Transverse shifts of a reflected light beam from the air-chiral interface

Based on a general beam-propagation model, we derive in the paraxial approximation the analytical expressions for the spatial transverse shift (TS) and the angular TS of a beam reflected from the air-chiral interface. The results show that the TSs are closely related to the propagation behaviors of the right-circularly polarized (RCP) and the left-circularly polarized (LCP) waves in the chiral medium. When the L(R)CP wave becomes evanescent, the left (right)-elliptically polarized incident beam suffers a smaller TS. When the L(R)CP wave propagates, the left (right)-elliptically polarized incident beam experiences a greater TS. When the total internal reflection happens, neither of the two elliptically polarized incident beams suffers TSs. TSs can be controlled not only by adjusting the central angle of incidence, but also by tailoring the permittivity, permeability, and chirality parameter of the chiral medium.

Figure 1

Geometry of the beam reflection from and transmission through the air-chiral interface.

Figure 2

Dependences of the reflectivity, the spatial TS and the angular TS (in seconds) on $\theta$ for $n=0.6$. The reflectivity curves for (a) $\gamma =0.3$ and (d) $\gamma =0.6$, where the solid, dashed, and dash-dot lines correspond to $\left|r_{\mathit{ss}}\right|{}^2$, $\left|r_{\mathit{pp}}\right|{}^2$, and $\left|r_{\mathit{sp}}\right|{}^2(|r_{\mathit{ps}}|{}^2)$, respectively. (b) and (e) for the spatial TS curves, and (c) and (f) for the angular TS curves, where the solid and dashed lines correspond to the right- and left-elliptically polarized incident beams with $\sigma =\pm 2/3$, reflectivity; (b) and (c) for $\gamma =0.3$; (e) and (f) for $\gamma =0.6$.

Figure 3

Dependences of the reflectivity, the spatial TS, and the angular TS (in seconds) on the central angle of incidence for $n=1.5$. The reflectivity curves for (a) $\gamma =0.4$ and (d) $\gamma =1.1$, where the solid, dashed, and dash- dot lines correspond to $\left|r_{\mathit{ss}}\right|{}^2$, $\left|r_{\mathit{pp}}\right|{}^2$, and $\left|r_{\mathit{sp}}\right|{}^2(|r_{\mathit{ps}}|{}^2)$, respectively. (b) and (e) for the spatial TS curves, and (c) and (f) for the angular TS curves, where the solid and dashed lines correspond to the right- and left-elliptically polarized incident beams with $\sigma =\pm 2/3$, reflectivity; (b) and (c) for $\gamma =0.4$; (e) and (f) for $\gamma =1.1$.

Figure 4

Dependences of the spatial TS and the angular TS (in seconds) on $n$ for the right-elliptically polarized incident beam with $\sigma =2/3$. The spatial TS curves for (a) $\gamma =0.5$ and (c) $\gamma =1$, and the angular TS curves for (b) $\gamma =0.5$ and (d) $\gamma =1$, where the solid, dashed, and dash-dot lines correspond to $\theta ={20}^{°},{30}^{°}$, and ${40}^{°}$, reflectivity. In Figs. 4a and 4b, A and C are the critical points where the total internal reflections happen for the LCP wave; B is the traditional limitation point, where $n_{-}=0$.