The spectral response of microcombbased MWP transversal filters in Eq. (1) is influenced by both theoretical limitations and experimental system errors. The former is caused by the finite tap number, which limits the filter spectral response with respect to resolution, rolloff rate, and maintosecondary sidelobe ratio. The latter refers to distortions in the filter response caused by the imperfect response of experimental components in Fig. 3, most notably those arising from the noise of microcomb, chirp of the EO modulator, TOD of SMF, OSS shaping errors, and BPD noise. In addition to the limitations in the filters’ spectral response mentioned above, in practical signal filtering the system’s limited operation bandwidth can also introduce additional errors when filtering wideband signals. In this section, we will provide a thorough analysis of the above factors on the performance of microcombbased MWP transversal filters. In Section 3.1, we will investigate the theoretical limitations in the filter response that arise from finite tap numbers. In Section 3.2, we will analyze the distortions in the filter spectral response due to the imperfect response of experimental components. In Section 3.3, we will assess the influence of limited operation bandwidth on the experimental performance of the signal filtering. We note that an onchip microcombbased MWP transversal filter comprised entirely of integrated components has also been demonstrated recently [29]. Our analysis and discussion also apply to it since it operates on the same principle as the filter in Fig. 3.
In our following analysis, we use lowpass filters (LPFs), bandpass filters (BPFs) with fcenter = 10 GHz, and highpass filters (HPFs) with a cutoff frequency of 10 GHz as examples to characterize the performance of the microcombbased MWP transversal filter in Fig. 3. These filters are designed based on Eq. (1) but with different tap coefficients. The tap coefficients for a LPF can be given by [14]
$${\text{a}}_{\text{LP}\text{,}\text{ }\text{n}}\text{ }\text{=}\text{ }\text{1, }\text{n}\text{ }\text{=}\text{ }\text{0,}\text{ }\text{1,}\text{ }\text{2,}\text{ }\text{…}\text{ }\text{, }\text{M}\text{1}$$
4
which features equal weighting for different taps. Based on the LPF, a BPF can be realized by modifying the tap coefficients as follows [14]
$${\text{a}}_{\text{BP}\text{,}\text{ }\text{n}}\text{ }\text{=}\text{ }{\text{a}}_{\text{LP}\text{,}\text{ }\text{n}}\text{cos}\frac{{\text{f}}_{\text{center}}\text{πn}}{\text{FS}{\text{R}}_{\text{MW}}}{\text{e}}^{\text{}\frac{{\left(\text{n}\text{}\text{b}\right)}^{\text{2}}}{\text{2}{\text{σ}}^{\text{2}}}}\text{, }\text{n}\text{ }\text{=}\text{ }\text{0,}\text{ }\text{1,}\text{ }\text{2,}\text{ }\text{…}\text{ }\text{,}\text{ }\text{M}\text{1}$$
5
where fcenter is the center frequency of the filter, and Gaussian apodization is applied to improve the MSSR, with σ and b denoting the root mean square width and the peak position of the Gaussian function, respectively.
To achieve an HPF, R sets of tap coefficients corresponding to BPF with increasing center frequencies, given as aBP,n,r (r = 1, 2, …, R), are summed up together, which can be expressed as [14]
$${\text{a}}_{\text{HP}\text{,}\text{ }\text{n}}\text{ }\text{=}\text{ }\sum _{\text{r}\text{=1}}^{\text{R}}{\text{a}}_{\text{BP}\text{,}\text{ }\text{n}\text{,}\text{ }\text{r}}, \text{n}\text{ }\text{=}\text{ }\text{0,}\text{ }\text{1,}\text{ }\text{2,}\text{ }\text{…}\text{ }\text{,}\text{ }\text{M}\text{1}$$
6
3.1 Theoretical limitations of the filter response
In this subsection, we analyze the theoretical limitations of microcombbased MWP transversal filters resulting from finite tap numbers. We focus on evaluating three primary parameters that determine the filter performance, including resolution, rolloff rate (ROR), and maintosecondary sidelobe ratio (MSSR). The definitions of these three parameters are provided in Table II For the analysis in this section, we assume all the components in Fig. 3 have a perfect response. For comparison, we also assume the filters have the same comb spacing (∆λ = 0.4 nm, i.e., ~ 50 GHz) as well as length and dispersion for the SMF (L = 2.1 km and D2 = 17.4 ps/nm/km), which allow for an FSRMW of ~ 68 GHz.
Table II Definitions of performance parameters of microcombbased MWP transversal filters.
Parameters

Lowpass filter (LPF)

Bandpass filter (BPF)

Highpass filter (HPF)

Resolution

3dB bandwidth of the passband or stopband a)

Rolloff rate
(ROR)

The rate at which the filter’s amplitude response changes with frequency

Maintosecondary sidelobe ratio
(MSSR)

The ratio between the amplitude of the main lobe and the first side lobe
in the RF amplitude response

a) For the LPF and BPF, the resolution is the 3dB bandwidth of the passband. For the HPF, the resolution is the 3dB bandwidth of the stopband.
Although increasing the number of taps can improve the performance for all three types of filters when the comb spacing ∆λ is much larger than the input microwave signal’s bandwidth, it is important to note that practical MWP transversal filters are still subject to limitations induced by the number of taps. While microcombs with a large spectral bandwidth can provide a large number of taps, most components in real systems, such as the EDFA and EOM, operate within the telecom Cband (1530–1565 nm). Consequently, increasing the number of taps can only be achieved by reducing the comb spacing ∆λ. Nevertheless, as per the Nyquist sampling theorem, the comb spacing must be at least twice as wide as the bandwidth of the input RF signal to avoid overlap between modulated RF replicas on different wavelength channels. This implies that a narrow comb spacing would restrict the operation bandwidth of the filter. Thus, for practical applications, one must strike a proper balance between the tap number and comb spacing. In Fig. 4, it can be observed that there is only a slight improvement in both the RORs and the MSSRs when the tap number M is increased beyond 80. As a result, M = 80 has been widely adopted for microcombbased MWP transversal filters [14, 15, 27], corresponding to a comb spacing of ~ 0.4 nm (i.e., ~ 50 GHz).
3.2 Distortions in the filter response induced by imperfect components
In addition to theoretical limitations, the imperfect response of components in experimentally realized systems gives rise to discrepancies in the filter response. To characterize these, we use average distortion (AD) to describe the difference in the MWP transversal filter’s RF amplitude response with and without errors, which is defined as
$$\text{AD}\text{ }\text{=}\text{ }\sqrt{\sum _{\text{i}\text{=1}}^{\text{k}}\frac{{\left({\text{Y}}_{\text{i}}\text{}{\text{y}}_{\text{i}}\right)}^{\text{2}}}{\text{k}}}$$
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where k is the number of sampled points, Y1, Y2, …, Yn are the RF amplitude response without error sources, and y1, y2, …, yn are the RF amplitude response with errors induced by imperfect sources. Here, we discuss the influence of different components in real systems on the response of microcombbased MWP transversal filters. In Sections 3.2.1 ‒ 3.2.4, we investigate the influence of specific error sources, assuming the other sources are errorfree. In Section 3.2.5, we compare the contributions of the different error sources to the overall filter performance. In the analysis of this section, we assume that the tap number M = 80 and all the other parameters are the same as those in Section 3.1.
3.2.1 Influence of microcomb noise
Microcomb imperfections induce both intensity and phase noise in the wavelength channels. The intensity noise refers to power fluctuations of the comb lines and the intensity noise floor, which mainly arise from photon shot noise and spontaneous emission beat noise [58]. A consequence of imperfect microcombs is that the accuracy of tap coefficients is degraded, leading to a deviation from the ideal filter response that would result from a perfect microcomb.
To characterize the intensity noise of microcombs, the optical signaltonoise ratio (OSNR) is introduced, which is defined as the ratio of the maximum optical signal power to the noise power in each comb line. Figures 5(aⅰ) − (cⅰ) show the RF amplitude response of LPFs, BPFs, and HPFs, respectively, with varying OSNR ranging from 10 dB to ∞. The OSNR of ∞ represents the condition of a perfect microcomb with zero intensity noise. As the OSNR increases, the response of all the filters gets closer to that of the filters that would result from a perfect microcomb. We also note that the OSNR has a more significant influence on the response of the BFPs and HPFs.
The ADs for the three types of filters are shown in Figs. 5(aⅰⅰ) − (cⅰⅰ). As expected, the ADs decrease with OSNR, which agrees with the trend in Figs. 5(aⅰ) − (cⅰ). The OSNR induces small distortions (< 0.012) in the response of the LPFs, while giving rise to relatively large distortions for the BPFs and HPFs. The ADs exhibit a sharp decrease when the OSNRs are below 20 dB. As the OSNR increases, the decrease in AD becomes more gradual, and there is only a very small reduction (< 0.005) in AD beyond an OSNR of 20 dB for all the filters.
On the other hand, the phase noise of microcombs, which results in a broadened linewidth, multiple repetitionrate beat notes, and a reduced temporal coherence [59], is difficult to quantitatively analyze. The microcomb’s phase noise is influenced by multiple factors, including the noise of the CW pump and the mechanical and thermal noise of the MRR [60, 61]. Using modelocked microcombs with low phase noise is crucial for achieving stable and longterm operation of microcombbased MWP transversal filters. This can be accomplished through a variety of modelocking approaches that have been demonstrated [45, 46].
3.2.2 Influence of the EO modulator
In Fig. 3, an EO modulator is employed to modulate the input microwave signal onto each wavelength channel. Due to their high modulation efficiency, large operation bandwidth, and low insertion loss, MachZehnder modulators (MZMs) are commonly used [62]. In addition to intensity modulation, as the existence of an asymmetry in the electric field overlap at each electrode [63], experimental MZMs also result in undesired phase modulation, namely modulation chirp. In this subsection, we analyze the influence of modulation chirp on the filter performance. The chirp can be characterized by the chirp parameter defined as [64]
$$\text{α}\text{ }\text{=}\text{ }\frac{{\text{γ}}_{\text{1}}\text{+}{\text{γ}}_{\text{2}}}{{\text{γ}}_{\text{1}}\text{}{\text{γ}}_{\text{2}}}$$
8
where γ1 and γ2 are the voltagetophase conversion coefficients for the two arms of the MZM. When α = 0 (i.e., γ1 = −γ2), pure intensity modulation is achieved, whereas for α = ∞ (i.e., γ1 = γ2), it is pure phase modulation. Modulation chirp causes distortions in the optical signals after modulation, thus leading to distortions in the response of the filters.
Figures 6(aⅰ) − (cⅰ) show the RF amplitude response of LPFs, BPFs, and HPFs, respectively, with varying chirp parameters α ranging from 0 to 1. The α = 0 corresponds to the condition of chirpfree modulation. As α decreases, the response of all the filters approaches that of the filter with chirpfree modulation, indicating less distortion caused by a low modulator chirp. The calculated ADs as a function of α are shown in Figs. 6(aⅰⅰ) − (cⅰⅰ), where the ADs increase with α, showing agreement with the trend in Figs. 6(aⅰ) − (cⅰ). The impact of modulation chirp on the filter response is more pronounced for the BPFs than for the LPFs and HPFs. It is worth noting that the modulation chirp can result in various types of distortions in the filter response, depending on the type of filter. For the LPFs, although the ROR is increased from 10.76 dB/GHz to 38.12 dB/GHz, the MSSR is degraded from 13.26 to 6.96. For the BPFs, the ROR is almost unchanged, but the MSSR is decreased from 64.79 to 52.9. For the HPFs, the ROR is decreased from 7.0 dB/GHz to 0.67 dB/GHz, and the MSSR is almost unchanged.
3.2.3 Influence of the SMF
In Fig. 3, A spool of SMF is used as the dispersive module to introduce time delays between adjacent wavelength channels. The secondorder dispersion (SOD) of the SMF is desired to produce uniform time delays, while the existence of thirdorder dispersion (TOD) introduces nonuniform time delays, hence giving rise to undesired phase errors. In this subsection, we analyze the distortions induced by TOD of the SMF.
The additional nonuniform time delays of the nth tap induced by TOD can be expressed as [44]
$$\text{Δ}{\text{T}}_{\text{TOD}}\text{ }\text{=}\text{ }{\text{D}}_{\text{3}}\text{ }\text{L}\text{ }\text{Δ}{\text{λ}}^{\text{2}} {\text{n}}^{\text{2}}$$
9
where D3 is the TOD parameter. Figures 7(aⅰ) − (cⅰ) show the RF amplitude response of LPFs, BPFs, and HPFs, respectively, with varying D3 ranging from 0 to 0.2 ps/nm2/km. The D3 = 0 corresponds to the condition of an SMF with zero TOD. For all the filters, as D3 decreases, their response gets closer to that corresponding to the SMF with zero TOD. This indicates decreased distortions in the filter response for a smaller TOD parameter. When D3 increases from 0 to 0.2 ps/nm2/km, the MSSR for the LPFs is slightly degraded from 13.26 to 13.06. For the BPFs and HPFs, the RORs are decreased from 4.42 dB/GHz to 3.85 dB/GHz and from 7.0 dB/GHz to 6.39 dB/GHz, respectively.
Figures 7(aⅰⅰ) − (cⅰⅰ) show the ADs versus D3, where the ADs increase with D3 for all the filters, showing agreement with the trend in Figs. 7(aⅰ) − (cⅰ). We also note that the influence of the TOD on the BPFs’ response is more significant as compared to the LPFs and HPFs, indicating that the BPFs require a higher level of phase accuracy among different wavelength channels.
3.2.4 Influence of the optical spectral shapers and BPDs
Here we analyze the distortions induced by the OSS and BPD in Fig. 3. The OSS is employed to apply the designed tap weights to the delayed signals across different wavelength channels, and the BPD is used to sum the delayed and weighted signals and generate the microwave output. The OSS can induce shaping errors, which give rise to inaccurate tap weights, consequently resulting in distortions in the filter response. In addition, the distortions can be caused by the presence of noise and uneven transmission response of the BPD.
We introduce the concept of shaping error range, which refers to the random tap weight shaping errors within a certain percentage range, to characterize the shaping errors of the OSS. The RF response of all filters is shown in Figs. 8(aⅰ) − (cⅰ), respectively, with different shaping error ranges varying from 0–10%. A shaping error range of 0% represents the condition of an OSS without any shaping errors. As the shaping error range increases, the response for all the filters deviates further from an errorfree OSS, suggesting an increase in the distortions caused by larger shaping errors of the OSS. When the shaping error range increases from 0–10%, for the LPFs, the ROR and MSSR have shown negligible differences. For the BPFs, the MSSR decreases from 64.79 to 32.15. For the HPFs, the MSSR reduces from 73.1 to 64.79. When shaping error range increases from 0 to 10%, the ROR and MSSR for the LPFs show insignificant differences. However, for the BPFs, the MSSR decreases from 64.79 to 32.15, while for the HPFs, the MSSR decreases from 73.1 to 64.79.
Figures 8(aⅰⅰ) − (cⅰⅰ) show the ADs as a function of the shaping error range. The ADs increase with the shaping error range for all the filters, which agrees with the trend in Figs. 8(aⅰ) − (cⅰ). Moreover, it is worth noting that the impact of OSS shaping errors on the response of the HPFs is more significant compared to that of the LPFs and BPFs, reflecting that the HPFs require a higher level of shaping accuracy.
Through using a BPD, the commonmode noise of optical signals is effectively suppressed. Therefore, the distortions induced by the BPD mainly stem from its limited response bandwidth and uneven transmission response, which causes additional tap weight errors after spectral shaping. Similar to the BPD, the EO modulator’s limited response bandwidth and uneven transmission response also give rise to tap weight errors. These tap weight errors and the OSSs’ shaping errors can be effectively alleviated via feedback control. We also note that, shot noise of the BPD introduces stochastic power fluctuations, which constrains the lowest achievable phase noise floor [65]. Distortions induced by this are similar to those induced by the phase noise of the microcomb and can be reduced by using a BPD with an improved sensitivity [66].
3.2.5 Contributions of different error sources
Here, we analyze the contributions of the error sources discussed above to the overall distortions in the response of microcombbased MWP transversal filters. Figures 9(a) – (c) show the simulated RF amplitude response for all three types of filters, after accounting for the distortions induced by different error sources including the (I) OSNR of microcombs, (II) chirp of EO modulator, (III) TOD of SMF, and (IV) errors of the OSS and BPD. The ideal filter response without any errors is also shown for comparison. Based on the measured parameters of the components in our previous experiments [17, 22, 67], the chirp parameter of the EO modulator, the TOD parameter of the SMF, and the random shaping error range are set to α = 0.5, D3 = 0.083 ps/nm2/km, and shaping error range = 5%, respectively. In our simulations, we also used the OSNRs of the comb lines of a practical microcomb that were measured by an optical spectrum analyzer. Consistent with expectations, the overall distortions in the filter response exhibit a noticeable increase as a consequence of the accumulation of the errors induced by the imperfect response of components mentioned above.
To quantify the contributions of different error sources, we calculate the ADs from the results in Figs. 9(a) − (c) and plot them in Fig. 9(d). The ADs increase with the accumulation of the errors induced by sources I ‒ IV, showing agreement with the trend in Figs. 9(a) − (c). For all the three filters, the main source of distortions is the EO modulator chirp. Compared to the LPF and HPF, the distortions of the BPF are more significantly influenced by the TOD of the SMF. In addition, the distortions caused by the imperfect microcomb affect the HPF more significantly than the LPF and BPF.
3.3 Influence of signal bandwidth
For signal filtering, according to the theory of signals and systems [68], the ideal filter output can be given by
$${\text{S}}_{\text{out}}\text{(}\text{ω}\text{)}\text{ = }{\text{S}}_{\text{in}}\text{(}\text{ω}\text{) ‧ }\text{H}\text{(}\text{ω}\text{) }$$
10
where Sin(ω) and Sout(ω) are the input and output microwave signals in the spectral domain, respectively, and H(ω) is the spectral transfer function in Eq. (1). In Sections 3.1 and 3.2, we have analyzed the influences of both theoretical limitations and experimental system errors on H(ω). In practical signal filtering, the system’s limited operation bandwidth introduces additional errors when filtering wideband signals, thus leading to deviations between the filter output signal and the ideal output signal in Eq. (10). In this section, we analyze the errors introduced in microcombbased MWP transversal filters when processing microwave signals with varying spectral bandwidths.
As discussed in Section 3.2, a microcombbased MWP transversal filter has a periodic spectral response due to its finite impulse response. The FSR of the spectral response is equal to the inverse of the time delay between adjacent wavelength channels. On the other hand, according to the Nyquist sampling theorem, a continuoustime signal that is bandwidthlimited requires sampling at a rate greater than twice its maximum frequency component to prevent aliasing. This limitation sets a maximum allowable bandwidth for the RF signal to be processed, which should not exceed half of the microcomb’s comb spacing. Therefore, the operation bandwidth of a microcombbased MWP transversal filter can be expressed as:
OBW = min {∆λ/2, FSRMW/2} (11)
where ∆λ is the comb spacing, FSRMW is the FSR of the RF response in Eq. (3), and min {} represents taking the minimum value between the two.
Based on the physical processes of signal delay and summation in the transversal filter, assuming all the components in Fig. 3 are errorfree, the filter output can be expressed as
$${\text{S}}_{\text{out}}\text{(}\text{ω}\text{)}\text{ =}\text{ }\text{IFT[}{\text{s}}_{\text{out}}\text{(}\text{t}\text{)}\text{]}\text{ }\text{=}\text{ }\text{IFT[ }\underset{\text{n}\text{=0}}{\text{∑}^{\text{M}\text{1}}}{\text{a}}_{\text{n}}{\text{s}}_{\text{in}}\text{(}\text{t }\text{ }\text{n}\text{Δ}\text{t}\text{)] }$$
12
where sout(t) and Sout(ω) are the output microwave signal in the time and spectral domain, respectively, sin(t) is the input microwave signal in the time domain, and IFT[] denotes the operation of the inverse Fourier transform. In Eq. (12), a1,2,…,M−1 and ∆t are the designed tap coefficients and time delay in Eq. (1), respectively. Due to the limited operation bandwidth of an experimental system, the output signal calculated from Eq. (12) differs from that calculated from Eq. (10). After accounting for the influence of the imperfect response of real components on the tap coefficients and time delay, the filter output with experimental errors can be given by
$${\text{S}}_{\text{out}}{\prime }\text{(}\text{ω}\text{)}\text{ = IFT[}{\text{s}}_{\text{out}}{\prime }\text{(}\text{t}\text{)}\text{]}\text{ }\text{=}\text{ }\text{IFT[ }\underset{\text{n}\text{=0}}{\text{∑}^{\text{M}\text{1}}}{\text{a'}}_{\text{n}}{\text{s}}_{\text{in}}\text{(}\text{t }\text{ }\text{n}\text{Δ}\text{t'}\text{)] }$$
13
where sout'(t) and Sout'(ω) are the experimental output microwave signals in the time and spectral domain, respectively, a\({\prime }\)1,2,…,M−1 and ∆t\({\prime }\) are the tap coefficients and time delay with errors induced by the imperfect response, respectively.
To quantify the errors in the filter outputs, the root mean square errors (RMSE) is introduced. Similar to the definition of AD, the RMSE is defined as [45]
$$\text{RMSE = }\sqrt{\underset{\text{r}\text{=1}}{\text{∑}^{\text{R}}}\frac{\text{(}{\text{Y}}_{\text{r}}\text{ }\text{– }{\text{y}}_{\text{r}}\text{)}\text{2}}{\text{R}}}$$
14
where R is the number of sampled points, Y1, Y2, …, Yr are the values of ideal output signal spectra calculated based on Eq. (10), and y1, y2, …, yr are the values of the filter output spectra calculated based on either Eq. (12) or Eq. (13). Figures 10(aⅰv) − (civ) show the calculated RMSEs for the filter outputs as a function of the input signal bandwidth. As expected, the RMSEs increase with the spectral bandwidth of the input signal for all filters, and the RMSEs for the filter outputs with experimental errors are higher than those for the filter outputs without experimental errors. It should also be noted that when the spectral bandwidth of the input signal is larger than FSRMW / 2 = 15 GHz but lower than ∆λ / 2 = 25 GHz, the filter can still work but with notable inaccuracies. However, if the spectral bandwidth of the input signal increases beyond 25 GHz, the filter cannot function properly. This is because there will be aliasing between the modulated signals of adjacent wavelength channels due to the comb spacing being 50 GHz. This work has wide applicability to microwave photonic devices [69–96] based on optical microcombs [97–119] with potential applications to quantum optical devices [120–132] as well.