# A Review of Super-Resolution Imaging through Optical High-Order Interference [Invited]
## 3 Super-Resolution with Scanning Focused Beam  
In 2009, [PERSON] et al. proposed that n-photon coincidence can be employed for realizing sub-Rayleigh quantum imaging in lens-assisted imaging schemes [31], giving rise to the standard quantum limit (SQL) resolution with an enhancement factor \(\sqrt{n}\). Nevertheless, direct n-photon detection within the conventional lens-assisted imaging scheme does not guarantee the realization of SQL super-resolution imaging. The key is to employ proper illumination, such that the nth-order cumulant function for super-resolution is obtained. The nth-order cumulant function is the filtered/efficient nth-order correlation function with the contributions related to photons from different point sources of the object being eliminated [33]. To achieve this goal, entangled n-photon illumination and scanning focused-beam illumination are introduced, leading to coherent SQL super-resolution imaging  
\[C_{coh}^{(n)}(\mathbf{x}^{\prime})\,\propto\,\Big{|}\int[O(\mathbf{x})]^{n} \cdot[\mathrm{PSF}(\mathbf{x}^{\prime},\mathbf{x})]^{n}\,\mathrm{d}\mathbf{x} \Big{|}^{2}, \tag{3}\]  
Figure 1: (**a**) Schematic of lens-assisted imaging system. The divergence of light wave originating from a point of the object is reversed by the lens, converging to a point on the image plane; (**b**) The point-to-point correlation between the object and the image plane is not ideal. The image spot is of finite size that is determined by the wavelength of the light wave \(\lambda\) and the numerical aperture of the imaging lens \(n\sin(\theta/2)\) (refractive index \(n=1\) for air). and incoherent SQL super-resolution imaging  
\[C_{inc}^{(n)}(\mathbf{x}^{\prime})\propto\int|O(\mathbf{x})|^{2n}\cdot[\text{ PSF}(\mathbf{x}^{\prime},\mathbf{x})]^{2n}\,\mathrm{d}\mathbf{x}, \tag{4}\]  
respectively. Here, \(O(\mathbf{x})\) is the object aperture function, \(\text{PSF}(\mathbf{x}^{\prime},\mathbf{x})\) is the point-spread function in an imaging scheme. We see that the effective PSF of the nth-order cumulant function is nth power of the original PSF of conventional intensity imaging, leading to an enhancement factor of \(\sqrt{n}\) in the spatial resolution. Please note that with these two kinds of illumination mentioned above, there is no contributions to the nth-order correlation function that is related to n-photon paths originating from different positions of the object, i.e., the cross talk between different positions of the object is prohibited [42]. Therefore, the nth-order correlation function is exactly the nth-order cumulant function. From the result, we see that the SQL super-resolution is a result of constructive interference of n-photon paths when the n photons are from the same position of the object. Figure 2: **(I)**: Schematic of the experimental setup for realizing SQL super-resolution imaging with scanning-focused-beam illumination. **(II)**: Super-resolution image realized in the experiment for a scheme with Rayleigh resolution limit 1.86 mm. The triple-slit object with slit width 125 \(\upmu\)m is marked by the red arrow (**a**). The intensity image realized with conventional lens-assisted imaging scheme is presented in **(b)**, and the super-resolution image realized with the n-photon detection (\(n=23\)) is shown in **(c)** and **(d)**, where **(d)** is the three-dimensional image of **(c)** by clipping the event counts at 800. The figure is reproduced with permissions from Ref. [32] of APS. With the scanning-focused laser beam illumination, [PERSON] et al. have demonstrated the sub-Rayleigh imaging effect by taking n-photon detection [32]. In the experiment, a 532 nm laser mounted on an xy translation stage was used for providing scanning focused-beam illumination as shown in Figure 2I. The \(N\)-photon detection is done with a detector consisting of 32 \(\times\) 32 single-photon avalanche diode (SPAD) array [46].