Complex-Frequency Chirped Pulses for Trajectory-Resolved Scattering

Abstract

We introduce complex-frequency chirped pulses as finite-energy analytic waveforms designed to directly measure scattering responses along prescribed trajectories in the complex-frequency plane. Although poles and zeros of analytically continued scattering responses govern resonant photonic devices, experiments typically access only finite-band, real-frequency data. Our engineered waveforms establish an instantaneous complex frequency,ω~(t)=ωr(t)+iωi(t), where ωr(t) dictates the local phase slope and ωi(t) defines the logarithmic envelope-growth rate. A finite burst window contributes a deterministic imaginary-frequency correction. By calculating a local least-squares input–output ratio, we can extract the analytically continued response of stable, linear, time-invariant (LTI) devices along the measured trajectory. This extraction requires a sufficiently slow traversal, an adequate signal-to-noise ratio, and a trajectory well-separated from system poles in the resolvent sense. Rather than locating individual poles or zeros from a single burst, this technique supplies targeted off-axis samples to validate pole–zero models inferred from standard real-frequency or time-domain data. Furthermore, when tracing closed loops under quasi-steady conditions, the measurement yields a phase-winding diagnostic that directly evaluates the net number of enclosed zeros minus poles. We validate this extraction approach using temporal coupled-mode theory (TCMT) simulations, demonstrating the influence of pole proximity and traversal speed while distinguishing true analytic phase winding from finite-pulse tracking limitations.