Diffraction-induced enhancement of optical spin Hall effect in a dielectric grating

Electrons that possess opposite spin may undergo transverse shift along the opposite direction even in the absence of an external magnetic field.^1,2 This spin-dependent deflection of electrons known as the spin Hall effect is not a unique phenomenon that can only be found in electronic systems, but has analogies in photonics in terms of spin-dependent phenomena, such as transverse splitting,^3,4 angular splitting,^5,6 and splitting of exciton–polaritons.^7,8 Among them, the optical spin Hall effect (OSHE), also known as the photonic spin Hall effect and spin Hall effect of light, manifests itself as a spin-dependent and transverse splitting of circularly polarized light at an interface between two media,^3,4,9 and is also known as the Imbert–Fedorov effect.^10,11 The physical origin of the OSHE lies on the finite beam waist of the incident beam, which contains not only central wave vector components, but also other wave vectors slightly deflected from the central beam axis in both in-plane and out-of-plane directions with respect to the incident plane. The deflected wave vectors have different incident angles and incident planes. Therefore, the wave vector deflection results in the deviation of polarization basis, and thereby induces a positive geometric phase for one circular polarization and a negative geometric phase for the other.

OSHE is a universal phenomenon that occurs at an interface between natural media^4,10–13 and random media.¹⁴ In general, the amount of shift is small corresponding to a fraction of the wavelength. OSHE can be increased by exploiting structured media that transmit or reflect s- and p-polarization differently, such as metamaterials,^15–19 metasurfaces,^20–22 and two-dimensional (2D) materials.^23–27 Alternatively, OSHE can be significantly increased under a specific circumstance in which the transmission or reflection of only one polarization vanishes. One example is reflection near the Brewster angle.^12,13 In contrast, previous studies to increase OSHE in the transmission type have mainly focused on a small incident angle, on the order of milli-radians, in which OSHE is significantly enhanced as the incident angle goes to zero.^17,18,28,29 The enhancement of OSHE at a large incident angle has not been proposed in a transmission type.

Here, we present OSHE in a dielectric grating on a metal film by maximizing the difference between the transmission coefficients of s-polarization and those of p-polarization. The metal film impedes light transmission except the p-polarized first-order diffracted mode that is coupled to surface plasmon polaritons (SPPs). We numerically demonstrate that these polarization-dependent transmission coefficients lead to high OSHE at a large incident angle, whereas the OSHE is enhanced at a small incident angle in previous instances. We further show that the operating wavelength or incident angle of the OSHE can be tuned by changing materials or geometrical parameters of a device. The diffraction-induced OSHE will be advantageous in optical applications, including filters and sensors.

A vertically polarized incident beam that transmits at an optical interface is shifted by an amount of δ given as²⁸ [Fig. 1(a)]

assuming that the beam waist ω₀ is sufficiently large [$(k1ω0/cotθi)2≫1$]. Here, k₁ is a wave vector in medium 1, θ_i is an incident angle, t_s and t_p are transmission coefficients for s- and p-polarization, respectively, + denotes left circular polarization and − denotes right circular polarization.

Equation (1) implies that OSHE can be increased in two ways, either by reducing θ_i or by increasing the difference between t_s and t_p. However, reducing θ_i does not always make δ increase. At an interface between natural materials, δ decreases as θ_i decreases⁴ because t_s/t_p (or t_p/t_s) converges more rapidly to unity than cot θ_i diverges. In contrast, OSHE generally increases at small θ_i (≪π/2) in artificially structured media^17,18,28,29 as a result of the diverging behavior of cot θ_i. This term is associated with the short length of the adiabatic path in the momentum space [Fig. 1(b)]. OSHE can be understood to be a consequence of the geometric phase Φ_B that is acquired by a deflected wave vector in the polarization space.^9,30,31 The deflected wave vector dk that is perpendicular to the incident plane alters both the incident plane and the polarization basis [Fig. 1(b)]. For a given dk, the angular deflection Φ = dk/(k sin θ_i) in the momentum space increases as θ_i decreases [insets in Figs. 1(b) and 1(c)]. Increase in Φ directly yields the increased geometric phase as³² Φ_B = Φ cos θ_i ∝ cot θ_i. Thus, δ, which is closely related to the gradient of the geometric phase,^33,34 has the cot θ_i term, which diverges as θ_i converges to zero. Therefore, the remarkable enhancement of OSHE has been demonstrated in a narrow range of θ_i within a fraction of a degree of 0°,^17,18,28,29 in which δ changes drastically in response to θ_i. Despite the significant increase of δ, the high sensitivity complicates the task of fine controlling the shift. Moreover, sometimes a large OSHE at large θ_i is preferable [Fig. 1(c)], for example, for applications such as polarization-dependent filters and beam splitters. Thus, a new route to attain large OSHE without relying on the short path length in the momentum space is in demand.

To increase OSHE without reducing θ_i, we focus on the other term t_p/t_s. We propose a dielectric grating on a metal film [Fig. 2(a)] to achieve polarization-dependent diffraction. The grating and substrate are silicon dioxide (SiO₂) and the metal is gold (Au). The geometric parameters of the grating are as follows: period p = 700 nm, width $w$ = p/2, and height H = 100 nm. The thickness of the metal layer is t_m = 60 nm, which is much thicker than the skin depth of metal in the visible spectrum. We first examine the SPPs of the SiO₂–Au–air interface. Their dispersion curve can be obtained by solving two boundary conditions of p-polarized modes at the three-layer structure as³⁵ 

Here, ε denotes relative permittivity, $kn=β2−k02εn$⁠, where the subscripts denote n = m for metal, n = d for dielectric, n = a for air; β is the propagation constant and k₀ is the wave vector in the free space. The dispersion relationship of SPPs can be obtained by solving Eq. (2) in terms of β.

The dispersion curve of SPPs has a higher wave vector component than light [Fig. 2(b)]. Thus, light that is incident obliquely from the substrate can excite the SPPs, but they cannot be coupled to air and remain evanescent. Grating can compensate for this phase mismatch and allow the transmission of the p-polarized incidence. The phase matching condition of grating is applied to p-polarization as³⁵ 

where m is the diffraction order, G = 2π/p is the reciprocal vector of the grating, θ_t is a transmitted angle, and k₂ is a wave vector in the outer medium, which is air in this case. Whereas SPPs have a wave vector that is higher than that of light, m = −1 mode [red dashed curve, Fig. 2(b)] falls inside the light line and can be coupled to air. Because the phase compensation using a grating only occurs for p-polarization and the SPPs exist only for p-polarization, the transmission of m = −1 mode appears for p-polarization, whereas most of the s-polarized incidence cannot be transmitted by any mode because of the thick metal layer and lack of the coupling mechanism. This polarization-dependent transmission contributes strongly to the increase in OSHE at large θ_i.

To numerically demonstrate polarization-dependent transmission, the transmission coefficients of diffraction order m = −1 for p- and s-polarization were calculated using COMSOL Multiphysics (Fig. 3). The kinks highlighted as blue curves are not numerical artifacts, but Rayleigh anomalies that universally appear in grating structures at the transition between propagating and evanescent modes.³⁶ To simulate transmission under oblique incidence, 2D simulation was performed in the x–y plane, in which the boundaries that are normal to the x-axis are set as the Bloch boundary condition. To obtain m = −1 mode, we focus on a large incident angle in the range of 50° ≤ θ_i ≤ 80°. In this range, Eq. (3) has no real solution of θ_t for m = 0, while has a real solution for m = −1. The polarization-dependent phase compensation leads to a transmission peak of diffracted light (m = −1) only for p-polarization [Figs. 3(a) and 3(b)]. For a given θ_i, the p-polarized incidence is converted to SPPs at a wavelength that satisfies the phase matching [Eq. (3)]. The SPPs, which are localized at the interfaces, become radiative because of the phase compensation of the grating. The peak has a maximum value of 0.52 and appears in a broad range of visible wavelengths [green solid curve, Fig. 3(b)]. In contrast, under s-polarized incidence [Fig. 3(d)], SPPs are not excited and phase compensation by the grating does not occur. Thus, transmission is small (|t_s| < 0.1) and has no prominent feature [Figs. 3(d) and 3(e)]. This polarization-dependent transmission increases the term 1 − t_p/t_s in Eq. (1), thereby resulting in the enhancement of OSHE. The phases of transmission coefficients [Figs. 3(c) and 3(f)] are plotted because their phase difference also influences δ according to Eq. (1).

We calculate δ by using the simulated transmission coefficients and Eq. (1) [Fig. 4(a)]. As expected from Eq. (1), δ exhibits a similar peak with t_p at a broad range of 50° ≤ θ_i ≤ 80°. The maximum shift in a given parameter space is approximately 3.7 µm at λ = 730 nm and θ_i = 63°, which corresponds to 5.1λ for δ > 0, and −3.5 µm at λ = 800 nm and θ_i = 50°, which corresponds to 4.4λ for δ < 0. To see more clearly, δ at specific wavelengths are shown in Fig. 4(b). Wavelengths are selected in the range of 600 nm ≤ λ ≤ 660 nm to avoid the kinks that originate from Rayleigh anomalies [blue curves, Fig. 4(a)]. δ exhibits the resonant feature near the wavelength satisfying the phase matching condition. The operating incident angle can be tuned by changing the wavelength of interest. The results of the three-dimensional full-wave simulation at λ = 640 nm and λ = 660 nm can be found in the supplementary material.

The efficiency of the optical spin Hall effect is low as a result of the low transmission of s-polarized incidence. The amplitude of electric field of the transmitted beam divided by that of the incidence is equal to |t_s|, which has a maximum value of 0.074 in the given parameter space. The efficiency can be increased by reducing t_m. The transmission of the proposed system and a system with a thinner metal layer can be found in the supplementary material.

The operating wavelength and incident angle of OSHE are closely related to the structural peak of |t_p|. The shape of the peak is determined by the phase matching condition of SPP dispersion, in which the refractive indices of metal and dielectric and geometrical parameters are important. Therefore, changing the materials or the geometrical parameters may alter the operating wavelength and incident angle. Because δ shares the same structural peak with |t_p|, we plot the peak of |t_p| in three additional systems to demonstrate the tunability of OSHE (blue curves, Fig. 5).

For the first additional system, we replaced Au with silver (Ag) while keeping the other materials and geometrical parameters the same [Fig. 5(a)]. The wavelength range of the enhanced OSHE widened, ranging from 400 nm to 800 nm. The peak θ_i of a given wavelength also differed as a result of the change in SPPs dispersion. At a fixed wavelength in the visible regime, the real part of permittivity is lower for Ag than that for Au. Thus, considering that SiO₂ has constant-like permittivity, the excitation of SPPs and their conversion to radiative waves occurs at a lower wavelength for a given θ_i. Second, we changed the dielectric from SiO₂ to aluminum oxide (Al₂O₃) for both the grating and the substrate [Fig. 5(b)]. The dispersion of SPPs at the Al₂O₃–Au–air interface is similar to that at the SiO₂–Au–air interface according to Eq. (2). Instead, the higher refractive index of Al₂O₃ than that of SiO₂ results in a peak at the smaller θ_i for a fixed wavelength. Last, we changed the grating width from $w$ = 350 nm to $w$ = 500 nm while keeping the other parameters fixed [Fig. 5(c)]. Although the role of the duty cycle cannot be explained by the phase matching [Eq. (3)], larger $w$ increases the contact area of the Au film with the dielectric grating and alters SPPs dispersion to have larger β for a given wavelength. The increase in $w$ led to a red shift of |t_p|.

In conclusion, we numerically demonstrate that diffraction induces the enhancement of OSHE at a large incident angle by exploiting a dielectric grating on a metal film. The polarization-dependent behavior of the grating and surface plasmon polaritons that only occur for p-polarization results in amplified shift that reaches several wavelengths. We demonstrate that the enhancement of the optical spin Hall effect does not necessarily require a short path length in the momentum space. The large OSHE at a large incident angle will facilitate polarization-sensitive optical devices, such as switches and beam splitters.

See the supplementary material for the three-dimensional direct simulation results and analysis on the efficiency.

J.R., M.K., and D.L. conceived the idea and initiated the project. M.K. conducted numerical simulations. M.K. and D.L. mainly wrote the manuscript. M.K., D.L., and B.K. analyzed the data. All authors participated in the discussion and confirmed the final manuscript. J.R. guided the entire project. M.K. and D.L. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This work was financially supported by the National Research Foundation (NRF) (Grant Nos. NRF-2019R1A2C3003129, CAMM-2019M3A6B3030637, and NRF-2019R1A5A8080290) funded by the Ministry of Science and ICT (MSIT) of the Korean government. M.K. acknowledges the Global Ph.D. fellowship (Grant No. NRF-2017H1A2A1043204) from NRF-MSIT of the Korean government.