1. Introduction

2. Generation of Optical Microcombs

2.2. Device Architectures

In addition to developing new material platforms and fabrication technologies, significant attention has been devoted to the modification of device architectures. Table 1 summarizes typical architectures of resonators for generating optical microcombs, including toroid [10,63,72,73,74,75,76,109,141], wedge [47,142,143,144], ridge [138], sphere [70], disk [21,51,60,90], rod [22,23,24,71,145,146,147], single ring [68,93,135,148], and multiple rings [149,150,151]. The resonators with different architectures show differences in the Q factors as well as the repetition rates for the generated microcombs. The device structures are also closely related to the material platforms, with different device architectures showing different compatibility for on-chip integration. In this section, we review and discuss the modification of device architectures for microcomb generation, including dispersion engineering, coupling control, and multi-mode interaction. Since the different device architectures for generating optical microcombs have already been reviewed in previous literature [11,12,83], here we focus on recent progress after 2018, which has improved the microcomb performance in terms of spectral bandwidth, power consumption, conversion efficiency, and stability.

Waveguide dispersion in particular is an important parameter that determines the spectral bandwidth of optical microcombs. For optical resonators, their resonance frequencies can be expanded in a Taylor series around the pumped mode as follows [11,12]

(1)

where l (= 0, 1, 2, …) are the labelled mode numbers of the mode spectrum, l₀ is the eigennumber of the pump mode, ω_l are eigenfrequencies of the resonator, and N is the order of truncation for the expansion. In Eq. (1), the Taylor expansion coefficients D_n are related to the dispersion coefficients of the propagation constant β_n as below [11]

(2)

where L is the round-trip length, ω₀ is the pump frequency, v_g = 1/β₁ is the group velocity, and Δf is the FSR. In Eq. (2), D₁ = 2πΔf is related to the FSR, D₂ is the dispersion parameter widely used for characterizing the waveguide dispersion. Another frequently used parameter is the group velocity dispersion β₂, and the relation between D₂ and β₂ can be given by

(3)

where c and λ are the light speed in vacuum and light wavelength, respectively. Waveguides with normal dispersion correspond to D₂ < 0 or β₂ > 0, whereas those with anomalous dispersion correspond to D₂ > 0 or β₂ < 0. For materials with positive Kerr coefficients n₂ (e.g., silicon, silicon nitride, and silica), waveguides with anomalous dispersion are preferable since they can mitigate the phase mismatch for nonlinear optical processes such as four-wave mixing (FWM), self-phase modulation (SPM), and cross-phase modulation (XPM) [153,154]. It should also be noted that the high-order (n ≥ 3) dispersion items in Eq. (2) should be taken into account for optical microcombs with large spectral bandwidths.

To obtain a large comb spectral bandwidth, the waveguide dispersion needs to be engineered to achieve broadband phase matching. Octave-spanning dissipative Kerr soliton (DKS) states covering the wavelength window for biological imaging (700 nm – 1400 nm) have been generated by engineering the waveguide dispersion of a Si₃N₄ MRR (Figure 6(a)), which had a cross section of ~1300 nm × 740 nm, resulting in anomalous dispersion within the wavelength range of interest [104]. Based on precise control of dispersive waves via dispersion engineering (Figure 6(b)) [100], DKS states with a spectral bandwidth (from ~150 THz to ~350 THz) exceeding a full octave have also been generated. By engineering the dispersion of a Ge-on-Si MRR to achieve both dispersion flattening and dispersion hybridization, two-octave spanning mid-infrared microcombs from ~2.3 μm to ~10.2 μm have been generated at a low pump power of ~180 mW [155]. Increased comb flatness and reduced pump power can be achieved by engineering the waveguide zero-dispersion wavelengths. Octave spanning microcombs with increased spectral flatness and decreased pump power enabled by engineering the zero dispersion wavelengths of strip / slot hybrid MRRs have been theoretically investigated in Refs. [156,157].

Engineering the waveguide dispersion can also improve comb stability. By tapering the waveguide width of a Si₃N₄ MRR (Figure 6(c)), thus creating group velocity dispersion (GVD) oscillation, the frequency detuning stability zone has been improved by over an order of magnitude relative to microcombs generated by comparable MRRs but with a uniform waveguide width [152].

Engineering the coupling between microresonators and the access, or bus, waveguides to achieve broadband light extraction with high efficiency is also needed for realizing optical microcombs with large spectral bandwidths.

By using a pulley coupling configuration, up to 20-dB enhancement of light extraction at short wavelengths has been achieved compared to devices using directional couplers [158]. By slightly deforming a SiO₂ microtoroid resonator (Figure 6(d)) to create a chaotic tunneling channel [109], broadband collection of the generated optical microcombs via a nanofiber has been achieved, yielding a comb spectral bandwidth exceeding two octaves (from ~450 nm to ~2008 nm).

Amongst all the resonator structures, ring resonators have shown the highest compatibility with integrated platforms as well as the highest accuracy in tailoring the resonance modes. This makes it possible to generate localized anomalous dispersion for microcomb generation by engineering the interaction between different ring resonators that feature normal dispersion.

Microcomb generation based on two coupled Si₃N₄ MRRs with normal dispersion was first demonstrated in Ref. [149], where different comb states were achieved by using a micro-heater to adjust the mode interactions between the two MRRs. Recently, DKS has been generated in photonic diatomic molecules formed by coupled Si₃N₄ MRRs with normal dispersion (Figure 6(e)) [151], showing attractive advantages in achieving high conversion efficiency, uniform power distribution, and low-power operation. Coherent microcombs have also been generated by using a 300-nm-thick Si₃N₄ concentric MRR (Figure 6(f)), where a tapered concentric racetrack structure was chosen to selectively excite the resonant mode having anomalous dispersion [150].

2.3. Soliton Classes

According to the classical theory of microcombs [11,83], for microresonators with low cavity loss and considering a single mode and monochromatic driving, the evolution of the intracavity optical complex field envelope can be expressed by the mean-field Lugiato-Lefever (LLE) equation as below [11]

(4)

where t_R is the round-trip time, E is the envelope of intracavity field, E_in is the external driving field, τ is time expressed in a reference frame ‒ moving with the group velocity of light at the pump frequency, t is the slow time of the microresonator related to the number of round-trips m as E(t = mt_R, τ), α describes the total losses of the microresonator, δ₀ is the phase detuning of the pump field, L is the round-trip length of the microresonator, κ is the power coupling coefficient of the input/output coupler, γ is the nonlinear parameter, and β_n are the dispersion coefficients of the propagation constant in Eq. (2).

Different types of solutions of Eq. (4) correspond to different microcomb operation regimes, such as primary combs, chaotic states, and DKS states. Amongst them, DKS states, with high coherence and low noise resulting from the dual balance between the nonlinearity and dispersion as well as loss and gain, are the most widely used for practical applications [42,46,47]. In Table 2, we compare different types of soliton states generated by optical microcombs, including bright solitons, dark solitons, breather solitons, Dirac solitons, Stokes solitons, Brillouin-Kerr solitons, soliton crystals, soliton molecules, and laser cavity-solitons.

Bright and dark solitons are two basic soliton types determined by the dispersion of microresonators. Bright solitons are generated in microresonators with anomalous dispersion and excited by a red-detuned pump, in contrast to requirement for microresonators to have normal dispersion and a blue-detuned pump in the case of dark solitons. Unlike the stochastic nature of the number of peaks generated for bright solitons, there is a deterministic pathway towards mode-locking in the normal-dispersion region, which allows for the repeatable generation of dark solitons [99]. Dark solitons also display much higher power-conversion efficiency [43,172], which is compelling for coherent optical communications. One reason for these advantages lies in the fact that dark solitons, being in some sense the inverse of single soliton states, have a naturally high power punctuated by dark solitons – and so the overall intracavity power of these states is very similar to the chaotic background. Hence there is very little photo-induced resonance shift when generating these states, which is the chief challenge for bright single solitons. For Si₃N₄ MRRs, the challenge of achieving anomalous dispersion to generate bright solitons has confronted the field since its beginning since this requires very thick (typically > 700 nm) Si₃N₄ layers which, because Si₃N₄ needs to be annealed at high temperatures to reduce the loss, often induces cracking [173,174]. In contrast, the generation of dark solitons based on microresonators with normal dispersion avoids the need to grow thick Si₃N₄ layers [99].

On the basis of bright and dark solitons, derivative soliton types can be realized by exploiting other effects such as dynamical instabilities, mode-coupling induced dispersion, stimulated Raman scattering, and stimulated Brillouin scattering. These include breather solitons, Dirac solitons, Stokes solitons, and Brillouin-Kerr solitons.

Breather solitons are generated in presence of dynamic instabilities, where bright or dark solitons experience a periodic evolution in amplitude and duration [163]. The study of breather solitons is useful for understanding the soliton dynamics and provides guidance for generating stable soliton microcombs [163]. The typical operation regime of breather solitons is between the modulation instability regime and the steady soliton regime. Inter-mode breather solitons, which can be triggered by avoided mode crossings, have been demonstrated in the conventionally stable soliton regime [175].

Dirac solitons are generated by engineering dispersion induced by nonlinear mode-coupling in the visible band [166]. They normally have asymmetric soliton comb spectra due to different mode compositions on the different sides of the spectra.

Stokes solitons are derived from the Raman effect when the primary soliton is generated. They have frequency separation with primary solitons, which has significant potential for mid-infrared comb generation [167].

Brillouin-Kerr solitons are formed by exciting red-detuned Brillouin lasing in a microresonator via a blue-detuned pump. They are easier to generate due to long soliton steps and stable access to single soliton states. They have shown advantages in achieving narrow-linewidth comb lines and stable repetition rates, which are beneficial for generating low-noise microwave signals [168].

Interactions amongst co-existing solitons within the same resonant cavity can also be engineered to form solution groups with new capabilities, such as soliton crystals and soliton molecules.

Soliton crystals are self-organized ensembles of multiple co-propagating solitons with a crystal-like profile in the angular domain [102]. They are easy to generate, even with slow manual pump wavelength tuning, and are naturally robust [102]. As in dark solitons, the reason for this lies in the fact that the intracavity power for soliton crystals is much higher than single soliton states and so when forming them out of the chaotic background, there is very little photo or thermal induced resonance shift, meaning that rapid pump detuning is not required. Although anomalous dispersion and mode crossings are needed, they are not challenging to realize via dispersion engineering [44]. Soliton crystals feature scalloped shaped spectra which, although has been considered a disadvantage for practical applications, has in fact been shown not to present a significant drawback [44], partly due to the much higher efficiency soliton crystals achieve compared to single soliton states.

Soliton molecules are the bound states of solitons achieved when there is a balance between attractive and repulsive effects. They can be generated by engineering the interaction amongst co-existing solitons in the same resonant cavity and using a discrete pump in the red-detuned regime. Soliton molecules are reproducible with high coherence and high conversion efficiency [171], making them appealing for optical communications and precision measurements.

Laser cavity-soliton (LCS) microcombs are a new class of microcomb-based solitons realized by nesting a microresonator in a fiber loop with gain [106]. They are generated based on filter-driven FWM [176] and offer many highly attractive features, such as a high mode efficiency (defined as the fraction of optical power residing in the comb modes relative to the most powerful line) and a reduced average pump power. The LCSs have a theoretical mode efficiency that can approach 100%. A key reason for this high efficiency is the fact that LCSs do not require a CW background state for the solitons to operate, which frees up all of the optical energy to be able to reside in the optical pulses and hence in the comb wavelengths. Another important advantage of LCSs is the flexibility in adjusting their repetition rates, which can be tuned by more than a megahertz through simply adjusting the overall fiber loop laser cavity length.

Featuring high coherence and low noise, soliton microcombs have underpinned many recent breakthroughs in microwave photonics [83,177], optical communications [42,44], precision measurements [46,60], neuromorphic computing [55,56], and quantum optics [57,178]. Moreover, the rich variety of different types of solitons, each having its own distinctive characteristics, significantly increases the range of performance for soliton microcombs that can be used for practical applications. For applications in microwave photonics and optical communications, two widely used families of soliton states are dark solitons [43,99] and soliton crystals [44,102], mainly because they are intrinsically easier to generate than bright single soliton states and are more energy-efficient and stable. As mentioned above, the reasons for all of these advantages stem from the same issue – the fact that the intracavity energy for these states is much higher than for single solitons. Hence there is very little power drop when they are formed, making them much easier to generate and stabilize, and also yielding much higher power levels in the cavity that allows for higher energy efficiency.

3. Microwave Photonics Based on Optical Microcombs

3.1. Frequency Synthesizers

The synthesis of microwave or optical signals with accurate control of their frequencies is a fundamental requirement for microwave photonic applications. Optical microcombs provide a new solution to realize broadband frequency synthesizers on a chip scale. Photodetection of mode-locked microcombs to obtain microwave beat notes corresponding to the microcombs’ repetition rates provides a method to directly generate microwave signals [26]. Another widely used method for microwave frequency synthesis is optical-to-microwave frequency division, which uses a microcomb locking to a stable reference light source to divide optical frequency down to microwave frequency [205]. The reverse process of the optical-to-microwave frequency division enables up-conversion of microwave frequency to optical frequency, which can be used for optical frequency synthesis [21].

Generally, frequency stability and spectral purity are two basic performance characteristics of a frequency synthesizer. Allan deviation has been widely used for the quantitative analysis of the frequency stability [21,22,24,26,27,28,144,191]. Figure 8(a) compares the Allan deviations of frequency synthesizers based on optical microcombs as well as LFCs generated by solid-state lasers and mode-locked fiber lasers. As can be seen, although the frequency synthesizers based on optical microcombs show a relatively low stability (i.e., high values of Allan deviations) for the synthesized microwave signals, they are capable of generating signals at high frequencies above 10 GHz [24,27,28]. On the other hand, the spectral purity of the generated signals is affected by a number of factors such as relative-intensity noise of pump lasers, shot noise, Q factors of microresonators, thermal fluctuations, quantum vacuum fluctuations, pump-resonance detunings, and microcomb driving mechanisms [11,191], and is normally characterized by the single-sideband (SSB) phase noise [191]. Figure 8(b) compares the SSB phase noise at different offset frequencies (1 kHz, 10 kHz, and 10 MHz) of frequency synthesizers based on optical microcombs as well as LFCs generated by solid-state lasers and mode-locked fiber lasers. As can be seen, the frequency synthesizers based on optical microcombs have already achieved high spectral purities that are comparable with their bulky counterparts.

Theoretically, the phase noise floor after photodetection can be given by [191]

(5)

where q is the elementary charge, Z is the resistance of the PD load, R is the responsivity of the PD, P_DC is the DC optical power reaching the PD, and P_RF is the power of the generated radio frequency (RF) signal. The SSB phase noise at different offset frequencies is affected by different factors. For the offset frequency below 1 kHz, SSB phase noise is mainly induced by microresonators’ frequency fluctuations [11]. For the offset frequency above 10 MHz, the SSB phase noise is dominated by the shot noise, which can be reduced by employing a narrow-band RF filter after the PD [191]. The SSB phase noise at offset frequencies between 1 kHz and 10 MHz is mainly due to the transfer of laser relative-intensity noise (RIN) to microwave phase modulation by microcomb dynamics [191]. The phase noise induced by the RIN of the pump laser can be expressed as [23,92]

(6)

where D₁/2π is the FSR, κ/2π is the cavity energy decay rate, η is the coupling impedance of the resonator, c is the speed of light, V_eff is the mode volume, n₂ is the Kerr nonlinearity, n₀ is the refractive index, f is the pump frequency, P_in is the pump power, and S_RIN(f) is the RIN of the pump laser.

Recent advances in microcomb-based frequency synthesizers have shown significant increase in performance, with the frequency range extending from the microwave to the optical domain. Table 4 summarizes the performance metrics of state-of-the-art frequency synthesizers based on optical microcombs.

Note that for different works some of the performance parameters (e.g., phase noise and Allan deviation) cannot be directly compared, since their experimental configurations are very different, with significant associated differences in complexity, volume, and cost.

A key challenge has been to reduce the frequency spacing, or FSR, of the microcombs into mainstream microwave bands, well below the 100 ‒ 600 GHz range where they were first demonstrated [11]. A recent breakthrough achieved an FSR of ~ 49 GHz [212], which enabled microwave frequency conversion from the L band (1‒2 GHz) to the U band (40‒60 GHz) [25], achieving an SSB phase noise of −90 dBc/Hz at 10⁴ Hz. Even lower FSRs have since been reported, including a microwave frequency synthesizer generating signals in the X (8‒12 GHz) and K (18‒27 GHz) bands (Figure 9(a)) [26], produced by photodetection of DKS states formed in Si₃N₄ MRRs with centimeter-scale circumferences and an ultralow propagation loss of ~1.4 dB/m. The relative Allan deviation was on the order of 10^-8 at an integration time of 1s, together with a low SSB phase noise of ~-110 dBc/Hz at 10⁴ Hz.

A 100-GHz microwave frequency synthesizer with improved signal-to-noise ratio as well as microwave signal linewidth has been demonstrated (Figure 9(b)) [27], The constructive interference among the RF beat notes generated by a mode-locked soliton microcombs from a Si₃N₄ MRR was exploited to achieve a 5.8-dB improvement of the microwave signal power compared to that using two-laser heterodyne detection. A 100-fold reduction in the microwave signal linewidth over the pump laser was also achieved after the transfer of the pump frequency variation to the comb repetition rate.

Optical-to-microwave frequency division via LFCs has proven to be an effective way to generate microwave signals with extremely high purity and frequency stability [207,213]. Recently, a ~300-GHz frequency synthesizer has been demonstrated based on frequency division using a soliton microcomb [28], where photomixing between a ~3.6-THz optically carried reference generated by stimulated Brillouin scattering in a fiber ring cavity and a ~300-GHz soliton microcomb generated by a Si₃N₄ MRR was achieved in a broadband uni-travelling-carrier photodiode, achieving a phase noise of ~-100 dBc/Hz at 10⁴ Hz.

Another microwave frequency synthesizer based on optical-to-microwave frequency division has been demonstrated using a soliton microcomb with a repetition rate of ~14 GHz generated by a MgF₂ crystalline resonator (Figure 9(c)) [23]. The frequency division was achieved by using a transfer oscillator method based on electronic division and mixing, yielding a low phase noise below ~-135 dBc/Hz at 10⁴ Hz for the generated ~14-GHz microwave signal.

Based on a new frequency division method that transfers the spectral purity of a cavity soliton oscillator into the subharmonic frequencies of the microcomb’s repetition rate, a low-noise microwave frequency synthesizer has been demonstrated (Figure 9(d)) [24], where a soliton microcomb was injected into a GSL driven by a sinusoidal current with a frequency equaling to the subharmonic frequency of the microcomb’s repetition rate, leading to the generation of a dense GSL comb dividing the microcomb’s line spacing. Successful microwave signal generation was demonstrated by dividing the repetition rates of soliton microcombs generated by a ~14.09-GHz-FSR MgF₂ crystalline resonator and a ~100.93-GHz-FSR Si₃N₄ MRR.

A microwave frequency synthesizer with reduced phase noise has been demonstrated by creating intracavity potential gradient via injection locking to trap soliton microcombs and control their repetition rate [22], achieving a low SSB phase noise of ~-130 dBc/Hz at 10⁴ Hz for ~14-GHz microwave signals generated by a MgF₂ crystalline resonator.

Recently, a palm-sized microwave frequency synthesizer has been reported [144], where a fiber delay line spool with a centimeter-scale diameter was employed as a time reference to stabilize the repetition rate of a microcomb generated by a silica wedge resonator. The generation of ~22-GHz microwave signals was experimentally demonstrated, achieving a phase noise of ~-110 dBc/Hz at 10³ Hz and ~-88 dBc/Hz at 10² Hz. The measured Allan deviation was on the level of 10⁻¹³ at 1 s. By employing a voltage-controlled oscillator and an acousto-optic frequency shifter, a tunable repetition rate over 100 kHz bandwidth was achieved.

A low-noise microwave frequency synthesizer has been realized through photodetection of soliton microcombs generated by a ~6-mm-diameter silica microrod resonator with a spherical cross section [146]. High-purity fiber preform was lathe machined to form the large-mode-volume microrod resonator with a Q factor exceeding 4 × 10⁹, enabling an extremely low power of ~110 μW for comb initiation. The synthesized microwave signal in the X band (~11.4 GHz) had a phase noise of ~-107 dBc/Hz and ~-133 dBc/Hz at 10³ Hz and 10⁴ Hz, respectively.

In addition to microwave frequency synthesizers, an optical frequency synthesizer has also been demonstrated (Figure 9(e)) [21], where two soliton combs generated by a Si₃N₄ MRR and a SiO₂ microdisk created a phase-coherent link between microwave and optical frequencies, enabling programmable optical frequency output with an ultrahigh resolution of ~1 Hz across 4 THz band around 1550 nm. The output optical frequency was highly stable, with a synthesis error < 7.7 × 10⁻¹⁵. The measured Allan deviation was (9.2 ± 1.4) × 10⁻¹⁴ at 1 s ‒ about two orders of magnitude lower than previous microcomb-based frequency synthesizers [191].

3.2. Microwave Photonic Filters

Microwave photonic filters are indispensable components in microwave photonic systems. Microcomb-based microwave photonic filters are typically implemented in the transversal filter structure based on discrete-time processing of microwave signals [83], where the microcomb serves as a multi-wavelength source to provide discrete-time channels or taps. The spectral transfer function of the transversal filter can be engineered via the design of the tap coefficients for different channels, thus allowing for highly reconfigurable filter shapes. In the following, we provide quantitative analysis and review of recent advances of microcomb-based microwave photonic filters.

For a microwave filter based on discrete-time processing, the system function can be given by [214]

(7)

where z^-1 is the basic delay between the sampled signals, and a_n and b_k are the filter coefficients. In Eq. (7), the numerator and the denominator correspond to the finite and infinite impulse parts, respectively, with R and M denoting their orders. When b_k = 0 for all k (=1, 2, 3, … , M), the filter is a transversal filter, and the corresponding impulse response can be expressed as [214]

(8)

where Δt is the time delay between two adjacent taps. The time difference between different channels is a featured characteristic of transversal filters, and they are also termed “delay-line filters” in many literatures [83,212,215]. The progressive time delays can be engineered to implement true time delay lines [216], which are fundamental building blocks in phased array antennas [212,217] and microwave beamformers [218,219]. After Fourier transformation from Eq. (8), the spectral transfer function of a transversal filter can be given by [214]

(9)

where N = R + 1 is the tap number. Due to the discrete-time processing nature of the transversal filter, the filter shape corresponding to Eq. (9) will appear periodically in the RF transmission spectrum with an FSR given by

(10)

Eq. (9) forms the basis for the implementation of microcomb-based microwave photonic filters with arbitrary filter shapes through designing the tap coefficients a_n (n = 0, 1, 2, 3, …, N−1) for different wavelength channels. For example, a low-pass (sinc) filter can be realized by shaping microcombs to have the same tap coefficient for each tap as below [30]

(11)

On the basis of the low-pass filter in Eq. (11), Gaussian apodization can be applied to improve the main-to-secondary sidelobe ratio (MSSR), where the tap coefficients a_{sinc, n} (n = 0, 1, 2, 3, …, N−1) need to be modified to the product of itself and a Gaussian function as below [30]

(12)

where σ and b are the root mean square width and the peak position of the Gaussian function, respectively. A band-pass filter can be realized by multiplying a_{gau, n} in Eq. (12) by a sinc function as below [30]

(13)

where f_center is the center frequency of the band-pass filter. In principle, for N → ∞ in Eq. (9), reconfigurable filters with arbitrary filter shapes can be realized by applying corresponding tap coefficients to each wavelength channel. For practical systems with limited numbers of available channels, negligible deviations between the practical and the ideal filter shapes can be achieved when N is sufficiently large.

Figure 10 provides a quantitative analysis for the influence of the number of comb lines (i.e., tap number of the transversal filter system) on the performance of the microcomb-based microwave photonic filters, using low-pass and band-pass filters designed based on Eqs. (11) and (13) as examples. Figure 10(a-ⅰ) shows the simulated transmission spectra of the low-pass filters with the same cut-off frequency of 1 GHz but different tap numbers ranging from 10 to 80. The filtering resolution (i.e., 3-dB bandwidth) and the MSSR extracted from Figure 10(a-ⅰ) are shown in Figures 10(a-ⅱ) and (a-ⅲ), respectively. Figure 10(b) shows the corresponding results for the band-pass filters with the same center frequency of 10 GHz but different tap numbers. As can be seen, both the filtering resolution and the MSSR of the low-pass and band-pass filters can be improved by increasing the tap number. This reflects the advantage of implementing transversal filter systems based on optical microcombs, which can simultaneously provide a large number of discrete taps using a compact device in chip scale. Another important advantage for the transversal filter systems is their high degree of reconfigurability. By changing the tap coefficients, a diverse range of spectral responses can be realized based on the same system. This is very challenging to realize for microwave photonic filters based on passive optical filters [220,221].

In one of the first microwave applications of low FSR (< 100GHz) microcombs, an 80-tap microwave photonic transversal filter was demonstrated based on soliton crystal microcombs generated by a 49-GHz-FSR Hydex MRR (Figure 11(a)) [30]. The increased number of taps, ‒ about 4 times that of previous work based on a similar structure [217], yielded a filtering resolution that was 4 times higher. Many different filter shapes, together with tunable center frequency and bandwidth, have also been demonstrated.

To improve the frequency resolution of the microcomb-based transversal filter systems, bandwidth scaling was introduced [31], where another MRR was used as a passive Vernier comb filter to slice and sample the shaped comb before photodetection, yielding a high resolution of ~117 MHz and a large microwave instantaneous bandwidth of ~4.64 GHz (Figure 11(b)).

Reconfigurable microwave photonic filters without the need for external pulse shaping have been demonstrated (Figure 11(c)) [32], where the filter shape was reconfigured all-optically by exploiting the rich family of soliton states formed in a Si₃N₄ MRR. Perfect soliton crystals were triggered at different resonances to multiply the comb spacing, thus dividing the center frequency of bandpass filters. A bandpass filter with tunable center frequency from ~0.85 GHz to ~7.51 GHz was also demonstrated based on spectral interference between two solitons in the same cavity.

Recently, a microcomb-based microwave photonic filter with a significantly improved degree of integration has also been demonstrated (Figure 11(d)) [33], where soliton microcombs generated by a heterogeneously integrated source chip were employed to drive a silicon-on-insulator (SOI) shaping chip. The source chip, which contained an InP distributed feedback (DFB) laser and an AlGaAs MRR, generated turnkey dark soliton microcombs with a spacing of ~180 GHz. The SOI shaping chip was composed of a Mach-Zehnder modulator, an MRR shaping array, and spiral delay lines. Reconfigurable bandpass filtering with tunable 3-dB bandwidth from ~1.97 GHz to ~2.42 GHz was demonstrated. The filtering performances of using on-chip and off-chip delay lines were also compared.

3.3. Microwave Photonic Signal Processors

Microcomb-based transversal filter systems can be used not only for spectral filtering but also for temporal signal processing. Different temporal signal processing functions can be realized by engineering the discrete impulse response of the transversal filter system, i.e., the inverse Fourier transform of the spectral transfer function in Eq. (9). Although implemented with photonic hardware, the transversal filter systems essentially perform digital processing, which can not only maintain the high processing speed of optical processing but also achieve improved processing accuracy compared to optical analog processing [83,222,223].

In Table 5, we summarize and compare the different functions that have been demonstrated using microcomb-based microwave photonic signal processors. On the basis of previously demonstrated Hilbert transform [34] and temporal differentiation [224], many new signal processing functions have been demonstrated after 2018, including fractional-order differentiation [37], fractional-order Hilbert transform [35], integration [38], phase encoding [40], and arbitrary waveform generation [39].

Differentiation and integration are basic signal processing functions, and more complex functions such as phase encoding and arbitrary waveform generation can be realized based on them. The spectral transfer function of Nth-order temporal differentiation is given by [224]

(14)

where j = $\sqrt{-1}$, ω is the angular frequency, and N is the differentiation order that can be either an integer or a fraction. The spectral transfer function of Nth-order temporal integration is the inverse of the Nth-order differentiation, which can be given by [38]

(15)

Hilbert transform has been widely used for signal processing in radar systems, measurements, speech processing, and image processing [226]. The transfer function of a Hilbert transformer is given by [35]

(16)

where ϕ = P × π/2 is the phase shift, with P denoting the order of the Hilbert transformer. When P = 1, Eq. (16) corresponds to a classical integral Hilbert transformer. According to Eq. (16), a Hilbert transformer can be regarded as a phase shifter with phase shifts of ±ϕ around a center frequency. The corresponding impulse response can be given by [34]

(17)

For microcomb-based transversal filter systems, differentiation, integration, and Hilbert transform can be realized by designing filters in Eqs. (14) ‒ (16) according to Eq. (9). In principle, the operation bandwidth of a microcomb-based transversal filter system is limited by the Nyquist zone, i.e., half of the comb spacing [214]. Microcombs provide a large Nyquist zone typically on the order of 10’s or 100’s of GHz, which is well beyond the processing bandwidth of electronic devices and challenging to achieve for the LFCs generated by solid-state lasers and mode-locked fiber lasers. For practical systems, the operation bandwidth can be expressed as

(18)

where Δt is the time delay between adjacent taps, Δλ is the comb spacing, β and L are the dispersion coefficient and length of the dispersive medium (e.g., single-mode fibers), respectively. Note that the OB in Eq. (18) is the same as the FSR_RF in Eq. (10). This is because both the spectral filter and the temporal signal processor are based on the same transversal filter system. According to Eq. (18), a broader operation bandwidth can be achieved by decreasing the comb spacing or the length of the dispersive medium. It is also worth mentioning that such broadening of the operation bandwidth is not unlimited, and it is still subject to the limitation of the Nyquist zone.

For signal processors, the root-mean-square error (RMSE) is widely used to characterize the processing accuracy, which can be expressed as [227]

(19)

where Y₁, Y₂, …, Y_n are the values of ideal signal processing result, y₁, y₂, …, y_n are values of practical signal processing result, and n is the number of sampled points. Figure 12 provides a quantitative analysis for the influence of the tap numbers on the RMSEs of microcomb-based microwave photonic signal processors, using integral and fractional differentiation as examples. Note that the RMSEs here are the best values that can be theoretically achieved. Figure 12 (a-ⅰ) and (a-ⅱ) show the simulated amplitude and phase responses of first-order differentiators with different tap numbers ranging from 10 to 80. Figure 12 (a-ⅲ) shows the corresponding output waveforms for the Gaussian input signal. The calculated RMSEs between the output waveforms and the ideal differentiation result are shown in Figure 12 (a-ⅳ). Figure 12 (b) shows the corresponding results for 0.5-order differentiators. As can be seen, the processing accuracy increases with tap number for both the integral and fractional differentiators. Similar to that shown in Figure 10, this reflects the fact that the increase of the tap number can also improve the performance of the transversal filter systems for temporal signal processing.

The high reconfigurability of the transversal filter system also enables the realization of versatile processing functions based on a single system ‒ simply through comb shaping according to specific tap coefficients and without any changes of the hardware [34,35,37,38,39,40,83,224]. This is usually challenging for signal processing based on spatial light devices [228,229,230,231] and passive integrated photonic devices [232,233,234,235], making the microcomb-based microwave photonic signal processors attractive for practical applications with diverse processing requirements.

A fractional-order differentiator with reconfigurable orders from 0.15 to 0.9 has been demonstrated (Figure 13(a)) [37], where 27 wavelength channels of soliton crystal microcombs generated by a ~49-GHz-FSR Hydex MRR were employed as discrete taps for the transversal filter system, resulting in an operation bandwidth of ~15.49 GHz. By using 81 wavelength channels generated by the same comb source, an integrator has also been demonstrated (Figure 13(b)) [38], which had an integration time window of ∼6.8 ns and an operation bandwidth of ~11. 9 GHz, yielding a large time-bandwidth product of ~81.

A fractional-order Hilbert transformer with reconfigurable orders from 0.17 to 0.83 has been demonstrated (Figure 13(c)) [35], where 17 taps were used, achieving an operation bandwidth up to 9 octaves, in contrast to 5 octaves achieved for previous integral-order Hilbert transformer [34]. By changing the tap coefficients, tunable bandwidth (from ~1.2 GHz to ~15.3 GHz) and center frequency (from baseband to ~9.5 GHz) of the fractional-order Hilbert transformer were also demonstrated (Figure 13(d)) [36].

Microwave phase-encoded signals, with low power densities and random phases designed to avoid their hosts being tracked, have been widely used for secure communications in radar systems. Recently, high-speed microwave phase encoding has been realized using a microcomb-based transversal filter system (Figure 14(a)) [40], showing reduced system size, complexity, and instability compared to previous methods based on polarization modulators [236,237], dual parallel modulators [238,239], and Sagnac loops [240,241]. The operation principle of microwave phase encoding was the same as that of the microwave photonic integrator in Figure 13(b), except that phase flipping of delayed replicas was introduced via differential photodetection to assemble a phase-encoded sequence in the time domain. By using 60 wavelength channels of a soliton crystal microcomb generated by a Hydex MRR, microwave phase encoding with a high encoding rate of ~6 Gbit/s and a high pulse compression ratio of ~29.6 was achieved. A reconfigurable encoding rate was also demonstrated by changing the length of each phase code.

Microcomb-based transversal filter systems have also been used for broadband arbitrary waveform generation (AWG), with wide applications to radar, remote sensing, and communication systems [242,243,244]. Similar to phase encoding, the waveform generation was achieved based on temporal integration, except that the flattened comb lines were replaced by shaped comb lines that correspond to different temporal waveforms [39]. By changing the tap coefficients, the generation of square waveforms with different duty ratios, sawtooth waveforms with different slope ratios, and various chirp waveforms was demonstrated using the same system with 81 wavelength channels (Figure 14(b)).

A microwave AWG based on coherent dual-microcomb sampling has also been demonstrated [225], which avoids the need for long optical delay lines to generate sufficient delays between the comb lines. Microwave frequencies were generated by beating a signal soliton microcomb with a local soliton microcomb, both of which were generated by Si₃N₄ MRRs with a repetition rate offset of ~150 MHz. Amplitude and phase shaping of the signal soliton microcomb was mapped onto the generated microwave frequency comb, enabling the synthesis of triangle, square, and “UVA”-like waveforms in the experimental demonstration. A tunable repetition rate of the microwave frequency comb was also achieved by adjusting the repetition rate of the local soliton microcomb (Figure 14(c)).

4. Optical Communications Based on Optical Microcombs

4.1. Coherent Optical Communications

Coherent optical communications have been the mainstream application of microcombs in optical communication systems, where coherent optical microcombs with comb spacings matching the ITU WDM spectral grid standards act as both optical carriers at the transmitters and local oscillators (LOs) at the receivers [253,254]. Compared to the IM-DD optical communications, coherent optical communications provide an attractive advantage in achieving high communication capacities, resulting from the fact that the amplitude, phase, and polarization of optical carriers are all employed to carry information [42]. The spectral efficiencies in coherent optical communications can be significantly increased by adopting advanced modulation formats [41,42], which enables ultra-dense optical data transmissions. Another important advantage of coherent optical communication systems is the improvement of the receiver sensitivity by using heterodyne or homodyne detection [2,255,256]. The noise in coherent receivers mainly comes from the LO-induced shot noise, as compared to the dark current and thermal noise for the IM-DD systems [256,257].

The data rate of a microcomb-based WDM optical communication system can be expressed as [41,42]

(20)

where SR is the symbol rate of the modulated electrical signal, BPS is the number of bits per symbol, N is the number of WDM channels that equals to the number of available comb lines, and M is the number of polarization states.

In addition to the raw data transmission rate, another important performance parameter is spectral efficiency (SE), or the number of bits/s transmitted for a given bandwidth (bits/s/Hz), which can be given by [41,42]

(21)

where CS is the comb spacing. The SE is critical since the optical bandwidth of the telecommunications band is fixed, and so this determines the ultimate achievable transmission data rate. According to Eq. (21), improved SE can be achieved by employing advanced modulation formats to increase the BPS. In practical communications, error correction coding technologies such as forward error correction (FEC) are usually employed to control the errors occurred during the communication process, which results in a net SE that is slightly lower than that calculated based on Eq. (21).

The bit error ratio (BER), which is defined as the number of error bits divided by the total number of bits transferred within a specific time interval, is widely used for characterizing the quality of data transmission [258,259]. For M-ary quadrature amplitude modulation (QAM) signals that are widely used in coherent optical communication systems, assuming the optical additive white Gaussian noise is the dominant source of errors and the reception is data-aided, the BER can be given by [42,258]

(22)

where M is the number of symbols in the complex constellation plane, OSNR is the measured optical signal to noise ratio, and erfc is the complementary error function [258]. For bright soliton microcombs used for coherent optical communications, the powers of the comb lines can be approximately given by [42]

(23)

where μ = (ω – ω₀)/D₁ is the comb mode index relative to the pump mode, ω and ω₀ are the angular frequencies of the comb and pump modes, respectively, and D₁/2π is the FSR of the microresonator. The total cavity loss-rate is given by κ = κ₀ + κ_ex, which includes the internal loss rate κ₀ and the coupling rate κ_ex, D₂ is the dispersion parameter, ħ is the reduced Planck constant, g/2π is the nonlinear coupling constant, and P_in is the input pump power. According to Eq. (23), the power degradation at comb lines far away from the pump would lead to reduced OSNRs and hence increased BER. Therefore, in order to achieve massively parallel data transmission in broad bandwidths, the dispersion and coupling of the microresonators used for generating microcombs need to be properly engineered.

The first demonstration of coherent data transmission using a multi-wavelength microcomb source was reported in 2014 [41], where 20 comb lines of a low-noise microcomb generated by a Si₃N₄ MRR were modulated with quadrature phase-shift keying (QPSK) and 16-QAM signals, achieving a maximum data rate of ~1.44 Tbit/s and a SE of ~6 bits/s/Hz.

Massively parallel coherent optical communications have been demonstrated using two interleaved soliton microcombs generated by Si₃N₄ MRRs (Figure 15(a)) [42], where the microcombs with channel spacings of ~100 GHz served as both a multi-carrier source at the transmitter and a LO at the receiver, achieving a maximum data rate of ~55.0 Tbit/s for 16-QAM signals modulated on 179 parallel optical carriers spanning over the C and L bands.

Mode-locked dark-pulse microcombs generated by a Si₃N₄ MRR with normal dispersion have been used for ~4.4 Tbit/s coherent optical data communication (Figure 15(b)) [43]. The high internal power conversion efficiency of the dark-pulse microcombs (> 30%) yielded a low pump power < 400 mW and a high transmitted OSNR > 33 dB for 64-QAM signals modulated on 20 optical carriers in the C band.

Recently, coherent optical communication with a record high SE has been demonstrated using a soliton crystal microcomb generated by a Hydex MRR (Figure 15(c)) [44], where the soliton crystal microcomb with a small comb spacing (~49 GHz), high conversion efficiency (~40%), and high stability enabled a maximum data rate of ~44.2 Tbit/s and a maximum SE of ~10.4 bit/s/Hz, enabled by the use of a higher modulation format of 64-QAM across 80 comb lines in the C band. This work also featured a live field trial demonstration over an installed optical fiber network in the metropolitan Melbourne area.

With no need for filtering or guard bands within the channel, the use of broadband optical superchannels consisting of multiple densely spaced subchannels (with an overall bandwidth larger than that of the PDs) in coherent communications can improve the SE. Coherent superchannel communication based on soliton microcombs generated by a silica wedge resonator has been demonstrated [251], where 52 comb lines with a spacing of ~22.1 GHz were employed for each superchannel. Experimental demonstrations of ~12 Tbit/s transmission of a 256-QAM signal over ~82 km and ~8 Tbit/s transmission of 16-QAM signal over ~2100 km were carried out, achieving spectral efficiencies of ~10.4 and ~6.7 bits/s/Hz, respectively.

Coherent communication with reduced power consumption and system complexity has also been demonstrated based on coherence-cloned soliton microcomb regeneration over a long distance (Figure 15(d)) [252], where the frequency and phase of 20 comb lines of a soliton microcomb generated by a Si₃N₄ MRR at transmitter were regenerated at the receiver 50 km away through sharing pump laser and two-point locking, yielding totally saved frequency offset estimation and substantially reduced carrier phase estimation in coherent detection. The demonstration of 1.68 Tbit/s transmission of 16-QAM signal over 50 km was reported, achieving a BER on the order of 10^-4.

5. Precision Measurements Based on Optical Microcombs

5.1. Dual-Comb Spectroscopy

Dual-comb spectroscopy (DCS) based on multi-heterodyne detection of two Vernier LFCs is a key approach that can achieve broadband sampling of optical spectra. In DCS, acquisition time, i.e., the time to acquire the optical spectra, is an important parameter determined by the repetition rates of the LFCs. Optical microcombs with large repetition rates provide a powerful solution for realizing broadband DCS with low acquisition times and fast measurements [60,148,260]. In the spectral domain, the output of a DCS system is an RF comb consisting of a series of heterodyne beats corresponding to different pairs of Vernier comb teeth. The minimum time τ to resolve the RF comb lines and acquire a single spectrum can be given by [264]

(24)

where Δf_rep is the difference between the repetition rates of the two combs. For practical DCS systems, the acquisition time T should satisfy

(25)

According to Eqs. (24) and (25), a larger comb spacing allows for a larger Δf_rep and hence a smaller acquisition time T. This reflects the advantage of realizing DCS based on optical microcombs compared to conventional LFCs [262,272,273].

In DCS, the wavelength ranges of microcombs are a critical parameter since they need to match the absorption bands of the materials being sensed. Hence, this motivates the need for micro-combs at both shorter and particularly longer (e.g., mid-infrared) optical wavelengths compared to more traditional ones that operate in the telecom band.

In 2016, DCS based on microcombs generated by two compact silica disk resonators was first reported for measuring molecular absorption spectra in the telecom band [60], where the large comb spacing of the microcombs (~22 GHz) yielded significantly improved acquisition rates as compared with mode-locked fiber lasers.

By using thermo-optic microheaters, the simultaneous generation of two microcombs from two Si₃N₄ MRRs on the same chip was realized with a single pump laser at ~1561 nm (Figure 17(a)) [260], which significantly reduced the system complexity. Real-time DCS measurement of dichloromethane’s absorption across a range of 170 nm near the pump wavelength was demonstrated, achieving an acquisition time as short as ~20 μs.

Subsequently, mid-infrared DCS based on optical microcombs generated by silicon MRRs has been demonstrated (Figure 17(b)) [148]. A single CW light at ~3 μm was employed to pump two silicon MRRs, generating two mutually coherent mode-locked frequency combs across a wavelength range of 2.6 μm ‒ 4.1 μm. The silicon MRRs had p-i-n junctions to sweep out the free carriers and were fabricated using an etchless method. DCS measurement of acetone in the liquid phase covering 2.9 μm – 3.1 μm was performed, achieving a spectral resolution of 127 GHz and an acquisition time of ~2 μs.

To improve the spectral resolution of mid-infrared DCS based on optical microcombs, a technique termed interleaved difference-frequency generation (iDFG) was employed (Figure 17(c)) [261], where mid-infrared combs were generated from two near-infrared combs. The microwave signal derived from a 1.5-μm soliton microcomb was used to drive a 1.0-μm electro-optic comb, enabling a densified comb spacing of the generated mid-infrared combs for finer spectral sampling. A spectral resolution of ~1.4 GHz was achieved, representing a 100-fold improvement compared with previous reports [148].

The iDFG technique has also been used for mid-infrared DCS based on 1.55-μm counter-propagating soliton microcombs in the same silica wedge resonator (Figure 17(d)) [265]. Here, the system complexity was significantly reduced as a result of the mutual coherence of the counter-propagating soliton microcombs in the same microcavity. DCS measurements of ethane and methane around 3.3 μm were experimentally demonstrated, achieving a high normalized precision of ~1.0 ppm⋅m⋅s^1/2.

Gas molecule detection based on multiple Stokes solitons generated by a silica microsphere asymmetrically coated with monolayer graphene has also been demonstrated (Figure 17(e)) [274]. By using a single CW pump to drive the over-modal resonant cavity, multiple Stokes solitons corresponding to different cavity modes with different center frequencies, at the same repetition rate locked to the co-generated Kerr soliton, were generated. This produced sensitive beat notes that allowed for sub-Hz spectral resolution for single molecule detection of NH₃ and the identification of NH₃, CO₂, and H₂O in a mixture.

5.5. Spectrum Channelizers

In precision measurements, spectrum channelizers are often employed to slice the spectra of wideband signals with instantaneous bandwidths that are beyond the capability of available instruments, thereby enabling high-resolution measurement to be performed in multiple frequency channels with smaller bandwidths [288]. Recently, microcomb-based microwave spectrum channelizers have been demonstrated [270,271], where the microcombs served as compact sources providing numerous wavelength channels, yielding improved slicing resolution as well as reduced SWaP and complexity compared to microwave spectrum channelizers based on fiber gratings [289], nonlinear fibers [290], acousto-optic crystals [291], and discrete laser arrays [292].

For spectrum channelizers, channelized frequency step and slicing resolution are two important performance parameters. The former is used for characterizing the frequency difference between channelized RF spectral segments, and the latter for the bandwidth of channelized RF spectral segments. Considering N comb lines with a comb spacing of Δf_MC, the optical frequency of the k_th comb line can be given by [271]

(26)

where f(1) is the frequency of the first comb line on the red side. After modulation, the input RF signal is multicast onto each comb line, and the spectrum of the RF signal can be spectrally sampled, or sliced by using another passive microresonator with a slightly different FSR of Δf_MR. Therefore, the k_th channelized RF spectrum can be expressed as [271]

(27)

where f_MR(k) is the kth center frequency of the passive microresonator and [f_MR(1) – f(1)] is the relative spacing between the first comb line and the adjacent resonance of the passive microresonator (i.e., the offset of the channelized RF frequency). According to Eq. (27), the channelized frequency step between adjacent comb lines is Δf_MR−Δf_MC, i.e., the difference between the FSRs of the two resonators. On the other hand, the slice resolution is determined by the filtering bandwidth of the passive filter, i.e., the resonance bandwidth of the passive resonator.

A microwave spectrum channelizer with a broad operation bandwidth of ~90 GHz has been demonstrated based on microcombs generated by a Hydex MRR, with an FSR of ~200 GHz, resulting in 20 channels in the C band (Figure 21(a)) [270]. An additional Hydex MRR with an FSR of ~49 GHz was employed as a passive Vernier filter to slice microwave signals modulated on the microcomb at every 4th resonance, achieving a channelized frequency step of ~4.43 GHz and a slice resolution of ~1.04 GHz.

To further improve the performance, a microwave spectrum channelizer using two MRRs with similar FSRs has been demonstrated [271], where soliton crystal microcombs generated by a 49-GHz-FSR Hydex MRR provided 92 channels in the C band (Figure 21(b)), yielding significantly improved channelized frequency step of ~87.5 MHz and slice resolution of ~121.4 MHz.

6. Neuromorphic Computing Based on Optical Microcombs

6.1. Single Neurons

Single neurons, or neuron perceptrons, which are fundamental building blocks in the ANNs, determine whether the input belongs to some specific classes [54,295]. For a single neuron in an ANN, the output can be regarded as the dot product of an input vector and a synaptic weight vector being biased and mapped by a nonlinear function onto a desired range [296,297]. As the basic computing function for single neurons, vector dot product (VDP) has a general form given by

(28)

where A and B are vectors including k elements, a_i and b_i are the i^th elements in A and B, respectively.

Recently, a reconfigurable single optical neuron has been realized on the basis of the microcomb-based transversal filter system [54], which unlocks the applications of microcombs for neuromorphic computing [298,299,300]. Optical microcombs can provide many wavelength channels that serve as synapses, with different values of synaptic weights being obtained by shaping the power of the comb lines. Moreover, the transversal filter system enables the delay of different wavelength channels and the sum of the delayed replicas in the same time slot. These form the basis of VDP computing and hence neuron perceptrons based on optical microcombs. In contrast to processing analog signals for differentiation, integration, or Hilbert transformation in Section 3, the realization of VDP enables processing digital signals such as digital figures by converting them into vectors, which greatly broadens the application scope of optical microcombs.

Figure 22(a) shows the experimental setup of the reconfigurable single neuron based on optical microcombs [54], where the synapses are mapped onto 49 wavelength channels of a soliton crystal microcomb generated by a Hydex MRR. The input data were first flattened and converted to vectors, and then the vectors were time-multiplexed via a digital-to-analog converter (DAC), with each symbol occupying a single time slot. The transversal filter system simultaneously performed synapse weighting in the spectral domain and vector scaling in the temporal domain, yielding a high processing speed of up to ~95.2 Gbit/s. The performance of the perceptron was tested by classifying handwritten digits and cancer cells, achieving accuracies of ~93.75% and ~86.67%, respectively (Figure 22(b)).

6.2. Neural Networks

By connecting multiple single neurons to implement neural networks, complex neuromorphic computing functions can be realized. To date, many different architectures of ONNs have been reported, mainly including feed-forward neural networks [299,301], spiking neural networks (SNNs) [297], recurrent neural networks (RNNs) [302], reservoir computing [303], and convolutional neural networks (CNNs) [55,56,304]. Optical microcombs, which offer a large number of wavelength channels that can be used for vector computing, have been mainly employed in CNN architectures based on vector convolution [55,56].

CNNs are an important class of ANNs that have been widely used for visual imagery analysis such as image recognition, classification, and segmentation [308,309]. The implementation of optical CNNs can greatly increase the processing speed and reduce the power consumption over their electronic counterparts [55,56]. In Table 8, we compare the performance of optical CNNs based on different photonic hardware. In 2016, an optical CNN was experimentally demonstrated [304], where optical computing was performed in the first layer using angle-sensitive pixels, yielding up to a ~90% reduction of power consumption in the image sensor. Subsequently, optical CNNs based on Mach-Zenhder interferometer arrays [299], diffractive optical surfaces [305], digital micromirrors [306], and spatial light modulators [307] have been investigated. Recently, microcomb-based optical CNNs have also been demonstrated [55,56], providing a promising path towards compact on-chip CNNs with high processing speed and low power consumption for data-heavy artificial intelligence applications.

In CNNs, the diverse computing functions are realized based on vector convolution [309,310], which can be achieved on the basis of VDP. For two vectors A = [a₁ a₂ … a_m] and B = [b₁ b₂ … b_n], their convolution A * B is a new vector C = [c₁ c₂ … c_m+n−1], where c_i is the i^th element in C that can be expressed as

(29)

In Eq. (29), u ranges over all legal subscripts for a_u and b_i-u+1. For vector convolution, the processing speed is normally dictated by the number of accomplished floating-point operations (FLOPs) per second, or operations per second (OPS). For microcomb-based transversal filter systems, the OPS can be expressed as [56]

(30)

where Δ t is the time delay between adjacent taps (the same as that in Eq. (18)), R is the number of multiply-accumulate (MAC) operations accomplished in Δ t, and K is the length of the input vector. The effect of the term (K−R+1)/(K+R−1) can be omitted considering K is generally much larger than R in practical neuromorphic processing. Since there are two FLOPs in each MAC operation, i.e., a multiply operation and an accumulate operation, there is a factor of 2 in Eq. (30). Note that the OPS can be scaled up by dividing the overall available comb lines into several groups, with Eq. (30) working for each group and the overall OPS being scaled by a factor equaling to the number of the divided groups.

Figure 23(a) shows a tensor core capable of performing parallel convolutional computing for optical CNNs based on a soliton microcomb generated by a Si₃N₄ MRR [55]. The vector convolution was realized by transmitting shaped microcombs through a passive waveguide network patched onto phase-change materials, where the input vector was encoded on the amplitudes of different comb lines and the kernel vector was encoded on the phase states (i.e., amorphous or crystalline) of the phase-change materials. An experimental demonstration of a 9 × 4 waveguide network with four multiplexed input vectors was reported at a modulation speed of 14 GHz, achieving a maximum overall processing speed of 2 trillion MAC operations per second. The classification of handwritten digits (0 to 9) was performed to test the performance of the tensor core, achieving an accuracy of ~95.3% agreeing well with theoretical predictions (96.1%).

Figure 23(b) shows another optical CNN realized by a microcomb-based transversal filter system [56], where the same hardware was used in both the convolutional layer and the fully connected layer. In the convolutional layer, 90 wavelength channels of a soliton crystal microcomb generated by a Hydex MRR were divided into 10 groups for parallel processing, each included 9 channels for a kernel and was modulated with flattened vectors converted from images to be processed. Convolution of the flattened input vectors and the shaped comb lines as kernel vectors was realized by computing the VDP in each time slot, which formed one of the elements of the output convolution vector. A computing speed of ~1.13 trillion operations per second (TOPS) was achieved for each kernel, representing a total of ~11.3 TOPS for 10 kernels. The output data of the convolutional layer were then pooled electronically and flattened into vectors to form the input data for the fully connected layer, where VDP computing was performed for predictions. The performance of the optical CNN was tested by classifying the full set of handwritten digits (0 to 9), achieving an accuracy of ~88%, which was very close to the theoretical accuracy for the system of 90%.

7. Quantum Optics Based on Optical Microcombs

7.1. Generation of Single / Entangled Photons

Single and entangled photons are of fundamental importance for quantum photonic devices, both of which can be produced by spontaneous nonlinear parametric processes [6]. The generation of quantum microcombs involves interactions among individual photons, which features distinctive quantum characteristics [316]. The spontaneous nonlinear parametric processes can produce entangled photons via either second- (χ⁽²⁾) or third-order (χ⁽³⁾) optical nonlinearities. The former is based on spontaneous parametric down-conversion (SPDC, i.e., quantum counterpart of classical difference-frequency generation), whereas the latter is enabled by spontaneous four-wave mixing (SFWM, i.e., quantum counterpart of classical FWM). For Kerr microcombs, the SFWM process can be triggered when the resonator is pumped below the optical parametric oscillation threshold, where two pump photons annihilate, resulting in simultaneous and spontaneous generation of the signal and idler photons. The photonic interaction can be expressed as [11]

(31)

where ħ is the reduced Planck constant, ω_p, ω_s, and ω_i are the pump, signal, and idler angular frequencies, respectively. Since SFWM intrinsically arises from the coupling between the pump photons and the vacuum fluctuation of the side modes [11], it has a purely quantum nature and can be analyzed from a fully quantum perspective [316].

In 2014, the generation of heralded single photons at different wavelengths of a microcomb was demonstrated based on a Hydex MRR embedded in an external fiber cavity (Figure 24(a)) [317]. Subsequently, a SFWM scheme employing two orthogonally-polarized excitation pumps was introduced to produce complex quantum states (Figure 24(b)) [318], achieving direct generation of orthogonally-polarized photon-pairs. In 2016, the generation of bi- and multiphoton-entangled qubits based on a microcomb generated by a Hydex MRR was demonstrated (Figure 24(c)) [59], showing attractive features such as high scalability enabled by operation over multiple modes, high-purity for the generated entangled photons, and high compatibility with fiber technology.

In 2017, the generation of energy-time entangled states covering multiple resonances was demonstrated based on biphoton microcombs generated by a Si₃N₄ MRR (Figure 24(d)) [319]. By using these quantum frequency combs, nonlocal dispersion cancellation has also been demonstrated. Subsequently, the successful generation of high-dimensional entangled quantum states was achieved based on SFMW in a Hydex MRR (Figure 24(e)) [320], where a coherent manipulation scheme was introduced to control the entangled states, allowing for deterministic high-dimensional gate operations.

In 2018, qubit and qutrit frequency-bin entanglement was demonstrated based on a microcomb with 40 mode pairs generated by a Si₃N₄ MRR with an FSR of ~50 GHz (Figure 24(f)) [321]. In 2019, the generation of cluster states, i.e., multipartite entangled states where each particle is entangled with more than one other particle, was achieved based on SFWM in a Hydex MRR [322]. Experimental characterization of the noise sensitivity of the generated three-level, four-partite cluster states was performed, together with demonstrations of high-dimensional one-way quantum operations.

7.2. Generation of Squeezed Light

In quantum theory, the energy exchange between physical systems is quantized. As a result, the sensitivity of a measurement system is intrinsically limited by the quantum noise [323]. Squeezed states of light, which show ‘squeezed’ fluctuations with less uncertainty than coherent states [324], open up new avenues to improve the measurement sensitivity beyond the quantum noise limit [325]. A notable example is their application to GEO 600 – an optical interferometer employed for gravitational wave detection [323].

In a Kerr microcomb, the out-coupled photon flux is the same for the two symmetric sidemodes ±l from the pump in the semi-classical limit. As a result, the difference between the photon numbers of the two sidemodes given by

(32)

is anticipated to be null [11]. The photon number difference in Eq. (32) can be experimentally characterized by measuring the power difference of the two sidemodes. From a quantum viewpoint, the operator corresponding to Eq. (32) can be expressed as [11]

(33)

which is not null and results in fluctuations since there is a shot-noise floor level after photodetection [61]. When specific conditions are satisfied, squeezed light with a noise level below the photocurrent shot-noise level can be generated [11], and the noise reduction (NR) relative to the shot-noise level is usually employed to characterize the squeezing performance. Table 9 compares the performance of state-of-the-art squeezed light sources based on optical microcombs.

In 2015, all-optical squeezing was experimentally realized based on a Si₃N₄ MRR pumped at ~90 mW (Figure 25(a)) [326], achieving a NR of ~1.7 dB measured by the intensity-difference squeezing method that corresponded to ~5-dB squeezing after correcting system losses. Subsequently, the generation of squeezed vacuum states has been demonstrated based on a LiNbO₃ WGM resonator with high second-order optical nonlinearity (Figure 25(b)) [327], achieving a measured NR of ~1.4 dB at a low pump power of ~300 μW in degenerate single-mode operation.

In 2020, the generation of quadrature-phase squeezed states in the RF carrier sideband was demonstrated based on a Si₃N₄ MRR with dual pumps [325], achieving a NR of ~1.34 dB characterized by homodyne measurements that corresponded to ~3.09-dB squeezing after loss correction. Subsequently, both quadrature squeezing and photon number difference squeezing were demonstrated based on SFWM in a Si₃N₄ MRR pumped at ~104.9 mW [328], achieving NRs of ~1.0 dB and ~1.5 dB, respectively (Figure 25(c)).

In 2021, strongly squeezed light was generated based on a photonic molecule resonator consisting of two coupled Si₃N₄ MRRs pumped at ~70 mW (Figure 25(d)) [329]. By using micro-heaters to tune the photonic molecule resonator, the unwanted parametric processes were suppressed, yielding a measured NR of ~1.65 dB that corresponded to ~8-dB squeezing after loss correction. Subsequently, the generation of a deterministic, two-mode-squeezed quantum microcomb was demonstrated based on a silica wedge resonator pumped at ~120 mW (Figure 25(e)) [330], where 40 frequency multiplexed qumodes were obtained, achieving a measured NR of ~1.6 dB that corresponded to ~3.1-dB squeezing after correcting system losses.

8. Challenges and Perspectives

9. Conclusions

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