Beyond Chu’s Limit with Floquet Impedance Matching

Chu’s limit determines the minimum radiation quality factor $Q$ of an electrically small resonator, and hence its maximum operational bandwidth, which is inversely proportional to its volume. This bound imposes severe restrictions in several areas of technology, from wireless communications to nanophotonics and metamaterials. We show that a suitably tailored temporal modulation of the matching network, combined with proper detuning of the feeding impedance, can overcome this limit and enable radiation over broader bandwidths, which scale as $1/\sqrt{Q}$, ensuring at the same time stability. Our findings open opportunities for communication systems, nanophotonics, and sensor technology.

Figure 1

Schematic of the Floquet matching network for an electric dipole, which is modeled as a series capacitance-resistance circuit. Floquet matching is achieved with a time-varying inductor $L(t)=L_0[1+m\cos ({\omega }_{M}t)]$ (see the dashed line).

Figure 2

(a) Transducer gain $T_0$ as a function of the (normalized) excitation frequency $\tilde{\omega }$ of the source for (relative) modulation amplitude $m=0$, 0.02, 0.05. (Inset) Maximum value of $T_0$ versus $m$. (b) Stability phase diagram of the Floquet matching system in Fig. 1 in the parameter space $(m,r)$. Numerical (green dashed line) and theoretical (green circle) evolution of $r=r_1(m)$ for $\max [T_0({\omega }_r)]=1$. Here, $Q=100$, ${\tilde{\omega }}_M=2$, and three typical scenarios for different $m$ are marked with different symbols.

Figure 3

Breaking Chu’s limit using Floquet impedance matching with parametric gain. (a) Transducer gain $T_0$ versus (normalized) excitation frequency $\tilde{\omega }$ of the source when the (relative) modulation amplitude $m=0.0$, 0.2, 0.6. Numerical and theoretical results for $r_1(m)$ are adopted from Fig. 2. (b) Bode-Fano integral in Eq. (1) versus $m$, showing operation beyond the Bode-Fano bound (the red solid line). As $m$ increases, the resistance ratio $r=R_s/R_r$ is correspondingly tuned as $r=r_1(m)$. Other common parameters are the same as in Fig. 2.

Figure 4

Realistic simulation of Floquet matching of a loop antenna. (a) Geometry of a small loop antenna with electrical length $k_{r}a\approx 0.22$ at the design frequency ${\omega }_r/2\pi \approx 300\mathrm{MHz}$, matched via a variable capacitor. The quality factor $Q\approx 224$ of the passive antenna is larger than Chu’s limit $Q_{\mathrm{Chu}}=1/{(k_{r}a)}^3\approx 94$. (b) Simulated transducer gain $T_0$ as a function (the black dashed line) of the normalized excitation frequency $\tilde{\omega }=\omega /{\omega }_r$ of the source, significantly surpassing passive matching (the red dashed line). As a reference, the result for an ideal Chu-limited antenna is also shown (red solid line).