Giant Photonic Spin Hall Effect by Anisotropic Band in Photonic Crystal Slabs

When a paraxial light beam is reflected from/ transmitted through an optical element with rotation symmetry, a tiny small incident angle can arise wave-vector-varying Pancharatnam-Berry (PB) phases, resulting in high-performance photonic spin Hall effect (PSHE). Differently, here we investigate the PSHE in photonic crystal slabs of square lattices constructed with air holes, and demonstrate that the C4v symmetry of the slabs enables giant PSHE in both isotropic and anisotropic band structures with a small incident angle. The guided-mode resonances of the subwavelength-thickness slab enhance the PSHE, allowing the spin separation to exceed the incident beam waist. Via changing the incident angle, wavelength and even the beam waist, the spin separation can be adjusted in a wide range, which makes our proposal more flexible in manipulation of spin photons.


I. INTRODUCTION
P HOTONIC Spin Hall Effect (PSHE) associated with the geometric phase is caused by spin-orbit interactions when a paraxial light beam projected onto an oblique plane is reflected or transmitted at an interface [1], [2], [3]. Typically, PSHE causes photons with opposite spin components to split spatially into left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) light [4], [5], [6], [7]. In other words, featured by spin-dependent displacements, PSHE introduces additional spin degrees of freedom for the flexible manipulation of light, and has attracted much attention in various realms, including optical sensing, precision metrology and nanophotonic devices [8], [9], [10], [11]. However, the weak spin-orbit interactions result in a generally weak PSHE [12], [13], [14], [15], [16], [17]. Although various methods have been demonstrated to enhance the PSHE, the enhancements sacrifice the energy efficient. Recently, a novel PSHE was proposed based on the wave-vector-varying Pancharatnam-Berry (PB) phases, allowing us to address the contradiction between spin separation and energy efficiency in the conventional PSHE [18]. The wavevector-varying PB phases arise naturally when a paraxial light beam is reflected from or transmitted through an optical element with rotation symmetry with a tiny small incident angle. With a uniaxial yttrium vanadate crystal, the opposite-spin photons of transmitting beam can be completely separated with a spin separation of 2.2 times the incident beam waist with a high energy efficiency of ∼70% [19]. However, to form the rotation symmetric system, the optical axis of the uniaxial crystal is perpendicular to its input and output interfaces. The needed crystal length is up to 10 mm, which hinders the development of integrated spin-photonic devices.
Here, we investigate the PSHE in subwavelength-thickness photonic crystal (PhC) slabs of square lattices with air holes. Consisting of a periodic modulation of the refractive index at the wavelength scale, the PhC slabs can be used to realize many novel optical applications including filters [20], [21], [22], [23], [24], [25], [26], [27], optical vortex generation [28], structural color generation [23], and polarization tailoring [29]. The PhC slabs support both isotropic and anisotropic band structures [30]. We demonstrate here that giant PSHE from wave-vector-varying PB phases can be obtained in both isotropic and anisotropic band structures.
II. THEORY Fig. 1 gives the structure of the proposed PhC slab. The PhC slab is a free-standing silicon nitride (Si 3 N 4 , refractive index 2.02) with periodic circular holes, the unit cell of which is shown in the inset of Fig. 1(a). The period of the lattice a = 380 nm, the thickness t = 120 nm, and the radius of the air hole r = 140 nm. The transmission spectra for s-plane wave are calculated with the finite element method (COMSOL Multiphysics 5.6) and shown in Fig. 1   be coupled to by free-space illumination. Thus at these bands, guided-mode resonances occur. Especially, the Γ point at 530.6 nm is bound states in the continuum (BIC), with infinite life times and ideally Dirac transmission dip. On contrast, the transmission dips around 495 nm are wide, since Band 2 and 3 are very close to each other. Fig. 2(a), (b) give a deep insight on the band structures near λ = 532 nm and 493 nm. Near λ = 532 nm, the band structure is in rotation symmetric, while which is in four-fold rotational symmetry for λ = 493 nm. The band structure makes influence to the transmission coefficients, as shown in Fig. 2(c)-(j), where the coefficients functions of transverse wavevector (k x , k y ) are shown. At λ = 532 nm, coefficients |t ss | and |t pp | are rotationally symmetric, and |t ps | and |t sp | vanish identically. Thus, s and p polarizations are decoupled. At λ = 493 nm, however, the |t ps | and |t sp | are nonzero, except for at the lines of k x = 0 and k y = 0. The coupling of s and p polarizations is because the Band 2 and 3 are close to each other very much, they will be excited simultaneously when the obliquely incident plane waves are not along a high symmetry plane.
In the following, we focus our attention on the PSHE of the transmitted beam from the PhC slab. The PhC slab can be considered as modulator in the momentum space. When a paraxial light beam illuminates obliquely on a slab, the local incident plane changes with the transverse wavevector κ y : φ = κ y /sinθ, where φ is the azimuthal angle between the local incident plane and the x-axis. The local incident plane (azimuthal angle) increases linearly with κ y . Before the transmission process, the incident field should be decomposed into the s and p waves, whose transmission coefficients are given in Fig. 2. According to Zhu et al. [18], the decomposed matrix under small incident angle approximation is given bỹ The transmission matrix for s and p plane waves arẽ For the isotropic band, |t ps | and |t sp | are zero. For the anisotropic band, all the transmission coefficients vary with the azimuthal angle. Thus, the coefficients can be written into Taylor series with respective to incident θ and azimuthal φ angles. Under the first order approximation, we have The angular spectrum of the transmitted field is thereforẽ E t =Û +FÛẼ i . Here, we consider a simple case with a vertically-polarized one-dimensional (1D) Gaussian beam incident along the x-axis,Ẽ i = exp[−k 2 y w 2 0 /4]êy, with w 0 being the beam waist. The transmitted beam thus can be calculated as: t ij = ∂t ij (θ, φ)/∂φ. The phases ∓2iφ are the wave-vectorvarying PB phases. The spin-dependent displacements are defined as Δ ± = Ẽ ± |i∂/∂ k yẼ ± / Ẽ ± |Ẽ ± , which are One can find from (5) that the spin-displacements origin from two parts: one is from the wave-vector-varying PB phases and the other from the azimuthal-dependent polarization conversion effect. It is worth to point out that the spin-dependent displacements are zero for tiny small incident angles, when a paraxial beam transmitting through a uniaxial slab with its optical axis within the input interface. It is because the spin-displacements induced by the wave-vector-varying PB phases and azimuthaldependent cross polarization conversion effect are canceled out. Moreover, based on Bloch modes that rely on only symmetry and periodicity, our proposal for spin separation is substrate independent [27], [29], [31], [32].

III. SIMULATION
To verify the PSHE in PhC slabs, we perform 3D numerical simulations with the finite element method (COMSOL Multiphysics 5.6). Here, the incident interface is selected as the xz plane, and the angle between the incident direction and z-axis is θ as shown in Fig. 1(a). For simplicity, 1D Gaussian beams are employed in the simulation, thus the period boundary condition can be used in the x-direction. To save the simulation time, the beam waists of the incident beams are set as 5 μm, and the simulation region is a × 150a×(3λ + 5t). For the s-polarized incident beams, the simulated intensity distributions in the yz plane are given in Fig. 3(a1), (b1) for the the wavelength and incident angle (λ, θ) being (493 nm, 2°) (a1) and (542 nm, 4°) (b1), respectively. The peak intensities are found in within the PhC slab owing the local confinement effect of light fields. The RCP and LCP components of the transmitted light field will be moved toward opposite direction.
The transmitted light field are monitored at the z = (3λ+5t)/2 plane, where the local confinement effect can be neglected (see Fig. 3(a1), (b1)), thus the separation of the RCP and LCP components can be well observed. As shown in Fig. 3(a2), (b2), the intensity profiles of the RCP and LCP components are moved along the ∓y directions for the both cases. The movements of the RCP and LCP components of the transmitted light fields are also evident in the Stocks parameter S 3 , as shown in Fig. 3(a3), (b3). The positive and negative values of S 3 are separated by the y = 0 line, indicating that the transmitted light fields are in opposite handness polarization states for the ±y regions. Fig. 4(a) gives the Spin-dependent displacements as functions of the incident angle for the operation wavelength of 493 nm, where the solid and dotted lines denote the theoretical and simulated results. The theoretical solid lines are calculated according to (5). The theoretical and simulated results are in  good agreement. The small disagreements are mainly due to the small beam waist (5 μm), which cannot ensure the paraxial approximation. For case of λ = 493 nm, the spin-displacements reach peak values at θ = ±2°and ±12°. At θ = ±2°, the spin separation Δ = |Δ + −Δ − | = 5 μm, identical to the incident beam waist. Fig. 4(b) shows the transmittances of light beams. The peaks of the spin-displacements are usually accompanied with the low transmittances, lower than 0.01. At the peaks of θ = ±12°for λ = 493 nm, the transmittances are up to 0.3. Fig. 5 shows the spin-displacements and transmittance |t s | 2 as functions of the wavelength for the incident angles θ = 2°, 4°, 6°, and 8°, respectively. From the transmittance, three dips can be noticed around the wavelengths of 460, 495 and 550 nm, among which the dip near 495 nm hardly changes with the incident angle, while the other two dips show a great dependence on the incident angle. The dip of transmittance at short wavelength (460 nm) moves to shorter wavelengths as the incident angle increases, while the dip at long wavelength (550 nm) behaves conversely.
Near the 460 nm, there are two close dips in the transmittance when θ = 2°. These two dips merge gradually when the incident angle increase, and only one single dip can be found when θ = 8°. According to (5) the spin-displacements can be enhanced near the dips of |t ss | for y-polarized incident 1D Gaussian beams. At exactly the dip positions, however, no spin-dependent displacement occurs (see Fig. 5). Near the dip positions, the spin-displacements are very sensitive to the wavelength. The spin-displacements change their signs when the operation wavelength crosses the the dip of |t s | 2 , except for the wide dips near 460 nm when θ = 4°, 6°. More noteworthy is that the spin-displacements around the wavelength of 495 nm exceed w 0 (5 μm), reaching 5.3 μm for the incident angles θ = 4°and 6°. And for the incident angle θ = 2°and 8°, the spin-displacements around the wavelengths of 535 nm and 495 nm are also larger than w 0 , reaching 5.1 μm.
The dependence of the spin-dependent displacements on the incident beam waist is investigated. As shown in Fig. 6, the spin separation Δ increases from 5 to 10 μm, when the incident beam waist increase from 5 to 10 μm. It is worth to note that, for each incident beam waist, the spin separation can approach close to the beam wait by optimizing the incident angle and operation wavelength.

IV. CONCLUSION
In conclusion, a giant PSHE is demonstrated in a subwavelength-thickness PhC slab with a square lattice of air holes. The C 4v symmetry of the slab enables it to support both isotropic and anisotropic band structures. With the anisotropic band structures, the spin-displacements caused by the wavevector-varying PB phases and azimuthal-dependent cross polarization conversion effect. Both the theoretical prediction and numerical simulation results show that giant spin separations larger than the incident beam waist can be obtained. The spin separation can be adjusted in a wide range via changing the incident angle, wavelength and even the beam waist. These findings deepen the understanding of the PSHE and offer a flexible way for designing nanoscale spin-photonic devices.