1. Introduction

In recent years, the orbital angular momentum (OAM) of light, which is characterized as a multi-degree freedom, has been used to generate entangled states [1,2] and perform micromanipulations [3] as well as induce higher-dimensional entangled states [4]. One of the commonly used light fields with OAM is the so-called Laguerre-Gaussian (LG) mode beam with a spiral phase structure along the propagation direction [5]. An optical vortex (OV) is a typical case of the LG mode and possesses a singular point, where the field intensity becomes zero and around which the phase screws up like an n-armed spiral, with n being the topological charge [6]. Studies on the interaction between OV and matters have investigated covering second-harmonic generation [7], parametric down-conversion [8], Raman-resonant four-wave mixing (FWM) [9,10], light scattering [11,12], structured photonic media [13], and electromagnetically induced transparency (EIT) [14,15]. Also, by exploiting light–matter interactions, OV beams are extended to carry out studies in quantum computations [16], improving astronomical imaging [17]. With respect to various interactions that occur between the OV and the adopted medium, nonlinear processes easy accessible in optics are effective and feasible in unveiling interesting physical mechanisms involved. Meanwhile, it should be noted that the optical nonlinearity also modulates the dynamic behaviors of vortex beams. Recently, propagation behaviors of vortex in an atomic system without dressing effect are investigated [18], and more types of energy-level systems about exchanging optical vortices to FWM process are predicted theoretically [19], vortex FWM applied in CNOT-gate has been demonstrated [20]. However, to the best of our knowledge, there is no report on dressing vortex FWM that are modulated by Kerr-nonlinearity and the spiral phase of vortex light.

To address this gap in the research, in this study, we theoretically and experimentally demonstrate the Kerr-nonlinearity-modulated propagating behaviors of a dressed vortex FWM in a rubidium atomic vapor cell with a photonic band gap (PBG) structure. By adding a dressing field, the behaviors of vortex FWM could be effectively modulated, once the phase-matching condition (PMC) had been satisfied. During the propagation of the vortex beam in the third-order nonlinear medium, the spatial effects, including focusing/defocusing, shifting, and the incompleteness of the vortex shape are obtained experimentally and analyzed theoretically. The results should be applicable both for dressed vortex FWM modulation and applications involving an OV in nonlinear optical processes.

2. Experimental setup

The experiment was performed in a cell filled with rubidium vapor. The corresponding inverted-Y type four-level atomic configuration is shown in Fig. 1(b), which is composed of two hyperfine states, F = 2 (state |3〉) and F = 3 (state |0〉), of the ground state 5S_1/2; the first-excited state 5P_3/2 (state |1〉); and the higher-excited state 5D_5/2 (state |2〉). The probe laser beam, E₁ (frequency  ω₁, wave vector k₁, and Rabi frequency G₁) connects the transition between levels |0〉 ↔ |1〉. A pair of coupling laser beams, E₃ (ω₃, k₃, G₃) and E₃’ (ω₃, k₃’, G₃’), drive transition |1〉 ↔ |3〉. A helical phase is added onto the probe laser using a spatial light modulator (SLM) to obtain the vortex probe which is shown in Fig. 1(a). The dressing field, E₂ (ω₂, k₂, G₂), couples with the upper transition, |1〉 ↔ |2〉. In the experimental setup used (see Fig. 1(a)), the coupling fields, E₃ and E₃′ (780.2 nm), which propagate through the Rb vapor in opposite directions, establish a standing wave. These results are generated in the electromagnetically induced grating (EIG) shown in Fig. 1(c). The weak probe, E₁ (780.2 nm), propagates through the Rb vapor in the same direction as E₃′ but with a small angle θ of 0.3°, and the two beams intersect at the center of the cell. The dressing field E₂ propagates in the same direction as E₃, and the angle between them is small. In the case of phase matching condition (k_F = k₁ + k₃’ - k₃) is satisfied, the corresponding reflect component can be viewed as a FWM signal and the transmitted component of the incident probe beam conducted by the induced EIG are injected into APD2 and APD1, respectively. In this light-matter-interaction manner, the OAM of the probe field can also be transferred to the generated FWM [21]. CCD2 and CCD1 are used to capture images of the vortex probe and vortex FWM, respectively. During the transferring of the OAM through such a third-nonlinear process, both the momentum and the OAM of the photons involved are conserved as k_F = k₁ + k₃’ - k₃ and L₁ = L_F + L_T, respectively, where L_i represents the OAM of the corresponding photons.

 

Fig. 1. (a) Experimental setup and probe vortex image before injecting cell sample for this study. PBS: polarization beam splitter, CCD: charge-coupled device camera, L: convex lens, APD: avalanche photodiode. (b) Inverted-Y type four-level atomic configuration. (c) Schematic of electromagnetically induced grating formed by coupling beams E₃ and E₃′.

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3. Theoretical model

The OV exhibits a spiral phase distribution of exp(imφ) and carries an OAM of mh, where m is an integer denoting the azimuthal mode index (topological charge) and φ is the azimuthal coordinate. With the probe beam E₁ and the coupling beam E₃′ turned on, an EIT window satisfying Δ₁ − Δ'₃ = 0 is created under the Doppler-free conditions. Here, frequency detuning is defined as Δ_i = ω_ij - ω_i, where ω_ij is the transition frequency between states |i〉 and |j〉. Similarly, with E₁ and E₂ on, a second EIT window (Δ₁ + Δ₂ = 0) can also be generated. As a result, we can get the vortex EIT signal from the probe channel when the spiral phase is appended onto the probe field by the SLM. Firstly, when strong fields E₃’ and E₂ drive level |1〉 together, according to the energy system and based on perturbation chain [22] $\rho _{00}^{(0)}\mathop \to \limits^{\omega 1} \rho _{10}^{(1)}$, the considering spiral phase first-order density matrix element for describing the coexisting EIT windows can be written as

The propagation behaviors (induced by the self- and cross-phase modulation (SPM and XPM, respectively)) of the vortex probe and vortex FWM can be understood based on the following equations [24]:

The Kerr nonlinear coefficients can be described using the following form

4. Experimental results

First, we study the vortex FWM in the absence of the dressing effect of E₂. The spectrum and spatial transmission behaviors of the FWM versus Δ₁ based on a three-level configuration (|0〉 ↔ |1〉 ↔ |3〉) when E₂ is blocked are shown in Fig. 2. The spectrum of the PBG FWM (received by APD1) versus Δ₁ is given in Fig. 2(a) under the phase-matching condition (k_F = k₁ + k₃ - k₃′), and one can be seen clearly that the strongest position appears near two-photon resonance, with Δ₁ - Δ₃ = 0 is strictly satisfied. The changes in the intensity of the FWM images, shown in Fig. 2(c), are in line with the intensity profile in Fig. 2(a). Next, we show the spatial evolutions of the Gaussian FWM (without carrying the OAM) and the vortex FWM (carrying the OAM) with respect to Δ₁. One can clearly see from Figs. 2(c1)–2(c2) that the Gaussian FWM images in Fig. 2(c1) exhibit a slight shift along with y-direction with discrete increases in the probe detuning value, Δ₁; however, the vortex FWM images in Fig. 2(c2) do not exhibit this shift but split appear. On the one hand, the cross-Kerr nonlinearity coefficient n₂^X3, which corresponds to E₃, varies with Δ₁, as shown in Fig. 2(b). When Δ₁ < 55 MHz and Δ₁ > 105 MHz, the nonlinear refractive index coefficient, n₂, tends to zero. Thus, the spatial movement is not obvious. With the Kerr nonlinearity coefficient is n₂ < 0 by properly setting the detuning value Δ₁ ≈ 80 MHz, the Gaussian FWM and vortex FWM are all repulsed [16] by E₃ (arranged slightly above the FWM). When the value of n₂ reaches to the highest, the repulsive force is the largest, and so the spot position move most. As n₂ decreases, the spot gradually returns back to its original point, causing Gaussian FWM to shift and vortex FWM to split. On the other hand, the spiral phase (in the ranges of 0-2π) carried by the vortex FWM interacts with the nonlinear phase shift caused by E₃, and the resulted nonlinear phase shift may be different in both sign and value for different points on the vortex beam. Thus, the vortex FWM does not move, as the attractive and repulsive forces cancel each other. Then, with a change in the power proportion of E₃ and E₃′, the Kerr nonlinearity arising from E₃ increases; the corresponding results are shown in Fig. 2(c3). One can see from the figures that, over the range of Δ₁, the spiral phase only balances a part of the nonlinear phase shift. Thus, the cross-Kerr nonlinearity attributable to E₃ causes the singularity position of the vortex FWM to move downwards along the y-direction and then move up.

 

Fig. 2. (a) Spectrum of Gaussian FWM versus Δ₁. (b) Simulated curves of nonlinear refractive index n₂^XE3 from field E₃ versus Δ₁ according to Eq. (4). (c1, c2) Experimental images of Gaussian FWM and vortex FWM, respectively, versus Δ₁ at Δ₃ = 80 MHz without dressing field, E₂. (c3) Vortex FWM versus Δ₁ at Δ₃ = 80 MHz when the power proportion of generating fields, E₃ and E₃′, is changed.

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Next, when the dressing field E₂ is turned on, the Kerr nonlinearity modulates the properties of dressed spatial vortex probe and vortex FWM transmission in different manners, owing to the changes in the power of E₂, as shown in Fig. 3. In this part, the spectrum signals of Gaussian probe and FWM are presented in Figs. 3(a) and 3(b) to reflect the vortex probe and FWM intensity variation versus different detuning and power. In Fig. 3(a), the strongest transparency window appears almost at resonance, that is, at Δ₂ = − 80 MHz, where the EIT condition Δ₁ + Δ₂ = 0 is well satisfied. Further, with the gradual decrease of E₂ power, the dressing effect of E₂ also weakens gradually. Thus, the transparency windows appearing on the probe field also decrease gradually. According to Ref. [25], the generated transparency windows are Autler-Townes splitting (ATS) [26] or EIT. If E₂ field is strong, the transparency windows are ATS due to the frequency shift induced by strong control field. With E₂ field decreasing, a transition from ATS to EIT can be induced so that there is just EIT caused by Fano interference existing under condition of weak E₂ field. Further, for the FWM, as shown in Fig. 3(b), the spectra are suppressed in the term of the dip owing to the dressing effect of E₂. With a decrease in the power of E₂, the degree of suppression of the FWM is gradually reduced, and its transmitted intensity increases. The background (the straight line) of each of the FWM spectral signals represents the non-dressed FWM produced by E₁, E₃, and E₃′. Correspondingly, switching probe field from Gaussian to Laguerre-Gaussian, with a decrease in the power of E₂ from 26.6 mW (see Figs. 3(d1) and 3(e1)) to 0.01 mW (Figs. 3(d5) and 3(e5)), the changes in the intensities of the vortex probe and vortex FWM, shown in Figs. 3(d) and 3(e), well match with those in Figs. 3(a) and 3(b), respectively. It can be seen from the spatial vortex probe images in Fig. 3(d) that the output vortex intensity distribution is nonuniform, which can be attributed to the uniform absorption of medium but undesirably average atomic density. One can see clearly in Fig. 3(d1) from the left to the right that the probe vortex singularity shifts along the y-direction. From the top to the bottom, the slight shift in the vortex gradually disappears with a decrease in the power of E₂. These results indicate that the shifting phenomenon is caused by the Kerr nonlinearity induced by E₂. Owing to the fact that the cross-Kerr nonlinearity n₂^X2 arising from E₂ varies with Δ₂ (see Fig. 2(c)), with a change in Δ₂ to values smaller than −105 MHz and greater than −55 MHz, the nonlinear refractive index coefficient, n₂, becomes close to zero. Thus, the phenomenon of spatial movement cannot be observed. When Δ₂ is close to −80 MHz, the sign of the Kerr nonlinear coefficient related to E₂ is n₂ < 0 such that the vortex probe is repulsed. Thus, the spot moves for a larger Δ₂ value as n₂ decreases, eventually returning to the point with the highest value. As the power of E₂ is decreased gradually, the dressing effect of E₂ on the vortex probe is also weakened. Thus, one can see that, from the top to the bottom, the slight shift in the vortex disappears gradually. On the other hand, the dressed vortex FWM, shown in Fig. 3(e), does not exhibit a shift. The reason for the generation of a stable vortex is that the spiral phase and the nonlinear phase ascribable to E₃ balance each other, and the Kerr nonlinearity from E₃’ balances the Kerr nonlinearity from E₂. The absence of both a shift and splitting in these dressed vortex FWMs confirms that the symmetry of the photon-excited structure is not sensitive to the effects of an external field.

 

Fig. 3. (a, b) Gaussian probe and FWM spectral signals versus Δ₂ at Δ₁(Δ₃) = 80 MHz with increase in power of E₂ (from bottom to top). (c) Value of nonlinear refractive index according to Eq. (4) varies with detuning of E₂. (d, e) Switching probe field from Gaussian to Laguerre-Gaussian, images of vortex probe and vortex FWM corresponding to (a, b), respectively.

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Finally, we show the evolution of intensity of the stable vortex FWM when Δ₃ is discretely reduced from 110 MHz to 50 MHz, as shown in Fig. 4. As is evident from the dressed FWM spectrum shown in Fig. 4(a), the dressing effect of E₃ suppresses the FWM and the corresponding suppression pit is generated. When Δ₃ is discretely reduced from 110 MHz to 50 MHz, the spectrum shape of the FWM is first strengthened and then weakened from the top to the bottom. Further, when Δ₃= 80 MHz and Δ₂= −80 MHz, the two dressing conditions (Δ₁ + Δ₂ = 0 and Δ₁ - Δ₃ = 0) are satisfied. As a result, the suppression pit initially becomes deeper and then shallower. The intensity evolution of the FWM in Fig. 4(b) is in keeping with the intensity profile in Fig. 4(a). To allow for better visualization of the spatial evolution of the dressed vortex FWM, pink dashed lines indicating the vortex have been added at the center of each image. It can be seen clearly from Fig. 4(b) that the dressed vortex FWM profiles remain constant and that the vortex singularities do not shift; however, their intensity distributions around vortex singularity vary with the Kerr nonlinearity caused by E₃. This is the reason the spiral phase, φ, and nonlinear phase, ϕ₃, induced by E₃ interact with each other. As a result, the periodic spiral phase shift in combination with the nonlinear phase shift (mφ+ϕ₃) may result in variations in the nonlinear phase shift at different points in the spiral phase.

 

Fig. 4. (a, b) Fixing Δ₁ at 80 MHz, spectral signals and images of FWM versus Δ₂ for (1) Δ₃= 110 MHz, (2) Δ₃= 90 MHz, (3) Δ₃= 70 MHz, and (4) Δ₃= 50 MHz.

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5. Conclusion

In summary, we investigated the Kerr-nonlinearity-modulated spatial characteristics and spectral signals of a vortex probe and a vortex FWM with the dressing effect in a four-level rubidium atomic system. By adjusting the frequency detuning value and the power of the dressing field, we could elucidate the spatial propagation characteristics. It was found that the nonlinear phase shift and the spiral phase of the vortex light mutually affect the generation and dynamic of the vortex FWM. The results of this study should help further the applicability of optical vortices in applications such as optical computing and information processing.