1. Introduction

Reflection happens when the light impinges on the surface of two homogeneous isotropic media, and this phenomenon can be perfectly described by the Fresnel formulas [1]. For confined fields, the center of the reflected- light “spot” on the boundary surface between the two media is not coincident with that of the incident. There are two kinds of shifts. The one parallel with the incident plane is called the Goos-Hänchen shift (GHS) [2], and the other one perpendicular to the incident plane is named the Imbert-Fedorov shift [3], or transverse shift (TS). Interestingly, these two kinds of shift can be described in an unified theory [4–7]. The prediction of TS is based on the nonvanishing transverse energy flux of the evanescent field inside the media [8]. In fact, the TS can be found in reflection or refraction of an incident beam when it has a non-zero spin momentum (i.e., is not of linearly polarized) [9, 10]. The reason for such universality is that the TS is a corollary of the conservation of spin angular momentum and orbital angular momentum [11]. On the other hand, TS can be understood as a different version of spin Hall effect of light (SHEL) [11] which is a phenomenon about splitting of a linearly polarized incident beam into two circularly polarized light in reflection or refraction.

Recently, manipulation of GHS in the fixed configuration or device got a lot of attentions. One way to control GHS is to modify the refractive index of the media by applying a strong laser field [12, 13]. Then electromagnetically induced transparency [14] (EIT) seems to be a promising tool. Take the three-level Λ system as an example, EIT can be realized by applying a coupling field and a probe field driving respectively two adjacent transitions that share a common exited atomic level. The commonly used media for EIT are atomic gas [15, 16], and solid material, such as Pr³⁺-doped Y₂SiO₅ or Pr:YSO [17, 18], and nitrogen-vacancy (N-V) centers in diamond [19–21]. In the transparency window, the probe field experiences nearly zero absorption and steep positive dispersion [22]. Coherent control of GHS using EIT has been discussed in [23]. And most recently, M. S. Zubairy and his coworkers discussed controlling the GHS and TS in a cavity containing three-level atomic media via both pump and driving fields [24].

For the EIT system, in the middle of the transparency window (double-photon detuning δ = 0), the refractive index n = 1. For δ ≠ 0, the refractive index becomes larger, so does the absorption. The system is powerful for suppressing the absorption, but week for manipulating the refractive index. When the TS of the reflected field becomes our object of interest, we certainly wish that the absorption can be limited. To accomplish this, the high refractive index with zero absorption (HRIOA) [25–28] is a promising tool. One can introduce a microwave field (6.8 GHz for ⁸⁷Rb atomic gas) into the system driving the two lower atomic levels. In such system, one can find both reduced and enhanced refractive index with zero absorption so that the reflected or refracted light beams do not suffer too much absorptive losses; Such reduced refractive index can be all-optically manipulated to get an enhanced difference in refraction coefficients for a high background refractive index so that a large enough transverse shift can be observed; Since the refractive index is tunable, and the absorption is limited, this phenomenon seems perfect for manipulating the TS.

In this article we present the research on controlling TS of the reflected field by HRIOA, which is realized by introducing a microwave field driving the lower atomic levels in the three-level Λ system. We adopt the formulas given in [9, 10] to present the analytical result of TS. The formulas are experimentally confirmed in [29]. The incident field is assumed to be the right-handed circularly polarized light. In Sec. 2, we present the model of three-level Λ system, and the equations describing the TS of the reflected field. Sec. 3 deals with manipulation of TS via the strength and the phase of the microwave field. We summarize our results and discussion in Sec. 4.

2. Atomic media and transverse shift of the reflected light

Let us consider a two-layer system composed by the media with the fixed permittivity ε₁ and a layer of cold ⁸⁷Rb atomic gas with the tunable permittivity ε₂, as shown in Fig. 1(a). Both the GHS and the TS are marked out. The atomic media is modelled by the three-level Λ system. The schematic is presented in Fig. 1(b). For example, the three atomic levels could be |1〉 = |5S_1/2, F = 1〉, |2〉 = |5S_1/2, F = 2〉, and |3〉 = |5P_1/2, F′= 2〉. When the Zeeman structure is considered, a static magnetic field should be applied to lift the degeneracy, In order to suppress the redundant transition [30]. The coupling field Ω_c drives the transition |3〉 − |2〉, and a weak field Ω_p probes the transition |3〉 − |1〉. The transition between the lower levels is driven by a microwave Ω_d. The Rabi frequencies of the fields are defined as the complex numbers ${\tilde{\mathrm{\Omega }}}_c={\mathrm{\Omega }}_c{e}^{i{\varphi }_c}=\left({\wp }_{32}{E}_c/2\hslash \right)e^{i{\varphi }_c}$, ${\tilde{\mathrm{\Omega }}}_d={\mathrm{\Omega }}_d{e}^{i{\varphi }_d}=\left({\wp }_{21}{E}_d/2\hslash \right)e^{i{\varphi }_d}$, ${\tilde{\mathrm{\Omega }}}_p={\mathrm{\Omega }}_p{e}^{i{\varphi }_p}=\left({\wp }_{31}{E}_p/2\hslash \right)e^{i{\varphi }_p}$. Where ℘_ij is the matrix element of the dipole moment between levels |i〉 and |j〉. And E_x (x ∈ {c, d, p}) is the amplitude of corresponding field, ϕ_x denotes the phase of the field Ω_x. The density equations that describes the atomic system are:

 

Fig. 1 (a) Schematic of the transverse shift and the Goos-Hänchen shift of the reflected field on the surface of EIT media. (b) The three-level Λ system of ⁸⁷Rb with a microwave field coupling the lower levels.

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Here ϕ = ϕ_d + ϕ_c − ϕ_p. The detunings of the fields are defined as Δ_c = ω₃₂ − ω_c, Δ_p = ω₃₁ − ω_p, δ = Δ_p − Δ_c with ω_c (ω_p) being the frequency of the coupling (probe) field, and ω₃₂ and ω₃₁ as the resonant frequencies of the corresponding optical transitions. Γ_ij is the population decay rate between |i〉 and |j〉. We assume that the microwave Ω_d resonates with the transition |2〉 − |1〉. γ_m is the coherence decay rate of the spin transition |2〉 − |1〉. For both optical transitions (|3〉 − |1〉 and |3〉 − |2〉), the coherence decay rates are assumed to have the same value denoted by γ. To the first order of the probe and the microwave fields, the density element σ₃₁ can be written as

 

Fig. 2 Reσ₃₁ (red solid line) and Imσ₃₁ (dark-blue dashed line) plotted as the function of Δ_p. The data is calculated from Eqs. (2) under the condition of ${\dot{\sigma }}_{ij}=0$. The parameters are Ω_c = 3γ, Ω_d = 0.5γ, Ω_p = 0.01γ, γ = 2π × 6MHz, Δ_c = 0, γ_m = 0, ϕ = π/2, The horizontal magenta dashed line indicates σ₃₁ = 0. And the vertical lines denote Δ_p = −2.9γ and Δ_p = 2.9γ at which Imσ₃₁ = 0. P₁ and P₂ are the intersection points.

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Without the microwave Ω_d, EIT is realized. The important feature of EIT is that, with a resonant coupling field Ω_c, there are two absorption peaks located at Δ_p = ±Ω_c, and Im σ₃₁ approaches zero at Δ_p = 0 [14]. When the microwave Ω_d is applied, the closed-loop transitions |1〉 → |2〉 → |3〉 → |1〉 and |1〉 → |3〉 → |2〉 → |1〉 may occur with the coherence of two ground states being perturbed to result in either constructive or destructive interference. Consequently, enhanced or suppressed EIT peaks can be obtained depending especially on the relative phase ϕ between the optical fields and the microwave field. From another point of view, for the proper values of the parameters, the enhancement of Ω_p takes place as a consequence of the Raman process |1〉 → |2〉 → |3〉 → |1〉 while the probe absorption naturally exists in a nearby spectral region. At the “junction” of the attenuation and the amplification section, HRIOA can be realized. Under the assumption Ω_c > γ > Ω_d ≫ Ω_p, and ϕ = π/2, the extrema of Reσ₃₁ are ±Ω_d/γ approximately, and located at Δ_p ≈ ∓Ω_c. With the vanishing absorption and tunable refractive index, this effect is very promising for controlling the TS of the reflected light. The permittivity of the atomic media can be calculated by ${\epsilon }_2=1+\left({\wp }_{31}^{2}N{\sigma }_{31}\right)/\left(\hslash {\epsilon }_0{\mathrm{\Omega }}_p\right)$, and the introduction of this relation and a comprehensive description of refractive index enhancement without absorption can be found in the famous textbook [31].

The TS of the reflected field can be calculated through the formulas given in [9, 10], and the general form is s = s_t + s_a with

Where ξ = ε₂ − ε₁ sin² θ. The shift s_a is the angular transverse shift, it vanishes for the circularly polarized light. TS s_t given in Eq.(7) is what we are interested. The dimensionless transverse shift (s/λ) is given in Fig. 3. The Rabi frequencies are chosen as Ω_c = 3γ, Ω_d = 0.5γ, Ω_p = 0.01γ. Then HRIOA happens at Δ_p = ±2.9γ ≈ ±Ω_c. Due to the absence of absorption (Im ε₂ = 0), β is a real number for partial reflection. It only becomes complex when total reflection occurs (ε₂−ε₁ sin² θ < 0). In Fig. 3, a red-solid (blue-dashed) curve corresponding to reduced (enhanced) refractive index with zero absorption is plotted for Δ_p = 2.9γ (Δ_p = −2.9γ) with ϕ = π/2. The maximal transverse shift is seen to occur at the critical angle of total reflection in both red-solid and blue-dashed curves because a largest extrinsic orbital angular momentum is required to satisfy the conservation law of total angular momentum when the transmitted field vanishes [32]. The maximal shift in the red-solid curve is higher than that in the blue-dashed curve, this is due to the fact that the contrast ϵ₁/ϵ₂ is larger in the case of reduced refractive index than that in the case of enhanced refractive index.

 

Fig. 3 The dimensionless transverse shift s/λ with different probe detunings, Δ_p = 2.9γ (red solid line) and Δ_p = −2.9γ (dark-blue dashed line). The incident angles for the maximum shift (θ = 0.056π, and 0.212π rad) are marked by the magenta dashed line. ε₁ = 2.2², and $N{\wp }_{31}^2/{\epsilon }_0\hslash =0.05\gamma$. Other parameters are identical with that in Fig. 2.

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3. Manipulating the transverse shift

In addition to realize HRIOA in the EIT media, The microwave Ω_d provides effective ways to control the TS through the strength (Ω_d) and the relative phase (ϕ).

3.1. Tuning the strength of microwave field

Fixing the intensity of the probe field and coupling field (Ω_p = 0.01γ, Ω_c = 3γ), with the previous assumption Δ_c = 0, Reσ₃₁ depends on Ω_d, Δ_p, and ϕ. Here we set ϕ = π/2, the values of Re σ₃₁ for P₁ and P₂ are given in Fig 4(a) as the function of Ω_d. For weaker microwave field (Ω_d < 0.2γ), Re σ₃₁ ∝ Ω_d which is consistent with the properties of the HRIOA we mentioned in the previous section. The values of Δ_p for P₁ and P₂ are presented in Fig. 4(b) as a function of Ω_d as well. As we can see that Δ_p approximately remains constant as Ω_c for P₂, and −Ω_c for P₁. This is due to the feature of EIT.

 

Fig. 4 (a) The value of Reσ₃₁ for P₁ (dark-blue dashed line) and P₂ (red solid line) plotted as the function of Ω_d. The relations between Δ_p and Ω_d at points P₁ and P₂ are given in (b). And (c) shows the largest TS of the reflected light when HRIOA is achieved. The incident angle is the critical angle for total reflection. Other parameters are Ω_c = 3γ, Ω_p = 0.01γ, γ = 2π × 6MHz, ϕ = π/2, ε₁ = 2.2², and $N{\wp }_{31}^2/{\epsilon }_0\hslash =0.05\gamma$.

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The relation given in Fig. 4(b) guarantees that the atomic system stays in HRIOA states. Without absorption or amplification of the incident light, ε₂ remains a real number. In this subsection, we assume that the system is always in HRIOA states. The maximal TS of the reflected field corresponding to P₁ and P₂ are calculated, and shown in Fig. 4(c). Defining $n=\sqrt{{\epsilon }_2/{\epsilon }_1}$, and the maximum of the shift for partial reflection can be easily calculated as

3.2. Tuning the phase of microwave field

The system of HRIOA is very sensitive to the relative phase ϕ [27]. Taking advantage of this feature, the TS can be effectively manipulated. Fig. 5(a) shows the real and imaginary parts of density matrix element σ₃₁ with ϕ = −π/2. Roughly speaking, σ₃₁ (ϕ = −π/2) ≈ −σ₃₁ (ϕ = −π/2) comparing the results shown in Fig. 2 where ϕ = π/2. Set Ω_c = 3γ, Ω_d = 0.5γ, Ω_p = 0.01γ, and Δ_p = 2.9γ, the TS with ϕ = π/2 and ϕ = −π/2 are presented in Fig. 5(b). The result is much similar with Fig. 3, however it reveals how effectively the TS can be manipulated by the relative phase ϕ.

 

Fig. 5 (a) Re σ₃₁ (red solid lines) and Im σ₃₁ (dark-blue dashed lines) versus the probe detuning Δ_p with ϕ = −π/2, (b) Transverse shifts with ϕ = π/2 (dark-blue dashed line) and ϕ = π/2 plotted as the incident angle. Δ_p = 2.9γ. The other parameters are identical with Fig. 2.

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Before considering the manipulation of TS continuously by tuning ϕ, we first focus on the absorption and amplification of the system. With certain Ω_c, Ω_d, and set Δ_p = −Ω_c, we can treat σ₃₁ (which is from the solution of Eq.(2) with ${\dot{\sigma }}_{ij}=0$) as the function of ϕ. And by solving the equation Imσ₃₁ (ϕ) = 0, we get ϕ = ±ϕ₀, with ϕ₀ = arccos (Ω_p/Ω_d) under the condition of Ω_c > γ > Ω_d ≫ Ω_p. The process of obtaining the above solution is trivial and complicated, therefore we shall not present it here. Set Ω_d = 0.5γ and Ω_p = 0.01γ, we have ϕ₀ = ±0.49π. In Fig. 6(a), we present the numerical solution of σ₃₁ obtained from Eq. (2). The imaginary part of σ₃₁ corresponds to the absorption or amplification of the probe field. Here the positive Im σ₃₁ means absorption, while the negative one means amplification. In the previous discussion, the parameter β becomes a complex value when total reflection happens. Here when the non-zero Im σ₃₁ is considered, β becomes complex as well since it relate to ε₂ (which is determined by σ₃₁). In order to obtain β, we have to calculate the term $\sqrt{\xi }$ in Eqs. (2). This value actually contributes to the wave vector of the probe light in the EIT media. Set k_n as the projection of the wave vector onto the normal direction of the boundary surface, then $k_n=\sqrt{\xi }k=\sqrt{\left|\xi \right|}k{e}^{i{\theta }_{\xi }/2}$. As we already knew that, when light is totally reflected by the gain media, it get amplified [33], therefore θ_ξ should be in the range −π/2 < θ_ξ ≤ 3π/2 [34]. For total reflection $\left(\theta >\text{arcsin}\sqrt{\left(Re{\epsilon }_2\right)/{\epsilon }_1}\right)$ the means Re k_n < 0 (evanescent wave) and Im k_n > 0 (gain). Fig. 6(b) shows the reflection coefficient of the circular light (m = −i), $R=\frac{1}{2}{\left(R_{\perp }^2+R_{\parallel }^2\right)}^{1/2}$. As we can see that R > 1 when the total reflection takes place with the negative Im σ₃₁.

 

Fig. 6 (a) Re σ₃₁ (red solid line) and Im σ₃₁ (dark-blue dashed line) plotted as the function of ϕ. (b) The reflection coefficient of the circularly polarized light (m = −i). The parameters are Ω_c = 3γ, Ω_p = 0.01γ, Ω_d = 0.5Ω_c, Δ_p = −Ω_c, γ = 2π × 6MHz, ε₁ = 2.2².

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We stimulate the TS of the reflected light for different relative phases ϕ and the incident angles θ. The result is given in Fig. 7(a). The detailed behavior of TS in the yellow rectangle is presented in Fig. 7(b). There are two peaks of TS in the ϕ − θ plane, corresponding to the total reflection with HRIOA. The TS is a result of conservation law for normal-direction component of total angular momentum [10]. When tuning ϕ with the other parameters fixed, the EIT medium might become active. The absorption or amplification can break this conservation, and result in this two-peaks structure.

 

Fig. 7 (a) The TS of the reflected light versus the relative phase ϕ and the incident angle θ. The detailed behavior of the TS in the yellow rectangle is presented in (b). $N{\wp }_{31}^2/{\epsilon }_0\hslash =0.05\gamma$. The other parameters are identical with that in Fig. 6.

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It is of interest to discuss what will happen when the medium is a hot atomic vapor. In the presence of Doppler broadening, EIT windows become much shallower to exhibit more residual absorption. Fortunately, a so-called Doppler-free technique [35] can cancel the difference of Doppler shifts on different transitions, yielding roughly same EIT spectra as in cold atomic gases. Similarly, we expect to attain roughly same results on HRIOA when the microwave Ω_d is applied. EIT can also be realized in the solid materials, such as Pr:YSO, and N-V centers in diamond. The transition frequency between the lower levels are 10.2 MHz for Pr:YSO [17, 18] which means a radio-frequency field should be used to couple those two levels. and for negatively charged N-V centers, the microwave of frequency 2.88 GHz [36] is reasonable to realize HRIOA.

4. Conclusion

In this article we investigated the transverse shift (TS) on the surface of EIT media. A microwave field is employed to couple the two lower atomic levels, in order to construct high refractive index with zero absorption (HRIOA). Using this effect, we can control the refractive index, and furthermore, the TS of the reflected field. We briefly investigated the properties of HRIOA. When all the intensities of the fields are fixed, normally Ω_c > γ > Ω_d ≫ Ω_p, and the coupling field and the microwave field drive the corresponding transition resonantly. There are two value of Δ_p that leads to HRIOA. Consequently, there exists two type of the TS. The TS of the reflected field can be controlled by the strength and the phase of the microwave field. We maintain the system in the HRIOA state, and change the strength of the microwave field to investigate the maximum shift that can be obtained. The result is given in Fig. 4. For the phase of the microwave, the EIT media can become active (leave the state of HIROA) as the phase changed continuously meanwhile the other parameters fixed. Two peaks of TS are found in the ϕ − θ plane, correspond to the two states of HRIOA of the EIT system. And this means that the TS can respond to the phase of microwave fields sensitively. Our investigation is done for a two-layer system composed by a medium with a fixed refractive index and a medium with a controllable refractive index. To implement a relevant experimental research, however, more details need to be taken into account like transverse shifts at the interface to the containing vessel and polarization changes under an oblique incident angle. With these considerations, the polarization giving a maximum shift at the critical incidence may be not exactly circular [37] as we show here.