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.

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The zero refractive index properties of two-dimensional photonic crystals (PCs) are studied theoretically. Three necessary conditions for PCs to mimic the zero index materials (ZIMs) are obtained. In addition, through a comparative study of the properties for two representative PC structures with different types of Dirac cones, we find that the PC with a Dirac-like cone which meets the three necessary conditions does not behave as a ZIM in some cases. Further analysis shows that its non-zero index properties originate from the flat dispersion band. These findings clarify the fundamental physical issue of which type of Dirac cone PC can mimic a real ZIM.