## Abstract

Considering the fact that the orbital angular momentum of light can be transferred through light-matter interactions, we experimentally induced a dressed vortex four-wave mixing (FWM) with the interaction between a vortex probe beam and an inverted Y-type four-level atomic system with a photonic band gap. Further, the Kerr-nonlinearity-modulated propagation behaviors of the probe and the dressed FWM vortices are investigated, including the spatial shift, splitting, and incompleteness of the vortex shape. Strikingly, the propagation behaviors of the vortex beams can be influenced by the interaction between the nonlinear phase and the spiral phase. This study would promote the development of optical computing and information processing science related to the interactions between optical vortices and samples.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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, *E*_{1} (frequency * ω*_{1}, wave vector *k*_{1}, and Rabi frequency *G*_{1}) connects the transition between levels |0〉 ↔ |1〉. A pair of coupling laser beams, *E*_{3} (*ω*_{3}, *k*_{3}, *G*_{3}) and *E*_{3}** ’** (

*ω*

_{3},

*k*_{3}

**,**

*’**G*

_{3}

*’*), 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,

*E*_{2}(

*ω*

_{2},

*k*_{2},

*G*

_{2}), couples with the upper transition, |1〉 ↔ |2〉. In the experimental setup used (see Fig. 1(a)), the coupling fields,

*E*_{3}and

*E*_{3}′ (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,

*E*_{1}(780.2 nm), propagates through the Rb vapor in the same direction as

*E*_{3}′ but with a small angle θ of 0.3°, and the two beams intersect at the center of the cell. The dressing field

*E*_{2}propagates in the same direction as

*E*_{3}, and the angle between them is small. In the case of phase matching condition (

*k*_{F}=

*k*_{1}+

*k*_{3}’ -

*k*_{3}) 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}=

*k*_{1}+

*k*_{3}’ -

*k*_{3}and

*L*

_{1}=

*L*

_{F}+

*L*

_{T}, respectively, where

*L*_{i}represents the OAM of the corresponding photons.

## 3. Theoretical model

The OV exhibits a spiral phase distribution of exp(*imφ*) and carries an OAM of *m*h, where *m* is an integer denoting the azimuthal mode index (topological charge) and *φ* is the azimuthal coordinate. With the probe beam *E*_{1} and the coupling beam *E*_{3}′ turned on, an EIT window satisfying Δ_{1} − Δ'_{3} = 0 is created under the Doppler-free conditions. Here, frequency detuning is defined as Δ* _{i}* =

*ω*-

_{ij}*ω*, where

_{i}*ω*is the transition frequency between states |

_{ij}*i*〉 and |

*j*〉. Similarly, with

*E*_{1}and

*E*_{2}on, a second EIT window (Δ

_{1}+ Δ

_{2}= 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

*E*_{3}’ and

*E*_{2}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

*d*

_{1}= Γ

_{10}+

*i*Δ

_{1},

*d*

_{2}= Γ

_{20}+

*i*(Δ

_{1}+Δ

_{2}), and

*d*

_{3}= Γ

_{30}+

*i*(Δ

_{1}-Δ

_{3}), with Γ

*being the transverse relaxation rate. Here, the Rabi frequency is ${G_i} = {\mu _{ij}}{E_i}/\hbar$, where*

_{ij}*μ*is the transition dipole moment for transition |

_{ij}*i*〉 ↔ |

*j*〉 and

*E*is the electric-field intensity of field

_{i}

*E**. Then, with all the beams turned on, the third-order density matrix element [23] (according to perturbation chain $\rho _{00}^{(0 )} \to \rho _{10}^{(1 )} \to \rho _{30}^{(2 )} \to \rho _{{G_3}{G_3}{G_2}}^{(3 )}$) is responsible for describing the dressed vortex FWM can be expressed as*

_{i}*ϕ*is the nonlinear phase shift caused by strong beams

_{i}

*E**(*

_{i}*i*=2, 3) and |

*G*

_{3s}|

^{2}=

*μ*

_{13}

^{2}(

*E*

_{3}

^{2}+

*E*

_{3}′

^{2}+2

*E*

_{3}

*E*

_{3}′cos2

*k*

_{3}

*x*)/

*h*

^{2}. The corresponding intensity of vortex FWM is ${\mathop{I}\nolimits} \propto {|{\rho_{\textrm{10}}^{\textrm{(3)}}} |^2}$. Here, the first- and third-order nonlinear susceptibilities are given as

*χ*

^{(1)}=

*Nμ*

_{10}

*ρ*

_{10}

^{(1)}/

*ε*

_{0}

*E*

_{1}and

*χ*

^{(3)}=

*Nμρ*

_{10}

^{(3)}/

*ε*

_{0}

*E*

_{1}

*E*

_{2}

*E*

_{3}, respectively, where

*N*and

*ε*

_{0}are the atomic density and dielectric constant, respectively.

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]:

*z*is the longitudinal coordinate;

*k*

_{1}=

*k*

_{F}=

*ω*

_{3}

*n*

_{1}/

*c*;

*n*

_{1}is the linear refractive index;

*n*

_{2}

^{S1, S2}is the self-Kerr coefficient of

*E*_{1, F}; and

*n*

_{2}

^{X1, X2}is the cross-Kerr coefficient of

*E*_{1}induced by

*E*_{3}′ and

*E*_{2}. Further,

*n*

_{2}

^{X3, X4}is the cross-Kerr coefficient of

*E*_{F}induced by

*E*_{3}and

*E*_{2}.

The Kerr nonlinear coefficients can be described using the following form

*E*_{2}or

*E*_{3}can be expressed as where

*ξ*is the center coordination of

*E*_{2}or

*E*_{3}in the transverse dimension relative to the center coordination of

*E*_{1, F}as the original point,

*I*is the intensity of

_{i}

*E**field (*

_{i}*i*= 2, 3). Therefore, the additional transverse vortex propagation wavevector is $\delta {{\mathop{\rm k}\nolimits} _ \bot } = \partial {\phi _{NL}}({z,\xi } )/\partial \xi $, which determines the

*E*_{3}- and

*E*_{2}-induced spatial characteristics of

*E*_{1, F}in the transverse dimension. The dressed FWM expressed by the perturbation chain (Eq. (2)) can also be considered as the reflected part of the vortex probe launched from the PBG. Consequently, this FWM can also be called the PBG FWM. Meanwhile, considering that the OAM of the probe beam can be transferred to the FWM, the spiral phase,

*φ*, and nonlinear phase,

*ϕ*(

_{i}*i*= 2, 3), induced by

*E*_{2}and

*E*_{3}can interact with each other. As a result, the spatial dynamics of the vortex beams can be governed by the Kerr nonlinear coefficient

*n*

_{2}; the intensities of the coupling fields; and the interactions between the spiral phase,

*φ*, and the nonlinear phase,

*ϕ*. The sum of the circular spiral phase shift and the nonlinear phase shift (m

_{i}*φ + ϕ*) gives the nonlinear phase shift for the different points in the spiral phase.

_{i}## 4. Experimental results

First, we study the vortex FWM in the absence of the dressing effect of *E*_{2}. The spectrum and spatial transmission behaviors of the FWM versus Δ_{1} based on a three-level configuration (|0〉 ↔ |1〉 ↔ |3〉) when *E*_{2} is blocked are shown in Fig. 2. The spectrum of the PBG FWM (received by APD1) versus Δ_{1} is given in Fig. 2(a) under the phase-matching condition (*k _{F}* =

*k*

_{1}+

*k*

_{3}-

*k*

_{3}′), and one can be seen clearly that the strongest position appears near two-photon resonance, with Δ

_{1}- Δ

_{3}= 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 Δ

_{1}. 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, Δ

_{1}; 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

*n*

_{2}

^{X3}, which corresponds to

*E*_{3}, varies with Δ

_{1}, as shown in Fig. 2(b). When Δ

_{1}< 55 MHz and Δ

_{1}> 105 MHz, the nonlinear refractive index coefficient,

*n*

_{2}, tends to zero. Thus, the spatial movement is not obvious. With the Kerr nonlinearity coefficient is

*n*

_{2}< 0 by properly setting the detuning value Δ

_{1}≈ 80 MHz, the Gaussian FWM and vortex FWM are all repulsed [16] by

*E*_{3}(arranged slightly above the FWM). When the value of

*n*

_{2}reaches to the highest, the repulsive force is the largest, and so the spot position move most. As

*n*

_{2}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

*E*_{3}, 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

*E*_{3}and

*E*_{3}′, the Kerr nonlinearity arising from

*E*_{3}increases; the corresponding results are shown in Fig. 2(c3). One can see from the figures that, over the range of Δ

_{1}, the spiral phase only balances a part of the nonlinear phase shift. Thus, the cross-Kerr nonlinearity attributable to

*E*_{3}causes the singularity position of the vortex FWM to move downwards along the y-direction and then move up.

Next, when the dressing field *E*_{2} 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 *E*_{2}, 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 Δ_{2} = − 80 MHz, where the EIT condition Δ_{1} + Δ_{2} = 0 is well satisfied. Further, with the gradual decrease of *E*_{2} power, the dressing effect of *E*_{2} 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 *E*_{2} field is strong, the transparency windows are ATS due to the frequency shift induced by strong control field. With *E*_{2} 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 *E*_{2} 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 *E*_{2}. With a decrease in the power of *E*_{2}, 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 *E*_{1}, *E*_{3}, and *E*_{3}′. Correspondingly, switching probe field from Gaussian to Laguerre-Gaussian, with a decrease in the power of *E*_{2} 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 *E*_{2}. These results indicate that the shifting phenomenon is caused by the Kerr nonlinearity induced by *E*_{2}. Owing to the fact that the cross-Kerr nonlinearity *n*_{2}^{X2} arising from *E*_{2} varies with Δ_{2} (see Fig. 2(c)), with a change in Δ_{2} to values smaller than −105 MHz and greater than −55 MHz, the nonlinear refractive index coefficient, *n*_{2}, becomes close to zero. Thus, the phenomenon of spatial movement cannot be observed. When Δ_{2} is close to −80 MHz, the sign of the Kerr nonlinear coefficient related to *E*_{2} is *n*_{2} < 0 such that the vortex probe is repulsed. Thus, the spot moves for a larger Δ_{2} value as *n*_{2} decreases, eventually returning to the point with the highest value. As the power of *E*_{2} is decreased gradually, the dressing effect of *E*_{2} 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 *E*_{3} balance each other, and the Kerr nonlinearity from *E*_{3}** ’** balances the Kerr nonlinearity from

*E*_{2}. 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.

Finally, we show the evolution of intensity of the stable vortex FWM when Δ_{3} 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 *E*_{3} suppresses the FWM and the corresponding suppression pit is generated. When Δ_{3} 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 Δ_{3 }= 80 MHz and Δ_{2 }= −80 MHz, the two dressing conditions (Δ_{1} + Δ_{2} = 0 and Δ_{1} - Δ_{3} = 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 *E*_{3}. This is the reason the spiral phase, *φ*, and nonlinear phase, *ϕ _{3}*, induced by

*E*_{3}interact with each other. As a result, the periodic spiral phase shift in combination with the nonlinear phase shift (m

*φ+ϕ*) may result in variations in the nonlinear phase shift at different points in the spiral phase.

_{3}## 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.

## Funding

National Key Research and Development Program of China (2017YFA0303700); National Natural Science Foundation of China (11604256, 11804267, 11904279, 61975159); The New Star Team of Xi'an University of Posts & Telecommunications.

## Disclosures

The authors declare no conflicts of interest.

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