Source and Receiver in a Modulated Medium: The Half-Period Shift in Green's-Function Reciprocity

@misc{mark2026source,
  title = {{Source and Receiver in a Modulated Medium: The Half-Period Shift in Green's-Function Reciprocity}},
  author = {László Márk},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = sep,
  doi = {10.5281/zenodo.22981902},
  url = {https://doi.org/10.5281/zenodo.22981902},
  note = {CC BY},
}

TY  - UNPB
TI  - Source and Receiver in a Modulated Medium: The Half-Period Shift in Green's-Function Reciprocity
AU  - László Márk
PY  - 2026
DA  - 2026/09/26
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.22981902
UR  - https://doi.org/10.5281/zenodo.22981902
AB  - In a medium that does not change in time, a source and a receiver can be exchanged when the constitutive tensors obey Lorentz reciprocity, and the adjoint medium is the natural object of that statement. A medium modulated in space and in time breaks this picture: the response depends on when the source fires, and the exchange of source and receiver is no longer an exchange of positions alone. This article states what replaces it. For any linear medium with a real, causal kernel — anisotropic, gyrotropic, bianisotropic, dispersive, lossy or active, spatially nonlocal and time-varying — the Green function equals the transposed Green function of the time-reversed adjoint medium. That medium is Kong’s adjoint medium with the time arguments of the kernel reflected, and the map keeps loss as loss and a causal kernel causal. When the time-reversed adjoint is the original medium shifted by half a spatial period and reflected in time, source and receiver can be exchanged, but only together with the half-period shift and the reflection of time. The operation is the family operation of the earlier records, whose quotient of space–time is a Klein bottle; it acts on the Green function by transposition, not by complex conjugation. A parity rule follows: the parts of the constitutive relation that are reciprocal in a static medium — symmetric permittivity and permeability, omega and chiral couplings — must be even under the family operation, and the parts that break static reciprocity — gyrotropic, Tellegen and moving-medium couplings — must be odd. A magnetized medium with alternating domains, or with a bias odd in time, therefore obeys the exchange and has symmetric bands without any reversal of the external bias. In a homogeneous, lossless medium the rule reduces to the known classification of temporal reciprocity, which the theorem reproduces; for a uniformly modulated omega medium it locates the reported nonreciprocity in the exchange at a fixed pump phase. Extended to rotations and reflections of the transverse plane, the result contains, as its time-invariant corner, the classical reciprocity theorem for orthogonal mappings of the adjoint medium. The theorem carries over to acoustic and elastic waves, where the even and the odd Willis coupling obey the same parity rule and a loss modulated in time breaks the relation. For the scattering matrix, a finite slab is reciprocal not with itself but with its twin, the same medium cut half a period away: a side that does not reflect is inherited by the twin, with loss as well, and the converted channels carry a phase fixed by the symmetry time. In the zone-edge sector, where the family operation squares to minus one, a port cannot reflect into its glide partner, a receiver at the glide partner of a source, timed symmetrically, is blind to the co-polarized field, and every Floquet multiplier is doubled, with loss and with gain. Time-reversal focusing is exact when each recording is re-emitted half a period away and reversed about the symmetry time; the field then refocuses half a period from the source with the recorded energy. Two effects often read as consequences of reciprocity have a different origin. Green-function retrieval by correlation, and the pairing of coherent perfect absorption with lasing, rest on the symplectic form of a lossless medium: they hold in lossless space–time modulated media, reciprocal or not, and fail with loss, a family-symmetric medium included. The family operation governs the transposing relations, with loss and with gain; the symplectic form governs retrieval and the absorber–laser pair. Every statement is measured with deterministic scripts deposited with this record, each with the inputs on which it must fail. Beyond periodic modulation, a disordered realization and its family image are twins, and an ensemble whose statistics are invariant under the family group is reciprocal in the glide sense on average, while a drift leaves a residue. A switched modulation keeps the relation when its envelope is symmetric about a symmetry time of the carrier, and a sequence of reciprocal steps is reciprocal when it reads the same backwards. A modulated two-port whose harmonics share no symmetry time isolates even without loss, the reverse wave being diverted rather than absorbed; the isolation is odd in the phase of the second pump harmonic and first order in its amplitude, so that one phase sets its size and direction. In a switched antenna array the receive pattern of the reverse conversion is the transmit pattern mirrored unless the pulses share a centre. These results give design rules and refutable predictions for magnet-free circulators, time-division MIMO, wavelength-division links, radar, magnetic resonance and laser spectroscopy.
ER  -