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

We present a photon amplifier that operates as a port–parametric transformer with label-preserving out-coupling. The device combines three functional stages: a boundary port cavity that captures a weak optical seed, an intracavity gain stage that provides amplification through either parametric processes or population inversion, and a structured mode converter that emits the amplified field while preserving orbital angular momentum (OAM) and polarization labels. Using standard quantum input–output theory, we derive the conditions required for successful amplification, including gain thresholds, stability criteria for guided port modes, and optimal coupling near the critical balance between intrinsic and external cavity losses. We further analyze quantum noise limits for both phase-sensitive and phase-insensitive amplification. Phase-sensitive parametric gain can approach the quantum limit in the amplified quadrature, while phase-insensitive amplification obeys the universal Caves noise bound and is further degraded by coupling loss. We show that structural labels can be preserved through appropriate pump selection and cavity alignment, with polarization leakage scaling quadratically with misalignment angle and orbital angular momentum conserved under zero-OAM pumping. The theoretical framework maps directly to microcavity implementations using current photonic platforms such as lithium-niobate or silicon-nitride resonators. A practical design recipe and simulation workflow are provided to guide experimental realization. The resulting architecture provides a controlled platform for amplifying structured optical modes while maintaining high label fidelity, with applications in quantum communication, high-dimensional optical encoding, and integrated photonic systems.