Perfect nonreciprocity by loss engineering

Realization of nonreciprocal transmission with low insertion loss and high contrast simultaneously is in great demand for one-way optical communication and information processing. Here we propose a generic approach to achieving perfect nonreciprocity that allows lossless unidirectional transmission by engineering energy losses. The loss of the intermediate mode induces a phase lag impinging on the indirect channel for energy transmission, which does not depend on the energy transmission direction. When the direct transmission channel coexists with the indirect lossy transmission channel, the dual-channel interference can be tuned to be destructive simultaneously for backward transmission from the rightmost mode to the leftmost mode, and for forward transmission from the leftmost mode to the intermediate mode. The former interference outcome corresponds to 100% nonreciprocity contrast, and the latter guarantees zero insertion loss for forward transmission. Additionally, our scheme also allows a nonreciprocity response over a wide bandwidth by increasing the losses while keeping perfect nonreciprocity at resonance. The robustness against loss indicates that our scheme is advantageous in the implementation of nonreciprocal optical devices with high performance.

Figure 1

(a) A system composed of two resonance modes $a_1$ and $a_2$ coupled with a lossy mode $b$. The coupling constant is denoted by $g_l$ ($g_r$). $a_1$ and $a_2$ are also coupled with each other directly. The coupling coefficient is represented as the constant $g_a$ with a phase factor $e^{i\theta }$. The energy decay rates of the resonance modes $a_1$, $a_2$, and $b$ are represented as ${\gamma }_1$, ${\gamma }_2$, and $\kappa$, respectively. (b) and (c) Energy-domain illustration of dual-channel interference for implementing perfect nonreciprocity. The solid horizontal lines represent the resonance frequencies of the modes, and the dashed horizontal lines are respective detunings. Engineering the interference of the two channels for backward energy transmission (b), i.e., for direct channel $a_2\to a_1$ and indirect channel $a_2\to b\to a_1$, to be destructive, zero energy occupation on $a_1$ warrants that input from $a_2$ cannot be transmitted to $a_1$, indicating the blockade of backward energy transmission. Applying this interference mechanism to the forward energy transmission (c), when the interference between the direct channel $a_1\to b$ and indirect channel $a_1\to a_2\to b$ is tuned to be destructive, zero energy occupation on $b$ makes it transparent for forward transmission ($a_1\to a_2$), indicating zero insertion loss for forward transmission. Hence lossless unidirectional forward energy transmission can be implemented by combining the interference mechanism depicted in (b) and (c).

Figure 2

(a) Forward transmission efficiency ($T_{\to }$, blue solid curve) at the resonant point ($\omega =0$) as a function of the coupling strength ratio $g_a/g$, where $g_l=g_r=g$ is assumed for simplicity. The backward transmission efficiency under the condition of unidirectional forward transmission [Eq. (6)] is 0 ($T_{\leftarrow }$, red dotted curve). (b) The corresponding insertion loss for unidirectional forward transmission and nonreciprocity contrast as a function of the coupling strength ratio $g_a/g$. (c) The transmission efficiencies ($T_{\to }$, blue solid curve; $T_{\leftarrow }$, red dotted curve) at the resonant point ($\omega =0$) as functions of the coupling phase $\theta$, where the coupling strengths are chosen to fulfill the condition given in Eq (7). (d) The corresponding energy occupation of the intermediate mode $b$ at the resonant point ($\omega =0$) as a function of the coupling phase $\theta$ when considering input from $a_1$ (blue solid curve) and $a_2$ (red dotted curve). The left parameters are fixed as ${\gamma }_{1/2}=\kappa$, ${\mathrm{\Delta }}_{1/2}=\delta =0$.

Figure 3

(a) The unidirectional forward transmission efficiency $T_{\to }$ at the resonant point ($\omega =0$) as a function of the detuning-to-loss ratios $\delta /\kappa$ and $\mathrm{\Delta }/\gamma$, where the coupling strengths are tuned to satisfy the condition of perfect nonreciprocity [Eq. (7)]. The white dashed line illustrates the condition of perfect nonreciprocity for the detuning-to-loss ratios, i.e., $\mathrm{\Delta }/\gamma =\delta /\kappa$. (b) The corresponding insertion loss for unidirectional forward transmission as a function of the ratio $\mathrm{\Delta }/\gamma$ when $\delta /\kappa =3$.

Figure 4

(a) The insertion loss of the unidirectional forward transmission and the required coupling strengths [determined by the condition of perfect forward nonreciprocity given in Eq. (7)] as functions of the energy loss rate $\gamma$. (b) Nonreciprocity bandwidth ($\mathrm{\Delta }\omega$) as a function of the loss rate $\gamma$. The red dotted curve illustrates the asymptotic scaling of the nonreciprocity bandwidth in the limit of $\gamma /\kappa \ll 1$. The inset plots the nonreciprocity spectrum function $I\left(\omega \right)=T_{\to }\left(\omega \right)-T_{\leftarrow }\left(\omega \right)$ when choosing $\gamma /\kappa =20$.