## 4 Coherent Manipulation on Light Fields via Atomic Coherence Gratings
### Manipulation on Optical Vortexes  
An optical vortex carrying an optical topological charge (OTC), \(m\), with \(m\) being an integer, is of an azimuthal phase structure in the form \(e^{i\mu\phi}\), which forms a phase singularity [82; 83; 84]. [PERSON] et al. [85] pointed out that each photon in such an optical vortex beam with an OTC \(m\) carries an orbital angular momentum of \(m\hbar\). The transfer of the orbital angular momentum of photons to the micro/nanoparticles during the light-matter interaction can be used to trap and manipulate the micro/nanoparticles. The eigenmodes of the orbital angualr momentum, i.e., the Laguerre-Gaussian modes, are mutually orthogonal and can be used for mode-multiplexing to increase the signal transmission capacity in fiber optical communications, and these eigenmodes also span an infinite dimensional discrete Hilbert space for quantum information processing and quantum computation. Optical vortex beams have many potential applications, such as optical signal processing, optical trapping, optical communications and quantum information processing, and the generation of and manipulation on an optical vortex beam has become one of the hot topics recently [86]. Several methods have been developed to generate and manipulate the optical vortex beam, for example, through linear optical elements, such as \(q\) plates, Dove prisms, mode converters by cylindrical lens, and computer-generated holograms with a forklike dislocation, or through nonlinear optical processes, such as four-wave mixing and spontaneous parametric down-conversion [82; 83; 84; 86]. Similar to the conventional holographic gratings, the EIT-based atomic coherence gratings can also record both the amplitude and the phase information of a light field, therefore, it is possible to record and recover special phase structures carried by the light field. [PERSON] et al. [41] reported the storage of an optical vortex based on the EIT effect in a vapor of Rb atoms, which proved to be topologically stable to atomic diffusion. More recently, the storage and retrieval of optical vortexes in the single-photon level was also demonstrated in atomic vapor [27; 28; 87]. We demonstrated the transfer of OTC and mode conversion of optical vortexes through EIT-based light pulse buffer memory in Pr:YSO crystal [43]. The experimental setup was similar to that in Figure 1 except that the two lenses, L\({}_{1}\) and L\({}_{2}\),were of the same focal length \(f\), and the Pr:YSO crystal was placed on the confocal plane of two lenses. The signal, \(E_{S}\), the coupling, \(E_{C}\), and the readout, \(E_{R}\), were able to carry OTC and were focused into the Pr:YSO crystal by lens L\({}_{1}\), and then the retrieved output signal pulses, \(E_{St}\), were mapped onto the iCCD to record the intensity profile. An additional plane wave was introduced to interfer with the retrieved output signal pulses to show the phase structure of the retrieved output signal pulses. The optical vortex carried by the signal pulse, \(E_{S}\), could be stored in Pr:YSO crystal based on the EIT effect and then retrieved with the help of a Gaussian coupling beam, \(E_{C}\), and a Gaussian readout beam, \(E_{R}\). Although the EIT-based storage and retrieval of the optical vortexes are temporally separated, the phase-matching condition should be satisfied, and the retrieved output signal pulses, \(E_{St}\), can be expressed as [43]  
\[|E_{S^{\prime}}|e^{i\phi_{S^{\prime}}}\propto\int|E_{S}E_{C}^{*}E_{R}|e^{i( \phi_{S}-\phi_{C}+\phi_{R})}e^{i\Delta\mathbf{k}\cdot\mathbf{r}}d\mathbf{r}^{ 3}\,, \tag{5}\]  
where \(\phi_{j}\,(j\,=\,S,\,C,\,R,\,S^{\prime})\) is the transverse profile of the phase of the corresponding beam.