Reconfigurable radiofrequency filters based on versatile soliton microcombs

The rapidly maturing integrated Kerr microcombs show significant potential for microwave photonics. Yet, state-of-the-art microcomb-based radiofrequency filters have required programmable pulse shapers, which inevitably increase the system cost, footprint, and complexity. Here, by leveraging the smooth spectral envelope of single solitons, we demonstrate microcomb-based radiofrequency filters free from any additional pulse shaping. More importantly, we achieve all-optical reconfiguration of the radiofrequency filters by exploiting the intrinsically rich soliton configurations. Specifically, we harness the perfect soliton crystals to multiply the comb spacing thereby dividing the filter passband frequencies. Also, the versatile spectral interference patterns of two solitons enable wide reconfigurability of filter passband frequencies, according to their relative azimuthal angles within the round-trip. The proposed schemes demand neither an interferometric setup nor another pulse shaper for filter reconfiguration, providing a simplified synthesis of widely reconfigurable microcomb-based radiofrequency filters.

Recently, an all-integrated RF photonic filter has been shown in a monolithic platform 14 . Among others, RF filters constructed from tapped delay line (TDL) structures attract great attention. They can be classified into two types 18 , depending on whether the filtering profile is given by the physical path delays 9,14 or the light source spectra 7,8,[15][16][17][19][20][21][22] . While the former approach is straightforward, a multiwavelength source combined with dispersive propagation also functions as a TDL filter. This greatly simplifies the structural complexity of a finite impulse response filter, as only a single dispersive delay line is needed. Nevertheless, the main complexity is then shifted to the multiwavelength source. Electro-optic (EO) combs 8,19,20 or mode-locked lasers 21,22 are generally adopted as light sources, which remain expensive and bulky options.
Integrated optical Kerr combs (microcombs) have appeared as an interesting alternative. Microcombs have already been applied not only to filter RF signals [15][16][17] but also for various RF photonic processing, such as true-time delay beamforming 23 , RF channelization 24 , and analog computation 25 . The large comb spacing of microcombs also enhances RF filters with broader Nyquist zone (spur-free range), lower latency 15 , and less dispersion-induced fading, as well as larger number counts of equivalent delay lines 16 , unparalleled by other approaches. However, so far all these microcomb-based RF filters have been implemented on either dark pulses 15,26 or complex soliton crystal states 16,17 . Additional programmable pulse shaping modules are inevitably required to equalize or smooth the comb spectral shape. Thus, the system complexity is significantly increased while the potential for lowcost and high-volume applications is compromised. To date, harnessing the smooth sech 2 spectral envelope of single soliton [27][28][29] or other regulated dissipative Kerr soliton (DKS) states for photonic RF filtering is yet to be investigated. Because single-soliton microcombs have facilitated a myriad of applications, ranging from Tb/s coherent communication 30 , ultrafast and long-distance ranging 31,32 , dual-comb spectroscopy 33 , and astronomical spectrometer calibration 34 , to microwave synthesis 35 , a great benefit for RF filtering can be expected.
In this paper, we demonstrate soliton microcomb-based RF photonic filters without any external pulse shaping. In addition, the synthesized RF filters can be all optically reconfigured through the internal versatile soliton states. Specifically, we trigger, in a deterministic fashion, the perfect soliton crystals (PSCs) to multiply the comb spacing 36,37 , thereby dividing the RF passband frequencies. Moreover, we achieve filter reconfiguration based on versatile two-soliton microcombs (TSMs). The spectral interference of two solitons is functionally equivalent to an interferometric setup, shifting the filter passband frequency via modification of the angle between them. A proof-of-concept filter reconfiguration experiment is also shown using TSM-based RF filters. The internal exploitation of abundant and regulated soliton formats of microresonator effectively bypasses the need of another programmable pulse shaper and interferometric setup for RF filter tuning 15,16 . Thus, the proposed scheme dramatically reduces the system complexity and form factor of microcomb-based RF filters, and are readily applicable to the current radar systems, 5G wireless, and satellite communications.

Results
Principle of soliton microcomb-based RF filters. Figure 1a illustrates the conceptual setup for soliton microcomb-based RF filters. First, a telecom C-band continuous wave (CW) laser initiates microcomb generation, where each comb line serves as the RF filter tap. By modulating the RF signals from a vector network analyzer (VNA) on an electro-optic Mach-Zehnder modulator (MZM), the RF signals are broadcast to each microcomb mode. Then, the upconverted signals are propagated through a spool of single-mode fiber (SMF) to acquire incremental delay between filter taps. Finally, the signals are converted back to the RF domain in a fast photodetector (PD). The detailed experimental setup is described in the "Methods". This arrangement exactly corresponds to a TDL filter, where the power of each comb line p k is the tap weight, and the delay is determined by the comb spacing f m and the accumulated dispersion ϕ 2 = −β 2 L (the product of SMF second-order dispersion β 2 and fiber length L). When the filter tap weights take the sech 2 envelope of a singlesoliton comb (case 1), the RF filter response is given as (see Supplementary Note 1): where f FSR and G(f RF ) are, respectively, defined as 1/(2πϕ 2 f m ) and Þ, with T = 1/f m the repetition period, and T 0 the soliton pulse width. f RF denotes the RF frequency. Note that higher-order dispersion of SMF is neglected here to give a more intuitive picture. The overall RF filter response can be seen as a periodic function of lineshape G(f RF ) with RF free spectral range (FSR) of f FSR , modulated by an envelope due to the doublesideband (DSB) modulation scheme being used. As the tap weights are all positive, the passband frequencies of the RF filters are at every multiple of f FSR , including a DC response. Throughout this article, we focus on the first passband.
Besides, by exploiting the rich soliton states of microresonator, the RF filters can be easily reconfigured at no additional cost nor complexity. Among soliton crystal structures 38,39 , the defect-free PSC is of particular interest, as N(N ∈ N + |N ≥ 2) equally spaced solitons (case 2) within one round-trip time simply multiplies the initial comb spacing by N times. This imparts N times division of the filter passband frequencies while preserving the filter bandwidth. The automatic PSC control is equivalent to the Talbot-based processer for discrete programming the RF filters in ref. 22 . Less intuitively, all-optical reshaping of the RF filters can also be achieved via versatile TSM spectra (case 3). Two solitons residing in one period induce sinusoid interference on the sech 2 spectral shape of a soliton, modulating the tap weights of the TDL filter. This rewrites the RF filter response as (see Supplementary Note 1): where α is the relative azimuthal angle between two solitons (expressed in radian for calculation). Clearly, new RF passbands of halved amplitude appear due to two-soliton interference, which is displaced at both sides from the initial response according to the azimuthal angle between them. Thus, the RF filter passbands can slide inside f FSR by modifying the relative soliton angles. Unlike ref. 16 in which the authors artificially introduce the sinusoidal modulation via programming the spectral carving, we alleviate the need for a pulse shaper and realize sinusoidal modulation by directly generating a series of TSM spectra. This novel scheme achieves wideband reconfiguration of RF filters without either interferometric configuration or additional pulse shaping.
Experimental implementation. The soliton microcombs used for RF filtering are generated from a 103.9 GHz ultra-low loss integrated silicon nitride (Si 3 N 4 ) microresonator (Q~1 × 10 7 ), fabricated by the photonic Damascene reflow process 40 . By employing the frequency-comb-assisted diode laser spectroscopy, the detailed properties of resonances and the integrated group velocity dispersion of the microresonator are measured (see Supplementary Note 2). Strong avoided mode crossings (AMXs) are observed~1565 nm, which lead to the modulation of intracavity CW background, thereby resulting in the ordering of the DKS pulses 41 and the formation of soliton crystals 36,38,39 . Figure 1b shows the simulated stability diagram (see "Methods"), which consists of modulation instability (MI), breathers, chaos (spatio-temporal chaos and transient chaos), and stable DKS states. In addition, it has been revealed that the pump power level is critical for whether the PSCs or stochastic DKS states are formed 36 (see Supplementary Note 3). In our case, the threshold pump power P th is found to be~20 mW in the bus waveguide. When the laser scanning route is operated below threshold pump power, PSC states can be accessed without crossing the chaos region. Contrarily, DKS states with stochastic soliton number are accessed above the threshold power. Experimentally, the singlesoliton and TSM states are obtained by either directing falling to the states or backward tuning from the states with higher soliton number 42 . Note that switching from PSC to single soliton or TSM is prohibited unless the pump power is increased to be over the threshold 36 . Thus, through controlling the pump power and resonance frequency, various soliton microcombs (single soliton,   Fig. 1c shows three distinct optical spectra obtained from the resonance of 1555.1 nm: single soliton, PSC (N = 4), and TSM (α = 132.7°, see "Methods"), respectively. It is also worthwhile to mention that although this type of RF filter does not require coherent comb states, the high-intensity noise of MI combs is certainly undesirable 15 . As the intensity noise will be transferred to the synthesized RF filters, mode-locked comb states are preferred to minimize the noise at the RF output. Figure 2 depicts the RF photonic filters using single soliton and various PSC microcombs. The single-soliton-based RF filter (Fig. 2a) is centered at 16.24 GHz, with main-to-sidelobe suppression ratio (MSSR) of 23.2 dB. Further, various PSC states are deterministically obtained at different resonances under the threshold power, thereby all optically reconfiguring the corresponding RF filters. The comb spacing multiplication via PSC results in the division of the corresponding RF passbands. RF filters centered at 8.12, 5.42, and 4.06 GHz (Fig. 2b-d) are experimentally synthesized through 2, 3, and 4 equally spaced solitons, with MSSR of 22.6, 25.6, and 20.4 dB, respectively. All these RF filters achieve MSSR over 20 dB without additional programmable spectral shaping. The MSSR here is limited by the smoothness of the optical spectra 19 , as several AMX can be seen in the microcombs. Nevertheless, all these microcombs preserve well the sech 2 envelope, and remained smooth after amplification. In addition, the measured RF filter responses are in excellent agreement with simulations, by taking into account of third-order dispersion (β 3 ) of SMF (see "Methods"). Note that the bandwidths of experimentally synthesized RF filters broaden slightly with their center frequencies, also due to the third-order dispersion of SMF 15 . Figure 3a shows the TSM spectra and their corresponding RF filter responses, pumped at resonance of 1556.0 nm. According to Eq. (2), the first passband frequencies of RF filters scale linearly with the relative angles between two solitons, so that the filter reconfiguration is achieved. In the experiment, TSM spectra with relative angles of 19. Importantly, the possible angles between two solitons are determined by the overall AMX profile, and are rather robust to both laser power and frequency detuning, thereby deterministically dictating the filter passband frequencies to be either one of those shown in Fig. 3a. To gain insights into the relative angles between two solitons, we also perform perturbed Lugiato-Lefever equation (LLE) simulation to investigate the TSM formations (see "Methods"). The blue curve in Fig. 3b shows one example of the steady-state two-soliton temporal intracavity profile. Due to the AMX effect, periodic intensity modulation is observed upon the CW background. It is clearly seen that the soliton can only be excited at specific parameter gradients 41 , as manifested by the green dashed lines, which correspond to the stationary solutions obtained in the simulation. These possible soliton angles are in good agreement with the experimental results, which is indicated as red dashed lines. To further test the robustness of the angle between two solitons, an external perturbation is deliberately introduced on their relative angle. Figure 3c illustrates the  ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18215-z dynamical evolution of the two-soliton formation. The simulation is initiated as a standard laser scanning scheme to kick out two solitons. Once the simulation reaches a stable two-soliton solution (relative angle of 168.0°), one of the solitons is dragged from its original position by a 10.0°on purpose. After a period of free-running, the two solitons converge back to their original relative positions, again at 168.0°apart. This confirms the regulation of two solitons under AMX background modulation.
RF filter reconfiguration experiment. A proof-of-concept RF filter reconfiguration experiment is also illustrated in Fig. 4 using TSM-based RF filters (see "Methods"). Two superimposed phaseshift keying (PSK) signals in which a 40 Mb/s modulation at 1.96 GHz tone and a 20 Mb/s modulation at 3.05 GHz tone are prepared as input test signals. The RF filters are then, respectively, reconfigured at 1.96 and 3.05 GHz to filter the input signals, by triggering the TSM spectra of corresponding soliton angles. At the output of the RF filters, the nearly complete rejection of either one of the PSK signals is observed on the electrical spectra (Fig. 4a), where the extinction ratios exceed 35 dB for both cases. Figure 4b, c shows the filtered output RF waveforms. The periodicity of the output temporal traces corroborates the filtering of the original RF signals.

Discussion
A comparison of the soliton RF filters presented in this work with other RF photonic filters based on various multiwavelength sources is shown in Table 1. First, we see that the MSSR achieved by soliton microcomb is comparable or even better than other implementations that do not require pulse shapers, such as combs   19 , laser arrays 7 , or mode-locked lasers 21,22 , while none of them is integrated solutions regarding the light sources. Second, all of the past microcomb-based RF filters require, to our knowledge, two high-fidelity pulse shapers [15][16][17] , in order to equalize the largely unbalanced comb line intensity in the complex microcomb states. The achievable MSSR is indeed higher as expected, due to the fine equalization of comb amplitudes. Without pulse shapers, these approaches are not able to achieve similar MSSR nor tunability. In our case, the MSSR is mainly limited by the spectral roughness induced by mode crossings. One could envision combining our current implementation with an integrated spectral shaper as previously demonstrated 8 , although the device is based on indium phosphide platform. The integrated shaper can enhance our filters with improved MSSR and faster reconfiguration, while the soliton microcombs can greatly simplify the EO combs incorporating seven cascaded modulators used in ref. 8 . It is worth noting that using integrated shaper for comb equalization in complex microcomb spectra would be inefficient, due to the shaper's limited extinction ratio. Thus, the ability to control the soliton states among microcombs is truly essential for practical implementation. From the table, it is also evident that only a few past works report the link performances of comb-based RF filters. We carried out simple optimization of the RF filter link specifically based on 4-PSC as an example, where we achieve simultaneous positive link gain and noise figure within 30 dB (see Supplementary Note 5). Although the link performance is not yet as good as that of pure microwave solutions, it is already very close to the optimized performances of the state-of-the-art comb-based RF filters 43 . In addition, we achieve widely reconfigurable RF photonic filters from 0.8 to 16.2 GHz, taking into account both passbands obtained from TSM spectra. This multioctave filter operation is generally considered challenging by pure RF engineering 44,45 , especially at higher carrier frequency towards millimeter wave or  terahertz wave. However, there is no such limitation for microwave photonic filters. It is worth mentioning the recent work on RF photonic filters synthesized from passive cascaded ring resonators with a wide continuous tuning capability 12 . However, in order to configure the filters, a number of power-consuming thermal heaters have to be controlled with fine resolution voltage sources. Although our filters cannot be tuned continuously, the exploration of multiple resonances dramatically enriches the achievable RF filters, while the possible controlled mode interaction 46 or using dichromatic pumps 47 may empower our filters with continuous tuning ability (see Supplementary Note 6). Moreover, additional functionalities, such as photonic downconversion of microwave signals, can be simultaneously realized in the microcomb-based filter setup, without electrical mixer and local oscillators 48 . The microcomb FSR will act as the local oscillator, which is promising for on-chip terahertz wave signal processing.
The RF filters obtained in this study are based on microcombs generated in a 103.9 GHz microresonator. If one targets RF filters with narrower bandwidth, a microresonator with larger size generating smaller FSR comb will be a better choice, as the filter bandwidth scales with the comb FSR given a certain soliton pulse width (see Supplementary Note 1). The recent ultra-low loss Si 3 N 4 technology has allowed soliton formation down to the 10 GHz range 35 , which would enable RF filters of only tens of MHz bandwidth without additional pulse shaping. However, in order to maintain the filter center frequency when using small FSR combs, longer dispersive fiber is needed, while the envelope fading becomes more severe under the DSB modulation scheme 20 . In this case, more sophisticated modulation schemes are required, such as single-sideband modulation with broadband electrical 90°hybrid, in order to eliminate the fading effect. In comparison, microcombs of large comb spacing allow for a simple RF uploading method. Also, since we are interested in RF filters up to~16 GHz (limited by the measurement range of VNA), the underlying combs with repetition rate over 32 GHz are preferred to avoid spurs generated from the signal beating with other comb lines 19 . Since obtaining a narrow RF photonic filter is not the primary interest in the current study, we utilize the 103.9 GHz microresonator in the experiment. The techniques presented here are compatible with other microcomb FSR.
In summary, we demonstrate reconfigurable soliton-based RF photonic filters using simplified approaches. Contrary to previous demonstrations where pulse shapers are necessary to obtain descent passband responses [15][16][17] , the proposed schemes are intrinsically well shaped with the smooth spectral envelopes of solitons. More importantly, we harness various intrinsic DKS states of microresonator, like PSC and TSM, for RF filter reconfiguration at no additional cost. The diversity and regularization of soliton formats in microresonator are investigated in the favor of RF photonic filters. To a certain extent, these inherent soliton states could be in place of substantial efforts made in the past for reconfiguring the comb-based RF filters, such as using interferometric architecture 15,19 , programmable pulse shaping 16,20 , or Talbot-based signal processor 22 . Although our soliton-based RF filters are much simpler than previous microcomb-based RF filters 15-17 as we eliminate the use of two bulky pulse shapers, the form factor is not yet as compact as their electrical counterparts. Nevertheless, the basic components of our filters can be integrated 7 . The recent advancements on the integration between laser chip and microresonator 28,29 , as well the possibility to replace the SMF with a highly dispersive integrated waveguide 7 , can be further connected to the current work for miniaturization. To conclude, our work significantly reduces the system complexity, size, and cost of the microcomb-based RF filters, while preserving their wide reconfigurability. The proposed schemes set as a stepping stone for chip-scale, cost-effective, and widely reconfigurable microcomb-based RF filters.

Methods
Experimental setup. A C-band tunable CW laser is amplified by an Erbium-doped fiber amplifier (EDFA) with amplified spontaneous emission (ASE) filtered, polarization aligned at the TE mode, and then coupled to the Si 3 N 4 microresonator for soliton microcomb generation. The input and output coupling of the chip is achieved via lensed fibers of~30% fiber-chip-fiber coupling efficiency. The soliton microcombs are initiated by scanning the pump over the resonances, with the assistance of an arbitrary function generator 27 . The residual pump of generated microcombs is then filtered by a tunable fiber Bragg grating, while a circulator is inserted in between to avoid back-reflection. 10% of light is tapped to an optical spectrum analyzer to record the microcomb spectra. The other 90% of the light is amplified, and polarization managed, before sending to a 30 GHz bandwidth MZM. RF signals from the VNA are applied to the MZM in DSB modulation format. The modulated spectra are then propagated through a spool of 4583.8m SMF to acquire dispersive delays, and finally beats at an 18 GHz PD to convert the signals back to the RF domain. The length of SMF is measured by a commercial optical time-domain reflectometer.
For the system demonstration, a 12 GSa/s arbitrary waveform generator (AWG) is used to prepare the input RF signals. A 40 Mb/s PSK signal modulated at 1.96 GHz tone and 20 Mb/s PSK signal modulated at 3.05 GHz tone are generated separately from the two channels of the AWG. After adding the two streams of signals in a combiner, the composite signal is then sent through the TSM-based RF filters, tuned at 1.96 and 3.05 GHz, respectively. The output spectra are measured by an electrical spectrum analyzer, while the waveforms are measured using a highspeed real-time oscilloscope.
Si 3 N 4 microresonator. The Si 3 N 4 microresonator used in the experiment is a ring structure with a radius of 217 μm. Its waveguide cross-section (width × height) is made to be 1500 nm × 750 nm. The microresonator is coupled with a bus waveguide, which possesses the same cross-section as the ring to realize high coupling ideality 40 . To achieve critical coupling for the resonances, the gap distance between the ring and bus waveguide is designed to be 690 nm. In our experiment, the pumped resonances are around 1556 nm, where both the intrinsic linewidths and coupling strengths are approximately 20 MHz (see Supplementary Note 2). With respect to the reference resonance of ω 0 /2π = 192.8 THz, the dispersion parameters of microresonator are measured: FSR of microresonator D 1 /2π ≈ 103.9 GHz, second-order dispersion term D 2 /2π ≈ 1.28 MHz, and negligible third-order dispersion term D 3 =2π $ Oð1Þ kHz (see Supplementary Note 2).
where A(ϕ, t) is the temporal envelope of the intracavity field. κ = κ ex + κ 0 is the total cavity loss rate, where κ ex is the coupling rate, and κ 0 is the internal loss rate. ω 0 and ω p denote the angular frequencies of the pumped resonance and the CW pump laser, respectively. g is the Kerr frequency shift per photon, defined as g ¼ _ω 2 0 cn 2 =n 2 0 V eff , where n 0 is the effective group refractive index, n 2 is the nonlinear optical index, and V eff is the effective mode volume. D 2 corresponds to the second-order dispersion term, and |s in | 2 is the pump power. To involve the AMX effect, an additional frequency detuning Δ k is introduced at the k-th mode, so that the frequency of the k-th mode becomes ω k = ω 0 + D 1 k + D 2 k 2 /2 + Δ k (see Supplementary Note 2). Here in simulation, the dispersion is limited to D 2 , and the Raman and thermal effects are not taken into account. According to the dispersion measurement and the generated microcomb spectra, the parameters for the AMX in the simulation are set as k = 15 and Δ k /2π = 100 MHz, enabling the modulation of the CW intracavity background for the trapping of soliton temporal positions. Note that the strength of the AMX here is estimated to introduce the regularizability of solitons, but without disturbing their formations 36 . Other parameters used in the simulation are retrieved from the characterization, that is, D 1 /2π = 103.9 GHz, D 2 /2π = 1.28 MHz, κ ex /2π = κ 0 /2π = κ/4π = 20 MHz.
The simulation of the stability chart is obtained by initializing the numerical model with single-soliton solution at various pump power and detuning conditions 36 . Four different states are found: MI, breathers, spatio-temporal and transient chaos, and stable DKS states. The threshold pump power, separating the PSC and stochastic DKS formations, is estimated from both the simulation and experimental conditions. For the TSM simulation, the numerical model is operated under standard CW laser pump scanning from blue-detuned to red-detuned side, until it reaches the stable TSM states. All the possible angles of TSM are recorded. To test the robustness of the TSM azimuthal angle, the model is initialized with one of the exact two-soliton solution, but deliberately perturbed by 10.0°angle deviation. Two solitons are gradually re-stabilized at its original azimuthal angle after a period of free running.