The adjoint medium beyond locality – how far the reciprocity criterion reaches

A medium modulated in space and time can be built to pass a wave one way and block it the other. The opposite question is asked less often: what forbids it. For a medium described by a real scalar permittivity one answer is known — if the modulation is unchanged by a shift of half a cell in space combined with a reversal of time, the band structure is symmetric in the direction of propagation. For a medium whose constitutive relation is tensorial but local, that symmetry does not preserve the symmetry of the bands directly; it ties the medium to its adjoint, and the bands are symmetric exactly when the medium is its own adjoint. Both results assume that the medium responds to the field at a point. Real media respond to the field in a neighbourhood, and the resulting dependence on the wavevector is not a small correction: modulating a scalar permittivity in time produces an effective medium whose dominant dispersion is spatial rather than temporal. A criterion that assumes locality is therefore narrower than it appears. We show that the criterion survives. The argument rests on a parity rule: under the reversal of the harmonic indices, a dependence on the wavevector carries the sign of its parity alongside the transpose — an even dependence one way, an odd one the other. The rule follows from two exact identities and holds for any medium; it is an identity rather than a restriction, and on its own it forbids nothing. What it supplies is the recipe for building the adjoint of a medium whose response depends on the wavevector: by parity, with the corresponding sign. With that construction the band criterion holds unchanged, and for a local response it reduces to the known one. Expanding the dependence in powers of the wavevector is how the rule is proved, and is not a condition on the medium. A rational dependence with a pole is even and obeys the rule exactly, although no finite combination of orders reproduces it and the expansion does not converge near that pole — and that is the shape the effective medium takes where the spatial dispersion is strongest. Nor is a definite parity required: every dependence splits uniquely into an even and an odd part, and the adjoint is built from the two with their own signs. Giving the whole dependence a single sign is what fails, and that is a mistake in the construction rather than a property of the medium. The parity rule makes one case odd, and it is the case for which the answer is hardest to decide in advance: the magnetoelectric coupling described by an antisymmetric dyadic, for which nonreciprocity under uniform temporal modulation has been reported. That coupling separates the two polarisations into independent problems, each of which is chiral and therefore its own adjoint, so the criterion covers it. The separation is a property of the coupling rather than of the calculation, and is measured as such before anything is concluded from it; it survives a transverse wavevector, the independent problems growing from two components to three, provided the longitudinal components are carried rather than eliminated. Eliminating them replaces the permeability by a quantity involving the frequency operator, which belongs to the reduction and not to the medium, and to which the condition on the modulation does not apply. None of this needs the medium to be periodic in a single direction. On a lattice periodic in any number of directions the closure of the family operation admits one offset for each non-empty subset of the lattice generators, and the criterion holds at every one of them. The criterion is tested there against a coupling that is not its own adjoint, and the test has to be made in two dimensions or more: in one dimension that control happens not to discriminate, and a reader who ran it only there would be left believing the criterion was carrying no weight. Reciprocity here means the symmetry of the band structure of an unbounded medium. A second criterion, the symmetry of the response to sources, is not equivalent to it in a medium that does not conserve energy, and the reported nonreciprocity of the modulated coupling above is of that second kind. The two concern different quantities; which one is meant is fixed at the outset rather than left to be inferred. Numerical verification accompanies every claim, each measuring test paired with an input on which the claim must fail.