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

Transcription

1 Spin-to-orbital angular momentum onversion in fousing, sattering, and imaging systems Konstantin Y. Bliokh, 1,,* Elena A. Ostrovskaya, 3 Miguel A. Alonso, 4 Osar G. Rodrígue-Herrera, 1 David Lara, 5 and Chris Dainty 1 1 Applied Optis Group, Shool of Physis, National University of Ireland, Galway, Galway, Ireland A. Usikov Institute of Radiophysis and Eletronis, 1 Ak. Proskury St., Kharkov 6185, Ukraine 3 Nonlinear Physis Centre, Researh Shool of Physis and Engineering, The Australian National University, Canberra ACT, Australia 4 The Institute of Optis, University of Rohester, Rohester, New York, 1467, USA 5 The Blakett Laboratory, Imperial College London, SW7 BW, London, United Kingdom *k.bliokh@gmail.om Abstrat: We present a general theory of spin-to-orbital angular momentum (AM) onversion of light in fousing, sattering, and imaging optial systems. Our theory employs universal geometri transformations of nonparaxial optial fields in suh systems and allows for diret alulation and omparison of the AM onversion effiieny in different physial settings. Observations of the AM onversions using loal intensity distributions and far-field polarimetri measurements are disussed. 11 Optial Soiety of Ameria OCIS odes: (6.543) Polariation; (6.64) Singular optis; (35.137) Berry s phase Referenes and links 1. L. Allen, S. M. Barnett, and M. J. Padgett, eds., Optial angular momentum (Taylor and Franis, 3).. S. Franke-Arnold, L. Allen, and M. Padgett, Advanes in optial angular momentum, Laser & Photon. Rev. (4), (8). 3. G. Biener, A. Niv, V. Kleiner, and E. Hasman, Formation of helial beams by use of Panharatnam-Berry phase optial elements, Opt. Lett. 7(1), (). 4. A. Ciattoni, G. Cinotti, and C. Palma, Angular momentum dynamis of a paraxial beam in a uniaxial rystal, Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt ), (3). 5. L. Marrui, C. Mano, and D. Paparo, Optial spin-to-orbital angular momentum onversion in inhomogeneous anisotropi media, Phys. Rev. Lett. 96(16), (6). 6. G. F. Calvo and A. Pión, Spin-indued angular momentum swithing, Opt. Lett. 3(7), (7). 7. E. Brasselet, Y. Idebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Dynamis of optial spin-orbit oupling in uniaxial rystals, Opt. Lett. 34(7), (9). 8. M. Y. Darsht, B. Y. Zel dovih, I. V. Kataevskaya, and N. D. Kundikova, Formation of an isolated wavefront disloation, Zh. Eksp. Theor. Fi. 17, 1464 [JETP 8, 817 (1995)]. 9. Z. Bomon, M. Gu, and J. Shamir, Angular momentum and geometri phases in tightly-foused irularly polaried plane waves, Appl. Phys. Lett. 89(4), 4114 (6). 1. Z. Bomon and M. Gu, Spae-variant geometrial phases in foused ylindrial light beams, Opt. Lett. 3(), (7). 11. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. MGloin, and D. T. Chiu, Spin-to-orbital angular momentum onversion in a strongly foused optial beam, Phys. Rev. Lett. 99(7), 7391 (7). 1. Y. Zhao, D. Shapiro, D. MGloin, D. T. Chiu, and S. Marhesini, Diret observation of the transfer of orbital angular momentum to metal partiles from a foused irularly polaried Gaussian beam, Opt. Express 17(5), (9). 13. T. A. Nieminen, A. B. Stilgoe, N. R. Hekenberg, and H. Rubinstein-Dunlop, Angular momentum of a strongly foused Gaussian beam, J. Opt. A, Pure Appl. Opt. 1(11), 1155 (8). 14. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, Observation of the spin-based plasmoni effet in nanosale strutures, Phys. Rev. Lett. 11(4), 4393 (8). 15. P. B. Monteiro, P. A. M. Neto, and H. M. Nussenveig, Angular momentum of foused beams: Beyond the paraxial approximation, Phys. Rev. A 79(3), 3383 (9). 16. O. G. Rodrígue-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Optial nanoprobing via spin-orbit interation of light, Phys. Rev. Lett. 14(5), 5361 (1). 17. M. R. Foreman and P. Török, Spin-orbit oupling and onservation of angular momentum flux in non-paraxial imaging of forbidden radiation, New J. Phys. 13(6), 6341 (11). # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 613

2 18. A. Dogariu and C. Shwart, Conservation of angular momentum of light in single sattering, Opt. Express 14(18), (6). 19. C. Shwart and A. Dogariu, Baksattered polariation patterns, optial vorties, and the angular momentum of light, Opt. Lett. 31(8), (6).. C. Shwart and A. Dogariu, Baksattered polariation patterns determined by onservation of angular momentum, J. Opt. So. Am. A 5(), (8). 1. D. Haefner, S. Sukhov, and A. Dogariu, Spin hall effet of light in spherial geometry, Phys. Rev. Lett. 1(1), 1393 (9).. Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, Observation of optial spin symmetry breaking in nanoapertures, Nano Lett. 9(8), (9). 3. L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbah, Eletromagneti spin-orbit interations via sattering of subwavelength apertures, Phys. Rev. Lett. 14(8), 8393 (1). 4. Y. Gorodetski, S. Nehayev, V. Kleiner, and E. Hasman, Plasmoni Aharonov-Bohm effet: Optial spin as the magneti flux parameter, Phys. Rev. B 8(1), (1). 5. E. Hasman, G. Biener, A. Niv, and V. Kleiner, Spae-variant polariation manipulation, Prog. Opt. 47, (5). 6. L. Marrui, E. Karimi, S. Slussarenko, B. Piirillo, E. Santamato, E. Nagali, and F. Siarrino, Spin-to-orbital onversion of the angular momentum of light and its lassial and quantum appliations, J. Opt. 13(6), 641 (11). 7. S. M. Barnett and L. Allen, Orbital angular-momentum and nonparaxial light-beams, Opt. Commun. 11(5-6), (1994). 8. C.-F. Li, Spin and orbital angular momentum of a lass of nonparaxial light beams having a globally defined polariation, Phys. Rev. A 8(6), (9). 9. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, Angular momenta and spin-orbit interation of nonparaxial light in free spae, Phys. Rev. A 8(6), 6385 (1). 3. N. B. Baranova, A. Y. Savhenko, and B. Y. Zel'dovih, Transverse shift of a foal spot due to swithing of the sign of irular-polariation, JETP Lett. 59, 3 34 (1994). 31. B. Y. Zel dovih, N. D. Kundikova, and L. F. Rogaheva, Observed transverse shift of a foal spot upon a hange in the sign of irular polariation, JETP Lett. 59, (1994). 3. V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojo, and D. Petrov, Mie sattering distinguishes the topologial harge of an optial vortex: a homage to Gustav Mie, New J. Phys. 11(1), 1346 (9). 33. E. Brasselet, N. Muraawa, H. Misawa, and S. Juodkais, Optial vorties from liquid rystal droplets, Phys. Rev. Lett. 13(1), 1393 (9). 34. F. Manni, K. Lagoudakis, T. Paraïso, R. Cerna, Y. Léger, T. Liew, I. Shelykh, A. Kavokin, F. Morier-Genoud, and B. Deveaud-Plédran, Spin-to-orbital angular momentum onversion in semiondutor miroavities, Phys. Rev. B 83(4), 4137 (11). 35. S. J. van Enk and G. Nienhuis, Spin and orbital angular momentum of photons, Europhys. Lett. 5(7), (1994). 36. S. J. van Enk and G. Nienhuis, Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields, J. Mod. Opt. 41(5), (1994). 37. E. Wolf, Eletromagneti diffration in optial systems. I. An integral representation of the image field, Pro. Roy. So. London, Ser. A 53, (1959). 38. B. Rihards and E. Wolf, Eletromagneti diffration in optial systems. II. Struture of the image field in an aplanati system, Pro. R. So. Lond. A Math. Phys. Si. 53(174), (1959). 39. M. V. Berry, Interpreting the anholonomy of oiled light, Nature 36(611), (1987). 4. P. Török, P. D. Higdon, and T. Wilson, On the general properties of polaried light onventional and onfoal mirosopes, Opt. Commun. 148(4-6), (1998). 41. A. Bekshaev, K. Y. Bliokh, and M. Soskin, Internal flows and energy irulation in light beams, J. Opt. 13(5), 531 (11). 4. R. Bhandari, Polariation of light and topologial phases, Phys. Rep. 81(1), 64 (1997). 43. M. A. Alonso and G. W. Forbes, Unertainty produts for nonparaxial wave fields, J. Opt. So. Am. A 17(1), (). 44. M. A. Alonso, The effet of orbital angular momentum and heliity in the unertainty-type relations between foal spot sie and angular spread, J. Opt. 13(6), 6416 (11). 45. N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, Investigation of polariation effets for high-numerialaperture first-order Laguerre-Gaussian beams by D sanning with a single fluoresent mirobead, Opt. Express 13(6), (5). 46. Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, Investigation of the enter intensity of first- and seond-order Laguerre-Gaussian beams with linear and irular polariation, Opt. Lett. 3(16), (7). 47. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, Coriolis effet in optis: unified geometri phase and spin-hall effet, Phys. Rev. Lett. 11(3), 344 (8). 48. M. Born and E. Wolf, Priniples of Optis, 7th edn. (Pergamon, 5). 49. G. Moe and W. Happer, Conservation of angular momentum for light propagating in a transparent anisotropi medium, J. Phys. B 1(7), (1977). # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6133

3 5. V. Rossetto and A. C. Maggs, Writhing geometry of stiff polymers and sattered light, Eur. Phys. J. B 9(), (). 51. D. Laoste, V. Rossetto, F. Jaillon, and H. Saint-Jalmes, Geometri depolariation in patterns formed by baksattered light, Opt. Lett. 9(17), 4 4 (4). 5. M. V. Berry, Optial urrents, J. Opt. A, Pure Appl. Opt. 11(9), 941 (9). 53. A. Bekshaev and S. Sviridova, Mehanial ation of inhomogeneously polaried optial fields and detetion of the internal energy flows, arxiv: O. G. Rodrígue-Herrera, D. Lara, and C. Dainty, Far-field polariation-based sensitivity to sub-resolution displaements of a sub-resolution satterer in tightly foused fields, Opt. Express 18(6), (1). 55. Y. A. Kravtsov, B. Bieg, and K. Y. Bliokh, Stokes-vetor evolution in a weakly anisotropi inhomogeneous medium, J. Opt. So. Am. A 4(1), (7). 56. M. V. Berry, Paraxial beams of spinning light, Pro. SPIE 3487, 6 11 (1998). 1. Introdution Together with energy and momentum, angular momentum (AM) is one of the most important dynamial harateristis of light [1,]. For paraxial fields in free spae, the eigenmodes of the AM operator are irularly-polaried vortex beams, where polariation heliity 1 speifies the value of spin AM (SAM) per photon, whereas the vortex harge, 1,,... yields the orbital AM (OAM) per photon. Generation of the heliity-dependent vorties in optial systems signifies the spin-to-orbital AM onversion. This phenomenon is attrating notieable attention in reent years and ours in two basi situations: (A) upon interation of paraxial light with anisotropi media possessing ertain aimuthal symmetries [3 8] and (B) in essentially non-paraxial optial fields in loally-isotropi media [8 4]. (Optial fibers [8] an be regarded as an intermediate ase.) In the ase (B), nonparaxial AM states of light appear mostly upon (i) tight fousing [9 17], (ii) sattering by small partiles or apertures [16 4], and (iii) in high numerial aperture (NA) imaging of small partiles [16,17]. Importantly, the ase (iii), involving a ombination of fousing and sattering proesses, represents a fundamental mehanism for translating the fine spin-orbit effets at miro- and nano-sales to the far-field. The spin-to-orbital AM onversion in anisotropi paraxial systems is an extrinsi phenomenon produed by the aimuthally-dependent phase differene between ordinary and extraordinary modes, whih is well-studied and reviewed [5,6]. On the other hand, the AM onversion in nonparaxial fields owes its origin to the intrinsi properties of light, geometri Berry phases, and fundamental separation of the SAM and OAM in the generi nonparaxial ase [7 9]. The spin-to-orbital onversion in nonparaxial light has been onsidered for different systems using different ad ho methods, suh as Debye-Wolf theory for fousing or Mie theory for sattering. Furthermore, the spin-orbit interation in a variety of similar imaging shemes is asribed either to fousing [3,31], or to sattering [3], or to anisotropy [33,34]. Obviously, all these mehanisms o-exist and their unifying desription and disrimination is neessary. In the present paper we examine the spin-to-orbital AM onversion that appears in nonparaxial optial fields interating with loally-isotropi media upon (i) fousing, (ii) sattering, and (iii) imaging. We develop a unifying theory of these effets based on universal, purely geometrial transformations of the fields and fundamental AM operators. Our approah highlights the ommon geometri origin of AM onversion due to different optial proesses and allows us to ompare the onversion effiieny in different physial settings depending on the aperture angles and properties of the inoming light.. Basi equations We onsider monohromati wave eletri fields r,t harateried by their omplex i t amplitudes Er : r, t Re Er e. Similar relations are implied when the wave magneti field Hr is involved. The Fourier or angular spetra of the fields are denoted by # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6134

4 Ek and Hk. Throughout the paper we onsider optial systems with axial symmetry about the -axis, and, for the AM desription, it is onvenient to use the basis of irular polariations with respet to the optial axis. Given that ux, u y, u are the basi vetors of the laboratory Cartesian frame and L Ex, Ey, E T E is the vetor of omplex amplitudes of a monohromati eletri field in this basis, the basi vetors and eletri field omponents in the irular basis are: so that E C E, E, E the unitary transformation T ux iuy Ex ie y u, E, (1). Thus, transition from the Cartesian to irular basis is realied via 1 1 ˆ ˆ 1 EL V E C, V i i. () Throughout the paper we mostly analye the fields in the irular basis, and the subsripts C are omitted. The operators of the -omponent of the OAM and SAM of light in the Cartesian basis are [1]: L ˆ i / and S ˆ i ij, where is the aimuthal angle (either in oordinate ij or momentum spae, depending on representation) and ijl is the Levi-Civita symbol. In the irular basis, these operator beome, orrespondingly ˆ L, ˆ ˆ ˆ ˆ i V SV diag 1, 1,. Note that the vortex fields E exp i are eigenmodes of L ˆ with the eigenvalues the same time, the irularly polaried paraxial field E 1,, e and E,1, are eigenmodes of ˆ with the eigenvalues 1, whereas the third eigenvetor of ˆ E,,1, with the eigenvalue. There are orresponds to the -polaried field, some fundamental mathematial diffiulties in using anonial operators ˆL and Ŝ, Eq. (3), for generi nonparaxial fields [9,35,36], but they do not affet the OAM and SAM expetation values whih will be alulated for different optial systems below. 3. High-NA fousing Let us onsider the Debye Wolf theory of fousing with a spherial lens [37,38], Fig. 1. The inident field E E, E, T. E r is paraxial, and one an neglet its -omponent: Entering partial rays with the wave vetor k ku are marked by oordinates at the entrane pupil, whih an be expressed via spherial angles, defined with respet to the origin in the foal point: x f sinos, y f sinsin, and f os ( f is the foal distane of the lens). After refration, the partial rays onverge at the foal point and have the nonparaxial k-vetors harateried by spherial oordinates,, (3). At e in the # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6135

5 momentum spae: k ksinos, k ksinsin, and k kos (see Fig. 1) [38]. The x rays are distributed in the range,,, the lens, and arry the eletri fields Ek. Thus, y, where is the aperture angle of x y kx k, ky k, k k, (4) f f f and the lens performs a sort of Fourier transform translating the initial real-spae distribution Ek in the image spae. In doing so, the angles E r into the momentum distribution, serve both as real-spae oordinates for the inident field E and momentum-spae oordinates for the refrated field E. Fig. 1. Fousing of light by a spherial high-na lens. The inident paraxial field xy E is refrated in the meridional plane and transformed to a spetrum of plane waves (rays) E, with the k -vetors distributed on the sphere in k-spae:,,,,. The eletri field vetors are parallel-transported along eah partial ray with the loal heliity being onserved. The Debye Wolf theory assumes that partial waves do not hange their polariation state in the loal basis attahed to the ray, and the eletri fields experiene pure meridional rotations by the angle together with their k-vetors. This is an adiabati approximation whih neglets the polariation dependene of the refration oeffiients, f [39]. As a result, the foused field spetrum transformation [16,4] a Ek an be written using purely geometrial rotational Uˆ, Vˆ Rˆ ˆ ˆ ˆ Ry R V, where Rˆ exp ˆ a isa x, y,, is the matrix of rotation about the a-axis by the angle [ S ˆa i aij ij, are the SO(3) generators of rotations, i.e., spin-1 matries]. Expliitly, the transformation of the field in the irular basis takes the form [16,41]: i i a be abe ˆ os, ˆ i i E U E, U, be a abe, (5) i i abe abe a b # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6136

6 where a os /, b sin /, and os is the apodiation fator that ensures the onservation of the energy flow [38]. Remarkably, the same unitary transformation Uˆ, desribes transition from the global irular basis (1) to the loal heliity basis attahed to the partial wave vetor for a generi nonparaxial field [9]. This reflets the fat that the refrated field does not hange its polariation state in the heliity basis (labelled by subsript H ): H ˆ. H E U E E E (6) It follows from here that the high-na fousing of a paraxial irularly-polaried field generates a nonparaxial pure heliity state of light. Reently, some of us onsidered nonparaxial vetor Bessel beams with well-defined heliity [9], whih ould be generated, e.g., via fousing by an axion lens with fixed. The fousing with a spherial lens differs only by a smooth -distribution of the produed plane-wave spetrum. Rˆ, Rˆ y, and R The three suessive rotations, ˆ, in the transformation U ˆ indiate, respetively: (i) aimuthal rotation of the oordinate frame superimposing the x, - plane with the loal meridional plane, (ii) refration of the field on the angle therein, and (iii) the reverse aimuthal rotation ompensating the first one. Beause of the nonommutativity of the rotations, this transformation is aompanied by a generation of geometrial phases whih appear in the form of vorties in the off-diagonal elements of Eq. (5). These elements desribe effetive transitions between different AM modes. For instane, if the inident wave is -irularly polaried, E e, i.e., has only the E omponent, the refrated wave is Uˆ T i i be, a, abe T i i E E, i.e., a, be, abe E and E. This indiates the generation of the E omponent with the harge- vortex e i and the oppositely-polaried omponent E with the harge- vortex e i. We emphasie one again that the atual heliities of partial waves remain unhanged, Eq. (6), and these omponents appear beause of the observation of the redireted rays in the same laboratory frame. Nonetheless, the aimuthal phases signify real generation of the OAM in the laboratory frame, i.e., the spin-to-orbital AM onversion. Let the inident wave be a paraxial vortex beam with irular polariation and vortex harge, and let us denote the AM state of light with respet to the -axis by the OAM and SAM quantum numbers:,. Then, the polariation transformation (5) an be symbolially written as, os a, b, ab,. Equation (7) exhibits the onservation of the total AM quantum number in eah term: onst, where the last term, orresponds to the longitudinal field E whih arries no SAM. Both the transverse field with the opposite polariation (b-term) and the longitudinal field ( ab -term) ontribute to the AM onversion. However, in the paraxial approximation, 1, one has b /4 and ab /, so that the main ontribution is due to the -omponent of the field [11]. First, we alulate the loal (angle-resolved) OAM and SAM densities, l and s, without integration over the, -distribution of the field. Using operators (3) in the irular basis, one an write (7) # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6137

7 * * E E E ˆ E * * l, i, s,. E E E E The eletri field of the inident irularly-polaried vortex beam an be written as i E e E,, E F e, (9) and F sin for the vortex beams with the waist muh larger than the entrane pupil. From Eq. (5) the refrated field is ˆ os U, E, E e, and Eqs. (8) brings about l B, s 1 B, B 1 os b. (1) Here stands for the spin-rediretion Berry phase (see, e.g., [39,4]) B B assoiated with the aimuthal distribution of partial rays with a fixed polar angle. Suh distribution orresponds to the irular ontour onst,, diretions S /k 1 os d 1 os (8) on the sphere of k and the Berry phase gained after traversing this ontour is B [9]. Equations (1) haraterie a -dependent spinto-orbital AM onversion whih vanishes as B / in the paraxial limit 1. This onversion is desribed by the Berry-phase term, whose appearane is explained in [9] in terms of phase mathing of the geometrial-optis rays and quantiation of austis. To alulate the OAM and SAM expetation values, L and S (throughout the paper we imply values per photon), one has to integrate both numerators and denominators of Eqs. (8) over the spherial angles,. Implying integration d d d sin, the OAM and SAM expetation values are For the field * * de ˆ E de E L i, S. * * de E de E E, this results in (f. [15,9]) L (11), S 1, 1 os, (1) B B B where the averaged polar angle represents a measure of the diretional spread of the field and is defined as [43,44]: F os sin d P os. W F os sin d (13) Here P and W are the expetation values of the longitudinal momentum and energy of the foused field, whih are based on the operators pˆ kos and ŵ. Evidently, both angle-resolved densities and integral values of the OAM and SAM satisfy the onservation of the total AM per one photon: # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6138

8 about l s L S. (14) Evaluation of Eq. (13) for F ( ) sin (i.e., the vortex-ore distribution) brings (1 )! os os F1,, ;os. sin 3!! 3 (15) where F 1 (,, ; ) is the Gauss hypergeometri funtion. In the simplest ase of the inident plane wave with F, Eqs. (15) yield 3 and 1 os 1 os / 3sin. Then, for small aperture angle, 1, os 1 / 4 and L /4. For maximal aperture /, one has os / 3, L /3, and the effiieny of the spin-to-orbital AM onversion reahes the value of 1/ 3. For higher values of, the effiieny of the onversion inreases as the field beomes onentrated at higher angles (see Fig. 8 below). Let us ompare Eqs. (1) and (13) with other alulations of the OAM and SAM of the tightly foused irularly-polaried light [9,13,15,8,9]. First, our results differ from alulations [9] based on approah of [7] with a nononserved total AM: L S. At the same time, the post-paraxial estimation L /4 is analogous to the multipole expansion of a strongly foused Gaussian beam [13]. Equations (1) and (13) are similar to alulations in [15] and [8], but differ from the final result in [15], apparently due to an arithmeti inauray therein. Finally, Eq. (1) is entirely analogous to the OAM and SAM of nonparaxial Bessel beams with well-defined heliity [9] modified by averaging over the polar angles. So far, we used only the plane-wave spetrum of the foused field, Ek. The atual realspae eletri field near the foal point is determined by the interferene of the partial plane waves and is given by the Debye integral similar to Fourier transform [37,38]: i,, r E r de, e, k xsin os ysin sin os. (16) Here k r R kf k r R / f is the optial phase gained along the path from the refration point R f sin os,sin sin, os to the observation point r x, y, ( r f / k and the ommon phase kf is subtrated). Calulating the Debye integral (16) * and then the intensity, I E E, for the inident paraxial beam (9) with the fousing transformation (5), one an derive (f [15,9].): I, a J b J ab J. (17) Here we denoted k sin ( is the radial ylindrial oordinate) and... sin os ik os d F e. Note that for pure heliity states under onsideration the magneti-field intensity in real spae oinides with the eletri-field intensity, so that Eq. (17) an be regarded as the total intensity of the eletromagneti field. # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6139

9 Fig.. Foal intensity distributions I,, Eq. (17), in omparison with the radii R, Eq. (18), (vertial lines) for 1,,3, 1, and different values of the aperture angles. It is seen that the struture of radii (18) underlies positions of the first maxima of intensity (17), f [9]. It is seen that the intensity distribution of the foused field depends on both the OAM (vortex) and SAM (heliity) quantum numbers, and. It is invariant with respet to the,,,, nor transformation but neither with respet to,,, whih reflets symmetry typial of spin-orbit interation. For a salar wave, the intensity (17) would be I, J. Nonparaxial vetor terms proportional to b B / desribe spin-orbit oupling between the main mode of the order to the modes of the order and whih appear owing to the aimuthal geometri phases in the matrix U ˆ, Eq. (5). Dependene of the intensity distribution (17) on and is losely related to the spin-to-orbital AM onversion and the OAM expetation value L, Eq. (1). Namely, generaliing the Bessel-beam results of [9] for the averaged polar angle, Eq. (13), the mehanial radius R orresponding to the orbital angular momentum L and underlying the foal intensity (17) an be estimated as L 1os R. (18) ksin ksin Indeed, the transverse momentum P ksin represents the aimuthal momentum in the foal spot (the mean radial momentum vanishes there), and using mehanial definition L PR one an write L P R. Comparison of the -dependent fine struture of radii (18) with the wave intensities (17) is given in Fig.. Similar polariation-dependent properties of the foal intensities have appeared in [1,11,14,15]. In partiular, it follows from # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 614

10 Eq. (17) that the intensity of the beams with antiparallel OAM and SAM,, does not vanish on the axis for 1,. This was experimentally verified in [45,46]. In general, the sie of the foal spot for a strongly foused beam has a fundamental lower bound that depends not only on the beam s diretional spread but also on its OAM and SAM, suh that beams with antiparallel OAM and SAM an ahieve tighter foal spots than those for whih these AM are parallel [44]. Note that asymmetri fousing with a trunated lens or distribution of the inident light brings about the - and -dependent transverse shifts of the foal-spot intensity entroid, whih is proportional to L, i.e., orbital and spin Hall effets [9,3,47]. 4. Dipole sattering Sattering of paraxial light by a nano-partile loated at the origin essentially represents the spherial rediretion of partial plane waves, Fig. 3. We examine the simplest dipole approximation when the sattered spherial wave is generated by the dipole moment E proportional to the inident field E at the origin. Sine higher-order paraxial beams (9) with have ero field in the enter, E, below we onsider an inident plane wave with irular polariation, E e. The eletri far field of the sattered wave is [48] r r E E,, r, (19) r where r r / r is the unit radial vetor with spherial oordinates, of the observation point. Akin to fousing, the real-spae spherial angles, serve as the oordinates in momentum spae for the sattered far field. In this manner, the r -vetor plays the role of the k / k -vetor for sattered waves and transformation (19) is reminisent of the vetor transformation k k E in free-spae Maxwell equations k double vetor produt E of a nondiagonal operator ˆ, Uˆ ˆ, ˆ U, k k E E. The r r E E r r E represents a spherial projetion of onto the r -sphere and an be written, in the laboratory irular basis (1), as the ation [16]: i i 1 a1 b1 e a1b 1e 1 1, ˆ,, ˆ i i E E b1e 1 a1 a1b 1e. () r i i a1b 1e a1b 1e b 1 Here ˆ diag 1,1, is the projetor onto the orthogonal plane, a1 os, and b1 sin. # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6141

11 Fig. 3. Akin to fousing, Fig. 1, dipole sattering of the inident paraxial field xy, small spherial partile transforms it into a spetrum of plane waves, spherially-distributed k -vetors. E by a E with the Projetion () is a nonunitary transformation, and it does not preserve the loal heliity of the sattered field. Indeed, in the heliity basis attahed to the sphere of partial k -vetors, the sattered field is [f. Equation (6)]: E Uˆ E ˆUˆ E ˆUˆ E (1) H H Other than that, Eq. () demonstrates features that are quite similar to those of the fousing transformation (5). In partiular, the off-diagonal geometri-phase elements of the matrix ˆ produes spin-to-orbital AM onversion whih an be symbolially written similarly to Eq. (7): 1, (1 a1 ), b1, a1b 1,. The AM onversion upon sattering of light on various objets was analyed in several papers [16 4]. While in the ase of fousing the polariation is not hanged in the heliity basis, the transformation (1) of the sattered field an be written as the following heliity transition: i a be H H H (). (3) In the linear approximation in (for small sattering angles 1), the heliity (3) is onserved [19,], and the sattering transformation () beomes similar to the fousing one, Eq. (5). Assuming an inident plane wave with irular polariation, E e, we determine the sattered field () and alulate the OAM and SAM angle-resolved densities using Eqs. (8): 1 os os l, s. 1os 1os (4) Idential results were obtained for the field radiated by a rotating dipole [49], and similar results [but without 1 os 1 fator] have appeared for the Rayleigh sattering [18]. Equations (4) yield l for the paraxial angles 1 and the total onversion l # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 614

12 at / (see Fig. 8 below). The integral OAM and SAM values are determined via Eqs. (11) with the spherial integration d d d sin, whih results in L 1 os sin d os sin d 1 1, S. 1 os sin 1 os sin d d (5) Thus, the spin-to-orbital AM onversion has the effiieny 1/ [18,49] (see Fig. 8 below). Equations (4) and (5) demonstrate onservation of the loal and integral total AM per photon, Eq. (14). It is worth remarking that the onverted part of the AM in Eq. (4) an be expressed as sin l P, s 1 P, P, 1os (4') where a and b stand for loal intensities of the heliity omponents, Eq. (3). Here the quantity P oinides with the degree of polariation of the sattered unpolaried light [48]. Thus, the hanges in the degree of polariation upon sattering are also onneted with the above-onsidered geometri transformations of the wave field. In partiular, the depolariation of multiply sattered polaried light and typial four-fold polariation patterns of the baksattered light are intimately related to the spin-to-orbital AM onversion [19,]. In the weak-sattering approximation of small sattering angles ( 1 in a single sattering), these depolariation effets an be explained via Berry-phase aumulation along the partial sattering paths [19,,5,51]. This establishes a geometri-phase link between the AM onversions in fousing and sattering proesses. For strong single sattering event ( ~1), the Berry-phase (adiabati) approximation is not appliable beause the geometri transformation () represents a projetion rather than parallel-transport rotation (5) of the field. 5. Imaging of nanopartiles Strongly foused or sattered fields are essentially nonparaxial, whih is the main soure of the spin-orbit phenomena. Aordingly, the AM onversion an be deteted via various nearfield methods: e.g., traing motion of testing partiles [11,1] or using near-field probes [14,1,4,47]. It should be noted that the use of testing partiles is somewhat ambiguous as they an undergo orbital motion due to both orbital and spin energy flows, i.e., OAM and SAM [5,53]. At the same time, traditional paraxial-optis detetors are unable to measure adequately the spin-orbit phenomena in nonparaxial fields with strong longitudinal field omponent. It turns out, however, that a standard imaging system suessfully resolves this dilemma by transfering the spin-orbit oupling into a paraxial far-field, where it an be easily deteted. There were several experimental observations [16,31 34,54] of the spin-orbit interations of light using imaging sheme: (i) fousing of the inident paraxial light with a high-na lens; (ii) sattering by a small speimen; (iii) olletion of the sattered light by another high-na lens transforming it to the outgoing paraxial light, see Fig. 4. In most ases the observed effets were asribed either to fousing or to sattering proess, although all the three elements of the imaging system ontributed to the effet. The lens-satterer-lens system (Fig. 4) represents the basis for optial mirosopy and it is important to desribe the spin-orbit effets in suh imaging system taking into aount all its elements [16,17]. # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6143

13 Fig. 4. Sheme of the lens-satterer-lens imaging system. First, the inident paraxial light E is foused by a high-na lens, then a small speimen in the fous satters the nonparaxial foused field, and finally the sattered light is olleted by the seond high-na lens. The E, has a spae-variant polariation distribution and bears output paraxial field information about the spin-orbit oupling inside the system. This offers an effiient tool to retrieve fine subwavelength information about the speimen. Sine both the input and output fields in the imaging system are paraxial, the transformation of the field by the system an be desribed by an effetive Jones matrix whih gives the angle-resolved polariation state of the output field. Using suessive appliations of the geometri transformations of the first lens, Eq. (5), together with the Debye integral (16), dipole-sattering transformation (), and the inverse transformation (5) of the olletor lens, one an obtain the 3D transformation operator of the system [16], E, Tˆ, E, : Here 1 T U e d U e os ˆ ˆ i,, ˆ,, ˆ, rs i rs os,. d d d sin, the angles, and, entrane and exit pupils, respetively (Fig. 4), x, y, s s s s partile near the ommon fous of the two lenses, and the phases are (6) mark the partial rays at the r is the position of the sattering k r R/ f k x sin os y sin sin os, s s s s k r R/ f k x sin os y sin sin os. s s s s Here R f sin os,sin sin, os and f sin os,sin sin, os (7) R are the refration points at the fousing and olleting lenses, respetively. The projetor ẑ in Eq. (6) ensures transversality of the output field, and the upper left setor of the operator T ˆ,, provides the effetive spae-variant Jones matrix onneting the transverse (6), omponents of the input and output fields in the basis of irular polariations. In what follows we denote the transverse omponents of the vetors and matries by the subsript : E E, E T, et. Note that even if the inident light represents a homogeneous plane # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6144

14 wave E, bearing information about the spin-orbit oupling in the system and speimen properties. First, let us onsider the symmetri ase when the satterer is loated preisely in the fous: r s. If the inident field is a homogeneous plane wave E (whih implies ), one an evaluate the integral (6) analytially. The resulting Jones operator is [16]: E, the output field will have nonuniform polariation i A a be ˆ (), i T os be a (8) 15 3/ where the aperture-dependent oeffiient is A 8 os 5 3os A at 1. The Jones matrix (8) is proportional to the transverse part of U ˆ, Eq. (5), and it desribes the spin-to-orbital AM onversion between two paraxial states of light:, a, b,. (9) Comparing this equation with Eq. (3), one an see that, seemingly, the imaging sheme transfers the sattering-indued AM onversion whih appears in the loal heliity basis to the paraxial field and laboratory irular basis. AM onversion (9) strongly resembles paraxial spin-to-orbital onvertors based on loally-anisotropi axially symmetri strutures [3 7,5,6]. In our ase, the -dependent helial phases arise from purely geometrial 3D transformations of the field inside the imaging system, whih demonstrate effetive birefringene of the radially (TM) and aimuthally (TE) polaried modes. Assuming - irularly polaried inident wave, E e, the output field is E i T a, be / os and E i be a loal OAM and SAM densities at the exit pupil: T E ˆT () and e, i.e.,, / os. Using Eqs. (8), we determine the os os l 1, s. 1os 1os From Eqs. (11), implying integration over the exit pupil, dxdy d d d sin os, one an obtain the integral OAM and SAM values: L 13os 3os os 3sin,, 4 3os os 4 3os os 3 S 3 3 whih satisfies the total AM onservation (14). The effiieny of the spin-to-orbital AM 4 onversion is tiny in the imaging system with small aperture: L / 4 at 1, and it reahes the value L /4 at /. Sine the output field, (3) (31) E is paraxial, its polariation properties an be harateried by spae-variant Stokes parameters. The normalied Stokes vetor 1,, 3 alulated using the Pauli matries ˆ ˆ, ˆ, ˆ [55]: 1 3 S S S S an be * E ˆ E S,, (3) * E E # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6145

15 Note that the spin density s, Eq. (3), oinides with the loal degree of irular polariation, i.e., the third omponent of the Stokes vetor: S 3 s. Indeed, for paraxial fields it is determined by the transverse part of the operator ˆ : ˆ diag 1, 1 ˆ 3. The omplete desription of the polariation ontains more information than the spin AM of light [56]. In partiular, the first two omponents of the Stokes vetor (3) an evidene onversion to the OAM, Eq. (9). Indeed, alulating Eq. (3) for the inident irularly-polaried plane wave, we obtain S 1, ab os, absin, a b. a b (33) This distribution of the Stokes parameters is shown in Fig. 5. Parameters S 1 and S demonstrate typial four-fold patters whih signify generation of the oppositely polaried omponent with the optial vortex e i [16,19,,33,5]. Fig. 5. Distributions of the Stokes parameters,, S S S S, Eqs. (3) and (33), in the exit 1 3 pupil of the optial mirosope (Fig. 4) in the ase of the on-axis loation of the speimen, r, and left-hand irular polariation of the inident light, 1. The normalied s oordinates x x / f sin and y y / f sin are used. The four-fold patterns in the S 1 and S distributions are the signature of the generation of the right-hand polaried omponent with optial vortex 3 /8. e i, i.e., the spin-to-orbital AM onversion (9). The aperture angle is The ylindrial symmetry is broken in the imaging system if the speimen is transversely shifted in the foal plane. In this ase, the system is not rotationally-invariant about the - axis, and the total AM is no longer onserved. This results in the AM-dependent orthogonal shift of the enter of gravity of the output field, i.e., the Hall effet of light. Considering a small subwavelength displaement of the speimen in the xy, plane, r x, y,, k r s s s s 1, the effetive Jones matrix an be evaluated analytially from Eq. (6) as a () (1) orretion to Eq. (8) [16]: Tˆ Tˆ Tˆ, B sin e e i i ˆ (1) s s ik. i i os se se T Here we introdued s xs iys and the aperture-dependent oeffiient is B 3/ 8 os 11 3os 1, with 4 B /4 at 1. The Jones matrix (34) reveals oupling between the position of the satterer and polariation of light, whih (34) # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6146

16 results in the spin Hall effet. Taking the inident irularly polaried plane wave E e, we assume s xs, determine the output field E ˆT e, and alulate the transverse position of entre of gravity of the field. It is determined as R d * R E E d E where R X, Y f sin os,sin sin * E through the Jones matrix (8) and (34) into Eq. (35), we obtain, (35). Substituting the output field alulated 4, s. 3 A 4 3os os X Y f kx 3Bsin The -dependent shift, orthogonal to the displaement of the speimen, represents the spin Hall effet of light. It is also aompanied by a nonero mean momentum Px. The shift 4 (36) behaves as Y f kx s /8 at 1 and reahes value Y.13 f kxs at /. Thus, for high-na systems and subwavelength displaements of the partile, kx ~1, the shift of the entre of gravity is of the order of a fration of the foal length s f. In other words, high-na optial mirosope dramatially magnifies the spin Hall effet from the typial subwavelength sale to the sale of the exit pupil [16]. Figure 6 * displays the intensity distributions I E E and shifts of the field entroid at different positions x s of the satterer. Similar -dependent deformations of intensity in the imaging system with inident paraxial vortex beams, i.e., orbital Hall effet, were observed in [3]. Another striking manifestation of the spin Hall effet is a strong separation of loal SAM densities in the inident linearly polaried light [16]. (36) Fig. 6. Distributions of the output field intensity I, in the imaging system Fig. 4 for the inident right-hand irularly polaried light ( 1 ) at different subwavelength displaements of the sattering partile: x, / 3, / 3. The transverse shift of the enter of gravity of the field, Y s s x signifies the spin Hall effet of light. Parameters are the same as in Fig. 5. Alongside the Stokes polarimetry, whih reveals onversion to different polariation and OAM modes, the state of the output light an be fully desribed via intensities and phases of the two polariation omponents following from the Jones matries (8) and (34). Distributions of the intensities and phases of the right- and left-hand irularly polaried field omponents for the on-axis and off-axis satterer are shown in Fig. 7. It is seen that the nonero intensity and the vorties in the oppositely polaried (with respet to the inident light) omponent indiates the spin-to-orbital AM onversion (9). At the same time, the # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6147

17 intensity redistribution and phase gradient in the main polariation omponent are responsible for the shifts (36), Y, P for an off-axis satterer. x Fig. 7. Intensities and phases of the right- and left-hand irularly polaried omponents in the output field for the on-axis ( r s ) and off-axis ( x.8 ) positions of the satterer (Fig. 4). The inident field is right-hand irularly polaried ( 1 ), the aperture of the system orresponds to s sin.9. The harge- optial vortex and nonero intensity in the lefthand polaried omponent is learly seen for the on-axis partile. Displaement of the satterer indues splitting of the harge- vortex into two harge-1 vorties (f [34].), strong deformation of the intensity of the right-hand omponent (responsible for the spin-hall effet and Y ), and smooth gradient of the phase in the right-hand omponent whih yields P. 6. Conlusions x We have examined the spin-to-orbital AM onversion in basi optial systems involving nonparaxial fields. The AM onversion originates from geometri transformations of the wave field: parallel-transport rotations in the ase of fousing and spherial projetions in the ase of sattering. Despite the fat that problems of spin-to-orbital onversion were onsidered in reent years both in fousing and sattering systems, a number of ontroversies and the use of dissimilar approahes did not allow one to obtain solid quantitative results and ompare the effiieny of the onversion in different systems. Our approah unifies various treatments of the AM in different nonparaxial optial systems and is based on the geometri field transformations and anonial AM operators. As a result, one an ompare the effiieny of the spin-to-orbital onversion in high-na lens fousing, Rayleigh (dipole) sattering, and in far-field lens-satterer-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 omparison we also plot there the OAM and SAM of nonparaxial Bessel beams alulated in [9]. One an see that for the plane inident wave ( ) the onversion effiieny dereases in a sequene: Bessel beam sattering fousing imaging. Curiously, the orresponding maximum effiienies reahed at the aperture angle / are equal to 1, 1/, 1/3, and 1/4. The total spin-to-orbital onversion for Bessel modes with / was observed in a plasmoni experiment [14]. # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6148

18 Fig. 8. Left: OAM angle-resolved density l onverted from SAM for the ases of fousing (red) [Eq. (1)], sattering (blue) [Eq. (4)], and imaging (blak) [Eq. (3)], with irularly polaried inident light, 1,. Right: The integral OAM, L, onverted from SAM, vs. the aperture angle for fousing [Eqs. (1) and (13)], sattering [Eq. (5)], and imaging systems [Eq. (31)]. The solid, dashed, and dotted urves for fousing represent inident paraxial beams with, 1, and ( 1 everywhere). For sattering, the dependene is obtained by formally replaing the upper limit of integration in Eqs. (5): ; analytial results above are reovered at semitransparent urves indiate the orresponding quantities for the SAM, i.e., S or /. The s and, satisfying the onservation law (14). For omparison, the right-hand panel also displays the OAM and SAM values for nonparaxial vetor Bessel beams (yellow) [9], whih ahieve the total spin-to-orbital onversion at the aperture angle /, see [14]. Although the OAM and SAM are important dynamial harateristis of light, it is diffiult to measure them diretly in optial systems. In partiular, orbital motion of small testing partiles [11,1] annot be used as a measure of the OAM beause both the spin and orbital energy flows an be responsible for it [5,53]. However, the lose onnetion between the intensity distributions (in partiular the radius of the foal spot) and the OAM values, Eq. (18) and Fig., enables one to quantify the AM onversion via spin-dependent intensity profiles. Also, as we have shown, the angle-resolved polarimetry in the far-field imaging systems enables one to reonstrut the field distribution and unambiguously haraterie its AM features [16] (Setion 5). Importantly, universal mehanisms of the spin-to-orbital AM onversion manifest themselves not only in optial but also in plasmoni [14, 4,47] and semiondutor [34] systems. Therefore, the presented unified geometri and operator desription of the AM onversions provides a useful link between spin-orbit phenomena in nonparaxial wave fields of various nature. Aknowledgements This work was supported by the European Commission (Marie Curie Ation), Siene Foundation Ireland (Grant No. 7/IN.1/I96), and the Australian Researh Counil. # $15. USD Reeived 3 Aug 11; aepted 6 Ot 11; published 7 De 11 (C) 11 OSA 19 Deember 11 / Vol. 19, No. 7 / OPTICS EXPRESS 6149