## 4 Coherent Manipulation on Light Fields via Atomic Coherence Gratings
### First-Order Subwavelength Interference  
Subwavelength interference is capable of breaking through the classical Rayleigh spatial resolution limitation of a light field [88]; therefore, it has many potential applications, such as super-resolved optical lithography and precision measurement [89; 90; 91; 92]. There are various ways to achieve subwavelength interference, for example, by using the higher-order optical coherence of an entangled quantum NOON state [93; 94; 95] or phase-correlated classical light sources [96; 97; 98; 99; 100; 101]. For the quantum light sources, interfering fringes with a visibility up to 100% can be reached theoretically,  
Figure 6: Transfer of OTCs from the coupling beam, \(E_{C}\), to the retrieved output signal pulse, \(E_{St}\). Here, \(m_{S}=m_{C}=0\) and \(m_{R}=1\), and the OTC of the retrieved output signal pulse is \(m_{St}=-1\). The first row shows the intensity distribution of each beam, while the second row shows the interference fringe of each beam with a plane wave (not shown in Figure 1). The forklike interference fringe indicates the phase singularity of the respective beam. Note that the forklike fringe of the retrieved output signal pulse points in opposite direction with respect to that of the coupling vortex, indicating that the sign of OTC is flipped. Figure partially adapted from reference [43]. Figure 7: Mode conversion of the optical vortexes achieved through the EIT-based light pulse storage and retrieval process. Here, \(m_{S}=m_{R}=1\) and \(m_{C}=0\), and the OTC of the retrieved output signal pulse is \(m_{St}=2\), in good accordance with Equation (6). The first row shows the intensity distributions of the respective interacting beams, and the second row depicts the interference fringes of the corresponding beams with the plane wave. The forklike interference fringes indicate the OTCs carried by each optical vortex beam. Figure partially adapted from reference [43]. whereas for practical applications, such as quantum lithography, the main challenge is the conflicting requirements on the light field strength--it should be strong enough for efficient multi-photon nonlinearity but weak enough to maintain the quantum features [92]. For classical light sources, the fringe visibility of subwavelength interference is usually relatively low in comparison with that of quantum light sources. In general, efficient \(n\)-photon nonlinear optical processes and recording materials are required to achieve a subwavelength spatial resolution with an effective wavelength of \(\lambda/n\), which is extremely challenging in practice. Therefore, in view of practical applications, high visibility first-order subwavelength interference that corresponds to the one-photon absorption process and is linear in light intensity is extremely in demand [102]. We demonstrated that first-order subwavelength interference with high visibility through a spatial frequency doubling process can be achieved using the light pulse storage and retrieval technique based on the EIT effect in Pr:YSO crystal [103]. As shown in Figure 1, following an EIT-based light pulse storage and retrieval process, the retrieved signal pulse field, \(E_{S^{\prime}}(x,y)\), can be written as  
\[E_{S^{\prime}}(x,y)\propto E_{S}(x,y)E_{\mathcal{C}}^{\star}(x,y)E_{R}(x,y)\,, \tag{7}\]  
under the paraxial approximation. By inserting a double-slit mask, O, with a slit separation distance, \(d\), and a slit width, \(a\) (see Figure 8a), on the front focal plane of lens L1, the interference fringes or the spatial frequency spectrum of the double-slit O, \(E_{S}(x)\propto\cos(\pi dx/\lambda f_{1})\text{sinc}(\pi ax/\lambda f_{1})\), are stored in Pr:YSO crystal placed on the rear focal plane of lens L1 based on the EIT effect by adiabatically switching off the coupling beam. The same interfering fringes can be retrieved when one launches a plane wave or a Gaussian readout beam under the phase-matching condition.