Trajectory probing of complex-frequency scattering with chirped analytic pulses

Abstract

Characterizing resonant scatterers is challenging because their poles and zeros usually lie away from the real-frequency axis, whereas most measurements sample only real frequencies and infer off-axis behavior from fitted models. Here we introduce complex-frequency chirped pulses: finite-energy analytic waveforms that probe a device continuously along a prescribed contour in the complex-frequency plane. We give a direct synthesis rule for an in-phase/quadrature (I/Q) waveform and show that finite-duration windowing deterministically distorts the realized trajectory, which makes it necessary to analyze only a central time interval where the window contribution is small. For stable linear time-invariant devices, we extract a time-local least-squares input--output ratio and identify when it follows the continued complex-frequency response, with errors that grow at higher traversal speeds and near resonant poles. Numerical tests on a coupled-mode resonator validate the method and show that closed contours enable an integer phase-winding consistency check. We also outline an implementation based on standard arbitrary waveform generation, I/Q modulation, coherent reception, and digital signal processing.

Figures (4)

• Figure 1: Ordinary chirps and complex-frequency chirped pulses in a rotating frame. Real part of the rotating-frame signal $s_{\mathrm{rot}}(t)=s_{\mathrm{in}}(t)e^{+i\omega_c t}$ (blue), its envelope $A_{\mathrm{total}}(t)=|s_{\mathrm{rot}}(t)|$ (orange), and the finite-duration window $w(t)$ (black dashed) for (a) a windowed carrier, (b) a real-frequency chirp defined by a programmed $\omega_r(t)$ with an approximately flat envelope over the central analysis interval, and (c) a complex-frequency chirp with the same $\omega_r(t)$ as in (b) but with a prescribed $\omega_i(t)$ that reshapes the envelope throughout the burst. Panel (d) shows the programmed laws used for (b,c): real-frequency detuning $\Delta f_r(t)\equiv[\omega_r(t)-\omega_c]/(2\pi)$ (solid, left axis) and imaginary frequency $f_i(t)\equiv\omega_i(t)/(2\pi)$ (dashed, right axis).
• Figure 2: A closed complex-frequency trajectory and its finite-energy waveform. (a) Prescribed loop $\mathcal{C}$ traced by $\tilde{\omega}(t)=\omega_r(t)+i\omega_i(t)$, plotted as detuning $\Delta f_r(t)\equiv[\omega_r(t)-\omega_c]/(2\pi)$ and imaginary frequency $f_i(t)\equiv \omega_i(t)/(2\pi)$ (both in GHz). The orange marker indicates the start/end point on the real axis ($f_i=0$), the arrow indicates the traversal direction, and the "zero" marker denotes an example scattering zero enclosed by the loop. (b) Corresponding rotating-frame input waveform $\Re\{s_{\mathrm{in}}(t)e^{+i\omega_c t}\}$ (blue), envelope $|s_{\mathrm{in}}(t)|$ (orange), and finite-duration window $w(t)$ (black dashed). The central analysis interval is chosen from the flat part of $w(t)$. (c) Programmed laws $\Delta f_r(t)$ (solid, left axis) and $f_i(t)$ (dashed, right axis) used to realize the loop over one traversal.
• Figure 3: Numerical validation of trajectory probing and its speed dependence. (a) Map of $\log_{10}|r(\tilde{\omega})|$ for a one-port resonator in normalized detuning units. The contour $\mathcal{C}$ (white) encloses a zero (green) while avoiding a pole (red). (b) Complex-$r$ trajectory for a baseline loop with $T=40$ arb. units and $\nu=2\pi/T\simeq0.157$: continued response $r(\tilde{\omega}(t))$ (solid) and time-local least-squares estimate $\hat{r}(t)$ (dashed). (c) Unwrapped phase for the same loop, giving $\Delta\phi=6.26$ rad, $N_{\mathrm{est}}=\Delta\phi/(2\pi)=0.997$, and $N=1$. (d) RMS tracking error versus traversal speed for fixed contour geometry, evaluated over the central interval; annotations show $N_{\mathrm{est}}$, illustrating increased mismatch and small departures from integer-$2\pi$ phase accumulation at higher speed.
• Figure 4: Implementation workflow for trajectory probing. A target contour $\mathcal{C}$ in the complex-frequency plane defines the desired instantaneous complex frequency $\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$ and the corresponding baseband waveform $s_{\mathrm{bb}}(t)=I(t)+iQ(t)$. This waveform drives the transmitter, which launches the signal toward the device under test. The incident signal is monitored at the input, giving $s_{\mathrm{in}}(t)$, while the device response is recorded as $s_{\mathrm{out}}(t)$. Digital processing then aligns the input and output records in time, selects a central interval $[t_1,t_2]$ where window-induced distortion is small, estimates the realized complex-frequency trajectory $\tilde{\omega}(t)$ from the monitored input, and extracts the time-local scattering ratio $\hat{r}(t)$ from the synchronized records. For closed contours, the recovered ratio can also be used for an optional phase-winding check.