# Reconfigurable microwave intensity differentiator based on integrated microcombs

David Moss

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# Reconfigurable microwave intensity differentiator based on integrated microcombs

David J. Moss

Centre for Micro-Photonics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia

dmoss@swin.edu.au

We propose and experimentally demonstrate a microwave photonic intensity differentiator based on a Kerr optical comb generated by a compact integrated micro-ring resonator (MRR). The on-chip Kerr optical comb, containing a large number of comb lines, serves as a high-performance multi-wavelength source for the transversal filter, which will greatly reduce the cost, size, and complexity of the system. Moreover, owing to the compactness of the integrated MRR, up to 200-GHz frequency spacing of the Kerr optical comb can be achieved, enabling a potential operation bandwidth of over 100 GHz. By programming and shaping individual comb lines according to the calculated tap weights, a reconfigurable intensity differentiator with variable differentiation orders can be realized. The operation principle is theoretically analyzed, and experimental demonstrations of first-order, second-order, and third-order differentiation functions based on the principle are presented. The radio frequency (RF) amplitude and phase responses of multi-order intensity differentiations are characterized, and system demonstrations of real-time differentiations for Gaussian input signal are also performed. The experimental results show good agreement with theory, confirming the effectiveness of our approach.

## I. INTRODUCTION

With the ever-increasing demand for processing speed and throughput in modern communication systems, optical information processing technologies have attracted great interest due to their advantages in overcoming the intrinsic bandwidth bottleneck of electronic processing [1–6]. As one of the basic building blocks in optical signal processing and computing systems [7], photonic differentiators are a key requirement in high-speed signal analyzing, waveform shaping, pulse generation, and system control [8–10].

To implement photonic differentiators, a number of schemes have been proposed, which can be classified into two categories, namely, field differentiators and intensity differentiators [11]. Field differentiators based on apodized fibre Bragg gratings [8, 10, 12] and integrated silicon photonic devices [13–16] have been recently demonstrated. These types of differentiators yield the derivative of a complex optical field, having the ability to shape ultra-short optical pulses that could find applications in optical pulse generation and advanced coding [17–19]. On the other hand, some other applications such as ultra-wideband (UWB) frequency generation, radio frequency (RF) measurement, and RF filters, require intensity differentiators that could provide the derivative of the temporal intensity profiles associated with RF signals [20–22]. A photonic intensity differentiator based on a dual-drive Mach-Zehnder modulator (DDMZM) together with a RF delay line was reported in Ref. [11], but the processing speed was intrinsically limited by the operation bandwidth of the RF delay line. Photonic intensity differentiators based on semiconductor
optical amplifiers (SOAs) and optical filters (OFs) have also been reported [23–24], featuring high processing speeds up to 40-Gb/s. This approach, however, works only for a fixed differentiation order and lacks reconfigurability, whereas processing systems with variable differentiation orders are desired in practical applications, in order to meet diverse computing requirements. To implement highly reconfigurable intensity differentiators, transversal filter schemes based on discrete microwave photonic delay-lines have been investigated [9, 25]. However, these approaches had limitations of one form or another, such as the need of generating the taps using discrete laser arrays, thus significantly increasing the system cost and complexity.

In this paper, a reconfigurable microwave photonic intensity differentiator based on an integrated Kerr comb source is proposed and experimentally demonstrated. By employing an on-chip nonlinear MRR, we generate a broadband Kerr comb with a large number of comb lines and use it as a high-quality multi-wavelength source for the transversal filter. Moreover, the large frequency spacing of the integrated Kerr comb also yields an increased Nyquist zone, thus leading to a potential operation bandwidth of over 100 GHz, well beyond the processing bandwidth of electronic devices. By programming and shaping the power of individual comb lines according to corresponding tap weights [26–27], reconfigurable intensity differentiators with variable differentiation orders can be achieved. We present a detailed analysis of the operation principle, and perform experimental demonstrations of first-, second-, and third-order intensity differentiations using the proposed device. The RF amplitude and phase responses of the intensity differentiator are experimentally characterized, and system demonstrations of real-time differentiations for Gaussian input pulses are also carried out. The experimental results are consistent with theory, verifying the feasibility of our approach as a solution to high-speed reconfigurable microwave photonic intensity differentiators.
## II. OPERATION PRINCIPLE

Based on the classical theory of signals and systems [28], the spectral transfer function of an N-th order temporal differentiator can be expressed as

$$H(\omega) \propto (j\omega)^N, \qquad (1)$$

where $j = \sqrt{-1}$, $\omega$ is the angular frequency, and N is the differentiation order. According to the above transfer function, the amplitude response of a temporal differentiator is proportional to $|\omega|^N$, while the phase response has a linear profile and a zero or $\pi$ jump at zero frequency for N even and odd, respectively. The ideal RF amplitude and phase responses of first-order, second-order, and third-order microwave differentiators are shown in Figs. 1(a)–(c), respectively.

In this paper, we employ a versatile approach towards the implementation of microwave photonic differentiators based on transversal filters, where a finite set of delayed and weighted replicas of the input RF signal are produced in the optical domain and combined upon detection [29–31]. The transfer function of a typical transversal filter can be described as

$$H(\omega) = \sum_{n=0}^{M-1} a_n e^{-j\omega nT} \qquad (2)$$

where M is the number of taps, $a_n$ is the tap coefficient of the n-th tap, and T is the time delay between adjacent taps. It should be noted that the differentiator design based on Eq. (2) is an intensity differentiator, i.e., the

<figure><img src="image_3.png" /><figcaption>Fig. 1. 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.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic illustration of the reconfigurable microwave photonic intensity differentiator. TLS: Tunable laser source. EDFA: erbium-doped fibre amplifier. PC: polarization controller. BPF: optical bandpass filter. TCS: temperature controller stage. MZM: Mach-Zehnder modulator. SMF: single mode fibre. OC: optical coupler. PD: photodetector. OSA: optical spectrum analyzer. VNA: vector network analyzer. AWG: arbitrary waveform generator.</figcaption></figure>

output RF signal after being combined upon detection yields an exact differentiation of the input RF signal, in contrast to field differentiators that yield the derivative of a complex optical field [8, 10, 12–16].

To implement the temporal differentiator in Eq. (1), we calculate the tap coefficients in Eq. (2) based on the Remez algorithm [32]. The corresponding amplitude and phase responses of the first-, second-, and third-order differentiators as a function of the numbers of taps are plotted in Figs. 1(a)–(c). When the number of taps is increased, it is clear that the discrepancies between the amplitude responses of the transversal filters and the ideal differentiators are improved for all the three orders, whereas the phase response of the transversal filters is identical to that of the ideal differentiators regardless of the number of taps. To quantitatively analyze the discrepancies in the amplitude responses, we further calculate the root mean square errors (RMSEs) for the first-, second-, and third-order differentiators as a function of the number of taps, as shown in Fig. 1(d). One can see that the RMSE is inversely proportional to the number of taps, as reasonably expected. In particular, we note that when the number of taps increases, the RMSE decreases dramatically for a small number of taps, and then decreases more gradually as the number of taps becomes larger.

Figure 2 shows the schematic illustration of the reconfigurable microwave photonic intensity differentiator. It consists of two main blocks: one is a Kerr comb generation module based on a nonlinear MRR and the other is a transversal filter module for reconfigurable intensity differentiation. In the first module, the continuous-wave (CW) light from a tunable laser source (TLS) is amplified by an erbium-doped fibre amplifier (EDFA) followed by a tunable optical bandpass filter (BPF) to suppress the amplified spontaneous emission (ASE) noise. A polarization controller (PC) is inserted before the nonlinear MRR to make sure that the polarization state matches the desired coupled mode. When the wavelength of the CW light is tuned to a resonance of the nonlinear MRR and the light power is large enough for sufficient parametric gain, the optical parametric oscillation (OPO) process in the nonlinear MRR can be initiated, which generates a Kerr optical comb with nearly equal line spacing [33–34]. The nonlinear MRR is mounted on a highly precise temperature controlled stage (TCS) to avoid resonance drifts and maintain the wavelength alignment of the resonances to the CW light. Owing to the compact size and ultra-high quality factor of the
nonlinear MRR, the generated Kerr comb provides a large number of wavelength channels with narrow
linewidths for the subsequent transversal filter module. As compared with conventional intensity
differentiators based on laser diode arrays, the cost, size and complexity can be greatly reduced. The
generated Kerr comb then is directed to the second module. After being amplified by another EDFA, the Kerr
comb is processed by a waveshaper to get weighted taps according to the coefficients calculated by means of
the Remez algorithm. Considering that the generated Kerr comb is not flat, a real-time feedback control path
is introduced to read and shape the comb lines' power accurately. A 2×2 balanced Mach-Zehnder modulator
(MZM) is employed to generate replicas of the input RF signal. When the MZM is quadrature-biased, it could
simultaneously modulate the input RF signal on both positive and negative slopes, thus yielding modulated
signals with opposite phases and tap coefficients with opposite algebraic signs. After being modulated, the
tapped signals from one output of the MZM are delayed by a dispersive fibre. The time delay between
adjacent taps is co-decided by the frequency spacing of the employed comb source and the dispersion
accumulated in the fibre. Finally, the weighted and delayed taps are combined upon detection and converted
back into RF signals to form the differentiation output.

It is worth mentioning that due to the intrinsic advantages of transversal filters, our scheme features a
high degree of reconfigurability in terms of processing functions and operation bandwidth, thus offering a
reconfigurable platform for diverse microwave photonic computing functions. By simply programming the
waveshaper to shape the comb lines according to the corresponding tap coefficients, our scheme can also
apply to other computing functions such as Hilbert transforms and differential equation solving [35–36]. The
operation bandwidth can also be changed by adjusting the time delay between adjacent taps or employing
specific tap coefficients. An increased operation bandwidth can be achieved by simply employing a
dispersive fibre with shorter length. The operation bandwidth is fundamentally limited by the Nyquist zone,
which is determined by the comb spacing. In our case, the frequency spacing of the Kerr comb generated by
the nonlinear MRR reaches 200 GHz, thus leading to a potential operation bandwidth of over 100 GHz.

### III. EXPERIMENTAL RESULTS

In our experiment, as Fig. 3 (a) shows, the nonlinear MRR used to generate the Kerr comb was fabricated on a high-index doped silica glass (Hydex) platform using CMOS compatible fabrication processes [33]. First, high-index (n = 1.7) doped silica glass films were deposited using standard plasma enhanced chemical vapour deposition (PECVD), then photolithography and reactive ion etching (RIE) were employed to form

<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Hydex MRR. (b) SEM image of the cross-section of the Hydex MRR before depositing the SiO₂ upper cladding.</figcaption></figure>
waveguides with exceptionally low surface roughness. Finally, the waveguides were buried in a lower index overcladding. The radius of the Hydex MRR is ~135 µm. The compact integrated MRR has a large free spectral range (FSR) of ~1.6 nm, i.e., ~200 GHz. Such a large FSR enables an increased Nyquist zone of ~100 GHz, which is challenging for mode-locked lasers and externally-modulated comb sources [37–38]. The advantages of the Hydex platform for nonlinear OPO include ultra-low linear loss (~0.06 dB·cm⁻¹), a moderate nonlinearity parameter (~233 W⁻¹·km⁻¹), and in particular a negligible nonlinear loss up to extremely high intensities (~25 GW·cm⁻²) [33–34]. After packaging the input and output ports of the device with fibre pigtails, the total insertion loss is ~3.5 dB. A scanning electron microscope (SEM) image of the cross-section of the Hydex MRR before depositing the SiO₂ upper cladding is shown in Fig. 3 (b). By boosting the power of the CW light from the TLS via an EDFA and adjusting the polarization state, multiple mode-spaced combs were first generated, in which the primary spacing was determined by the parametric gain. When the parametric gain lobes became broad enough, secondary comb lines with a spacing equal to the FSR of the Hydex MRR were generated via either degenerate or non-degenerate four wave mixing (FWM). In our experiment, the power threshold for the generation of secondary comb lines was ~500 mW.

The resulting Type II Kerr optical comb [39] as shown in Fig. 4(a) is over 200-nm wide, and flat over ~32 nm. Since the generated comb only served as a multi-wavelength source for the subsequent transversal filter in which the optical power from different taps was detected incoherently by the photodiode, the coherence of the comb was not crucial and the proposed differentiator was able to work under relatively incoherent conditions. In the experiment, the numbers of taps used for first-, second-, and third-order

<figure><img src="image_6.png" /><figcaption>Fig. 4. (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.</figcaption></figure>
<table><caption>TABLE I. Tap coefficients for first-order, second-order, and third-order differentiation</caption><thead><tr><th>Order of differentiation</th><th>Number of taps</th><th>Tap coefficients</th></tr></thead><tbody><tr><td>First-order</td><td>8</td><td>[-0.0226, 0.0523, -0.1152, 1, -1, 0.1152, -0.052, 0.0226]</td></tr><tr><td>Second-order</td><td>6</td><td>[0.0241, -0.1107, 0.0881, 0.0881, -0.1107, 0.0241]</td></tr><tr><td>Third-order</td><td>6</td><td>[0.0450, -0.4076, 1, -1, 0.4076, -0.0450]</td></tr></tbody></table>

differentiation demonstrations were 8, 6, and 6, respectively. The choice of these numbers was made mainly by considering the power dynamic range of the comb lines, i.e., the difference between the maximum power of the generated comb lines and the power associated with the noise floor. The power dynamic range was determined by the EDFA before waveshaping, and in our case, it was ~30 dB. An increased number of taps requires a broader power dynamic range, which can be achieved by using a further EDFA with a lower noise floor. As analysed in section II, 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 get better performance with a limited number of taps, we decreased the operation bandwidth of the second- and third-order differentiators to half of the transversal filter's Nyquist frequency when designing the response function with the Remez algorithm. It should be noted that the actual bandwidth of the differentiator is not limited by this design since the FSR of the transversal filter can be enlarged. The calculated tap coefficients for first-, second-, and third-order differentiation are listed in Table I. The selected comb lines of the generated optical comb were processed by the waveshaper based on these coefficients. Considering that the generated Kerr comb was not flat, we adopted a real-time feedback control path to increase the accuracy of comb shaping. The comb lines' power was first detected by an optical spectrum analyzer (OSA) and compared with the ideal tap weights, and then an error signal was generated and fed back into the waveshaper to calibrate the system and achieve accurate comb processing. The shaped optical combs are shown in Figs. 4(b)–(d). A good match between the measured comb lines' power (red solid line) and the calculated ideal tap weights (green cross) was achieved, indicating that the comb lines were successfully shaped. They were then divided into two parts according to the algebraic sign of the tap coefficients and fed into the 2×2 balanced MZM biased at quadrature. The modulated signal after the MZM was propagated through ~2.122-km single mode (dispersive) fibre (SMF). The dispersion of the SMF is ~17.4 ps/(nm·km), which corresponds to a time delay of ~59 ps between adjacent taps and yields an effective FSR of ~16.9 GHz in the RF response spectrum.

After the weighted and delayed taps were combined upon detection, the RF responses for different differentiation orders were characterized by a vector network analyser (VNA, Anritsu 37369A). Figures. 5(a-i), (b-i), and (c-i) show the measured and simulated amplitude responses of the first-, second-, and third-order intensity differentiators, respectively. The corresponding phase responses are depicted in Figs. 5(a-ii), (b-ii), and (c-ii). It can be seen that all the three differentiators feature the responses expected from ideal differentiation. The FSR of the RF response spectra is ~16.9 GHz, which is consistent with the time delay
<figure><img src="image_7.png" /><figcaption>Fig. 5. 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.</figcaption></figure>

between adjacent taps, as calculated before measurement. Note that by adjusting the FSR of the proposed transversal filter through the dispersive fibre or by programming the tap coefficients, a variable operation bandwidth for the intensity differentiator can be achieved, which is advantageous for meeting diverse requirements in operation bandwidth.

We also performed system demonstrations of real-time signal differentiation for Gaussian input pulses with a full-width at half maximum (FWHM) of ~0.12 ns, generated by an arbitrary waveform generator (AWG, KEYSIGHT M9505A), as shown in Fig. 6(a). The waveform of the output signals after differentiation are shown in Figs. 6(b)–(d) (blue solid curves). They were recorded by means of a high-speed real-time oscilloscope (KEYSIGHT DSOZ504A Infiniium). For comparison, we also depict the ideal differentiation
<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) Measured temporal waveforms of a Gaussian input pulse. Theoretical (red dashed) and experimental (blue solid) responses of the (b) first-, (c) second-, and (d) third-order differentiators.</figcaption></figure>

results, as shown in Figs. 6(b)–(d) (red dashed curves). The practical Gaussian pulse in Fig. 6(a) is used as the input RF signal for the simulation. One can see that the measured curves closely match their theoretical counterparts, indicating good agreement between experimental results and theory. Unlike the field differentiators in Refs. [8, 10, 12–16], temporal derivatives of intensity profiles can be observed, indicating that intensity differentiation was successfully achieved. For the first-, second-, and third-order differentiators, the calculated RMSE between the measured and the theoretical curves are ~4.15%, ~6.38%, and ~7.24%, respectively.

To further investigate the imperfections associated with the experimental results, we employed commercial software (VPIphotonics) to simulate the RF amplitude and phase responses of the various differentiators by considering the tap weight error during the comb shaping, the chirp induced by the MZM, and the third-order dispersion (TOD) of the fibre. Based on the measured error signal from the feedback control path and the empirical chirp value in previous experiments [35], the tap weight error, chirp coefficient, and TOD in our simulation were set to be 0.5 dB, 0.5, and 0.083 ps/(nm²·km), respectively. The simulated amplitude and phase responses are plotted in Figs. 5(a)–(c). One can see that after incorporating these two effects, the simulated RF responses fit more closely to the experimental results, thus confirming that the degradation of the RF responses can be attributed to both of these effects. By employing a low-chirped MZM, further increasing the resolution of the feedback control path, and compensating the TOD, we can anticipate improved performance of the differentiators. This work has broad implications for microcombs [41-67] and their applications to microwave photonics. [68-131] The addition and use of 2D materials [132-165] will
add extra functionality to microcomb chips for potential applications to quantum photonics [166 - 181] and other areas [182-191]

## IV. CONCLUSION

We propose and demonstrate a reconfigurable microwave photonic intensity differentiator based on an integrated Kerr comb source. The Kerr optical comb was achieved via a CMOS-compatible nonlinear MRR, which greatly increased the processing bandwidth and has the potential for a low cost, size, and complexity processing system. By programming and shaping the individual comb lines' power according to the calculated tap weights, we successfully demonstrate the first, second, and third-order intensity differentiation of the RF signal. The RF amplitude and phase responses of the proposed differentiator are characterized, and system demonstrations of real-time differentiations are performed for Gaussian input pulses. We achieve good agreement between theory and experimental, thus verifying the effectiveness of our approach. In particular, our scheme provides a new way to implement microwave photonic intensity differentiators with high processing bandwidth and re-configurability, thus holding great promise for future ultra-high-speed computing and information processing.

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