Cavity quality factor and loss channels

For a weakly damped mode, define quality factor as angular frequency times stored energy divided by cycle-averaged power leaving that mode. The energy includes both electric and magnetic contributions. At resonance these exchange through the cycle; using only average electric energy while comparing it with total loss power misses half the total storage. In a lossless ideal cavity Q would be unbounded, but a real cavity has wall currents, dielectric loss, radiation leakage, and measurement coupling.

The energy-over-power definition gives a decay timescale when the loss power is proportional to the instantaneous slowly varying modal energy. This assumes a linear mode with approximately fixed spatial shape and decay rates that do not change appreciably over the narrow resonance. Strong nonlinearity, heating, mode mixing, or frequency dependent baths can make one constant Q inadequate. The quality factor is consequently a property of a specified mode under stated conditions, not a universal material constant attached to an enclosure.

Suppose each independent loss channel removes power proportional to the same stored energy. Their powers add, so their energy decay rates add and their inverse quality factors add. Intrinsic or unloaded Q includes losses within the cavity system, such as conductor and dielectric dissipation, while external Q quantifies escape through a selected port. Loaded Q includes both. A low external Q means strong coupling, which shortens the measured lifetime even though it need not damage the intrinsic cavity.

For a single ideal measurement port, critical coupling means that external loss equals intrinsic loss. At the resonance frequency, interference between the promptly reflected drive and the cavity-radiated field can then cancel the net reflection. That cancellation is not evidence of zero internal loss: in the simple one-port model all accepted steady-state power is dissipated internally. Multiple ports and direct transmission paths change how critical coupling appears, so one should not apply a one-port reflection criterion unmodified to every transmission notch.

After the drive is switched off, modal energy decays with rate kappa while field amplitude decays with half that rate. An energy detector therefore gives a lifetime half the amplitude-envelope lifetime. For an isolated resonance with slowly varying coupling and a Lorentzian stored-energy response, the full width at half maximum in angular frequency equals the energy decay rate. Dividing frequency by full width gives the loaded Q when ordinary frequencies are used consistently.

A notch or peak measured through a microwave network can sit on a complex background and have interference asymmetry. Reading two arbitrary half-depth points from that trace need not recover the intrinsic linewidth. Use a response model and calibration consistent with the geometry, or use an independently measured ringdown in its valid regime. Separating loaded and intrinsic Q additionally requires knowledge of external coupling. A narrower line alone does not prove a better dielectric, since reducing port coupling also narrows the loaded response.