# Supporting Information of. Observation of enhanced optical spin Hall effect in a. vertical hyperbolic metamaterial

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1 Supporting Information of Observation of enhanced optical spin Hall effect in a vertical hyperbolic metamaterial Minkyung Kim 1, Dasol Lee 1, Tae Hak Kim 2, Younghwan Yang 1, Hui Joon Park 2, and Junsuk Rho 1,3 1 Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea 2 Department of Organic and Nano Engineering, Hanyang University, Seoul 04763, Korea 3 Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea This document contains 7 pages with 6 figures. S1

2 Transmission coefficients from effective medium theory s p k y z x ε 2 = Medium 1 (ε 1 ) ε x Medium 2 ε y z=0 Medium 3 (ε 3 ) z=d Figure S1: Schematics Transmission coefficients of a slab of a hyperbolic metamaterial are calculated analytically by solving Maxwell s equations at interfaces (Fig. S1). If transmission and reflection coefficients at two interfaces (z = 0 and z = d) are given, then total transmission coefficients can be obtained by taking multiple reflections into account. We first consider the first interface (z = 0). Transmission and reflection coefficients of s- and p-polarization across the single interface are r p = εx cos θ ε 1 1 ε 1 sin 2 θ εx cos θ + ε 1 1 ε 1 sin 2 θ t s = r s + 1 (S1) (S2) r s = ε1 cos θ ε y ε 1 sin 2 θ ε1 cos θ + ε y ε 1 sin 2 θ (S3) t p = (r p + 1) ε 1 ε1 ( 1 1 ε x ) sin 2 θ + 1 ε x (S4) where θ is an incident angle. Substituting ε 3 for ε 1 in Eqs. S1 to S4 gives transmission and reflection coefficients when light is injected from medium 3 to medium 2. Therefore, we can calculate transmission and reflection coefficients ( t j and r j where j = s, p) of the S2

3 second interface (z = d) by replacing ε 3 with ε 1 and applying the Stokes relation. Then the transmission coefficient of the hyperbolic metamaterial slab can be calculated as T j = t j t j exp(ik j d) 1 + r j r j exp(2ik j d) (S5) where k j is a z-component of the wave vector given as k p = k 0 ε x ε 1ε x sin 2 θ (S6) k s = k 0 ε y ε 1 sin 2 θ (S7) where k 0 is the wave vector in free space. Effective permittivities and dispersion of a vertical hyperbolic metamaterial consisting of gold and air Effective permittivities of gold and air with the same volume fraction are calculated using Eq. 3 in the main manuscript. Equi-frequency contours shown in Fig. S2(b) and S2(c) prove type-ii hyperbolic dispersion. a b c Figure S2: (a) Effective permittivities, (b) equi-frequency contour and (c) equi-frequency curve in k x -k z plane of a vertical hyperbolic metamaterial composed of gold and air. Black curve in (c) represents equi-frequency curve of air. Metal filling ratio is 0.5. S3

4 Optical spin Hall effect in a type-i hyperbolic metamaterial In the main text, we demonstrate the enhancement of optical spin Hall effect in a type- II vertical hyperbolic metamaterial. To clarify that the enhancement is not restricted to a specific type of hyperbolicity, optical spin Hall effect of type-i hyperbolic metamaterial is presented in this section. A hyperbolic metamaterial consisting of silver and titanium dioxide, whose optical properties and shift are presented in Fig. 2 in the main manuscript, exhibits type-i hyperbolicity in the smaller wavelength. Around 450 nm, ε < 0 and ε > 0 (Fig. S3(a)). Therefore, equi-frequency surface of the hyperbolic metamaterial is composed of two branches with a momentum gap along the stacked direction (Fig. S3(b) and (c)). Transmission coefficients and shift of the horizontal and vertical hyperbolic metamaterial are plotted in Fig. S3(d)-(f). Thickness of the hyperbolic metamaterials is set as 5 nm to avoid nearly zero transmission originating from high optical losses. Gigantic optical spin Hall effect with high wavelength-selectivity is obtained. a b c d e f t p (hhmm) t p (vhmm) t s (hhmm) t s (vhmm) Figure S3: (a) Effective permittivities. Equi-frequency contour of a (a) horizontal and (b) vertical hyperbolic metamaterial composed of silver and titanium dioxide at 450 nm. (d) Amplitude and (e) phase of transmission coefficients and (f) shift when incident angle is 5 and thickness of hyperbolic metamaterials are 5 nm. S4

5 Shift and shift enhancement at small incident angle Since cotangent function diverges as its argument goes to zero, the assumption k1w cot 2 θ i is not true for sufficiently small θ i. Then, Eq. (1) in the main manuscript cannot be simplified to Eq. (2). We compare the transverse shift calculated by the exact and approximate formula (Eq. (1) and Eq. (2) in the main manuscript respectively) at small θ i. Fig. S4(a) and S4(b) show that when θ i is sufficiently small (< 0.2 here), the shift calculated by the exact form deviates from that by the approximate form and converges to zero as θ i goes to zero, whereas the shift calculated by the approximate form diverges. For completeness, the shift enhancement when θ i 5 is shown in Fig. S4(c). Shift enhancement, which is defined as δ vhmm /δ hhmm, increases as θ i decreases. a b c Figure S4: Shift and shift enhancement at small θ i. Shift calculated by the approximate (dashed lines) and exact (solid lines) formula when (a) θ i 20 and (b) θ i 0.2 (c) Shift enhancement when θ i 5 Comparison between vertically- and horizontally-stacked dielectric multilayer In the main manuscript, we present an analysis based on effective medium theory to explain the enhanced optical spin Hall effect in a vertical hyperbolic metamaterial. However, a vertical multilayer of two different dielectric materials also exhibits enhanced optical spin Hall effect compared to a horizontally stacked dielectric multilayer (Fig. S5). Due to the S5

6 weak anisotropy in the horizontally stacked dielectric multilayer, the absolute value of the shift is smaller than that of a vertical hyperbolic metamaterial. a b c t ph t sh t pv t sv d e Figure S5: Dispersion, transmission coefficients and shift of dielectric multilayer composed of titanium dioxide and silicon dioxide. (a) Equi-frequency contour at 600 nm. (b) Amplitude and (c) phase of transmission coefficients and (d) shift when incident angle is 5. (e) Shift enhancement at four incident angles. Effect of diffraction in the measurement β t Air (index: 1) θ t k t Substrate (index: n) k i θ i β i p Figure S6: A schematic of diffraction in grating In order to prove that the shift originates solely from the optical spin Hall effect, we consider the diffraction which may be caused by the grating pattern (Fig. S6). The trans- S6

7 mitted angle θ t of the diffracted beam can be obtained by applying boundary equation. Since tangential component should be continuous at the interface, β t = β i + 2πm/p (S8) where m is an integer characterizing the diffractive modes; β i and β t are transverse wave vectors of incident and transmitted beams defined as β i = k i sin θ i = 2πn λ sin θ i (where n is a refractive index of the substrate) and β t = k t sin θ t = 2π λ sin θ t respectively. Therefore, θ t can be expressed as ( θ t = sin 1 n sin θ i + m λ ) p (S9) Because λ/p > 2 and incident angle is small (θ i < 10 ) in our experiment, the argument of sin 1 in Eq. S9 has an absolute value larger than unity for non-zero m. Therefore, one can conclude that the diffraction does not occur (m = 0), and the transverse shift indeed results from the optical spin Hall effect. S7

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