Unconventional time-bandwidth performance of resonant cavities with nonreciprocal coupling

The time-bandwidth limit is a mathematical tenet that affects all reciprocal resonators, stating that the product of the spectral bandwidth that can couple into a resonant system and its characteristic energy decay time is always equal to 1. Here, we develop an analytical and numerical model to show that introducing nonreciprocal coupling to a generalized resonator changes the power balance between the reflected and intracavity fields, which consequently overcomes the time-bandwidth limit of the resonant system. By performing a full evaluation of the time-bandwidth product (TBP) of the modeled resonator, we show that it represents a measure of the increased delay imparted to a light wave, with respect to what the bandwidth of the reciprocal resonant structure would allow to the same amount of in-coupled power. No longer restricted to the value 1, we show that the TBP can instead be used as a figure of merit of the improvement in intracavity power enhancement due to the nonreciprocal coupling.

Figure 1

Layout of a Gires-Tournois resonator with a nonreciprocal front mirror, whose transmission coefficients depend on the direction of wave propagation, and a fully reflective rear mirror.

Figure 2

Graphs in color scale of the total intracavity (top row) and reflected (bottom row) power encased in one FSR, normalized to the total input power and $\mathrm{\Delta }{\omega }_{\mathrm{FSR}}$, plotted as a function of the in-coupling and out-coupling transmittances. The values are related to a resonator with (a) 0.1, (b) 1, and (c) 2 dB of internal loss, while the red dashed line indicates the points relative to the reciprocal coupling.

Figure 3

Left-hand side: comparison of the values obtained from the simulations with those retrieved from Eq. (4) of the total power enhancement and reflected power, encased in one FSR and normalized to it. The graphs are related to the first (a) and the second (b) case of study and plotted as a function of the degree of nonreciprocity and the in-coupling transmittance, respectively. Right-hand side: spectral distribution over one FSR of the reflected power and the intracavity power enhancement. The curves are related to a degree of nonreciprocity equal to 0.4 (violet solid panel) and 1 (green dotted-dashed panel), and to the case with $|{\mathbf{t}}_{21}{|}^2=0$ and $|{\mathbf{t}}_{12}{|}^2=0.8$ (yellow dashed panel).

Figure 4

Derivation of the TBP of a resonant system starting from its loading and decay processes. The TBP is given by the ratio between the acceptance and the cavity bandwidth which are the FWHM of the Lorentzian function associated, through the Fourier transform, to the loading and decay curve, respectively.

Figure 5

(a) Comparison between the TBP of a reciprocal and a nonreciprocal resonator plotted as a function of the in-coupling transmittance. The curves of the nonreciprocal cases are relative to resonators with different absorption losses. (b) Graph of the TBP in color scale as a function of the in- and out-coupling transmittance for a resonator with 0.1 dB of internal loss. The red dashed line indicates the values relative to the reciprocal coupling (i.e., $\mathrm{TBP}=1$), which corresponds to the time-bandwidth limit.

Figure 6

Comparison between the values of the ratio $G_{\mathrm{cav}}^{\mathrm{NR}}/G_{\mathrm{cav}}^R$ calculated using the values of $G_{\mathrm{cav}}$ obtained from the simulations, and the TBP as a function of the degree of nonreciprocity of the system.

Figure 7

Schematic representation of the setup used in vpiphotonics for the numerical simulations. The phase modulation in combination with the nonreciprocal coupler both integrated in a Sagnac interferometer ensures a total control of the transmission coefficient of the fiber loop, allowing us to emulate the nonreciprocal front mirror of the Gires-Tournois resonator.

Figure 8

Schematic representation of the Gires-Tournois resonator in the form of a figure-nine cavity for (a) $t_1\le t<t_2$ and for (b) $t\ge t_2$. PM: phase modulator; NRC: nonreciprocal coupler; RE: reflective element.