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Abstract
We present a scheme for enhancing Goos-Hänchen shift of light beam that is reflected from a coherent atomic medium in the Kretschmann-Raether configuration. The complex permittivity of the medium can be coherently controlled and has significant influence on the surface plasmon resonance (SPR) at the metal-medium interface. By tuning the atomic absorption, the internal damping of SPR system can be modulated effectively, thereby leading to giant positive and negative lateral displacements. The refractive index of medium determines the SPR angle. Thus the peak position of the beam shift becomes tunable. As the optical response of the coherent medium depends on the intensity and detuning of the controlling fields, we are able to conveniently manipulate the magnitude, the sign, and the angular position of Goos-Hänchen shift peaks.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When a light beam impinges upon the interface of two different media around the total reflection angle, the reflected beam undergoes a lateral displacement from the position predicted by the geometric optics [1]. This lateral shift was experimentally observed by Goos and Hänchen [2], and then explained by Artmann [3]. Since then it has attracted considerable interest due to its potential applications in optical switching [4], sensors, and precision measurements [5–8]. For a single dielectric interface, the Goos-Hänchen (GH) shift is relative small. In the past few decades, much attention has been paid to enhance the GH shift. Large lateral displacement can be obtained on the boundaries of various materials and structures, such as single- or double-negative materials [9–11], weakly absorbing or active media [12,13], dielectric-metal interface [14], graphene [15–17], dielectric slab [18,19], photonic crystal [20,21], layered configuration with surface plasmon resonance (SPR) [22–25], waveguide [26,27] and quantum well [28]. Recently, special light fields were utilized to achieve large GH shift, such as vortex beams [29,30], Bessel beam [31]. The technique of weak measurements is also proposed to observe the lateral displacement [32–34].
The manipulation of GH effect is of great interest for practical applications. In recent years, many investigations have been carried out on the control of lateral shift with an optical cavity containing coherent media [35–43]. The cavity resonance as well as the phase difference of incident light and reflected/transmitted light is closely related to the absorption of intracavity medium. Therefore, the lateral shift can be coherently controlled by modifying the susceptibility of medium based on atomic coherence and quantum interference effects.
At the same time, much interest has been focused on the interaction between surface plasmon polaritons (SPPs) and coherent medium [44–48]. Based on electromagnetically induced transparency (EIT) and relevant coherent effects [49,50], SPPs can be effectively manipulated. The excitation of SPR is accompanied by a sharp dip in the reflection spectrum. Within the dip the phase difference of light beam experiences a dramatic variation, and thus leading to large lateral shift. The magnitude and sign of GH shift can be tuned by the thickness of metal layer [22,24]. There is a critical thickness above (below) which a negative (positive) beam displacement occurs. However, this adjustment is not convenient for a fixed configuration. Therefore, we turn to the manipulation of GH shift in a SPR structure via quantum coherence effect.
In this paper, we investigate the lateral displacement of a three-layer SPR system in the Kretschmann-Raether configuration where the metal film is surrounded by N-type coherent atoms. The absorption of medium is closely related to the internal damping which together with the radiation damping determines the magnitude and sign of the lateral shift. Giant positive and negative beam shifts can be observed by adjusting the imaginary part of susceptibility of atoms. In addition, as the SPR angle is dependent on the refractive index of medium, we can also control the angular position of GH shift peak. The high tunability of lateral shift may have applications in SPR-based optical devices.
2. Model and equations
We consider a layered structure in the Kretschmann-Raether configuration [see Fig. 1(a)]. A metal film of thickness d is sandwiched between a prism and a coherent medium of which the complex refractive index can be manipulated via quantum interference. The permittivities of the three layers are ${\varepsilon _1}$, ${\varepsilon _2}$ and ${\varepsilon _3}$, respectively. When the thickness of the metal is appropriate, SPR can be excited by a transverse magnetic (TM) polarized probe field at an incident angle $\theta $. As a result, attenuated total reflection (ATR) takes place with an enhancement of GH shift in the reflected light. According to the stationary phase theory [3], when the incident light beam has a narrow angular spectrum ($\Delta k\;< <\;k$), the GH shift for the reflected beam can be expressed as
with ${\phi _r}$ being the reflection phase described by $r = |r |\exp (i{\phi _r})$. Here $\lambda $ is the light wavelength in prism. The complex reflective coefficient is given by
Fig. 1. (a) Schematic diagram of three-layered surface plasmon resonance structure: prism, metal and coherent medium. (b) Four-level N-type configuration of the coherent medium.
As is well known, SPR is sensitive to the complex permittivity of medium on the metal surface. Here, an N-type atomic system is considered as the coherent medium [see Fig. 1(b)]. A weak probe field with amplitude ${E_p}$ and two strong fields (coupling field ${E_c}$ and driving field ${E_d}$) couple the transitions $|1 \rangle \leftrightarrow |3 \rangle $, $|2 \rangle \leftrightarrow |3 \rangle $ and $|2 \rangle \leftrightarrow |4 \rangle $, respectively. The corresponding Rabi frequencies are ${\Omega _p} = {{{\mu _{13}}{E_p}} \mathord{\left/ {\vphantom {{{\mu_{13}}{E_p}} {(2\hbar )}}} \right.} {(2\hbar )}}$, ${\Omega _c} = {{{\mu _{23}}{E_c}} \mathord{\left/ {\vphantom {{{\mu_{23}}{E_c}} {(2\hbar )}}} \right.} {(2\hbar )}}$ and ${\Omega _d} = {{{\mu _{24}}{E_d}} \mathord{\left/ {\vphantom {{{\mu_{24}}{E_d}} {(2\hbar )}}} \right.} {(2\hbar )}}$ with ${\mu _{ij}}$ being the electric dipole momentum of transition $|i \rangle \leftrightarrow |j \rangle $. Under the electric-dipole and the rotating-wave approximations, the Hamiltonian of the system in the interaction picture is given by
The atomic-optical response of the probe field is determined by the complex permittivity ${\varepsilon _3}$, which can be expressed as ${\varepsilon _3} = 1 + \chi $ with $\chi $ being the susceptibility. According to the polarization of the medium $\vec{P} = {\varepsilon _0}\chi {\vec{E}_p} = N{\vec{\mu }_{13}}{a_3}a_1^{\ast }$, the susceptibility in the limits of steady-state (${\dot{a}_i} = 0$) and weak probe field (${a_1} \approx 1$) can be derived from Eqs. (5a)–(5d) as
3. Results and discussions
The susceptibility of medium depends strongly on the intensities and detunings of the applied optical fields. Then the reflection phase ${\phi _r}$ can be modulated. As a result, we are able to manipulate the GH effect conveniently. In the numerical calculations, we consider a glass prism (${\varepsilon _1} = 2.25$) and a gold film with thickness of $d = 40\textrm{ nm}$. In addition, the four-level atomic system can be realized in 87Rb D1 line, where the hyperfine energy levels $|{5{S_{1/2}},F = 1} \rangle $, $|{5{S_{1/2}},F = 2} \rangle $, $|{5{P_{1/2}},F = 1} \rangle $ and $|{5{P_{1/2}},F = 2} \rangle $ correspond to $|1 \rangle $, $|2 \rangle $, $|3 \rangle $ and $|4 \rangle $, respectively. At the resonant wavelength of D1 line (λp=795 nm), the dielectric constant of Au is ${\varepsilon _2} ={-} 21.74\textrm{ + }1.744i$. The decay rates of the excited states are 5.75MHz and the electric dipole moment of the transition is $2.538 \times {10^{ - 29}}\textrm{C} \cdot \textrm{m}$. Here, we assume that ${\Gamma _3} = {\Gamma _4} = 1$, ${\gamma _2} = 0.001$ and $\beta = 0.05$ which corresponds to an atomic density of $8.3 \times {10^{11}}\textrm{c}{\textrm{m}^{\textrm{ - 3}}}$. All the other parameters, such as Rabi frequencies and detunings, are scaled by ${\Gamma _3}$.
In the past decades, the N-type system has been extensively investigated both theoretically and experimentally [51–53]. When the driving field is on-resonance or off-resonance to the $|2 \rangle \leftrightarrow |4 \rangle $ transition, enhanced two-photon absorption or giant Kerr nonlinearity can be realized respectively. Therefore we consider two different cases to investigate how the tunable complex susceptibility of medium affects the GH shift in reflected light beam.
3.1 Case I: driving field is on-resonance
Firstly, we consider the case when the driving field is on-resonance, i.e. ${\Delta _d} = 0$. The lateral shift ${S_r}$ as a function of the incident angle $\theta $ for various values of driving field Rabi frequency ${\Omega _d}$ is shown in Fig. 2(a). The corresponding reflectivity is plotted in Fig. 2(b). Due to the excitation of SPR, the incident energy penetrates into the structure, thereby resulting in a sharp dip in the reflection spectrum. Within the resonance dip, the reflected beam undergoes a large lateral displacement. With the increase of ${\Omega _d}$, ${S_r}$ firstly increases and transits from positive to negative, then its magnitude decreases. Moreover, it can be seen that large lateral shift is accompanied by low reflectivity and narrow SPR dip. Figure 2(c) illustrates the dependence of complex susceptibility on ${\Omega _d}$. In the case of ${\Delta _p} = {\Delta _c} = 0$, the two-photon resonance condition is fulfilled, and then the system evolves into the dark state which can be written as $|{dark} \rangle = {{({\Omega _c}|1 \rangle - {\Omega _p}|2 \rangle )} \mathord{\left/ {\vphantom {{({\Omega _c}|1 \rangle - {\Omega _p}|2 \rangle )} {(\Omega _p^2 + \Omega _c^2}}} \right.} {(\Omega _p^2 + \Omega _c^2}}{)^{1/2}}$. This makes the medium transparent to the probe field. Driven by the coupling field, the upper level $|3 \rangle $ splits into two dressed state $|\pm \rangle = {{(|2 \rangle \mp |3 \rangle )} \mathord{\left/ {\vphantom {{(|2 \rangle \mp |3 \rangle )} {\sqrt 2 }}} \right. } {\sqrt 2 }}$ with eigenenergies ${E_ + } ={\pm} {\Omega _c}$. Then the inhibition of the single-photon absorption is also a result of destructive quantum interference between transitions $|1 \rangle \leftrightarrow |+ \rangle $ and $|2 \rangle \leftrightarrow |- \rangle $. As the resonant driving field is applied, two-photon process occurs between the two pathways $|1 \rangle \leftrightarrow |+ \rangle \leftrightarrow |4 \rangle $ and $|1 \rangle \leftrightarrow |- \rangle \leftrightarrow |4 \rangle $ [51]. Owing to the asymmetric coupling, constructive quantum interference takes place, leading to two-photon absorption. Therefore, the probe susceptibility $\chi $ is purely imaginary and increases monotonically with ${\Omega _d}$ [see Fig. 2(c)]. The tunable absorption of medium can effectively affect the internal damping ${\Gamma ^{{\mathop{\rm int}} }}$, which originates from the intrinsic loss of surface plasmon and is positively related to ${\mathop{\rm Im}\nolimits} (\chi )$ in the case of weak absorption [54]. In the SPR structure, there is also radiation damping which results from the back-coupled radiation loss owing to finite metal film thickness. The radiation damping, ${\Gamma ^{\textrm{rad}}}$, is almost independent of ${\mathop{\rm Im}\nolimits} (\chi )$ and can be regarded as a constant. The difference between these two damping determines the amplitude and sign of GH shift [24,26]. Therefore, we can manipulate the beam displacement by adjusting the driving field. The dependence of ${S_r}$ and $|r |$ at resonance angle ${\theta _{\textrm{res}}}$ on ${\Omega _d}$ is plotted in Fig. 2(d). When ${\Omega _d}\;<\;0.16$ and ${\Omega _d}\;>\;0.16$, we can achieve ${\Gamma ^{{\mathop{\rm int}} }}\;<\;{\Gamma ^{\textrm{rad}}}$ and ${\Gamma ^{{\mathop{\rm int}} }}\;>\;{\Gamma ^{\textrm{rad}}}$, which correspond to positive and negative lateral shifts, respectively. As ${\Omega _d} \to 0.16$, the angular width of SPR decreases significantly, thus leading to a dramatic enhancement of beam displacement in the attenuated-total-reflection dip. Particularly, when ${\Omega _d} = 0.16$, the absorption of coherent medium approaches the optimal value at which the two damping equals to each other (${\Gamma ^{{\mathop{\rm int}} }} = {\Gamma ^{\textrm{rad}}}$) and the reflectivity at ${\theta _{\textrm{res}}}$ becomes zero. In this case, ${S_r}$ goes to delta function with an infinite value owing to the abrupt change of ${\phi _r}$. However, the GH shift has no physical meaning in this case.
Fig. 2. (a) and (b) GH shift ${S_r}$ and reflectivity $|r |$ as a function of incident angle $\theta $ for different values of the driving field Rabi frequency ${\Omega _d}$. (c) Probe susceptibility versus ${\Omega _d}$. (d) ${S_r}$ and $|r |$ at resonance angle ${\theta _{\textrm{res}}}$ versus ${\Omega _d}$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Delta _d} = 0$ and ${\Omega _c} = 1$.
From Figs. 2(a) and 2(b), we can see that the resonance angle ${\theta _{\textrm{res}}}$ does not change with ${\Omega _d}$. The expression of resonance angle of SPR is given by [54]
In addition to the driving field, the probe absorption can be controlled via the coupling field as well. With the increase of ${\Omega _c}$, the ac-Stark splitting between dressed states $|+ \rangle $ and $|- \rangle $ becomes large, thus reducing the two-photon absorption. As illustrated in Fig. 3, the results are similar to those in Fig. 2, however, the variation of ${S_r}$ with ${\Omega _c}$ is different from that with ${\Omega _d}$. In this case, the internal damping is larger than the radiation damping for small value of ${\Omega _c}$, and the corresponding lateral displacement is negative. While for large ${\Omega _c}$, the reflected beam suffers a positive shift [see Figs. 3(a) and 3(d)]. In the vicinity of ${\Omega _c} = 0.63$ where ${\Gamma ^{{\mathop{\rm int}} }} = {\Gamma ^{\textrm{rad}}}$, both positive and negative GH shifts can be dramatically enhanced. Meanwhile, the resonance angle ${\theta _{\textrm{res}}}$ remains unchanged.
Fig. 3. (a) and (b) GH shift ${S_r}$ and reflectivity $|r |$ as a function of incident angle $\theta $ for different coupling field Rabi frequency ${\Omega _c}$. (c) Probe susceptibility versus ${\Omega _c}$. (d) ${S_r}$ and $|r |$ at resonance angle ${\theta _{\textrm{res}}}$ versus ${\Omega _c}$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Delta _d} = 0$ and ${\Omega _d} = 0.1$.
Therefore, by tuning the intensity of the driving or the coupling fields, we can coherently manipulate the absorption of medium. Accordingly, the absorption-dependent internal damping varies with ${\Omega _d}$ and ${\Omega _c}$, thereby resulting in tunable and giant GH shift.
3.2 Case II: driving field is off-resonance
Next we consider the case when the driving field is far from resonance. In Fig. 4, the dependence of GH shift and reflectivity on the incident angle is shown. With the increase of the driving field Rabi frequency, ${\Omega _d}$, ${S_r}$ increases and stays positive. In this situation, the reflectivity at resonance is decreased. The GH shift peak and reflection dip become narrower accordingly. As ${\Omega _d}$ exceeds a certain value, the lateral displacement transits from a large positive value to a large negative value. Then the magnitude decreases gradually and the reflection dip becomes shallow. In addition, we can find that the peak position of ${S_r}$ and $|r |$ shift toward small angle. As can be seen from Fig. 4(b), the anglular shift $\Delta \theta $ is not linear to ${\Omega _d}$. Large value of ${\Omega _d}$ corresponds to large $\Delta \theta $.
Fig. 4. (a) GH shift ${S_r}$ and (b) reflectivity $|r |$ as a function of incident angle $\theta $ with various $\Omega _d^2$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Delta _d} = 50$ and ${\Omega _c} = 1$.
In the case of large detuning ($|{{\Delta _d}} |\;> >\;{\Gamma _3},{\Gamma _4}$), the driving field leads to two asymmetric ac-Stark splittings to the ground state $|2 \rangle $, which can be written as $|{2 + } \rangle \approx {{({\Delta _d}|2 \rangle - {\Omega _d}|4 \rangle )} \mathord{\left/ {\vphantom {{({\Delta _d}|2 \rangle - {\Omega _d}|4 \rangle )} {{{(\Omega _d^2 + \Delta _d^2)}^{1/2}}}}} \right.} {{{(\Omega _d^2 + \Delta _d^2)}^{1/2}}}}$ and $|{2 - } \rangle \approx {{({\Omega _d}|2 \rangle + {\Delta _d}|4 \rangle )} \mathord{\left/ {\vphantom {{({\Omega _d}|2 \rangle + {\Delta _d}|4 \rangle )} {{{(\Omega _d^2 + \Delta _d^2)}^{1/2}}}}} \right.} {{{(\Omega _d^2 + \Delta _d^2)}^{1/2}}}}$ with the corresponding eigenenergies ${E_{2 + }} = {{\Omega _d^2} \mathord{\left/ {\vphantom {{\Omega _d^2} {{\Delta _d}}}} \right.} {{\Delta _d}}}$ and ${E_{2 - }} ={-} {\Delta _d}$, respectively. As the sublevel $|{2 - } \rangle $ is far from level $|2 \rangle $, it has little contribution to the near-resonant interaction. Then a new $\Lambda $ system is formed by levels $|1 \rangle $, $|{2 + } \rangle $ and $|3 \rangle $. The driving field makes the system deviate from two-photon resonance condition by an intensity-dependent detuning ${E_{2 + }}$. This small deviation then results in a non-zero susceptibility. In the transparency window, the dispersion is linear so that the refractive part of $\chi $ is proportional to $\Omega _d^2$ [see Fig. 5(a1)]. This also refers to the cross Kerr nonlinearity [52,53]. According to Eq. (8), the anglular shift $\Delta \theta $ and the resonance angle ${\theta _{\textrm{res}}}$ decrease almost linearly with $\Omega _d^2$ [see Fig. 5(a2)].
Fig. 5. (a1) Susceptibility $\chi $, (a2) resonance angle ${\theta _{\textrm{res}}}$, (a3) GH shift ${S_r}$ and reflectivity $|r |$ at ${\theta _{\textrm{res}}}$ as a function of $\Omega _d^2$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Delta _d} = 50$ and ${\Omega _c} = 1$. (b1) Susceptibility $\chi $, (b2) resonance angle ${\theta _{\textrm{res}}}$, (b3) GH shift ${S_r}$ and reflectivity $|r |$ at ${\theta _{\textrm{res}}}$ as a function of $\Omega _d^2$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Delta _d} ={-} 50$ and ${\Omega _c} = 1$.
In the case I, the absorptive part of susceptibility is proportional to $\Omega _d^2$ as shown in Fig. 2(c). This relates to the third-order two-photon absorption process. In this case, large driving field makes higher order nonlinearity dominant. We can see from Fig. 5(a1) that the probe absorption is approximately proportional to $\Omega _d^4$, i.e. ${\mathop{\rm Im}\nolimits} (\chi ) \propto I_d^2$ where ${I_d}$ represents the intensity of the driving field. By modulating the driving field, the internal damping, which is closely related to the absorption of coherent medium, can be controlled effectively. When ${\Omega _d}$ increases from 0, enhanced positive beam shift can be achieved. In the case of $\Omega _d^2 = 15.2$, we arrive at the optimal absorption which corresponds to zero reflection. When $\Omega _d^2\;>\;15.2$, a negative spatial displacement is achieved. With the further increase of driving field, the magnitude of GH shift decreases gradually [see Fig. 5(a3)]. As can be seen from Eq. (8), the shift of resonance angle due to the coherent medium is related to the real part of susceptibility ${\mathop{\rm Re}\nolimits} (\chi )$. This intensity dependent Kerr nonlinearity is due to the ac-Stark effect by the driving field and is proportional to the Stark shift ${E_{2 + }}$. If we change the detuning ${\Delta _d}$ to $- {\Delta _d}$, the variation trend of ${\mathop{\rm Re}\nolimits} (\chi )$ with $\Omega _d^2$ is then opposite. However, the absorptive part of the susceptibility ${\mathop{\rm Im}\nolimits} (\chi )$ is the same in the case of ${\Delta _d}$. This behavior is illustrated in Figs. 5(b1)–5(b3). We observe that the resonance angle increases monotonically with $\Omega _d^2$, while the absorption related lateral shift and reflectivity remain almost the same as seen in Fig. 5(a3).
The detuning of driving field can be also utilized to manipulate the lateral displacement. In Fig. 6, we plot the probe susceptibility, resonance angle of SPR, GH shift and reflectivity at resonance as a function of ${\Delta _d}$. As mentioned above, ${\mathop{\rm Re}\nolimits} (\chi )$ is proportional to the Stark shift ${E_{2 + }}$ and is inversely proportional to ${\Delta _d}$ [see the solid line in Fig. 6(a)]. As a result, the SPR angular shift increases almost linearly with $\Delta _d^{ - 1}$. The driving field detuning also has influence on the absorption of the probe field. With the decrease of ${\Delta _d}$, the interaction between the driving field and the coherent medium is enhanced, thus leading to the increase of probe absorption. Based on this feature, the amplitude and the sign of lateral displacement can be effectively controlled. With certain driving field detuning, we can achieve giant and tunable positive or negative GH shift.
Fig. 6. (a) Susceptibility $\chi $, (b) resonance angle ${\theta _{\textrm{res}}}$, (c) GH shift ${S_r}$ and reflectivity $|r |$ at ${\theta _{\textrm{res}}}$ as a function of $\Delta _d^{ - 1}$. Parameters are ${\Delta _p} = 0$, ${\Delta _c} = 0$, ${\Omega _c} = 1$ and ${\Omega _d} = 4$.
In the above discussions, large tunable lateral shift of the reflected beam is investigated by using the stationary-phase theory, in which the width of incident light beam is assumed to be infinity. In the following, we will examine the validity of stationary-phase theory by considering a Gaussian light beam. As shown in Fig. 1(a), the electric field of a Gaussian beam at incident angle ${\theta _0}$ on the plane of $z = 0$ is given by
Fig. 7. Numerical simulations of the lateral shift of a light beam with Gaussian profile reflected from the structure. The half-width of beam waist are (a) $\sigma = 1000{\lambda _p}$ and (b) $\sigma = 500{\lambda _p}$. The light beam is incident at the resonance angle ${\theta _{\textrm{res}}} = {43.236^ \circ }$. Other parameters are the same as those in Fig. 2.
Fig. 8. Dependence of lateral shift on the beam waist of light beam for ${\Omega _d} = 0.13$. The light beam is incident at the resonance angle ${\theta _{\textrm{res}}} = {43.236^ \circ }$. Other parameters are the same as those in Fig. 2.
In the following let us have a look at the potential applications of the enhanced GH shift via SPR assisted by quantum coherence. Firstly, it can be used for beam steering. When a light beam is incident on an interface, there are spatial and angular lateral shifts which all depends on the reflective coefficient and can be controlled by the coherent medium. Therefore, this beam steering is all-optical and there is no need of electro-optical or acoustic-optical effects. Secondly, it can be exploited as a biochemical SPR sensor if we extend this structure to a four-layer prism-metal-dielectric-solid coherent medium structure where the dielectric layer is designed as a sample cell. Because the accurate control of the thickness of nano film is difficult, the optimal SPR with extremely low reflectivity and high sensitivity is hardly to be achieved. Fortunately, we can tune the internal damping to be closed to the radiation damping via the coherent medium, and then the phase of reflected light varies more rapidly as a function of environment parameters compared with the change in reflectance [5]. As the GH shift is a direct response to the phase change, the sensitivity of the sensor will be more superior and it is also immune from the power fluctuation of light source [6]. Moreover, the lateral displacement can be directly measured by a position sensitive detector which is simpler than other phase measurement methods, such as interferometer, differential ellipsometric method.
4. Conclusions
In conclusion, we have investigated the lateral displacement in a Kretschmann-Raether configuration composing of prism, gold film and four-level coherent medium. When surface plasmon resonance is properly excited at the metal-medium surface, a large lateral displacement occurs at resonance angle. Its magnitude and sign depend strongly on the internal damping of SPR system, which is positively related to the absorptive part of susceptibility of medium. As a result, the GH effect can be effectively enhanced based on quantum coherence effect. Around the value of zero absorption, a giant enhancement of positive and negative beam shifts can be achieved. The dispersive part of susceptibility determines the resonance angle. Therefore, the peak position of GH shift can be coherently adjusted as well. The proposed scheme is much more efficient than the conventional SPR systems where lateral shift is modulated by tuning the thickness of metal film or guiding layer [22,24]. These findings can find applications in the development of new optical devices for optical switching and beam steering, etc.
Funding
National Natural Science Foundation of China (11204367); Natural Science Foundation of Shaanxi Province (2018JQ1051); China Scholarship Council (201806875036); Fundamental Research Funds for the Central Universities (GK202003021).
Acknowledgment
King Abdulaziz City for Science and Technology (KACST).
Disclosures
The authors declare no conflicts of interest.
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