Transverse spin dynamics in structuredelectromagnetic guided wavesPeng Shia1 Luping Dua12 Congcong Lia Anatoly V Zayatsbc2 and Xiaocong Yuana2

aNanophotonics Research Center Shenzhen Key Laboratory of Micro-Scale Optical Information Technology amp Institute of Microscale OptoelectronicsShenzhen University Shenzhen 518060 China bDepartment of Physics Kingrsquos College London London WC2R 2LS United Kingdom and cLondon Centrefor Nanotechnology Kingrsquos College London London WC2R 2LS United Kingdom

Edited by J B Pendry Imperial College London London United Kingdom and approved December 29 2020 (received for review September 9 2020)

Spinndashmomentum locking a manifestation of topological proper-ties that governs the behavior of surface states was studied in-tensively in condensed-matter physics and optics resulting in thediscovery of topological insulators and related effects and theirphotonic counterparts In addition to spin optical waves may havecomplex structure of vector fields associated with orbital angularmomentum or nonuniform intensity variations Here we derive aset of spinndashmomentum equations which describes the relationshipbetween the spin and orbital properties of arbitrary complex elec-tromagnetic guided modes The predicted photonic spin dynamicsis experimentally verified with four kinds of nondiffracting surfacestructured waves In contrast to the one-dimensional uniform spinof a guided plane wave a two-dimensional chiral spin swirl is ob-served for structured guided modes The proposed framework opensup opportunities for designing the spin structure and topologicalproperties of electromagnetic waves with practical importance in spinoptics topological photonics metrology and quantum technologiesand may be used to extend the spin-dynamics concepts to fluidacoustic and gravitational waves

angular momentum of light | spinndashorbit coupling | transverse spin |spinndashmomentum locking

Spinndashmomentum locking characterized by unidirectional sur-face spin states has been extensively studied in topological

insulators (1) superconductors (2) magnons (3) and cold-atom(4) and BosendashEinstein condensates (5) The photonic analogyof unidirectional surface spin states was demonstrated with thepseudospin by engineering an ldquoextrinsicrdquo spinndashorbit interaction andbreaking the time-reversal symmetry in artificial photonic struc-tures (6ndash8) On the other hand due to an ldquointrinsicrdquo spinndashorbitcoupling governed by the Maxwellrsquos field theory the spinndashmomentum locking of light was reported and linked to the modeswith the evanescent field components such as surface waves orwaveguided modes (9ndash11) For example surface plasmon polar-itons (SPPs) as surface modes propagating at an insulatorndashmetalinterface (12) exhibit features of spinndashmomentum locking that areanalogous to the behavior of surface state of a topological insulator(6ndash8) Although photons are bosons with integer spin and surfaceand waveguided electromagnetic modes suffer from backscattering(13) in contrast to the helical fermion behavior of surface Diracmodes they possess the topological Z4 invariant and hence cantransport spin unidirectionally (9) This intrinsic optical spinndashmomentum locking is a basis for many intriguing phenomena suchas spin-controlled unidirectional excitation of surface and wave-guided modes and offers potential applications in photonic inte-grated circuits polarization manipulation metrology and quantumtechnologies for generating polarization entangled states (14ndash20)Optical transverse spin plays a key role in the intrinsic spinndash

momentum locking effect in evanescent waves (11) In contrastto a conventional longitudinal spin of light with the spin vectorparallel to the propagating direction the orientation of trans-verse spin is perpendicular to the propagating direction enablingmany important phenomena and applications (21ndash28) An em-pirical procedure to identify the optical spin direction includes

calculating the spin angular momentum (SAM) S and comparingthe spin orientation to the wave vector k This empirical per-spective provides an intuitive way to identify the optical trans-verse spin in various optical configurations involving plane wavesbut cannot be generalized to more complex scenarios for ex-ample when structured waves with an arbitrary trajectory andorbital angular momentum need to be considered Although onecan define a ldquolocalrdquo wave vector which is related to the orbitalenergy flow density Po it cannot describe quantitatively an opticaltransverse spin associated with a structured vector wave for whichthe spin part of the Poynting vector (Ps) is also important (29)Here we overcome this limitation and extend the under-

standing of the spinndashmomentum locking and spin dynamics fromplane evanescent waves to a two-dimensional (2D) chiral spinswirl associated with the structured guided modes thereforegeneralizing the optical spinndashmomentum locking to arbitraryguided vector fields From the perspective of energy flow density(P = Ps + Po) we derive four equations describing dynamictransformations of spin and momentum of the electromagneticwave that are analogous to the Maxwellrsquos equations for elec-tromagnetic fields The proposed framework is verified experi-mentally on the example of four structured surface waves andopens up opportunities for understanding and designing the spindynamics and topological properties of electromagnetic wavesfrom the radiofrequency to ultraviolet spectral ranges and forapplications in spin optics topological photonics polarizationmeasurements metrology and quantum technologies Since theenergy flow density can be represented through a current densityterm in the Hertz potential (SI Appendix section VI) the proposeddescription allows also extending the concepts of the dynamics of

Significance

We formulate and experimentally validate a set of spinndashmomentum equations which are analogous to the Maxwellrsquosequations and govern spinndashorbit coupling in electromagneticguided waves The Maxwell-like spinndashmomentum equationsreveal the spinndashmomentum locking the chiral spin texture ofthe field Berry phase and the spinndashorbit interaction in theoptical near field The observed spinndashmomentum behavior canbe extended to other classical waves such as acoustic fluidgas and gravitational waves

Author contributions LD designed research PS LD and CL performed research PSLD AVZ and XY analyzed data and PS LD AVZ and XY wrote the paper

The authors declare no competing interest

This article is a PNAS Direct Submission

This open access article is distributed under Creative Commons Attribution License 40(CC BY)1PS and LD contributed equally to this work2To whom correspondence may be addressed Email lpduszueducn azayatskclacuk or xcyuanszueducn

This article contains supporting information online at httpswwwpnasorglookupsuppldoi101073pnas2018816118-DCSupplemental

Published February 1 2021

PNAS 2021 Vol 118 No 6 e2018816118 httpsdoiorg101073pnas2018816118 | 1 of 5

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

July

29

202

1

transverse spin from electromagnetic waves to fluid acoustic andgravitational waves (30ndash32)

Results and DiscussionFor an arbitrary electromagnetic wave propagating in a homo-geneous medium the curl of the energy flow density can bepresented as (SI Appendix section I)

nabla times Ρ = v2nabla times p = ω2S minus 14Reminus(nablaotimesEp) middotH minus (nablaotimesE)Τ middotHp

+(nablaotimesHp) middotE + (nablaotimesH)Τ middotEp [1]

where p is the kinetic momentum density of the field which islinearly related to the Poynting vector in a homogeneous me-dium p = Pv2 v is the speed of light in the medium ω representsthe angular frequency of the electromagnetic field and E and Hindicate the electric and magnetic field respectively Here thesymbol otimes indicates a dyad vector and p denotes the complexconjugate The second part on the right-hand side of Eq 1 hasthe same structure as the quantum 2-form (33) that generates theBerry phase associated with a circuit which indicates a spinndashorbit interaction in the optical system (SI Appendix section II)In particular for electromagnetic waves with an evanescent fieldsuch as surface or guided waves an intrinsic spinndashmomentumrelationship can be derived from the Maxwellrsquos theory

S = 12ω2nabla times P = 1

2k2nabla times p [2]

where k = ωv is the wave number of the electromagnetic wave inthe medium Since curl of a vector field can be regarded as itscurrent vortices Eq 2 reveals that the optical spin of an evanes-cent field is associated with the local vorticity of the electromag-netic energy flow density and is source-less (nablaS = 0) The SAMin this case is related to the transverse gradient of the energyflow density At the same time the longitudinal optical spin doesnot fulfill the above spinndashmomentum relationship For examplea monochromatic circularly polarized plane wave bears the SAMaligned parallel to the wave vector while the curl of the Poyntingvector vanishes because of the uniformity of the energy flowdensity over the space Therefore the spinndashmomentum law inEq 2 only describes the dynamics of optical transverse spin pre-sent in the evanescent waves It also reveals that in addition to

the optical spin oriented along the surface (in-plane transverseSAM) which has been recently studied intensively there existsanother category of the transverse spin of an evanescent fieldoriented out of the surface plane This SAM can be induced bythe in-plane energy flow density of the structured guided or sur-face wave while the in-plane transverse spin is due to the gradientof energy flow density normal to the interface The appearance ofa transverse spin indicates to the rotation of polarization andhence the phase difference between all the field components ofthe waveThe spinndashmomentum locking in an evanescent plane wave

(Fig 1A) as demonstrated in previous work (9) is a special one-dimensional (1D) case of spinndashmomentum locking with the SAMvector aligned along the interface Assuming the guided modepropagating along the y direction and evanescently decayingalong the z direction one can deduce the Poynting vector P =yωlaquo(2β)exp(minus2kzz) and the SAM S = xlaquokz(2ωβ)exp(minus2kzz)where laquo denotes the permittivity of the medium and β and ikzstand for the in-plane and out-of-plane wave vector componentsrespectively The energy flow density and SAM of the evanescentplane wave are connected through the generalized spinndashmomentum relation S = nabla times P(2ω2) = ‒x(partPypartz)(2ω2) Forstructured evanescent modes with spatially varying intensity dis-tribution the inhomogeneity of energy flow density can induceseveral SAM components in different directions The variation ofthe energy flow density in the z direction induces an in-planecomponent of the SAM while its in-plane variations induce a zcomponent Both are perpendicular to the local energy propaga-tion direction The relationship between the two componentsleads to a chiral spin texture with spin vectors swirling around theenergy flow (Fig 1B) More importantly its tendency of direc-tional variation (ie the chirality) is locked to the momentumdirection This is a manifestation of the generalized optical spinndashmomentum locking associated with an electromagnetic evanescentwaveIt should be noted that the transverse spin discussed here is

different from the ldquospinsrdquo in conventional topological photonicstypically called a ldquopseudospinrdquo For a pseudospin the spinndashmomentum locking is achieved by engineering the spinndashorbitinteraction in artificial photonic structures in order to break thetime-reversal symmetry (8) In the case of the optical transversespin of an evanescent wave the generalized spinndashmomentum

Fig 1 Generalization of spinndashmomentum locking for structured guided modes (A) In unstructured plane guided wave optical spinndashmomentum lockingresults in the transverse spin (S) uniformly distributed and parallel to the interface The spin vector direction is perpendicular to the wave vector k and flips ifthe propagation direction flipped from +k to minusk (B) In an arbitrary structured guided wave the optical spin is related to the vorticity of the energy flowdensity P The transverse spin vector varies from the ldquouprdquo state to the ldquodownrdquo state around the energy flow density remaining perpendicular to the localwave vector This forms a chiral swirl of the 2D transverse spin which is locked to the energy propagating direction and fulfills a right-handed rule Thedirection of the local transverse spin vector flips if the energy flow density flipped from forward (+P) to backward (minusP)

2 of 5 | PNAS Shi et alhttpsdoiorg101073pnas2018816118 Transverse spin dynamics in structured electromagnetic guided waves

Dow

nloa

ded

by g

uest

on

July

29

202

1

locking is an ldquointrinsicrdquo feature of the spinndashorbit interaction gov-erned solely by the Maxwellrsquos theory The nonzero spin Chernnumber for the structured waves (SI Appendix section IV) impliesthe existence of nontrivial helical states of electromagnetic waveswhich are strictly locked to the energy propagation directionHowever since the topological Z2 invariant of these states vanishesowing to the time-reversal symmetry of the Maxwellrsquos equationsthere is no protection against (back)scattering Although thetransformation of the two helical states of evanescent waves is nottopologically protected against scattering the spinndashmomentumlocking and the induced unidirectional excitation and propagationare the intrinsic feature of the Maxwellrsquos theory and are topolog-ically nontrivial possessing the Z4 topological invariantTo demonstrate the spinndashmomentum locking features described

by Eq 2 four types of the electromagnetic modes exhibiting ev-anescent field with inhomogeneous spatial energy distributionwere investigated including the solutions of a wave equation inCartesian coordinates (Cosine beam) (34) in cylindrical coordi-nates (Bessel beam) (35) in parabolic coordinates (Weber beam)(36) and in Cartesian coordinates but with a parabolic path (Airybeam) (37) (SI Appendix section V) The spatial distributions oftheir energy flow densities are shown in Fig 2 AndashD while thebeamsrsquo propagation directions can either be forward (+P) orbackward (ndashP) The corresponding cross-sections along the dashedlines are shown in Fig 2 EndashH and IndashL for the beams with opposite

propagation directions respectively together with the SAM dis-tributions and the spin vector variations For all four differenttypes of the beams the spin orientation varies progressively fromthe ldquouprdquo state to the ldquodownrdquo state when the energy propagatesalong the forward direction (Fig 2 EndashH) The intrinsic spinndashmomentum locking present in evanescent waves ensures the to-pological protection in terms of spin vector swirl being completelydetermined by the energy flow density Thus to observe the re-versal of the spin swirling from the ldquodownrdquo state to the ldquouprdquo statethe propagation direction must be reversed (Fig 2 IndashL) Thisspinndashmomentum locking is preserved even for surface modessuffering from the ohmic losses (12) which influence only theintensity of the wave but not the orientation of photonic spinvector Note that the spin vector has orientation along the inter-face at the maxima of the energy flow density and is normal to it atthe nodes Therefore a period of spin variation can be definedbetween the two adjacent nodes of energy flow density whichexhibits a similar feature to a topological soliton (38ndash42)In order to experimentally observe the spinndashmomentum

locking features associated with the structured surface waves andout-of-plane transverse SAM the experiments were performedon the example of SPPs (SI Appendix section VII) SPPs wereexcited under the condition of a total internal reflection using amicroscope objective with high numerical aperture (NA) = 149Spatial light modulator and amplitude masks were employed to

Fig 2 Spinndashmomentum locking in various surface structured waves (AndashD) The spatial distributions of the energy flow density for different structuredsurface waves (A) surface Cosine beam (B) surface Bessel beam with topological charge l = plusmn1 (C) surface Weber beam and (D) surface Airy beam Thesebeams can either propagate in the forward (labeled +P) or backward (labeled ndashP) directions (EndashH) Transverse SAM components Sz and Sx and the cross-sections of the energy flow density distributions along the green dashed lines in AndashD for the beams propagating in the direction indicated with the arrowlabeled +P (IndashL) The same as EndashH for the beams propagating in the opposite direction indicated with the arrow labeled ndashP in AndashD The inserts at the top of EndashLshow the local transverse spin vector orientations (cf Fig 1) The spin vectors are swirling around the energy flow density and their local orientations varyfrom the ldquouprdquo to the ldquodownrdquo states fulfilling the right-handed rule These orientations are seen inverted for the waves with the opposite direction of theenergy propagation Note that for the beams with curved trajectory the spin variation is considered in the plane perpendicular to the local tangential di-rection of the energy flow density The distance unit is the wavelength of light in vacuum

Shi et al PNAS | 3 of 5Transverse spin dynamics in structured electromagnetic guided waves httpsdoiorg101073pnas2018816118

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

July

29

202

1

modulate the phase and wavevector of the excited SPPs to gen-erate the desired plasmonic modes A scanning near-field opticalmicroscope which employs a dielectric nanosphere to scatter theSPPs to the far field and a combination of a quarter waveplateand a polarizer to extract the two circular polarization components(IRCP and ILCP) of the far-field signal were used to measure theout-of-plane SAM component Sz = laquoβ2[4ωkz

2(IRCP ‒ ILCP)] Thecorresponding in-plane spin components were also reconstructedfrom the measurements (SI Appendix section VIII) The measureddistributions of the SAM components are shown in Fig 3 and SIAppendix Figs S16ndashS19 for the four types of structured SPP wavespropagating in the forward and backward directions All the pre-dicted SAM and spinndashmomentum locking features are observed ex-perimentally 1) the variation of SAM from the positivenegative stateto the negativepositive across the beam profile and 2) the reversal ofspin variation when inversing the beam propagation directionSince the kinetic momentum density can be divided into the

spin (ps) and orbital (po) parts p = ps + po where ps = nabla times S2and obeys the spinndashmomentum relationship (Eq 2) we canformulate a set of the Maxwell-like equations linking the trans-verse spin and the momentum density (and the Poynting vector)of evanescent electromagnetic fields (Table 1) This formulationprovides comprehensive and intuitive understanding of theboundary conditions and the dynamical properties of the spinmomentum and energy flow in guided waves (SI Appendixsection III) For example the flip of the out-of-plane spin andthe in-plane Poynting vector of a SPP wave across the metaldielectric interface immediately follows from the boundaryconditions due to the opposite sign of the permittivities on thedifferent sides of the guiding interface The same as variations ofE field induces H field in the Maxwellrsquos equations equation nabla times

p = 2k2S indicates that the spatial variations of the momentumenergy flow density induce the transverse SAM In the samemanner equation nabla times S = 2ps = 2(p minus po) tells us that the spinvariation in turn contributes to the momentumenergy flowdensity with the remainder provided from the orbital part (po)Consolidating spinndashmomentum equations results in an analog ofa Helmholtz equation nabla2S + 4k2S = 2nabla times po which describesspinndashorbit interaction in evanescent waves linking transversespin and orbital part momentum density In both the Helmholtzequation and the last Maxwellrsquos equation in Table 1 current J isan external source of magnetic field similarly in the corre-sponding spinndashmomentum equations po which determines theorbital angular momentum influences the spin Since an elec-tromagnetic wave in a source-free and homogenous medium canbe described with Hertz potential (Ψ) satisfying the Helmholtzequation and the Poynting vector can be calculated from the Hertzpotential as P prop (ΨnablaΨminusΨnablaΨ) (43) one can obtain the spin andorbital properties of the electromagnetic guided waves directlyfrom the spinndashmomentum equations without any knowledge on theelectric and magnetic fields (SI Appendix section VI)

ConclusionWe have demonstrated an intrinsic spinndashmomentum law whichgoverns the transverse spin dynamics of guided electromagneticwaves It was shown that the 1D uniform spin of surface planewave evolves in a 2D chiral spin swirl for structured guidedmodes providing a manifestation of the generalized photonicspinndashmomentum locking Four types of structured surface wavesincluding the Cosine beam Bessel beam Weber beam and Airybeam have been investigated both theoretically and experimentallyto demonstrate the concept of the generalized spinndashmomentumlocking Applying this spin-momentum locking we obtained a set

Fig 3 Experimental validation of the spinndashmomentum locking The measured out-of-plane SAM components (Sz) for (A E and I) surface Cosine beam (B Fand J) surface Bessel beam (C G and K) surface Weber beam and (D H and L) surface Airy beam the spatial distributions of Sz spin components for thebeams with (AndashD) forward (+P) and (EndashH) opposite (minusP) energy propagating direction (IndashL) the cross-sections of AndashH The direction of the out-of-planetransverse SAM is seen inverted for the waves propagating in opposite directions The distance unit is the wavelength of light in vacuum

4 of 5 | PNAS Shi et alhttpsdoiorg101073pnas2018816118 Transverse spin dynamics in structured electromagnetic guided waves

Dow

nloa

ded

by g

uest

on

July

29

202

1

of spinndashmomentum equations that are analogous to the Maxwellrsquosequations and the boundary conditions This optical spin frame-work can be used to evaluate the spinndashorbit coupling in the elec-tromagnetic guided waves and for designing specific transverse spinstructures without a priori information on the electric and magneticfields The generalized intrinsic spinndashmomentum features could alsoappear in other types of waves with evanescent field such as fluidsurface elastic acoustic and gravitational waves The effect couldbe of importance to the development of spin optics for quantumtechnologies and topological photonics

Materials and MethodsExperimental Setup The experimental setup for studies of the optical spinndashmomentum locking is shown in SI Appendix Fig S9 The experiment wasperformed on the example of SPPs which are TM-mode evanescent wavessupported on a metalndashdielectric interface An HendashNe laser beam with awavelength of 6328 nm was used as a light source After a telescope systemto expand the beam a combination of linear polarizer half-wave platesquarter-wave plates and vortex wave plates was employed to modulate the

polarization of the laser beam A spatial light modulator was then utilized tomodulate the phase of the beam The structured beam was then tightlyfocused by an oil-immersion objective (Olympus NA = 149 100times) onto thesample consisting of a thin silver film (45-nm thickness) deposited on acoverslip to form the desired SPP beams at the airsilver interface

A polystyrene nanosphere was immobilized on the silver film surface as anear-field probe to scatter the SPPs to the far field The sample was fixed ona piezo scanning stage (P-545 Physik Instrumente) providing resolutiondown to 1 nm A low-NA objective (Olympus NA = 07 60times) was employedto collect the scattering radiation from the nanosphere A combination ofquarter-wave plate and linear polarizer was used to extract the right-handed(RCP) and left-handed (LCP) circular polarization components of the collectedsignals Finally the intensities of RCP and LCP components are measured by aphotomultiplier tube (R12829 Hamamatsu) As the piezo scanning stage rasterscanned the near-field region the distributions of RCP and LCP componentscan be mapped and used to reconstruct the longitudinal SAM component

Numerical Simulation The numerical simulations were performed with a customprogram inMATLAB Thedetails canbe found in SIAppendix sectionsVVII andVIII

Data AvailabilityAll study data are included in the article andor SI Appendix

ACKNOWLEDGMENTS This work was supported in part by Guangdong MajorProject of Basic Research No 2020B0301030009 National Natural Science Founda-tion of China grants U1701661 61935013 62075139 61427819 61622504 and61705135 leadership of Guangdong province program grant 00201505 NaturalScience Foundation of Guangdong Province grant 2016A030312010 Science andTechnology Innovation Commission of Shenzhen grants JCYJ20200109114018750JCYJ20180507182035270 Shenzhen Peacock Plan KQTD2015071016560101 andKQTD20170330110444030 the Engineering and Physical Sciences Research Counciland European Research Council iCOMM project 789340 LD acknowledgesthe support given by the Guangdong Special Support Program We thankProf Michael Somekh for proofreading the manuscript

1 C L Kane E J Mele Z2 topological order and the quantum spin Hall effect Phys Rev

Lett 95 146802 (2005)

2 K Gotlieb et al Revealing hidden spin-momentum locking in a high-temperature

cuprate superconductor Science 362 1271ndash1275 (2018)

3 N Okuma Magnon spin-momentum locking Various spin vortices and Dirac mag-

nons in noncollinear antiferromagnets Phys Rev Lett 119 107205 (2017)

4 M Aidelsburger Cold atoms twisting spin and momentum Science 354 35ndash36 (2016)

5 Y-J Lin K Jimeacutenez-Garciacutea I B Spielman Spin-orbit-coupled Bose-Einstein conden-

sates Nature 471 83ndash86 (2011)

6 F D M Haldane S Raghu Possible realization of directional optical waveguides in pho-

tonic crystals with broken time-reversal symmetry Phys Rev Lett 100 013904 (2008)

7 Z Wang Y Chong J D Joannopoulos M Soljacic Observation of unidirectional

backscattering-immune topological electromagnetic states Nature 461 772ndash775 (2009)

8 T Ozawa et al Topological photonics Rev Mod Phys 91 015006 (2019)

9 K Y Bliokh D Smirnova F Nori Quantum spin Hall effect of light Science 348

1448ndash1451 (2015)

10 T Van Mechelen Z Jacob Universal spin-momentum locking of evanescent waves

Optica 3 118ndash126 (2016)

11 A Aiello P Banzer M Neugebauer G Leuchs From transverse angular momentum

to photonic wheels Nat Photonics 9 789ndash795 (2015)

12 A V Zayats I Smolyaninov A Maradudin Nano-optics of surface plasmon polar-

itons Phys Rep 408 131ndash314 (2005)

13 X Chen Z-C Gu Z-X Liu X-G Wen Symmetry-protected topological orders in in-

teracting bosonic systems Science 338 1604ndash1606 (2012)

14 P Lodahl et al Chiral quantum optics Nature 541 473ndash480 (2017)

15 F J Rodriacuteguez-Fortuntildeo et al Near-field interference for the unidirectional excitation

of electromagnetic guided modes Science 340 328ndash330 (2013)

16 M Neugebauer P Wozniak A Bag G Leuchs P Banzer Polarization-controlled

directional scattering for nanoscopic position sensing Nat Commun 7 11286 (2016)

17 M F Picardi A V Zayats F J Rodriacuteguez-Fortuntildeo Janus and Huygens dipoles Near-field

directionality beyond spin-momentum locking Phys Rev Lett 120 117402 (2018)

18 A Bag M Neugebauer P Wozniak G Leuchs P Banzer Transverse Kerker scat-

tering for Angstrom localization of nanoparticles Phys Rev Lett 121 193902 (2018)

19 F J Rodriacuteguez-Fortuntildeo N Engheta A Martiacutenez A V Zayats Lateral forces on cir-

cularly polarizable particles near a surface Nat Commun 6 8799 (2015)

20 T V Mechelen Z Jacob Unidirectional Maxwellian spin waves Nanophotonics 8

1399 (2019)

21 K Y Bliokh F J Rodriacuteguez-Fortuntildeo F Nori A V Zayats Spinndashorbit interactions of

light Nat Photonics 9 796ndash808 (2015)

22 K Y Bliokh F Nori Transverse spin of a surface polariton Phys Rev A 85 061801

(2012) (R)

23 K Y Bliokh A Y Bekshaev F Nori Extraordinary momentum and spin in evanescent

waves Nat Commun 5 3300 (2014)

24 F Khosravi C L Cortes Z Jacob Spin photonics in 3D whispering gallery mode

resonators Opt Express 27 15846ndash15855 (2019)

25 A Y Bekshaev K Y Bliokh F Nori Transverse spin and momentum in two-wave

interference Phys Rev X 5 011039 (2015)

26 K Y Bliokh F Nori Transverse and longitudinal angular momenta of light Phys Rep

592 1ndash38 (2015)

27 A Aiello P Banzer The ubiquitous photonic wheel J Opt 18 085605 (2016)

28 L Peng et al Transverse photon spin of bulk electromagnetic waves in bianisotropic

media Nat Photonics 13 878ndash882 (2019)

29 M V Berry Optical currents J Opt A Pure Appl Opt 11 094001 (2009)

30 K Y Bliokh F Nori Spin and orbital angular momenta of acoustic beams Phys Rev B

99 174310 (2019)

31 Y Long J Ren H Chen Intrinsic spin of elastic waves Proc Natl Acad Sci USA 115

9951ndash9955 (2018)

32 S Golat E A Lim F J Rodriacuteguez-Fortuntildeo Evanescent gravitational waves Phys Rev

D 101 084046 (2020)

33 M V Berry Quantal phase factors accompanying adiabatic changes Proc R Soc

Lond A Math Phys Sci 392 45ndash57 (1984)

34 J Lin et al Cosine-Gauss plasmon beam A localized long-range nondiffracting sur-

face wave Phys Rev Lett 109 093904 (2012)

35 L Du A Yang A V Zayats X Yuan Deep-subwavelength features of photonic

skyrmions in a confined electromagnetic field with orbital angular momentum Nat

Phys 15 650ndash654 (2019)

36 P Zhang et al Nonparaxial Mathieu and Weber accelerating beams Phys Rev Lett

109 193901 (2012)

37 A Minovich et al Generation and near-field imaging of Airy surface plasmons Phys

Rev Lett 107 116802 (2011)

38 M Gilles et al Polarization domain walls in optical fibres as topological bits for data

transmission Nat Photonics 11 102ndash107 (2017)

39 S Tsesses et al Optical skyrmion lattice in evanescent electromagnetic fields Science

361 993ndash996 (2018)

40 T J Davis et al Ultrafast vector imaging of plasmonic skyrmion dynamics with deep

subwavelength resolution Science 368 eaba6415 (2020)

41 Y Dai et al Ultrafast microscopy of a plasmonic spin skyrmion arXiv191203826 (9

December 2019)

42 S Tsesses K Cohen E Ostrovsky B Gjonaj G Bartal Spin-orbit interaction of light in

plasmonic lattices Nano Lett 19 4010ndash4016 (2019)

43 E Wolf A scalar representation of electromagnetic fields II Proc Phys Soc 74

269ndash280 (1959)

Table 1 Spinndashmomentum equations and the analogy to theMaxwellrsquos equations

Maxwellrsquos equations Spinndashmomentum equations

nabla middot E frac14 0 nabla middotp frac14 0nabla middotH frac14 0 nabla middot S frac14 0nabla3E frac14 iωμH nabla3p frac14 2k2Snabla3H frac14 J iωlaquoE nabla3S frac14 2ethp poTHORN

Helmholtz equationnabla2Hthorn k2H frac14 nabla3J nabla2Sthorn 4k2S frac14 2nabla3po

Shi et al PNAS | 5 of 5Transverse spin dynamics in structured electromagnetic guided waves httpsdoiorg101073pnas2018816118

APP

LIED

PHYS

ICAL

SCIENCE

S

Dow

nloa

ded

by g

uest

on

July

29

202

1