1. Introduction

Optical superchannels [1–3] are used to optimise bandwidth usage in optical communications experiments, which has lead to record high demonstrated data rates [3–5]. Optical frequency combs are an interesting light source for optical superchannels, and combs from micro-optic resonators (microcombs) potentially provide a compact and energy efficient platform for superchannel transceivers [4–10], to meet size, weight and power requirements for next-generation optical communications systems [11]. However, the individual comb lines from microcomb devices can suffer from low per-line power, and amplification unavoidably contaminates the signal with optical noise from amplified spontaneous emission (ASE). This in turns can degrade the performance of microcomb-based communication systems [6].

To mitigate this problem, several techniques to rejuvenate the comb lines against the surrounding noise have been explored, including optical injection locking (OIL) [12,13] or Stimulated Brillouin scattering (SBS) [14,15]. However, these approaches have potential drawbacks in complexity and/or energy consumption. OIL requires multiple free running lasers, while SBS requires high laser power to trigger the required frequency-selective amplification. These issues compound when addressing an increasing number of comb lines. Hence, we recently proposed and demonstrated an alternative approach, using a microring resonator (MRR) as a narrowband filter for multiple comb lines [16]. In short, comb lines contaminated with broadband noise were transmitted through through to the ‘drop’ port of a dual-bus MRR, which has an inverse-notch frequency domain transfer function [17]. The distilled lines were then employed in both ends of the communication systems with the 64QAM format, contributing to $\sim$ 9-12 dB OSNR penalty reduction compared to those without comb distillation.

Here, to further study the dependence of this system on individual device design parameters, we turn to simulations to search for the best device design for comb distillation. We simulate comb distillation according to the physical setup in [16]. We adjust the coupling factors on the simulated MRR device, with a set propagation loss for the platform, to minimise insertion loss while varying the resonance bandwidth. We find that an MRR on the chip platform investigated, with the bandwidth of 100 MHz and fibre-to-fibre device loss of 3 dB, could offer OSNR penalty reduction of up to 16 dB under symmetric coupling, regardless of the modulation format used. This benefit reduces with increases in resonance bandwidth, coupling loss, and the line spacing mismatch between the comb and the MRR resonances. We find that in the case of line spacing mismatches, the OSNR benefit exhibits the identical profile as the distilling resonance, i.e. a Lorentzian shape with a bandwidth matched to that of the distilling resonance. These results provide an insight into how to optimise MRR device parameters for optical frequency comb distillation, and may inform the co-design of microcomb devices with matching MRRs for distillation, to push microcomb performance close to large benchtop-scale optical frequency combs.

2. Microring resonator (MRR) as a filter for comb lines

We have previously demonstrated, in experiment, that MRRs could be used as frequency periodic filters to reduce broadband optical noise stemming from microcomb amplification [16]. When considering MRRs as a filter, the MRR device parameters affect filter performance. On the device side, and considering a fixed comb line spacing (free-spectral range - FSR), waveguide loss will tend to be fixed by the chip platform used, while coupling to and from the ring can be adjusted in design. This coupling is parameterised by the coupling ($\kappa$) and transmission ($r$) coefficients. Both ($\kappa$) and ($r$) affect filter performance, as they determine both the insertion loss of the device and the MRR resonance bandwidths ($\Delta f$), for a given waveguide loss and FSR [17–19]. We base our simulations on the Hydex waveguides with a loss of 0.06 dB/cm [20,21] and use a 20 GHz FSR, linking to our experimental work [16]. Given these assumptions, we then treat ($\kappa$) and ($r$) as free parameters, to enable us to vary the MRR resonance bandwidth as a critical system parameter for MRR comb distillation. Changing the MRR resonance bandwidth then also corresponds to a changing insertion loss. This can be calculated from the drop port transfer function $H_d$ of a dual-bus MRR [17]:

Here, the subscripts, 1 and 2, denote the input and drop ports, respectively. $a_{rt}$ is the roundtrip field loss, and $\phi _{rt}$ is the roundtrip phase shift of the input light. For lossless couplers, $r^{2}+\kappa ^{2}=1$, this can be rearranged so that the required coupling can be calculated for a desired MRR resonance bandwidth $\Delta f$ and FSR [17]:

This then means that the products of the couplings ($r_1$ and $r_2$) and the round trip loss ($a_{rt}$) will have a set value for the target resonance widths ($\Delta f$). So, this can be rewritten as:

We can then find optima for $r$ or $\kappa$, by searching for minimums in the on-chip insertion loss at a given bandwidth ($\Delta f$). The power transfer function at the resonance peak, for a lossless coupling condition (i.e. $r^{2}+\kappa ^{2}=1$) is given as:

Given that we need to constrain the product of $r_1$ and $r_2$ to match the resonance bandwidth $\Delta f$ and the set $FSR$, we then constrain $|H_d(\phi _{rt}=2m\pi )|^{2}$ as per Eq. (2b), by setting the denominator as a constant (i.e. $C=r_1r_2a_{rt}$). By equating the derivative of Eq. (3), respective to $r_1$, to zero (i.e. $\frac {d|H_d(\phi _{rt}=2m\pi )|^{2}}{dr_1}=0$), we can find the relationship between couplings $r_1$ and $r_2$ to minimise loss.

We can treat the right-hand side in Eq. (2b) as a constant, $C=r_1r_2a_{rt}$, at the given $FSR$ and $\Delta f$. Therefore:

Substituting this back into Eq. (4):

So, from this we find that loss is minimised when $r_1=r_2$ (and so $\kappa _1=\kappa _2$,) i.e. the coupling coefficients at the input and drop ports are identical (symmetric coupling). Interestingly, the symmetric coupling is the subset of undercoupling ($r_1>a_{rt}r_2$ with the field loss $a_{rt}<1$). This is different to the critical coupling often used in MRR designs [22–25], which minimises on-resonance transmission to the through port.

Fig. 1 (a) shows device insertion loss against coupling $r_1$, for a 100-MHz MRR, which is the highest device insertion loss over the range of resonance bandwidths we investigate. Compared with symmetric coupling ($r_1=r_2$), critical coupling ($r_1=a_{rt}r_2$) gives about 3 dB larger insertion loss. While this results in larger on-resonance nulling at the through port (see Fig. 1 (b)), in our application we are more interested in transmitting comb lines through the drop port by minimising insertion losses. Fig. 1 (c) backs up the choice of symmetric coupling for minimising insertion loss, and is more important where smaller $\Delta f$ is achieved by reducing coupling overall. The relationship between reduced coupling and lower $\Delta f$ is shown in Fig. 1 (d), for a fixed waveguide loss for the fabrication platform. As such, we can expect diminishing returns from reducing resonance bandwidth, trading off greater broadband noise removal with higher device insertion loss.

 

Fig. 1. (a) Response at a resonance peak ($|H_d(\phi _{rt}=2m\pi )|^{2}$) against transmission coefficient ($r_1$) at the input port of 100-MHz-Bandwidth MRR. Here, the black dash line represent the maximum response of MRR with symmetric coupling ($r_1=r_2$), whereas the red and green dash lines indicate the conditions at critical coupling and its reciprocal, respectively., (b) Insertion loss at various resonance bandwidths due to symmetric and critical couplings, (c) Through and drop port profiles under symmetric and critical couplings with double arrows showing insertion loss for each case for $\Delta f = 100$ MHz, (d) Transmission coefficient ($r$) in blue and the corresponding coupling coefficient ($\kappa$) in red for the symmetric coupling

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3. Simulation setup

3.1 System configuration

To simulate comb distillation, we wrote MATLAB scripts emulating our previous experiment [16], shown in Fig. 2 (a). To model the EDFAs, we took the output power to be either set as $P_{set}$, or saturated at the product of maximum gain ($G_{max}$) and input power ($P_{in}$) with the corresponding optical noise ($P_{ASE}(G_{max}$)):

Here $P_{set}$ can be separated into the amplified signal and noise powers as:

 

Fig. 2. (a) Simulation setup for comb distillation based on the configuration in [16] with part (i) representing distillation setup for a comb line and part (ii) and (iii) representing exploitation of comb distillation at the transmission and reception sides, respectively.; here, ECL: External cavity laser, VOA: Variable optical attenuator, S: Power scaler without added noise, L: total loss at transmission side, MRR: Microring resonator., (b) Spectrum of amplified comb line after a simulated amplifier (EDFA1). Here, $B_0=64$ GHz is the FFT window of the simulation and $B_n$ is the bandwidth for noise in OSNR calculation. We note that the PSD of noise is flat for the entire window with $PSD_{ASE}=2n_{sp}hf_0(G-1)$., (c) Example of spectrum from the raised-root-cosine (RRC) signal with the symbol rate of 18 GBd (bandwidth of 18 GHz) modulated with the comb line

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The gain ($G\equiv \frac {P_{set}-P_{ASE}}{P_{in}}$) required to achieve the set power affects power of optical noise, $P_{ASE}(G)$ as in standard EDFA models [26,27] ($n_{sp}$ is the inversion factor of an EDFA, $f_0$ is the operating frequency, and $h$ is Planck’s constant). Since the sampling frequency (FFT window) in our simulation was 64 GHz, the bandwidth, $B_0$, was chosen to be 64 GHz so that the power spectral density (PSD) of optical noise was $2n_{sp}(G-1)hf_{0}$ throughout the window (Fig. 2(b)). However, we used a standard 12.5 GHz bandwidth ($B_n$) for noise when assessing OSNR as illustrated in Fig. 2(b). We note that the noise figure (NF) was reported to be dependent on gain according to specifications, ranging from 5 dB (at $G \geq 25$ dB) to 8 dB (at $G < 5$ dB). We therefore interpolated the NF for the corresponding gain from the available data sheets.

When amplifying optical frequency combs, comb line OSNR is degraded. This is particularly important when low power microcombs [4,7,10] are to be using in optical communications systems. Here, we simulated a single comb line with a power between -18 to -50 dBm, input to the system that then amplifies this line to between 11 and 14 dBm. As per our experimental work [16], we set the EDFA1 (before the MRR) with $G_{max}=25$ dB and achieved the per-line power of 3 dBm (therefore we set $P_{set}=3$ dBm), then calculated OSNR using the standard 12.5-GHz noise bandwidth definition. The comb line was then either passed to an MRR filter model and amplified with EDFA2 ($G_{max}=25$ dB and $P_{set}=3$ dBm), or directly to a wavelength filter model (WSS-Waveshaper). For the MRR, we modelled the transfer function using Eq. (1) and Eq. (2) for any bandwidths with FSR = 20 GHz under the symmetric coupling scheme, with a 11 dB coupling loss as per measured device performance [16]. The WSS represented by a 10-GHz bandpass filter was modelled with the convolution of rectangular and gaussian functions. The single comb line was amplified to 11 dBm and 14 dBm (EDFA3, ($G_{max}=25$ dB)) for the transmitter and receiver, respectively.

At the transmitter side, the line was modulated with the digitized dual-polarization data (64 GSa/s synchronized with the sampling frequency), with the loss of 17 dB representing total loss of transmitter-side devices, as shown in Fig. 2 (a. ii). The digitized data in our simulation were generated with a 2.5% root-raised cosine shaped under 64-, 128-, 256- and 512-QAM at 18 GBd with 18-GHz bandwidth (Fig. 2(c)). The signal was then amplified to 14 dBm (EDFA4, ($G_{max}=35$ dB)) to represent amplification of a full superchannel, then attenuated to -6 dBm (VOA2) before demodulation with the ECL laser signal (as an LO) at the reception side to the baseband frequency [28]. We note that EDFA4 had a negligible impact on performance. At the receiver side, we simulated transceiver (i.e. transmitter and receiver) noise by adding white noise to the baseband signal, set to a single noise power level which was determined by a fit to experimental data (Fig. 3 (a)-(b)). The signal was resampled at the rate of 80 GSa/s followed by DSP for data recovery (as per in [16]), using multi-modulus algorithms (MMA) to process high-order QAMs. We then calculated SNR or signal quality factor, $Q^{2}$, from the error-vector-magnitude [29]: $Q^{2}=1/EVM^{2}$, bit-error ratio (BER) and generalized mutual information (GMI) indicating the achievable information rates [30,31]. GMI is also the more accurate metric than BER when soft-decision decoder is employed [31]. The metric can be estimated with the formula in Eq. (9) as:

Here, $m$ is the number of bits per symbol with a particular modulation format. $n_s$ is the time length or total length of transmitted symbols. $c_{k,l}$ is the transmitted bit at $k^{th}$ position in the demapped transmitted symbol at time instant $l$, and $\lambda _{k,l}$ is the corresponding log-likelihood ratio (LLR) calculated from soft-decision demapping of received symbols [31]. For the comb distillation at the receiver side, we swapped the positions of the ECL laser and the output from part (i) as shown in Fig. 2 (a. iii). The output power of VOA2 in the setup (Fig. 2 (a. iii)) was adjusted from $\geq -6$ dBm to maintain the beating power at the receiver, as per prior experiment [16].

 

Fig. 3. (a)-(b) Comparisons of signal quality factor, $Q^{2}$, between the simulation results (sim) from Fig. 2(a) with resonance widths of 300-500 MHz and the previous experimental results in [16] when the un- and distilled comb lines were deployed as carrier and LO, respectively. The term $OSNR_{before}$ indicates OSNR measured after the noise loading stage.

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Since the MRR in [16] has the quality factor, $\frac {f_0}{\Delta f}\sim 4\times 10^{5}-5\times 10^{5}$ at C-band and coupling loss at 11 dB, we plotted the simulation results from the corresponding resonance widths of $\sim$ 300-500 MHz at this coupling loss to compare with the previous results in Fig. 3(a)-(b). Our simulation results broadly align with the experiment. In the plots, $Q^{2}$ is linearly proportional to OSNR where optical noise dominates SNR, whereas at high OSNRs $Q^{2}$ is limited due to the transceiver noise. We note that the simulation results when using a noisy comb line at either the transmitter and receiver sides are identical, and we suspect that the differences of the experimental traces seen in Fig. 3 (a)-(b) might be due to non-ideal operation of the EDFAs after the MRR. With the confidence that our model broadly matches experimental results, we adjusted the transceiver noise to enable a maximum $Q^{2} = 25 dB = Q^{2}_{max}$, to enable investigation of higher order QAM (64-, 128-, 256- and 516-QAM, here), and to align our simulation with the range of $Q^{2}$ that state-of-the-art demonstrations have achieved (24-26 dB) [32–34]. In system performance analysis, we did not use an indicative FEC threshold as a benchmark, as this should be the limit of total system performance including impairments from transmission. Instead, we assume that a 5$\%$ reduction in achievable GMI is an acceptable limit for degradation caused by noisy optical carriers, and use this as an indicative benchmark for comparison. This then leaves margin for FEC to be applied in a transmission scenario.

4. Results for communication systems with comb distillation

4.1 Effects of MRR resonance widths and coupling losses

After calibrating the setup and adjusting transceiver noise to achieve $Q^{2}_{max}$ = 25 dB, we tested system performance with all tested QAM formats when the MRR resonance widths varied from 100 to 1000 MHz with coupling loss ranging from 3 to 13 dB. To assess the impact of device parameters on the systems, we plotted GMI from various resonance widths and the undistilled case against OSNR before distillation ($OSNR_{before}$) of each MRR coupling loss and modulation format. For high OSNRs (> 35dB), we observe GMIs of 12, 13.83, 15.1 and 15.75 b/symbol for 64-, 128-, 256- and 512-QAMs, respectively. We benchmark performance where GMI is reduced by 5% from the high OSNR value, and define the required $OSNR_{before}$ from where the GMI traces cross this benchmark, as illustrated in Fig. 4 (a)-(b). On Fig. 4, we plotted GMI against $OSNR_{before}$ for the highest and lowest order modulation formats we tested (i.e. 64- and 512-QAM), for the undistilled case, and for the highest and lowest MRR resonance bandwidths we tested. By finding the difference between the $OSNR_{before}$ required to reach the 5% GMI reduction benchmark, we can then define an improvement in require $OSNR_{before}$ due to MRR distillation. This improvement is indicated as black and red double headed arrows, for resonance widths of 100 and 1000 MHz, respectively. The required $OSNR_{before}$ increases with higher-order modulation, as expected. However, the differences in required $OSNR_{before}$ (i.e. the OSNR benefit) for the different modulation formats is constant, indicating that the distillation process produces a modulation-format-independent effect. This is somewhat different to measurements in experiment, which showed some small modulation format dependence [16]. To gain more insight into the required $OSNR_{before}$ and OSNR benefit, Fig. 5 (a)-(b) show the pattern of the required $OSNR_{before}$ and OSNR benefit from the tested modulation schemes. Figure 5 (a) shows that for 64- and 512-QAM, the change in required OSNR against MRR device parameters (i.e. resonance bandwidth and loss) is very similar regardless of modulation format. This similarity is present for the other tested modulation schemes (i.e. 128- and 256-QAM). We then plotted OSNR benefit against device parameters only, as there is negligible difference in performance against modulation format.

 

Fig. 4. GMI when 64QAM and 512 QAM is deployed for transmission against line OSNR before distillation ($OSNR_{before}$) of 100 and 1000 MHz MRR resonance widths with coupling loss of (a) 3 dB, (b) 13 dB. The black and red double headed arrows indicate required $OSNR_{before}$ for distilled cases respected to those without distillation, shown as $\Delta$OSNR

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Fig. 5. (a) Required $OSNR_{before}$ at the 5% reduction of $GMI_{max}$ against resonance widths for the coupling loss of 3 and 13 dB when 64- and 512-QAMs are employed, (b) OSNR benefit against the resonance bandwidths with coupling loss ranging from 3 to 13 dB

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Figure 5 (b) shows the (modulation format independent) overall OSNR benefit against resonance bandwidths when the coupling loss varies from 3 to 13 dB. The suggests that distillation by MRR can save up to 16 dB in required $OSNR_{before}$ at our smallest MRR bandwidth and coupling loss (100 MHz and 3 dB, respectively). By looking closer at the OSNR benefit patterns, we witness that the OSNR benefit generally decreases with resonance widths when coupling loss. Intuitively, this is because more noise is included in the increasing bandwidths. Counter to this intuition, we also find that the OSNR benefit deteriorates for low resonance bandwidths when the MRR coupling loss increases.

For lower bandwidth resonances to be achieved on the material platform we are investigating, the coupling to the rings needs to be reduced. This results in a higher loss at resonance through the MRR device for the lowest resonance bandwidths, as shown in Fig. 1(c). In cases where the overall device loss is low (e.g. 3dB coupling loss), the noise from the pre-ring amplifier (EDFA1) dominates, and there is still an increase in OSNR benefit with lower resonance bandwidths, as one would intuitively expect. As the device loss increases, though, the noise from the post-ring amplifier (EDFA2) becomes significant. The benefit from tighter filtering of the noise from EDFA1 is then lost as EDFA2 must operate at higher gain to counteract the losses from the ring device, adding in more noise in the process. Effectively, higher loss in the ring device pushes the filtered lines into the noise floor of EFDA2. This leads to a wider resonance bandwidth, with lower loss, being optimal for high fibre-to-chip coupling loss - in our case, 200 MHz for higher coupling losses (i.e. 6 dB or greater).

4.2 Distillation with line spacing mismatches between comb lines and resonances

Practically, it may be difficult to perfectly align comb lines to MRR resonances. The comb line spacing may be different from the FSR of the MRR, thermal shifts may detune all resonances away from the comb [35], or chromatic dispersion in the MRR waveguides may shift resonances away from an equidistant grid [36]. This will effect the performance of comb line distillation.

To emulate these scenarios, we introduced a frequency mismatch between the single comb line and the MRR resonance to the simulations of transmission. We then calculated the required $OSNR_{before}$ of both undistilled and distilled cases for each frequency mismatch. The OSNR benefit against frequency mismatch was plotted with symbol points in Fig. 6, for various MRR bandwidths of the MRR. Figure 6 (a) is with coupling loss of 3 dB, regarded as the lowest loss for the Hydex rings [21], and Fig. 6(b) with the 11 dB loss for a previously used device [16]. These results suggest that distillation with the large bandwidth MRRs is more tolerant to the resonance shifting factors than those with smaller bandwidth, but this comes with a trade-off for the achievable maximum OSNR benefits.

 

Fig. 6. OSNR benefit against frequency mismatch between the comb line and resonance of different bandwidth when coupling loss is (a) 3 dB, (b) 11 dB. The symbol points represent results from simulations and solid lines indicate Lorentzian shapes.

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We note that the OSNR benefit trends against frequency mismatch resemble MRR resonance shapes, and so we then overlaid the plot in Fig. 6 with Lorentzian functions of the corresponding bandwidths and show a close match of the measured OSNR benefit to the appropriate Lorentizian distribution, implying that degradation of OSNR benefits with frequency offset can be calculated from the resonance profiles. This is due to the fact that varying the frequency misalignment resembles scanning a laser across the MRR resonance, if the laser linewidth is significantly smaller than the resonance bandwidth (here, <100 kHz, compared with resonance bandwidths of $\geq$100 MHz). Physically, the line power changes with the frequency offset according to the resonance profile, while the noise carved by the MRR remains the same regardless of the frequency mismatch. The line power (‘signal’) is then attenuated with the resonance while the noise level stays the same. As a result, the OSNR of the distilled line after the MRR should manifest the Lorentzian shape with the frequency offset. This in turns shapes the OSNR benefit profile in the same way.

5. Discussion and conclusion

We have simulated frequency comb distillation with the Hydex-based MRR for optical communications systems. In the simulations, we have explored the impacts of various parameters (i.e. coupling scheme, MRR bandwidths, external coupling loss, modulation formats and frequency mismatch between comb lines and resonances) on transmission performance. We find a maximum of 16 dB improved tolerance to incoming comb line OSNR when using 100 MHz resonance bandwidths on the platform we investigate, using a benchmark of a $5\%$ reduction of GMI from optima for different modulation formats.

From the results and OSNR benefit being independent to modulation schemes, we could infer that MRR-based comb purification is more effective to save OSNR budget for high-order modulations where line OSNR of carriers or LOs becomes stringent for achieving any acceptable thresholds (e.g. Forward Error Correction (FEC)). This could then support metro-area links in which the high-order modulations could be employed to serve increasing capacity requirements.

While our approach conforms to specific experimental conditions in order to give confidence of the real-world applicability of our findings, we are also able to make several general observations that would hold irrespective of changes to the system set-up. We find that when the MRR device is designed to minimise the resonance bandwidth given the limits of propagation loss for the waveguide platform used, symmetric coupling has advantages as a result of low insertion loss to the drop port, particularly when contrasted with the commonly used critical-coupling design. This is an important consideration, as it is possible to improve the performance of comb distillation by reducing the resonance bandwidth, even when ring-coupling-induced loss is increased. We also predictably show that the MRR device should be designed to have low external coupling loss so that a higher OSNR benefit could be attained.

In terms of system performance, two significant trends emerged. One is a device platform dependent optimum for resonance bandwidth, where coupling loss to the ring begins to dominate over noise-rejection benefits from narrowed filtering, where reduced coupling ratios in the MRR (small $\Delta f$) are used to enable narrower filtering. This effect is reduced when the total device insertion loss is minimised, enabling a given platform to beneficially access sharper filtering if fibre-to-waveguide coupling is minimised. Second is that the filtering benefit seems to both be modulation format independent and work equally well for comb lines employed at either the transmitter or receiver. This suggests that comb distillation would be beneficial for higher order QAM where it is difficult to achieve the required high comb line OSNR with low power microcomb generation schemes.

However, the OSNR benefit can reduce when resonance bandwidths expand, coupling loss increases or there is frequency misalignment between MRR resonances and comb lines, due to included noise inside resonances or added noise after distillation being more comparable to the signal. Although deployment of the MRR with wide bandwidths might not achieve maximum OSNR benefit, the approach is tolerant against hard-to-avoid frequency mismatch, e.g. due to temperature fluctuation or chromatic dispersion, as confirmed by the fitted Lorentzian shapes. Therefore, this resonance bandwidth trade-off should be considered when applying this noise filtering approach to the comb-based transmission systems.

When considering costs of comb distillation with MRR, the extra EDFA to compensate for the total loss of the MRR (EDFA2 here) adds some extra energy cost into the frequency comb system. The relative energy costs of frequency comb communication systems, and their scaling with number of lines, are discussed in [8]. For a large number of lines (e.g. 50-100 lines), this would seem to be a minimal cost to a superchannel system, and is likely also offset by decreased signal processing load in transceivers (e.g. [9]).

To more deeply characterize the characteristics of noise in these distillation systems, appropriate mathematical models should be established. Such models would help to find general optima for comb distillation, ultimately providing some idea of both the benefits and limitations of using low-power microcombs at both ends of an optical communications link.