Transverse spinning of light with globally unique handedness

Access to the transverse spin of light has unlocked new regimes in topological photonics and optomechanics. To achieve the transverse spin of nonzero longitudinal fields, various platforms that derive transversely confined waves based on focusing, interference, or evanescent waves have been suggested. Nonetheless, because of the transverse confinement inherently accompanying sign reversal of the field derivative, the resulting transverse spin handedness experiences spatial inversion, which leads to a mismatch between the densities of the wavefunction and its spin component and hinders the global observation of the transverse spin. Here, we reveal a globally pure transverse spin in which the wavefunction density signifies the spin distribution, by employing inverse molding of the eigenmode in the spin basis. Starting from the target spin profile, we analytically obtain the potential landscape and then show that the elliptic-hyperbolic transition around the epsilon-near-zero permittivity allows for the global conservation of transverse spin handedness across the topological interface between anisotropic metamaterials. Extending to the non-Hermitian regime, we also develop annihilated transverse spin modes to cover the entire Poincare sphere of the meridional plane. Our results enable the complete transfer of optical energy to transverse spinning motions and realize the classical analogy of 3-dimensional quantum spin states.

Gaussian focused wave propagating in a homogeneous isotropic bulk medium (ε x = ε y > 0 for all x), and (b) an evanescent plasmonic wave confined at an interface between an isotropic metal (ε x = ε y < 0 for x < 0) and a dielectric medium (ε x = ε y > 0 for x > 0). (c) Proposed T-spin potential between two inhomogeneous anisotropic media around the uniaxial ENZ medium. The overlapping fields in (a-c) indicate each spin density distribution.
In the context of the inverse design of optical potentials from the target eigenmode, the T-spin mode of globally 'pure' handedness in the entire space (ψ-= 0 for all regions) can be molded. We assume electrically anisotropic materials having 1-dimensional (1D) variations of εx,y(x) and a nonmagnetic feature (μ = μ0). The electric field then has the form of E(x,y) = Ψ(x)·e -iβy , where Ψ(x) is the wavefunction envelope and β is the y-propagating wavevector. In the T-spin representation of Ψ(x) = ψ+(x)·e + + ψ-(x)·e -for the z-axis spins e ± = (e x ± i·e y )/2 1/2 'transverse' to y-propagating waves, a globally positive T-spin mode Ψ(x) = ψ+(x)·e + from ψ-(x) = 0 is then achieved with the anisotropic and inhomogeneous optical potential (see Supplementary Note S1 for the detailed derivation) (1) where k0 = 2π/λ0 is the free space wavenumber and εe is the effective permittivity of the +y-propagating (β > 0) T-spin eigenmode from β 2 = εe·k0 2 (see Supplementary Note S2 for the dependence on the propagation direction and mirror symmetry). Equation (1) can be applied to obtain potential landscapes for any nodeless spatial profiles of the target ψ+(x).
First, we consider the trivial case of a T-spin 'plane wave' where xψ+ = 0. The required potential profile from Eq. (1) then becomes constant as εx(x) = εe and εy(x) = 0. This solution reveals that zero longitudinal permittivity εy = 0, which corresponds to uniaxial ENZ 20-22 materials, imposes the full degree of freedom on the longitudinal electric field Ey and thus allows the emergence of the ideal T-spin plane wave (Dy = εy·Ey = 0). However, from εy(x) = 0, any Ey(x) is able to satisfy Maxwell's equations, thereby hindering the exclusive excitation of the pure T-spin state. We thus explore the condition of nontrivial "confined" eigenmodes that have T-spin polarizations distinct from that of the trivial plane wave case.
Without a loss of generality, we consider an example of the Gaussian profile ψ+(x) = exp(-x 2 /(2σ 2 )) while keeping ψ-(x) = 0 (Fig. 2a, σ = 5λ0). The required optical potential solution (Fig. 2b) is then 'inhomogeneous' anisotropic media (see Supplementary Note S3 for the field profiles and power flow of the target eigenmode Ψ(x) = ψ+(x)·e + with different modal sizes). Remarkably, although εx(x) has its distribution near the target effective permittivity εe (upper black dashed line in Fig. 2b), εy exhibits an intriguing transition from negative to positive values near x = 0, thus realizing the topological transition 23 of the isofrequency contour (IFC) between hyperbolic 24 and elliptic materials (Fig. 2c).
Compared with the plane wave solution of constants εx(x) = εe and εy(x) = 0, the spatial transition of the IFC near uniaxial ENZ (εy (x < 0) > 0 and εy (x > 0) < 0) not only allows the pure T-spin handedness of the target eigenmode ( Fig. 2d) but also yields a momentum mismatch (yellow regions in Fig. 2c), which 6 leads to the confinement of the eigenmode near x ~ 0 (Fig. 2e). We also note that the target T-spin eigenmode is separated from the other eigenmodes in both the spatial and momentum domains (see Supplementary Note S4 for details on the nearby eigenmodes), which enables the exclusive excitation of the T-spin without the other polarization states. Although the result in Fig. 2 demonstrates the existence of a confined T-spin mode with a globally 'pure' handedness, the spatially varying condition of anisotropic materials may hinder practical implementations. Because the modal profile of e -αx allows for 'homogeneous' realizations (Eq. (1) of εx(x) = εe and εy(x) = -α 2 /k0 2 ), we now explore the condition of potential landscapes for the confined modal profiles of ψ+(x) = exp(-|x| g /2σ g ) (Fig. 3a,b, g = 2 to 1, from Gaussian to exponential). As expected, the potential becomes homogeneous for g ~ 1 except for the discontinuity in xψ+ at x = 0 and the corresponding singularity of εy(x). By neglecting the contribution of the second-order derivative  (d) Envelope functions of the electric field ψ x,y (E x,y (x,y) = ψ x,y (x)·e -iβy ) and (e) net spin density |ψ + | 2 -|ψ -| 2 of the T-spin interface mode. (f,g) Excitation of the T-spin interface mode implemented in the platform of hyperbolicand elliptic-layered metamaterials through the oblique incidence (φ = 43.2°): (f) excited ψ + and (g) ψfields. For all cases, the transverse permittivity is ε x (x < 0) = 2.5 and ε x (x  0) = 1.5, and the longitudinal permittivity is ε y (x < 0) = -0.5 and ε y (x  0) = 0.5, corresponding to ε e = 2 and Δε = 0.5. The detailed design of the layered metamaterials and excitation conditions in (f,g) are shown in Supplementary Notes S6,7. The results of (f,g) are obtained from the stable scattering matrix calculation 27 . Extending the realization of globally pure T-spin waves further, we challenge the feasibility of obtaining a fundamental but unexplored state of polarization (SOP): the zero-T-spin mode corresponding to a linear polarization 'oblique' to the propagation direction on the Poincaré sphere of the meridional plane (Fig. 4a), which we call the T-spin Poincaré sphere. In isotropic Hermitian media for plasmonic or dielectric confinements of light, the emergence of a longitudinal field with an intrinsic π/2 phase difference with respect to the transverse field always accompanies the T-spin component. To annihilate the T-spin (Fig. 4b), the T-spin modes with different signs should be degenerate with the same amplitude and α1 2 = α2 2 = α 2 = (Δε 2 /εe)·k0 2 . We can then obtain the ratio between the longitudinal (Ey) and transverse (Ex) electric fields, which results in the T-spin mode Ψ(x) = ψ+(x)·e We note that Δε defines the trade-off between the strong confinement (large α) and pure handedness

Author Contributions
X.P. conceived the presented idea. X.P. and S.Y. developed the theory and performed the computations.
N.P. encouraged X.P. to investigate the pure existence of transverse spins while supervising the findings of this work. All authors discussed the results and contributed to the final manuscript.

Competing Interests Statement
The authors declare that they have no competing financial interests.

Note S1. Derivation of anisotropic optical potentials for T-spin modes
Consider Maxwell's wave equation of the electric field vector E = Ex·e x + Ey·e y , for 2-dimensional (2D) in-plane transverse magnetic (TM) modes in electrically anisotropic materials (εx,y): where k0 = 2π/λ0 is the free space wavenumber. For 1-dimensional (1D) variations of optical potentials εx,y = εx,y(x), the electric field can be set to E(x) = Ψ(x)·e -iβy as Ex,y(x,y) = ψx,y(x)·e -iβy , where β is the wavevector propagating toward the y-axis. By employing the T-spin representation Ψ(x) = ψ+(x)·e + + ψ-(x)·e -to describe z-axis "transverse" spins to y-propagating waves as e ± = (e x ± i·e y )/2 1/2 , Eq. (S1) then becomes (S2) From Eq. (S2), the interaction between optical potentials and T-spin states of light can be represented Considering the +y-propagation only (β > 0), Eq. (S3) further reduces to We note that contrary to the standard eigenvalue representation 1 of the wave equation deriving eigenstates from the given potential, Eq. (S3,S4) generates the required optical potential εx,y(x) for the given set of the nodeless eigenmode Ψ and its eigenvalue εe. Figure S1 shows the schematics of the detailed inverse design procedure of optical potentials defined by Eq. (S4). For the y-axis-propagating eigenmode of the arbitrary electric field profile (Fig.   S1a,b), its envelope Ψ can be separated for each T-spin state as Ψ(x) = [ψ+(x),ψ-(x)] T (Fig. S1c,d).
The corresponding anisotropic optical potential εx,y(x) is then obtained (Fig. S1e,f) from Eq. (S4) using Ψ(x) and the predefined eigenvalue εe. We note that the propagating mode with real-valued εe leads to the dielectric-dominant field profile for the transverse permittivity εx(x) (Fig. S1e) while the longitudinal permittivity εy(x) has a rapidly varying distribution dependent on the spin states of light (Fig. S1f).
The lower-index mode (εe = 2.16) is concentrated in the definite material (Fig. S4a), whereas the higher-index mode (εe = 2.39) is concentrated in the indefinite material (Fig. S4e). Such a spatial separation for different eigenmodes can be understood through the spatial distribution of the IFC (Fig. 2c in the main text), which shows a low k for the definite region and high k for the indefinite region. The polarization of nearby eigenmodes is close to linear (Fig. S4b,f), demonstrating the selective excitation of the target T-spin eigenmode (Fig. S4d).

Note S5. Trade-off relation between the confinement and spin handedness
The modal size of evanescent waves can be evaluated from 2 A = ∫W(x)dx / max{W(x)}, where W(x) = (D·E + B·H)/2 is the electromagnetic energy of the mode. Figure S5a shows the variation of the modal size of T-spin interface modes with respect to Δε. The increased discontinuity by |Δε| exponentially reduces the modal size, which is expected from the dispersion relation of the decay constant α 2 = (Δε 2 /εe)·k0 2 in the Methods Summary.
As shown in Eq. (3) in the Methods Summary, Δε breaks the pure spin condition Ey = ± iEx. We quantify the T-spin purity of the T-spin interface mode by S = ∫(|ψ+| 2 -|ψ-| 2 )dx / ∫(|ψ+| 2 + |ψ-| 2 )dx, where S = +1 for purely positive spin and S = -1 for purely negative spin. Although the increase of the discontinuity by |Δε| decreases the spin purity, nearly ideal T-spin waves can be achieved except in the case of deep-subwavelength confinement (Fig. S5b, S ~ ± 0.993 for A ~ λ0 and S ~ ± 0.829 for The confinement of the T-spin density can also be defined as Sλ0/A (Fig. S5c), which represents the 'concentration' of the spin state for each wavelength. For a given value of εe, an optimum point occurs for T-spin confinement (Δε = 1.42 in Fig. S5c).
for the elliptic region, where fH and fE denote the filling ratios of the media ε1H and ε1E, respectively. Figure S6 represents the required permittivities of each layer in hyperbolic (Fig. S6a) and elliptic (Fig. S6b) materials for Δε = 0.5. We adopt the case of fH = 0.7 (ε1H = 1.5 and ε2H = -5.2) and fE = 0.9 (ε1E = 1.3 and ε2E = -6.9) for the results of Fig. 3 in the main text, which allows for the utilization of dielectric and metallic layers with experimentally accessible material parameters.

Note S7. Excitation conditions of the T-spin interface mode in metamaterials
For an oblique incidence (Fig. 3c in  (S9) where εb is the permittivity of the incident region. Without introducing structural modulations 5 or near-field techniques 6 , the permittivity of the incident region is restricted by εb > εe·[1 -(Δε/εe) 2 ], which is similar to the prism coupling technique for the excitation of surface plasmon polaritons 7 . Figure S7 presents the S-matrix-calculated 8 angular dependency of the excited T-spin mode energy at the interface for unit cell lengths of λ0/5, λ0/10, λ0/20, and λ0/40. As shown, the deepsubwavelength layer construction of metamaterial leads to an optimum angle close to that of the theoretical value. The trade-off relation between the excited energy and angular bandwidth is also observed.