K1: photon Amplifier Port–Parametric Transformer with Label-Preserving Out-Coupling

We introduce a photon amplifier that functions as a port–parametric transformerwith a label-preserving out-coupler. The architecture consists of three stages: (i)a boundary port cavity that phase-matches a weak photon seed to an intracavitymode, (ii) an internal gain stage that amplifies the seed either through phase-sensitiveparametric processes or population inversion, and (iii) a mode converter and out-couplerthat emit a bright photon or beam while conserving orbital angular momentum andpolarization. We derive the input–output equations, net gain and threshold conditions,and selection rules for label preservation, and outline an experimental implementationin a microcavity platform. Quantum-noise limits are analyzed for both phase-sensitiveand phase-insensitive amplification. This design provides a practical approach for label-preserving optical amplification, relevant to quantum communication, high-dimensionalencoding, and integrated photonics.