Adjustable RF photonic fractional Hilbert transformer based on Kerr microcombs

— We demonstrate an RF photonic fractional Hilbert transformer based on an integrated Kerr microcomb source featuring a record low free spectral range of 49 GHz. By programming and shaping the comb lines according to calculated tap weights for up to 39 wavelengths across the C-band, we achieve tunable bandwidths ranging from 1.2 to 15.3 GHz as well as variable center frequencies from baseband to 9.5 GHz, for both standard integral and arbitrary fractional orders. We experimentally characterize the RF amplitude and phase response of the tunable bandpass and lowpass Hilbert transformers with 90 and 45-degree phase shifts. The experimental results show good agreement with theory, confirming the effectiveness of our approach as a powerful way to implement standard and fractional order Hilbert transformers with broad and variable bandwidths and center frequencies, with high reconfigurability and greatly reduced size and complexity.


I. INTRODUCTION
he Hilbert transform is a fundamental mathematical function for signal processing systems. It has found wide applications in radar systems, signal sideband modulators, measurement systems, speech processing, signal sampling, and many others [1][2][3]. Fractional Hilbert transforms can meet specific requirements for secure single sideband communications [2] as well as hardware keys [3] since they provide an additional degree of freedom in terms of a variable phase shift. In practical applications such as multiplexing and demultiplexing signals, analyzing individual sub-channel spectral components, etc., Hilbert transformers are typically realized as a truncated or windowed version of the ideal Hilbert transform impulse response [4][5][6]. Thus, Hilbert transformers that cover a wide range of different bandpass regions are in high demand. While electronic approaches to Hilbert transformers are subject to the well-known electronic bandwidth bottleneck [7], photonic RF techniques have attracted wide interest due to their numerous advantages, including optical bandwidths that can reach 10's of THz, a strong immunity to electromagnetic interference, and low transmission loss.
A significant amount of research has been devoted to developing and extending the capability of photonic RF Hilbert transformers, including the use of free-space optics [ although many approaches suffer from a tradeoff between performance and system complexity. Photonic approaches for both standard and fractional Hilbert transforms offer broad operational bandwidths through the use of novel approaches such as fibre Bragg gratings [8][9][10][11][12][13][14], macroing / microdisk resonators [15,16], and integrated reconfigurable microwave processors [17]. However, most of these approaches focus only on generating the Hilbert transform of the complex optical field, and not the actual RF signal. To implement highly reconfigurable RF photonic Hilbert transformers, transversal schemes have been investigated. Fractional Hilbert transformers based on transversal structures have been demonstrated [18,19] that offer high reconfigurability. However, methods that employ multiple discrete laser sources present limitations in the overall system footprint, processing performance, and the potential for full monolithic integration.
Here, we report a highly reconfigurable and versatile fractional Hilbert transformer based on a soliton crystal Kerr microcomb with a 49 GHz comb spacing. The device has the ability to switch both its bandwidth and center frequency simultaneously and independently. This versatility, together with its high performance, is enabled by the use of 39 comb lines generated by the microcomb over the C-band, representing more than twice the number of wavelengths, or taps, used in our previous work [63]. We experimentally demonstrate a Hilbert transformer with an RF amplitude and phase response at bandwidths ranging from 1. feasibility of our approach towards the realization of high-speed reconfigurable Hilbert transformers with reduced footprint, lower complexity, and potentially reduced cost.

II. PRINCIPAL OF OPERATION
The spectral transfer function of a general fractional Hilbert transformer is given by [62,71]: where j = √-1, φ = P × π / 2 denotes the phase shift, P is the fractional order (when P = 1, it becomes a standard Hilbert transformer). The corresponding impulse response is given by a continuous hyperbolic function: This hyperbolic function is truncated and sampled in time with discrete taps for digital implementation. The null frequency is given by: where ∆t denotes the sample spacing. The coefficient of the tap at t = 0 can be adjusted to achieve a tunable fractional order [3]. The normalized power of each comb line is: where N is the number of comb lines, or taps, and n = 0, 1, 2, … , N-1 is the comb index.
In order to scale the bandwidth of the standard and fractional Hilbert transformer, we multiplied the corresponding impulse response of the spectral transfer function of the Hilbert transformer with a cosine function to vary the operation bandwidth (designed by the Remez algorithm [72]). Hence, the resulting discrete impulse response becomes: where fBW is the scalable bandwidth. To further switch the center frequency of the Hilbert transformer, the tap coefficients were multiplied by a sine function to shift the RF transmission spectrum. The corresponding discrete impulse response is given by Here, we use this transversal approach to achieve Hilbert transformers with both variable bandwidths as well as RF centre frequencies. The transfer function of the transversal structure [74][75][76][77][78][79][80][81][82][83][84] can be described as: where M is the number of taps, ω is the RF angular frequency, T is the time delay between adjacent taps, and h(n) is the tap coefficient of the n th tap. Figure 1 illustrates the schematic diagram of the microcombbased RF photonic Hilbert transformer. The microcomb was generated by pumping a nonlinear high-Q MRR, fabricated on a high-index doped silica glass platform, with a CW laser, amplified with an erbium-doped fibre amplifier (EDFA), with the polarization state aligned to the TE mode of the MRR. When the pump wavelength was swept manually across one of the MRR's resonances and the pump power was high enough to generate sufficient parametric gain, optical parametric oscillation occurred, ultimately generating Kerr microcomb with a spacing equal to the free spectral range of the MRR. The generated microcomb was then spectrally flattened and 3 modulated by the WaveShapers to achieve the designed tap weights.
Following this, the shaped comb lines were then imprinted with the RF signal via an intensity modulator, yielding replicas of the RF input waveform in the optical domain, transmitting the wavelength through a dispersive medium to acquire wavelength-dependent delays. The weighted and delayed signals were summed together upon photodetection and converted back into the RF domain.

III. EXPERIMENTAL RESULTS
The MRRs used here were fabricated on a platform based on Hydex glass [30,33,34,38,39] with CMOS compatible fabrication processes. First, Hydex films (n = ~1.7 at 1550 nm) were deposited using plasma enhanced chemical vapour deposition (PECVD), then patterned by deep ultraviolet (UV) photolithography and etched via reactive ion etching [29] to achieve waveguides with very low surface roughness. Finally, an upper cladding layer composed of silica (n = ~1.44 at 1550 nm) was deposited. We often employ a vertical coupling scheme between the ring resonator and bus, where the gap (approximately 200 nm) can be controlled by film growth, which is much more accurate than using lithographic techniques. The advantages of our platform, particularly in relation to optical microcombs, include ultra-low linear loss (~0.06 dB · cm -1 ), a moderately high optical nonlinear parameter (~233 W -1 · km -1 ), and in particular, a negligible nonlinear loss -even up to extremely high intensities of about 25 GW · cm 2 . As a result of the ultra-low loss of the platform, our MRR features narrow resonance linewidths corresponding to Q factors of up to ~1.5 million for the 49 GHz MRR. The insertion loss of the through-port was as low as 0.5 to 1 dB/facet a result of very efficient on-chip mode converters that allowed packaging the device with fibre pigtails. The radius of the MRR was ~592 µm, corresponding to an optical FSR of 0.393 nm or 48.9 GHz. This small FSR greatly increased the number of   wavelengths (channels) to as many as 75 over the C band, of which we used up to 39 wavelengths in this work, more than twice the number of wavelengths over our previous work [63].
To generate microcombs with the 49 GHz device, the CW pump power was amplified to ~30.5 dBm, and the wavelength swept from blue to red near one of the TE resonances of the MRR at ~1553.2 nm [51]. When the detuning between the pump wavelength and cold resonance of the MRR became small enough, such that the intra-cavity power reached a threshold, modulation instability (MI) driven oscillation occurred [22]. Primary combs were then generated with the spacing determined by the MI gain peakmainly a function of the dispersion and intra-cavity power. As the detuning was changed further, distinctive 'fingerprint' optical spectra eventually appeared (Fig. 2). The spectra are similar to what has been reported from spectral interference between tightly packed solitons in the cavityso-called "soliton crystals" [26][27][28]. The soliton crystal states provided the lowest RF noise states of all our microcombs and have been used as the basis for an RF oscillator with low phase-noise [57].
The soliton crystal microcomb was first flattened via an optical spectral shaper (Finisar, WaveShaper 4000S) and then modulated with the RF input signal to multicast the RF waveform onto all wavelength channels simultaneously, yielding up to 75 replicas. These were then passed through ~3.84 km of standard single mode fibre (SMF) to provide the progressive time delays between wavelengths. The dispersion of the SMF was ~17.4 ps/nm/km, corresponding to a time delay ∆t = 26.25 ps between adjacent wavelengths. Next, the second WaveShaper accurately shaped the comb power according to the designed tap coefficients, with the shaped comb spectrum shown in Fig. 3. The wavelength channels for both positive (solid red line) and negative (solid gray line) taps were separately measured by an optical spectrum analyser (OSA), achieving good agreement with theory (blue dot). Finally, the weighted and delayed replicas were combined and converted back into the RF domain via a balanced photodetector (Finisar BPDV2150R).
The system RF frequency response was characterized with a calibrated vector network analyser (VNA, Agilent MS4644B) to measure the RF transmission and phase response. Fig. 4 (ae) presents the simulated (dashed curves) and measured (solid curves) RF frequency response for both the magnitude and phase of the standard Hilbert transformer with variable bandwidths ranging from 3.4 to 15.3 GHz. The centre frequency of the Hilbert transformer was set into half of the FSRRF, which was 19/2 = 9.5 GHz in our case. Figure 5 shows the measured results for the RF amplitude and phase response of the lowpass Hilbert transformer, showing tunable bandwidths ranging from 1.2 to 7.1 GHz that match closely with the simulated results. We also performed a demonstration of a fractional Hilbert transformer with switchable RF bandwidths ranging from 3.5 to 15.2 GHz. The simulated and measured RF amplitude and phase responses are shown in Fig. 6. We achieved a fractional order of 0.5, which corresponds to a 45-degree phase shift. We used up to 39 wavelengths or taps, which represents an increase of a factor of two from our previous work. This resulted in a significant reduction in the overall root-mean-square error (RMSE) as well as an increase in the potential operational bandwidth, as was predicted [63]. Note, in order to achieve Hilbert transforms with tunable fractional orders we use an additional tap in the middle of the comb lines that we adjust in weight, resulting in an effective tap spacing of twice the FSR: 49 × 2 = 98 GHz. Hence, for the fractional Hilbert transformer this resulted in 39 of the 75 comb lines generated across the C-band being available. For consistency we used the same number of taps for the standard Hilbert transformer as well.  We demonstrate a broadband RF photonic Hilbert transformer with variable bandwidths as well as RF center frequency, based on a Kerr soliton crystal microcomb. Up to 39 wavelengths, or taps were used of the 75 that were generated, resulting in a tunable bandwidth ranging from 1.2 to 15.3 GHz as well as a switchable center frequency from baseband to 9.5 GHz, achieved by programming and adjusting the tap weights. Microcombs are a powerful approach to achieving a fully integrated photonic signal processor for future ultra-high-speed RF systems.