1. Introduction

The reflection of light at an interface can be simply described by the Fresnel equation and Snell’s law [1]. However, when a light beam that has a distributed angular spectrum incidents at an interface, the reflected light beam will undergo in-plane and out-plane spatial shifts, which are so called Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts [2–8]. GH and IF shifts have been verified to exist in many experiments [9,10] and have stupendous potential applications in diverse areas such as integrated optics [11,12], optical sensors [13,14], plasma physics [15,16], and quantum mechanics [17,18]. The amplitude and the direction of GH and IF shifts are not only largely determined by the incident light including Airy beams [19], X-Waves [20], and Laguerre-Gaussian beams [21,22]but also affected by the medium, for instance, graphene-dielectric medium [23], gradient metasurfaces [24], silicene [25,26], and Weyl semi-metal [27]. Moreover, the variation of the beam shifts can mirror the change in the properties of the medium so that the beam shifts are expected to be large enough to be easily detected and controlled.

However, most relevant studies have been on the fixed structure with which the GH and IF shifts are not easy to adjust. Hence, how to flexibly control the shifts has attracted more and more research interest. For instance, Nabamita et al. [28] studied GH and IF shifts in the Kretschmann-Raether geometric structure with a ZnSe prism and an E44 liquid crystal layer between two silver metal layers. It is found that GH and IF shifts can be enhanced near resonance angle due to the excitation of surface plasmon. They also realized the manipulation of GH and IF shifts from positive to negative by applying an external electric field. In 2017, Farmani et al. [16] studied GH and IF shifts on graphene plasmonic metasurfaces for the reflection of terahertz beams where the incident light experienced repetitive reflections. In each reflection, the incident light can be coupled with the surface modes of the graphene metasurface that enhanced GH and IF shifts. In addition, a coherent atomic medium is a good candidate for controlling GH and IF shifts due to its flexible changes in absorption and dispersion by the trigger field. Asiri et al. [6] considered a three-level atomic medium with a pump field and a coherent driving field. Their results show that the amplitude and direction of the GH and IF shifts of incident light can be changed by turning on and off the pump field. This modulation of GH and IF shifts mainly attribute to quantum coherence and interference effects in the atomic medium.

In recent years, quantum coherence has pushed forward developments in optical and quantum communication [29–33], and by tradition, can be created by the coherent interaction of a multi-level atom with coupling laser fields. However, the quantum coherence in the atomic medium can also occur in spontaneous emission which was viewed as the decoherence process. This coherence which is on account of spontaneous emission is normally named spontaneously generated coherence (SGC) [34]. The SGC effect has also been studied extensively, including magnification without inversion [35,36], reduction of spontaneous emission [37–40], and all-optical switching [41]. Recently, Wan et al. [42] studied electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence. The nonlinear absorption or refraction can be enhanced when the SGC effect exists due to quantum coherence between two decay channels from two upper close levels. Subsequently, Tariq et al. [43] investigated a four-level N-type atomic ensemble in a cavity with the SGC effect. It was found that the augment of the transparency window is proportional to the strength of the SGC effect at resonance conditions.

Considering the absorption and dispersion of coherent atomic medium can be modified dramatically by the SGC effect, we set the atomic medium at the bottom of the waveguide to expect flexible control for GH and IF shifts. In addition, the phase of the incident light, such as the vortex phase of the LG beam, is also a key factor for controlling the beam shifts. So, in this paper, we investigate the GH and IF shifts of a high-order LG beam on a guided-wave SPR (GWSPR) structure backed by a four-level double-ladder atomic medium. Thus, in this paper, we investigated the GH and IF shifts for a high-order LG beam by applying a four-level double-ladder atomic medium with SGC effect in a guided-wave SPR (GWSPR) structure. The linear absorption of the atomic medium with the SGC effect vanishes, while its nonlinear absorption can be enhanced and coherently controlled. As a result, both GH and IF shifts can be simultaneously enhanced and flexibly controlled by the Rabi frequency of the trigger field. Especially, the switch of the direction of GH shifts corresponds to the nontrivial change in the loci of the reflection coefficient $r_{p}$ which can be explained by the competitive mechanism between the inherent damping and the radiative damping [52]. Compared with other fixed structures, not only can we get larger beam shifts, but we can also manipulate them flexibly. Our results are helpful for the fields like integrated optics, optical sensors, and plasma physics.

This paper is organized as follows: In Sec. 2, we introduce the theoretical model of a guided-wave surface plasmon resonance (GWSPR) structure backed by a four-level double-ladder atomic medium with the SGC efftect, and then we derive the general expressions for GH and IF shifts for the reflected light. In Sec. 3, we investigate the influence of Si-layer thickness on the Fresnel reflectance of a GWSPR structure and the excitation of some modes in the GWSPR structure. In Sec. 4.1, we get the analytical expression for beam shifts and show that IF shifts can be enhanced and controlled by the orbital angular momentum in the LG beam. In Sec. 4.2, the results and discussion are presented to show that both GH and IF shifts can be enhanced by the SGC effect and controlled by adjusting the complex susceptibility of the atomic medium through the trigger field. In Sec. 4.3, we investigate the change in the direction of GH shifts and interpret it as the competitive mechanism between the inherent damping and the radiative damping corresponding to the nontrivial change in the loci of the reflection coefficient. Finally, we present our conclusions in Sec. 5.

2. Model

The GWSPR we consider here is a Kretschmann configuration as shown in Fig. 1(a). It consists of four layers, the top layer is a prism, a thin gold film is coated on the prism with thickness $d_{2}$ which leads to the excitation of surface plasmon modes, the next Si-layer with thickness $d_{3}$ constitutes a waveguide, and the bottom is the four-level double-ladder atomic medium with the SGC effect. Different from the general surface plasmon resonance(SPR) structure [15], a GWSPR structure has an extra waveguide film between the metal layer and bottom material [44]. Given that a arbitrarily linear polarized Laguerre-Gauss beam incident on gold film from the prism, its angular spectrum in local coordinates can be described as $\widetilde {E}_{i}=[a \hat {e}_{ix}+b\hat {e}_{iy}]\tilde {\phi }_{L}$, where $a=cos\varphi$, $b=sin\varphi$, with $\varphi$ representing the angle between the linear polarization vector and the $\hat {e}_{ix}$ axis. $\hat {e}_{ix,iy}$ denote the polarization unit vectors parallel and perpendicular to the incidence plane. The scalar spectrum of the LG profiles is $\tilde {\phi }_{L}\propto \lbrack w_{0}(-ik_{ix}+sgn[L]k_{iy})/ \sqrt {2}]^{\left \vert L\right \vert }\exp [-(k_{ix}^{2}+k_{iy}^{2})w_{0}^{2}/4]$ with $w_{0}$, $L$ and $k_{ix,iy}$ being the beam waist, orbit angular momentum and lateral wave vectors. For a arbitrarily linear polarized incident light, SPR modes can be excited by the TM-polarized component. The evanescent field caused by SPR modes will be strongly enhanced and bounded at the Si-atoms interface. The strong localization of the electromagnetic field can dramatically enhance the interaction between light and the bottom medium which is a four-level double ladder atomic medium as shown in Fig. 1(b). Then, to deal with the interaction between light and atomic medium, we use the density matrix method. By utilizing the Weisskopf-Wigner theory of spontaneous emission, probability amplitudes of different states can be described by equations given as

 

Fig. 1. (a) Schematic diagram of GH and IF shifts in a GWSPR structure composed of a prism, gold, and a silicic waveguide. The waveguide is backed by an atomic medium. (b) Four-level double ladder atomic system for atomic medium with two closely spaced levels $\left \vert 2\right \rangle$ and $\left \vert 3\right \rangle$ coupled by spontaneous emission. The transition $\left \vert 2\right \rangle$ $\leftrightarrow$ $\left \vert 4\right \rangle$ and $\left \vert 3\right \rangle$ $\leftrightarrow$ $\left \vert 4\right \rangle$ are driven by a trigger field $E_{t}$, while a weak probe field $E_{p}$ couples the ground level $\left \vert 1\right \rangle$ to $\left \vert 2\right \rangle$ and $\left \vert 3\right \rangle$. The absorption and refractivity of the medium can be coherently controlled.

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For the GWSPR structure we proposed here, the reflection coefficients $r_{p}$ and $r_{s}$ can be written as

Finally, GH($\Delta _{X}$) and IF($\Delta _{Y}$) shifts of the reflected beam can be obtained in momentum space as: $\Delta _{X,Y}=\left \langle \widetilde {E} _{r}\left \vert i\partial _{krx,ry}\right \vert \widetilde {E}_{r}\right \rangle /\left \langle \widetilde {E} _{r}\mid \widetilde {E}_{r}\right \rangle$ [48]. After some straightforward calculation, we arrive at

Here, $M$=$(r_{p}+r_{s})\cot \theta$, $r_{p}$ and $r_{s}$ are the reflective coefficients of TM and TE polarized wave, respectively. $r_{p}^{\prime }$=$\partial r_{p} /\partial \theta$, and $k_{1}$ denotes the wave vector in the prism.

3. Si-layer in the GWSPR structure

For the GWSPR structure, a TM-polarized incident light can be coupled with some modes in the structure. The Si-layer, as a component of the waveguide structure, significantly affects the modes and can cause changes in the optical properties of the structure. In this section, we demonstrate that a Si-layer of a specific thickness can result in significant absorption of the incident field and the excitation of certain modes. First, we investigate the influence of Si-layer thickness on the Fresnel reflectance of the GWSPR structure. We consider a TM-polarized LG incident light $\varphi$=0, with the wavelength $\lambda$=780$nm$. The dielectric constants of materials are: $\varepsilon _{1}$=2.22, $\varepsilon _{2}$=-20.327+1.862i, $\varepsilon _{3}$=14.3663+0.0784i and $\varepsilon _{4}$=1+$\chi$. The thickness of gold film is $d_{2}$=46$nm$. For simplicity, we set $\Gamma _{2}$=$\Gamma _{3}$=$\Gamma _{4}$=$\gamma$ and all the other parameters are scaled by $\gamma$, while $p$=1, $q$=-1, and $\eta$=1. The Rabi frequency of the trigger field $\Omega _{t}$=0.16($\gamma$), while both the detuning of the probe and the trigger field are zero. In Figs. 2(a) and 2(b), we give the Fresnel reflectance $|r_{s}|^{2}$ and $|r_{p}|^{2}$ changing with the incident angle when the Si-layer thickness $d_{3}$ are 0, 13, 17 and 22$nm$, respectively. It can be found that $|r_{s}|^{2}$ changes very little with the increasing incident angle and even keeps almost stable as the incident angle increasing. For the TM-polarized incident light, $|r_{p}|^{2}$ has a narrow resonant dip near the resonant angle due to the excitation of the SPR. When $d_{3}$ is increased, there are no significant changes of reflectance $|r_{s}|^{2}$. However, the Si-layer can significantly affects the reflectance $|r_{p}|^{2}$. Here, as $d_{3}$ increased, the dip of $|r_{p}|^{2}$ becomes deeper initially and shallower later. As the thickness of the Si-layer is chosen appropriately, such as $d_{3}$=17$nm$, the GWSPR structure has a stronger optical absorption, and the incident field will be coupled to the surface modes [49]. The strong absorption of the incident field can be advantageous for flexible manipulation of beam shifts using atomic medium. Significantly, the SPR mode can be excited when the energy penetrates into the GWSPR structure, however, there may be exists other modes, called waveguide (WG) modes. To analyse these modes in the GWSPR structure, we study the criterion for the four-layer waveguide mode of TM-polarized incident light which can be written as follows according to [50]

 

Fig. 2. (a) Reflectance $|r_{s}|^{2}$, (b) Reflectance $|r_{p}|^{2}$ as functions of incident angle with the different thickness of Si-layer, where the Si-layer thickness ($d_{3}$) is fixed to the four values: 0, 13, 17 and 22nm, respectively. (c) Dependence of effective refractive index $N^{'}$ on the waveguide thickness for TM-polarized light. Parameters are $\Omega _{t}$=0.16($\gamma$), $\Delta _{t}$= $\Delta _{p}$=0($\gamma$), $p$=1, $q$=-1, and $\Gamma _{2}$=$\Gamma _{3}$=$\Gamma _{4}$=$ \gamma$, accordingly.

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4. Tunable giant GH and IF shifts

4.1 Via the orbital angular momentum

We consider the beam shifts of a TM-polarized LG incident light $\varphi$=0, with the wavelength $\lambda$=780$nm$ since the surface plasmon resonance cannot be excited by TE-polarized light in the GWSPR structure. The thickness of Si-layer is chosen as $d_{3}=$17$nm$ based on Figs. 2(b) and 2(c). However, the enhancement and control of beam shifts in a fixed structure have many limitations. Thus, in this section, we show that beam shifts can be enhanced and controlled by the orbital angular momentum carried by the LG beam. In our calculation, the beam waist of the incident beam $w_{0}$ is 1000 $\mu m$. With the TM-polarized incident light, Eq. (11) and Eq. (12) can be rewritten as

Figures 3(a) and 3(b) display the effects of orbital angular momentum of the incident light on GH and IF shifts, near the SPR angle, respectively. From Fig. 3(a), we find that GH shifts have peaks at 57.31${{}^\circ }$ for different vortex charge $L$. It’s worth noting that, the peaks of GH shifts (57.31${{}^\circ }$) do not exactly corresponding to surface plasmon polaritons(SPPs). The SPPs here is corresponding to the dip in the reflectance $|r_{p}|^{2}$ (57.32${{}^\circ }$) in Fig. 2(b). Actually, from Eq. (19), GH shifts are proportional to $\partial \phi /\partial \theta$, while $\partial \phi /\partial \theta$ is very large but not at its maximum value at the dip of in the reflectance $|r_{p}|^{2}$ (57.32${{}^\circ }$) in Fig. 2(b). This leads to the fact that the peaks of GH shifts are not strictly corresponding to SPPs. The direction of GH shifts is always positive and independent of L, while its amplitude decreases as $L$ increases.

 

Fig. 3. (a) GH shifts and (b) IF shifts as functions of incident angle for different vortex charge $L$ with the insets are intensity distribution of a Laguerre-Gauss beam with different orbital angular momentum $L$=0, $L$=1, $L$=2 and $L$=3. The black-solid line corresponds to $L$=0. The blue-solid(dashed) line, green-solid(dashed) line and red-solid(dashed) line correspond to $L$=1(-1), $L$2=(-2) and $L$=3(-3), respectively. Other parameters are the same as those in Fig. 2.

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Actually, the effect of orbital angular momentum $L$ on GH shift is dependent on the polarization of the incident light. When the incident light is either TE or TM-polarized light, the orbital angular momentum is not beneficial to GH shifts. For further discussion we recall the GH shifts written as Eq. (11), the third part of the numerator $L\cdot ab\text { Im}(M^{\ast }(r_{p}+r_{s}))$ is angular IF shifts due to the geometric phase and is related to the orbital angular momentum $L$. However, when the incident beam is a TM-polarized beam the third part of the numerator vanishes, and angular IF shifts do not contribute to GH shifts.

However, IF shifts can be greatly influenced by the orbital angular momentum, as shown in Fig. 3(b). It is seen that when the incident angle $\theta$ passes through the SPR angle, the value of IF shifts change sharply from positive to negative only if $L\neq$0, while IF shifts vanish exactly at the SPR angle. It’s worth noting that, the vanish of IF shifts is corresponding to SPPs which is different from the condition in [51] where IF-shifts peak or dip corresponds to the surface polariton. In fact, from Eq. (20), IF shifts are proportional to $\partial \left | r_{p} \right | /\partial \theta$, while $\partial \left | r_{p} \right | /\partial \theta$ is exactly equal to zero at the dip of the reflectance $|r_{p}|^{2}$ (57.32${{}^\circ }$) in Fig. 2(b). This leads to the vanish of IF shifts when SPPs is excited. The result shows that IF shifts can be greatly influenced by the SPR mode. Furthermore, as $L$ increases, IF shifts will be amplified accordingly. For example, the maximum IF shift can be up to $\pm 1250\lambda$ for $L=\pm$3. We conclude that IF shifts can be enhanced greatly by vortex charge $L$, and the direction is determined by the sign of $L$. The influence of orbit angular momentum on IF shifts can be understood as the angular momentum conservation during reflection [45].

4.2 Via SGC effect in a coherent medium

Since the GH and IF shifts are closely related to the phase of the reflection coefficient sensitive to the dielectric constant of the medium, we can manipulate these shifts by controlling the susceptibility of the coherent atomic medium. For a four-level double-ladder atomic medium as shown in Fig. 1(b), if the dipole matrix elements $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d}_{12}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d}_{13}$ are parallel, a perfect SGC effect will manifest, but if the dipoles are orthogonal, the effect is lost. Figure 4 shows GH, IF shifts, reflectivity $|r_{p}|$ and the phase of reflection coefficient $\phi$ varies with $\theta$ for no SGC effect ($\eta$=0) and perfect SGC effect($\eta$=1), respectively. The orbital angular momentum discussed above is fixed at $L=$2 and the other parameters are the same as in Fig. 3. It is clearly to be seen that the maximum magnitude of the GH shifts is enhanced from 43.3$\lambda$ to 443.3$\lambda$ due to the SGC effect, the direction of the GH shifts switches from negative to positive, and there exists a slight deviation in the resonant angle of the GH shift from 57.32${{}^\circ }$to 57.31${{}^\circ }$. Moreover, from Fig. 4(b) we can see that, with the SGC effect, the magnitude of positive and negative peaks of IF shifts are enhanced from 45.7$\lambda$ to 981.1$\lambda$ and 43.1$\lambda$ to 979.9$\lambda$ respectively. However, the direction of IF shifts is independent of the SGC effect, and the resonant angle of positive and negative peaks undergoes large deviations from 57.10${{}^\circ }$to 57.30${{}^\circ }$and 57.52${{}^\circ }$ to 57.32${{}^\circ }$. Meanwhile, as shown in Figs. 4(c) and 4(d), the reflectivity dip becomes deeper and the change of $\phi$ is steeper in the presence of the SGC effect accordingly. It indicated that the SGC effect has a major impact on $|r_{p}|$ and $\phi$, and we conclude that both GH and IF shifts are largely enhanced with deviation of the peaks around the resonant angle due to the SGC effect. However, GH and IF shifts are related to the properties of medium which are actually the dielectric constant of the medium. Thus, it is necessary to investigate the influence of the SGC effect on the susceptibility of the atomic medium. In fact, the susceptibility of the atomic medium can be expanded as [42]

We note that the absorption of the probe field caused by $\chi ^{(1)}$ is independent of the trigger field while the absorption induced by $\chi ^{(3)}$ is up to the Rabi frequency of the trigger field.

 

Fig. 4. (a) GH shifts, (b) IF shifts, (c) Reflectivity $|r_{p}|$ and (d) the phase of reflection coefficient $\phi$ as functions of incident angle. The blue-dashed line and red-solid line correspond to no SGC effect( $ \eta$=0) and perfect SGC effect($ \eta$=1), respectively. Parameter $L$=2 and the others are the same as Fig. 2.

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In Figs. 5(a) and 5(b), we depict the absorptive parts of the first-order linear susceptibility $\chi ^{(1)}$ and third-order nonlinear susceptibility $\chi ^{(3)}$ as the functions of the detuning of the probe field ($\Delta _{p}$) with perfect SGC effect (red solid line) and no SGC effect (blue dashed line), respectively. It can be seen that in the absence of the SGC effect (dashed line), there exist two incoherent Lorentzian curves. For this case, the absorption spectrum has a dip at the zero detuning and two symmetric peaks about the zero detuning. However, in the presence of the SGC effect (solid line), the dip in the absorption spectrum reaches zero point at the zero detuning. The atomic medium is transparent to the probe light as a result of this phenomenon. This phenomenon is owing to the destructive quantum interference between the spontaneous emission of the two upper states $|2\rangle$ and $|3\rangle$. We note that the vanishing of [Im$\chi ^{(1)}$] helps to enhance the beam shifts, and this is the basic idea behind improving beam shifts with the SGC effect. Moreover, the absorptive parts of the third-order nonlinear susceptibility $\chi ^{(3)}$ shown in Fig. 5(b) are dependent on the Rabi frequency of the trigger field, according to Eq. (21). It is evident that in the absence of the SGC effect (dashed line), the absorption spectrum has a peak at the zero detuning and two dips around the zero detuning. However, in the presence of the SGC effect (solid line), the absorption at the zero detuning in the absorption spectrum is enhanced. In fact, the atom and field system at the zero detuning is in a dark state due to the SGC effect. As a result, the two-photon absorption is caused by coherent trapping between the two nearby upper states $|2\rangle$ and $|3\rangle$. Thus, we can manipulate the GH and IF shifts by changing the Rabi frequency of the trigger field in the presence of the SGC effect.

Figure 6 shows GH, IF shifts, reflectivity $|r_{p}|$, and the phase of reflection coefficient $\phi$ as functions of incident angle $\theta$ for different Rabi frequencies of the trigger field $\Omega _{t}$. The orbital angular momentum $L$ is fixed at 2. Other parameters are the same as those in Fig. 2. As shown in Figs. 6(a) and 6(b), when $\Omega _{t}$ is increased, the magnitude of GH and IF shifts are increased initially and then decreased. Meanwhile, the direction of GH shifts switches from negative to positive as $\Omega _{t}$ increased, but IF shifts still keep the same direction at the fixed incident angle. Essentially, the magnitude of GH and IF shifts are related to the phase of reflection coefficient $\phi$ and reflectivity $|r_{p}|$, accordingly. As shown in Fig. 6(c), as $\Omega _{t}$ increased, the reflection dip becomes deeper initially and shallower later in the same way as the magnitude of IF shifts. Meanwhile, in Fig. 6(d), when $\Omega _{t}$ increased, the change of $\phi$ is steeper initially and then flat in the same way as the magnitude of GH shifts. So both the magnitude of GH shifts and IF shifts can be controlled by the Rabi frequency of the trigger field $\Omega _{t}$.

 

Fig. 5. Absorptive parts of $ \chi ^{(1)}$ (a) $ \chi ^{(3)}$ (b) versus probe field detuning $\Delta _{P}$ in the absence (blue-dashed line) and presence (red-solid line) of SGC effect. Parameters are $\Delta _{t}$=0($\gamma$) and $\Omega _{t}$=0.16($\gamma$). Other parameters are the same as those in Fig. 2.

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Fig. 6. (a) GH shifts, (b)IF shifts,(c)$|r_{p}|$ and (d)$\phi$ as functions of incident angle $ \theta$ for different Rabi frequency $\Omega _{t}$ of the trigger-field. Parameter is $L$=2. Other parameters are the same as those in Fig. 2.

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4.3 Direction of GH shifts

To further understand the direction change of GH shifts, we investigate the effect of $\Omega _{t}$ on the loci of $r_{p}(\theta )$. In Fig. 7, we plot the loci of $r_{p}(\theta )$ in the complex plane as functions of incident angle from 45${{}^\circ }$to 65${{}^\circ }$ for different $\Omega _{t}$. The arrows which are tangent to the loci represent the direction of the increasing incident angle, the x-axis is the real part of $r_{p}$ and the y-axis is the image part of $r_{p}$, respectively. The phase of the reflection coefficient can be written as $\phi =\arctan [$Im$(r_{p})/$Re$(r_{p})]$, and $\phi$ is not well defined when Im$(r_{p})=$Re$(r_{p})=0$. In Fig. 7, we find that the distance between the loci of $r_{p}(\theta )$ is increased initially and then decreased with $\theta$. Especially, the loci of $r_{p}(\theta )$ will cross the origin as $\Omega _{t}$ increase from 0.1($\gamma$) to 0.3($\gamma$). Meanwhile, according to Eq. (19), $\partial \phi /\partial \theta$ changes from negative to positive which serve as a switch in the direction of GH shifts.

 

Fig. 7. Loci of reflection coefficient $r_{p}$ of incident angle from 45${{}^\circ }$ to 65${{}^\circ }$ in complex plane for different trigger-field Rabi frequency.

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In fact, the reflection coefficient $r_{p}$ of a GWSPR structure around the SPR angle can be written as follow [52]:

The sign of Re$(r_{p})$ and Im$(r_{p})$ are determined by $\Gamma ^{int}-\Gamma ^{rad}$. When the loci of the reflection coefficient cross the origin, the sign of Re$(r_{p})$ and Im$(r_{p})$ will change, while the value of $\Gamma ^{int}-\Gamma ^{rad}$ will switch between negative and positive, and so is the direction of GH shifts. When $\Gamma ^{int}=\Gamma ^{rad}$, the resonance of two attenuation channels will make the amplitude of $r_{p}$ become zero and the loci of the reflection coefficient will intersect the origin. It is this competitive mechanism between the inherent damping $\Gamma ^{int}$ and the radiative damping $\Gamma ^{rad}$ that leads to the direction switch of the GH shifts. In fact, the inherent damping $\Gamma ^{int}$ and the radiative damping $\Gamma ^{rad}$ can be written as [52]

 

Fig. 8. (a) Imaginary and real part of $ \chi$ and (b) $\Gamma ^{int}$ versus Rabi frequency $\Omega _{t}$ of the trigger-field. We choose $c_{1}$=1, and other parameters are the same as those in Fig. 2.

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5. Conclusion

In summary, we proposed a scheme to enhance and manipulate the IF and GH shifts for a reflected Laguerre-Gauss beam from the structure of a GWSPR with a four-level double-ladder atomic medium. By increasing the orbital angular momentum of the incident Laguerre-Gauss beam, IF shifts can be largely enhanced around the SPR angle but not for GH shifts. However, in the presence of the SGC effect in the atomic medium, both the GH and IF shifts can be simultaneously enhanced and well controlled. When the two electric-dipole moments $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d}_{13}$ and$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {d}_{12}$ are parallel in the atomic medium, the perfect SGC effect can occur, which results in the vanishing of the Im$\chi ^{(1)}$ and the enhancement of the Im$\chi ^{(3)}$. Such the SGC-induced vanish of Im$\chi ^{(1)}$ allows the simultaneous enhancement of GH and IF shifts. Moreover, the SGC-enhanced Im$\chi ^{(3)}$ is dependent on the Rabi frequency $\Omega _{t}$ of the trigger field. Thus, we can realize the manipulation of GH and IF shifts, especially the direction of GH shifts, by coherent controlling Im$\chi ^{(3)}$. When $\Omega _{t}$ increases, there exists a switch in the direction of GH shifts which is corresponding to the nontrivial change in the loci of the reflection coefficient. It can be understood as the competition between the inherent damping and radiative damping of the medium.

We can enhance and manipulate the GH and IF shifts in a flexible manner without changing the system structure, in contrast to other schemes where the beam shifts are governed by the fixed thickness and refraction index of the medium. The results can help us better understand how light interacts with atoms under the SGC effect, which will enhance how optical shifts are used in integrated optics, optical switches, optical sensors, and weak measurement.