Silicon photonics have widespread applications in optical communications, photonic sensors, and quantum information processing systems. Different photonic integrated circuits often require similar basic functional elements such as tunable filters, optical switches, wavelength de-multiplexers, optical delay lines, and polarization crosstalk unscrambling. Other optical signal processing functional elements may be needed in specific applications, for example, the differentiation with respect to time of time-varying optical signals and the implementation of very high extinction interferometers in some integrated quantum photonic circuits. Just as reconfigurable electronic processors in microelectronics have advantages in terms of ready availability and low cost from large-volume generic manufacturing and are useful for configuration into different functionalities in the form of field-programmable gate arrays, here, we show how an integrated coherent network of micro-ring resonators can be used in reconfigurable photonic processors. We demonstrate the implementation of optical filters, optical delay lines, optical space switching fabric, high extinction ratio Mach–Zehnder interferometer, and photonic differentiation in a reconfigurable network where the control of the phase in the different arms of the coherent network can determine the implemented functionality.

## I. INTRODUCTION

Programmable integrated circuits have advantages of function diversity, low-cost, and rapid turn-around time for developing initial prototypes when compared with custom-developed application-specific integrated circuits. These advantages also hold for programmable photonic integrated circuits. Cascaded Mach–Zehnder Interferometer (MZI) lattices have been proposed and demonstrated^{1–3} for universal matrix transformations, which have found applications in deep learning,^{4} integrated quantum photonic circuits,^{5} and optical communication systems.^{6} Having no recirculating propagation, the MZI-based feedforward configurations are inherently limited in optical feedback and are less efficient for implementing functions that need recursion. Inspired by the field-programmable gate arrays (FPGAs) in electronic integrated circuits, there has been rapid progress in reconfigurable photonic circuits in recent years, as described in the recent review by Bogaerts *et al.*^{7} Two-dimensional (2D) unit cells implemented by MZI waveguide-based square, hexagonal, and triangular-shaped meshes were used to demonstrate both the finite impulse response (FIR) and the infinite impulse response (IIR) elements on the reconfigurable photonic circuit.^{8} However, the mesh network of MZI requires more complicated control to implement feedback circuits: for example, 7 phase shifters must be controlled to implement one ring resonator. The size of such reconfigured circuits to form optical resonators must also be multiple times the size of the tunable directional couplers,^{8} and thus, the size of the ring resonator is large, with a correspondingly small free spectral range (FSR). Since micro-ring resonators (MRRs) inherently have IIR,^{9} in this paper, we evaluate the possible use of a coherent network of MRR instead of the previously explored MZI lattices in reconfigurable photonic circuits.

This paper proposes the use of MRR-based coherent networks and demonstrate their use as multi-functional processors. This paper organizes the MRR network applications into two distinct sets: (i) applications that use the incoherent response of the MRR network (i.e., there is no need for controlling the phase in the waveguide connecting neighboring MRR) and (ii) applications that use the coherent response of the network, requiring additional phase shifters between each MRR. The network for coherent applications is different from that for incoherent functions, but both of the network topologies are based on Reck’s architecture^{10} as illustrated in Fig. 1(a). The main difference lies in the basic fabric as shown in Figs. 1(b) and 1(c). There are two phase shifters in the basic fabric for coherent applications, while there is only one phase shifter on MRR for incoherent application.

The incoherent network shows the advantages of the MRR basic fabric for implementing tunable filters, optical switches, optical delay lines, and photonic differentiators and has been explored previously.^{11,12} The coherent MRR fabric has not been previously explored and can be used to perform matrix operations for the realization of photonic networks, which can be configured for signal processing. In this paper, we demonstrate two different application examples of the coherent MRR network: a polarization crosstalk descrambler and a high extinction ratio (ER) MZI.

## II. INCOHERENT APPLICATIONS

### A. Filters and de-multiplexers

Figure 2(a) shows the layout of an example implementation of the MRR mesh network with four channels for a proof-of-concept demonstration. The network can be further scaled up based on the same topology as shown in Fig. 1(a). The silicon photonic circuit was fabricated on a silicon-on-insulator (SOI) multi-project wafer shuttle run by *IMEC*. The silicon waveguides were fully etched on the 220 nm thick silicon layer, and the 2 *µ*m buried oxide (BOX) and 2 *µ*m silicon oxide over-cladding provided the optical waveguide confinement. The sample network shown in Fig. 2(a) has a footprint of 1.28 × 0.45 mm^{2}. The optical phase control in the network was realized using tungsten wires embedded in the oxide cladding as thermal heaters. The preliminary prototype was based on MRR with a radius and gap of 20 and 0.2 *µ*m, respectively. The MRRs demonstrate highly consistent filtering profiles, showing a FSR of 4.5 nm, a 3-dB optical bandwidth of 30 GHz, and over 25 dB out-of-band rejection ratio. The quality factor (Q) of the fabricated MRRs was 16 000 as illustrated in Fig. 2(d). Making use of a single MRR, we can implement both a passband and stopband tunable filter for communication purposes. To illustrate, Fig. 2(e) displays the transmission spectra when the thermal heater is applied with different power. Tuning of the MRR resonance is achieved by applying a bias on the thermal phase shifter on the top of the partial waveguide in the MRR. A linear trend line was fitted on the resonant wavelength and the tuning efficiency was extracted to be 0.04 nm/mW*.* The tuning efficiency could be further improved by thermally crosstalk mitigation using deep trenches and substrate undercut.^{13,14} Figure 2(b) shows the configuration when the MRR-based network acts as a wavelength de-multiplexer. Three MRRs were biased to different resonances. Figure 2(f) shows the spectrum of the three wavelength channels with a wavelength spacing of 1 nm and crosstalk below −23 dB. The example implementation may be scaled up to form an MRR network that can support up to (N − 1) different wavelength channels, where N is the number of I/O waveguides. Illustration in Fig. 2(b) shows the case when N = 4. The wavelength spacing was limited by the FSR of MRR and determined by the radius.

### B. Switching

The crossbar MRR can be configured as 4 × 4 hitless path routers. By switching the MRR between “OFF” (off-resonance) and “ON” states (on resonance), the optical path can be switched to remain in or change the original propagation direction. For example, in Fig. 3(a), if all the MRR are “OFF,” the light path will direct from I1 → O1. If we turn on R1, R2, and R3 each time, the light path will direct from I1 → O2, I1 → O3, and I1 → O4, separately. To scale up, *N* × *N* switches could be realized using *N* × (*N* − 1)/2MRRs. Generally, there are N optical routes when all the resonators are “OFF.” MRRs will give rise to other *N* × (*N* − 1) optical routes with each MRR contributing to two routes.

A continuous-wave tunable laser is used to launch transverse-electric (TE) polarization light into the chip. The MRRs have a FSR of 4.5 nm, a Q of 16 000, 3-dB bandwidth of 30 GHz, and >25 dB out-of-band rejection ratio as shown in Fig. 3(b). For applications that require wider optical bandwidths, the design of the MRRs could be changed to have lower loaded Q by over coupling, or the use of higher-order MRR to meet the bandwidth requirements for future 200 and 400 G communication signals.^{15} In our preliminary test structures, the off-resonance loss, on-resonance loss, and path-dependent loss are estimated at 0.24, 1.5, and 1.74 dB per MRR including waveguide propagation loss and the insertion loss (IL) introduced by the waveguide crossing. Figure 3(b) shows the schematic of the 4 × 4 switch fabric, which can connect one input port (I1) to four possible output ports (O1, O2, O3, or O4). For example, switching from port I1 to port O2 can be achieved by thermally tuning the phase of R1 to maximize the signal intensity measured at the O1 port. Then, we measured the crosstalk by characterizing the transmission spectra at the O2, O3, and O4 ports as illustrated in Fig. 3(b). The crosstalk level under −20 dB was observed and is plotted in Fig. 3(c). Data transmission through the switch fabric at 12.5 Gbps non-return-to-zero (NRZ) on-off-keying using pseudorandom binary sequence (PRBS) is performed to examine four switch paths.

The optical carrier is launched from a tunable laser at 1549.7 nm and modulated by a Mach–Zehnder modulator (MZM). The intensity-modulated optical signal is then guided to the silicon photonic MRR-based switch via a polarization controller (PC). Figure 3(a) illustrates the experimental setup for measuring the four different switch paths. Figure 3(d) shows the received eye diagrams. The distortion on the eye diagram of path2/3/4 is due to the limited bandwidth of the MRR in this design. The bandwidth can be further improved by reducing the loaded Q of the MRR.

### C. Delay line

We show in this section that the MRR-based network can also be configured as a programmable optical delay line. MRRs are also attractive candidates for optical delay lines or buffer in photonic integrated circuits^{12–17} because of their more compact size compared with long spiral waveguides. Instead of the long propagation distance, the MRR delay lines take advantage of the strong dispersion or large group delay near the optical resonance of the MRR. By cascading multiple low extinction ratio MRR, optical buffers have been previously realized with relatively large bandwidth and low insertion loss.^{15} Xia *et al.* demonstrated two of the most popular structures for delay line applications: optical delay lines consisting of 100 orders of all-pass filter and a coupled-resonator optical waveguide.^{15,18}

Similar to the architecture constructed by the cascaded all-pass filter, our proposed network can also be configured as a programmable optical delay line. To demonstrate this, we show the optical group delays generated by first, second, and third order MRRs. The configurations are shown in Fig. 4(a). Temporal delays of 25, 34, and 40 ps were obtained and are shown in Figs. 4(b)–4(d) for the three different configurations with IL of 20 dB on resonance and maximum 48 GHz bandwidth. The nonlinear increase in temporal delay is a result of slight misalignment of resonances despite our use of thermal tuning. Since we kept the optical power at a low level (less than −10 dBm), nonlinear effects in the relatively low-Q MRR (Q ∼ 16 000) may be neglected. There are two possible contributions for the slight broadening of the “on” state pulse in Fig. 4(b). One contribution comes from the time-bandwidth product: when the pulse passes through a narrow filter, the pulse may be bandwidth limited and be broadened. A second contribution is present even if the pulse is not spectrally filtered by the MRR, but passes near the slope of the resonance where the group delay dispersion (GDD) near resonance can broaden the pulse.^{15} A longer delay time could be achieved by cascading more stages and the delay time should scale linearly with the number of MRRs theoretically. However, the increase in tunable delay steps that can be added will be limited by the IL per MRR. The IL can be further reduced by optimizing parameters of the MRR, i.e., using lower Q MRR^{15} or compensated by amplification using III–V-on silicon MRR with optical gain.^{19} With the increase in cascading stages, the accumulated group delay dispersion will also limit the working bandwidth. Ultimately, there always exists a trade-off between the delay time and the bandwidth.^{20}

### D. Differentiator

Differentiation (DIFF) and integration are two fundamental operations for optical signal processing. DIFF has been previously used for pulse shaping, ultra-fast signal generation, and solving differential equations. Various approaches for optical differentiators and integrators have been reported by different groups.^{16–23} Among all the works, MRRs are configured as differentiators using the through port response^{23} while the MRR can be configured as integrators using the drop port response.^{22} The order of differentiation can be increased by using higher-order interferometers.^{24} In this section, we show that our proposed network can also be configured as a multi-functional differentiator. The schematic of the programmable differentiator is illustrated in Fig. 5(a).

The pulse generator drives the MZMs to generate a Gaussian pulse train with a pulse width of 50 ps. Then, we apply voltages to finely tune the resonance of the MRR, and align one, two, or three resonances with the wavelength of the tunable laser. We measure the temporal waveforms of DIFFs of first, second, and third order MRRs. The 3 dB operational bandwidth of different order MRRs is 18, 24, and 48 GHz, respectively. The fractional orders are calculated to be 1.05, 1.43, and 1.60 as presented in Figs. 5(c)–5(e), respectively. It can be found that the measured differentiated pulses match the calculated pulses except for some higher-order components and slight broadening. The broadening is due to the limited device operation bandwidth and finite notch depth.^{24}

## III. COHERENT APPLICATIONS

### A. Unitary mode converter

In this section, we will show that the proposed MRR-based coherent network can also be configured to realize universal unitary matrix operations. As an example, we proposed a polarization descrambler, illustrated in Fig. 6(a), which was composed of a 2D-grating coupler, four MRR, and six thermal phase shifters. The 2D-grating coupler splits the input optical field into two orthogonal spatial modes that couple the orthogonal (arbitrary orientated X and Y polarizations) in the input fiber into the TE modes of two silicon waveguides. We will show that MRR-based lattices (1) and (2) can be configured to realize a universal 2 × 2 unitary matrix similar to that of MZI-based lattices.^{25} MRRs (3) and (4) are combined to implement 2 × 2 diagonal matrix transformation. Since any 2 × 2 matrix can be divided into a product of two unitary matrices and a diagonal matrix through singular value decomposition,^{3,10} the network we proposed in Fig. 6(a) is thus able to achieve arbitrary 2 × 2 matrix transformation. The light is coupled from the fiber through a single 2D waveguide grating coupler (formed by a 2D array of nano-holes), which couples the light into the TE modes of two orthogonally orientated waveguides. By properly adjusting the value of phase shifters in the coherent MRR network, the chip can be configured to automatically unscramble the mixed data channels in a polarization-multiplexed optical fiber transmission system.

Usually, the two polarization-multiplexed data channels in the transmitter are launched into the optical fiber in orthogonal polarizations, but arrive at the receiver with arbitrary orientations that do not coincide with the X and Y polarizations selected by the 2D waveguide grating coupler at the receiver. Polarization crosstalk due to the optical transmission link will change the state of polarization (SOP). Nevertheless, the interference of light waves in the communication systems is a linear process that can be described by matrix multiplications.^{26} Therefore, we can exploit the network proposed in Fig. 6(a) and configure it to be the inverse of the matrix. In this way, we can mathematically undo the mixing of the two data channels in the polarization division multiplexing (PDM) fiber transmission.

We will show that the MRR lattice [Fig. 6(c)] is equivalent to the MZI-based basic fabrics [Fig. 6(b)] firstly proposed in Ref. 4 under certain circumstances. And similar to the MZI fabric, the MRR-based fabric can also be configured as Reck *et al.*^{10} and Clements *et al.*^{3} proposed architectures to achieve the universal N × N unitary matrix. The transfer matrix *T* was used to relate the electric field between input and output ports $EO1EO2=TEi1Ei2$. Transfer matrix *T*_{MZI} of MZI in Fig. 6(b) is $TMZI=cos\u2061\theta sin\u2061\theta sin\u2061\theta \u2212cos\u2061\theta $. The transfer function *T*_{MRR} of MRR in Fig. 6(c) can be derived as follows:

We assume that the MRR in Fig. 6(c) has coupling coefficients *k* and transmission coefficient *t* related by *t*^{2} + *k*^{2} = 1, roundtrip loss *α*, roundtrip accumulated phase *θ*, and insertion loss of waveguide crossing *α*_{c}. As proposed in our preliminary work,^{27} we can simplify (1) by introducing the equivalent phase *θ′* under low loss approximation that *α* = 1 and *α*_{c} = 1. In the design of a linear coherent optical network, the relative phase between different paths is important. The equivalent phase *θ*′ was used to substitute the amplitude of the entries in (1), let

Substituting Eq. (2) into Eq. (1), we can obtain the simplified version of transfer matrix *T*_{MRR} as

Numerical simulation results show that the two relative phase terms δ_{1} and δ_{2} satisfy that *δ*_{1} + *δ*_{2} = *π* when *α* = 1 and *α*_{c}*=* 1.^{27} And thus, *T*_{MRR} and *T*_{MZI} can be related by

whereΔ*δ* = *π* + *δ*_{2}. Therefore, the MZI-based coherent lattice with 2 phase shifters [Fig. 6(b)] can be equivalent to the MRR-based coherent lattice with one more phase shifter (red) as shown in Fig. 6(c). Using the MRR-based coherent lattice, we can build the network as shown in Fig. 6(a) to realize arbitrary 2 × 2 matrix transformation after removing the redundant phase shifters.

Furthermore, instead of using the ideal MRR without loss, we introduced loss in the MRR model and implement it in the mode unscrambling and switching network. For a given unitary matrix, we used the simulated annealing method^{28} to derive the required phase shift at each phase shifter based on the architecture as shown in Fig. 6(a). We chose the Mean Squared Error (*MSE*) as a figure of merit (FOM) to evaluate the difference between the configured N × N transfer matrix *T*′ and the desired matrix *T*, $MSE=1N2\u2211i=1,j=1N,N|Tij\u2032\u2212Tij|2$, where *i* (*j*) is the row (column) mode number of *T*′ and *T*. Using *MSE* as the FOM, we repeated the simulated annealing algorithm for 25 different random unitary matrices. In the MRR model, we set the parameters of MRR as *k*^{2} = 0.5, α = 0.98, α_{c} = 0.98, and radius R = 10 *µ*m. Figure 6(f) shows the *MSE* (blue) of the unitary mode converter based on the MRR network after optimizing the phase shifter conditions using the simulated annealing method. The error bar shows the variation of 25 random optimization processes. We can see that the *MSE* maintains lower than −30 dB. We also investigate the mode crosstalk level when utilizing this MRR-based network as a polarization descrambler. In a low mode-dependent loss PDM communication system, mode mixing can be described by a unitary matrix *U*. The coherent network can be configured to be inverse of unitary matrix *U* to retrieve the original signals. Even if the mode mixing and coupling are not unitary (loss exists for desired modes), *U*^{−1} can be decomposed as multiplication of two unitary and one diagonal matrices,^{30,31} which still can be realized by the combination of two MRR-based coherent networks (unitary) and one modulator array (diagonal).^{4} To simplify, we assume that the insertion loss during propagation is negligible. Because the mode coupling inside a single fiber in a real system is random,^{31} we generate a random unitary matrix to investigate the performance of the MRR-based polarization descrambler network. Specifically, we optimized the phase shifters to minimize the MSE between configured transfer matrix *T*′ and desired matrix *U*^{−1}, and then extract the maximum off-diagonal element of *T*′ · *U* as the crosstalk level (red) plot in Fig. 6(f). Figures 6(d) and 6(e) plots the squared absolute of all the 2 × 2 elements in the matrix *U* and *T*′ · *U*. Mode mixing and coupling cause a significant level of off-diagonal entries [Fig. 6(d)]. After optimizing, the MRR-based coherent network effectively suppressed the off-diagonal value below −28 dB [Fig. 6(e)].

We also verify that any pure SOP can be recovered to the transmitting state after unscrambling by the MRR-based network. If the two orthogonal polarization channels used to load data (x and y) are coherent, i.e., when the two channels are prepared by the same laser source, the Jones matrix to describe the initial (transmitting) state of the two channels can be written as $P=cos\u2061\theta ,ej\delta \u2061sin\u2061\theta T$, where $\theta \u22080,2\pi $ and *δ* ∈ [0, *π*]. cos *θ* and sin *θ* represent the amplitude of x and y polarization and *δ* is the phase difference. The unscrambled SOP will be *P*′ = *T*′ × *U* × *P*. We can extract the recovered polarization state represented by *θ*′ and *δ*′ using $P\u2032=[cos\theta \u2032,ej\delta \u2032\u2061sin\theta \u2032]T$. Because we assume that the two channels are coherent, *P* and *P*′ are pure states that could be described by the Poincare sphere. In the Poincare sphere, the state of *P* can be represented by the point related to Stokes parameters (*S*_{0}, *S*_{1}, *S*_{2}, and*S*_{3}). Thus, we can compare the SOP before transmitting and recovered by the polarization descrambler. Figure 7(a) shows the Poincare sphere containing a polarization state before transmitting and after going through the descrambler network. The original SOPs (red crosses) are recovered accurately with maximum deviation Δ*θ*, Δ*δ* < 1.2°. Even when the two polarization data channels were incoherent with each other, the SOP is a mixed state described by the addition of pure states, which can be recovered accurately. Thus, our proposed system can work as a polarization descrambler in the optical communication system.

We define the bandwidth as where the crosstalk is lower than −10 dB. The upper bound of bandwidth could be estimated by the FSR of the MRR while the lower bound of bandwidth (blue) with relationship to Q of the MRR is plotted in Fig. 7(b). The bandwidth is inversely proportional to the Q of the MRR. When Q is 500, the bandwidth could reach 0.8 nm, which is adequate for 100Gbaud optical communications as illustrated in Fig. 7(c). In Fig. 7(c), the add-up power of two ports (i.e., orange and blue) was assumed to be 0 dB. Besides, the periodic transmission spectrum could be utilized to unscramble multi-wavelength channels simultaneously.

### B. High extinction ratio MZI

In this section, we show how a high extinction ratio (ER) MZI can be implemented using the MRR as a variable beam splitter (VBS). This method consists of using two additional MRRs acting as VBS and finding their optimal configuration, thereby circumventing the requirements for perfectly fabricated beam splitters. Monitoring the transmission spectrum of the MRR assisted MZI (MMR-MZI) system, the splitting ratio of two VBS can be configured to match with each other and results in a high extinction ratio MZI. Using a silicon photonic device, we measured more than 60 dB ER in the interferometric fringes, corresponding to an improvement of 28 dB over the non-optimized case.

Figure 8(a) shows the schematic of the MMR–MZI consisting of three fundamental parts, each with its parameter: three thermo-optical phase shifters (*H*_{L}, *H*_{R}, and *H*_{MZI}) and four passive directional couplers with a fixed coupling coefficient. The phase shifters are realized using resistive heating elements and induce a local change in the refractive index of the waveguide core by varying the local temperature. This allows us to optimize and control the MZI by applying electrical voltage onto each phase shifter. To achieve an optimized MZI, we need to find the voltage settings for the outer phase shifters, *H*_{L}*,* and *H*_{R} that construct the VBS to be beam splitters with a matched splitting ratio. Once this has been determined, the *H*_{L} and *H*_{R} can be set to the constant phase and the central *H*_{MZI} varied as a usual phase shifter in the MZI. The transfer function of the whole system can be written as

where *t*^{R}_{11} and *t*^{R}_{12} represent the transmission and reflectance coefficients of the VBS_{R} and can be approximated by $t11=1\u2212t212$.

To find the optimum voltage settings for *H*_{L} and *H*_{R}, we monitor the transmission spectrum at output port 3 that allows us to set the splitting ratio of VBS_{L} and VBS_{R} to match with each other. If we inject light from port 1, the optical power measured at port 3 can be written as

*θ* is the phase difference that is tuned by *H*_{MZI} at $\varphi =0,2\pi $ and determined by $\theta =angletR11tL11+\varphi \u2212angletR12tL21$. The maximum and minimum powers, respectively, correspond to the case when *θ* = 0 and *θ* = *π*, with the cosine of *θ*, therefore, giving a sign change in the last term of Eq. (6). To achieve a high ER, perfect destructive interference is necessary to obtain $Pmin3=0$. One can readily show that, to achieve perfect destructive interference, the required condition is

At the settings of *θ* = *π*, we determine the optimal voltages for *H*_{L} and *H*_{R}. The problem of configuring the VBS is thus reduced to a two-dimensional optimization problem over the transmission coefficient, *t*^{R}_{11} and *t*^{L}_{11}, of the two VBS. The VBS needs to satisfy Eq. (7). For example, when the front VBS has a splitting ratio of 60:40, the rear VBS should have a splitting ratio of 40:60 to achieve a high ER. In the ideal 50:50 case, $tR12/tR11=tL11/tL21=1$.

We use the algorithm of Ref. 32 to optimize the ER, while the algorithm can be made more robust by using the thermal crosstalk compensating method.^{33} Figure 8(b) depicts the calculated *P*^{(3)} when*θ* = *π*, subject to variation in the phase shifter voltages, *V*_{HL} and *V*_{HR}. If we set the wavelength of the laser source to *λ*_{M} and also align the resonances of the two VBS to *λ*_{M} by adding the initial bias, the optimization will start from point M. The dark blue contour indicates perfect destructive interference when Eq. (7) holds. We randomly selected three possible working points (A, B, and C) along the perfect destructive interference contour and plot the corresponding high-ER interference fringes in Fig. 8(c). All three interference fringes show high-ER more than 60 dB for the 300 kHz linewidth tunable laser.

With the use of the optimized VBS_{L} and VBS_{R}, we then implement the MZI. The MZI yielded an ultrahigh ER of more than 60 dB [Fig. 9(a)]. Compared with the un-optimized MMR–MZI, the optimized MMR–MZI shows an increase of 28 dB ER by compensating for fabrication imperfections. With optimized VBS_{L}, VBS_{R}, and V_{MZI}, the MMR–MZI can also be configured as a high-ER filter. Using the proposed scheme, we demonstrated three high-ER filters with different center wavelengths as shown in Fig. 9(b). Although the current bandwidth at −60 dB is less than 0.1 nm, the bandwidth can be improved by decreasing Q. Figure 9(c) depicts the bandwidth (FWHM) and fineness with relationship to Q. One can see that when Q is larger than 10 000, the FWHM is lower than 0.2 nm and simulation matches our experimental results. The simulation also shows that when Q is 3000, the FWHM will become 0.5 nm, and the linewidth of −40 dB is 5.8 pm. Figure 9(d) illustrates the simulated spectrum. Moreover, our demonstration shows that MRR instead of MZI can be used as VBS in coherent networks but with a more compact footprint.

This unique implementation of the high-ER filter demonstrated is wavelength-dependent and different from that constructed by MZI lattices. The high-ER filter could potentially serve as pump rejection filters for narrowband laser sources in integrated quantum photonic applications.^{34}

## IV. DISCUSSION

The advantages of MRR include its improved energy efficiency (less refractive index change is needed to tune across the resonance), lower latency (shorter through path lengths for off-resonance light), and more compact size. Assisted with the laser trimming process,^{35} it is possible to manufacture rings that have the same resonances without active thermal tuning. The tuning efficiency can be potentially improved by a factor F (finesse of MRR). Lower latency results naturally from the more compact size of the ring network. High-speed tuning can be achieved by replacing the thermal heater with a high-speed PN diode for fast reconfigurable implementation without sacrificing footprint. The disadvantage of the MRR comes from its increased sensitivity to thermo-optic changes. Because of the narrower transmission spectrum, the transmission of MRR is F times more sensitive to thermal crosstalk/noise than that of MZI. There is thus a trade-off between tuning efficiency and thermal stability. The use of smaller tuning power for each MRR, in turn, leads to a lower overall thermal load for the whole chip, which may mitigate the thermal problem. Improved energy efficiency by using MRR is normally deemed preferable if the thermal stability is ensured (for example, with a thermoelectric cooler to maintain constant chip temperature). Alternatively, SiN platforms or silica waveguide platforms have smaller thermo-optic coefficients compared with silicon and will alleviate the thermo-optic sensitivity and reduce thermal crosstalk, but at the cost of increasing tuning power. Another limitation of MRR lies in the relatively small bandwidth of MRR. This is a disadvantage in some wideband applications. For instance, the MRR network may be inappropriate for high-speed communication systems >200 GHz optical bandwidth. In other applications, for example, narrowband RF filters implemented in a photonic network, the narrower bandwidth might be advantageous.

Another problem that needs discussion is the concern that MRRs are sensitive to fabrication error and temperature/voltage fluctuation. In this paper, the resonances of MRRs are close to each other due to their compact footprint. We monitored the transmission spectrum to actively tune the resonances. However, when the MRR array scales up, the fabrication error and environmental temperature fluctuation introduce phase and resonance deviations. This is a well-known and also well-studied problem. There are three typical solutions to cope with fabrication errors. One is the post-fabrication trimming of MRR resonances^{35} by laser annealing of germanium implanted waveguides. Besides, higher resolution lithography (<45 nm) can reduce the fabrication error and reduce the variation in the MRRs resonances.^{36} Although these methods can make the resonances of MRRs more consistent, thermal heaters are nonetheless needed to realize reconfigurable circuits. There are now well-established techniques^{37–39} using feedback control loops to adaptively tune the MRRs to overcome the environmental temperature fluctuations and lock the resonances to the input laser wavelength. One such technique is to use low-frequency pilot tones and contactless integrated photonic probe (CLIPP) sensors^{40} to lock the resonance. Another example is to use impedance monitoring^{39} to monitor and achieve continuous weighting of MRRs.

## V. SUMMARY

In conclusion, we have proposed and demonstrated the use of MRR-based coherent network for implementing different photonic processing functions. Taking advantage of its inherent spectral characteristics, the MRR-based network can be readily configured as both passband and stopband filters, wavelength de-multiplexers, optical switches, optical delay lines, and differentiators. For other optical processing functions that rely on coherent combination of different paths in the mesh network, we show that the MRR-based lattice network can be considered as being equivalent to the MZI-based lattice and can be used as a polarization crosstalk descrambler in polarization-multiplexed optical communications systems. Besides, we also experimentally demonstrate that the MRR-based lattices can be configured as variable beam splitters that can match the intensities of two branches of a MZI to achieve an ER of 60 dB, which could be used to perform high-fidelity quantum manipulation. The MRR lattice network has the potential for compact implementations of reconfigurable photonic circuits.

## ACKNOWLEDGMENTS

The authors would like to thank IMEC and AMF for chip fabrication. This work was funded by CUHK research Contract No. TH1913551. Y.W. is thankful to the ITF Research Talent Hub for financial support.

## AUTHOR DECLARATIONS

### Conflict of Interest

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.