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1 M. Tan, X. Xu, J. Wu, and D. J. Moss are with the Optical Sciences Centre, Swinburne University of Technology, Hawthorn, VIC
3122, Australia. (Corresponding e-mail: dmoss@swin.edu.au).
2 R. Morandotti is with INRS -Énergie, Matériaux et Télécommunications, 1650 Boulevard Lionel-Boulet, Varennes, Québec, J3X 1S2,
Canada; visiting professor, Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China.
3A. Mitchell is with the School of Engineering, RMIT University, Melbourne, VIC 3001, Australia.
* X. Xu current address: Electro-Photonics Laboratory, Dept. of Electrical and Computer Systems Engineering, Monash University,
VIC3800, Australia.
Abstract— Integrated Kerr micro-combs are a powerful source of multiple wavelength channels for photonic
radio frequency (RF) and microwave signal processing, particularly for transversal filter systems. They offer
significant advantages featuring a compact device footprint, high versatility, large numbers of wavelengths, and
wide Nyquist bands. We review progress photonic RF and microwave high bandwidth temporal signal
processing based on Kerr micro-combs with comb spacings from 49GHz to 200GHz. We focus on integral and
fractional Hilbert transforms, differentiators as well as integrators. The future potential of optical micro-combs
for RF photonic applications in terms of functionality and ability to realize integrated solutions is also discussed.
Index Terms—Microwave photonics, micro-ring resonators.
I. INTRODUCTION
All-optical signal processing based on nonlinear optics has proven to be extremely powerful, particularly when
implemented in photonic integrated circuits based on highly nonlinear materials such as silicon [1-3]. All optical signal
processing functions include all-optical logic [4], demultiplexing at ultra-high bit rates from 160Gb/s [5] to over 1Tb/s
[6], optical performance monitoring (OPM) using slow light [7,8], all-optical regeneration [9,10], and others [11-16].
Complementary metal oxide semiconductor (CMOS) compatible platforms are centrosymmetric and so the 2rd order
nonlinear response is zero. Hence, nonlinear devices in these platforms have been based on the 3rd order nonlinear
susceptibility including third harmonic generation [11,17-21] and the Kerr nonlinearity (n2) [1,2]. The efficiency of
Kerr nonlinearity based all-optical devices depends on the waveguide nonlinear parameter (γ). Although silicon-on-
insulator nanowire devices can achieve extremely high nonlinearities (γ), they suffer from high nonlinear optical losses
due to two-photon absorption (TPA) and the resulting generated free carriers [2]. Even if the free carriers are swept out
by p-i-n junctions, silicon’s relatively poor intrinsic nonlinear figure of merit (FOM = n2 / (β λ), where β is the TPA
and λ the wavelength) of around 0.3 in the telecom band is too low to achieve good performance. While TPA can be
turned to advantage for some all-optical functions [22-24], for the most part, silicon’s low FOM in the telecom band is
a limitation. This has motivated research on a range of alternate nonlinear platforms including chalcogenide glasses
[25-34]. However, while offering many advantages, these platforms are not compatible with CMOS processing – the
basis of the silicon computer chip industry.
In 2007/8 new CMOS compatible platforms for nonlinear optics were reported that exhibited extremely low two-
photon absorption in the telecommunications wavelength band. These included silicon nitride [35, 36] as well as high-
index doped silica glass (Hydex) [37-47], similar in its optical properties to silicon oxynitride. In addition to negligible
nonlinear absorption, these platforms displayed a moderate Kerr nonlinearity, resulting in an extremely high nonlinear
figure of merit as well as a nonlinear parameter that was high enough to support substantial parametric gain. Following
the first report of micro-ring resonator (MRR) based frequency comb source driven by the Kerr optical nonlinearity in
2007 [48], the first fully integrated optical parametric oscillators were reported in 2010 [36, 37] that were also CMOS
compatible. Since then the field of integrated micro-combs, or “Kerr combs” has become one of the largest fields in
RF and microwave photonic high bandwidth
signal processing based on Kerr micro-comb
sources
Mengxi Tan,1 Xingyuan Xu,1* Jiayang Wu,1 Roberto Morandotti,2 Arnan Mitchell,3
and David J. Moss1
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optics and photonics [47]. Integrated optical Kerr frequency comb sources, or “micro-combs” are a fundamentally new
and powerful tool to accomplish many new functions on an integrated chip, due to their ability to provide highly
coherent combs of multiple wavelength channels, at the same time offering a very high degree of wavelength spacing
control. Optical micro-combs are produced via optical parametric oscillation driven by modulational instability gain in
monolithic MRRs. They offer significant advantages over more traditional multi-wavelength sources. Many innovative
applications have been reported based on integrated Kerr micro-combs, from filter-driven mode-locked lasers [49-52]
to quantum optical photonic chips [53-61], ultrahigh bandwidth optical fiber data transmission [62-64], optical neural
networks [65-67], integrated optical frequency synthesizers [68]. Micro-combs have been extensively reviewed [47,
69 - 76]. The success of the new CMOS platforms silicon nitride and Hydex motivated the search for even higher
performing CMOS compatible platforms such as amorphous silicon [77] and silicon rich silicon nitride [78] that display
a combination of low linear loss, high nonlinearity and very low nonlinear loss.
All-optical signal processing based on nonlinear optics has attracted significant interest over the years for its ability
to achieve ultrahigh bandwidth without needing optical to electronic (or visa versa) conversion. It has been particularly
important for a wide range of signal processing functions for both telecommunications and RF/microwave applications.
RF photonic applications range from radar systems to signal generation and processing [79-128], and are attractive
because of the ultra-high bandwidths that can be achieved as well as low transmission loss and strong electromagnetic
interference immunity. There is a very wide range photonic RF approaches including techniques that map the optical
filter response in the optical domain onto the RF domain. This is perhaps best represented by integrated devices based
on stimulated Brillouin scattering [88-95], which has resulted in extremely high performance for RF resolution —
being able to resolve features as fine as 32 MHz, at the same time achieving a stopband rejection > 55 dB. A key
approach to achieving reconfigurable transfer functions for adaptive signal processing has been based on the transversal
filter method [96-100] that operate by generating weighted, progressively delayed replicas of an RF signal imprinted
on an optical carrier and then summing them via photo-detection. Transverse filters can realize a wide range of RF
signal processing functions by only adjusting the tap weights, and so this method is very attractive to realize advanced
RF filters that need to be dynamically adaptive. Both discrete laser diode arrays [101] as well as integrated and fibre
Bragg grating arrays and sampled gratings [103] have been successful at supplying the required taps. However, while
offering many advantages, these approaches suffer from significantly increased complexity as well as reduced
performance due to limitations in the available number of taps. Alternative approaches, such as employing optical
frequency comb sources achieved by electro-optic (EO) or acousto-optic (AO) modulation [102,104,105], can help
overcome this problem, but they require multiple high frequency modulators that rely on high-frequency RF sources.
Of these approaches, EO combs have been used the longest and have achieved significant success at realizing many
powerful functions.
Integrated optical Kerr micro-combs are a relatively recent innovation and have been very successful - offering many
advantages over other approaches that use multi-wavelength sources for RF applications. They have been demonstrated
to be capable of achieving extremely high bit rate telecommunications systems as well as a very wide range of RF
signal processing functions [107-128]. They can generate frequency combs with a much larger spacing than electro-
optic combs. EO combs and micro-combs are in fact complementary in many ways – EO combs focuses on finer
wavelength spacings from 10’s of MHz to 10 - 20 GHz, while integrated micro-combs typically have much wider
spacings from 10’s to 100’s of GHz and even into the THz regime. Larger comb spacings yield much wider Nyquist
zones that are needed to achieve large RF bandwidths, whereas finer spacings provide much larger numbers of
wavelengths or RF “taps”, but at the cost of a much smaller Nyquist zone, or RF bandwidth. Micro-combs can provide
higher numbers of wavelengths together with still being able to provide a large FSR, in a compact footprint. For RF
transversal filters the number of taps, or wavelengths, dictates the available number of channels for, for example, RF
true time delays as well as determining the performance of RF filters [85, 121]. Other systems such as beamforming
devices [112] can also be greatly improved in terms of their quality factor and angular resolution. Other innovative
approaches to filtering include techniques such as RF bandwidth scaling [125] that provide a certain bandwidth for
each wavelength channel, and so the total operation bandwidth (maximum RF signal bandwidth that can be processed)
will depend on the number of wavelengths, and will therefore be significantly enhanced by the use of micro-combs.
Recently [121], we reviewed transversal filtering and bandwidth scaling methods based on Kerr micro-combs applied
to RF and microwave spectral filters. Here, we focus on RF and microwave high bandwidth temporal signal processing
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based on integrated Kerr micro-comb devices. We review both integral and fractional order Hilbert transformers and
differentiators, as well as RF integrators. We discuss the trade-offs involved between using wide spaced micro-combs
with an FSR of 200GHz [109-111] with recently reported record low FSR spaced micro-combs at 49 GHz, based on
soliton crystals [122-128]. We highlight their potential and future possibilities, contrasting the different methods and
use of the differently spaced micro-combs. While 200GHz Kerr micro-combs have proven to be a powerful source for
RF transversal filters, enabling high versatility as well as dynamic reconfigurability, the relatively large comb spacing
FSR of ~1.6 nm restricts the number of taps to typically less than 21 wavelengths within the 30nm wide C-band. This
is important since this transversal filters require optical amplifiers and spectral shapers that are typically only available
in the telecommunications bands (1530-1620nm). This limitation in the tap number has restricted the performance of
micro-comb based transversal RF filters in frequency selectivity, bandwidth, and dynamic versatility. To improve on
this, we reported micro-comb-based photonic RF transversal filters operating with a record high number of taps − up
to 80 wavelengths over the C-band [117]. This is the highest number reported for micro-comb RF transversal filters,
and has been enabled by a record low 49GHz FSR integrated Kerr micro-comb source. For filter applications [121]
this enabled QRF factors for RF bandpass filters 4 times higher than that achievable with 200GHz combs. For temporal
signal processing, this yields dramatic improvements in both bandwidth and reconfigurability. Our results confirm the
feasibility of achieving high performance advanced as well as adaptive RF transversal filters for use in high bandwidth
RF signal processing systems, at the same time providing reduced cost, footprint, and complexity.
II. INTEGRATED KERR MICRO-COMBS
The generation of micro-combs is a complex process that relies on a combination of high nonlinear optical
parameters, low linear and nonlinear loss as well as engineered dispersion. Diverse platforms have been developed for
micro-combs [47] such as silica, magnesium fluoride, silicon nitride, and doped silica glass [47, 70, 85]. In 2008 [39]
we reported efficient four-wave mixing at milliwatt power levels based on the 575GHz FSR devices that had a
relatively low Q-factor of 60,000. This represented the first report of low power CW nonlinear optics in silica glass
based MRRs. This was followed in 2010 by the first integrated micro-combs [37, 38] in Hydex and silicon nitride - the
first optical micro-combs that were realized in an integrated photonic chip, that were inspired by the micro-combs
demonstrated in toroidal resonators [48]. Another breakthrough came with the report of [116, 117] integrated micro-
combs with record low spacings of < 50GHz which greatly expanded the number of wavelengths over the
telecommunications band to 80 or more. Apart from a record low FSR, these micro-combs operated via a different
process to single soliton states [66-73], called soliton crystals [129, 130]. Many other recent breakthroughs have been
reported in micro-combs, such as ultralow pump power combs [131], dark solitons [132], laser-cavity solitons [133]
and others [134-139].
The MRRs used in the work that is reviewed here were fabricated on a platform based on Hydex glass [37, 38] using
CMOS compatible fabrication processes. Ring resonators with Q factors ranging from 60,000 to > 1.5 million and with
radii ranging from ~592 μm to ~135 μm and even smaller at ~48 μm, corresponding to FSRs of ~0.4 nm (~49 GHz),
~1.6 nm (~200 GHz), and ~4.5 nm (~575 GHz), respectively. The RF signal processing devices that are reviewed here
were based on micro-combs with FSR spacings of 200GHz (Fig. 1a) and 49GHz (Fig. 1b). Hydex glass films (n = ~1.7
at 1550 nm) were deposited via plasma enhanced chemical vapour deposition, and then patterned by deep UV stepper
based photolithography. The waveguides were etched by inductively coupled plasma reactive ion etching to achieve
waveguides with very low surface roughness. Finally, an upper cladding layer composed of silica glass (n = ~1.44 at
1550 nm) was deposited. We typically a vertical coupling scheme between the bus and ring resonator, where the gap
of about 200nm can be controlled by film growth, that is much more accurate than lithography. The advantages of the
Hydex platform, particularly with respect to optical micro-combs, include its very low linear loss (~0.06 dB‧cm−1), its
moderately high optical nonlinear parameter (~233 W−1‧km−1), and lastly, in particular its negligible nonlinear optical
loss, even up to extremely high intensities of 25 GW‧cm−2. We reported narrow resonance linewidths corresponding to
Q factors of up to ~1.5 million (Fig. 1) for both the 49GHz and 200GHz FSR MRRs. After packaging the devices with
fiber pigtails, the through-port insertion loss was typically 0.5 to 1 dB per facet – a result of very efficient on-chip
mode converters.
To generate micro-combs with the 200GHz FSR devices, the continuous-wave (CW) pump power was typically
amplified to over ~ +30 dBm and then the wavelength was swept from blue to red near one of the TE resonances near
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~1550 nm of the MRRs. When the detuning between the pump wavelength and MRR’s cold resonant wavelength
became small enough, thus allowing the intracavity power to reach a threshold, modulation instability gain induced
oscillation occurred [47]. Primary combs were generated with the spacing determined by the MI gain peak — mainly
a function of the intra-cavity power and dispersion. As the detuning was changed further, finally single-FSR spaced
micro-combs eventually appeared. While the states that were reached by the micro-combs did not display soliton
behaviour, we note that achieving rigorous single soliton states such as the dissipative Kerr solitons (DKS) [66] is not
required. This is significant since, while much is now understood about DKS solitons and significant recent progress
has been made [139], they still require complex pump dynamics involving simultaneous frequency and amplitude
sweeping in both directions in order to “capture” or “kick” the soliton states out of the chaotic state. Our initial work
on micro-comb applications was based on the 200GHz micro-combs operating in this partially coherent state that,
while not exhibiting rigorous soliton behaviour, was nonetheless a relatively low noise state that managed to avoid the
chaotic regime [47] and which we found was adequate for microwave applications. The spectra shown in Figs. 1b and
1c are typical of the 200GHz combs in this initial microwave work, and clearly show that they are not a rigorous soliton
states. They were successful at demonstrating a wide range of RF signal processing functions [85, 107, 109 -111].
Subsequently, we reported Kerr solitons with a record low FSR of 49GHz, based on a new class of soliton [129, 130]
called soliton crystals. Soliton crystals result from mode crossings, in our case in the C-band, and are easier and more
robust to generate than DKS states, and indeed even easier than the partially coherent 200GHz FSR comb states. They
can be generated even with manual tuning of the pump laser wavelength. The underlying physics behind this is that
the internal cavity optical energy of the soliton crystal state is very close to that of the chaotic state. Therefore, when
transitioning from chaos to the soliton crystal state, there is only a very small shift in internal energy and hence very
little resonant wavelength jump. It is this “jump” in both internal energy and resonant wavelengths of the micro-
resonator that makes DKS states more challenging to achieve. At the same time, this same issue makes the efficiency
of the soliton crystal states (energy in the comb lines relative to the pump wavelength) much higher than DKS states
(the single soliton states in particular). There is a trade-off with soliton crystal microcombs, though, in that their spectra
are not flat but exhibit characteristic “curtain-like” patterns. While this can sometimes require spectral flattening, it has
not posed a fundamental barrier to achieving a wide range of high performance RF and microwave processing
functions. Soliton crystals also require mode-crossings in the resonator, and this can be somewhat challenging to
design, over and above the requirement of anomalous dispersion that DKS states need. This, however, has not posed a
barrier to fabricating many chips with a high yield [64].
When generating soliton crystals, we typically tuned the pump laser wavelength manually across a resonance (TE
polarized). When the pump (with sufficient power) was aligned well enough with the resonance, a primary comb was
generated, similar to the 200GHz states. In this case, however, as the detuning was changed further, distinctive
‘fingerprint’ optical spectra were observed (Fig. 2), arising from spectral interference between tightly packed solitons
in the cavity – solitons that formed the soliton crystals [129, 130]. A small but noticeable and abrupt step in the
measured intracavity power (Fig. 2(b)) was observed at the point where these spectra appeared. More dramatic was a
a very significant reduction in the RF intensity noise (Fig. 2(c)) at this point. Together with the spectra shapes, these
observations indicate soliton crystal formation [38], although to conclusively demonstrate this would need one to
perform time-resolved autocorrelation pulse measurements. The key issue for the RF experiments was not the specific
micro-comb state that was reached, but whether or not low RF noise and high coherence states were achieved, and we
found that these were relatively straightforward to achieve through adiabatic pump wavelength sweeping. We found
that overall soliton crystals provided by far the lowest noise states of all micro-combs, and have formed the basis for a
microwave oscillator with low phase-noise [120].
III. RF TRANSVERSAL FILTER: THEORY AND EXPERIMENT
The transfer function of a general transversal structure can be described as [85]
1
0
() Mj KT
k
k
F h e
(1)
Where j =
1
, ω is the angular frequency of the input RF signal, M is the number of taps, T is the time delay between
adjacent taps, and hk is the tap coefficient of the kth tap, which is the discrete impulse response of F(ω). The discrete
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impulse response can be calculated by performing the inverse Fourier transform of F(ω), and then temporally
windowing it with a short cosine bell. The transversal structure is essentially equivalent to a finite impulse response
digital filter, and both are fundamental tools for RF signal processing in general and photonic RF signal processing in
particular. By properly designing the tap weights hk for each wavelength channel, the transfer function for different
signal processing functions can be realized, including Hilbert transforms, differentiators and integrators. The Nyquist
frequency of the response function, which determines the maximum RF frequency that the devices can operate at, is
given by fNyquist = 1/2T. Figure 3 shows the schematic of a transversal filter based on a multi-wavelength source, showing
that the principle consists of several functions: First, all of the wavelengths from the microcomb source are imprinted
with the RF signal via an electro-optic modulator – essentially broadcasting, or multicasting, the RF signal onto all the
comb lines. This is then followed by transmitting the wavelengths through a dispersive medium to acquire wavelength-
dependent delays. Second-order dispersion, which yields a linear relationship between the wavelength and the delay,
is generally employed to progressively separate the replicas. The delayed signals can be either separately converted
into electronic RF signals or summed together upon photodetection. Separately converting the signals can be achieved
with wavelength demultiplexers and photodetector arrays for applications such as RF true time delays [116]. Summing
the signals on all of the wavelengths forms the basis for RF transversal filters for signal processing functions [85]. The
large number of wavelengths supplied by the micro-combs has significant advantages for enhancing the transversal
signal processor performance. The number and amplitude of progressively delayed RF replicas, or taps, determines the
performance of the system. By setting the tap coefficients, a reconfigurable transversal filter with virtually any transfer
function can be achieved, including both integral and fractional order Hilbert transformers [107, 126], integral and
fractional differentiators [109,127], bandpass filters [111,117, 121], integrators [123], waveform generators [128] and
much more.
Figure 4 shows the experimental setup for a transversal filter based on an integrated Kerr optical comb source with
an FSR of 200GHz (Fig. 4(a)) as well as the setup based on a 49GHz (Fig. 4(b)) comb. In both cases the micro-combs,
generated by the on-chip MRR, are amplified and fed to either one (200GHz comb) or two (49GHz comb) waveshapers
for channel equalization and weighting. A 2nd waveshaper was used for the 49GHz FSR comb to pre-flatten the non-
uniform comb spectra of the soliton crystals. The power of each comb line is then set by the waveshaper according to
the required tap weights. To increase the accuracy, real-time feedback control paths are used to read and shape the
power of the comb lines. The comb lines are then spatially divided into two paths according to the sign of the tap
coefficients – one path for negative weights and one for positive. Next, for the 200GHz micro-comb experiments the
signal was passed through a 2×2 balanced Mach-Zehnder modulator (MZM) that was biased at quadrature. The 2×2
balanced MZM simultaneously modulated the input RF signal on both positive and negative slopes, thus yielding
replicas of the input RF signal with phase and tap coefficients having both algebraic signs. For experiments with the
49GHz FSR microcomb, both positive and negative taps were achieved by spatially separating the wavelength channels
according to the designed tap coefficients and then feeding them into the respective positive and negative input ports
of a balanced photodetector (Finisar BPDV2150R) (Figure 4b).
For the system based on the 200GHz comb, the signal that was modulated by the MZM then went through ~2.122-
km of standard single mode fibre (SMF), where the dispersion was ~17.4 ps/(nm‧km), corresponding to a time delay T
of ~59 ps between adjacent wavelengths, or taps (the channel spacing of the time delay lines equalled the FSR of the
micro-combs), which resulted in a Nyquist frequency of ~8.45 GHz. The bandwidth of the systems was determined by
the Nyquist frequency, which could be easily enlarged by decreasing the time delay which, because of the large FSR
of the compact MRR, could reach over ~ 100 GHz. Finally, the weighted and delayed taps were combined upon
photodetection with a high-speed photodetector (Finisar, 40 GHz bandwidth) which converted them back into RF
signals.
For the transversal system based on the 49GHz FSR microcomb, the signal went through about 5km of standard
SMF to provide the progressive delay taps. The dispersion of the fibre was the same (17 ps/nm/km) as the 200GHz
system, which yielded a different time delay T of ~34.8 ps between adjacent taps. This yielded an operational bandwidth
(Nyquist frequency, half of FSRRF) of about 14.36 GHz for the transversal filter. As before, this can be increased by
decreasing the time delay (using a shorter spool of SMF), at the expense of having a reduced resolution in tuning. The
maximum operational bandwidth of the transversal filter is limited by the comb spacing. For the 49 GHz comb,
significant crosstalk between adjacent wavelength channels occurs for RF frequencies that are beyond 24.5 GHz —
half of the micro-comb’s spacing. This issue can be addressed by employing a micro-comb source with a larger comb
spacing, although this comes at the expense of providing fewer comb lines/taps across the C-band. Note that although
standard optical fibre is often used to produce the dispersive delays, this can readily be done much more efficiently
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using other approaches. Typically, only about 2-km of SMF, yielding a dispersion of 34 ps/nm, is needed. This is
within the range of multi-channel tunable dispersion compensators [140-143], which can also be designed with a built-
in dispersion slope offset. This approach would not only yield a compact and latency free delay but would enable
tunability to adjust the Nyquist zone. In the following sections, we will review progress made in high bandwidth
temporal RF and microwave signal processing functions beginning with Hilbert transforms, followed by differentiation
and then integration, all based on Kerr micro-combs.
IV. HILBERT TRANSFORMS
The Hilbert transform (HT) is a fundamental signal processing operation with wide applications in radar systems,
measurement systems, speech processing, signal sampling, single sideband modulators, and others [144]. Standard
integral HTs perform a ± 90° phase shift around a central frequency, with an all-pass amplitude transmission. Fractional
Hilbert transforms (FHTs) that yield a variable phase shift represent a powerful extension to the standard HT, with
applications to secure single sideband communications [145], hardware keys [145, 146], and in forming images that
are edge enhanced relative to the input object, where one can select the edges that are enhanced as well as the degree
of edge enhancement [147]. Electronic fractional Hilbert transformers are limited in operation bandwidth [145, 148],
whereas photonic technologies offer broadband operation as well as strong immunity to EMI. Hilbert transformers
based on free-space optics have been demonstrated [146, 149] that achieve high performance but are bulky and
complex. Phase-shifted fibre Bragg grating based Hilbert transformers typically have bandwidths of a few 100GHz
[150-153] but only yield a precise FHT for signals with specific bandwidths, fixed fractional orders, and only operate
on the complex optical field (not the actual RF signal). This also holds for integrated reconfigurable microwave
processors [154] as well Bragg grating approaches [155]. In practice, the FHT of the RF and microwave signals - not
the complex optical fields − is what is most desired for RF measurement and signal reshaping [156-163].
Integrated Hilbert transformers realized using micro-ring or micro-disc resonators, Bragg gratings in silicon, and
integrated InP-InGaAsP photonic chips yield compact devices with high stability and mass-producibility [154, 155,
159-161]. Nevertheless, they only operate on the complex optical field and generally only provide a limited phase shift
tuning range. Transversal filter approaches offer high reconfigurability [162, 163] but need many discrete lasers, thus
increasing system size, cost, and complexity, and limiting the number of taps and therefore performance. Optical
frequency combs (OFCs) provide a single high-quality source for many wavelengths, including mode-locked lasers
[164], electro-optical modulators [165], and micro-resonators [166]. Of these, Kerr micro-combs generated by micro-
resonators provide many wavelengths with greatly reduced footprint.
The spectral transfer function of a general fractional Hilbert transformer can be expressed as [80, 126]:
(2)
where j =
1
, φ = P × π / 2 is the phase shift, and denotes the fractional order. From Eq. (2), a FHT can be regarded
as a phase shifter with ± φ phase shifts centered around a frequency ω, which becomes a classical integral HT when P
= 1. The corresponding impulse response is a continuous hyperbolic function:
(3)
For digital implementations, this hyperbolic function is truncated and sampled in time by discrete taps. The sample
spacing Δt determines the null frequency fc = 1/Δt. The order of the FHT is continuously tunable by only adjusting the
coefficient of the tap at t = 0 while keeping the same coefficients for the other taps [146].
To achieve a Hilbert Transform the normalized power of each comb line is given by:
1
/ 2 0.5
n
pnN
(4)
where N is the number of comb lines, or filter taps, used in the filters, and n = 0, 1, 2, …, N-1 is the comb index.
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In 2015, the first Kerr micro-comb based Hilbert transformers was demonstrated, with up to 20 taps [107] from a
200GHz comb that achieved a record RF bandwidth (5 octaves). Figures 5(a) and 5(b) show the shaped optical micro-
combs measured using an optical spectrum analyzer for the 12 and 20 tap filter cases, respectively. The designed tap
weights, or wavelength target powers, are also shown as green crosses in Figs. 5(a) and 5(b). The waveshaper was used
to shape the powers of all comb lines successfully to within +/-0.5 dB of the target powers. This could easily be
achieved since the waveshaper resolution of 10 GHz was much smaller than the comb spacing which was 200GHz at
the time. Once the comb lines were attenuated in order to provide the correct tap coefficients of the impulse response
associated with a HT, with the unused comb lines attenuated below the noise floor, the system RF frequency response
both in amplitude and phase was then characterized with a vector network analyzer (VNA).
Figure 6a shows the measured photonic Hilbert transform filter RF amplitude frequency response for 12, 16 and 20
taps, respectively. All filters agree well with the theoretical calculations and show very low (< 3 dB) amplitude ripple.
All three filters have the same null frequency centred at 16.9 GHz, which corresponds to the tap spacing of
t = 1/fc =
59 ps. This matches the difference in delay between the comb lines, equal to the micro-ring FSR of 1.6 nm, produced
by propagation of the signal through the 2.122-km long spool of SMF that has the standard dispersion parameter of D
= 17.4 ps/nm/km. The null frequency could be varied by using different fiber lengths to adjust the tap delay spacing or
by using tunable dispersion devices. As expected, increasing the number of filter taps increases the filter bandwidth.
With a 20-tap filter, the Hilbert transformer exhibited a 3-dB bandwidth extending from 0.3 GHz up to 16.4 GHz - a
range of more than 5 octaves - which was a record for any approach to HTs at the time. It is possible to increase this
bandwidth even further by using more comb lines in the filter and in fact only a small portion of the generated comb
lines was actually used to realize the filter taps. The number of filter taps that could be achieved was in fact limited by
the bandwidth of the waveshaper, which was restricted to the C-band, as well as the gain bandwidth of the erbium-
doped fiber amplifiers (EDFAs). The amplitude ripple within the passband can be reduced even further by apodizing
the tap coefficients compared to the ideal hyperbolic function. Figure 6(b) shows the measured phase response of filters
with different numbers of taps, showing a very similar response. Each shows a relatively constant phase of near -90o
within the passband. There are some deviations from the ideal -90o phase at frequencies close to zero and particularly
for the null frequency fc = 16.9 GHz.
Recently, [126] Tan et. al used a Kerr microcomb having a significantly finer comb spacing of 49GHz that provided
a much larger number of comb lines (80 in the C band) to demonstrate a fractional Hilbert transformer (FHT). By
programming and shaping the comb lines according to calculated tap weights, FHTs with phase shifts of 15°, 30°, 45°,
60°, 75°, and 90° corresponding to fractional orders of 0.17, 0.33, 0.5, 0.67, 0.83, and 1, respectively, were achieved.
A maximum of 17 wavelengths were used, with the space between the 8th and 9th taps and 9th and 10th taps being 0.8
nm, while the remaining spacings were 1.6 nm. This was done in order to maximize the Nyquist zone. The delay line
was provided by a 2.1-km length of single mode fibre (β = ~17.4 ps / nm / km) that generated a time delay between
channels of τ = L × β × ∆λ = ~29.4 ps, corresponding to a FSRRF of 1/2τ = ~17 GHz. This bandwidth can be increased
by using a shorter length of fibre.
The theoretical amplitude and phase response of the FHTs with 15°, 30°, 45°, 60°, 75°, and 90° phase shifts are
shown in Figs. 7 (a)-(f) solid black curves for different numbers of taps. Figures 8(a) and (b) show the RF amplitude
and phase frequency response for a FHT with a 45° phase shift with 5, 9, 13 and 17 taps, respectively. Figure 8c) shows
the 3dB bandwidth as a function of the number of taps. As expected, the bandwidth increases with the number of taps.
With 17 taps, the operating 3-dB bandwidth for base-band RF signals was 480 MHz to 16.45 GHz - more than 5 octaves
– and ±0.07 rad phase variation within the 3-dB bandwidth. These numbers represent record performance for FHTs.
For a standard 90° HT the soliton crystal micro-comb provided more than 40 taps, thereby reducing the root-mean-
squared error (RMSE) and amplitude ripple within the passband [126]. As seen in Fig. 8 (c), the theoretical 3-dB
bandwidth increases rapidly with the number of taps up to 17 taps, after which it levels off, so that 17 to 20 taps is
close to the optimum number.
Figure 9 (ii)-(iii) presents the simulated (red dashed curves) and measured (blue solid lines) response of the fractional
Hilbert transformers with variable orders ranging from 0.166 − 0.833 (phase shifts 15° − 90°). The normalized
frequency response for both the magnitude and phase of the FHTs are shown together with the temporal response (Fig.
(iv)). The amplitude variations are less than +/- 1.5dB between 480MHz and 16.45GHz and the phase variations are
about ±0.07 rad within the 3-dB passband. System demonstrations of a real-time FHTs applied to Gaussian input pulse
signals were also performed. Fig. 9 (iv) shows a pulsed waveform after processing by the fractional Hilbert transformer
along with the simulated results (Fig. 9 (iv), red dashed curves). The detailed performance parameters are listed in
8
Table I of Ref. [126]. As can be seen, the measured curves closely match the theoretical counterparts, achieving good
performance that agrees well with theory.
In comparing the earlier results achieved with the 200GHz FSR microcomb with the more recent FHT results
achieved with the soliton crystals having a 49-GHz FSR spacing, it is clear that the soliton crystal based results yield
superior performance over the 200-GHz comb-based transformer in terms of bandwidth, mainly because of the larger
number of available taps. Nonetheless, we note that it is not always the case that a smaller FSR comb is necessarily
better. The 200-GHz devices are capable of reaching extremely high RF bandwidths, particularly in tunability,
approaching 100GHz, since this represents the Nyquist zone of these combs, whereas the 49GHz devices are limited
to about 25GHz. In summary, our good agreement with theory confirms that the approach of using microcombs is a
very effective way to implement high-speed reconfigurable HTs and FHTs with lower complexity, reduced footprint,
and lower cost.
V. DIFFERENTIATION
The fundamental operation of differentiation is a key function in RF and microwave signal processing. It has wide
applications for RF spectral analysis, ultra-wideband (UWB) generation, and RF spectral filters [167-171]. Significant
progress has been made on RF integral differentiators via photonic-based technologies [172-184] using self-phase
modulation (SPM) [172,173] and cross-phase modulation (XPM) [177-179] methods, frequency discriminators and
cross-gain modulation in semiconductor optical amplifiers (SOA) [176]. Compared to conventional integral
differentiation, fractional differentiation has many more unique and powerful applications [127,182,183]. It has played
an important role in many different fields including electricity and electronics, mechanics, chemistry, biology, and
economics. Perhaps its most important applications have been in image edge detection, control theory, and
mechatronics [182,183]. Interestingly though, despite these wide benefits and applications, photonic based fractional
differentiators have received little attention compared to other functions.
For the most part, photonic differentiators – both integral and fractional - have focused on generating the derivative
of the complex optical field, rather than the actual RF signal. For example, a photonic differentiator based on a dual-
drive MZM together with an RF delay line, [174] although successful, was limited in processing speed by the
bandwidth of the RF delay line. Photonic differentiators based on optical filters [175] feature high speeds of up to 40-
Gb/s, although they only work for a fixed (and typically integral) differentiation order and generally lack flexibility or
reconfigurability. To implement highly reconfigurable differentiators, transversal schemes have been investigated
using discrete laser arrays [169, 180, 181]. While this is a powerful and attractive approach, it also presents some
limitations, chiefly in the number of wavelengths that can be supplied. Increasing the number of lasers to provide more
taps dramatically increases the system size, cost, and complexity, thus limiting the processing performance. Using a
single source that can simultaneously generate a large number of high-quality wavelength channels would be highly
advantageous. Recently [109], a photonic RF integral differentiator with 8 taps, achieved using a 200-GHz FSR micro-
comb source was demonstrated, that generated the 1st, 2nd, and 3rd order derivatives operating at RF bandwidths up to
17 GHz with a potential bandwidth of 100 GHz.
An Nth-order temporal fractional differentiator can be considered as a linear time-invariant system with a transfer
function given by [168]
( ) ( )N
N
Hj
(5)
where j =
1
, ω is the RF angular frequency, and N is the differentiation order which can be fractional and even
complex [184,185]. According to Equation 5, the amplitude response of a temporal differentiator is proportional to
|ω|N, whereas the phase response has a linear profile, with a phase jump of Nπ at null frequencies.
Using the transverse filter approach to RF photonic differentiators yields a finite set of delayed and weighted
replicas of the input RF signal in the optical domain, which is then combined upon detection. The transfer function
from Eq. (5) becomes
(ω)an
M1
n=0 ejωnT (6)
where M is the number of taps, an is the tap coefficient of the n-th tap, and T is the time delay between adjacent taps. It
should be noted that differentiators that achieve the function based on Eq. (6) are intensity RF differentiators, i.e., the
output RF signal after being combined upon detection yields an exact differentiation of the input RF signal, in contrast
to optical field differentiators that produce the derivative of complex optical fields [167,171,153,154,185-187].
9
Recently, [109] the first use of a micro-comb to achieve RF differentiation was reported, demonstrating integral
1st, 2nd, and 3rd order intensity differentiators. That demonstration was achieved with a 200GHz FSR Kerr micro-comb
using 8, 6, and 6 taps, respectively. It achieved a dynamic range, determined by the EDFA, of ~30 dB. More taps are
needed when the differentiation order increases, and for a fixed number of taps, increasing the order of differentiation
also increases the required power dynamic range. In order to obtain better performance for a given number of taps, the
operation bandwidth of the integral differentiators was decreased to half of the Nyquist frequency. To implement the
temporal differentiator in Eq. (5), tap coefficients in Eq. (6) were calculated based on the Remez algorithm [188]. The
calculated tap coefficients for 1st, 2nd, and 3rd order differentiation are listed in Table I of [109]. As for demonstrations
of other RF functions, since the 200GHz Kerr comb was not flat, real-time feedback control was used to increase the
accuracy of the comb shaping. Both the phase and amplitude responses of the 1st, 2nd, and 3rd order differentiators are
shown in Figs. 10 (a)–(c) for different numbers of taps. As the number of taps increases, the discrepancies between the
ideal and realized amplitude responses of the transversal filter based differentiators are improved for all three orders.
In contrast the phase response of the transversal filters is identical to that of the ideal differentiators regardless of the
number of taps.
The shaped optical comb spectra shown in Figs. 11 (b)–(d) reveal a good match between the measured comb lines’
power (red solid line) and the calculated ideal tap weights (green crosses). Figures 12 (a-i), (b-i), and (c-i) show the
measured and calculated amplitude response of the differentiators. The corresponding phase response is depicted in
Figs. 12 (a-ii), (b-ii), and (c-ii) where it is clear that all three differentiators agree well with theory. The FSR of the RF
response spectra was ~16.9 GHz, consistent with expectations given the time delay between adjacent taps. By adjusting
the FSR of the transversal filter through varying the dispersive fibre delay or programming the tap coefficients, a
variable operational bandwidth for the intensity differentiator can be achieved, which is advantageous for the changing
requirements of different applications. In particular, the ability to change the order of the differentiator simply by
changing the tap weights offers a powerful way of dynamically changing the functionality of the system in real time
without needing to modify any hardware. This is a feature that is not common to other approaches to differentiation,
particularly optical filter-based systems.
Systems demonstrations of real-time signal differentiation were also performed for Gaussian input pulses having a
full-width at half maximum (FWHM) of ~120 ps, as shown in Fig. 13 (a). The generated waveforms of the output
signals, representing the different order derivatives, are shown in Figs. 13 (b)–(d) (blue solid curves), together with the
theoretical results (red dashed curves). The experimental Gaussian pulse shown in Fig. 13 (a) was used as the input RF
signal for the theoretical calculations. The measured curves closely match theory. Unlike the optical field differentiators
[167,171,153,154,185-187], the temporal derivatives of the RF intensity profiles can be observed, indicating that
intensity differentiation was successfully achieved. For the 1st, 2nd, and 3rd order differentiators, the calculated RMSE
between the measured and theoretical curves are ~4.15%, ~6.38%, and ~7.24%, respectively.
Recently, [127] Tan et.al reported the first fractional-order photonic RF differentiator that operates directly on the
RF signal rather than the complex optical field. It was based on a record-low 49-GHz-FSR Kerr micro-comb source,
which provided a large number of comb lines (80 in the C band), resulting in a broad operation bandwidth of 15.49
GHz. By programming and shaping the comb lines according to the calculated tap weights, reconfigurable arbitrary
fractional orders ranging from 0.15 to 0.9 were demonstrated. System demonstrations of real-time fractional
differentiation of Gaussian input pulses were also performed. The good agreement between theory and experiment
confirmed that this approach as an effective way to implement high-speed reconfigurable fractional differentiators with
reduced footprint, lower complexity, and potentially lower cost.
The success of the differentiator resulted from both the use of soliton crystals combined with the low FSR of 48.9
GHz, that generated over 80 wavelengths, or taps, in the C-band. The performance of the system was greatly aided by
the soliton crystal’s extremely robust and stable operation, ease of initiation and generation and as well as its much
higher intrinsic efficiency, all of which are enabled by the integrated CMOS-compatible platform. The relationship
between the number of taps and the performance of the fractional differentiator was also investigated. Figure 14 shows
the theoretical transfer function of the fractional order differentiator for six different orders and for varying numbers
of taps, compared to the ideal theoretical results. As can be seen, the operational RF bandwidth of the differentiator —
within which the slope coefficients of the simulated and ideal amplitude responses match closely—increases with the
number of taps. Figure 15 (a) shows that the number of taps is critical to the operational bandwidth for fractional
differentiators, which is evident in the experimental results for the fractional order of 0.45 (Figs. 15 (b) and (c)). When
the number of taps was increased to 27, the bandwidth of the fractional differentiator reached 15.49 GHz, equivalent
10
to over 91% of the Nyquist band. In the experiments, up to 29 comb lines were employed to generate the taps. To
increase the number of taps, the optical signal-to-noise ratio, which was subject to the optical amplifiers’ noise,
would need to be increased.
The shaped comb spectra used to generate the derivatives are shown in Fig. 16, where it is evident that the optical
power for each comb line matched the designed tap coefficients well.. The transmission frequency response of the
fractional differentiator was characterized with a VNA. The power response of the differentiator showed a bandwidth
ranging from DC to 15.49 GHz. As shown in Fig. 17, the range of fractional orders that the system was able to
achieve was a result of the close agreement between theory and experimental in terms of the slope coefficients of
the power response and phase shift in the phase response. Figure 18 shows the time domain performance of the
fractional differentiator for all orders, using a baseband RF Gaussian pulse with a FWHM of ~200ps (Fig. 18) as the
input signal, generated by an arbitrary waveform generator. The good agreement with theory reflects the range of
orders that are achievable with our approach to realizing a reconfigurable fractional differentiator.
VI. INTEGRATION
Temporal integration is another key fundamental function in signal processing systems. Photonic techniques offer a
broad bandwidth, in contrast to electrical integrators that are subject to the electronic bandwidth bottleneck, along with
a strong immunity to EMI, as well as low loss [79,80,189]. Significant work has been reported on photonic integrators,
including gratings based approaches [190-192] and methods using micro-ring resonators (MRRs) [15,42,193]. These
approaches have achieved optical integration with a time resolution as short as 8 ps [193] as well as offering a large
time-bandwidth product due to the availability of very high-Q resonant structures. However, these approaches still face
limitations for RF signal processing, including the fact that they are not adjustable in their performance parameters
such as temporal resolution or length of integration window. Hence, they lack the flexibility to process RF signals with
different bandwidths and integration time windows. Probably more importantly, those approaches rely on processing
the optical signals - they cannot directly process RF signals without electrical to optical interfaces. Another important
approach to photonic integrators is based on transversal structures, that offers a high degree of reconfigurability and
accuracy owing to the parallel scheme where each path can be controlled independently [194-196]. By tailoring the
progressive delay, RF signal integration with a reconfigurable operational bandwidth can be achieved [194-196]. Still,
however, these still face limitations arising from the limited number of channels in cases where diode laser arrays or
electro-optical comb sources were employed. These approaches have a significant trade-off between the number of
wavelengths and system complexity and ultimately they result in a limited number of channels which, and in turn,
limits the time-bandwidth product.
Integrated Kerr micro-combs [38, 47, 68-70,118,197] offer many advantages for RF integration, including a very
high higher number of wavelengths as well as a greatly reduced footprint and complexity. Recently, a highly
reconfigurable photonic RF temporal integrator was demonstrated with an integrated soliton crystal micro-comb source
[123], where the RF signal was imprinted onto the micro-comb lines and progressively delayed via dispersion, and
then summed upon detection. The large number of taps, or wavelengths (81) used in those experiments resulted in a
large integration time window of ~6.8 ns with a time resolution as short as ~84 ps. The integrator was successfully
tested with different input signals, with experimental results matching well with theory.
Figure 19 shows the operation principle of the transversal filter based photonic RF integrator. The integration process
can be achieved via a discrete time-spectrum convolution operation between an RF input signal f (t) and a flattened
micro-comb. With a delay step Δt, the operation is:
1
( ) ( + )
N
k
y t f t k t
(7)
where j =
1
, N is the total number of wavelengths. After the replicas of f(t) are progressively delayed and then
summed, the temporal integration of the function f (t) results, with a time resolution given by [194] Δt, an operation
bandwidth of 1/Δt, and a total integration time window of T = N × Δt. The frequency-domain transmission response of
an ideal continuous, or non-discrete, integrator is linear to 1/(j
ω
), where
ω
is the angular frequency. The frequency
transmission response of the discrete integrator is a low pass sinc filter [111,117], given by
11
(8)
The experimental setup of the photonic RF integrator was similar to that shown in Figure 4, and was based on a
soliton crystal micro-comb with an FSR of ~49 GHz. The soliton crystal micro-comb featured very low intensity noise
to the extent that it has even been used as a photonic local oscillator [120]. Thus, the noise of the micro-comb source
did not introduce any observable deterioration in the overall noise performance of the integrator. After flattening the
micro-comb spectral lines with a WaveShaper, the RF signal was imprinted onto the comb lines via an electro-optical
modulator (EOM). The replicas were progressively delayed by a ~13km spool of standard optical fibre SMF and
summed upon photodetection with a high-speed photodetector. The delay between the adjacent wavelengths (Δt) was
set by the dispersion, length of fibre and spacing between the comb lines, and this in turn determined the resolution of
the integrator [194]. The temporal resolution was ~ 84 ps and in theory can be made arbitrarily small by reducing the
amount of dispersion in the delay line, although with a trade-off that the integration window also decreases. Sixty comb
lines in the C-band were selected, yielding an integration time window (T = N × Δt) of 60 × 84 ps = 5.04 ns (yellow
shaded region in Fig. 20). The operation bandwidth of the integrator was 1/Δt =11.9 GHz. Signal integration for
different RF input signals was performed. The red curves in Figs. 20(a)−(c) show the results for Gaussian pulses
(blue curves) with a FWHM from 0.20 ns to 0.94 ns, with the integration window T (~5 ns) matching with theory
well (5.04 ns). Figures 20(d)−(e) show the results for dual Gaussian pulses with time intervals of 1.52 ns and 3.06
ns, where it is clear that the experimental results (red curves) illustrate the performance of the integrator by exhibiting
three distinct intensity steps in the integration waveforms. The left step corresponds to the integration of the first
pulse while the middle step reflects the integration of both of the two initial pulses, and finally the right step shows
the integration of only the second pulse since it is beyond the integration window of the first pulse. The integrator
was further tested with a rectangular input waveform having an equal width to the integration window (5 ns). The
measured integrated waveform exhibited a triangular shape that matched well with theory (gray curve).
Figure 20 shows small residual discrepancies between experiment (red curves) and theory (grey curves). The errors
were a result of the non-ideal impulse response of the system, caused by a number of relatively small effects that
include gain variation with wavelength of the optical amplifier, modulator and photodetector. To verify this, the
impulse response of the system was measured with a Gaussian pulse input. Since the time resolution of the system (~
84 ps) was much less than the pulse width, the wavelength channels were separated into multiple subsets (each with a
much larger spacing between the adjacent comb lines and thus obtained a temporal resolution larger than the input
pulse duration), and their impulse responses measured sequentially. Figure 21(a) shows the measured impulse response
of the system, which was not flat even when the comb lines were perfectly flattened. The measured impulse response
and the input RF signal in Fig. 20 was used to calculate the corresponding integral output, with the results matching
the experiment well, showing that the errors were induced by the non-ideal impulse response of the system.
These errors are in fact straightforward to compensate for. To reduce the errors, a more accurate comb shaping was
used where the error signal of the feedback loop was generated directly by the measured impulse response, instead of
the optical power of the comb lines. Thus this implicitly had the wavelength dependence of the components built-in.
A flattened impulse response was obtained (Fig. 21(b)), which was much closer to the ideal impulse response than Fig.
21(a). The integration was then performed with the same RF inputs, the results of which are shown in Fig. 22. Note
that during this measurement, 81 wavelength channels were enabled by the impulse response shaping process, so that
the integration time window (T = N × Δt) increased to 81 × 84 ps = 6.804 ns, resulting in an operation bandwidth of
1/84ps = 11.9 GHz and a time-bandwidth product of 6.804 ns × 11.9 GHz = ~81 (approximately equal to the number
of channels N). The measured integrals (red curves, Fig. 22) show significantly better agreement with theory, indicating
the success of the compensating impulse response shaping method as well as the overall power and feasibility of this
approach to photonic RF integration.
VII. DISCUSSION
We have reviewed recent progress on the use of Kerr micro-combs for high bandwidth temporal RF and
microwave signal processing, focusing on integral and fractional order Hilbert transformers, high order and fractional
differentiators, both with continuously tunability in their order, and finally RF integrators. Micro-combs produce a
large number of comb lines that greatly increase the performance and processing bandwidth of RF systems in terms of
both the amplitude and phase response. They have shown their powerful capabilities for RF signal processing due to
12
their large number of comb lines, their compact footprint, and large comb spacings that can be generated compared to
all other approaches such as those using electro-optic combs. The devices reached operation bandwidths of ~ 16 GHz,
for the 49 GHz devices and ~ 25 GHz for 200GHz FSR combs, evaluated both in the frequency domain using Vector
Network Analyzers and in the time domain with Gaussian pulse input signals.
For transversal frequency-domain signal processors such as Hilbert transformers and differentiators, the number of
wavelengths determines the number of taps, which in turn directly determines the system performance. We have
highlighted the advantages of using Kerr micro-combs as the basis for RF photonic microwave transversal filters,
illustrating the tradeoffs between using widely spaced micro-combs (200 GHz) and record-low spaced micro-combs
(49 GHz) in terms of performance. The greater number of lines supplied by the 49GHz comb (80 versus 20 for the
200-GHz device) yielded significantly better system performance. On the other hand, the 49-GHz device is more
limited in operational bandwidth, being restricted to roughly the Nyquist zone of 25 GHz, whereas the 200-GHz devices
were able to reach RF frequencies well beyond what conventional electronic microwave technology can achieve.
Two types of combs employed to realize RF functions were qualitatively very different. Apart from the spacings
being very different, the widely spaced 200GHz FSR microcomb operated in a quasi-coherent state that did not feature
solitons, whereas the low spaced microcomb with an FSR of 49GHz operated in a state referred to as soliton crystals.
Within the optical C band, the 49GHz spaced microcomb enabled up to 90 comb lines, yielding significantly better
performance for the temporal signal processors than the than the 200GHz spaced microcomb which only allowed 20
taps in the C-band. Further, the soliton crystal states offered many other advantages such as much lower noise, higher
stability, and much easier generation including the ability to even initiate soliton crystals through simple manual tuning
of the pump wavelength. However, on the other hand the 49GHz combs generally yielded much smaller Nyquist
bandwidths typically near ~16 GHz. For the transversal temporal signal processors such as the integrators, the large
number of wavelengths brought about a large number of broadcasted RF replicas, thus yielding a large integration
window for the integrator. The temporal response of the integrator was measured with a diverse range of RF inputs,
verifying a integration window of 6.8 ns and a time feature as fast as 84 ps. The good agreement with theory and
experiment verified that this approach is an effective way to implement high-speed reconfigurable signal processing
featuring high processing bandwidths, for future ultra-high-speed microwave systems.
We believe that micro-combs will bring further benefits to RF photonic signal processing in many respects. First,
the coherent nature of the soliton states will enable more advanced RF functions such as wideband frequency
conversion and clock generation. Maturing nanofabrication techniques have allowed the realization of high-Q MRRs
for comb generation with FSRs ranging from 10s’ of GHz [119] up to the THz regime [47], covering the full RF bands
of interest for almost any integrated RF system. The wide available FSR spacing of optical micro-combs allows for a
large Nyquist zone up to 100’s of GHz. This is well beyond the processing bandwidth of electronic devices and is also
challenging for mode-locked lasers and OFCs obtained by EO modulation. With tailored dispersion, ultra-wideband
bandwidth micro-combs - even up to octave-spanning and beyond, [71] can be achieved, thus enabling a very large
number of wavelength channels with a single on-chip source, to work in conjunction with broadband opto-electronic
equipment. This approach enables significantly enhanced wavelength-division parallelism for massive data
transmission and processing [62-64] for many applications including radio-over-fibre systems, for example.
Perhaps most importantly, the high reconfigurability of micro-comb based programmable RF photonic transversal
filters enables one to achieve highly versatile processing functions with dynamic real-time reconfigurability so that the
same system can be dynamically reprogrammed to perform a wide range of different functions without having to alter
any hardware. By simply programming and shaping the comb lines according to different specific tap coefficients, the
same physical equipment can be employed to achieve multiple signal processing functions. Such a high degree of
reconfigurability is unprecedented for photonic RF signal processors and enables highly reconfigurable processing
functions and high processing accuracy that typically cannot be obtained by passive photonic integrated circuits or via
optical analogue signal processing. Further, very wide operational RF bandwidths can be achieved for diverse
computing requirements of many practical applications. Photonic transversal filters are essentially equivalent to filters
achieved via purely digital signal processing, but implemented by photonic hardware.
There are a number of factors that can lead to tap errors during the comb shaping, thus leading to non-ideal impulse
responses of the system as well as deviations between the experimental results and theory. These factors mainly include
the instability of the optical micro-combs, the accuracy of the waveshapers, the gain variation with wavelength of the
13
optical amplifier, the chirp induced by the optical modulator, and the third-order dispersion of the dispersive fibre. To
combat this, real-time feedback control paths can be employed to effectively reduce the errors. In this approach, the
power of the comb lines is first detected by an optical spectrum analyser (OSA) and then compared with the ideal tap
weights, generating an error signal that is fed back into the waveshaper to calibrate the system and achieve accurate
comb shaping. To further reduce the tap errors and improve the accuracy of comb shaping, the error signal of the
feedback loop can be generated directly by the measured impulse response, instead of the raw optical power of the
comb lines. In this approach, replicas of an RF Gaussian pulse are measured at all wavelengths to obtain the impulse
response of the system, whose peak intensities are further extracted to obtain accurate RF-to-RF wavelength channel
weights. Following this, the extracted channel weights are subtracted from the desired weights to obtain an error signal
that is used to program the loss of the Waveshaper. After several iterations of the comb shaping loop, an accurate
impulse response that compensates the non-ideal impulse response of the system can be obtained, thus significantly
improving accuracy of the RF photonic transversal filter based signal processors. This only requires a one-time
calibration process.
Finally, although the results reviewed here relied on some benchtop components, such as the commercially available
Waveshaper, RF photonic signal processors based on Kerr micro-combs have significant potential to achieve much
higher levels of integration with current nanofabrication techniques. To begin with, the most important component is
the micro-comb source itself, which not only is already integrated but is fabricated with CMOS compatible processes.
Globally established CMOS fabrication foundries can achieve advanced hybrid integration of microcomb sources and
Ⅲ-Ⅴ devices, and this will potentially enable monolithic integration of the entire RF system. Other key components
have been realized in integrated form with state-of-art nanofabrication techniques [198-200], including the optical
pump source [131, 139], LiNO3 modulators [200], optical spectral shapers [198], large dispersion media [199] and
photodetectors. Further, advanced integrated microcombs have been demonstrated [139] that can generate soliton
crystals reliably in turn-key operation. Monolithically integrating the whole RF processing system would greatly
strengthen the performance, compactness and energy efficiency of the system. Even without this, however, using the
discrete integrated comb sources to replace discrete laser arrays already yields significant benefits for RF and
microwave systems in terms of performance, size, cost, and complexity.
VIII. CONCLUSION
We review recent work on the applications of Kerr micro-combs to photonic RF and microwave temporal signal
processing based on transversal filters. Optical micro-combs bring a new generation of compact multi-wavelength
sources to the RF photonics community and offer enormous possibilities for achieving high-performance RF signal
processing with higher performance, reduced footprint, lower complexity, and potentially lower cost. By programming
and shaping the comb lines according to calculated tap weights, diverse signal processing functions can be designed
and have been experimentally demonstrated. We focus on fractional and integral order Hilbert transforms and
differentiators as well as an RF integrator, based on integrated Kerr micro-comb sources that operate on the RF signal
rather than the complex optical field. Real-time system demonstrations are also performed and show good agreement
with theory. These results verify that the Kerr micro-comb is a highly effective tool to implement high-speed
reconfigurable signal processing for future ultra-high performance RF systems
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Fig. 1. Schematic illustration of the integrated MRRs for generating the Kerr micro-comb for both (a-c) 200GHz FSR combs and
(d-j) 49GHz combs. (b, c) Optical spectra of the micro-combs generated by 200GHz MRR with a span of (b) 100 nm and (c) 300
nm. (j) Optical spectra of the micro-combs generated by 50GHz MRR with a span of 100 nm. (e) Drop-port transmission spectrum
of the integrated MRR with a span of 5 nm, showing an optical free spectral range of 49 GHz. (f) A resonance at 193.294 THz
with full width at half maximum (FWHM) of 124.94 MHz, corresponding to a quality factor of 1.549×106.
Figure 1
24
Fig. 2. (a, d, e) Optical spectra of various soliton crystal micro-combs. (b) Optical power output versus pump tuning, showing the
very small power jump at the onset of soliton crystal combs. (c) Transition from high RF noise chaotic state to low noise state of
the soliton crystal comb.
Figure 2
25
Fig. 3. Theoretical schematic of the principle of transversal filters using wavelength multiplexing. MZM: Mach-Zehnder
modulators. PD: photo-detector.
Figure 3
26
Fig. 4. Experimental schematic for RF transversal filters based on 200GHz microcomb (top) and 49GHz microcomb (bottom).
TLS: tunable laser source. EDFA: erbium-doped fiber amplifier. PC: polarization controller. BPF: optical bandpass filter. TCS:
temperature control stage. MRR: micro-ring resonator. WS: WaveShaper. OC: optical coupler. SMF: single mode fibre. OSA:
optical spectrum analyzer. AWG: arbitrary waveform generator. VNA: vector network analyser. PD: photodetector. BPD:
balanced photodetector.
Figure 4
27
Fig. 5. Shaped optical spectra showing the weight of each tap for: (a) 12 tap filter, and (b) 20 tap filter
Figure 5
28
Fig. 6. Measured system RF frequency response for different number of filter taps: (a) amplitude; and (b) phase response.
Figure 6
29
(Maybe separate the top and bottom parts a bit)
Figure 7
Fig. 7. Theoretical RF amplitude and phase response of FHTs with (a, d) 15º, (b, e) 30º, (c, f) 45º, (g, j) 60º, (h, k) 75º, and (i, l)
90º phase shifts. The amplitude of the fractional Hilbert transformers designed based on Eq. (3) (colour curves) are shown according
to the number of taps employed.
30
Figure 8
Fig. 8. (a) and (b) Simulated and measured amplitude and phase response for the FHT for different numbers of taps for a FHT phase
shift of 45°. (c) Simulated and experimental results of 3-dB bandwidth with different taps for a phase shift of 45°.
Separate figure c into its own figure
31
Figure 9
Fig. 9. Simulated (dashed curves) and experimental (solid curves) results of FHT with various phase shifts of (a) 15º, (b) 30º, (c)
45º, (d) 60º, (e) 75º, and (f) 90º. (i) Optical spectra of the shaped micro-comb corresponding with positive and negative tap weights
(ii) RF amplitude responses with fractional orders of 0.166, 0.333, 0.5, 0.667, 0.833, and 1. (iii) RF phase responses with phase shifts
of 15°, 30°, 45°, 60°, 75° and 90°. (iv) Output temporal intensity waveforms after the FHT.
32
Fig. 10. Simulated RF amplitude and phase responses of the (a) first-, (b) second-, and (c) third-order temporal differentiators. (d)
RMSEs between calculated and ideal RF amplitude responses of the first-, second-, and third-order intensity differentiators as a
function of the number of taps.
Figure 10
33
Figure 11
Fig. 11. (a) Optical spectrum of the generated Kerr comb in a 300-nm wavelength range. Inset shows a
zoom-in spectrum with a span of ~32 nm. (b)–(d) Measured optical spectra (red solid) of the shaped
optical combs and ideal tap weights (green crossing) for the first-, second-, and third-order intensity
differentiators.
34
Figure 12
Fig. 12. Measured and simulated RF amplitude and phase responses of (a-i)–(a-ii) the first-order, (b-i)–
(b-ii) second-order, and (c-i)–(c-ii) third-order intensity differentiators.
35
Figure 13
Fig.13. (a) Measured temporal waveforms of a Gaussian input pulse. Theoretical (red dashed) and experimental
(blue solid) responses of the (b) first-, (b) second-, and (c) third-order differentiators.
36
Figure 14
Fig. 14. Simulated transfer function of different fractional differentiation orders with varying number of taps.
37
Figure 15
Fig. 15. (a) Relationship between the number of taps and operation bandwidth. (b, c) Experimentally demonstrated fractional
differentiator with varying number of taps.
38
Fig. 16. Optical spectra of the shaped micro-comb for different fractional orders.
Figure 16
39
Figure 17
Fig. 17. Simulated and measured the transmission response of the fractional differentiator at different orders
ranging from 0.15 to 0.90.
40
Figure 18
Fig. 18. Simulated and measured RF Gaussian pulse output temporal intensity waveform after the fractional differentiator.
41
Fig. 19. Schematic diagram of the photonic RF integration.
Figure 19
42
Fig. 20. Experimental results of the micro-comb-based RF integrator after comb optical power shaping for input (a-c) Gaussian
pulses with FWHM of 0.20, 0.38 and 0.94 ns, (d-e) dual Gaussian pulses with time intervals of 1.52 and 30.6 ns, and (f) a triangular
waveform with a width of 5.00 ns. The blue curves denote the input signal, the red curves denote the measured integration results,
the gray curves denote the ideal integration results, and the green curves denote the integration results calculated with the measured
impulse response of the system.
Figure 20
43
Fig. 21. Measured impulse response of the integrator (a) after comb optical power shaping and (b) after impulse response shaping
using a Gaussian RF input pulse.
Figure 21
44
Figure 22
Fig. 22. Experimental results of the micro-comb-based RF integrator after impulse response shaping for input (a-c) Gaussian pulses with
FWHM of 0.20, 0.38 and 0.94 ns, (d-e) dual Gaussian pulses with time intervals of 1.52 and 3.06 ns, and (f) a triangular waveform with a
width of 5.00 ns. The blue curves denote the input signal, the red curves denote the measured integration results.
