Constructions of synthetic lattices in modulated ring resonators attract growing attention to interesting physics beyond the geometric dimensionality, where complicated connectivities between resonant frequency modes are explored in many theoretical proposals. We implement experimental demonstration of generating a stub lattice along the frequency axis of light, in two coupled ring resonators of different lengths, with the longer one dynamically modulated. Such a synthetic photonic structure intrinsically exhibits the physics of flat band. We show that the time-resolved band structure read-out from the drop-port output of the excited ring is the intensity projection of the band structure onto a specific resonant mode in the synthetic momentum space, where gapped flat band, mode localization effect, and flat-to-nonflat band transition are observed in experiments and verified by simulations. This work provides evidence for constructing a synthetic stub lattice using two different rings, which, hence, makes a solid step toward experimentally constructing complicated lattices in multiple rings associated with synthetic frequency dimensions.

Microwave phased array antennas (PAAs) have attracted considerable interest owing to their advantages of rapid and accurate beam steering. Traditional PAAs based on electronic phase shifters suffer from the disadvantages of steering angle and instantaneous bandwidth of the radio-frequency (RF) signal due to the beam squinting problem. Integrated optical beamforming networks (OBFNs) use optical true time delay lines (OTTDLs) to broaden the instantaneous bandwidth. The processing RF signal in the optical domain also brings the advantages of low loss, compact size, light weight, and zero electromagnetic interference. Therefore, they can play an important role in broadband large-scale PAA systems. Moreover, the OBFNs can be integrated with other microwave photonic systems, such as optoelectronic oscillators, photonic analog-to-digital converters, and optical channelizing filters to realize fully integrated microwave photonic radars. The most important components in the integrated OBFNs are the OTTDLs. Integrated OTTDLs with a broad bandwidth and large delay tuning range are in high demand. Among the various integrated OTTDL structures, cascaded tunable microring resonators (MRRs) are promising owing to their continuous and broadband delay tuning. Previously, we demonstrated a cascaded-MRRs-based OTTDL chip working at the anti-resonant wavelength. It exhibited the advantages of broadband response with low delay fluctuation, high scalability, and simple control schemes. In this work, we proposed a 1×N binary tree OBFN chip based on cascaded MRRs (Fig. 1). The binary tree topology interlaces the power splitters with the OTTDLs, which effectively reduces the number of delay elements. Due to the random phase errors of MRRs and the thermal crosstalk upon tuning, we proposed and demonstrated an automatic calibration method to facilitate the state calibration of OTTDLs.

First, we theoretically investigated the working principle of the OBFN using the transfer matrix method. From the simulated delay spectra, we determined that the anti-resonant MRRs have lower delay fluctuation over a broad bandwidth compared with the on-resonant MRRs (Fig. 2). Moreover, the bandwidth of the OBFN is dependent only on the bandwidth of the continuously tuned MRRs. Therefore, we presented a delay tuning method to increase the system bandwidth by reducing the number of continuously tuned MRRs (Fig. 3). In each stage of the binary tree OBFN, only one MRR was continuously tuned for the delay residual, and the other MRRs were digitally tuned to two specific states (coupling coefficient K=0 or K=1).

To correct the random initial states of the MRRs, we developed a control system to automatically perform delay state calibration and characterization (Fig. 5). The calibration procedure includes the following two parts: the optical calibration stage and microwave delay calibration stage (Fig. 6). In the optical calibration stage, we first measured the power efficiency of the MRR phase shifters. Next, the MRRs were calibrated to K=0 and K=1 using the resonance extinction ratio method and the spectrum mean square error method, respectively (Fig. 7). Thereafter, the continuously tuned MRRs were calibrated by aligning the anti-resonant wavelengths to the operating wavelength at various coupling coefficients. The linear power relationship of the MRR tunable coupler and the ring phase shifter was extracted. Considering the inter- and intra-MRR thermal crosstalk, the power relationship could be remedied using a point-slope method (Fig. 8). In the microwave delay calibration stage, the group delay responses of continuously and digitally tuned MRRs were measured. By iterative control of the phase shifters based on the measured delay spectra, we effectively eliminated the influence of thermal crosstalk between the MRRs and obtained a flat delay response in the operating bandwidth.

We experimentally verified the proposed algorithms by testing the longest path of a silicon nitride 1×8 OBFN chip. The chip was fabricated on the TriPLex? platform. The measured delay spectra of a continuously tuned MRR verify the accuracy of the optical calibration (Fig. 9). Due to the thermal crosstalk, the microwave delay calibration procedure is needed for multiple MRRs. Using the proposed method, all the 21 MRRs in the path are digitally tuned and the measured maximum delay is 560 ps with a delay fluctuation of less than 11.2 ps. Moreover, we successfully realize continuous delay tuning of three MRRs. The delay variation is less than 7.5 ps in the bandwidth of 08 GHz (Fig. 10). Comparing our work with other MRR-based OBFN and OTTDL chips, our OBFN controls the largest number of cascaded MRRs for a widely tunable delay range (Table 1). In addition, with the automatic calibration system based on the optical and microwave joint optimization, our proposed OBFN allows for a broad operating bandwidth and a low delay fluctuation.

We proposed a 1×N binary tree OBFN based on tunable cascaded MRRs working at the anti-resonant wavelength. The delay tuning method that utilizes one continuously tuned MRR in one stage expands the operating bandwidth of the OBFN. An automatic calibration system was experimentally verified by testing the longest path of a 1×8 OBFN chip. With the joint optimization algorithm, we realized the accurate calibration of 21 MRRs automatically. This method, with reduced system complexity, can be further used to calibrate the entire OBFN chip.