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Abstract
Spin Hall effect (SHE) of Laguerre-Gaussian (LG) beams reflected and transmitted in parity-time (PT) symmetric metamaterials are investigated near the coherent-perfect-absorption (CPA)-laser point and exceptional points (EPs). The numerical results show that large transverse shifts occur at the CPA-laser point regardless of the incident direction. But at EPs, the SHE increases at one side and disappears at the other side, thus achieving the intense SHE of the reflected light beams at the specified side incidence. In addition, it is found that Bragg oscillation can be generated by increasing the period number of PT symmetric metamaterial layers, thus increasing the number of formants in transverse displacement. In particular, the transverse shift peaks of the transmitted beams merge into a positive peak when the incident angle is close to $90^\circ$ and does not change basically with the increasing of Im(ɛ), which can also be considered as a strong tolerance to the variation of Im(ɛ). This feature is expected to realize a new type of optoelectronic devices with anti-interference performance. These results provide a feasible path for the modulation of spin Hall effect of light (SHEL) and provide the possibility for the development of new nanophotonic devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When a linearly polarized beam with finite width propagates in an inhomogeneous medium, the gradient change of refractive index makes left-circularly polarized (LCP) and right-circularly polarized (RCP) components separate equally with opposite directions [1,2]. This phenomenon is called spin Hall effect of light (SHEL), since it is similar to the spin Hall effect in an electronic system. It is also called the Imbert-Fedorov (IF) effect as it was initially predicted by Fedorov and then verified by Imbert in the experiment [3].
Onoda et al. predicted theoretically the spatial separation of the two optical spins in 2004, and Bliokh et al. proposed an accurate expression for the shift of SHEL in 2006 [4]. However, the displacement is usually on the subwavelength scale. It was not until 2008 that Hosten et al. observed this weak effect [5] for the first time by using quantum weak measurement technology, which proved the correctness of previous studies. Subsequently, some better ways to manipulate and enhance SHEL have been explored due to its promising applications in precision metrology. Such as in references [6–8], it was demonstrated that the transverse shift can be enhanced by launching a Gaussian beam near the Brewster angle. The excitation of the surface plasmon resonance (SPR) can significantly control the spin splitting of the reflected beams [9]. Taking advantage of $\varepsilon$-near-zero materials, the large spin dependent splitting of transmitted beams can be obtained [10], and reference [11] showed that loss enhances spin dependent displacement up to 24.676 micrometers.
The concept of parity-time (PT) symmetry [12] originates from quantum mechanics, it requires that the operator of every observable quantity must be a Hermitian operator in quantum mechanics, but the Hamiltonian under PT symmetry is allowed to have real eigenvalues even if it is non-Hermitian [13]. In order to satisfy PT symmetry, the optical system needs to satisfy $n(x)=n^*(-x)$, it means that the real part of the refractive index of the metamaterial is even symmetric in space, while the imaginary part is odd symmetric. In the field of optics, PT symmetry has been applied in coherent perfect absorbers, unidirectional Bloch oscillations, optical switches, asymmetric transmission and topological optics, which has injected new vitality into the research of traditional optical devices [14–16].
In this paper, we systematically study the asymmetric SHEL of the first-order LG beams at the interface of PT symmetric metamaterials. This asymmetry is manifested in two aspects, one is the asymmetric splitting of LCP and RCP components caused by the extra displacement which is result from the intrinsic orbital angular momentum (OAM) carried by the vortex beams. The second asymmetry is the transverse shifts of the reflected beams generated by the incidence from the gain and loss side of the PT symmetric metamaterials, it stems from the dependence of EPs on direction. It can be greatly enhanced at the CPA-laser point whether reflected or transmitted. In addition, it is found that Bragg oscillation can be generated by increasing the period number of PT symmetric metamaterial layers, thus increasing the number of transverse displacement peaks. This means that our results provide a feasible path for the modulation of SHEL.
2. Theory and models
We consider the transmission and reflection of a first-order LG beam at the interface of an air-PT symmetric metamaterial with the incident angle $\theta _i$, and the metamaterial is placed in the plane $z=0$, as shown in Fig. 1. Its background medium is air, the relative dielectric constant in air, gain and loss medium are denoted as $\varepsilon _{0}$, $\varepsilon _{1}$ and $\varepsilon _{2}$, respectively. Since we consider nonmagnetic medium, the relative permeability of each medium layer meets $\mu _{r}=1$. The thicknesses of both loss layer and gain layer are $d$.
When the beam propagates along the $z$ axis of the structure, taking TM wave as an example, based on the transfer matrix method (TMM), we can obtain [17]
With LG beam incidence, the angular spectrum of p(H) polarization and s(V) polarization can be expressed as $\tilde {\boldsymbol {E}}_{i}^{H}=\tilde {\phi }_{l}\hat {\boldsymbol {e}}_{ix}$ and $\tilde {\boldsymbol {E}}_{i}^{V}=\tilde {\phi }_{l}\hat {\boldsymbol {e}}_{iy}$, where, $\hat {\boldsymbol {e}}_{ix}$, $\hat {\boldsymbol {e}}_{iy}$ are the basis vectors parallel and perpendicular to the incident plane. The scalar spectrum of LG beam is $\tilde {\phi }_{l}\propto [w_{0}(-ik_{x}+s_lk_{y})/\sqrt {2}]^{|l|}\textrm {exp}[-(k_{x}^{2}+k_{y}^{2})w_{0}^{2}/4]$, with $w_0$ being the beam waist, and $k_{x}$, $k_{y}$ being the wavevector components, $l$ and $s_{l}=\textrm {sign}(l)$ denote the OAM and the sign of $l$, respectively. The rectangular coordinate systems $(x,y,z)$, $(x_{i},y_{i},z_{i})$, and $(x_{r},y_{r},z_{r})$ are the laboratory coordinate systems, the coordinate systems of incidence and reflection, respectively. The angular spectrum of the reflected beams can be obtained through the transformation matrix between the incident and reflected angular spectrum [19]
The peculiarity of PT symmetric system is that, as long as the system parameters are properly adjusted, the system can undergo phase transformation, and EPs can obviously affect the dynamic characteristics of the system. In order to better reveal the underlying phenomena of PT symmetry, the scattering matrix is used as $S=\begin {pmatrix} t & r_{L} \\ r_{G} & t \end {pmatrix}$, with $S_{\pm }=t\pm \sqrt {r_{L}r_{G}}$ being the eigenvalues. The eigenvalues of the S-matrix are either reciprocal moduli pair ($|S_{+}|=1/|S_{-}|\neq 1$) or unimodular pair ($|S_{+}|=1/|S_{-}|=1$). For the EPs, they are exactly at the transition points between reciprocal moduli pair and unimodular pair [23,24].
3. Results and discussion
First, we chose the PT symmetric metamaterial with a single period ($N=1$) to study the SHE of first-order LG beams, which is shown in Fig. 2. In order to facilitate research, the scattering matrix eigenvalue, reflection and transmission coefficients are the logarithm of the module, and the displacements are multiples of wavelength, these will not be repeated in the following text. The PT symmetric system is optimized as $\varepsilon _{1}=0.1-0.1i, \varepsilon _{2}=0.1+0.1i$, $d=0.785\lambda$, and $w_{0}=15\lambda$. We find that the transverse shifts are zero at the EPs (the transition point between non-zero and zero values), but can be dramatically enhanced in their vicinity. The underlying physics is associated with the near-zero value and abrupt phase jump of the reflection coefficients at EPs. Because of the spontaneous PT symmetry breaking across the EPs, the giant transverse shifts have opposite signs at the two sides of EPs. Therefore, it is reasonable to argue that the dependence of sign change of the giant transverse shift is related to the phase transition in PT symmetric systems. Furthermore, numerical results show that near the EPs, the reflection spectrum increases only at one side, and at the other side disappears. Since $\delta _{r\pm }$ with p(s) polarization inputting is closely related $|r_{s}|/|r_{p}|$($|r_{p}|/|r_{s}|$) [25], a significant value of $|r_{s}|/|r_{p}|$($|r_{p}|/|r_{s}|$) near EPs would directly leads to the appearance of the giant SHEL for the p(s)-polarized incident beam. Thus, we can conclude that the sign switch of transverse shift accross the EPs is dependent on the propagation direction of the incident beam. This is the reason why EPs are often used to implement unidirectional reflectionlessness [16,26]. While at the CPA-laser point, large transverse shifts can be obtained regardless of the incident direction. In addition, although transverse shift of the LG beam is caused by both the spin-orbit angular momentum interaction and the orbital-orbit interaction, however, the transverse displacement generated by the intrinsic OAM is dominant. As a result, LCP and RCP will move in the same direction, and even almost coincide when it strikes at certain angles. Therefore, when the LG beam propagates in the PT symmetric metamaterial, we can not only obtain the asymmetric displacements of LCP and RCP, but also achieve EPs and CPA-laser points by adjusting the incident angle, so as to realize the intense SHE of the reflected beam when it inputs from the specified side.
Fig. 2. Dependences of the eigenvalue of the scattering matrix (the first row), the reflectivity (the second row), and the transverse displacement of reflected light (the third row) on $\theta$ for the cases of H polarization (the first column), and V polarization (the second column), respectively. The red (blue) line represents incident from the gain (loss) layer, and the solid (dashed) line represents reflectivity and the transverse displacement of LCP (RCP), respectively. In our calculations, $N=1$, $\varepsilon _{1}=0.1-0.1i, \varepsilon _{2}=0.1+0.1i$, $\mu _{r}=1$, $d=0.785\lambda$, and $w_{0}=15\lambda$.
Next, we investigate the transverse displacement of transmitted light in the PT symmetric metamaterial with a single period ($N=1$), as shown in Fig. 3. Unlike the reflection case, the transverse shifts are the same regardless of the incidence from the loss or gain layer, since the transmission coefficients are the same for both sides. Similar to the reflected beam, The transverse shift of the transmitted beam is greatly enhanced at the CPA-laser point, the sign change of the enhanced transverse displacement of transmitted light across the laser mode is observed, which is caused by the phase change of $\pi$ at the laser mode [25]. But the difference is that it is not obvious near the EPs. Moreover, surprisingly, although the corresponding transmission coefficient is almost zero when the incident angle approaches $90^\circ$, the transverse displacement still reaches almost 6$\lambda$. This is a large lateral displacement caused by the intrinsic OAM carried by the vortex beam.
Fig. 3. Dependences of the transmission coefficient (the first row), and the transverse displacement of transmitted light (the second row) on $\theta$ for the cases of H polarization (the first column), and V polarization (the second column), respectively. The solid (dashed) line represents the transverse displacement of LCP (RCP), respectively. Other parameters are the same as Fig. 2.
The period number ($N$) of the metamaterial is an important parameter affecting the SHEL. In order to further study the influence of $N$ on the transverse displacement, we consider a periodic PT symmetric structure with $N=10$, and other parameters are the same as those in Fig. 2. The obtained results are shown in Fig. 4. With the increase of $N$ , we obtain more positive and negative transverse shift peaks than in the case of N=1, since the refractive index modulation along the propagation direction constitutes a PT Bragg grating structure. In this periodic PT symmetric structure, the incident light produces periodic transmission and reflection at the interface between the gain and loss media, the constant superpositions of these transmitted and reflected waves constitute the transmitted and reflected spectra of the Bragg grating, which lead to the oscillations of the positive and negative peaks of the transmitted and reflected transverse shifts. However, there is still only one EP in the case of s polarization, this is because the excitation of the surface mode at the interface between the gain and loss layers has a great effect on the p polarization, but has no effect on the s polarization. Even so, the transverse shifts on both sides still show more positive and negative peaks, this is because Bragg resonance causes positive and negative oscillations of reflection and transmission coefficients in the periodic medium. Therefore, we conclude that with the increase of $N$ we can obtain more severe Bragg oscillations, thus increase the number of formants in the transverse displacement.
Fig. 4. Dependences of the eigenvalue of the scattering matrix (first row), the reflectivity (second row), and the transverse displacement of reflected light (third row) on $\theta$ for the cases of H polarization (the first column), and V polarization (second column), respectively. The red (blue) line represents incident from the gain (loss) layer, and the solid (dashed) line represents reflectivity and displacement of LCP (RCP), respectively. Where, $N=10$, $\varepsilon _{1}=0.1-0.1i, \varepsilon _{2}=0.1+0.1i$, $\mu _{r}=1$, $d=0.785\lambda$, and $w_{0}=15\lambda$.
Now let’s adjust the system parameters appropriately, change Im($\varepsilon$) and observe its effects on transverse displacement. For the sake of research, we first explore the changing trend of the material’s scattering properties with respect to the incident angle when the Im($\varepsilon$) changes. As shown in Fig. 5, for both p and s polarization incidence, an EP (the transition point between non-zero and zero values) appears in the region with $\theta _{i} < 10^{\circ }$ and Im$(\varepsilon ) < 0.20$. In addition, a CPA-laser point and two other EPs appear in the region with $\theta _{i} > 10^{\circ }$ and move towards a larger angle with the increase of Im($\varepsilon$) in the p polarized incidence.
Fig. 5. The eigenvalue of the scattering matrix contour for different Im($\varepsilon$) with H (a) and V (b) inputs, respectively. Here we choose $N=1$, $\Re (\varepsilon _{1})=\Re (\varepsilon _{2})=0.1$, $\mu _{r}=1$, $d=0.785\lambda$, and $w_{0}=15\lambda$.
We have found that the incident angle corresponding to the CPA-laser point changes with the Im($\varepsilon$), taking LCP as an example, the peak of the transverse displacement also moves correspondingly. In the case of p polarization incidence, as shown in Fig. 6, the peak value of reflected light displacement appears near the CPA-laser point and EPs. At the segment with $\theta _{i}< 10^{\circ }$ and Im($\varepsilon$) $< 0.2$, it is worth noting that the displacement peak only occurs at the incidence from the loss side, this is because the propagation of light in the waveguide depends on the incident direction when the PT symmetry condition is introduced. Before the spontaneous PT symmetry breaking, the light intensity propagates periodically between the two waveguides in an oscillating manner, but when the gain and loss coefficients reach the spontaneous PT symmetry breaking, the light intensity always propagates locally in the gain waveguide, whether it inputs from the gain waveguide or the loss waveguide. At the same time, the energy in the waveguide increases exponentially and finally concentrates only in the gain waveguide. Jiang et al. have shown in [19] that there is a negative correlation between SHEL and energy, as a result, the huge difference in energy between the two sides results in a larger transverse shift on the loss side. However, when $\theta _{i} > 10^{\circ }$, the peak value of the displacement moves towards a larger angle as Im($\varepsilon$) increases, and roughly coincides with the CPA-laser point movement trend. This can more clearly proves that large transverse shifts occur at the CPA-laser point regardless of the incident direction. The same phenomenon occurs for s polarization incidence, and we can explain it by the same physical mechanism. Thus it can be concluded that SHEL can be enhanced by adjusting the parameters of Im($\varepsilon$) and incident angle simultaneously.
Fig. 6. Transverse shift contour (integer multiples of wavelength) of LCP reflected light for different Im($\varepsilon$) with H (a) and (c) and V (b) and (d) inputs, respectively. Here, the first (second) row is incident from the loss(gain) side. Other parameters are the same as Fig. 5.
Finally, we study the transverse displacement of transmitted light in a PT symmetric metamaterial with a single period ($N=1$). It can be seen from Fig. 7 that the peak of the lateral displacement will shift to a larger angle with the increase of Im($\varepsilon$) for both p and s polarization incidence, and there will be a sudden change from negative to positive (or from positive to negative) peak. In particular, as the incident angle continues to increase, we can observe that the transverse shift of the transmitted beam coalesces into a positive peak near $90^{\circ }$, and with the increase of Im($\varepsilon$), the peak does not change basically. This huge transverse shift is caused by the extra OAM carried by LG beams. It can also be considered as a strong tolerance to the variation of Im($\varepsilon$), which is expected to enable new optoelectronic devices with anti-interference performance.
Fig. 7. Transverse shift contour (integer multiples of wavelength) of LCP transmitted light for different Im($\varepsilon$) with H (a) and V (b) inputs, respectively. Other parameters are the same as Fig. 5.
4. Conclusion
In summary, we have shown the asymmetric SHE of the first-order LG beams at the interface of PT symmetric metamaterials. By adjusting the parameters of the structure and the incident angle of the light, we can not only obtain the asymmetric displacement of LCP and RCP, but also realize the intense SHE of the reflected beam when it incident on the specified side. In addition, with the increase of the number of periodic layers, due to the Bragg resonance, the transmission and reflection coefficients achieve violent oscillations. Therefore, the transverse shift peaks are increased and enhanced effectively. Finally, we have found that the peak of the transverse shift can always appear near the CPA-laser point, even though it will move to a larger angle as Im($\varepsilon$) increases. In particular, we can observe that the transverse shift peak of the transmitted beam coalesces into a positive peak near $90^\circ$, this is caused by the extra OAM carried by the LG beams and is largely unaffected by the variation of Im($\varepsilon$). Our results provide a feasible path for the modulation of SHEL and provide the possibility for the development of new nanophotonic devices in some relevant fields.
Funding
Guangzhou Municipal Science and Technology Project (201707010403); Program for University Key Laboratory of Guangdong Province (2017B030311003); Natural Science Foundation of Guangdong Province (2016A030313439, 2018A030313480).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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