Predictions from the Sector Framework for Space-Time Modulated Media

A medium whose properties are modulated in space and time can be built to pass a wave one way and block it the other, and the conditions that forbid this are known for a scalar permittivity, for a tensorial one, and for a response that depends on the wavevector. This article asks what those conditions say to someone designing a device. It first states, and where possible proves, what the framework implies for media that are lossy, dispersive and driven. In a medium with real material functions, the growth rate at a fixed quasimomentum can never depend on direction, so a directional amplifier must be designed from the spatial gain instead. A loss modulated together with the permittivity is harmless whenever the difference between it and a constant multiple of the permittivity is static and family-symmetric — in particular when the relaxation rate is constant; this gives an explicit sufficient criterion for a co-modulated loss and, when the permeability is modulated too, a space-time form of the distortionless-line condition. The residual left by the time-dependent remainder is linear in it. A symmetry combining a fraction of the period with a transformation of the field fixes which sectors can carry a fractional frequency offset. The criterion is sufficient, not necessary. A modulation even under spatial inversion stays reciprocal whatever its defects are; in a structure carrying several modes the involution that forbids a directional spectrum acts on the internal space as well, so a mirror-symmetric device can still be nonreciprocal; and an involution that reverses the frequency along with the wavevector forbids nothing. The glide symmetry of the branch labels and the fractional-period screw of the sector rule are shown to be cases of one construction, whose selection rule has a tolerance linear in the symmetry defect. A symmetry also decides where an exceptional point may form: degeneracies between sectors are protected rather than exceptional, and a ring of n one-way coupled elements fixes the order of one. The same reading gives further results. A plasma frequency, a resonance and a damping driven with one waveform at a constant mass keep reciprocity, locally and nonlocally, which covers an actively pumped medium. In one dimension a reciprocal medium, such as a family-symmetric one with a harmless loss, excludes a unidirectional skin effect; in two dimensions, at the zone edge, the family symmetry squares to −1 and protects a Z2 skin effect. A ring circulator obeys a screw relation whose forbidden sectors fill linearly in a phase error, and 2 elements in antiphase cannot isolate. Phase-sensitive gain lives only in the self-conjugate sectors, which is why the exceptional point of a photonic time crystal is phase-sensitive. The transition of a plasmonic photonic time crystal is located as a Jordan block of the half-integer sector, and its loss sets only the lasing threshold. A modulation that only changes the sector reads a trapped-mode resonance without the bright background. The article then turns these into predictions for devices, each with the chain it rests on, the quantity it asserts, and the measurement that would refute it, on two platforms: circuits with lumped elements, and materials in which the modulation acts on free carriers. Among them is a one-knob test of the family condition in the scattering matrix. A dictionary with a status column records what is proved, what is only measured, and what was tested and does not hold. Every number this article measures resolves to a test key in a deposited verification script; numbers taken from the literature or from a companion record are attributed where they appear. The derivations draw on a spectral analysis written for non-orientable quotients, and this article is its first application to modulated media: the space-time cell of an admissible medium is a Klein bottle, and the sector structure that follows carries the results on the dual criterion, on admissible media, on the sector rule, on exceptional points, on skin effects and on sensor readout. Results that follow from that reading are marked as such.