Limitations of the transmitted photonic spin Hall effect through layered structure

In this paper, we show theoretically that the spin-dependent transverse shift of the transmitted photonic spin Hall effect (SHE) through layered structure cannot exceed half of the incident beam waist. Exact conditions for obtaining the upper limit of the transmitted SHE are clarified in detail. In addition, different from the popular view in many investigations, we find that there is no positive correlation between the spin-dependent transverse displacement and the ratio between the Fresnel transmission coefficients (tp, ts). In contrast, the optimal transmission ratio is determined by the incident angle and the beam waist. Moreover, two conventional transmission structures are selected and studied in detail. The characteristics of the transverse displacements obtained are in very good agreement with our theoretical conclusions. These findings provide a deeper insight into the photonic spin Hall phenomena and offer a guide for future related research.

Re derivation and verification of analytical expressions of the transverse shifts Figure S1 illustrates a beam incident to a layered optical structure. The z axis of the laboratory Cartesian frame (x, y, z) is normal to the interfaces of the layered structure and the xz plane is the incident plane. We assume that the +z i, t direction of the local coordinate systems attached to the incident/transmitted beams is along the propagating direction of the central wave, as shown by the black lines. The red coordinates and the red lines are used for an arbitrary noncentral wave. We consider a monochromatic linearly polarized Gaussian beam passing through this layered structure. The angular spectrum of the incident Gaussian profile in the local coordinate system attached to the incident beam can be written as: where ω 0 is the beam waist.

Figure.S1
Schematic illustrating the central and local wave vectors when a beam incident upon a layered structure.
In Eq.(S1), k ix and k iy are the transverse wave vectors of the noncentral waves. These transverse wave vector distributions will induce change of the incident angle for a noncentral wave. For example, if the incident angle of the central wave is θ i , a small in-plane deflection of wave vector k ix will induce an incident angle change to θ k . Considering the incident angle of each individual wave of the incident beam, and considering coordinate transformations between the local coordinate system attached to the incident beam and that attached to the transmitted beam, the relationship of the angular spectra between transmitted beam and incident beam in each local coordinate system can be obtained as 30 : where E , E , E and E are the H and V components of the angular spectra of the transmitted and incident beams, respectively. t p and t s denote the Fresnel transmission coefficients for H and V polarization states. In the following operation, these transmission coefficients, which will exist in the integrand functions, should be transformed into explicit functions of k ix or k iy . Using a Taylor series expansion and a first-order approximation, considering the symmetry of integrand function in the y direction, we only performed the Taylor expansion in the k ix direction, , , where k ix =0, k iy =0 corresponds to the central wave of the incident beam. By substituting Eq.(S1) and Eq.(S3) into Eq.(S2), the angular spectrum of the transmitted field can be easily obtained. For an incident Gaussian beam with H polarization (E E , E 0), the transmitted angular spectrum components are: where η θ θ . Usually, the whole structure is immersed inside one homogeneous medium.
For simplicity, in the following deduction, η is set to 1. The complex amplitude of the transmitted beam in real space is then calculated via an inverse Fourier transform of the angular spectrum components. We obtain where E , , exp is the complex amplitude distribution of the incident Gaussian beam, and z k ω /2 is the Rayleigh length. Using a circular basis √ , the transmitted complex amplitude components can be decomposed into two circular components as: In our calculations, the parity of the integral factor and the Poisson integral formula are used in this step. Through careful calculation, the analytical expressions of the transverse displacements can be obtained as: Similarly, the transverse displacements with V polarization input can be obtained by making replacements of t p →t s , t s →t p in Eq.(S10), The displacements contain both transverse spatial shifts and an angular spin shift, where the z dependent term is the angular spin shift. For actual SHE structures, the thicknesses of the optical structures are usually much smaller than the Rayleigh length z 0 of the light source, thus, the angular spin shift can be ignored. Here we only consider the spatial shifts. Eq.(S10) and Eq.(S11) are similar to many expressions of the transverse displacements in the literature, but they are not exactly the same. Formally, these two formulae are closest to the expressions in Ref.
[10] and Ref. [22], except that the last term of the denominator is , ⋅ , * rather than , . It is apparent that when t s, p is a complex number, these two terms are different, especially when , varies rapidly with .
To distinguish which formula is correct, we used them separately to study an ENZ-air-ENZ structure. The permittivity of the three layers are set to be 0.01, 1, 0.01 in turn and the corresponding thicknesses take values of λ， 3λ， λ. The wavelength and the beam waist of the incident beam are set to be λ=0.633 µm and ω 0 =10λ, respectively. . Apparently, at some large resonant angles, the transverse shifts in Fig.S2 (b) are much greater than those in Fig.S2 (a). Moreover, if we increase the beam waist to 50λ, the largest transverse shift in Fig.S2 (b) almost reaches one thousand times of λ. It is apparently unreasonable. We also used a commercial software COMSOL Multiphysic to simulate the interaction between a Gaussian beam and this three-layer structure. In simulation all the parameters of the incident beam and the optical structure are exactly the same as those used in Fig.S2. Fig.S3 shows the electric field distributions in the plane 0.1 µm from the exit surface for three different incident angles, θ i =0, 15.3, 34 degree. It is seen that, for normal incidence, θ i =0 degree, the field distribution almost remains unchanged compared with the incident Gaussian beam, indicating that there is no photonic SHE for normal incidence. We swept the parameter of the incident angle, and the sweep step is modulated to be small enough to avoid missing the resonant peak. It is found that the largest transverse shift appears at θ i =15.3 degree, shown as Fig.S3(b), corresponding to the point A in Fig.S2. Fig.S3 (c) is the electric distribution at θ i =34 degree, corresponding to the point B in Fig.S2. The light spot is elongated along the y direction, however, the separation is much smaller than that at θ i =15.3 (point A), this is obviously contrary to the calculated results in Fig.S2(b). Therefore, only the results calculated by using , ⋅ , * are self-consistent. The transverse displacements calculated by using Eq.(S10) and Eq.(S11) are reliable.