1. Introduction

In the past several decades, the study of beam shifts has been one of the hot spots in the field of optics due to their application in optical sensor [1] and optical switching [2]. In $1947$, Goos and Hänchen first experimentally observed that the light beam totally reflected by the dielectric interface would suffer a lateral displacement in comparison with the prediction of geometric optics [3]. This displacement is the well-known Goos-Hänchen (GH) shift, which has been widely studied in different optical materials and structures, such as weak absorption medium [4], optical waveguide [5], graphene [6] and surface plasmon resonance (SPR) structure [7]. In recent years, many efforts have been made to realize the all-optical controllable GH shift in optical structures containing atomic vapors [8–11], quantum wells [12–14] and quantum dots [15,16]. Apart from GH shift, the wave packet of light would suffer a spin-dependent transverse shift perpendicular to the incident plane. This phenomenon is called as Imbert-Fedorov effect [17] or photonic spin Hall effect (SHE) [18–21]. It was first theoretically predicted by Fedorov [17] and experimentally demonstrated by Imbert [22]. When the finite light beam is linearly polarized light propagating in inhomogeneous medium, it can be regarded as the superposition of left-handed circular light and right-handed circular light, which is induced by the spin-orbit momentum coupling to conserve total angular momentum. When the beam is totally reflected at the interface, it can be divided into two spin elliptical polarization components: left-handed circular polarization and right-handed circular polarization. These two polarization components would split along the opposite direction [20,23], resulting a nonvanishing transverse energy flux inside the evanescent wave, and produce transverse shifts parallel and anti-parallel to the central vector, respectively. Previous experimental investigations have demonstrated that the splitting of the two spin components can be enhanced near the reflected Brewster angle [24–27]. Up to now, the photonic SHE has been widely studied in different optical materials and structures, such as near-zero materials [28,29], metal films [30], metamaterials [31,32], multilayer nanostructures [33–36], graphene [37–39], PT symmetric systems [40,41] and metal-clad waveguides [42,43]. In recent years, Krechman-Reiter (K-R) structure provides a promising platform for the enhancement of the photonic SHE [44–47]. In this structure, the transverse shifts of the two polarization components can be enhanced at the resonance angle due to the excitation of surface plasmon resonance (SPR) at the resonance reflection dip. However, these schemes are not equipped to optically control the SHE of light of the reflected beam. In order to overcome this problem, several schemes have been proposed to realize the all-optical control of photonic SHE in the interfaces or K-R structure containing cold atomic vapors [48–50].

On the other hand, Er$^{3+}$-doped yttrium aluminum garnet (YAG, chemical formula Y$_3$Al$_5$O$_{12}$) crystal has attracted extensive research. Er$^{3+}$-doped YAG crystal is an effective active medium for all-optically controlled solid-state lasers in the communication band. Besides, Er$^{3+}$-doped YAG crystal has been applied to optical communication and optical information processing due to its inherent advantages such as flexible design, wide parameter adjustment range and non-atomic diffusion. Zharikov et al. first observed the stimulated emission of Er$^{3+}$ ions in Er$^{3+}$-doped YAG crystal [51]. Based on the electromagnetically induced transparency (EIT) [52], Er$^{3+}$-doped YAG crystal provides a promising solid-state system to explore various quantum optical phenomena, such as electromagnetically induced gratings [53,54], flat gain [55], normal dispersion and anomalous dispersion [56], enhancing refractive index with no absorption [57], fast and slow light [58], optical bistability and multistability [59,60] and GH shifts [61,62]. It is worthy noting that Er$^{3+}$ ion concentration can alter the electric dipole moments and the energy level lifetimes of Er$^{3+}$ ions, and thereby leading to the change of the absorption and dispersion properties of Er$^{3+}$-doped YAG crystal. Thus, one question remains: how does Er$^{3+}$ ion concentration affect the photonic SHE of the reflected beam in a K-R structure?

In this paper, we investigate the effect of Er$^{3+}$ ion concentration on the SHE of light of the reflected beam in a K-R structure. Here, an Er$^{3+}$-doped YAG crystal with a three-level ladder-typed Er$^{3+}$ ion system acts as the substrate of the K-R structure. The absorption and refractive index of the Er$^{3+}$-doped YAG crystal can be modified by the electric dipole moment and energy level lifetime induced by Er$^{3+}$ ion concentration. Using the experimentally achievable parameters, it can be seen that the SHE of light is sensitive to Er$^{3+}$ ion concentration. Furthermore, it is found that the magnitude, sign and position of the transverse shift can be effectively manipulated via tuning the intensity and detuning of the control field for different Er$^{3+}$ ion concentrations. More importantly, the transverse shift can be significantly enhanced via adjusting the relevant parameters at 15${\%}$ Er$^{3+}$ ion concentration.

2. Models and equations

As shown in Fig. 1 (a), we consider a K-R structure and set the prism-silver interface on the plane $z=0$. The K-R structure consists of a BK$7$ glass prism, a silver film with a thickness of $d$ and a substrate medium. In our proposal, an Er$^{3+}$-doped YAG crystal with a three-level Er$^{3+}$ ion system is selected as the substrate of K-R structure. The dielectric constants of the glass prism, silver film and substrate medium are $\varepsilon _1$, $\varepsilon _2$ and $\varepsilon _3$, respectively. The dielectric constant $\varepsilon _3$ in Er$^{3+}$-doped YAG crystal is obtained by the relationship $\varepsilon _3=1+\chi$, where $\chi$ is the probe susceptibility. The refractive index formula is $n = \frac {c}{v} = \sqrt { \varepsilon _ {r} {\mu }_{r}}$ , which is equivalent to taking ${\mu }_{r}=1$. Therefore, the permeability has little effect on the shifts. In this case, it is not necessary to consider the optical activity, and the optical activity of the Er$^{3+}$ -doped YAG crystal has little effect on the shift of the beam during reflection, and there is no significant difference between the SHE of light based on the Er$^{3+}$-doped YAG crystal and the SHE of light based on ordinary non-magnetic materials. When the silver film thickness is appropriate, an incident TM-polarized Gaussian beam can be reflected by the prism-silver interface due to the excitation of SPR. Subsequently, the SHE of light of the reflected beam can be enhanced, which is caused by the attenuation of the reflection in the resonance dip. Accordingly, the left-handed circular and right-handed circular polarization components of the reflected beam split along the direction perpendicular to the incident plane. Each of the two circularly polarized components can be divided to a $p$-polarized beam and a $s$-polarized beam. Therefore, the corresponding transverse shifts $\delta _{p}^{+}$ and $\delta _{p}^{-}$ of the two circularly polarized components depend on the reflection coefficients $r_p$ and $r_s$ of the K-R structure, where $r_p$ and $r_s$ can be written as [49]

 

Fig. 1. Level structure of atomic system and the position relation between field and atomic system. (a) Sketch of the SHE of light in K-R structure composed of a BK7 prism, a thin Ag film and an Er$^{3+}$ ion -doped YAG crystal; (b) Sketch of diagram of the three-level ladder-type Er$^{3+}$ ion system interacting with the probe and control fields.

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In the proposed K-R structure, the transverse shifts $\delta _{p}^{+}$ and $\delta _{p}^{-}$ of the left-handed circular and right-handed circular polarization components can be derived as [33,44,45]

For the K-R structure, the $s$-polarized beam can be almost totally reflected, i.e., $| r_s |\approx 1$, while the $p$-polarized beam can excite SPR and leads to a resonance reflection dip. Note that the position and reflectivity of the resonance dip depends on the dielectric constant $\varepsilon _3=1+\chi$ of the Er$^{3+}$-doped YAG crystal, where $\chi$ represents the probe susceptibility. In the following, we derive the susceptibility of the Er$^{3+}$-doped YAG crystal. As shown in Fig. 1(b), we choose a three-level ladder-type Er$^{3+}$ ion system, where the levels $| 1 \rangle$, $| 2 \rangle$ and $| 3 \rangle$ correspond to the lowest Stark levels ${}^{4}I_{15/2}$, ${}^{4}I_{13/2}$ and ${}^{4}S_{3/2}$ of Er$^{3+}$ ion, respectively. A weak probe field $E_p$ with a central frequency $\omega _p$ is applied to the transition $| 2 \rangle \to | 1 \rangle$, while the transition $| 3 \rangle \to | 2 \rangle$ is driven by a strong control field $E_c$ with a central frequency $\omega _c$.

Under the electric-dipole and rotating-wave approximations, the interaction Hamiltonian of the three-level Er$^{3+}$ ion system can be written as ($\hbar =1$):

The equation of motion of the density matrix can be written as [63]

We assume that the intensity of the probe field is much smaller than the intensity of the control field, i.e., $\Omega _p\ll {\Omega _c}$. In this case, almost all Er$^{3+}$ ions remain in the ground state $| 1 \rangle$, i.e., $\mathop {\rho }_{11}=1$, $\mathop {\rho }_{22}=\mathop {\rho }_{33}=0$. Under steady-state condition ($\partial \rho _{ij}/\partial t=0$), the analytical solution of $\rho _{21}$ can be obtained as

Note that the polarization of the Er$^{3+}$-doped YAG crystal is given by $P=\varepsilon _{0}\chi E_p=N\mu _{21}\rho _{21}$. Therefore, the probe susceptibility $\chi$ can be written as

Previous studies [64,65] have demonstrated that both the electric dipole moment $\mu _{ij}$ ($i,j=1,2,3$) and energy level lifetime $\tau _{m}$ ($m=1,2$) are related to the doped Er$^{3+}$ ion concentration of the Er$^{3+}$-doped YAG crystal. The Y$^{3+}$ ions in YAG crystal can be replaced by Er$^{3+}$ ions of similar size to the Y$^{3+}$ ions, as this does not have much effect on the lattice structure. The percentage of the doped Er$^{3+}$ ion concentration of the Er$^{3+}$-doped YAG crystal indicates the percentage of Y$^{3+}$ ions substituted by Er$^{3+}$ ion in YAG crystal. The change of the doped Er$^{3+}$ ion concentration of the Er$^{3+}$-doped YAG crystal has little effect on the density of the medium. When we study the optical properties of Er$^{3+}$-doped YAG crystal, the change of the doped Er$^{3+}$ ion concentration will lead to the change of electric dipole moment and energy level lifetime, but the near dipole-dipole interaction and Lorentz local field correction on the dielectric constant can be ignored. To present the influence of Er$^{3+}$ ion concentration on the probe susceptibility $\chi$, we treat the Er$^{3+}$ ion density $N_0$ and electric dipole moments $\mu _{12_0}$, $\mu _{23_0}$ at $0.5{\%}$ Er$^{3+}$ ion concentration as the the reference values. Then we set $N=C{N_0}$, $\mu _{12}=k_{12}\mu _{{12}_0}$, $\mu _{23}=k_{23}\mu _{{23}_0}$, where $C$, $k_{12}$ and $k_{23}$ are the ratios of the parameters between the corresponding Er$^{3+}$ ion concentration and the $0.5{\%}$ Er$^{3+}$ ion concentration. Then, Eq. (13) can be rewritten as

Based on the experimental results [64,65], the values of the electric dipole moments ($\mu _{12}$, $\mu _{13}$ and $\mu _{23}$) and energy level lifetimes ($\tau _2$ and $\tau _3$) of Er$^{3+}$ ions for different concentrations of Er$^{3+}$ ion are listed in Table 1. Therefore, we have $C = (1, 6, 30, 66)$, $k_{12} = (1, 1.17, 0.87, 0.76)$, $k_{23}= (1, 1.09, 0.80, 0.70)$, $\gamma _{21}=( 0.5,0.36,0.36,1.63 )\gamma$ and $\gamma _{31}=( 0.23,0.31,3.25,6.5 )\gamma$ with $\gamma =1/\tau _{2_0}=0.154$MHz for $(0.5{\%}, 3{\%},15{\%},33{\%})$ Er$^{3+}$ ion concentrations. For the sake of the unit, the parameters $\beta$, $\Delta _p$ and $\Delta _c$ are scaled by $\gamma$.

Table 1. Electrical dipole moments and energy level lifetime in an Er$^{3+}$-doped YAG crystals containing four different concentrations of the Er$^{3+}$ ion [64,65]

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At the resonance wavelength of the transition ${}^4I_{{13}/2}\rightarrow {}^4I_{{15}/2}$ (i.e., $\lambda =1516$nm), the dielectric constants of the BK7 glass prism and the silver film are $\varepsilon _1=2.3059$ and $\varepsilon _2=-123.01+3.1354i$, respectively [66,67]. In the numerical calculation, we choose $d=15.16$nm, $\beta =0.05\gamma$. It is worth noting that the transverse shifts $\delta _p^+$ and $\delta _p^-$ have the same magnitude and opposite sign. In the following, we only analyze the transverse shift $\delta _p^+$ of the left-handed circular polarization component of the reflected beam.

3. Results and discussions

First, we investigate the influence of Er$^{3+}$ ion concentration on the SHE of light of the reflected beam when $E_c=35E_0$ and $\Delta _p=\Delta _c=0$. Here, $E_0/2\hbar =\gamma /\mu _{23_0}$ is treated as the unit of the electric intensity of the control field. For this assumption, we have $\Omega _{c_0}=\mu _{23_0}E_c/2\hbar =35\mu _{23_0}E_0/2\hbar =35\gamma$ at $0.5{\%}$ Er$^{3+}$ ion concentration and $\Omega _c=k_{23}\Omega _{c_0}$ at the other Er$^{3+}$ ion concentrations. It can be seen from Eq. (4) that the transverse shift $\delta _p^+$ of the left-handed circular polarization component is determined by the ratio of the reflection coefficients ${| r_s |}/{| r_p |}$ and the phase difference $\varphi _{s}-\varphi _{p}$. For the K-R structure, the $s$-polarized beam is totally reflected, i.e., $|r_s|\approx 1$, while the $p$-polarized beam excites SPR and results in a reflection dip at the resonance angle. The resonant shifts rely on a material resonance and, ultimately, the mechanism of weak measurements. When the total reflection of the incident beam occurs, the reflection spectrum decreases rapidly. This is because part of the incident energy spills into the structure, and the evanescent wave is converted in the optical medium and becomes a continuous radiation wave, which is coupled with the surface plasmon wave, indicating that the surface plasmon is excited. Near the SPR resonance angle, the phase difference changes drastically, which leads to the enhancement of the SHE of light. Whether SPR can be excited depends on the dielectric constant near the metal-dielectric interface. By adjusting the absorption of the medium, the SPR system is effectively adjusted, thereby greatly enhancing the SHE of light. Thus, the enhancement of the SHE of light can be realized via reducing the $|r_p|$. Figure 2(a) shows the reflection coefficient $|r_p|$ as a function of the incident angle $\theta$ for different values of Er$^{3+}$ ion concentration. The corresponding transverse shift $\delta _p^+$ of the left-handed circular polarization component is plotted in Fig. 2(b). As shown in Figs. 2(a) and 2(b), a reflection dip occurs due to the excitation of SPR at the resonance angle $\theta _{\textrm {res}}=41.67^\circ$, around which a negative or positive shift peak can be observed. In the case of $0.5{\%}$ and $3{\%}$ concentrations of Er$^{3+}$ ion, the two reflection curves overlap well and have the same reflectivity at the resonance angle [see the red-solid and blue-dashed lines in Fig. 2(a)]. Accordingly, the curves of the two transverse shifts coincide well and show a negative shift peak ($\sim -0.27\lambda$) around $\theta _{\textrm {res}}$ [see the red-solid and blue-dashed lines in Fig. 2(b)]. With the increase of Er$^{3+}$ ion concentration from $3{\%}$ to $15{\%}$, the reflection coefficient $|r_p|$ exhibits a narrower SPR dip and a lower reflectivity at $\theta _{\textrm {res}}$, which leads to the enhancement of the negative transverse shift $\delta _p^+$ [see the the pink-dotted lines in Figs. 2(a) and 2(b)]. When the Er$^{3+}$ ion concentration increases to 33${\%}$, the reflection dip becomes wider and the minimum reflectivity at $\theta _{\textrm {res}}$ increases [see the green-pecked line in Fig. 2(a)]. According, the magnitude of the spin splitting decreases in comparison with the case of $15{\%}$ Er$^{3+}$ ion concentration [see the green-pecked line in Fig. 2(b)]. However, the spin splitting $\delta _p^+$ transits from a negative peak ($\sim -0.83\lambda$) to a positive peak ($\sim 0.13\lambda$). In addition, it can be seen from Fig. 2 that the resonance angle $\theta _{\textrm {res}}$ remains unchanged for different Er$^{3+}$ ion concentrations.

 

Fig. 2. (a)The reflectivity $|r|$ and (b) the spin-dependent transverse shift $\delta _{p}^{+}$ as a function of incident angle $\theta$ for different values of doped Er$^{3+}$ ion concentration. The number represents the concentration of the doped Er$^{3+}$ ion. Other parameters are $\gamma =0.154$MHz, $\beta =0.05\gamma$, $E_c=35E_0$ ($E_0/{2\hbar }=\gamma /\mu _{23_0}$) and $\Delta _p=\Delta _c=0$.

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The above interesting feature of the photonic SHE of the reflected beam can be physically explained via the internal damping $\Gamma _{\textrm {int}}$ and the radiation damping $\Gamma _{\textrm {rad}}$ [49]. In the K-R structure, the excitation of SPR leads to a minimum reflectivity, which can be written as

In Fig. 3, we plot the spin-dependent transverse shift $\delta _{p}^{+}$ relative to the thickness $d$ of the metal film. When the film is relatively thin, each concentration can excite SPR by decreasing the reflectivity. When the incident angle is the resonant angle $\theta _{\textrm {res}} = 41.67^\circ$, SPR is resonantly excited, and the total reflection is attenuated, resulting in the spin-dependent transverse shift $\delta _{p}^{+}$ change. When the doping concentration of Er$^{3+}$ ion is $0.5 {\%}$ or $3{\%}$, as $d$ increases to 16.46 nm, $\Gamma _{\text {int}}= \Gamma _{\text {rad}}$. When the concentration of Er$^{3+}$-doped ion is $15{\%}$, the thickness of the metal film $d$ is 16.01 nm, and the maximum transverse shift $\delta _{p}^{+}$ can be obtained. When the concentration of Er$^{3+}$-doped ion is $33{\%}$, $\Gamma _{\text {int}}$ and $\Gamma _{\text {rad}}$ can be balanced when $d$ is 11.35 nm.We can conclude that changing the thickness $d$ of the metal film can obtain the optimal spin-dependent transverse shifts $\delta _{p}^{+}$ enhancement corresponding to different Er$^{3+}$ ion concentrations.

 

Fig. 3. The spin-dependent transverse shift $\delta _{p}^{+}$ as a function of the Ag film thickness $d$. The incident angle $\theta _{\textrm {res}}=41.67^\circ$ and other parameters are the same as in Fig. 2.

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In Fig. 4, we plot the imaginary part Im($\chi$) and real part Re($\chi$) of the susceptibility $\chi$ as a function of the probe detuning $\Delta _p$ for different Er$^{3+}$ ion concentrations, respectively. As we know, Er$^{3+}$ ion concentration can modify the electric dipole moment and coherent decay rate of Er$^{3+}$ ions and change the absorption and dispersion properties [52]. Actually, states $|1\rangle$, $|2\rangle$ and $|3\rangle$ construct a ladder-type EIT system. Under the drive of a strong control field, the excited state $|2\rangle$ splits into two dressed levels $\left | \pm \right \rangle ={\left ( \left | 2 \right \rangle \mp \left | 3 \right \rangle \right )}/{\sqrt {2}}$ with eigenenergies ${{E}_{\pm }}=\pm {k_{23}{\Omega }_{c_0}}$. For $0.5{\%}$ and $3{\%}$ Er$^{3+}$ ion concentrations, an ideal transparency window with little resonance absorption can be observed (i.e., Im$(\chi )\approx 0$ at $\Delta _p=0$) [see the red-solid and blue-dashed lines in Fig. 4(a)]. When the Er$^{3+}$ ion concentration increases from $3{\%}$ to $33{\%}$, the increase of Er$^{3+}$ ion density and the decrease of the level splitting lead to the increase of the resonance absorption [see the pink-dotted and green-pecked lines in Fig. 4(a)]. For the selected parameters, we have $\Gamma _{\textrm {int}}<\Gamma _{\textrm {rad}}$ for $0.5{\%}$, $3{\%}$ and $15{\%}$ Er$^{3+}$ ion concentrations and $\Gamma _{\textrm {int}}>\Gamma _{\textrm {rad}}$ for $33{\%}$ Er$^{3+}$ ion concentration, which correspond to the negative and positive transverse shifts, respectively. In the cases of $0.5{\%}$ and $3{\%}$ Er$^{3+}$ ion concentrations, the Er$^{3+}$-doped YAG crystal is nearly lossless medium. Thus, $\Gamma _{\textrm {int}}\ll \Gamma _{\textrm {rad}}$ and the corresponding negative transverse shift is very small. When Er$^{3+}$ ion concentration increases from $3{\%}$ to $33{\%}$, the resonance absorption increases, which leads to the increase of the $\Gamma _{\textrm {int}}$. Say concretely, $\Gamma _{\textrm {int}}$ is close to $\Gamma _{\textrm {rad}}$ and $|r_p|_{\textrm {min}}$ decreases when Er$^{3+}$ ion concentration increases from $3{\%}$ to $15{\%}$. Accordingly, the negative transverse shift $\delta _p^+$ is enhanced at $15{\%}$ Er$^{3+}$ ion concentration. In addition, the shift of the resonance angle $\theta _{\textrm {res}}$ is proportional to the real part Re$(\chi )$ of the probe susceptibility [49]. As shown in Fig. 3(b), we have $\textrm {Re}(\chi )=0$ at $\Delta _p=0$. Therefore, the peak position of the transverse shift remains unchanged for different Er$^{3+}$ ion concentrations.

 

Fig. 4. (a) Imaginary part Im($\chi$) and (b) real part Re($\chi$) of the susceptibility $\chi$ as a function of the probe detuning $\Delta _p$. The inset in (a) shows the enlarged view of the Im$(\chi )$ near $\Delta _p=0$ for $0.5{\%}$, $3{\%}$, $15{\%}$ and $33{\%}$ Er$^{3+}$ ion concentrations. Other parameters are the same as in Fig. 2.

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We further explore the influence of the intensity $E_c$ of the control field on the photonic SHE of the reflected beam under the conditions of different Er$^{3+}$ ion concentrations. Figure 5 shows the variation of the probe susceptibility with $E_c$ under the resonance condition (i.e., $\Delta _p=\Delta _c=0$). As the intensity of the control field $E_c$ increases from 25$E_0$ to 35$E_0$, the probe absorption is almost unchanged for $0.5{\%}$ and $3{\%}$ Er$^{3+}$ ion concentrations, while it monotonically decreases for $15{\%}$ and $33{\%}$ Er$^{3+}$ ion concentrations [see Fig. 5(a)]. Meanwhile, the refractive index of the crystal medium remains unchanged in the presence of different concentrations of Er$^{3+}$ ion [see Fig. 5(b)]. In Fig. 6, we plot the transverse shift $\delta _{p}^{+}$ as a function of the incident angle $\theta$ for different values of $E_c$. When the Er$^{3+}$ ion concentration is 0.5${\%}$ and 3${\%}$, the Er$^{3+}$-doped YAG crystal is always a nearly lossless medium. In this case, the internal damping $\Gamma _{\textrm {int}}$ is much smaller than the radiation damping $\Gamma _{\textrm {rad}}$. Therefore, the corresponding transverse shift is always negative and remains unchanged with a small value of $\delta _{p}^{+}=-0.26\lambda$ [see Figs. 6(a) and 6(b)]. As the concentration of Er$^{3+}$ ion arrives at 15${\%}$, the probe absorption crosses the critical value, where $\Gamma _{\textrm {int}}=\Gamma _{\textrm {rad}}$. Accordingly, the internal damping is lager than the radiation damping for a small value of $E_c$, and the corresponding transverse shift is positive. For a larger $E_c$, the left-handed circular polarization component of the reflected beam suffers a negative transverse shift. Especially in the vicinity of $E_c^c=28.9E_0$, where $\Gamma _{\textrm {int}}=\Gamma _{\textrm {rad}}$, the transverse shift can be significantly enhanced [see Fig. 5(c)]. Under the condition of 33${\%}$ Er$^{3+}$ ion concentration, the probe absorption suffers a large reduction at a high level, where $\Gamma _{\textrm {int}}>\Gamma _{\textrm {rad}}$ is always satisfied. So, the transverse shift is always positive, which can be slightly enhanced via increasing $E_c$. Therefore, the control intensity plays different roles for the magnitude and sign of the transverse shift at different Er$^{3+}$ ion concentrations. Besides, the resonance angle remains constant owing to the unchanged refractive index of the medium.

 

Fig. 5. (a) Imaginary part Im($\chi$) and (b) real part Re($\chi$) of the susceptibility $\chi$ as a function of the intensity $E_c$ of the control field. Other parameters are the same as in Fig. 2.

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Fig. 6. The spin-dependent transverse shift $\delta _{p}^{+}$ as a function of the incident angle $\theta$ and the control field intensities $E_c$ for different concentrations of doped Er$^{3+}$ ion. (a) $0.5{\%}$, (b)$3{\%}$, (c)$15{\%}$, (d)$33{\%}$. Other parameters are the same as in Fig. 5.

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Finally, we investigate the influence of the detuning $\Delta _c$ of the control field on the SHE of light at different Er$^{3+}$ ion concentrations. The variations of Im$(\chi )$ and Re$(\chi )$ with the control detuning $\Delta _c$ are plotted in Fig. 7. With the increase of $\Delta _c$ from $0$ to $60\gamma$, the probe absorption is almost unchanged for $0.5{\%}$ Er$^{3+}$ ion concentration, while it monotonically increases for Er$^{3+}$ ion concentrations of ($3{\%}$, $15{\%}$, $33{\%}$ ) [see Fig. 7(a)]. As shown in Fig. 7(b), the refractive index of the medium is linearly varied with $\Delta _c$ ,but the variation rate of the refractive index are different for different Er$^{3+}$ ion concentrations. Figure 8 plots the variation of the transverse shift $\delta _{p}^{+}$ with $\Delta _c$ for different Er$^{3+}$ ion concentrations. When Er$^{3+}$ ion concentration is $0.5{\%}$, the almost unchanged absorption at the low level (i.e., $\Gamma _{\textrm {int}}\ll \Gamma _{\textrm {rad}}$) leads to the negative transverse shift remaining almost unchanged ($\sim -0.26\lambda$) [see Fig. 8(a)]. As Er$^{3+}$ ion concentration increases to $3{\%}$, the absorption slightly increases at the low level with the increase of $\Delta _c$. Thus, the negative transverse shift is marginally enhanced [see Fig. 8(b)]. In the case of $15{\%}$ Er$^{3+}$ ion concentration, the probe absorption passes through the critical absorption value, where $\Gamma _{\textrm {int}}=\Gamma _{\textrm {rad}}$. Accordingly, we have $\Gamma _{\textrm {int}}<\Gamma _{\textrm {rad}}$ for $\Delta _c<\Delta _c^c=43.5\gamma$ and $\Gamma _{\textrm {int}}>\Gamma _{\textrm {rad}}$ for $\Delta _c>\Delta _c^c=43.5\gamma$, where the negative and positive transverse shifts can be observed, respectively. With $\Delta _c\rightarrow \Delta _c^c$, the transverse shift $\delta _{p}^{+}$ can be significantly enhanced [see Fig. 8(c)]. For $33{\%}$ Er$^{3+}$ ion concentration, the probe absorption monotonically increases at the high level, where $\Gamma _{\textrm {int}}>\Gamma _{\textrm {rad}}$ is always satisfied. Thus, the left-handed circular polarization component of the reflected beam always suffers a positive transverse shift. And the significant reduction of the positive transverse shift with $\Delta _c$ can be induced owing to the increase of probe absorption [see Fig. 8(d)]. In addition, the linear decrease of the refractive index with $\Delta _c$ leads to the resonance angle linearly shifting to the smaller incident angle. But the variation range of the resonance angle are different for different concentrations of doped Er$^{3+}$ ion [see the inset in Fig. 8]. These results implies that the control detuning $\Delta _c$ has different effects on the magnitude, sign and position of the transverse shift under different concentrations of doped Er$^{3+}$ ion. According to the above discussions, one can conclude that the photonic SHE of the reflected beam is sensitive to the intensity and detuning of the control field at 15${\%}$ Er$^{3+}$ ion concentration.

 

Fig. 7. (a) Imaginary part Im($\chi$) and (b) real part Re($\chi$) of the susceptibility $\chi$ as a function of the detuning $\Delta _{c}$ of the control field. $E_c=35E_0$ and other parameters are the same as in Fig. 5.

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Fig. 8. The spin-dependent transverse shift $\delta _{p}^{+}$ as a function of incident angle $\theta$ and the incident angle $\theta$ for different values of control field detuning $\Delta _c$ and concentrations of doped Er$^{3+}$ ion. (a) $0.5{\%}$, (b)$3{\%}$, (c)$15{\%}$, (d)$33{\%}$. The insets in (a)-(e) show the variation of the resonance angle $\theta _{\textrm {res}}$ with the control detuning $\Delta _c$ . Other parameters are the same as in Fig. 7.

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We present an experimental device for measuring transverse shift as shown in Fig. 9. When the laser beam generated by the tunable laser (DL100, local photon) passes through two irises (I), polarizer (P), beam splitter (BS), acousto-optic modulator (AOM) and reflector (M), a coherent control beam is formed. When the laser beam generated by another laser passes through two irises (I), polarizer (P), beam splitter (BS) and attenuator (A), a weak probe beam can be obtained. By adding a polarizer in front of the laser, the output beam has a linear polarization rate with high extinction ratio. Two irises are used to limit the divergence of the incident laser beam. The beam passes through an AOM, which changes the frequency of the light. In this scheme, the beam is completely reflected onto the prism K-R structure made of BK7 glass. The incident angle $\theta$ of the probe beam can be precisely controlled by a rotating stage. Finally, the transverse shift position of the beam after total reflection is detected on the position sensitive detector (PSD) adjusted to the appropriate position, and the corresponding shift is obtained by imaging the differential signal between the photocurrents generated by the PSD. In addition, we should explain that the Gaussian beam method is a better method to show the spatial distribution of the reflected light field when studying the beam shift. Using the Gaussian beam method to deal with the lateral shift is to transform the incident Gaussian beam into the frequency domain, calculate the reflectivity in the spatial frequency domain, and then do the Fourier transform, we will get the spatial distribution of the reflected light field. However, it can only be used to measure the lateral GH shift at present, and cannot give the field distribution when the spin Hall effect of light generates transverse shift. The method used in our article is already a very effective standard method. When the beam waist radius is wide enough, the method we used can accurately reflect the change of the spin Hall effect of light.

 

Fig. 9. The experimental setup of displacement sensor based on the SHE of light. BS: beam splitter, A: attenuator, AOM: acousto-optic modulator, PSD: position-sensitive detector, I: Iris, P: Polarizer, M: Mirror.

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4. Conclusions

In summary, we have theoretically investigated the effect of doped Er$^{3+}$ ion concentration on the photonic SHE of a reflected beam in the K-R structure backed by an Er$^{3+}$-doped YAG crystal with a ladder-type configuration. It is demonstrated that the photonic SHE of the reflected beam are sensitive to the concentration of Er$^{3+}$ ion. These features of the photonic SHE can be explained via the internal damping and radiation damping. Furthermore, it is found that both the intensity and detuning of the control field play different roles on the magnitude, sign and position of the spin-dependent transverse shift under different Er$^{3+}$ ion concentrations. More importantly, the transverse shift can be significantly enhanced via choosing the suitable values of the control intensity and detuning at 15${\%}$ Er$^{3+}$ ion concentration. Therefore, our scheme may provide a basis for selecting suitable Er$^{3+}$ ion concentration to enhance the SHE of light in future integrated systems.