Nonreciprocal components are of significance for both classical and quantum photon-based information processing. Optical nonreciprocity can be achieved by using magneto-optical materials [1–4], spatiotemporal modulation [5–9], chiral light–matter interactions [10–13], atomic microscopic Doppler effect [14–16], macroscopic Doppler effect of moving atomic lattices [17–19], optomechanical interactions [20–22], and optical nonlinearities [23–31]. Specifically, by using the nonlinearities of optical resonators, bias-free and integrated nonreciprocal devices can be realized [26–28]. Despite constraints imposed by dynamic reciprocity [32] under simultaneous excitation from opposite ports, nonlinear nonreciprocal devices attract intense research for situations involving pulsed signals because they have advantages of being compatible with an on-chip platform. However, besides dynamic reciprocity, the nonreciprocity of nonlinear optical devices is crucially dependent on the intensity of incident light. Nonlinear nonreciprocal devices are constrained by a narrow operating intensity range [23–26]. A number of nonlinear nonreciprocal devices are affected by a trade-off between the maximum forward transmission and the nonreciprocal intensity range (NRIR) [25–28]. Cascaded Fano–Lorentzian nonlinear resonators have been demonstrated to relax the limitation to some degree [27,28]. However, it remains a big challenge to realize nonlinear nonreciprocal devices with a high transmission contrast, broad bandwidth, wide NRIR, and low insertion loss simultaneously.

Feedback control is an important method to manipulate the dynamic evolution of the system [33,34], and it can generate and amplify nonlinearity in a resonator [35,36]. Here we demonstrate a novel and simple nonreciprocal optical system based on a cavity with resonance frequency tuned by a feedback control system via the electro-optic (EO) effect. A part of the information of the cavity can be detected by a detector. The output current of the detector is fed back to an electric circuit to generate effective Kerr nonlinearity in the cavity. In our designed system, the feedback current intensities are dependent on the incident directions of light, leading to directional, in other words direction-dependent, Kerr nonlinearities and subsequently nonreciprocal light transmissions. To obtain a broad NRIR, we explore intensity-adaptive feedback control to dynamically amplify and tune the induced Kerr nonlinearity. As a result, our designed nonreciprocal optical system exhibits a transmission contrast of 0.99 and a broad NRIR at the same time.

The basic system depicted in Fig. 1(a) consists of a Fabry–Perot (FP) cavity containing an EO nonlinear crystal, e.g., ${\text{LiNbO}}_3$, and an intensity-adaptive feedback circuit. The feedback circuit includes mainly an electric filter, an electric amplifier, and two photodetectors (PDs). For a forward input ${\widehat{a}}_{\mathrm{in}}$, a portion of the transmitted light through the beam splitter (BS) is reflected by the cavity and monitored by PD1. Its output is filtered and amplified to drive the EO nonlinear crystal in the cavity. Because of the Pockels effect, the modulated EO nonlinear crystal can induce a resonance frequency shift to the cavity. However, for a backward input ${\widehat{b}}_{\mathrm{in}}$, a fraction of the transmitted light through the cavity is partly reflected by BS and monitored by PD1. Through the feedback circuit, the output current of PD1 drives the EO nonlinear crystal and induces a resonance frequency shift different from the forward case. Thus, the opposite-direction inputs can induce different feedback current intensities, yielding different resonance frequency shifts. As a result, the direction-dependent feedback current leads to directional Kerr nonlinearity in the cavity and the nonreciprocal transmission of the system. Note that a certain feedback current from PD1 produces the same resonance frequency shift of the cavity in opposite input directions. The intensity of the feedback current caused by feedback light incident on PD1 is dependent on the direction of the input light. This dependence causes the directional resonance frequency shift of the cavity, as depicted in Fig. 1(b). The basic idea is the following: at the beginning of the input light entering the cavity, the input is off resonance with the cavity, and the detuning ${\mathrm{\Delta }}_{\mathrm{in}}$ is large. For the forward input, the feedback-induced resonance frequency shift of the cavity is close to the detuning but with opposite signs. Thus, the shift “pulls” the cavity to be near resonance with the input. As a result, the forward transmission is high. In contrast, the feedback-induced resonance frequency shift in the backward-input case is small and cannot compensate for the initial detuning. The off-resonance backward transmission is weak. In this arrangement, we obtain a transmission contrast of 0.99. It is worth noting that our system is still under the constraint imposed by dynamic reciprocity. Therefore, to break the time-reversal symmetry, the forward and backward inputs are applied to the system separately in time.

PD2 is used to monitor the intensity of the forward incident light. A fraction of forward incident light is reflected by BS and detected by PD2, whose output modulates the gain of the electric amplifier. In this way, the gain of the electric amplifier is adjusted according to the input intensity, which is proportional to $|⟨{\widehat{a}}_{\mathrm{in}}⟩{|}^2$. Utilizing this intensity-adaptive feedback control, the NRIR can be greatly broadened without reducing transmission contrast or insertion loss.

For the forward case, the feedback current produced by PD1 is $$i_a^f(t)=\xi \gamma ⟨{\widehat{a}}_r^{†}(t){\widehat{a}}_r(t)⟩,$$(1)where $\xi$ is the photoelectric conversion efficiency of PD1, and $\gamma$ is the reflection coefficient of BS. According to the input–output relation of an optical cavity [37,38], the reflected field operator for the cavity is given in terms of the input and intracavity field operators as $${\widehat{a}}_r(t)=-\sqrt{1-\gamma }{\widehat{a}}_{\mathrm{in}}+\sqrt{{\kappa }_{\mathrm{ex}1}}\widehat{a},$$(2)where $\widehat{a}$ is the annihilation operator for the cavity mode excited by forward incident light, ${\widehat{a}}_{\mathrm{in}}$ is the annihilation operator for the forward incident field, and ${\kappa }_{\mathrm{ex}1}$ is the decay rate caused by the cavity mirror (M1). According to Eqs. (1) and (2), the feedback current is $$i_a^f(t)=\xi \gamma [(1-\gamma ){|{\alpha }_{\mathrm{in}}|}^2+{\kappa }_{\mathrm{ex}1}{|\alpha |}^2-2\sqrt{{\kappa }_{\text{ex}1}(1-\gamma )}\mathrm{Re}({\alpha }_{\mathrm{in}}^{*}\alpha )],$$(3)where the coherent amplitudes of the input field and the intracavity field are given by ${\alpha }_{\mathrm{in}}=⟨{\widehat{a}}_{\mathrm{in}}⟩$ and $\alpha =⟨\widehat{a}⟩$, respectively. An input power $P_{\mathrm{in}}$ corresponds to an input photon flux ${|{\alpha }_{\mathrm{in}}|}^2=P_{\mathrm{in}}/\hslash {\omega }_{\mathrm{in}}$.

Now we derive the feedback current acting on the optical cavity after a low-pass electrical filter. The feedback current is filtered by a low-pass filter with an impulse response function $h(t)=(2{\omega }_c/\sqrt{3})\exp (-{\omega }_{c}t/2)\sin (\sqrt{3}{\omega }_{c}t/2)$, where ${\omega }_c$ is the cutoff frequency. When the incident light enters the cavity at the beginning time period, the detuning with the cavity is large and causes oscillation in the feedback. We use this low-pass filter to block the rapidly oscillating component in the feedback-induced frequency shift. The high-frequency components of the feedback current caused by a high-power input can be filtered. The filtered feedback current is then amplified and modulates the EO nonlinear crystal inside the cavity, yielding a resonance frequency shift: $${\mathrm{\Delta }}_a^f(t)=\chi G{\int }_0^t{i}_a^f(\tau )h(t-\tau )\mathrm{d}\tau ,$$(4)where $\chi$ is the coefficient of the EO nonlinear process, and $G$ is the gain of the electric amplifier. We define conversion-amplification coefficient $A\equiv \chi G\xi$ for later convenience. Combining Eqs. (3) and (4), the cavity-excitation-dependent (i.e., ${|\alpha |}^2$-dependent) resonance frequency shift can be written as $$\begin{gathered} {\mathrm{\Delta }}_a^f(t)=A\gamma {\int }_0^t\{(1-\gamma )|{\alpha }_{\mathrm{in}}(\tau {)|}^2+{\kappa }_{\mathrm{ex}1}|\alpha (\tau {)|}^2 \\ -2\sqrt{{\kappa }_{\mathrm{ex}1}(1-\gamma )}\mathrm{Re}[{\alpha }_{\mathrm{in}}^{*}(\tau )\alpha (\tau )]\}h(t-\tau )\mathrm{d}\tau . \end{gathered}$$(5)

Working in a frame rotating at the incident field frequency ${\omega }_{\mathrm{in}}$, the system can be modeled as an effective Kerr nonlinear cavity driven by the forward input ${\alpha }_{\mathrm{in}}$ described by the Hamiltonian $$H_{\mathrm{fw}}=({\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_a^f){\widehat{a}}^{†}\widehat{a}+i\sqrt{{\kappa }_{\mathrm{ex}1}(1-\gamma )}({\alpha }_{\mathrm{in}}{\widehat{a}}^{†}-{\alpha }_{\mathrm{in}}^{*}\widehat{a}),$$(6)where ${\mathrm{\Delta }}_{\mathrm{in}}={\omega }_0-{\omega }_{\mathrm{in}}$, ${\omega }_0$ is resonance frequency of the cavity in the absence of feedback, and $\kappa$ is the total cavity decay rate, including the decay rates (${\kappa }_{\mathrm{ex}1}/{\kappa }_{\mathrm{ex}2}$) caused by cavity mirrors (M1/M2) and the intrinsic decay rate ${\kappa }_{\mathit{i}}$ caused by the absorption and scattering of the EO nonlinear crystal, i.e., $\kappa ={\kappa }_{\mathrm{ex}1}+{\kappa }_{\mathrm{ex}2}+{\kappa }_{\mathit{i}}$.

For a classical field input, we can apply the mean-field approximation to the cavity mode $\alpha =⟨\widehat{a}⟩$. In this case, the feedback produces an effective Kerr nonlinearity to the cavity. The cavity mode amplitude $\widehat{a}$ in the mean-field approximation evolves with time according to $$\dot{\alpha }=-i({\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_a^f)\alpha +\sqrt{{\kappa }_{\mathrm{ex}1}(1-\gamma )}{\alpha }_{\mathrm{in}}-\frac{\kappa }{2}\alpha .$$(7)

According to the input–output relation ${\widehat{a}}_{\mathrm{out}}=\sqrt{{\kappa }_{\mathrm{ex}2}}\widehat{a}$, the forward transmission amplitude is defined as $t_{\mathrm{fw}}=⟨{\widehat{a}}_{\mathrm{out}}⟩/{\alpha }_{\mathrm{in}}$. Thus, the corresponding forward transmission coefficient is $T_{\mathrm{fw}}={|t_{\mathrm{fw}}|}^2$.

We can numerically solve Eqs. (5) and (7) by using the four-order Runge–Kutta method to find the time evolution of the resonance frequency shift and the forward transmission. Additionally, it is possible to obtain an analytical expression for the steady-state transmission. For the forward case, we assume the feedback-induced resonance frequency shift stably remains at a certain value ${\mathrm{\Delta }}_{a,\mathrm{ss}}^f$ when the system reaches the steady state. We define the residual detuning as ${\delta }_a\equiv {\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_{a,\mathrm{ss}}^f$. In this case, the steady-state solution of Eq. (7) is $${\alpha }_{\mathrm{ss}}=\frac{\sqrt{{\kappa }_{\mathrm{ex}1}(1-\gamma )}{\alpha }_{\mathrm{in}}}{i{\delta }_a+\kappa /2}.$$(8)

According to the input–output relation and $t_{\mathrm{fw}}=⟨{\widehat{a}}_{\mathrm{out}}⟩/{\alpha }_{\mathrm{in}}$, the steady-state forward transmission is $$T_{\mathrm{fw}}=\frac{4{\kappa }_{\mathrm{ex}1}{\kappa }_{\mathrm{ex}2}(1-\gamma )}{4{\delta }_a^2+{\kappa }^2}.$$(9)

For the backward case, the intensity of feedback current produced by PD1 is $$i_b^f(t)=\xi \gamma ⟨{\widehat{b}}_t^{†}(t){\widehat{b}}_t(t)⟩.$$(10)

According to the input–output relation, the transmitted field operator for the cavity is given by $${\widehat{b}}_t(t)=\sqrt{{\kappa }_{\mathrm{ex}1}}\widehat{b},$$(11)where $\widehat{b}$ is the annihilation operator for the cavity mode excited by backward incident light. Note that $\widehat{a}$ and $\widehat{b}$ express the same cavity mode with different denotations to distinguish the opposite incident directions. Combining Eqs. (10) and (11), the feedback current can be expressed as $i_b^f(t)=\xi \gamma ·{\kappa }_{\mathrm{ex}1}{|\beta |}^2$, with the coherent amplitude of the intracavity field $\beta =⟨\widehat{b}⟩$.

The same with the forward case, the filtered feedback current is amplified and modulates the crystal with the same conversion-amplification coefficient $A$. So the resonance frequency shift is $${\mathrm{\Delta }}_b^f(t)=A\gamma {\kappa }_{\mathrm{ex}1}{\int }_0^t|\beta (\tau {)|}^{2}h(t-\tau )\mathrm{d}\tau .$$(12)

The Hamiltonian for the backward case is given by $$H_{\mathrm{bw}}=({\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_b^f){\widehat{b}}^{†}\widehat{b}+i\sqrt{{\kappa }_{\mathrm{ex}2}}({\beta }_{\mathrm{in}}{\widehat{b}}^{†}-{\beta }_{\mathrm{in}}^{*}\widehat{b}),$$(13)where ${\beta }_{\mathrm{in}}$ is the backward incident amplitude. The input power $P_{\mathrm{in}}$ yields the input photon flux ${|{\beta }_{\mathrm{in}}|}^2=P_{\mathrm{in}}/\hslash {\omega }_{\mathrm{in}}$. In the mean-field approximation $\beta =⟨\widehat{b}⟩$, the cavity also includes Kerr nonlinearity due to the feedback control. This effective Kerr nonlinearity coefficient is different from the forward case. The evolution of the cavity mode $\widehat{b}(t)$ in the mean-field approximation is given by $$\dot{\beta }=-i({\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_b^f)\beta +\sqrt{{\kappa }_{\mathrm{ex}2}}{\beta }_{\mathrm{in}}-\frac{\kappa }{2}\beta .$$(14)

Because of the input–output relations ${\widehat{b}}_{\mathrm{out}}=\sqrt{1-\gamma }{\widehat{b}}_t$ and ${\widehat{b}}_t=\sqrt{{\kappa }_{\text{ex}1}}\widehat{b}$, the backward transmission amplitude is given by $t_{\mathrm{bw}}=⟨{\widehat{b}}_{\mathrm{out}}⟩/{\beta }_{\mathrm{in}}$, and the corresponding backward transmission coefficient is $T_{\mathrm{bw}}={|t_{\mathrm{bw}}|}^2$. The time evolution of the frequency shift and the transmission in the backward case can be found by numerically solving Eqs. (12) and (14).

We assume the resonance frequency shift remains at ${\mathrm{\Delta }}_{b,\mathrm{ss}}^f$, and define the residual detuning as ${\delta }_b\equiv {\mathrm{\Delta }}_{\mathrm{in}}+{\mathrm{\Delta }}_{b,\mathrm{ss}}^f$. Thus, we obtain the steady-state solution to Eq. (14) as $${\beta }_{\mathrm{ss}}=\frac{\sqrt{{\kappa }_{\mathrm{ex}2}}{\beta }_{\mathrm{in}}}{i{\delta }_b+\kappa /2}.$$(15)

Using the input–output relations and $t_{\mathrm{bw}}=⟨{\widehat{b}}_{\mathrm{out}}⟩/{\beta }_{\mathrm{in}}$, we obtain the steady-state backward transmission $$T_{\mathrm{bw}}=\frac{4{\kappa }_{\mathrm{ex}1}{\kappa }_{\mathrm{ex}2}(1-\gamma )}{4{\delta }_b^2+{\kappa }^2}.$$(16)

For simplicity, we assume ${\kappa }_{\mathrm{ex}1}={\kappa }_{\mathrm{ex}2}={\kappa }_{\mathrm{ex}}$, such that $\kappa =2{\kappa }_{\mathrm{ex}}+{\kappa }_{\mathit{i}}$ throughout the investigation below.

We numerically solve Eqs. (7) and (14) with the four-order Runge–Kutta method to find the time evolution of the state of the cavity and its transmission. As shown in Fig. 2, the system reaches the steady state ($\dot{\alpha }\approx 0$ and $\dot{\beta }\approx 0$) after evolving a time period such that ${\kappa }_{\mathrm{ex}}t\gg 1$. Excited from opposite sides with the input fields with equal power ($P_{\mathrm{in}}=830\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$) and the same detuning (${\mathrm{\Delta }}_{\mathrm{in}}=-15{\kappa }_{\mathrm{ex}}$), the system reaches a steady state after a hysteresis (${\tau }_{\mathit{h}}\approx 138{\kappa }_{\mathrm{ex}}^{-1}$). The steady-state forward and backward transmissions shown in Fig. 2(a) are $T_{\mathrm{fw}}\approx 0.70$ and $T_{\mathrm{bw}}\approx 0.005$, respectively. The system exhibits a transmission contrast of $\eta =(T_{\mathrm{fw}}-T_{\mathrm{bw}})/(T_{\mathrm{fw}}+T_{\mathrm{bw}})\approx 0.99$ [10,17] and an insertion loss of $ℒ=-10{\log }_{10}(T_{\mathrm{fw}})\approx 1.5\mathrm{dB}$.

According to Eqs. (5) and (12), we obtain the feedback-induced resonance frequency shifts [Fig. 2(c)]. For the forward case, the feedback-induced resonance frequency shift begins oscillating from an initial frequency shift determined by the input, and locks to a certain value close to the input detuning, i.e., ${\mathrm{\Delta }}_{a,\mathrm{ss}}^f\sim -{\mathrm{\Delta }}_{\mathrm{in}}$, and ${\delta }_a/{\kappa }_{\mathrm{ex}}\sim 0$, at the end of hysteresis. Once the feedback-induced shift is locked, the cavity mode is highly excited [Fig. 2(b)]. As a result, the feedback control “pulls” the detuned resonator to near resonance, yielding a high forward transmission. However, for the backward case, the feedback-induced frequency shift is negligible, compared with the incident detuning, i.e., ${\mathrm{\Delta }}_{b,\mathrm{ss}}^f\ll -{\mathrm{\Delta }}_{\mathrm{in}}$, and ${\delta }_b\sim {\mathrm{\Delta }}_{\mathrm{in}}$. Thus, the cavity is weakly excited, as shown in Fig. 2(b). It means that the feedback control hardly changes the initial detuned state of the system. As a result, the backward transmission is low.

According to the steady-state analytical solutions, we below investigate how the nonreciprocal transmissions of the system are affected by the reflection coefficient of the BS and the intrinsic loss of the cavity. From the numerical results shown in Fig. 2(c), we can find the steady-state feedback-induced resonance frequency shifts ${\mathrm{\Delta }}_{a,\mathrm{ss}}^f\approx 14.6{\kappa }_{\mathrm{ex}}$ and ${\mathrm{\Delta }}_{b,\mathrm{ss}}^f\approx 0.5{\kappa }_{\mathrm{ex}}$ for the forward and backward cases, respectively. We can set the optimized residual detunings ${\delta }_a=-0.4{\kappa }_{\mathrm{ex}}$ and ${\delta }_b={\mathrm{\Delta }}_{\mathrm{in}}$ to calculate the analytical steady-state transmissions with Eqs. (9) and (16). We also compare our analytic formula with numerical solutions to Eqs. (7) and (14). It can be seen in Fig. 3 that the numerical results of steady-state transmissions are in excellent agreement with the analytical solutions.

It can be seen in Figs. 3(a) and 3(b) that the system exhibits the high insertion loss for $\gamma >0.1$, but the transmission contrast is independent of $\gamma$, satisfying the steady-state analytical solutions of $\eta =2({\delta }_b^2-{\delta }_a^2)/[2({\delta }_b^2+{\delta }_a^2)+{\kappa }^2]$. As shown in Figs. 3(c) and 3(d), a stronger intrinsic loss causes a higher insertion loss but the transmission contrast remains nearly constant in spite of small changes in the intrinsic loss, owing to $2({\delta }_b^2+{\delta }_a^2)\gg {\kappa }^2$.

The forward and backward transmissions are dependent on the input light power ($P_{\mathrm{in}}$) and the detuning between the input and the cavity (${\mathrm{\Delta }}_{\mathrm{in}}$). We show the numerically calculated steady-state forward and backward transmissions as a function of the input power and detuning in Figs. 4(a) and 4(b). For the input detuning with a positive value, the feedback will “push” the cavity resonance farther away from the input, suppressing transmission in both directions. For a negative detuning ${\mathrm{\Delta }}_{\mathrm{in}}$ and a moderate input power, e.g., $P_{\mathrm{in}}=500\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$, if $|{\mathrm{\Delta }}_{\mathrm{in}}|$ is large, the feedback-induced frequency shift cannot be large enough to compensate for this negative detuning. In this case, the cavity will still be off resonance with the input, preventing the transmission in both directions. If the detuning is too small, the feedback in the forward case will pull the cavity frequency to go beyond the resonance, also blocking the transmission. But for the backward case, the feedback is always too small to compensate for the initial detuning ${\mathrm{\Delta }}_{\mathrm{in}}$, resulting in a low backward transmission.

For a specific value of the detuning ${\mathrm{\Delta }}_{\mathrm{in}}=-15{\kappa }_{\mathrm{ex}}$, the system exhibits nonreciprocal transmission versus incident power [Fig. 4(c)]. The maximal transmission contrast can be $\eta =0.99$, and the maximal forward transmission reaches $T_{\mathrm{fw}}=0.70$. Referring to Refs. [25,28], we define the NRIR as the ratio of input power from opposite propagation directions that meet a special transmission contrast. For instance, to meet $\eta =0.99$, $\mathrm{NRIR}(0.99)=10{\log }_{10}(P_{\mathrm{in}2}/P_{\mathrm{in}1})\approx 1.4\mathrm{dB}$, where $P_{\text{in},1}=634\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$ and $P_{\text{in},2}=876\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$ are the lower and upper power boundaries of the NRIR, respectively, as shown by the green area in Fig. 4(c).

The nonreciprocal transmission as a function of the detuning is shown in Fig. 4(d) for a fixed incident power of $P_{\mathrm{in}}=830\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$ as an example. As shown by the green area in Fig. 4(d), the nonreciprocal bandwidth meeting $\eta =0.99$ and $ℒ⩽3\mathrm{dB}$ is about $24.2{\kappa }_{\mathrm{ex}}$. Within this nonreciprocal bandwidth, the backward transmission and transmission contrast almost remain unchanged, but the forward transmission linearly decreases with the detuning $|{\mathrm{\Delta }}_{\mathrm{in}}|$.

When the conversion-amplification coefficient ($A$) is fixed, the system holds nonreciprocal transmissions only over a small range of the incident intensity, as shown in Fig. 4(c). To broaden the NRIR, we apply intensity-adaptive feedback control to the system. PD2 is used to monitor the power of incident light ($P_{\mathrm{in}}$) and control the gain of the electric amplifier ($G$), which are proportional to ${|{\alpha }_{\mathrm{in}}|}^2$ and $A$, respectively. As a result, the conversion-amplification coefficient adjusts with the incident intensity.

At the beginning of the light incident into the system, the cavity mode is not excited, i.e., $\alpha =0$, as shown in Fig. 2(b). According to Eq. (5), we define the input-induced resonance frequency shift $${\mathrm{\Delta }}_{a,\mathrm{in}}^f\equiv A\gamma (1-\gamma )P_{\mathrm{in}}/\hslash {\omega }_{\mathrm{in}},$$(17)where the input power $P_{\mathrm{in}}$ is determined by ${|{\alpha }_{\mathrm{in}}|}^2$ in the forward case and ${|{\beta }_{\mathrm{in}}|}^2$ in the backward case. We assume that the optimal forward transmission is $T_{\mathrm{fw}}$, corresponding to an optimal input power $P_{\mathrm{in}}^{\mathrm{opt}}$. The adaptive feedback circuit generates an optimized input-induced frequency shift ${\mathrm{\Delta }}_{a,\mathrm{in}}^{f,\mathrm{opt}}=A\gamma (1-\gamma )P_{\mathrm{in}}^{\mathrm{opt}}/\hslash {\omega }_{\mathrm{in}}$. To maintain the optimal forward transmission, we need to use the adaptive circuit to fix this input-induced resonance frequency shift to ${\mathrm{\Delta }}_{a,\mathrm{in}}^{f,\mathrm{opt}}$. To do so, we can set the intensity-adaptive amplification coefficient to $$\tilde{A}=\frac{\hslash {\omega }_{\mathrm{in}}{\mathrm{\Delta }}_{a,\mathrm{in}}^{f,\mathrm{opt}}}{\gamma (1-\gamma )P_{\mathrm{in}}},$$(18)for different input powers $P_{\mathrm{in}}$. In this way, for various input light intensities, the system beginning with the same input-induced frequency shift ${\mathrm{\Delta }}_{a,\mathrm{in}}^{f,\mathrm{opt}}$ will evolve to the same steady state, yielding the same forward transmission $T_{\mathrm{fw}}$. In the backward case, the feedback circuit shares the same amplification coefficient $\tilde{A}$ with the forward case. In intensity-adaptive feedback control, the coefficient $A$ in Eqs. (5) and (12) is replaced with $\tilde{A}$.

From numerical results shown as an example in Fig. 4(c), for a specific incident power range, the maximal forward transmission of $T_{\mathrm{fw}}\approx 0.70$ is obtained for optimal parameters $A=12$ and $\gamma =0.01$. We take the upper boundary $P_{\mathrm{in}}^{\mathrm{opt}}=876\hslash {\omega }_{\mathrm{in}}{\kappa }_{\mathrm{ex}}$ for optimal input power. To improve the NRIR, the system needs to adjust the coefficient $\tilde{A}$ according to Eq. (18) via the intensity-adaptive feedback circuit. In consideration of experimental feasibility, 1000-fold current gain of an electric amplifier has been achieved [39]. Here we set $1⩽\tilde{A}⩽100$, varying over 20 dB.

The steady-state forward and backward transmissions of the intensity-adaptive feedback control system are shown in Figs. 5(a) and 5(b). Compared with the non-adaptive feedback system shown in Figs. 4(a) and 4(b), the intensity-adaptive feedback system exhibits a broader NRIR for incident light with some certain frequencies. For instance, for a specific value of the detuning ${\mathrm{\Delta }}_{\mathrm{in}}=-15{\kappa }_{\mathrm{ex}}$, the $\mathrm{NRIR}(0.99)$ is greatly improved to 20 dB. Within the NRIR, the forward transmission $T_{\mathrm{fw}}$ and the transmission contrast $\eta$ maintain 0.70 and 0.99, respectively. At the same time, the backward transmission $T_{\mathrm{bw}}$ is also extremely low, as is shown in Fig. 5(c).

To improve the NRIR, the coefficient $\tilde{A}$, limited by the performance of the electric amplifier of the feedback circuit, adaptively adjusts to different incident intensities [Fig. 5(d)]. According to Eq. (18), $\tilde{A}$ is inversely proportional to the input power ($P_{\mathrm{in}}$). Therefore, the NRIR is determined by the lower boundary ${\tilde{A}}_{\mathit{L}}$ and the upper boundary ${\tilde{A}}_{\mathit{U}}$ of $\tilde{A}$, so $\mathrm{NRIR}=10{\log }_{10}({\tilde{A}}_{\mathit{U}}/{\tilde{A}}_{\mathit{L}})$. For instance, if $1⩽\tilde{A}⩽1000$, varying over 30 dB, $\mathrm{NRIR}(0.99)$ is broadened to 30 dB.

To avoid the dynamic reciprocal problem by temporal multiplex, we input pulsed signals into the system from two opposite directions. Rectangularly pulsed signals are incident in both forward and backward directions with a long enough delay ${\tau }_{\mathit{d}}$, so that the temporal overlap in the system between the forward and backward propagating pulses can be avoided. We assume the time delay of feedback circuit ${\tau }_{\mathit{f}}$ is short enough, and the pulse duration ${\tau }_{\mathit{p}}$ is longer than the feedback delay and hysteresis duration ${\tau }_{\mathit{h}}$ but shorter than the pulse delay, i.e., ${\tau }_{\mathit{f}},{\tau }_{\mathit{h}}\ll {\tau }_{\mathit{p}}<{\tau }_{\mathit{d}}$. We take the pulse repetition interval $400\pi {\kappa }_{\mathrm{ex}}^{-1}$, and ${\tau }_{\mathit{p}}=400\pi /3{\kappa }_{\mathrm{ex}}^{-1}$, ${\tau }_{\mathit{d}}=200\pi {\kappa }_{\mathrm{ex}}^{-1}$. The transmissions of pulsed signals in two opposite directions are shown in Fig. 6. The upper panel shows high transmissions of the pulse trains in the forward direction, while the lower panel indicates very low backward transmissions of the pulse streams.

We proposed a feedback-induced nonreciprocal optical system using the experimentally existing technique. The frequency-tunable cavity can be made by using a 2 cm long FP cavity consisting of two mirrors with reflectivity of $R=98\%$ and a 0.15 cm long ${\text{LiNbO}}_3$ nonlinear crystal. The external loss of the cavity caused by the two mirrors is calculated to be about ${\kappa }_{\mathrm{ex}}\approx 2\pi \times 22.1\mathrm{MHz}$. The ${\text{LiNbO}}_3$ crystal typically has an absorption loss of 0.3 m⁻¹. We assume that two ends of the crystal are coated with 99.9% anti-reflection coating, leading to a total internal cavity loss of ${\kappa }_{\mathit{i}}\approx 2\pi \times 5.4\mathrm{MHz}\sim 0.2{\kappa }_{\mathrm{ex}}$. In such design, the free spectral range (FSR) of the cavity is calculated to be $\mathrm{FSR}\approx 311{\kappa }_{\mathrm{ex}}$, which ensures that the input detunings ${\mathrm{\Delta }}_{\mathrm{in}}$ and the resonance frequency shifts ${\mathrm{\Delta }}_{a(b)}^f$ cannot exceed the FSR and retains the system in a single cavity mode. Note that all our calculation results meet the conditions of $\mathrm{FSR}>\{{\mathrm{\Delta }}_{\mathrm{in}},{\mathrm{\Delta }}_{a(b)}^f\}$. The higher the input power $P_{\mathrm{in}}$, the larger the feedback-induced shift ${\mathrm{\Delta }}_{a(b)}^f$ and the required cut-off frequency ${\omega }_c$. In some sense, the aforementioned conditions related to FSR also limit the applicable highest $P_{\mathrm{in}}$ and thus the available NRIR.

Alternatively, the tunable cavity can also be realized by utilizing an FP cavity attached to a piezoelectric element [40]. The feedback signals drive the piezo to produce the resonance frequency shifts. This setup with much smaller intrinsic cavity loss can achieve a lower insertion loss, as shown in Fig. 3(d).

Our proposed method can also be applied to a microwave system. In the microwave system, the frequency-tunable FP cavity can be replaced with a microwave resonator (transmission-line resonator) made from a superconducting electronic circuit [41,42]. The feedback signals can change the frequency of the electric field in transmission line resonators through the tunable inductance of a superconducting quantum interference device (SQUID) [35,43,44], leading to resonance frequency shifts. Thus, in the same way, nonreciprocal microwave transmission can be realized by utilizing direction-dependent feedback signals. Moreover, a microwave-frequency photon source, BS, filter, amplifier, and detectors can be integrated on a chip [45]. Thus, nonreciprocal transmission of pulsed microwave signals can also be realized on-chip with our protocol.

In principle, fast switching can also isolate the reflected pulses from the pulsed input if the arriving time is precisely known. However, conventional switching is reciprocal because it does not break the time-reversal symmetry in any sense. On the other hand, the realization of a high-speed photonic switch is changing [28,46].

In our design, the feedback current is generated by the optical signal itself and creates an effective Kerr nonlinearity to the optical subsystem. Our nonlinear nonreciprocal device breaks the time-reversal symmetry, and the scattering matrix is asymmetric when the input and reflected pulses do not arrive at the same time [47]. Our feedback approach takes a step towards isolation of pulsed signals and has the potential to bypass the constraint due to dynamic reciprocity, if the reflected pulses are delayed by a long enough time from the arrival of the input pulsed signal.

In conclusion, we have explored feedback control to induce a directional Kerr nonlinearity in the cavity to achieve nonreciprocal transmission. By using intensity-adaptive feedback, we have realized a broad NRIR maintaining the transmission contrast of 0.99. Our protocol can also be implemented in a superconducting electronic circuit for nonreciprocal microwave transmission. Despite our system being subject to dynamic reciprocity, we have presented a general method to solve the outstanding challenge of integrated nonreciprocal devices. The proposed scheme promises useful applications in situations involving pulsed signals.