Photonic radio frequency and microwave intensity differentiator based on an optical frequency comb source in an integrated micro-ring resonator

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 implementing a transversal filter, which will greatly reduce the cost, size, and complexity of the system. Moreover, owing to the compactness of the integrated MRR, frequency spacings of up to 200-GHz can be achieved, enabling a potential operation bandwidth of over 100 GHz. By programming and shaping individual comb lines according to 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-, second-, and third-order differentiation functions based on this 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 a Gaussian input signal are also performed. The experimental results show good agreement with theory, confirming the effectiveness of our approach.

overcoming the intrinsic bandwidth bottleneck of electronic processing. [1][2][3][4][5][6][7][8][9] As one of the basic building blocks in optical signal processing and computing systems, 10 photonic differentiators are a key requirement in analyzing high-speed signals, as well as in waveform shaping, pulse generation, and systems control. [11][12][13] To implement photonic differentiators, a number of schemes have been proposed, which can be classified into two categories -namely, field differentiators and intensity differentiators. 14 Field differentiators based on apodized fibre Bragg gratings 11,13,15 and integrated silicon photonic devices [16][17][18][19][20] have recently been demonstrated. These types of devices yield the derivative of a complex optical field, and have the ability to shape ultra-short optical pulses that could find applications in optical pulse generation and advanced coding. [21][22][23] On the other hand, some other applications such as ultra-wideband frequency generation, radio frequency (RF) measurement and filters, require intensity differentiators that provide the derivative of the temporal intensity profiles associated with RF signals. [24][25][26] A photonic intensity differentiator based on a dual-drive Mach-Zehnder modulator together with an RF delay line, was reported [14] 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, 27,28 featuring high processing speeds of up to 40-Gb/s. This approach, however, works only for a fixed differentiation order and lacks reconfigurability, whereas in practical applications processing systems with variable differentiation orders are desired to meet diverse computing requirements.
To implement highly reconfigurable intensity differentiators, transversal filter schemes based on discrete microwave photonic delay-lines have been investigated. 12,29 However, these approaches have had limitations of one form or another, such as the need for 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 optical frequency comb source is proposed and experimentally demonstrated. By employing an on-chip

II. OPERATION PRINCIPLE
Based on the classical theory of signals and systems, 32 the spectral transfer function of an N th order temporal differentiator can be expressed as where j = √−1, ω 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 |ω| N , while the phase response has a linear profile, with a zero and π jump at zero frequency for N even and odd, respectively. The ideal RF amplitude and phase response of first-, second-, 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. [33][34][35] The transfer function of a typical transversal filter can be described as 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 based on Eq. (2) are an intensity differentiators for baseband RF input signals, i.e., the combined output RF signal after detection yields an exact differentiation of the input RF signal, in contrast to field differentiators that yield the derivative of a complex optical field. 11,13,[15][16][17][18][19][20] We note that, while optical field differentiators can be used to directly operate on microwave photonic signals, our approach has important advantages. When an RF signal is modulated onto an optical carrier, the intensity of the optical carrier is proportional to the square of the RF field. Thus, techniques that differentiate the optical field will yield the derivative of the square of the RF function rather than the exact derivative of the RF function directly, as our technique does.
To implement the temporal differentiator in Eq. (1), we calculate the tap coefficients in Eq. (2) based on the Remez algorithm. 36 The corresponding amplitude and phase response of the first-, second-, and thirdorder differentiators as a function of the numbers of taps are also plotted in Figs. 1(a)-(c). When the number of taps is increased, it is clear that the discrepancies between the amplitude response of the transversal filters and the ideal differentiators are improved for all 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 (see 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 a schematic illustration of the reconfigurable microwave photonic intensity differentiator.
It consists of two main blocks: the Kerr optical frequency comb generation module based on a nonlinear MRR and a transversal filter module for reconfigurable intensity differentiation. In the first module, the continuous-wave (CW) light from a tunable laser source 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 pump power is high enough for sufficient parametric gain, the optical parametric oscillation (OPO) process in the nonlinear MRR is initiated, generating a Kerr optical comb with nearly equal line spacing. 37 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. 39,40 Note that the high reconfigurability of the proposed differentiator cannot typically be achieved by passive silicon counterparts [16][17][18][19][20] , thus making our approach more suitable for diverse computing requirements in practical applications. The operation bandwidth can also be changed by adjusting the time delay between adjacent taps or employing different tap coefficients. An increased operation bandwidth can be obtained by simply employing a dispersive fibre with a 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, which is well beyond electrical processing bandwidths and comparable with that associated with integrated-waveguide Bragg gratings. 20

III. EXPERIMENTAL RESULTS
In our experiment (see Fig. 3(a)), the nonlinear MRR used to generate the Kerr comb was fabricated on a high-index doped silica glass platform using CMOS compatible fabrication processes. 37,38,[41][42][43] First, highindex (n = ~1.70 at 1550 nm) 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 waveguides with exceptionally low surface roughness. Finally, silica glass (n = ~1.44 at 1550 nm) was deposited via PECVD as an upper cladding. Our CMOS compatible fabrication process makes our differentiators comparable to, in terms of fabrication maturity, those implemented by means of optoelectronic silicon devices [16][17][18][19][20] . In particular we note that, due to the ultra-low loss of our platform, the ring resonator has a quality factor of ~1.2 million. Our device architecture uses a vertical coupling scheme where the gap can be controlled via film growth -a more accurate approach than lithographic techniques 40,44 . The gap between the bus waveguide and the MRR is approximately 200nm. The compact integrated MRR has a radius of ~135 μm with a relatively 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. [46][47][48] The advantages of our platform for nonlinear OPOs 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 ). 37,38,[41][42][43] 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 MRR before depositing the SiO2 upper cladding is shown in Fig. 3(b). By boosting the power of the CW light from the tunable laser source via an EDFA and adjusting the polarization state, multiple FSR 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 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 49 ( Fig. 4(a) ) was 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 photo-detector, achieving rigorous comb coherence 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 differentiation demonstrations were 8, 6, and 6, respectively. The choice of these numbers was made mainly by considering the power dynamic range, i.e., the difference between the maximum power of the generated comb lines and the power associated with the noise floor. The dynamic range was determined by the EDFA before waveshaping, which in our case was ~30 dB. An increased number of taps requires a broader power dynamic range, which can be achieved by using an 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 that of the transversal filter's Nyquist frequency when engineering 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 increased. The calculated tap coefficients for first-, second-, and third-order differentiations 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 or absolutely stable, 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. Subsequently, 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 and (c-ii). It can be seen that all three configurations exhibit a response expected from ideal differentiation.
In Fig. 5(a-i), we also indicate the operating frequency range of the first-order intensity differentiator. Since our device was designed to perform intensity differentiation for baseband RF signals, the operating frequency range starts at DC and ends at half of the spectral range between DC and the notch centred at ~17 GHz. The FSR of the RF response spectra is ~16.9 GHz, which is consistent with the time delay between adjacent taps.
Note that by adjusting the FSR of 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.
To further investigate the imperfections associated with the device performance, we employed

IV. CONCLUSION
We propose and demonstrate a reconfigurable microwave photonic intensity differentiator based on an integrated Kerr comb source. By programming and shaping the individual comb line powers according to calculated tap weights, we successfully demonstrate first-, second-, and third-order intensity differentiation of RF signals. The RF amplitude and phase responses of the proposed differentiator are characterized, and systems demonstrations of real-time differentiations are performed for Gaussian input pulses. We achieve good agreement between theory and experiment, thus verifying the effectiveness of our approach. Our technique, based on a CMOS-compatible nonlinear micro-ring resonator, provides a new way to implement microwave photonic intensity differentiators featuring compact device footprint, high processing bandwidth, and high reconfigurability, thus holding great promise for future ultra-high-speed computing and information processing.