Reciprocity (electromagnetism) - Conditions and Proof of Lorentz Reciprocity

Conditions and Proof of Lorentz Reciprocity

The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator relating and at a fixed frequency (in linear media):

is usually a Hermitian operator under the inner product for vector fields and . (Technically, this unconjugated form is not a true inner product because it is not real-valued for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the permittivity ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices (symmetric rank-2 tensors) — this includes the common case where they are scalars (for isotropic media), of course. They need not be real—complex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via )—and because of this the reciprocity theorem does not require time reversal invariance. The condition of symmetric ε and μ matrices is almost always satisfied; see below for an exception.

For any Hermitian operator under an inner product, we have by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator : that is, . The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields and, integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

This identity is then applied twice to to yield plus the surface term, giving the Lorentz reciprocity relation.