Surface equivalence principle

Remarks

Proof idea: define a new field equal to $(\mathbf{E}, \mathbf{H})$ outside $V$ and to $0$ inside $V$. It satisfies the homogeneous Maxwell equations on both sides of $S$. Its jumps across $S$ are exactly those that the surface currents $\mathbf{J}_s$ and $\mathbf{M}_s$ produce in the interface conditions, with Maxwell's equations extended by a magnetic current. It is also radiating, so by uniqueness of radiating solutions it is the field of these currents. The principle is often attributed to A. E. H. Love and to S. A. Schelkunoff. Other choices of the interior field give other equivalent problems. For example, one may fill $V$ with a perfect electric conductor and keep only a magnetic current. The price is that the currents no longer radiate in a homogeneous space. Example: for an aperture antenna, one takes $S$ to be a plane just in front of the aperture and estimates $\mathbf{E}$ and $\mathbf{H}$ there. The radiated field then follows from free-space integrals of the surface currents. The homogeneity of the outside medium matters only for computing the radiation of the currents with the free-space Green's function; the jump argument itself is local.

Going deeper

For a student

The principle turns a radiation problem with complicated sources into an integral over a surface, which is the starting point of the Stratton–Chu formulas and of boundary integral equations. In antenna engineering it is used to compute far fields from measured or estimated near fields on a surface. In numerical methods it underlies surface integral equations for scattering and the near-to-far-field transformation used after a finite-difference time-domain computation.