Commercially available nonreciprocal devices, such as isolators and circulators, play a fundamental role in communication systems. Since they commonly rely on magnetic materials, they tend to become bulky, expensive, and difficult to be integrated in conventional microelectronic circuits. Here, we explore the functionality of a magnetic-free circulator where reciprocity is broken by suitable geometric asymmetries combined with tailored nonlinearities. We show that it is possible to operate a fully passive coupled resonator system without external bias like a circulator for pulsed signals impinging at its ports within a desired range of intensities. The functionality can be applied to a variety of physical systems, ranging from electronics to photonics and acoustics.

According to Lorentz reciprocity, signal transmission between two points in space is the same for opposite propagation directions.^{1–5} In other words, if a signal can propagate from point *A* to point *B* through a given channel, the reversed propagation path from point *B* to point *A* is also possible with equal strength. There are several situations in which we want to transmit signals breaking this symmetry. For example, in full-duplex communication systems,^{6} unidirectional signal transmission from the transmitter to the antenna and, at the same time, isolation of the receiver from self-interference are sought after. In optics, source protection from back reflections and interferences is very important to avoid detuning. These are examples for which we need to operate outside the domain of validity of Lorentz reciprocity theorem.^{7}

Reciprocity is valid under the assumption that the electric permittivity tensor $\u03f5$ and the magnetic permeability tensor $\mu $ of the involved materials in the propagation channel are: (a) time-reversal symmetric, (b) time-independent, and (c) linear. The most common approach to achieve nonreciprocal devices is therefore to break the symmetry of the electric permittivity tensor, which can be achieved by applying an external magnetic field to a ferromagnetic medium.^{8} As a result, circularly polarized electromagnetic waves with opposite rotation directions interact differently with such media, and reciprocity is broken. Although biasing the material with a magnetic field is the most common way to break reciprocity, it is not the only one. Recently, magnetic-free circulators and isolators based on the time modulation of the electric permittivity of the medium have been studied both theoretically and experimentally.^{9–12} While this approach relaxes the need of special materials supporting magneto-optical phenomena, it still requires sufficiently fast temporal modulations, which are not necessarily easy or convenient to implement. Exploiting nonlinearities can alleviate this need since through nonlinearities, the signal itself traveling through the device can impart a form of bias, breaking reciprocity. Indeed, nonreciprocal propagation in one-dimensional structures exploiting quadratic or cubic nonlinearities has been proposed in the past and recent times^{13–18} although these approaches have limitations in their operation stemming from thermodynamic considerations^{19} and are restricted to nonsimultaneous excitation from different ports.^{20}

The vast majority of nonlinearity-induced nonreciprocal devices proposed to date have been limited to two-port operation. Circulators, on the other hand, are three-port devices where an electromagnetic signal applied to Port 1 only comes out from Port 2 (or Port 3), a signal applied to Port 2 only comes out from Port 3 (or Port 2), and a signal applied to Port 3 only comes out of Port 1 (or Port 2). In other words, the electromagnetic signals circulate in a preferred direction, either clockwise or counterclockwise, an ideal operation to realize full-duplex communications, connecting the transmitter, receiver, and antenna to the three ports of the device.

Consistent with the previous discussion, commercial circulators rely on magnetic materials, which make them bulky, expensive, and not CMOS compatible. Here, we explore the functionality of bias-free, fully passive nonlinear circulators with isolation factors that exceed 20 dB. A nonlinear magnetless circulator was presented in Ref. 17, but required the use of active materials, in contrast to the fully passive design proposed in the following.

We start in Fig. 1(a) by sketching the generic layout of our circulator. The structure consists of a loop of six identical coupled resonators interlaced by alternating linear and nonlinear coupling coefficients.

Due to the rotational symmetry of the structure, a circulator response can be accomplished if, for excitation from any port (e.g., port 1), we achieve zero transmission to one of the other ports (e.g., port 2) and maximum transmission to the third one (e.g., port 3). Such a network of coupled resonators can be efficiently described through coupled mode theory^{21} as

*A*and

*B j-*th resonator, respectively, $\omega 0$ is the uncoupled resonators' resonant frequency, $\kappa 0>0$ is the linear coupling coefficient, $\kappa 1>0$ is the linear part of the nonlinear coupling coefficient, $\alpha $ is the nonlinear parameter, $\Gamma =1/\tau $ is the coupling loss term, where $\tau $ is the average lifetime (dwell time) of the radiation in the resonators before the energy is released to Port 2 and Port 3, and, finally, $2\Gamma e\u2212i\omega t$ is the input signal (forcing term) at Port 1 oscillating at frequency $\omega $. In our system, resonators $Aj$ with $j=1,\u20092,\u20093$ have a termination at Port-

*j*, while resonators $Bj$ have no port termination. We now look for steady-state solutions of Eq. (1) in the form $ajkt=cj(k)e\u2212i\omega t$, with $k=A,B$ and $j=1,\u20092,\u20093$, where $cj(k)$ is the time independent amplitude of the field. The reflectance at Port 1 ($R1$) and transmittances at Port 2 and Port 3 ($T2$ and $T3$) are given by $R1=|2\Gamma c1A\u22121|2$ and $T2=2\Gamma |c2A|2$ and $T3=2\Gamma |c3A|2$, respectively, and power conservation dictates that $R1+T2+T3=1$. For the case of linearly coupled resonators and no coupling loss ($\alpha =0\u2009$ and $\Gamma =0$), the system admits four resonant frequencies (eigenfrequencies), given by $\omega 0\xb1(\kappa 1+\kappa 0)$ and $\omega 0\xb1\kappa 12\u2212\kappa 1\kappa 0+\kappa 02$. When the electromagnetic signal applied to Port 1 oscillates with a frequency close to one of the eigenfrequencies of the system, then energy is efficiently coupled with the resonators and eventually is expected to be transmitted to Port 2 and Port 3; otherwise, the signal is mostly reflected at Port 1.

In the linear regime ($\alpha =0$), because of symmetry, the power output at Port 2 is equal to the power output at Port 3 ($T2=T3$), *regardless* of the specific values of linear coupling coefficients $\kappa 0$ and $\kappa 1$. In order to shed some light on the nonlinear behavior of the system, we look for nonlinear solutions that can support complete isolation at Port 2. In other words, we look for solutions that support the condition of zero field at Port 2, i.e., $c2(A)=0.$ For $\kappa 0=\kappa 1$ at the resonant frequency $\omega \xaf=\omega 0+\kappa 0$, we find the following solutions:

with $\delta =\alpha /2\Gamma \kappa 0$. It can be easily ascertained by direct substitution that Eq. (2) satisfies Eqs. (1a), (1b), (1d), and (1e) under the conditions specified above, i.e., $c2(A)=0$, $\kappa 0=\kappa 1$, and $\omega \xaf=\omega 0+\kappa 0$. Under the same conditions, Eq. (2) satisfy Eqs. (1c) and (1f) on the order of $\delta $ for $\delta \u226a1\u2009$ and $\Gamma /k0\u223c\delta 2$. This can also be easily verified by substituting Eq. (2) into Eqs. (1c) and (1f) and neglecting the terms of order $\delta 2$ and higher. Physically speaking, the condition $\delta \u226a1$ means that the nonlinearity is a perturbation, while $\Gamma /k0\u223c\delta 2$ amounts to require high-quality (Q) resonators. The solutions in Eq. (2) lead to $R1=|2\Gamma c1A\u22121|2=\delta 2,\u2009T2=2\Gamma |c2A|2=0$ and $T3=2\Gamma |c3A|2=1$, and $R1+T2+T3=1+\delta 2$, and therefore, power is conserved on the order of $\delta $, consistent with the functionality investigated here. Hence, our analysis suggests that at the resonant frequency $\omega 0+\kappa 0,$ the electromagnetic energy can be almost completely routed to Port 3, almost perfect isolation at Port 2 is achieved, and reciprocity is largely broken.

This theoretical result is indeed confirmed by a full numerical integration of Eq. (1) in the time domain, as shown in Figs. 1(b)–1(e). In the linear case [Fig. 1(b)], four transmission resonances are found at the eigenfrequencies of the system, and transmission resonances at Port 2 and Port 3 are exactly the same, as expected. By increasing the nonlinear parameter (other panels in Fig. 1), the two transmission resonances at $\omega 0+\kappa 0$ depart from each other. Eventually, transmission at Port 2 becomes almost completely quenched, and most of the energy is routed to Port 3, with an isolation factor of more than 20 dB. We also note that at higher frequencies, the system undergoes an abrupt transition, switching to another stable state of the nonlinear system, with energy mostly routed to Port 2. In order to further corroborate this analysis, in Fig. 2, we present the results of a numerical simulation in the parameter space ($\alpha ,\Gamma $). In particular, in Figs. 2(a)–2(c), we show the transmittance and reflectance coefficients as a function of ($\alpha ,\Gamma $) at the frequency $\omega 0+\kappa 0$ for $\alpha $ varying from $10\u22124\omega 02$ to $10\u22123\omega 02$ and $\Gamma $ varying from $10\u22123\omega 0$ to $2\xd710\u22122\omega 0$. The highest isolation of 23 dB is reached for $\alpha =10\u22124\omega 02$ and $\Gamma =3\xd710\u22123\omega 0$. Figure 2(d) shows the corresponding reflection and transmission curves vs frequency for this scenario.

So far, in our analysis, we have considered the particular case in which $\kappa 0=\kappa 1$ because in such a case, it is possible to solve analytically for the value of isolation at Port 2 at the eigenfrequency $\omega 0+\kappa 0$. Nevertheless, qualitatively similar results are obtained when $\kappa 0\u2260\kappa 1$. In this case, the isolation at Port 2 arises close to the eigenfrequency $\omega 0+\kappa 12\u2212\kappa 1\kappa 0+\kappa 02$. An example of this more general scenario is provided in Fig. 3. By increasing the nonlinearity coefficient further with respect to the case shown in Fig. 3(d), isolation factors exceeding 20 dB are obtained although in the latter case, the frequency where isolation is achieved blue-shifts with respect to the position of the eigenfrequency of the linear scenario.

In order to verify the proposed concept, a realistic radio-frequency circuit—shown in Fig. 4—has been designed, optimized, and numerically simulated for realistic circuit parameters, including parasitics, as detailed in the table in Fig. 4. The resonators are chosen in the parallel resonance configurations, while the coupling elements are realized through nonlinear capacitors and impedance inverters for the nonlinear and linear coupling elements, respectively, as sketched in the insets, while the corresponding coupling coefficients are found in Refs. 22 and 23. Impedance inverters^{8} were chosen to avoid that the coupling elements affect the resonator impedances, enabling a simpler optimization procedure. The table in Fig. 4 provides the correspondence between the CMT parameters and the circuit elements. Following the results in our approximate analytical model described above, first, we investigated the low-intensity scenario using a full-wave circuit simulator with circuit parameter values $\Gamma =0.0125,\u2009kk=0.043,$ and $kn=0.021$. Figure 5(a) shows the corresponding reflectance and transmittance, which follow well the predicted results in Figs. 1 and 3. Next, we considered realistic nonlinearities as the input intensity is increased. For the same circuit parameters and nonlinearity coefficient $\alpha =0.212,\u2009$ circulator operation is found around the first peak of the transmission coefficient, consistent with the previous analysis [Fig. 5(b)]. Finally, we further increased the input power, yielding in Fig. 5(c) an isolation above 17 dB and an insertion loss of 0.57 dB with an input power of 3 mW, in very good agreement with the coupled mode theory model for the case $\kappa 0\u2260\kappa 1$ shown in Fig. 3(d). We have also analyzed the circuit in the time domain to verify the stability and the speed of the response. The envelope of the instantaneous power at the two output ports is plotted in Fig. 5(d) after normalization to the total input power from the source. Very good agreement with the frequency domain analysis is observed, confirming the practical realizability of this concept.

In conclusion, in this work, we have shown the concept and realistic design of a magnet-free, nonbiased, fully passive nonlinear circulator. Nonlinearities are ubiquitous in several physical systems, and therefore, our analysis, based on temporal coupled mode theory for a resonator network, is broadly applicable to a wide variety of practical problems, from passive circulators for radio-wave communications and radars to nanophotonic and quantum computing systems and acoustics. Our realistic simulations of a passive nonlinear circuit confirm the accuracy of our model and the realistic possibility of realizing a low-loss, fully passive, largely nonreciprocal circulator in which the signal itself, interacting with nonlinearities, breaks reciprocity and induces large isolation. The limitations outlined for other nonlinear, self-biased nonreciprocal devices in the recent literature^{19,20} apply also here, especially in the context of simultaneous excitation from multiple ports. The results in this paper have been derived for continuous wave excitation from a single port at a time, but they are expected to change when multiple continuous waves excite the device from the different ports, because of the nonlinearities. In this sense, pulsed operation is required to operate the circulator in its described functionality. This obvious deficiency may be compensated for some application by the inherent advantage of not requiring any form of bias and its full passivity, making this device particularly appealing for pulsed operation applications, such as in radar or lidar, defense-related, automotive and imaging systems. In applications for which transmitted and received pulses may have different strengths, the present approach may be generalized to asymmetric designs with different intensity thresholds for the different ports. This work was supported by the Air Force Office of Scientific Research.