Analytic representation of the probe-induced field

Derive an analytic expression for the field component V_δ generated by the coated probe in the coupled sample–probe system, rather than relying on the formal series representation used in the proof of the cloaking result.

Background

The active-cloaking analysis decomposes the probe-related field into V_δ, generated directly by the core-shell probe, and T_δ, representing the interaction between the probe and the sample. The proof of Theorem 6 establishes that both terms vanish outside the critical radius as the loss parameter δ tends to zero, thereby showing that the probe becomes effectively invisible.

Although the authors derive a formal Fourier-series representation for V_δ, its coefficients depend on the coupled density Φ and therefore incorporate sample–probe interaction information. Consequently, the representation is not an explicit analytic formula. Obtaining such a formula would clarify the field structure underlying the cloaking mechanism and potentially improve rigorous and computational analyses of more general probe configurations.