Nonreciprocal Phase Shifts in Spatiotemporally Modulated Systems

Abstract

Materials and devices subject to spatiotemporal modulation of their effective properties have a demonstrated ability to support nonreciprocal transmission of waves. Most notably, spatiotemporally modulated systems can restrict wave transmission to only one direction; i.e. a very large difference in the energy transmitted between two points in opposite directions. Taking on a different perspective on nonreciprocity, we here present a response regime in spatiotemporally modulated systems that is characterized by equal transmitted amplitudes (energies) but different phases. The only contributor to nonreciprocity is therefore the nonreciprocal phase shift, the difference between the transmitted phases in the opposite directions. We develop a methodology for realization of nonreciprocal phase shifts based on the response envelopes. This includes a formulation that ensures the same transmitted waveform, along with a special case of near-reciprocal transmission. We focus primarily on steady-state vibration transmission in short, weakly modulated systems, but include a special case of nonreciprocal phase shifts for systems with arbitrary length and strength of modulation. We discuss the main limitations of our methodology, as well as a pathway to overcome it, to motivate further developments on strongly modulated systems. While showcasing a new way for controlling vibration information transmission, our findings highlight the potential role of phase as an additional parameter in nonreciprocal transmission in spatiotemporally modulated systems.

Figures (12)

• Figure 1: Schematic of the modulated system with $n$ DoF.
• Figure 2: Comparison between the output displacements computed using the averaging method (solid curves) and the Runge-Kutta method (dashed curves). Dash-dotted curves are plots of $\pm \,E_5(\tau)$ and $\pm \,E_1(\tau)$ in panels (a) and (b), respectively. (a) $n=5$, $\Omega_m=0.2$, $\phi=0.42\pi$, $\Omega_f=0.88$, $K_c=0.6$, $K_m=0.1$, $\zeta=0.02$, $P_1=1$ and $P_5=0$; (b) $n=4$, $\Omega_m=0.3$, $\phi=0.95\pi$, $\Omega_f=1.1$, $K_c=0.8$, $K_m=0.1$, $\zeta=0.02$, $P_1=0$ and $P_4=1$.
• Figure 3: Surface plots of reciprocity bias as a function of $\Omega_f$ and $\phi$ for $K_m=0.1$, $\zeta=0.02$ and $P=1$. Red dashed lines indicate the combinations of $\Omega_f$ and $\phi$ that lead to $R=0$. (a) $n=5$, $\Omega_m=0.1$ and $K_c=0.8$; (b) $n=4$, $\Omega_m=0.2$ and $K_c=0.6$.
• Figure 4: Plots of (a,b) output displacements, (c) response envelopes and (d) carrier waves for point $O$. Dash-dotted curves are plots of $\pm \,E_5^F(\tau)$ and $\pm \,E_1^B(\tau)$ in panels (a) and (b), respectively. System parameters: $n=5$, $\Omega_m=0.1$, $\phi=0.11\pi$, $\Omega_f=0.94$, $K_c=0.8$, $K_m=0.1$, $\zeta=0.02$ and $P=1$.
• Figure 5: The curves in each panel show the locus of system parameters that satisfy one of the three constraints in Eqs. \ref{['eq_PhaseNon']}. (a) $n=2$, (b) $n=3$ and (c) $n=4$.