Theory of Single Susceptibility for Near-field Optics Equally Associated with Scalar and Vector Potentials
A nonlocal response theory was developed to describe a many-electron system within the neighborhood of a nanostructure radiating the longitudinal and transverse electric fields, which are fundamentally reduced to the scalar and vector potentials (SP and VP). The coexistence of the SP and VP incidences distinguishes such a near-field optical system from the ordinary optical system, in which only the VP (under the Coulomb gauge) incidence survives far from the light source. This fact is the motivation for equal treatment of the SP and VP as the cause of the response in the near-field optical system. In the semiclassical treatment, the linear and nonlinear single susceptibilities are derived in the form of Heisenberg operators by the functional derivatives of the action integral of the matter with respect to the SP and VP. These single susceptibilities relate the SP and VP (as the cause) to the induced charge and current densities (as the result), and guarantee charge conservation and gauge invariance; this theory is free from gauge-fixing. It is necessary to consider the quantum many-electron effect (exchange-correlation effect) to make the ground state bounded in the non-perturbed system. This is done by employing the fundamental idea of density functional theory, instead of the ordinary unequal treatment of the SP and VP, that is, remaking the SP into a Coulomb interaction between electron charges. Applying the present linear response theory to the non-metallic material in a limited near-field optical system reveals that the electric field with the associated permittivity is not suitable quantity to describe the response, instead, the SP and VP with associate single susceptibility are essential.