In order to generate the 49 GHz spacing micro-comb, the pump power was boosted up to 30.5 dBm via an EDFA and the wavelength was swept towards the red side. When the detuning between pump wavelength and MRR’s resonance became small enough such that the intracavity field reached a threshold value, modulation instability was initiated, and primary combs arose with a spacing of multiple FSRs. As the detuning was further changed, the parametric gain lobes broadened and secondary comb lines with a spacing equal to the FSR of the MRR were generated by both degenerate and non-degenerate four wave mixing. This finally resulted in a single FSR spaced Kerr optical comb [44, 59] (Fig. 2(e)) that was over 100nm wide. A zoom-in view (Fig. 2(f)) indicates that potentially over 400 wavelengths were available for the TTDL. For our experiments, however, the wavelength range was limited to the C-band because of the waveshaper and EDFA, and so this resulted in 81 usable channels, or wavelengths.

While the spectral profile of our comb was not indicative of operation in the single cavity soliton regime, at the same time the observed stability of the comb spectrum made it clear that we were not operating in the chaotic regime of the Ikeda map [60, 61]. In fact, both the shape and stability of our comb, we believe, are indicative of possible operation in multiple soliton states such as recently reported soliton crystals [62]. Practically speaking, we found that as long as the chaotic states are avoided, the combs generally exhibit sufficient stability to feature low enough intensity noise to allow for their effective use in these applications.

The generated comb served as the multi-wavelength source for the TTDL and since the comb spectrum was stable, a waveshaper could be used to shape the comb lines in order to generate desired channel weights. The optical power difference between the comb lines could be compensated for via multi-stage spectral shaping combined with optical amplification [20] to meet practical systems’ requirements. Looking to the future, high-conversion-efficiency Kerr combs generated by dark pulses [63] may provide an attractive solution to produce appropriately shaped combs with much higher energy efficiency.

As Fig. 1 shows, the modulated signal after the MZM (EOSPACE) was propagated through ~2.122-km of single mode fibre with a dispersion of ~17.4 ps/(nm‧km), corresponding to a time delay of ~14.8 ps between adjacent wavelength channels. We subsequently employed a Waveshaper (Finisar 4000s) as a wavelength division multiplexer to separate these channels, and then convert them back into the RF domain for PAA applications, thus achieving a high-channel count (up to 81 around the pump wavelength) TTDL with a compact structure.

The RF phase response was characterized by a vector network analyser (Anritsu 37369A), Fig. 3(a)), in which the channel at the central pump wavelength (channel 40) was set as the reference. The time delays corresponding to the measured phase slopes are shown in Fig. 3(b), in which a time delay step of ~14.8 ps/channel can be observed. To evaluate the uniformity of the delay steps between adjacent channels, we fit the measured delays with a second-order polynomial function

where *µ* = 1, 2, 3, … , 81, is the channel number, and *p*0, *p*1, *p*2 are the fit coefficients. As Eq. 4 shows, uniform delay steps correspond to linear *τ*(*µ*), while the fit curves (*p*0 = 6.7687, *p*1 = 14.8, *p*2 = 0.011878) shown in Fig. 3(b) indicate the appearance of small second-order terms arising from the third-order dispersion (TOD), thus introducing small delay errors in the TTDL (as extracted in the inset). Deviation in the comb wavelength spacing is extremely small (<1MHz), and so any delay errors induced by non-equidistance of the comb lines (<10-3 fs/step) is negligible.

To investigate key performance parameters of the PAA driven by the micro-comb based TTDL, we calculated the array factors (AFs). Since many factors contribute to the performance of the PAA, such as delay errors, imprecision in channel weights, the number of radiating elements and rapid RF frequency variations, we adopted control variables during the investigation and calculated the AFs by varying only one parameter at a time.

In Fig. 3(c), the AFs are calculated with, and without, delay errors (uniform weights, *λ**RF* = 2.5 cm, *d**PAA* = *λ**RF* / 2, *M*=81, *m*=1). As can be seen, the delay errors induced by TOD shift the beam steering angle by 1.54°, and so this must be compensated for when designing the PAA system. We note that TOD compensation in fiber has been widely investigated using many approaches, and in our case this can easily be achieved by programming the phase characteristics of the waveshaper.

In Fig. 3(d), the AFs are calculated with generated weights (inset, where the weights of 5 comb lines are supressed for the calculation) and uniform weights (λRF = 2.5 cm, dPAA = λRF / 2, M=81, m=1, T=14.8ps). Both of the AFs calculated this way yield different beam patterns that are each useful for specific applications. We note that arbitrary beam patterns can be achieved by shaping the channel weights, as we previously reported [48, 49].

In Fig. 3 (e), AFs are calculated as as *M* increases from 4 to 81 (uniform weights, *λ**RF* = 2.5 cm, *d**PAA* = *λ**RF* / 2, *m*=1, *T*=14.8ps), where we see that the corresponding 3dB beamwidth *θ**3dB* significantly decreases from 26.8° to 1.2°, matching well with the prediction *θ**3dB* = 102/*M*, as shown in Fig. 3(f). Compared with previous results based on a 200GHz-FSR Kerr comb [49], the much larger channel number provided by the 49GHz-FSR Kerr comb significantly improves beam resolution for the PAA by more than a factor of four.

To achieve a tunable beam steering angle, every *m**th*(*m* = 1, 2, 3, …) wavelength of the TTDL can be selected by using the waveshaper. As well, the time delay (*τ* ) between the radiating elements could be varied with a step size of *T*. As shown in Fig. 4(a), with 6 radiating elements (*M*=6), as *m* varies from 1 to 15, a large tuning range from -69.7° to 72.9° can be achieved. We note that tuning by selecting every *m**th*wavelength is practical in our case because of the large number of channels. The beamwidth ( ∼*m* ) and steering angle tuning range (∼ 1/*m* ) represent a trade-off in practical systems. To reflect the relationships between those parameters, in Fig. 4(b–e), we present AFs of PAAs with varying *m* and *M*, where *M* = ⌊Channel Number / *m* ⌋. Considering the practical requirements for beam steering, *M* can be set to 3 at least, and thus for the 21-channel PAA based on the 200GHz-FSR Kerr comb [49], *m* could reach 7 at most, while for the 81-channel PAA based on the 49 GHz-FSR Kerr comb, *m* could reach as much as 27. Fig. 4(b–c) shows the corresponding *M*, *θ**3dB* and *θ**0* as a function of *m*. As can be seen, on the one hand, the 49GHz FSR Kerr comb enables a much larger *M* as *m* varies, thus leading to a significantly smaller *θ**3dB* and greatly enhanced angular resolution (Fig. 4(d)). On the other hand, finer tuning steps (from 1.0° to 14.7°) as well as a larger tuning range (142.7°) of the beam steering angle *θ**0* are available due to the larger *m* (Fig. 4(e)).

Given the practical requirements in beamwidth and steering angle tuning range, optimized sets of *m* and *M* can be clearly identified from these calculations. Further, the beam steering angle can be improved in terms of both tuning step and tuning range by employing dispersion tunable media [32, 34] in addition to varying the number of TTDL channels.

Moreover, the PAA can also achieve a wide instantaneous RF bandwidth without beam squint (variation in beam steering angle with RF frequency). Based on the proposed TTDL, as indicated in Fig. 4(f), the beam steering angle *θ**0* remains 20.7° while the RF frequency varies from 2 GHz to 17 GHz (with uniform weights, *d**PAA* = 1.25 cm, *m*=1, *M*=81,*T*=14.8ps). As a result, due to our large number of channels, the PAA features greatly enhanced angular resolution, large instantaneous bandwidth, and a wide tunable range of the beam steering angle.

This approach is applicable to a wide range of wavelengths even outside the telecom band to the mid IR. [64-70] Soliton crystal microcombs have been extremely successful at a wide range of microwave and RF functions. [71-118]