Constant-Amplitude $2\pi$ Phase Modulation from Topological Pole--Zero Winding

Resonant phase shifters inevitably mix phase and amplitude. We present a topological synthesis that guarantees a full $2\pi$ phase swing at a prescribed constant scattering magnitude $|S_{ij}|=C$ by winding a scattering zero around the operating point in the complex-frequency plane while avoiding pole windings. We realize this either by complex-frequency waveform excitation on an iso-$|S_{ij}|$ (Apollonius) loop or by adiabatic co-modulation of detuning and decay at fixed carrier, suppressing AM--PM conversion and quantizing $\Delta\phi$ by the Argument Principle. The approach targets integrated resonant modulators, programmable photonic circuits, and quantum/beam-steering interferometers that require amplitude-flat phase shifts.