By solving the Maxwell's equations in Fourier space, we find that the cross-polarized component of the dipole scattering field can be written as the second-order spatial differentiation of the copolarized component. This differential operation can be regarded as intrinsic which naturally arises as consequence of the transversality of electromagnetic fields. By introducing the intrinsic spatial differentiation into heralded single-photon microscopy imaging technique, it makes the structure of pure-phase object clearly visible at low photon level, avoiding any biophysical damages to living cells. Based on the polarization entanglement, the switch between dark-field imaging and bright-field imaging is remotely controlled in the heralding arm. This research enriches both fields of optical analog computing and quantum microscopy, opening a promising route toward a nondestructive imaging of living biological systems.