plane of constant z, can then be shown to be where M 0 = ±π 2 C 2 lm ε 0 f 2 2 /λ and the azimuthal integration has again been performed analytically. Integrated fluxes of the xand y-component of AM through a fixed transverse plane is zero by symmetry. Symmetry in the kernels furthermore ensures that M orbit zz = M spin zz = 0 when m = 0. Similarly, an integrated energy flux through a fixed plane can be found by integrating the z-component of the Poynting vector, viz Given equations (3)-(5) it quickly follows that which holds regardless of which transverse plane is considered (mathematically this ratio holds for any surface which is rotationally symmetric with respect to the z-axis e.g. a hemisphere, although a transverse plane is of greater physical interest here). The ratio R 1 can also be considered before imaging, i.e. in the object space of the multipole source, whereby the same result holds. Evidently the ratio R 1 is not only a conserved quantity upon propagation in both spaces, but is also preserved upon imaging. Finally, it is worthwhile to investigate the relative magnitudes of the SAM and OAM flux in non-paraxial imaging systems. Accordingly, the ratio R 2 = M spin zz /M orbit zz is plotted in figure 3 for different multipole orders (m = 0) as a function of collector lens NA (letting z 2 = 0). These results are invariant with sin α 2 , and hence also magnification of the imaging system. In the low NA limit (α 1 → 0) the SAM flux is always non-zero, accounting for a fraction of 1/|m| of the total AM flux in the z direction (or equivalently F z /ω orh per photon). In contrast, the OAM constitutes min(|m ± 1|) of the total axial flux (in units of F z /ω).
As the non-paraxiality of the system increases, so the relative proportion of SAM flux drops, ultimately to a value close to zero. Interestingly, the conversion of SAM flux to OAM flux for l = 1, m = ±1 multipoles is negligible with R 2 1 at α 1 ∼ π/2. Observation of the SOE for these sources hence requires use of immersion lenses. As the order l is increased, the SOE progressively becomes more apparent with the drop off to zero of R 2 occurring at lower NAs. Conversely, the reduction in SAM flux for higher |m| sources occurs at higher NAs. Qualitatively, this behaviour can be understood by noting that uniformly polarized fields possess only SAM, such that OAM is introduced by breaking the uniformity. Lower l corresponds to less directional, and hence inherently more uniform, radiation patterns, which is only broken by the spatially dependent polarization mixing introduced by lenses of high NA. Higher |m| furthermore implies that the direction of maxima in the radiation lobes form smaller angles with the z-direction, such that the field collected by the lens is more uniformly polarized. Consider for example a z oriented dipole (l = 1, m = 0) which gives rise to a (highly non-uniform) azimuthally polarized field distribution in the back focal plane of the collector lens, whilst the combination of out of phase xand y-oriented dipoles (m = ±1) yields a fairly uniform circularly polarized beam. Large |m| sources hence again require higher NA lenses to introduce polarization mixing.

Conclusions
In summary, closed form rigorous electromagnetic results have been derived pertaining to the imaging of multipole sources of arbitrary order by a high NA 4 f imaging system. Orbital, spin AM and energy fluxes have been calculated in the focal region, whereby it was shown that the ratio of the total integrated AM flux to energy flux in the z direction was a conserved quantity upon both propagation and imaging. The magnitude of the SOE in non-paraxial imaging was also quantitatively studied. Possible applications and future research directions founded on these results could include design and optimization of optical detection modalities based on the SOE, measuring sample induced spin-orbit interactions [26] or imaging of forbidden radiation. Consideration could also be given to more arbitrary sources represented by a superposition of multipoles. On a final note, it should be mentioned that although throughout this paper a classical treatment has been given, following the discussion in [15] the problem can equally be formulated in a quantum domain.