1. Introduction

Polarization is one of the most salient features of light [1]. Compared to homogeneously polarized beams, vector beams (VBs) exhibit space-variant polarizations and this unique feature has attracted significant interest [2]. VBs have been utilized in many fields, such as optical micro-fabrication, optical trapping, optical micro-manipulation and second-harmonic generation [3–7]. Utilizing the available multi-dimensionally polarization information, VBs have been applied to high-dimensional coding, quantum communication and quantum entanglement [8–10]. VBs have also been explored for interactions with atomic media based on the atomic quantum coherence effect [11], such as quantum memory of VBs [12], spatially structured transparency with VB [13–15], polarization shaping for control of nonlinear propagation [16], detecting and preparing the spin of atoms in a spatially dependent manner [17], manipulation of space-variant polarization beams by using external magnetic fields in atomic vapor [18], and studying diffusive properties of vapor in the presence of buffer gases [19]. Generally, two types of methods have been used for the generation of VBs. The first one is the intra-cavity generation technique [20–22] which includes the birefringence element method [23], the conjugate Brewster lens method [24]; and the multi-layer polarization grating and mirror method [25]. The second method is the extra-cavity technique [26–29] based on the cone mirror [30], grating [26], liquid crystal spatial optical modulator [31] and Q-plate method [32, 33]. The extra-cavity technique is more flexible and efficient, and is more widely adopted.

The manipulation of VBs is essential for these unique applications. So far, the manipulation of VBs has mainly been studied and demonstrated with linear optics. Linear optics usually realizes the manipulation of a VB by some polarizing components and phase modulator [34, 35]. However, polarizing components may cause a change of the polarization state of the target probe beam and, consequently, one only gets back the intensity distribution but looses polarization information. Reference [36] provide a scheme to get a “patterned” VB by manipulating the phase of right and left circular polarization components on the spatial light modulator. Such arbitrary “patterned” VBs can be obtained with this method, but it depends on the complexity of program and optical path. Here, we present a method to manipulate VBs with an atomic ensemble. Atomic polarization properties can be adjusted with an optical field [37]. However, compared to linear optics, non-linear properties of an atom vapor supply a rich platform for interaction with VBs. Utilizing light with spatial polarization degrees of freedom interacting with atomic vapor produces an ensemble of spatially polarized atoms that exhibit spatially related transmission features. Vice versa these can be used to manage the transmission properties of a VB. Light transmission features in atomic vapor under different polarized configurations have been well studied several decades ago [38, 39]. However, so far most cases only deal with uniformly polarized beams. In contrast, for a VB, the spatially varied polarization will lead to spatial anisotropy of the atomic polarization and provide an effective spatial manipulation for light fields. One can expect to research the transmission features of different light modes by changing the pump beam mode or polarization state. Moreover, a spatially polarized atomic ensemble can also be utilized to manipulate the polarization state of the probe beam which cannot be easily achieved with the linear optics or phase modulator. In this way, a VB, as a unique spatially structured beam, with its spatially varied polarization, will experience various transmission properties and provide a feasible way for light manipulation with atomic ensembles.

In this paper, we report the transmission properties of VBs by adjusting the spatially atomic polarization via the optical pumping effect. We first generate the radial VB as probe beam by a Q-plate, and choose the hybrid VB as pump beam by adding a quarter-wave plate to the radially polarized VB. In a counter-propagating configuration, we find that the probe beam exhibits a four-petal pattern after passing through the atomic medium, and the four-petal pattern intensity distribution can be rotated by rotating the quarter-wave plate of the pump beam. We experimentally analyze the probe beam’s behaviors by inspecting the polarization dependent absorption with uniformly polarized states of pump and probe beams. Then, we explain the light transmission properties under our polarization arrangements using the Jones matrix method. Our experiment may provide a convenient and high-dimensional polarization selection method for VB manipulation based on atomic ensembles.

2. Experimental setup and Stokes parameters measurement

The experimental setup is schematically shown in Fig. 1(a). An 780nm extended cavity diode laser (ECDL) is used for pump and probe beams to make up an optical pumping scheme in an atomic vapor. The laser beam is divided into two parts by a polarization beam splitter (PBS). One part is employed for frequency locking at the 5$S_{1/2}$, F =1 → 5$P_{3/2}$, F’=0, 1 crossover transition of the ⁸⁷Rb D₂ line. The other part is collected by a single-modefiber (SMF) in order to improve the spatial mode; the achieved waist width of the high quality Gaussian beam is about 3mm. The beam then passes through a half-wave plate (HWP) and a PBS to provide a suitable light intensity for the experiment. After the beam size is expanded to two times larger by a telescope, we produce the radially polarized beam by sending the horizontally linearly polarized beam through a Q-plate (q = 1/2) with its optical axis along the horizontal. The radially polarized beam is then divided by a beam splitter (BS). The transmitted part is chosen as the probe beam (shown in Fig. 2(b)), while the reflected part becomes a hybridly polarized beam after passing a quarter-wave plate (QWP) and serves as the pump beam (shown in Fig. 2(c)).

 

Fig. 1 (a) Experimental setup for vector beam manipulation via optical pumping in an atomic vapor. (b) Energy-level diagram of the optical pumping configuration, the ${\sigma }^{+}$(red arrow) refers to the pump beam and the π(purple arrow) denotes the probe beam, and so on for ${\sigma }^{-}$ case where $\mathrm{\Delta }{m}_F=-1$. SMF, single-mode optical fiber; L, lens; M, Mirror; QWP, quarter-wave plate; BS, beam splitter; PD, photo-detector; CCD, charge coupled device camera.

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The pump and probe beams with different polarized states are counter-propagating with each other and couple with the corresponding Zeeman sub-levels of ⁸⁷Rb. The energy-level configuration is shown in Fig. 1(b). Note that, the different polarization states of the laser beam couple with different Zeeman sub-levels of the Rb atom. The linearly polarized beam couples two sub-levels with $\mathrm{\Delta }{m}_F=0$, while the circularly polarized beam couples $\mathrm{\Delta }{m}_F=\pm 1$ energy levels. The probe (purple) and pump (red) beams couple with different hyperfine levels of an identical atom due to the Doppler effect. In Fig. 1(b), we only show the left-circularly polarized (${\sigma }^{+}$)pump beam case, and so on for ${\sigma }^{-}$ case where $\mathrm{\Delta }{m}_F=-1$.

 

Fig. 2 (a) The experimental setup of Stokes parameters measurement. GLP, Glan-laser polarizer. (b) and (c) Polarization distribution and Stokes parameters of radially vector probe beam and hybridly vector pump beam. The black curves inside first column of (b) and (c) outline the real polarization distribution, while the blue curves inside S₀ indicate the reconstructed polarization distribution from measured Stokes parameters.

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In the experiment, the probe and the counter-propagating pump beams are collinear and overlapping in the Rb cell. The Rb cell has a length of 50mm and is filled with enriched ⁸⁷Rb gas. A three-layer μ-metal magnetic shield is used to isolate the ambient magnetic fields. We use a bidirectional winding solenoid inside the inner layer of magnetic shield to heat the Rb cell, and the temperature of cell is kept to 65°C using a temperature controller. After passing through the cell, the spatial intensity distribution of the probe beam is recorded by a charge-coupled device camera (CCD). The power of the probe beam is about 40 μW, and the pump beam is about 2mW.

To effectively examine the polarization states of the VB used in our experiment, we firstly analyze the Stokes parameters of the probe and pump beams by a combination of a QWP, a Glan laser polarizer (GLP) and a CCD as shown in Fig. 2(a) [40]. The real polarization distributions of probe and pump beams are plotted in the first column of Figs. 2(b) and 2(c). The measured Stokes parameters for probe and pump beams are shown in the other columns of Figs. 2(b) and 2(c), respectively. S₀ stands for the total intensity. S₁ describes the linear polarization component in horizontal (X) and vertical (Y) direction. S₂ represents the $\pm {45}^{\circ }$ linear polarization components. S₃ is a measurement of the left- and right-circular polarization components. The measurement results are the same as we expected. The probe beam generated from Q-plates shows the linear polarization aligned in the radial direction without circular polarization components. The pump beam is the hybridly polarized which contains varied distribution of polarization such as linear, elliptical and circular polarization.

 

Fig. 3 The experimental results for the manipulation of vector beam. The blue line is the fast axis of the QWP, the curve lines in four lobes at θ=0° indicate the reconstructed polarization distribution from measured Stokes parameters. The fast axis rotation angle of the QWP is θ with respect to the horizontal direction.

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3. Experimental results for VBs and uniformly polarized beams

In Fig. 1(a), the fast axis of the QWP is set parallel to the X axis. In this situation, the probe beam is radially polarized with the linear polarization direction aligned in the radial direction. The pump beam is hybridly polarized with the linear polarization aligned in the horizontal (X) and vertical (Y) directions, the circular polarization aligned in the diagonal directions, and elliptical polarizations in other directions. After passing through the cell, a four-petal pattern of transmitted intensity distribution shown in Fig. 3 is recorded on the CCD for an original radially polarized probe beam. Comparing the polarization states of the probe with those of pump beams at different positions, we find that the absorption happens when the linearly polarized direction of pump beam same as the probe beam’s. In contrast, the probe beam is transmitted at the other positions for which the pump beam contains non-linearly polarized states. We also observe that the four-petal pattern rotates with the fast axis of the QWP. The variation of the four-petal against the rotation angle of the fast axis of QWP is shown in Fig. 3 and the blue line shows the fast axis direction of the QWP. The variation period is the same with the rotation period of the QWP. The rotation of the four-petal can be understood by considering how the polarization distribution of the pump beam is rotated with the QWP. The polarization distribution of the transmitted probe beam is reconstructed with Stokes parameter measurement and shown in Fig. 3 at θ=0°. Polarization distributions at other angles will rotate correspondingly. We will interpret this distribution in the discussion section.

 

Fig. 4 The schematic diagram of the polarization transformation for the probe and pump beams for the uniformly polarized beam experiment. (a) The polarized state selectivity for the pump beam. (b) The polarized state selectivity for the probe beam.

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To understand the transmitted intensity distribution of the radially polarized probe beam when the hybridly polarized pump beam is applied, we set up a simple experiment schematically shown in Fig. 4 to simulate the process that the probe beam undergoes. For this configuration, we use the uniformly polarized states for both probe and pump beams instead of using the VBs. A horizontally polarized beam passes through the BS and is divided into two parts. The transmitted part is selected as the probebeam that always holds linear polarization with the polarized direction adjusted by the HWP2. The reflected part is chosen as the pump beam of arbitrary polarization modulated by a combination of HWP1 and QWP1. We select four linearly polarized states for the probe beam by adjusting the orientation of HWP2 for linearly polarization along horizontal, +45°, vertical and -45° directions corresponding to X axis. These polarization direction are set by the angle of the HWP2 fast axis to the X axis at α=0°, ${22.5}^{\circ }$, 45° and ${67.5}^{\circ }$, respectively. The selected four linearly polarized states represent different wave-front position of former radially polarized probe beam. As for the pump beam, the rotation angle of HWP1 is synchronous with the HWP2, while the fast axis of QWP1 is rotated with an angle ϕ from 0° to 180° with interval of ${7.5}^{\circ }$. In such way, each polarization state carried by the former hybridly polarized pump can be simulated.

 

Fig. 5 The intensity variations of the probe beam at different pump-probe polarization configuration against ϕ. Pol. means polarization of pump beam. Blue dots are the experimental data and the red line is the theoretical simulation. (a)-(d) show the probe beam polarized states as horizontally polarized state, linear $+{45}^{\circ }$ polarized state, vertically polarized state and linear $-{45}^{\circ }$ polarized state, respectively.

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Figure 5 shows the experimental results of the transmission efficiency for the probe beam against rotation angle ϕ. The blue dots are the experimental data and the red line is the simulation. The purple line with arrows beneath the diagram block represents the polarization state of the pump beam corresponding to ϕ. Each point in Fig. 5 stands for a different pump-probe polarization configuration. In Fig. 5(a), the probe beam with horizontal linearly polarized state successively experiences the pump beam with horizontal linearly polarized state, left circularly polarized state, horizontal linearly polarized state, right circularly polarized state and finally back to horizontal linearly polarized state. We note that the maximum absorption occurs when the pump beam has the same linear polarization with the probe beam, while the maximum transmission occurs for circular polarization of the pump beam. The same situation can be found in Figs. 5(b)-5(d) for setting different directions of the linear polarization of the probe beam.

With the transmission efficiency obtained from Fig. 5 for the different polarization configurations, we simulate the transmission intensity distribution of the vector probe beam as shown in Fig. 6. The simulation shown in Fig. 6(a) is very similar to our experimental result in Fig. 6(b). We also calculate the transmission efficiency of the vector probe beam along the azimuthal distribution as denoted in Fig. 6(b). As we expect, the transmission efficiency results shown in Fig. 6(c) are coincident with the results shown in Fig. 5.

 

Fig. 6 (a) Theoretical simulation of the four-petal pattern of vector probe beam passing through the atomic medium. (b) Diagram for selecting azimuthal angle to calculate the transmission efficiency. (c) Transmission efficiency of azimuthal distribution, the purple line expresses the polarization states.

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4. Results analysis and discussion

The results in Fig. 5 for the uniformly polarized beam can explicitly explain the four-petal pattern shown in Fig. 3 for the VB. At θ = 0, the same linear polarization of the probe and pump beam locates at two coordinators axes, X and Y, while the circular polarization of the pump beam is at diagonal directions. Therefore a straight four-petal pattern is formed. As the QWP in Fig. 1(a) rotates, the linearly polarized components of the pump beam rotates together with the fast axis of the QWP. This causes the corresponding position of the probe beam to be absorbed due to its radially aligned linear polarization, and makes the four-petal rotates simultaneously.

Polarization dependent absorption has been investigated several decades ago [37–39]. Absorption happens when the linearly polarized probe and pump beams have parallel polarization direction, while transmission happens when their linearly polarization directions are orthogonal to each other [37]. As for the case that linearly polarized probe beam meets with the elliptically or circularly polarized pump beam, there always exist orthogonal linearly polarized components of the pump beam. Such orthogonal linearly polarized components can deplete some atomic sub-levels and make the probe beam transparent due to the optical pumping effect. The maximum transparency occurs when the pump beam holds the circular polarization state since this situation corresponds to the maximum orthogonal linearly polarized component in our experimental case. In this way, we can analyze the probe beam transmission properties by examining the orthogonal linearly polarized component carried by the pump beam.

In the following, we use the Jones matrix method to track the polarization transformation of the pump beam. The Jones matrix for a wave plate with the phase retardation of δ can be expressed as

In Eq. (4), the second row element represents the vertical linearly polarized component which is orthogonal to the original horizontally polarized direction. This component contributes to the transparency of the probe beam, and the transmitted intensity is proportional to the square of this component expressed as

The different ϕ can be regarded as the various polarized components distributed on the wavefront of the hybridly polarized pump beam, which causes the radially polarized probe beam to be transmitted as four-petal pattern in the atomic vapor. Correspondingly, as the QWP rotates with ϕ, the transmitted probe beam intensity also varies as the red line shown in Fig. 5. On the other hand, as shown in Fig. 3 at θ=0°, we can see the polarization distribution in the four-petal pattern sets along the radial direction with the elliptical polarization. The results coincide with our theoretical analysis. Each linearly polarized component of the hybridly polarized pump beam with its polarized direction orthogonal to the radially polarized probe beam will contribute the transmission of the probe beam. The elliptically polarized distributions are mainly due to the compound polarization states carried by the hybridly polarized pump beam.

Before drawing conclusions, we should address that our results for the linearly polarized probe and circularly polarized pump beam configuration can also be explained by the optical pumping scheme shown in Fig. 1(b), as has been studied in our previous work [14]. However this explanation is not suitable for the linearly polarized probe and pump beam configuration since a linearly polarized pump beam can be split into two orthogonal circularly polarized components. These two circularly polarized components always cause a linearly polarized probe beam transmission whatever the polarized direction is, in contrast to our observed results. For the full interpretation our results, we thus adopt the explanation explored in [37] as the basis of our theoretical analysis. In this way, more general polarization dependent absorption can be interpreted.

5. Conclusion

We report an experimental realization of manipulating an atom vapor medium by a VB via the non-linear optical pumping effect. We obtain the light intensity distribution of the probe beam as a four-petal pattern induced by the polarization dependent absorption. With changing the orientation of the hybridly polarized pump beam, the four-petal pattern rotates simultaneously. We also setup a simple experiment with uniformly polarized light to explain this phenomenon. The results show that the polarized dependent absorption for uniformly polarized beam still holds for the structured beam. The spatially varied polarization of the vector beam leads to a spatial anisotropy of the atomic polarization and provides an effective spatially manipulation for the VB. Compared to the linear optical polarizer, our experimental scheme may provide a convenient and high-dimensional polarization selection method for VB manipulation based on atomic ensembles.