# A Review of Super-Resolution Imaging through Optical High-Order Interference [Invited]
## 6 Super-Resolution via Quantum Entanglement
Instead of employing a lens for imaging, the super-resolution imaging process is assisted by a dynamic random scattering layer, although the random scattering is well known for destroying imaging process in conventional single-photon imaging systems. By taking two-photon coincidence measurement with the totally scrambled light through the scattering layer, a direct imaging of the object can be obtained on the image plane, and the imaging resolution can reach the HL as shown in Figure 8 III. By analyzing the superposition of two-photon paths [27; 60; 61], it is found that constructive two-photon interference for imaging is employed naturally with the dynamic scattering layer, as shown in Figure 8 II. In contrast to the synchronous-position Heisenberg-resolution imaging expressed in Equations (5) and (6), the Hanbury Brown-Twiss type two-photon interference leads to an imaging term expressed as  
\[H_{hbit}^{(2)}(\Delta\mathbf{x^{\prime}}_{21})\propto\sum_{\eta=\pm 1}\int|O( \mathbf{x})|^{2}\cdot[\text{PSF}_{2}(\eta\cdot\Delta\mathbf{x^{\prime}}_{21}/2, \mathbf{x})]^{2}\text{d}\mathbf{x}, \tag{7}\]  
where \(\Delta\mathbf{x^{\prime}}_{21}=\mathbf{x^{\prime}}_{2}-\mathbf{x^{\prime}}_{1}\) with \(\mathbf{x^{\prime}}_{1}\) (\(\mathbf{x^{\prime}}_{2}\)) being the coordinate on the image plane, and \(\text{PSF}_{2}(\eta\cdot\Delta\mathbf{x}_{21}/2,\mathbf{x})\) is the two-fold enhanced point-spread function (PSF). The two possible values of \(\eta\) (=\(\pm 1\)) is related to exchanging the positions of the two detectors (D1 and D2), which does not change the super-resolution imaging result in scheme. Figure 7: **(I)**: Schematic of experimental setup for realizing super-resolution imaging with entangled two-photon source illumination. The entangled two-photon source is generated by optical parametric down conversion with the betabarium-borate (BBO) crystal. **(II)**: Super-resolution images **(right column)** in comparison with the conventional intensity images **(left column)**. The object to be imaged is marked by the red lines in the figure. The figure is reproduced with permissions from Ref. [39] of AIP. Recently, [PERSON] et al. have reported a synchronous-position Heisenberg-resolution imaging in a lens-assisted imaging scheme [41]. The object is first illuminated by a continuous-wave pump laser at 405 nm, such that its information is encoded into the complex amplitude of pump light as shown in Figure 9a. The transmitted pump light is then projected to the Fourier plane via a lens \(L_{1}\), where the pump light is down-converted to the entangled two-photon light through SPDC process. As a result, the Fourier image of the object encoded in the complex amplitude of pump light is transferred to the two-photon complex amplitude. Consequently, the image of the object can be obtained by employing two-photon coincidence measurement on the Fourier plane of the SPDC source [17], and the imaging resolution breaks the Rayleigh limit related to the SPDC photons by a factor of 2. In the experiment, the Fourier plane of the SPDC source is in fact imaged to the detection plane with an additional imaging lens \(L_{3}\), and the numerical aperture is limited by the aperture size of the lens \(L_{3}\). Nevertheless, the HL imaging resolution has been demonstrated in the scheme as shown in Figure 9b. The result has been explained by considering the optical centroid measurement state [62; 63]. Notably, the super-resolution imaging here is related to the down-converted photons, but not to the directly illuminating light photons transmitted through the object. From this respect, it is different from the Heisenberg-resolution imaging discussed by [PERSON] et al. [31], which is related to the light directly illuminating the object. Figure 8: **(I)**: Schematic of the super-resolution imaging scheme assisted by a dynamic scattering layer. The object is placed on the top half of the Fourier plane of the output plane of the entangled source, while the bottom half of the Fourier plane is kept transparent.