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

Abstract

Resonant phase shifters inevitably mix phase and amplitude. We present a topological synthesis that guarantees a full $2π$ 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 $Δφ$ by the Argument Principle. The approach targets integrated resonant modulators, programmable photonic circuits, and quantum/beam-steering interferometers that require amplitude-flat phase shifts.

Figures (4)

• Figure 1: Concept of topological phase control in the complex-frequency plane. A resonant response landscape is characterized by a pole (peak) and a zero (dip). A closed trajectory constrained to an iso-$|S_{ij}|$ manifold and enclosing a net topological charge produces a quantized $2\pi$ phase accumulation while enforcing constant amplitude. Protocol (i) uses a waveform-synthesized complex-frequency excitation to trace an iso-amplitude loop around a static zero; Protocol (ii) co-modulates device parameters to move the zero around a fixed real drive while maintaining $|S_{ij}|=C$.
• Figure 2: Complex-frequency maps of the reflection coefficient $r(\tilde{f})$ for an illustrative parameter set $f_0\simeq 193.41~\mathrm{THz}$ ($\lambda_0\simeq 1.55~\mu\mathrm{m}$), $\Gamma_0=0.2~\mathrm{THz}$, and $\Gamma_c=0.5~\mathrm{THz}$ (overcoupled). (a) Amplitude $|r(\tilde{f})|$ with the zero $\tilde{f}_z=f_0+\mathrm{i}(\Gamma_0-\Gamma_c)\simeq 193.41-\mathrm{i} 0.3~\mathrm{THz}$ and pole $\tilde{f}_p=f_0+\mathrm{i}(\Gamma_0+\Gamma_c)\simeq 193.41+\mathrm{i} 0.7~\mathrm{THz}$ marked; white curves show representative iso-$|r|$ (Apollonius) contours used for constant-amplitude phase winding. (b) Phase $\phi(\tilde{f})$ exhibiting $+2\pi$ winding around the zero and $-2\pi$ winding around the pole.
• Figure 3: Approach 1: phase winding at constant amplitude by steering the excitation along a complex-frequency iso-$|r|$ contour. (a) Real-axis sweep ($f_I=0$) exhibits an incomplete phase excursion and significant amplitude variation. (b) A synthesized complex-frequency drive $\tilde{f}_{\mathrm{exc}}(t)$ traverses an Apollonius contour ($|r|=0.3$) enclosing $\tilde{f}_z$ (excluding $\tilde{f}_p$, inset), yielding a $2\pi$ unwrapped phase shift while maintaining constant $|r|$ (within numerical tolerance).
• Figure 4: Approach 2: phase winding at constant amplitude with a fixed real excitation. (a) Trajectories of the dynamically tuned zero $\tilde{f}_z(t)$ (solid) and pole $\tilde{f}_p(t)$ (dashed) in the complex plane; the fixed excitation $f_{\mathrm{exc}}$ is enclosed by the zero’s path but not the pole’s. (b) At $f_{\mathrm{exc}}$, the unwrapped phase accumulates $2\pi$ over one modulation cycle while $|r(f_{\mathrm{exc}},t)|$ is held at the target $C_d$ (within numerical tolerance).