Spatial sub-Rayleigh imaging via structured speckle illumination

In a lens-assisted imaging scheme with speckle illumination, the spatial resolution can surpass the Rayleigh resolution limit by a factor of $\sqrt{2}$ with second-order auto-correlation of light intensity. In this work, integrated with the nonlinear structured illumination after the speckle sinusoidally modulated, the second-order auto-correlation imaging can surpass the Rayleigh resolution limit by a factor of $2+\sqrt{2}$. In theory, a higher spatial resolution with the surpassing factor $N+\sqrt{N}$ is available by the $N$-order auto-correlation measurement. Our imaging scheme combined two super-resolution technologies not only enhances the spatial resolution of the lens-assisted imaging, but also promotes the practicality of the intensity correlation imaging.