Edge Coupling of the Analytic Path Photon to a Conducting Slit: Full Derivation

This note derives, step by step, the electromagnetic interaction between the displacement- current spine of the Analytic Path (AP) photon and a conducting slit edge. The photon’s spine carries an effective line current Ieff = ω ε0cℏ/2, determined entirely by the photon energy ℏω. By the image-current method, the edge exerts an attractive transverse force on the spine, giving a soft coupling function f(d) that depends on the distance d between the spine and the edge. Two regimes emerge: a near-field regime (d ≪λ) where f(d) = λ/(4d), and a far-field regime (d ≫λ) where f(d) = λ2/(16πd2). For typical diffraction experiments (a ≫λ), most of the slit is in the far-field regime and the magnetic coupling is negligible compared to the diffraction scale λ/a; only photons within∼λ of the slit edge experience significant coupling. This is consistent with the main paper’s finding that the effective aperture function A(x) ≈1 across the slit: the edge coupling produces the evanescent modes that heal within R0/√n2 −1 nanometres of the edge, and the far-field sinc2 pattern emerges from the coherent Huygens integral over the essentially-uniform aperture.