  Near-field microscopy is widely used for characterizing electromagnetic
fields at nanoscale, where nanoprobes afford the opportunity to extract
subwavelength optical quantities, including the amplitude, phase, polarization
and chirality. However, owing to the complexity of various nanoprobes, a
general and intuitive theory is highly needed to assess the vectorial field
response of the nanoprobes and interpret the mechanism of the probe-field
interaction. Here, we develop a general imaging theory based on the reciprocity
of electromagnetism and multipole expansion analysis. The proposed theory
closely resembles the multipolar Hamiltonian for light-matter interaction
energy, revealing the coupling mechanism of the probe-field interaction. Based
on this theory, we introduce a new paradigm for the design of functional
nanoprobes by analyzing the reciprocal dipole moments, and establish effective
design principles for the imaging of vectorial near fields. Moreover, we
numerically analyze the responses of two typical probes, which can
quantitatively reproduce and well explain the experimental results of
previously reported measurements of optical magnetism and transverse spin
angular momentum. Our work provides a powerful tool for the design and analysis
of new functional probes that may enable the probing of various physical
quantities of the vectorial near field.
