Asymmetric spin splitting of Laguerre-Gaussian beams in chiral PT-symmetric metamaterials

DongXue Liu; DongXue Liu; ChengKang Liang; ChengKang Liang; DongMei Deng; DongMei Deng; GuangHui Wang; GuangHui Wang; Li Zhang

doi:10.1364/OE.475024

1. Introduction

PT-symmetric optical systems with both gain and loss have drawn a great deal of interest. In 1998, Bender et al. found that true eigenvalues can be achieved in Non-Hermitian operators if the PT symmetry condition is satisfied. The conditions for the quantum mechanical Hamiltonian to be PT-symmetric are as follows: The relationship between $\hat {T}$ and $\hat {P}$ satisfies $[\hat {P} \hat {T}, \hat {H}]=0$, where $\hat {P}$ is a cosymmetric operator and $\hat {T}$ is a time-reversal operator: $\hat {P}: \hat {p} \rightarrow -\hat {p},\;\ \hat {x} \rightarrow -\hat {x},\;\hat {T}: \hat {p} \rightarrow -\hat {p},\;\ \hat {x} \rightarrow \hat {x},\ i \rightarrow -i$. Subsequently, Dorkey et al. proved and generalized this content theoretically [1–3]. The term chiral metamaterials was first coined by Jagadis Chandra Bose [4]. Chirality refers to the property that an object cannot coincide with its mirror image through translations, rotations, and scaling [5]. Chiral PT-symmetric metamaterials refer to the introduction of PT-symmetric properties in chiral metamaterials [6–8]. And it can combine the effects of PT-symmetric and chiral metamaterials, incorporating two special propagation properties that produce different characteristics than before.

In 2004, Onoda et al. initially ventured theoretically to propose the SHEL [9]. Bliokh et al. reinterpreted it in 2006 using electromagnetic theory [10]. It was later demonstrated experimentally by Hosten and Kwiat in 2008 using weak measurement enhancement techniques [11]. After that, the amplification effect of the spin Hall shift has been extensively studied. Since the SHEL is very sensitive to slight changes in many physical parameters, augmentation strategies and the adjustability of the spin Hall shift have been researched in a number of systems [12–20]. For example, the enhancement effect of SHEL near the Brewster angle [21,22], the Dirac point [23] and the critical angle [24,25]. Modulation of SHEL by parameters in laminar structures [26,27] and anisotropic materials [28–31]. Other systems include monolayer and multilayer graphene [32–39], black phosphorus metamaterials [40,41], PT-symmetric materials, and chiral materials [42–47]. In addition, the SHEL can be used to measure material parameters, such as measuring the number of graphene layers and the Fermi energy, and measuring the spin luminosity of chiral materials [48–55]. Among other things, optical systems that integrate PT symmetry with other materials can achieve many special effects [18,56–58]. However, many optical systems, such as the one that is the focus of our study, which combines chirality and PT symmetry at higher-order beam incidence, have not been investigated.

We investigate the asymmetric spin Hall effect of LG beams in chiral PT-symmetric metamaterials. Although SHEL in solely chiral or PT-symmetric metamaterials has been explored [47,56,57], it has been shown that these shifts are quite small. In addition, the asymmetric spin splitting of these two materials combined has not been thoroughly investigated. Here, we see that for ${w}_{0}$= 15${\lambda }$ ($\ell$ = 0), the greatest displacement may reach 6.2${\lambda }$ or even larger. In close proximity to point $\left | {r}_{pp} \right |$ $\ll$ $\left | {r}_{ss} \right |$, one may acquire the enormous asymmetric spin splitting. Asymmetric spin splitting arises from cross-polarization effects and IOAM in chiral PT-symmetric metamaterials. In addition, the asymmetric spin splitting with its peak can be adjusted by changing the values of the topological charge $\ell$, the chiral parameter $\kappa$, the dimensionless frequency $M$ and the incident angle $\theta$.

2. Theory and models

To derive the requirements for chiral PT-symmetric metamaterials, we substitute the chiral parameter ($\kappa$) into Maxwell’s equations, $\nabla \times \boldsymbol {E}=i \omega \boldsymbol {B},\:\nabla \times \boldsymbol {H}=-i \omega \boldsymbol {D}$ and get $\boldsymbol {D}=\varepsilon \varepsilon _{0} \boldsymbol {E}+i(\kappa / c) \boldsymbol {H}$ and $\boldsymbol {B}=\mu \mu _{0} \boldsymbol {H}-i(\kappa / c) \mathbf {E}$. In the formula, $\varepsilon$, $\mu$, $\kappa$, and c represent the permittivity, permeability, chiral parameter $\kappa$, the speed of light in a vacuum, respectively. While subscript 0 represents their values in a vacuum. To achieve PT symmetry, Hamiltonian is made to interchange with PT operators. While time reversal ($t \to t$) has no effect on their chirality, spatial reversal ($x \to -x, y \to -y, z \to -z$) causes a change in chirality. Therefore, make the wave propagate along the z-direction. We obtain the necessary conditions for a chiral system to satisfy the PT symmetry as follows: $\varepsilon (\boldsymbol {r})=\varepsilon ^{*}(-\boldsymbol {r}),\: \mu (\boldsymbol {r})=\mu ^{*}(-\boldsymbol {r}),\: \kappa (\boldsymbol {r})=-\kappa ^{*}(-\boldsymbol {r})$ [1–3].

As indicated in Fig. 1(a), we consider the reflection of the LG beams in a chiral PT symmetric metamaterial with incident angle $\theta$. Our model system consists of two chiral flat plates stacked vertically and placed in air. Figure 1(b) is the section diagram of the model in the $y$ direction. The length in the $x$ direction is infinite, the thickness of gain layer and loss layer is $d$, and the total thickness of the structure L = $2d$. The parameters of the loss and gain layer can be expressed as $\varepsilon _{B}, \mu _{B}$, and $\kappa _{B}$ ($\varepsilon _{C}, \mu _{C}$, and $\kappa _{C}$). The refraction angles of LCP and RCP components in the loss and gain layers can be written as $\theta _{L}^{(B)}$ , $\theta _{L}^{(C)}$, $\theta _{R}^{(B)}$ and $\theta _{R}^{(C)}$. The upper corner scale B applies to the loss layer, C applies to the gain layer, and the lower corner scale L is the refraction angle of the LCP and R is the refraction angle of the RCP. Assuming TM and TE waves at an incident angle $\theta$, the following equation can be obtained [6]:

(1)$$\mathrm{TE}:\left\{\begin{array}{l}\boldsymbol{E}_{i}^{(A)}=E_{is} \hat{y} e^{i \omega / c\left(z \cos \theta-x \sin \theta\right)} \\ \boldsymbol{H}_{i}^{(A)}={-}\frac{E_{is}}{\sqrt{\mu_{0} / \varepsilon_{0}}}\left(\cos \theta \hat{x}+\sin \theta \hat{z}\right) e^{i \omega / c\left(z \cos \theta-x \sin \theta\right)}\end{array}\right.,$$

and

(2)$$\mathrm{TM}:\left\{\begin{array}{l}\boldsymbol{E}_{i}^{(A)}=E_{ip}\left(\cos \theta \hat{x}+\sin \theta \hat{z}\right) e^{i \omega / c\left(z \cos \theta-x \sin \theta\right)} \\ \boldsymbol{H}_{i}^{(A)}=\frac{E_{i p}}{\sqrt{\mu_{0} / \varepsilon_{0}}} \hat{y} e^{i \omega / c\left(z \cos \theta-x \sin \theta\right)}\end{array}\right.,$$

where $E_{ip}$ and $E_{is}$ represent the TM and TE wave amplitudes, respectively. The electric field in a chiral plate is written as a linear combination of RCP waves and LCP waves. Using the boundary condition of the continuity of tangential components of electric and magnetic fields on three interfaces ($z =-d$ , $z=0$, and $z=d$ ), 12 linear equations are written in the form of 12 $\times$ 12 matrix equation [6]:

(3)$${\textbf{U}}^\mathrm{T}=\boldsymbol{P} {\textbf{Q}}^\mathrm{T},$$

with,

(4)$$\boldsymbol{U}=\begin{bmatrix} \begin{smallmatrix}\frac{E_{ip}}{B_{i}} & \frac{E_{is}}{B_{i}} & \frac{E_{is}}{B_{i}} & \frac{E_{ip}}{B_{i}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{smallmatrix}\end{bmatrix},$$

(5)$$\boldsymbol{P}= \begin{bmatrix} \begin{smallmatrix} 0 & -B_{i} & \frac{R_{B1}}{B_{1}} & \frac{R_{B2}}{B_{2}} & -R_{B1} {B_{1}} & -R_{B2} {B_{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ -B_{i} & 0 & \frac{i}{B_{1}} & -\frac{i}{B_{2}} & i {B_{1}} & -i {B_{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ B_{i} & 0 & \mathrm{i} \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} \frac{R_{B1}}{B_{1}} & -\mathrm{i} \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} \frac{R_{B2}}{B_{2}} & -\mathrm{i} \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} R_{B1} {B_{1}} & \mathrm{i} \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} R_{B2} {B_{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & B_{i} & \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} \frac{1}{B_{1}} & \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}} \frac{1}{B_{2}} & \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}}{B_{1}} & \frac{\eta_{\mathrm{A}}}{\mathrm{Z}_{\mathrm{B}}}{B_{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \cos \theta_{R}^{(B)} & \cos \theta_{L}^{(B)} & -\cos \theta_{R}^{(B)} & -\cos \theta_{L}^{(B)} & -\cos \theta_{R}^{(C)} & -\cos \theta_{L}^{(C)} & \cos \theta_{R}^{(C)} & \cos \theta_{L}^{(C)} & 0 & 0 \\ 0 & 0 & i & -i & i & -i & -i & i & -i & i & 0 & 0 \\ 0 & 0 & \frac{-i \cos \theta_{R}^{(B)}}{Z_{B}} & \frac{i \cos \theta_{L}^{(B)}}{Z_{B}} & \frac{i \cos \theta_{R}^{(B)}}{Z_{B}} & \frac{-i \cos \theta_{L}^{(B)}}{Z_{B}} & \frac{i \cos \theta_{R}^{(C)}}{Z_{C}} & \frac{-i \cos \theta_{L}^{(C)}}{Z_{C}} & \frac{-i \cos \theta_{R}^{(C)}}{Z_{C}} & \frac{i \cos \theta_{L}^{(C)}}{Z_{C}} & 0 & 0 \\ 0 & 0 & \frac{1}{Z_{B}} & \frac{1}{Z_{B}} & \frac{1}{Z_{B}} & \frac{1}{Z_{B}} & -\frac{1}{Z_{C}} & -\frac{1}{Z_{C}} & -\frac{1}{Z_{C}} & -\frac{1}{Z_{C}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & R_{C1} {C_{1}} & R_{C2} {C_{2}} & \frac{-R_{C1}}{C_{1}} & -\frac{R_{C2}}{C_{2}} & 0 & -C_{t} \\ 0 & 0 & 0 & 0 & 0 & 0 & {i} {C_{1}} & -{i} {C_{2}} & \frac{i}{C_{1}} & -\frac{i}{C_{2}} & -C_{t} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -\mathrm{i} \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} R_{C1}{C_{1}} & \mathrm{i} \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} R_{C2} {C_{2}} & \mathrm{i} \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} \frac{R_{C1}}{C_{1}} & -\mathrm{i} \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}}\frac{R_{C2}}{C_{2}} & C_{t} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} {C_{1}} & \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} {C_{2}} & \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} \frac{1}{C_{1}} & \frac{\eta_{\mathrm{D}}}{\mathrm{Z}_{\mathrm{C}}} \frac{1}{C_{2}} & 0 & -C_{t} \end{smallmatrix} \end{bmatrix},$$

(6)$$\boldsymbol{Q}=\begin{bmatrix} \begin{smallmatrix}E_{rs} & E_{rp} & E_{R}^{(B 1)} & E_{R}^{(B 2)} & E_{L}^{(B 1)} & E_{L}^{(B 2)} & E_{R}^{(C 1)} & E_{R}^{(C 2)} & E_{L}^{(C 1)} & E_{L}^{(C 2)} & E_{ts} & E_{tp}\end{smallmatrix} \end{bmatrix},$$

here, $\eta _{A}=\sqrt {\frac {\mu _{A} \mu _{0}}{\varepsilon _{A} \varepsilon _{0}}}, \eta _{D}=\sqrt {\frac {\mu _{D} \mu _{0}}{\varepsilon _{D} \varepsilon _{0}}}, Z_{B}=\sqrt {\frac {\mu _{B} \mu _{0}}{\varepsilon _{B} \varepsilon _{0}}}, Z_{C}=\sqrt {\frac {\mu _{C} \mu _{0}}{\varepsilon _{C} \varepsilon _{0}}}, k_{i}=\frac {\omega }{c} \sqrt {\varepsilon _{A} \mu _{A}}, k_{t}=\frac {\omega }{c} \sqrt {\varepsilon _{D} \mu _{D}}$, $k_{(L,R)}^{(B)}=\frac {\omega }{c}\left (\sqrt {\varepsilon _{B} \mu _{B}} \pm \kappa _{B}\right ), k_{(L,R)}^{(C)}=\frac {\omega }{c}\left (\sqrt {\varepsilon _{C} \mu _{C}} \pm \kappa _{C}\right )$, $B_{i} = {e}^{i k_{i} d \cos \theta }, B_{1} = {e}^{i k_{R}^{(B)} d \cos \theta _{R}^{(B)}}$, $B_{2} = {e}^{i k_{L}^{(B)} d \cos \theta _{L}^{(B)}}, C_{1} = {e}^{i k_{R}^{(C)} d \cos \theta _{R}^{(C)}}, C_{2} = {e}^{i k_{L}^{(C)} d \cos \theta _{L}^{(C)}}, C_{t} = {e}^{i k_{t} d \cos \theta _{t}}$, $R_{B 1}=\frac {\cos \theta _{R}^{(B)}}{\cos \theta }, R_{B 2}=\frac {\cos \theta _{L}^{(B)}}{\cos \theta },$ $R_{C 1}=\frac {\cos \theta _{R}^{(C)}}{\cos \theta }$, and $R_{C 2}=\frac {\cos \theta _{L}^{(C)}}{\cos \theta }$. Where the lower corner markers R and L indicate RCP (+) and LCP (-) waves, respectively. $\eta _{A}$, $\eta _{D}$, $Z_{B}$, $Z_{C}$ indicates the impedance of each layer of the medium, the lower corners A, B, C, D indicate the media of each layer. $k_{i}$ $k_{t}$, $k_{(L,R)}^{(B)}$, $k_{(L,R)}^{(C)}$ indicate the wave vectors of incident light, refracted light, LCP component, and RCP component in the loss layer and gain layer, respectively. $k_{i} d \cos \theta$, $k_{R}^{(B)} d \cos \theta _{R}^{(B)}$, $k_{L}^{(B)} d \cos \theta _{L}^{(B)}$, $k_{R}^{(C)} d \cos \theta _{R}^{(C)}$, $k_{L}^{(C)} d \cos \theta _{L}^{(C)}$, $k_{t} d \cos \theta _{t}$ indicates the phase of each layer. Since the above equation is too complicated, we obtain the numerical solution to the equation, and the relation between TM and TE polarized incident wave and reflection coefficients can be obtained:

(7)$$\begin{bmatrix} E_{rs} \\ E_{rp} \end{bmatrix}=R\begin{bmatrix} E_{is} \\ E_{ip} \end{bmatrix} =\begin{bmatrix} r_{ss} & r_{sp} \\ r_{ps} & r_{pp} \end{bmatrix}\begin{bmatrix} E_{is} \\ E_{ip} \end{bmatrix} .$$

In the above formula, according to Eq. (1) and Eq. (2), when $E_{is}=0$, it represents TM wave, and when $E_{ip}=0$, it represents TE wave. By solving the above equation, the expression of reflection coefficients can be obtained:

(8)$$r_{sp}=\frac{E_{rs}}{E_{ip}}, r_{pp}=\frac{E_{rp}}{E_{ip}}, r_{ss}=\frac{E_{rs}}{E_{is}}, r_{ps}=\frac{E_{rp}}{E_{is}},$$

where, $r_{sp}$ and $r_{ps}$ are the cross-polarization coefficients induced by the chiral parameter $\kappa$ [8].

Fig. 1. The asymmetric spin splitting of a LG beam reflected from the lossy side of a chiral PT-symmetric interface. (a) When the H-polarized LG beam is incident on the structure, the reflected beam splits into two opposite spin components (LCP, RCP) and generates a spin Hall shift along the y-axis direction. (b) The lossy side is illuminated by the obliquely incident TM or TE polarized wave.

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As shown in Fig. 1, the overall coordinate system is $\left (x_{g}, y_{g}, z_{g}\right )$, the incident coordinate system is $\left (x_{i}, y_{i}, z_{i}\right )$, and the reflection coordinate system is $\left (x_{r}, y_{r}, z_{r}\right )$. Since the light beams can be regarded as the superposition of plane waves of different angular spectral components, it is assumed that the incident spectrum and reflection spectrum are represented by $\tilde {E}_{i}$ and $\tilde {E}_{r}$, respectively. The relationship between angular spectra can be obtained after coordinate rotation: $\tilde {E}_{r}\left (k_{r x}, k_{r y}\right )=M_{R} \tilde {E}_{i}\left (k_{i x}, k_{i y}\right )$, where $k_{r x} = - k_{i x}$, $k_{r y} = k_{i y}$ is the transverse wave number, $M_{R}$ is the rotation matrix of coordinate transformation.

We consider a linear LG beam with IOAM that is incident at an oblique angle to the chiral PT-symmetric interface. The scalar spectrum of the LG beams is $\tilde {\phi }_{l}{\propto }\left [{w}_{0} \left (-i k_{ix}+s_{\ell } k_{iy}\right ) 2^{-1/2}\right ]^{|\ell |} \exp \left [-\left (k_{ix}^{2}+k_{iy}^{2}\right ) {w}_{0}^{2} / 4\right ]$. On the basis of circular polarization: $\hat {\boldsymbol {e}}_{r \pm }=\left [\hat {\boldsymbol {e}}_{r x} \mp i \hat {\boldsymbol {e}}_{r y}\right ] / 2^{1 / 2}$, the angular spectrum of reflected beams at $z_{r} = 0$ is [47]:

(9)$$\tilde{\mathbf{E}}_{r}^{{\pm}}=\tilde{\phi}_{\ell} \left\{\left[r_{p p}-\frac{k_{r x} r_{p p}^{\prime}}{k_{0}}\right] \mp i\left[r_{s p}-\frac{k_{r x} r_{s p}^{\prime}}{k_{0}}-\frac{H k_{r y}}{k_{0}}\right]\right\} \hat{\mathbf{e}}_{r \pm} .$$

Where $k_{0} =\dfrac {2\pi }{\lambda }$, $\ell$ is the topological charge, $r_{ab}^{'}$ stands for the derivative of $r_{ab}$ with respect to $\theta$, $a$ and $b$ stand for $s$ or $p$. $\Delta _{\pm }=\left \langle \tilde {\mathbf {E}}_{r}^{\pm }\left |i \partial _{k_{r y}}\right | \tilde {\mathbf {E}}_{r}^{\pm }\right \rangle /\left \langle \tilde {\mathbf {E}}_{r}^{\pm } \mid \tilde {\mathbf {E}}_{r}^{\pm }\right \rangle$ is defined in terms of the centroids of the RCP and LCP components of the reflected beams along the y-axis. After some calculations and simplifications, we get [47]

(10)$$\begin{aligned}\Delta_{{\pm}}=&\dfrac{1}{W_{{\pm}} k_{0}} \Big\{ -{\ell}\left[\textrm{Re}\left[r_{p p}^{\prime} r_{p p}^{*} + r_{s p}^{\prime} r_{s p}^{*}\right] \mp \textrm{Im}\left[r_{s p}^{*} r_{p p}^{\prime} \mp r_{s p}^{\prime} r_{p p}^{*}\right]\right]\\ & \mp \textrm{Re}\left[H r_{p p}^{*}\right]+\textrm{Im}\left[H r_{s p}^{*}\right] \Big\} , \end{aligned}$$

among them, $H=(r_{p p}+r_{s s})\cot \theta$, and $W{_{\pm }}$ is the energy of RCP and LCP components, respectively [47],

(11)$$\begin{aligned}W_{{\pm}}=&\frac{(1+|{\ell}|)\left\{{\pm} 2 \textrm{Im}\left[r_{s p}^{\prime} r_{p p}^{\prime*}\right]+ \left | r_{p p}^{\prime} \right |^{2} + \left | r_{s p}^{\prime} \right |^{2} + \left | H \right |^{2}\right\}}{k_{0}{ }^{2} w_{0}{ }^{2}}\pm 2 \textrm{Im}\left[r_{s p} r_{p p}^{*}\right] +\\ &\left | r_{p p} \right |^{2} + \left | r_{s p} \right |^{2}. \end{aligned}$$

The first term in Eq. (10) is closely related to the IOAM and stems from the vortex structure, i.e., the beam contains a helical phase factor $\exp \left [i {\ell } \phi \right ]$. And the second term is related to the spin-dependent, that originates from the Gaussian envelope. The energy $W_{\pm }$ is related to spin angular momentum and the size of the IOAM. Their synergistic effect leads to asymmetric spin splitting. At ${\ell } = 0$, the above equation can also represent the spin Hall shift of the incident Gaussian beams.

3. Result and discussion

First, the spin Hall effect of Gaussian beams (${\ell }=0$) is explored in chiral PT-symmetric metamaterials. As illustrated in Fig. 2, the first row, and the second row indicate the fluctuation of the $\left | r_{s s} / r_{p p} \right |$ and the displacement of LCP and RCP components with angle $\theta$, respectively. The first, second and third columns depict the displacement splitting at different values of chirality ($\kappa$ = -0.5, 0, 0.5), respectively. For $\kappa$ = 0, i.e., a medium without cross-polarization $({r}_{sp} = 0 )$, the LCP and RCP components split symmetrically ($\bigtriangleup _{+ } = - \bigtriangleup _{- }$). In particular, at $\left | {r}_{pp} \right | {\ll } \left | {r}_{ss} \right |$, their displacements are equal and the splitting disappears ($\bigtriangleup _{+ } = \bigtriangleup _{- } = 0$). The maximum value of the displacement obtained near $\left | {r}_{pp} \right | {\ll } \left | {r}_{ss} \right |$ is 6.2$\lambda$. At $\kappa \ne$ 0, the LCP and RCP components undergo asymmetric splitting, which is caused by the cross-polarized ${r}_{sp}$ (Eq. (10)). As the angle $\theta$ increases, the spin splitting displacement still peaking near $\left | {r}_{pp} \right | {\ll } \left | {r}_{ss} \right |$. When $\kappa$ is positive (negative), the displacement produced by the LCP (RCP) component obtains a great value. This is due to the fact that the sign of the chirality parameter $\kappa$ will affect the polarization state through the bilayer wave and then the spin Hall shift between LCP and RCP components. As shown in Fig. 3, the displacement of the reflected RCP becomes smaller when the chiral parameter $\kappa$ > 0 of the material, and the displacement of the reflected LCP becomes smaller when $\kappa$ < 0. However, the displacement values do not vanish when they are equal i.e. $\bigtriangleup _{+ } = \bigtriangleup _{- } \ne 0$. The LCP and RCP components transform into the opposing spin components of the reflected beams. This variation is different from the behavior in chiral or PT-symmetric materials alone [47,56,57].

Fig. 2. $\left | r_{s s} / r_{p p} \right |$ and the displacement of LCP and RCP components vary with $\theta$. The first, second, and third columns ($\kappa = -0.5,\:\kappa = 0,\:\kappa = 0.5$) are shown. Among them, $M=0.1$, $\:\varepsilon _{B,C} = 3.7\pm 0.01i,\:\mu _{B,C} = 3.7\pm 0.1i\:$. The chiral parameter $\kappa$ ($\kappa _{B}=\kappa +i \kappa$, $\kappa _{C}=-\kappa +i \kappa$) is the real and imaginary part of the two-layer medium, and the dimensionless frequency $M$ is denoted by $M=\frac {\omega L}{c}$.

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Fig. 3. The linearly polarized light splits into different asymmetric LCP and RCP components after passing through metamaterials with different chiral signs.

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In the preceding section, we look at the asymmetric splitting of LCP and RCP components with a fixed dimensionless frequency ($M = 0.1$). Next, we explore the variation of LCP and RCP components spin splitting with $\kappa = 0, \kappa = 0.5$. The asymmetry of the peak variations of LCP and RCP components has a tight relationship with the chiral parameter $\kappa$ and dimensionless frequency $M$. It is deduced from equation $M = \frac {\omega L}{c}$ [6,7]. The dimensionless frequency $M$ is a function of the angular frequency ($\omega$) and the total thickness ($L$), and they are proportional. Therefore, we use $M$ to characterize the synergistic effect of total thickness ($L$) and angular frequency ($\omega$) on the spin Hall displacement. In Fig. 4(a1) and (a2), the displacement and peak variation of LCP and RCP components may be seen to be exactly symmetrical. However, in Fig. 4(b1) and (b2), the peak variation of LCP and RCP components are clearly asymmetric. The asymmetric peak can be modulated by the angle $\theta$ and the dimensionless frequency $M$. The peak value varies linearly with the dimensionless frequency $M$ and angle $\theta$. By adjusting the value of $M$ during the shift in angle from $5^{\circ }$ to $12^{\circ }$, symmetric or asymmetric peaks are always achievable. As $M$ grows, however, the sign of the peak changes. The zero point of the spin Hall shift is also linear with angle $\theta$. This apparent change can be found in Fig. 4(a3) and (b3), which arises near $\left | {r}_{pp} \right | {\ll } \left | {r}_{ss} \right |$.

Fig. 4. The variation of spin splitting (a1, a2, b1, b2) and $\left | r_{s s} / r_{p p} \right |$ (a3, b3) with $\theta$ and $M$ for LCP and RCP components. In the first, second row, the chiral parameters $\kappa$ are 0 and 0.5, respectively.

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The displacement of the splitting of LCP and RCP components varies with the chiral parameter $\kappa$. Figure 5 shows the variation of the displacement of LCP and RCP components with the chiral parameter $\kappa$ for the incident angle of $\theta$ ($5^{\circ }, 10^{\circ }, 15^{\circ }$) and the dimensionless frequency of $M$ (0.1, 1, 1.8). It is clear that the LCP and RCP components in the image split asymmetrically at $\kappa \ne$ 0. Furthermore, the peak gradually becomes sharper when $\left | \kappa \right |$ is very small. At $M$ = 0.1, the maximum value of the displacement at $\ell$ = 0 is obtained as 6.2$\lambda$. In each column, with $M$ constant, we find that the extreme value of the displacement of the LCP and RCP components splitting becomes progressively smaller in the interval from $5^{\circ }$ to $15^{\circ }$ for the angle. The flat peak at $M$ = 0.1 indicates that it is easy to obtain and the displacement is more tolerant to $\kappa$ values at $M$ = 0.1, which provides a larger advantage for the application of chiral PT-symmetric metamaterials. Moreover, in Fig. 5, the splitting of LCP and RCP components are centrosymmetric about the point (0, 0).

Fig. 5. The spin Hall displacement (LCP, RCP) of the variation of the chiral parameter $\kappa$ at different incident angle $\theta$ when the dimensionless frequency $M$ is 0.1, 1.0, and 1.8 (from left to right). The angle of each row is fixed from top to bottom as $5^{\circ }$, $10^{\circ }$, $15^{\circ }$ respectively.

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We study the variation of the asymmetric displacement of the spin splitting with respect to the chiral parameter $\kappa$ for the incident angles of $5^{\circ }, 10^{\circ }, 15^{\circ }$. The asymmetric displacement of the spin splitting of the reflected beam is denoted by $\bigtriangleup _{a } = \left | \bigtriangleup _{+ } \right | - \left | \bigtriangleup _{- } \right |$. In Fig. 6, it can be clearly seen that the peak of $\bigtriangleup _{a }$ is distributed around $\kappa = 0$. Similar to Fig. 5, the peaks of $\bigtriangleup _{a }$ at different angles become sharp as $M$ increases, and $\bigtriangleup _{a }$ is more sensitive to the chiral parameter $\kappa$. In addition, $\bigtriangleup _{a }$ is centrosymmetric to the point (0, 0) when $\kappa$ is varied at the same angle. Therefore, the magnitude and sign of $\kappa$ can be determined by adjusting the value of $\bigtriangleup _{a }$.

Fig. 6. The asymmetric displacement $\bigtriangleup _{a }$ as functions of the chiral parameter $\kappa$. Other values are taken as in Fig. 5.

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Combining Fig. 2, Fig. 3, Fig. 4, Fig. 5, and Fig. 6 above. The modulation of the spin Hall displacement asymmetry and peak value by three physical factors (chiral parameter $\kappa$, angle $\theta$, dimensionless frequency $M$) when $\ell$ = 0 is investigated. Following that, we explore the influence of higher order LG beams on the LCP and RCP components.

The LCP and RCP components split asymmetrically when the topological charge $\ell$ takes different values. To analyze the causes of this asymmetric splitting, we fix the angle $\theta$ at $5^{\circ }$, the dimensionless frequency $M$ at 0.1, and the chiral parameter $\kappa$ at 0.5, respectively. As shown in Fig. 7, the displacement of LCP and RCP components varies with the topological charge $\ell$ for different chiral parameter $\kappa$, dimensionless frequency $M$ and angle $\theta$. It is obvious that asymmetric spin splitting occurs when any one of chiral parameter $\kappa$, dimensionless frequency $M$, angle $\theta$ is fixed as long as the condition of ${\ell } \ne$ 0 or $\kappa \ne$ 0 is satisfied. In particular, the displacement of the $\bigtriangleup _{+ }$ or $\bigtriangleup _{- }$ splitting is more tolerant to chiral parameter $\kappa$, dimensionless frequency $M$ and angle $\theta$ at $\ell$ = 1 or $\ell$ = −1. This is due to the fact that the IOAM-related terms cancel a large part of the spin angular momentum-related terms in Eq. (10).

Fig. 7. The variation of LG beams LCP and RCP components displacements with topological charge $\ell$, chiral parameter $\kappa$, dimensionless frequency $M$ and angle $\theta$. The angle $\theta$ in the first column is $5^{\circ }$ and the dimensionless frequency $M$ is 0.1. The angle $\theta$ in the second column is $5^{\circ }$ and the chiral parameter $\kappa$ is 0.5. The dimensionless frequency $M$ in the third column is 0.1 and the chiral parameter $\kappa$ is 0.5.

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The first, second, and third columns in Fig. 8 represent the fluctuation of the asymmetric displacement $\bigtriangleup _{a }$ of the spin splitting with the chiral parameter $\kappa$, dimensionless frequency $M$, and angle $\theta$ when the topological charge $\ell$ is brought to various values. As shown in Fig. 8, when ${\ell } = 0$, the sign of $\bigtriangleup _{a }$ changes at $\kappa = 0$, from $\bigtriangleup _{a } < 0$ to $\bigtriangleup _{a } > 0$. When ${\ell } \ne 0$ at −0.2 < $\kappa$ < 0.2, $\bigtriangleup _{a }$ only undergoes a shift in size and the change of $\ell$ and $-\ell$ is symmetric about the center of the origin (Fig. 8(a1), (a2)). The variation of $\bigtriangleup _{a }$ with angle at different $\ell$ is shown in Fig. 8(b1). Near $\theta$ = $10^{\circ }$, the asymmetric displacements with the same value of $\left | {\ell } \right |$ are equal. Figure 8(b2) shows that the splitting of LCP and RCP components becomes symmetric at $M$ = 0.42 and the sign of $\bigtriangleup _{a }$ changes around $M$ = 0.42 when the chiral parameter $\kappa$ and the angle $\theta$ are constant and the dimensionless frequency $M$ is varied. Therefore, the sign and magnitude of $\bigtriangleup _{a }$ can be adjusted by a variety of parameters.

Fig. 8. The relationship between $\bigtriangleup _{a }$ and chiral parameter $\kappa$, dimensionless frequency $M$, and angle $\theta$, at ${\ell } \ne$ 0. (a1,a2) $M$ = 0.1, $\theta$ = $5^{\circ }$ (b1) $\kappa$ = 0.5, $\theta$ = $5^{\circ }$ (b2) $\kappa$ = 0.5, $M$ = 0.1.

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4. Conclusion

In conclusion, a comprehensive examination of the asymmetric splitting of the LG beams in chiral PT-symmetric metamaterials has been conducted. The asymmetric spin splitting of the LCP and RCP components is caused by the cross-polarization $r_{s p} (r_{p s})$ and IOAM. The displacements and asymmetric splitting of the LCP and RCP components are effectively amplified by the topological charge $\ell$ and the chiral parameter $\kappa$, and regulated by the dimensionless frequency $M$ and the angle $\theta$. In addition, the maxima of LCP and RCP components spin splitting occur close to $\left | {r}_{pp} \right | {\ll } \left | {r}_{ss} \right |$, and their distribution is asymmetric under the influence of various factors. We anticipate that the current work will provide new approaches for the investigation of the asymmetric spin Hall effect.

Funding

Science and Technology Program of Guangzhou City, China (201707010403); National Natural Science Foundation of China (2017B030311003); Natural Science Foundation of Guangdong Province (2018A030313480, 2022A1515012377).

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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Asymmetric spin splitting of Laguerre-Gaussian beams in chiral PT-symmetric metamaterials

Abstract

1. Introduction

2. Theory and models

3. Result and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (11)

Optics Express