# A Review of Super-Resolution Imaging through Optical High-Order Interference [Invited]
## 2 Rayleigh Resolution Limit  
Lens-assisted imaging is the most widely used imaging scheme. A thin lens can be viewed as a thin static phase screen. It introduces a phase pattern on the wavefront that reverses the divergence of light wave originating from an object point, making light wave converge to a point on the observation plane as shown in Figure 1a. In this way, a point-to-point correlation between the object and the observation plane is built up, giving rise to direct imaging for the object. The point-to-point correlation is expressed as [45]  
\[I_{coh}(\mathbf{x}^{\prime})\,\propto\,\Big{|}\int O(\mathbf{x})\cdot\mathrm{ PSF}(\mathbf{x}^{\prime},\mathbf{x})\,\mathrm{d}\mathbf{x}\Big{|}^{2} \tag{1}\]  
for coherent imaging, and  
\[I_{inc}(\mathbf{x}^{\prime})\,\propto\,\int|O(\mathbf{x})|^{2}\cdot[\mathrm{ PSF}(\mathbf{x}^{\prime},\mathbf{x})]^{2}\,\mathrm{d}\mathbf{x} \tag{2}\]  
for incoherent imaging. Here, \(O(\mathbf{x})\) is the object aperture function, \(\mathrm{PSF}(\mathbf{x}^{\prime},\mathbf{x})\) is the point-spread function of an imaging scheme.  
However, due to the wave nature of light, the imaging process is not ideal, but of limited resolution determined by the wavelength \(\lambda\) and the numerical aperture \(NA=n\sin(\theta/2)\), i.e., the well-known Rayleigh limit \(\delta x_{R}=0.61\lambda/NA\)[45]. Here, \(n\) is the refractive index of the environment dielectric material (\(n=1\) for air), and \(\theta\) is the maximum cross angle for the diverging wave originating from the object point as shown in Figure 1a. Please note that the numerical aperture is expressed as \(NA=D/(2L)\) in the paraxial approximation, where \(D\) is the diameter of the lens, and L is the distance from the object to the lens. Within the Rayleigh Criterion, better imaging resolution is achieved by employing light wave of shorter wavelength and scheme of larger numerical aperture.  
The emergence of Rayleigh resolution limit can be understood by considering superposition of indistinguishable single-photon paths. The imaging process is in fact a result of constructive interference between alternative single-photon paths from the object point to the image point, since the phase differences caused by propagation with a single photon transmitting through different positions of the lens are compensated for by the static phase pattern of the lens. By scanning the lateral position on the image plane, the phase compensation process is gradually destroyed until complete destructive interference occurs. This leads to an image spot formed on the observation plane for an object point as illustrated in Figure 1b, and any two adjacent image spots with a separation smaller than the size of the image spot are thus indistinguishable. For a quantitative estimation on the resolution limit, we consider the one-dimensional (1D) geometry in the paraxial approximation for simplicity, in which the complete destructive interference occurs when the phase mismatch \(\delta\phi\) between the two single-photon paths shown in Figure 1b (solid lines) reaches \(2\pi\). With a position shift \(\delta x^{\prime}\) on the image plane, \(\delta\phi=k_{0}D\delta x^{\prime}/L^{\prime}\) with \(k_{0}\) being the angular wavenumber of light while \(L^{\prime}\) being the distance from the lens to the image plane, giving rise to an image spot of size \(\delta x^{\prime}=\lambda L^{\prime}/D\). Therefore, two adjacent image spots of separation \(\delta x^{\prime}\) is not separable. By considering the magnification factor \(m=L^{\prime}/L\), the imaging resolution limit for the object is therefore derived as \(\delta x_{R}=\lambda L/D=0.5\lambda/NA\) in the 1D case. The Rayleigh resolution limit in the 2D case can be explained with the same argument.  
Next, we will review several recently developed schemes that surpass the Rayleigh resolution limit based on optical high-order interference effect.