Chirped Pulse Analysis and Control in Non-Hermitian Scattering Systems using Complex Time Delay

Abstract

We theoretically and experimentally establish a connection between linearly chirped pulse propagation properties and complex time delay for both transmitted and reflected pulses in reverberant scattering systems. We demonstrate that the time shift of the chirped pulse depends on both the real and imaginary parts of the complex time delay of the scattering system. We also show that the chirped pulse experiences a center frequency shift that is directly proportional to the imaginary component of complex time delay, similar to that found in Giovannelli and Anlage (2025). Using these insights, we then demonstrate how complex time delay can be harnessed to systematically tune the propagation properties of a chirped pulse such that a near-zero time shift can be achieved for a wide range of pulse center frequencies in a resonant scattering system.

Figures (12)

• Figure 1: (a) Schematic of time domain experiment setup. The ring resonator depicted is the one that was used for the transmission case. (b) Complex time delays and transmission magnitude (in dashed orange) for the resonator depicted in (a). The red and dark purple solid lines respectively correspond to the real and imaginary parts of transmission time delay ($\tau_T$).
• Figure 2: Examples of the normalized chirped pulses sent through the resonator depicted in Fig. \ref{['Transmission SParam and Time Delays']}(a). The pulses sent into the resonator are plotted in blue and the output pulses are plotted in yellow. The measured center frequencies ($f_c)$ and transmission times ($t_c$) are shown as corresponding vertical lines. The insets in (b) show detailed oscillations of the pulses in equal-width (10 ns) time windows, illustrating the chirped nature of the pulse.
• Figure 3: Summary of measured and expected time delays and frequency shifts for chirped pulses having various $\Omega'\delta_t^2$ and $\delta_f$ values, as a function of center frequency. The predicted time shift is plotted in magenta and the measured pulse time shift are the green triangles. The predicted frequency shift is in light purple and the measured pulse frequency shift is plotted as green squares. The chirp rates and pulse widths used are as follows: (a) $\Omega'=0.2848$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=83.8$ ns, (b) $\Omega'=0.0712$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=3747.82$ ns, (c) $\Omega'=-28.48$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=8.38$ ns, (d) $\Omega'=-0.0071$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=3748$ ns
• Figure 4: (a) Schematic of reflection resonator. (b) The red and dark purple solid curves are the real and imaginary parts of reflection time delay respectively. The orange dashed curve corresponds to the magnitude of the reflection coefficient for the resonator in (a). (c-e) Summary of measured and expected time shifts and frequency shifts for chirped pulses after reflection from a ring resonator. The predicted center frequency shifts are shown in in light purple, and predicted time shifts in magenta with the measured frequency shifts represented as green rectangles and time shifts represented as green triangles. This is done for various chirped pulse frequency bandwidths ($\delta_f$) and $\Omega'\delta_t^2$ values. The chirp rates and pulse widths used are as follows: (c) $\Omega'=0$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=10.71$ ns, (d) $\Omega'=0.0498$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=3747.80$ ns, (e) $\Omega'=-0.005$$\frac{\text{MHz}\,\text{Rad}}{\text{ns}}$ and $\delta_t=3747.87$ ns.
• Figure 5: Results for the case where the chirped-pulse temporal shift is canceled by setting $\text{Re}[\tau]/\text{Im}[\tau]=\Omega'\delta_t^2$ at each choice of center frequency. Panels (a) and (b) are the results for transmission and reflection, respectively. The green triangles correspond to the measured chirped pulse transmission (a) and reflection (b) time (right axis). The light orange circles correspond to time-shift normalized to the width of the pulse (left axis). The real and imaginary parts of transmission (a) and reflection (b) time delay are plotted in red and dark purple respectively. The lower insets in (a) and (b) show the predicted and measured center frequency shifts plotted in light purple and as green squares, respectively, along with the value of the common pulse bandwidth $\delta_f$. The upper insets show the chosen values of $\delta_t$ and $\Omega'$ versus frequency utilized to achieve $D_t=0$ for the chirped pulse.