1. Introduction

Quantized Hall conductance in the quantum Hall effect (QHE) manifests as a topological property of matter [1]. Topological properties can also be obtained in integrated optical systems by an array of coupled microresonators (MRs) that mimics key features of topological systems, such as the Hofstadter’s model [2] or achieves robust edge states in the absence of an external magnetic field [3]. In the last example, the integrated photonic circuit provides an artificial gauge field for photons moving within each unit cell of the array, resulting in robust transport properties along the edges of the structure [3–13]. These topological properties have been observed in two-dimensional (2D) photonic structures having as unit cell a plaquette of four microresonators connected by four link microresonators [3,4]. In order to avoid problems associated with light scattering from surface wall roughnesses, such as the excitation of counterpropagating modes by the backscattering of the propagating modes [14], it was selected a regime where the ratio of backscattering to coupling coefficients is low, i.e. the strong coupling regime. This has led to promising implementations of topological physics, such as the realization of lasers with unidirectional light output under time-reversal symmetry breaking [15]. Such implementations aim to revolutionize semiconductor lasers by introducing robust optical structures and scatter-free light propagation [8,9].

One of the building blocks used in integrated 2D topological lasers is the taiji-microresonator; an MR with an S-shaped waveguide inside [8,16–18]. It allows the selection of a preferred optical chirality in topological laser systems and exhibits robustness against backscattering [19,20]. However, the implementation of these topological features in integrated photonic devices is challenging because of the imperfections caused by the fabrication processes. These errors alter the spectral response of the devices, requiring the fabrication of many test structures to obtain the desired response. One of the main problems for MRs originates from backscattering caused by the surface roughness of the waveguides [14], which can lead to a significant energy exchange between clockwise (CW) and counterclockwise (CCW) modes in the MR and, even, overcome the action of the S-shaped waveguide in taiji MR [21].

In this work, we investigate the response of a photonic structure that is designed to be the building block of more complex topological photonic systems. It behaves as a four-port system whose response depends in a non-trivial way on the selected input port: When a port is used for input, the output signal is observed in two other ports. Conversely, when another port is selected, the input signal is broadcast to all four output ports. This counter-intuitive functionality is achieved in a novel device geometry for a coupled resonator optical waveguide (CROW) structure (Fig. 1). The structure consists of a sequence of two taiji MRs coupled by a link MR. For this reason, we refer to it as a taiji-CROW device. In this study, we model, fabricate, and test taiji-CROW devices and demonstrate the effect of the physical parameters controlling its geometry on its spectral response. The transmission and reflection properties of the fabricated taiji-CROW structures are investigated in the linear optical regime as a function of the selected input port. The observed characteristics are analyzed in terms of the physical properties (coupling coefficients, loss rates, and backscattering coefficients) by using the Temporal Coupled Mode Theory (TCMT) to describe the light propagation in the taiji-CROW device. In addition, we show that the finite fabrication spatial resolution introduces defects and random geometrical variations that strongly modify the taiji-CROW device’s response, making the fabricated structure unsuited for observing topological effects. Finally, our study provides the minimum value of the ratio between the coupling and the backscattering coefficients needed to preserve the ideal behavior and avoid the stochastic surface roughness from shedding the desired chirality. Therefore, this study gives key insights for the design of more complex topological structures based on integrated MRs.

 

Fig. 1. The taiji-CROW device. Two taiji microresonators (labeled with numbers 1 and 5) are coupled by a link microresonator (number 3). Starting from the left taiji MR, the CW (CCW) mode is represented by an enumerated black (red) arrow placed below the resonator. The continuing modes in the sequential resonators follow the same color coding: $1\rightarrow 3\rightarrow 5$ are black, and $2\rightarrow 4\rightarrow 6$ are red. Enumerated triangles represent grating couplers which are the input/output device ports. $\Gamma$ letters refer to the coupling coefficient between the taiji MR and the bus waveguides ($\Gamma _{1}$ and $\Gamma _{4}$) or between the taiji MR and the link MR ($\Gamma _{2}$ and $\Gamma _{3}$).

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The paper is organized in the following way. Section 2 introduces the modeling based on the temporal coupled mode theory (TCMT) and describes the ideal optical spectra of the taiji-CROW device. Then, section 3 introduces the backscattering model and its non-Hermitian character. Section 4 shows a simplified modeling of the structure based on a hopping approximation which is valid when the link resonator resonance is off with respect to the taiji MR resonance. Section 5 describes the fabricated structures. Section 6 reports a comparison of the experimental transmission and reflection spectra of the taiji-CROW device and their fit with the TCMT model. A discussion of the actual fit parameters and of their spectral dependence is also presented. Finally, Section 7 concludes the paper by elaborating on the critical ratio to preserve the desired chiral properties of the taiji-CROW device. In the appendixes, a comparison between the TCMT and the transfer matrix method (TMM) modeling of the structure is performed to link the intermodal coupling to the coupling coefficient between different MRs. The design details and fitting parameters are there reported as well.

2. Design and theoretical model

Figure 1 shows the taiji-CROW device. Two taiji MRs are linked by an MR and coupled to two side bus waveguides where grating couplers define the four input/output device ports. The embedded S-shaped waveguide in taiji MR couples CW modes to CCW modes but not vice versa. This non-reciprocal intermodal coupling between the two taiji propagating modes makes the taiji MR also a non-Hermitian structure [16,22,23]. Remarkably, when the input signal is coupled to port 1, only CCW modes are excited (in taiji MRs) and the output signal is only detected in port 2 and port 3. On the other hand, when the input signal is coupled to port 2, the S-shaped waveguide allows coupling between the CW and CCW modes and therefore an output signal is observed from all four device ports. Indeed, the taiji-CROW device can generate different transmission spectra without violating the Lorentz reciprocity theorem at each output port when it is excited from any of the four input ports. In addition, due to the S-shape waveguide, the reflection spectra (output signal from the same port where the input signal is coupled to) show a strong unidirectionality depending on which port is excited (e.g. when port 1 is used for input, no reflection is observed; however, when port 2 is used a strong reflection is observed). The device geometry also allows controlling the phase difference between the counter-propagating modes by shifting the link MR in vertical direction with respect to the taiji MRs, without altering the resonance frequencies of the individual MRs.

Within the TCMT, the propagating modes obey the differential equation [22]:

Here $\Gamma _1$ and $\Gamma _4$ are the coupling coefficients (coupling rates) between the bus waveguide and the taiji MR (see Fig. 1). The 6x6 Hamiltonian $\textbf {H}$ in Eq. (1) can be written as

Since the TCMT does not relate the intermodal coupling coefficients ($\beta _{ij}$) to the coupling rates ($\Gamma _2$ and $\Gamma _3$) at the taiji MR-link MR coupling points, we also use the transfer matrix method (TMM) to derive the transmission/reflection spectra of the system (see Appendix A). A comparison between the TCMT and TM modeling yields:

Figures 2(a) and 2(b) show the computed output responses as a function of the frequency detuning ($\Delta \omega = \omega - \omega _{\text {1}}$) for $\omega _{\text {1}} = \omega _{\text {3}} = \omega _{\text {5}}$ when the input signal is coupled to port 1 or port 3. The taiji-CROW device behaves as expected: when the input signal is coupled to port 1 we observe an output signal at ports 2 and 3, while when the input signal is coupled to port 3 we observe an output signal from all the ports (1,2,3,4) due to mode propagation in the S-shape waveguide. Note that for ease of reading of Fig. 2 the through signals $1\rightarrow 2$ and $3\rightarrow 4$ are not plotted. In addition, we observe that the modal interaction causes a splitting of the MR resonances in three peaks even though $\omega _{\text {1}}=\omega _{\text {3}}=\omega _{\text {5}}$ (due to the degeneracy in the eigenvalues of $\textbf {H}$ in Eq. (3)). The peak separation depends on the $\Gamma _{2}$ and $\Gamma _{3}$ coupling coefficients, while $\Gamma _{1}$, $\Gamma _{4}$ and $\gamma _{\text {t},i}$ determine their widths. Note also that Eq. (3) assumes that the intermodal coupling coefficients are Hermitian (e.g., $\beta _{\text {13}}=-\beta _{\text {31}}^*$) and identical for both directions of propagation at any point couplings (e.g., $\beta _{\text {13}}=\beta _{\text {24}}$). This simplifying assumption was not imposed in the fit. The non-Hermitian nature of the modal coupling in the taiji MRs (i.e. $\beta _{\text {12}}=\beta _{\text {56}} \neq$ 0 and $\beta _{\text {21}}=\beta _{\text {65}}=0$) produces different transmission spectra at the output ports (Fig. 2(a)) and different reflection spectra (Fig. 2(b)) for different input signal ports. Note also that the transmission spectra fulfill the Lorentz reciprocity theorem since the same spectra are obtained for each pair of reciprocal ports (e.g., transmission $1\rightarrow 3$ and $3\rightarrow 1$ in Fig. 2).

 

Fig. 2. Output signal spectra for the taiji-CROW device as a function of the frequency detuning with respect to the taiji MR resonant frequencies ($\Delta \omega =\omega -\omega _1$). (a) Transmission and (b) reflection spectra for the different input and output ports reported in the legend as two numbers, the first being the input port and the second being the output port. The two spectra refer to the case where all MRs have the same resonant frequency ($\omega _1=\omega _{3}=\omega _5$). (c) Transmission (black lines) and reflection (red lines) spectra when the input port is port 3 for different values of the resonant frequency of the link MR with respect to the taiji MR resonant frequency. The values are reported in the different spectra as $\omega _3-\omega _1$. The black, red, and blue triangles on the transmission peaks label the different contributions. The inset shows the intensities of the individual transmission peaks as a function of $\omega _3-\omega _1$. The colors of the symbols in the inset correspond to the colors of the labels on the spectra. (d) Same as (c) when the backscattering in the taiji-CROW device is introduced: the same backscattering coefficient for the different MRs is used. Spectra were vertically shifted in (c) and (d) for clarity. We used: $\Gamma _{1} = \Gamma _{4} = 8~{\rm GHz}$, $\beta _{12} = \beta _{56} = 6~{\rm GHz}$, $\beta _{13} = \beta _{24} = \beta _{35} = \beta _{46} = 100.5~{\rm GHz}$, and $\beta _{\text {BS},1} = \beta _{\text {BS},3} = \beta _{\text {BS},5} = 10~{\rm GHz}$ only for (d), otherwise zero. “sf” in panel (d) stands for the scale factor: the y-values are multiplied by the given number.

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Figure 2(c) illustrates how the spectra change when the resonant frequency $\omega _{\text {3}}$ of the link MR differs from those of the taiji MRs ($\omega _{\text {1}} = \omega _{\text {5}}$). As the frequency difference increases, the spectral splitting of the transmitted signal at the taiji frequency decreases while the transmission peak at the resonant frequency of the link MR shifts to high frequencies (for $\omega _{3}-\omega _1= 600~{\rm GHz}$ the peak shift is larger than the frequency detuning shown in Fig. 2(c)). The inset in Fig. 2(c) shows that the intensities of the different transmission peaks scale with the resonant frequency difference. The intensity of the $\Delta \omega = 0~{\rm GHz}$ peak is constant for $\omega _{3}-\omega _1$ up to $1000~{\rm GHz}$, while the intensity of the peak at negative $\Delta \omega$ increases and the intensity of the peak at large positive $\Delta \omega$ (due to the link MR resonance) vanishes. For large resonant frequency differences, the two taiji related peaks merge at $\Delta \omega \simeq 0~{\rm GHz}$ and their intensities start to decrease when the two taiji MRs’ resonant modes become degenerate. This is a consequence of a decreased effective coupling between the two taiji MRs due to the missing transmission resonance in the link MR at those frequency detunings. The same behavior is observed for different inter-resonator coupling strengths but the $\Delta \omega$ value where the intensities start to drop shifts toward smaller $\omega _{3}-\omega _1$ for weaker couplings. Detuning values of the resonant frequencies mentioned here are much smaller than the free spectral range (FSR) of the structures as required by TCMT. The taiji-CROW device in the off-resonance condition for the link MR can also be described as two MRs coupled by a hopping term (see section 4) [4].

3. Backscattering

Side wall roughness backscattering might excite the counterpropagating optical mode in microresonators [24,25]. To account for this in our model simulation Eq. (1), we introduce off-diagonal-backscattering coupling terms into the Hamiltonian in Eq. (3). Let us consider a backscattering coupling coefficient $\beta _{\text {BS},ij}$ which couples the $\alpha _{i}$ and $\alpha _{j}$ modes in each MR and assume a Hermitian backscattering ($\beta _{\text {BS},ij} = -\beta _{\text {BS},ji}^{*}$) [14]. Then, we call $\beta _{\text {BS},1}$, $\beta _{\text {BS},3}$ and $\beta _{\text {BS},5}$ the backscattering coefficients associated with each MR as enumerated in Fig. 1. Finally, we define $\beta _{34}$ the non-Hermitian component of the counter-propagating mode couplings in the link MR while we consider that for the taiji MRs the coefficients $\beta _{12}$ and $\beta _{56}$ take into account both the intermodal coupling due to the S-shape waveguide and the non-Hermitan part of the backscattering coefficient. As a result, the Hamiltonian of the system with the backscattering becomes:

Figure 2(d) shows the transmission and reflection spectra of the taiji-CROW device when an equal backscattering coefficient is considered for the taiji and link MRs. To highlight the effect of the backscattering, we chose a relatively high value for the backscattering coefficient which is comparable to the intermodal coupling coefficient due to the S-shape waveguide in the taiji MRs. It is observed that the interference between the CW and CCW modes excited by the backscattering causes a spectrally resolvable split in the resonance frequencies [25,26]. Consequently, the single transmission peaks are split by the backscattering into distinctly resolvable pairs. As in Fig. 2(c), increasing the resonant frequency of the link MR causes the split responses of the taiji MRs to merge in transmission or reflection peaks which have now a more complex linewidth than in absence of the backscattering.

4. Hopping model

When the link MR is off-resonance with respect to the taiji MR resonance and for weakly coupled high-Q taiji MRs, the full TCMT model can be simplified by describing the coupling between the two taiji MRs by a hopping term which directly couples the taiji optical modes [4]. In this way, the action of the link MR is simplified by considering a further $\beta _{ij}$ coupling coefficient that links the CW or the CCW modes of the two taiji MRs. This is possible since the effective coupling becomes weaker when the link MR is off-resonance. Consequently, $\boldsymbol {\alpha }$ in Eq. (1) becomes a vector of $4$ elements $[\alpha _1,\alpha _2, \alpha _5, \alpha _6]$ and $\textbf {H}$ reduces to a $4\times 4$ matrix which we name $\textbf {H}_{\text {h}}$:

Figure 3 shows the $1 \rightarrow 3$ transmission spectra for different values of the resonant frequency of the link MR with respect to the taiji MR resonant frequencies. For each resonant frequency value, the best value of the hopping coupling coefficient was found by matching the transmission peak intensity and spectral positions in the two models. The other parameters are the same as those used in Fig. 2(d). It is observed that the models agree at resonant frequency differences where the hopping coefficient amplitude is found to be small. However, in the strong coupling regime (i.e. small resonant frequency differences) the spectra calculated by the two models differ as one would expect. In addition, these results show that the observed asymmetry with respect to $\Delta \omega =0$ in the full model is caused by the link MR. The asymmetry is evident for small resonant frequency differences and vanishes as the link MR resonance gets different from the taiji resonance frequency. It should be noted that we considered only one modal order in our calculations since the link MR has a large free spectral range (FSR $\sim 1900~{\rm GHz}$). However, for a more rigorous analysis, the effect of the consecutive modal orders should be incorporated, especially for small FSR of the link MRs. To give a more quantitative measure of the agreement between the two models, the area under each curve (AUC) was calculated and compared. The inset in Fig. 3 plots AUC versus the difference between the MRs resonant frequencies: for resonant frequency difference $\lesssim 500~{\rm GHz}$ the results of the two models start to deviate.

 

Fig. 3. Comparison of the 1 $\rightarrow$ 3 transmission spectra calculated with the hopping (red lines) and the full (black lines) models for various link MR resonant frequencies. Each panel shows the values of the resonant frequency difference and of the used hopping coupling coefficient amplitude $\beta _h$. The inset shows the area under each curve (AUC) versus the resonant frequency difference for the two models. The same parameters as in Fig. 2(d) were used for both models.

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5. Experimental details

Taiji-CROW devices were designed in silicon photonics with a waveguide cross section of 450 nm x 220 nm embedded in a silica cladding. Taiji MRs have a ring-shaped geometry with a radius of 20 $\mu$m and are point-coupled to a bus waveguide with a gap of 263 nm. Their inner S-shaped waveguide is formed by connecting two semicircles with a radius of 9 $\mu$m. The gap between the semicircles of the S-shaped waveguide and the external rim of the Taiji MR is 296 nm. At both ends of the S-shaped waveguide there is an inverse taper to ensure no back-reflections. The link MR has a racetrack geometry obtained by joining two pairs of straight sections with two semicircles. The radius of the semicircles and the length of the straight sections are chosen to obtain two configurations of the link MR: (i) with the same perimeter as the taiji MR and (ii) with twice the perimeter of the taiji MR. Specifically, in the first, the radius of the semicircle is 10 $\mu$m, while in the second, it is 25 $\mu$m. The gap between the taiji MRs and the link MR is 263 nm, the same as between the taiji MR and the bus waveguide, hence $\Gamma _1=\Gamma _2=\Gamma _3=\Gamma _4$ by design. In the two configurations of the link MR, the length of the straight sections has been changed to have the link MR on or off resonance with the taiji MRs (more details are provided in Appendix C). The devices were fabricated by IMEC/Europratice within a multi-project wafer run.

The optical responses of the devices were measured using an interferometric optical setup that allows simultaneous measurement of transmission and reflection in both excitation directions [22]. The output of a fiber-coupled continuous-wave (CW) tunable laser (Yenista OPTICS, TUNICS-T100S) is divided into two arms by a 50/50 fiber splitter. Each arm is equipped with a variable optical attenuator (VOA), a fiber polarization controller, an optical circulator, and a photodetector. At the end of each arm, light is coupled to the device input port through a single-mode stripped fiber. For spectral measurements, we operated the CW tunable laser at $3~{\rm mW}$ with the VOA set to ensure a linear response, resulting in a power of about 30 $\mu$W in the device. Further details of the optical setup are provided in [22].

6. Results and discussion

Measurements show a lack of correlation between the observed spectral feature and the design parameter variations, e.g. the relative position of the resonance frequencies of the MRs. In addition, distinct splitting of the spectral transmission peaks was observed due to the presence of a random strong backscattering. This indicates the challenge of making such structures, since they are extremely sensitive to fabrication processes that significantly determine the device performance. For these reasons, the experimental spectral features will be analyzed by using the fit parameters, not by the nominal design parameters.

Figure 4 shows the transmission and reflection spectra for three different devices with different resonance conditions. The optical signal was input from port 1 or port 3, i.e. in a pair of reciprocal ports. For each device (shown in Fig. 4(a), 4(b), and 4(c)), the transmission spectra (13 or 31, where the first number refers to the input port and the second the output port) are identical while the reflection spectra (11 or 33) differ. Three main peaks are observed, which are split or broadened by a significant backscattering from the waveguide surface-wall roughness [14,26]. Backscattering is also the cause of the strong reflection observed from port 1. In fact, in an ideal taiji-CROW device, no reflection from port 1 is expected when the input signal is injected in port 1. The asymmetry in the split resonance lineshape is attributed to the non-Hermitian part of the backscattering. A fit of the experimental spectra with the model (full Hamiltonian Eq. (6)) is shown as dashed red lines in Fig. 4. In the fit procedure, the four spectra (i.e., two transmission and two reflection spectra) were simultaneously fit with the same parameters to increase the overall confidence and to reduce the parameters’ mutual dependence. Moreover, once a parameter set is found for a device, we used the same fit parameters to successfully fit the spectra measured from other pairs of reciprocal ports (e.g. port 2 and port 4). The consistency of the results validates the proposed model and the extracted parameter values.

 

Fig. 4. Measured spectra (black line) and fit (dashed red line) for three different devices which have one resonant frequency very different from the others: $\omega _{\text {1}}$ for (a), $\omega _{\text {5}}$ for (b) and $\omega _{\text {3}}$ for (c). The top panels refer to the 11 reflection (input port 1 output port 1). The middle panels to the 33 reflection (input port 3 output port 3). The bottom panels to the 13 or 31 transmission (input port 1 or 3 and output port 3 or 1). The insets show the taiji-CROW device with an arrow pointing to the MR which is responsible for the low frequency peak which is also marked by an arrow. The vertical dashed lines indicate the resonant frequencies of each MR as they result from the fit. “sf” stands for the scale factor. The parameters derived from the fit are reported in Tables 2, 3, and 4.

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Each device presented in Fig. 4 has one of the resonance frequencies slightly different from the others: the associated peak is separated from the others and corresponds to $\omega _{\text {1}}$, $\omega _{\text {5}}$ and $\omega _{\text {3}}$ in the cases of Fig. 4(a), Fig. 4(b) and Fig. 4(c), respectively. The vertical dashed lines in Fig. 4 mark the isolated MR resonance frequencies which are different from the observed spectral peak positions since they correspond to the eigenvalues of $\textbf {H}$ due to the mode couplings in the taiji-CROW device. Slight variations in fabrication parameters cause differences in taiji MRs that would otherwise be identical. These variations are also observed in the fact that the link MR is closer in resonant frequency to one of the taiji MRs, i.e., the exchange of energy between MRs 3-5 is stronger than the one between MRs 1-3 for Fig. 4(a) and the opposite happens for Fig. 4(b) (i.e., $|\omega _5-\omega _3| < |\omega _1-\omega _3|$ and $|\omega _1-\omega _3| < |\omega _5-\omega _3|$, respectively). In addition, the inter-resonator mode coupling coefficients obtained from the fit are different from each other even if they are designed to be equal: $\beta _{\text {35}}, \beta _{\text {46}} > \beta _{\text {13}}, \beta _{\text {24}}$ in Fig. 4(a) and $\beta _{\text {13}}, \beta _{\text {24}} > \beta _{\text {35}}, \beta _{\text {46}}$ in Fig. 4(b). Although the spectra show similar high frequency peak separations ($\sim$125 GHz in Fig. 4(a) and $\sim$130 GHz in Fig. 4(b)), the differences between the isolated resonance frequencies differ: $|\omega _{\text {5}} - \omega _{\text {3}}|\;\sim$ 61 GHz in Fig. 4(a) and $|\omega _{\text {1}} - \omega _{\text {3}}|\;\sim$ 22 GHz in Fig. 4(b). These observations imply that the coupling between the link MR and one of the taiji MRs is stronger for the device in Fig. 4(b). On the contrary, Fig. 4(c) shows the spectra for a device where the link MR is off-resonance with the taiji MRs: indeed the weak exchange of energy reduces the signal-to-noise ratio of the 13 or 31 transmission signals compared to what observed for the other devices. Additional details on the estimation of the parameters as well as the fit parameter values are available in Appendix D and Appendix E.

Since the link MR is out of resonance with respect to the taiji MRs for the device in Fig. 4(c), we could use the hopping model to fit the data. Figure 5 shows the results. To simplify even further, we used the same parameters for the two taiji MRs during the fit and still obtained an excellent agreement. The extracted parameters are within the same range as those obtained from the full model.

 

Fig. 5. Measured (black line) and hopping model fit (red line) spectra for single side excitation for the device of Fig. 4(c). Numbers in the panel’s legend refer to the input/output ports as indicated in the inset. The inset shows the hopping geometry of the taiji-CROW device where the link MR is replaced by a hopping term. “sf” stands for the scale factor. The parameters derived from the fit are reported in Table 5.

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The spectral dependence of the taiji-CROW device parameters can be obtained by looking at different spectral resonance modes. While the structure shown in Fig. 4(c) is better suited for topological applications due to the off-resonance link MR, its transmission exhibits a lower signal-to-noise ratio compared to the other structures. Consequently, we utilized the spectra corresponding to Fig. 4(b) for a more accurate derivation of the taiji-CROW structure’s characteristic parameters. Figure 6 shows the wavelength dispersion of the backscattering coefficients (top panel) and the 1 $\rightarrow$ 3 transmission spectra (bottom panel) for the device of Fig. 4(b). Noticeably, although the backscattering coefficients are different from zero in the whole observed spectral range, backscattering causes a transmission peak splitting only for few modal orders (marked and shown as ${\#}3$, ${\#}4$ and ${\#}5$ in the bottom panel of Fig. 6). This is due to its random nature which affects the wavelength region where peak splitting is observed. In these cases, $\beta _{\text {BS,1}}$ is distinctly different from the other coefficients with a large value of about $19~{\rm GHz}$. It is also worth mentioning that when the fit was forced to have higher values for $\beta _{\text {BS,3}}$ and/or $\beta _{\text {BS,5}}$ rather than $\beta _{\text {BS,1}}$, no accurate fits were obtained. Overall, the backscattering coefficients vary in the $2-20~{\rm GHz}$ range for all the investigated devices as reported in [21].

 

Fig. 6. Wavelength dispersion of the backscattering coefficients (top panel) and 1 $\rightarrow$ 3 transmission spectra (bottom panel) for the same device of Fig. 4(b). Each data point in the top panel shows the average of at least eight fit runs where the error bars represent the standard error of the mean. The insets in the bottom panel are the zoomed-in spectra of the resonances marked with arrows. The transmission spectrum shown in Fig. 4(b) is the peak labeled #5.

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Figure 7 presents the spectral dependence of the other fit parameters for the device of Fig. 4(b). We observe that the coupling coefficients between the bus waveguide and the taiji MRs ($\Gamma _{\text {1}}$ and $\Gamma _{\text {4}}$) smoothly increase for longer wavelengths (Fig. 7(a)). This is coherent with the decreased modal confinement of the waveguide propagating mode for longer wavelengths [27]. Note that the values of $\Gamma _1$ and $\Gamma _4$ are similar for each modal order which matches the nominal identical values for the bus waveguide/taiji MR gaps (see section 5).

 

Fig. 7. Spectral dispersion of the estimated fit parameters for the device of Fig. 4(b). (a) Coupling coefficients between the bus waveguide and the taiji MRs. (b) Total loss coefficients for the taiji and link MRs. (c) Non-Hermitian intra-resonator mode coupling coefficients. (d) Inter-resonator mode coupling coefficients. Each data point shows the average of at least eight fit runs, and the error bars represent the standard error of the mean. $\Gamma _{\text {4}}$ in (a) and $\beta _{\text {24}}$ and $\beta _{\text {46}}$ in (d) are shifted by $0.8~{\rm nm}$ in x-axis for clarity.

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Figure 7(b) shows the wavelength dispersion of the total loss rates $\gamma _{\text {t,i}}$ for each MR. $\gamma _{\text {t},3}$ for the link MR varies around 2 GHz which reflects its internal losses. $\gamma _{\text {t},1}$ and $\gamma _{\text {t},5}$ for the two taiji MRs value about 11–18 GHz. They are similar and larger than $\gamma _{\text {t},3}$ due to the S-shape waveguide and bus waveguide couplings.

Figure 7(c) refers to the non-Hermitian intra-resonator mode coupling coefficients. $\beta _{\text {12}}$ and $\beta _{\text {56}}$ for the taiji MRs are mainly associated with the S-shape waveguide and lie in the range of 5–12 GHz. The small yet finite ($\sim$ 2 GHz) $\beta _{\text {34}}$ for the link MR is due to the non-Hermitian component of the backscattering. These findings are also confirmed by the asymmetric splitting observed in the transmission peaks (see Fig. 4 and Fig. 6) and are quantitatively consistent with previous studies [14,25].

Figure 7(d) shows the inter-resonator mode coupling coefficients $\beta _{\text {13}}$, $\beta _{\text {24}}$ and $\beta _{\text {35}}$, $\beta _{\text {46}}$ between resonators 1-3 and 3-5 (see also Fig. 1). While the coupling coefficients for counter-propagating modes are similar in each coupling node, i.e. $\beta _{\text {13}} \approx \beta _{\text {24}}$ and $\beta _{\text {35}} \approx \beta _{\text {46}}$, they differ between the two coupling nodes, i.e. $\beta _{\text {35}}, \beta _{\text {46}} \geq \beta _{\text {13}}, \beta _{\text {24}}$. This result is consistent with the estimated resonance frequencies where we deduced that the coupling between the link MR is stronger with the taiji MR 5 than with the taiji MR 1. The situation is reversed ($\beta _{\text {13}}, \beta _{\text {24}} \geq \beta _{\text {35}}, \beta _{\text {46}}$) for the device in Fig. 4(a) while $\beta _{\text {13}}, \beta _{\text {24}} \approx \beta _{\text {35}}, \beta _{\text {46}}$ for the device in Fig. 4(c). Note that this last case corresponds to the case where the link MR is out of resonance with respect to both taiji MRs. The $\beta _{ij}$ values of Fig. 7(d) are then used to calculate the $\Gamma _2$ and $\Gamma _3$ coupling coefficients with Eq. (5a) and Eq. (5b). In the calculation, we used $\tilde {f} = c ~ {\rm FSR} / \lambda _{0}^2$, where $c$ is the speed of light in vacuum, ${\rm FSR}$ is the free spectral range in wavelength and $\lambda _{0}$ is the mode resonant wavelength. It was found $\Gamma _2 = (5.5 \pm 0.9)$ GHz and $\Gamma _3 = (6.4 \pm 0.6)$ GHz. Even if they diverge slightly at higher wavelengths, these values are comparable with those found for $\Gamma _1$ and $\Gamma _4$ (Fig. 7(a)) and satisfy the design condition ($\Gamma _1=\Gamma _2=\Gamma _3=\Gamma _4$). It is worth mentioning here that all samples presented in this study show similar behaviors as the one discussed in Fig. 6 and Fig. 7.

The measurements show that the taiji-CROW devices we analyzed have two major problems caused by the fabrication errors: (i) a strong backscattering that couples the counter-propagating modes and generates a resonant splitting in the response, and (ii) the measured resonance frequencies differ from the design values, in particular for the two taiji MRs differ by $100-300~{\rm GHz}$ (see Fig. 4) although they are nominally identical by design.

Backscattering can degrade or even prevent the intended functionality of the device, and it is hard to control. To quantify its impact, we compared the backscattering coefficients with the inter-resonator coupling coefficients. From the experimental results, we find a ratio $|\beta _{\text {BS}}| / \Gamma \simeq 1.5 \pm 1.1$, where $\Gamma$ is the average of $\Gamma _2$ and $\Gamma _3$. To mitigate the effect of backscattering, one can either reduce it (e.g., by modifying the waveguide geometry, material, and the fabrication method [28,29]) or increase the coupling between the different MRs [4] but this means compromising on the quality factor. Specifically, we found that $|\beta _{\text {BS}}| / \Gamma \lesssim 0.17$ is the onset value for mode splitting which indicates strong backscattering (details are given in Appendix F). Since the observed backscattering coefficient is about $20~{\rm GHz}$ at its maximum, coupling coefficient $\Gamma {\gtrsim} 120~{\rm GHz}$ satisfies the mitigation condition. However, this would reduce the effects of the backscattering but will also broaden the resonances. The broadening of the resonances will reduce the quality factor of the microresonators, but at the same time will mitigate the fabrication error on the resonant frequencies. One may also compare the backscattering coefficient values with the total loss coefficient ($\gamma _{t}$), which could be particularly useful for active materials and unidirectional lasing applications [30]. At the condition for strong backscattering (i.e., the onset of the mode splitting), we obtained a ratio $|\beta _{\text {BS}}| / \gamma _{t} \sim 0.1$. This value is consistent with the threshold defined for small and large backscattering in [30]. Again, for the taiji MRs, we estimated $|\beta _{\text {BS}}|/\gamma _t \simeq 0.7 \pm 0.1$ showing that the taiji MR operates in the large backscattering regime in our experiment.

7. Conclusion

The modeling and testing of the taiji-CROW device show that this peculiar structure exhibits strong unidirectional reflection by design and a non-reciprocal intermodal coupling due to its non-Hermitian nature, which arises from the S-shaped waveguide in each taiji microresonator. Fit of the experimental results with the model yields an estimation of the characteristic physical parameters of the taiji-CROW device. A strong backscattering was observed that restrains its intended functionality. At the same time, this study provides a quantitative estimate of the minimum value for the coupling coefficient for a given backscattering coefficient needed to mitigate the surface wall backscattering influence. Quantitatively, a ratio $|\beta _{\text {BS}}| / \Gamma \lesssim 0.17$ should be fulfilled. Specifically to our devices where the backscattering coefficient is about 20 GHz at its maximum, this dictates a minimum coupling coefficient between MRs of about 120 GHz. Such a high value limits the quality factor of the taiji-CROW device and restricts the parameters of topologically integrated photonic devices based on this building block. Moreover, it leads to an increase in the width of the transmission resonances and to a decrease in the field enhancement of the microresonator. This results in lower system efficiency in active media laser applications, requiring higher system excitation power. Finally, this study underlines that a proper understanding of the device physical parameters and a high level of control in the fabrication processes are essential for realizing non-Hermitian topological photonic integrated circuits or other integrated photonic circuits for different applications.

Appendix

A. Theoretical modelling: TCMT vs TMM

Two analytical approaches have been used to model the system: the transfer matrix method (TMM) and the Temporal Coupled Mode Theory (TCMT) [14,16,31]. The main motivation was to describe the taiji-CROW device with the TCMT (suitable for more complex structures) and use the TMM as a control tool. The challenge with TCMT is to find out how the intermodal coupling ($\beta$) is related to the coupling strength ($\Gamma$) at the taiji MR-link MR coupling point. For this purpose, a simplified structure consisting of two simple MRs was modeled by using both methods (see Fig. 8). In the modeling with TMM, we used the notation given in Fig. 8(a) where the arrows with numbers indicate the electric fields $E_i$ in their propagation direction, $t_i$ and $\kappa _i$ are the transmission and coupling coefficients at each coupling points, respectively. The electric fields propagating in the opposite directions of the indicated arrows (in Fig. 8(a)) for each branch are denoted as $E_{ir}$ where $r$ stands for “reverse”. The system of equations can now be written as follows:

(8)$$\begin{matrix} \begin{array} {l} E_{0} = 1/0 ,\\ E_{1} = t_{1} E_{0} + i \kappa_{1} E_{5} ,\\ E_{2} = t_{1} E_{5} + i \kappa_{1} E_{0} ,\\ E_{3} = e^{i \delta z_1} E_{2} ,\\ E_{4} = t_{2} E_{3} + i \kappa_{2} E_{9} ,\\ E_{5} = e^{i \delta z_2} E_{4} ,\\ E_{6} = t_{2} E_{9} + i \kappa_{2} E_{3} ,\\ E_{7} = e^{i \delta z_3} E_{6} ,\\ E_{8} = t_{3} E_{7} + i \kappa_{3} E_{11} ,\\ E_{9} = e^{i \delta z_4} E_{8} ,\\ E_{10} = t_{3} E_{11} + i \kappa_{3} E_{7} ,\\ E_{11} = 1/0, \\ \delta = \frac{2 \pi}{\lambda} n_{\text{eff}} + i \alpha , \end{array} & \begin{array}{l} E_{0r} = t_{1} E_{1r} + i \kappa_{1} E_{2r} ,\\ E_{1r} = 1/0 ,\\ E_{2r} = e^{i \delta z_1} E_{3r} ,\\ E_{3r} = t_{2} E_{4r} + i \kappa_{2} E_{6r} ,\\ E_{4r} = e^{i \delta z_2} E_{5r} ,\\ E_{5r} = t_{1} E_{2r} + i \kappa_{1} E_{1r} ,\\ E_{6r} = e^{i \delta z_3} E_{7r} ,\\ E_{7r} = t_{3} E_{8r} + i \kappa_{3} E_{10r} ,\\ E_{8r} = e^{i \delta z_4} E_{9r} ,\\ E_{9r} = t_{2} E_{6r} + i \kappa_{2} E_{4r} ,\\ E_{10r} = 1/0, \\ E_{11r} = t_{3} E_{10r} + i \kappa_{3} E_{8r} ,\\ \end{array} \end{matrix}$$

 

Fig. 8. Simplified structure used for comparison of the (a) TMM model and (b) TCMT model with the corresponding parameters indicated. All the parameters are described in the text.

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Here $\alpha$ is the attenuation coefficient per unit length and $z_i$ are the distances traveled on each branch between two coupling points. For simplicity, we assumed no source of backscattering in the structure, no loss in the coupling points (i.e., $\kappa _i^2 + t_i^2 = 1$), identical MRs with perimeter $L$ and up-down physical symmetry in the system with respect to the horizontal axis passing through the middle (i.e., $z_1 = z_2 = z_3 = z_4 = L/2$). For a given incident field $E_{1r}$, the steady state solution for the transmitted field $E_{11r}$ gives

(9)$$\frac{E_{11r}}{E_{1r}} ={-}\frac{i e^{i \delta L} \kappa_1 \kappa_2 \kappa_3}{1 + e^{i \delta L}[e^{i \delta L}t_1 t_3 - t_2 (t_1 + t_3)]} ~ .$$

To describe the input/output signal in this simplified geometry with the TCMT, we followed the same method and notation reported in the main text. In short, $\alpha _i$ are the propagating modes, $\beta _{ij}$ are the intermodal coupling coefficients, $\Gamma _i$ are the coupling rates at each coupling point, and $\gamma _i$ are the intrinsic loss rate. Similarly, for a given incident field $E_{in,1}$, the steady state solution for the transmitted field $E_{out,4}$ gives

(10)$$\frac{\varepsilon_{out,4}}{\varepsilon_{in,1}}={-}\frac{2 \sqrt{\Gamma_1 \Gamma_3}~\beta_{24}}{\beta_{24} \beta_{42} + [\Delta_1 + i (\gamma_1 + \Gamma_1)][\Delta_2 + i (\gamma_2 + \Gamma_3)]} ~ ,$$

where $\Delta _1 = \omega - \omega _1$ and $\Delta _2 = \omega - \omega _2$ are the detuning frequencies from isolated resonance frequencies, $\omega _1$ and $\omega _2$, for each MR. The following substitutions are then used to compare the TMM and TCMT results [14,23,32];

(11)$$t \simeq 1 - \frac{\Gamma}{\tilde{f}} ~ ~ , ~ ~ k \simeq \sqrt{\frac{2\Gamma}{\tilde{f}}} ~ ~ , ~ ~ \tilde{f} = \frac{c}{n_g L} = \frac{c ~ {\rm FSR}}{\lambda_0^2} ~ ~ , ~ ~ \Delta = \tilde{f} ~ \Re[\delta] ~ ~ , ~ ~ e^{-\Im{[\delta]}L} \simeq 1-\frac{\gamma}{\tilde{f}} ~ ,$$

where ${\rm FSR}$ is the free spectral range, $\tilde {f}$ is the repetition frequency and $c$ is the speed of light in vacuum. Moreover, assuming $\tilde {f}\gg \Gamma _1,\Gamma _2,\Gamma _3,\gamma _1,\gamma _2$ we obtain

(12)$$\beta_{24} ={-}i \sqrt{2 \tilde{f} ~ \Gamma_2} \,.$$

When two different MRs are considered, this equation generalizes to:

(13)$$\beta_{24} ={-} i \sqrt{2 (\tilde{f_1} \tilde{f_2})^{1/2} ~ \Gamma_2}$$

where $\tilde {f_1}$ and $\tilde {f_2}$ are the inverse of the round-trip times for the MR-1 and MR-2, respectively. Note that Eq. (6) is written only for the intermodal coupling between modes 2 and 4 ($\alpha _2$ and $\alpha _4$) as shown in Fig. 8 but can be easily extended to the other intermodal coupling coefficients. We then considered a structure made by two MRs coupled by a link-MR. Accordingly, a comparison of the two models (i.e. TMM and TCMT) is made here by calculating both the field amplitudes (Fig. 9(a)) and the phases (Fig. 9(b)).

 

Fig. 9. Computed field amplitudes (a) and phase terms (b) by the TMM and TCMT model for the structure shown in the inset. The color codes for the numbers in the inset are: black - input/output ports, blue - microresonators, red - beam-splitting interfaces. We used: $L_1 = L_3 / 2 = L_5 = 40 \pi ~\mu$m, $\Gamma _{1} = \Gamma _{2} = \Gamma _{3} = \Gamma _{4} = 8~{\rm GHz}$, $\alpha = 2$ dB/cm, $\lambda _0 = 1550$ nm. These correspond to: $\tilde {f_1}=\tilde {f_5}=497~{\rm GHz}$ and $\tilde {f_3}=994~{\rm GHz}$, $t_1 = t_4 = 0.9920$, $t_2 = t_3 = 0.9886$, $\beta _{ij} = -106i~{\rm GHz}$.

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B. Hopping model

We used the simplified structure shown in Fig. 10 in the hoping model to study the taiji-CROW device when the link MR is off-resonance. Following the same notation as in the main text, the $2 \rightarrow 4$ transmitted ($t_{24}$) and the $2 \rightarrow 2$ reflected fields ($r_{22}$) are given by:

(14a)$$t_{24} ={-}\frac{2 e^{i\phi} \beta_{h} \Gamma}{-\beta_{h}^2 + (\Delta \omega + i\gamma_t)^2} , $$

(14b)$$r_{22} ={-}\frac{2 \Gamma \beta_{12}(-\beta_{h}^2e^{2i\phi} + (\Delta \omega + i\gamma_t)^2)}{(-\beta_{h}^2 + (\Delta \omega + i\gamma_t)^2)^2} , $$

where $\beta _{h}$ and $\phi$ are the coupling rate and hoping phase, $\Gamma$ is the coupling rate between the bus waveguide and the taiji MR, $\beta _{12}$ is the intra-resonator mode coupling due to the S-shape waveguide, $\gamma _t$ is the total loss rate, and $\Delta \omega$ is the frequency detuning as described in the main text. We assumed $\beta _{h}=|\beta _{15}|=|\beta _{62}|=|\beta _{51}|=|\beta _{26}|$ with $\beta _{15}=\beta _{62}=\beta _{h}e^{+i\phi }$ and $\beta _{51}=\beta _{26}=\beta _{h}e^{-i\phi }$ (for clockwise modes see Fig. 10(a) and for the counterclockwise ones see Fig. 10(b)). We also assumed identical taiji MRs (i.e., $\Gamma _1 = \Gamma _4$, $\beta _{12}=\beta _{56}$, $\Delta \omega = \omega - \omega _1 = \omega - \omega _5$, $\gamma _{t,1} = \gamma _{t,5}$) and neglected the backscattering for simplicity.

 

Fig. 10. Simplified taiji-CROW structure used for the hopping model: (a) clockwise and (b) counterclockwise modes of the resonators. The color and number coding are: blue arrows with numbers indicate the propagating fields, big black numbers on the sides refer to the input/output ports, red numbers identify the taiji MRs and red arrows show the hopping directions with the corresponding phase.

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The computed output responses as a function of the frequency detuning when $\omega _{\text {1}} = \omega _{\text {5}}$ are shown in Fig. 11. The model assumed that hopping couplings are Hermitian and identical for both directions of propagation. The non-Hermitian nature of intra-resonator mode couplings (i.e. $\beta _{\text {12}}=\beta _{\text {56}} \neq$ 0 and $\beta _{\text {21}}=\beta _{\text {65}}=0$) produce different transmission and reflection spectra at each output port depending on the input port. When the input is port 4 (Fig. 11(a)), apart from the obvious direct transmission to port 3, the signal travels only in one path and exits from port 2. However, when the input is port 2 (Fig. 11(b)), an output signal is observed from each port due to the coupling of the signal to the S-shape waveguide.

 

Fig. 11. Optical spectra of the Taiji-CROW device calculated with the hopping model. (a) Input signal in the port 4, and (b) input signal in the port 2. Numbers in legends indicate the signal input and output ports as shown in Fig. 10. We used: $\Gamma _{1} = \Gamma _{4} = \Gamma = 1 ~{\rm GHz}$, $|\beta _{12}| = |\beta _{56}| = 0.5 ~{\rm GHz}$, $|\beta _{h}| = \Gamma = 1 ~{\rm GHz}$, and $|\beta _{\text {BS},1}| = |\beta _{\text {BS},5}| = 0~{\rm GHz}$.

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C. Design details

Table 1 summarizes the different device characteristics. As mentioned in the main text, the length of the straight sections of the link-MR was changed to create a given phase relation between the counter-propagating modes. All taiji-MRs have a radius of $20 ~\mu$m (i.e., a perimeter $P = 40 \pi ~\mu$m). Some of the taiji MR were designed with a cut S-shaped waveguide (see the inset in Fig. 12(b)) to have an MR that has the same losses of the taiji, but without the coupling coefficient $\beta _{12}$.

Table 1. The main characteristics of the designed structures.^a

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D. Estimation of the parameters from a single taiji MR

There are many parameters in the fit of the spectra of a taiji-CROW device that make their reliable estimation difficult [33]. To predetermine some of the structural parameters we measured and fit the spectra of a single taiji-MRs. Both single taiji-MRs or cut S-shaped waveguide taiji MRs were used (see the inset in Fig. 12). While the former structure allows unidirectional counter-propagating mode coupling, the latter prevents any mode coupling since the interconnecting waveguide is cut, yet it allows to study more in detail the non-Hermitian and the Hermitian parts of the backscattering.

In Fig. 12, we present the spectral dependence of the taiji MR parameters estimated from the fit of the measured spectra. Overall, both the standard taiji and the cut taiji structures (hereafter referred to as taiji and cut, respectively, as indicated in the inset) exhibit the same spectral dependence. As shown in Fig. 12(a), the total loss rates ($\gamma _{t}$) and the coupling rates ($\Gamma$) are found to be almost equal for both taiji and cut structures. Note that the total loss rate includes the bus/MR, the S-shape waveguide/MR couplings, as well as the internal scattering and absorption losses. It varies between $10-16$ GHz throughout the spectral range. Despite the fact that both structures have similar total loss values, cut structure exhibits higher internal loss ($\gamma$) of around 4 GHz because of the truncated ends of the S-shape waveguide (see Fig. 12(b)). It is worth to note here that we are not able to distinguish the losses due to the cut S-shaped waveguide and the losses due to the absorption and scattering in the case of the cut structure. As for the taiji structure, $\gamma$ varies around 2 GHz. Figure 12(c) shows the distinct difference in the non-Hermitian mode coupling coefficients ($\beta _{12}$) for both structures. Since this parameter is primarily related to the intended operation of the S-shape waveguide, the taiji structure exhibits much higher values of $4-12$ GHz while it fluctuates in $0-2$ GHz for the cut. In terms of backscattering, we observed high values for both structures (Fig. 12(d)). These fit values were used as a starting point for the fit and to narrow down the parameter variation range in the modeling of the optical spectra of the taiji-CROW devices.

 

Fig. 12. Spectral dependence of the estimated fit parameters for single taiji MR (labeled as taiji) and for the taiji MR with an interrupted S-shape waveguide (labeled as cut): (a) Total loss rates and the coupling rates between the bus waveguide and MR, (b) intrinsic loss rates, (c) mode coupling coefficients, and (d) backscattering values. Each data point shows the average of at least six fitting runs, and the error bars represent the standard error of the mean. The inset schematically shows MRs having the S-shape waveguide intact and cut. Note: $\gamma _t$ and $\Gamma$ for the cut structure in (a) are shifted by 0.8 nm in the x-axis for clarity.

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E. Parameters derived from the fit

The parameters derived from the fit of the experimental data presented in the main text are reported in Tables 2, 3, 4, and 5. Note that the values in each table form a unique set of parameter values obtained for a specific spectrum and, therefore, do not reflect the uncertainties due to different fit runs.

Table 2. Parameters extracted from the fit for Fig. 4(a) of the main text.

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Table 3. Parameters extracted from the fit for Fig. 4(b) of the main text.

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Table 4. Parameters extracted from the fit for Fig. 4(c) of the main text.

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Table 5. Parameters extracted from the fit for Fig. 5 of the main text.^a

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F. Quantification of the backscattering

To quantify the impact of the backscattering with respect to the experimentally controllable parameters, we calculated the transmitted fields in the taiji-CROW device for different inter-resonator coupling rates while keeping the backscattering coefficient fixed. In the analysis, we considered (i) the link MR is off-resonance with respect to the taiji MRs, (ii) the same coupling rate for each point coupling, i.e., $\Gamma =\Gamma _1=\Gamma _2=\Gamma _3=\Gamma _4=2\Gamma _{\rm S}$, and (iii) the same backscattering coefficient for each MR, i.e., $\beta _{\text {BS}}=\beta _{\text {BS,1}}=\beta _{\text {BS,3}}=\beta _{\text {BS,5}}$. Figure 13(a) shows the intensity map of the $1 \rightarrow 3$ transmission as a function of coupling rate for $\beta _{\text {BS}}=$ 20 GHz. To make the peak positions of the split resonances visually traceable, each spectrum was normalized to 1 with respect to its maximum (in the color map the maximum value is represented by the black color). Figure 13(b) shows three representative spectra without normalization. It is clear that, for $\Gamma \lesssim 3|\beta _{\rm BS}|$, the increased coupling rate not only broadens each resonance but also widens the resonance separation for a given backscattering coefficient. Instead, for $\Gamma {\gtrsim} 3|\beta _{\rm BS}|$, while the resonances broaden, the resonance splitting decreases. The noticeable effect of backscattering on the spectral response (i.e., resonance splitting) becomes negligible at larger coupling rates. In the case of Fig. 13(a) where $\beta _{\text {BS}}=$ 20 GHz, the threshold value was found to be $\Gamma {\gtrsim} 115$ GHz. This leads to a ratio $|\beta _{\text {BS}}| / \Gamma \lesssim 0.17$ ($\Gamma {\gtrsim} 6 |\beta _{\text {BS}}|$). Note that this threshold value depends on the backscattering coefficient, but the ratio remains constant.

 

Fig. 13. (a) Normalized intensity of the $3 \rightarrow 1$ transmission in a taiji-CROW device as a function of the coupling rate for $\beta _{\text {BS}}=$ 20 GHz. The black color corresponds to 1 on the normalized power scale. (b) Three of the spectra given in (a) without normalization to exemplify the effect of the increase in the coupling rate. See text for more details.

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Funding

European Social Fund (DM1062/2021); Programma Operativo Nazionale Ricerca e Competitività (DM1062/2021); H2020 European Research Council (788793, BACKUP); Ministero dell'Istruzione e del Merito (20177 PSCKT, PRIN PELM).

Acknowledgments

We gratefully thank Dr. Iacopo Carusotto and Dr. Alberto Muñoz de Las Heras for useful suggestions and interesting discussions. We also thank Enrico Moser for technical support. S. Biasi acknowledges the co-financing of the European Union FSE-REACT-EU, PON Research and Innovation 2014–2020 DM1062/2021. S. Ali acknowledges the financing of Q@TN, the joint lab between University of Trento, FBK- Fondazione Bruno Kessler, INFN-National Institute for Nuclear Physics and CNR-National Research Council, and financed by PAT.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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