We present a photonic approach for generating low phase noise, arbitrary chirped microwave waveforms based on heterodyne beating between high order correlated comb lines extracted from frequency-agile optical frequency comb. Using the dual heterodyne phase transfer scheme, extrinsic phase noises induced by the separate optical paths are efficiently suppressed by 42-dB at 1-Hz offset frequency. Linearly chirped microwave waveforms are achieved within 30-ms temporal duration, contributing to a large time-bandwidth product. The linearity measurement leads to less than 90 kHz RMS frequency error during the entire chirp duration, exhibiting excellent linearity for the microwave and sub-THz waveforms. The capability of generating arbitrary waveforms up to sub-THz band with flexible temporal duration, long repetition period, broad bandwidth, and large time-bandwidth product is investigated and discussed.
© 2015 Optical Society of America
The generation of broad bandwidth linearly or arbitrarily chirped microwave (MW) and sub-THz signals has attracted extensive research interest for a wide range of applications such as frequency-modulated continuous-wave (FMCW) radar [1–3 ], ultra-wideband sensing [4, 5 ], bio-medical imaging , physical chemistry , noncontact sensing and non-destructive diagnosis [1, 8, 9 ]. Frequency-chirped radio-frequency (RF) signals can be generated through electrical approaches such as direct synthesis using electrical oscillators , and digital signal processing or direct digital frequency synthesizer [11, 12 ] together with frequency up-conversion chains. However, due to the limited speed and bandwidth of electrical circuits, these approaches may not take the full advantages of the chirped MW and sub-THz technology. Thus they may hardly satisfy the growing requirements on chirped RF signals with frequencies up to hundreds of gigahertz.
Compared with the conventional electrical approaches, photonic techniques, benefiting from the intrinsic advantages of modern photonics such as ultra-high bandwidth, high-speed, compactness, and electromagnetic interference immunity [13, 14 ], are promising approaches for generating low phase noise and broadband frequency-chirped MW and sub-THz signals. Many photonic techniques aiming at the generation of chirped signals have been proposed and demonstrated during the past few years [15–21 ]. One most popular method is based on the heterodyne beating between two optical carriers of different frequency sweeping features. Such optical carriers can be produced through various approaches such as underlying their corresponding dispersive elements with different dispersions [18–20 ] or directly from two independent frequency-swept lasers of varied wavelengths . However, for the former approach, the especially fabricated dispersive elements impair the tunability and flexibility in terms of arbitrary waveform, time-bandwidth product, chirp rate and temporal duration, while for the latter approach, besides the compromise among chirp rate and bandwidth, the uncorrelated noises characteristic and the instability of the independent carriers lead to drastically deterioration in noise performance. Thereby, phase correlated carriers with flexible chirp features are of significant interest.
Moreover, these heterodyne-based approaches suffer from phase noise deterioration ascribing to the separated optic paths before beating [22, 23 ]. Environmental perturbations such as strain, temperature, vibration, acoustic noise, and humidity along the two separated paths introduce uncorrelated phase fluctuations to the two optical carriers, impairing the phase noise, stability and repeatability of the underlying generated signals. Therefore, the rigorous stabilization requirement is of highly concern. Considering the ultra-high frequencies of the generated waveforms, it remains challenging to detect and discriminate the phase fluctuation of the chirped signals extending to MW or even sub-THz range. Recently, we have demonstrated a dual-heterodyne phase error transfer (DHPT) method  for detecting and suppressing the phase noises induced by the separate optical paths in photonic generation of stable RF signals at fixed frequencies.
In this paper, we experimentally demonstrate photonic generation of linearly chirped MW waveforms based on heterodyne beating between two correlated frequency-chirped optical comb lines which are extracted from our specially designed frequency-agile optical frequency comb (OFC) . Such OFC offers arbitrary continuous sweep capability of the comb spacing, thus providing phase correlated, arbitrary chirped high order frequency-agile comb lines. By linearly sweeping the baseband drive signals applied to the frequency-agile OFC, the beating of comb lines can generate frequency-chirped signals expanding to sub-THz region with flexible chirp rate and temporal duration, high repeatability and ultra-high TBWP. In addition, the separate-optical-path-induced phase noise is effectively suppressed by exploring and extending the DHPT scheme to chirped frequencies. Moreover, the fidelity in terms of frequency error is analyzed and evaluated by a simple optical delayed self-heterodyne structure followed by self-mixing, which eliminates the need of high-speed photo-detector (PD). The arbitrary characteristics of the generated signals such as flexible temporal duration, repetition period and large TBWP, which potentially make our approach applicable for photonic arbitrary waveforms generation, are investigated and discussed.
2. Operation principle and theory
The principle of the proposed photonic approach is exhibited in Fig. 1 . A narrow linewidth laser as the optical source is fed into the frequency-agile optical frequency comb generator (OFCG). As we have demonstrated in , the frequency-agile OFCG incorporates two cascaded phase and intensity modulators together with a tunable delay module (TDM). When fed with a swept frequency arbitrary drive signal, the OFCG will arbitrarily sweep the comb spacing accordingly, and thus a scaled arbitrary sweep at high order comb lines is obtained. Owing to the broadband phase matching of the drive signals using the TDM, the power fluctuation of all the comb lines is kept within 4 dB during the sweep. A 90% part of the OFCG output is passed through the acousto-optic frequency shifter 1 (AOFS1) and then filtered by two optical bandpass filters (OBPF) to select two arbitrary and comb lines at path a and b respectively. The filtering process is shown in the inset optical spectra of Fig. 1. The OBPFs with bandwidth a little less than the comb spacing, are appropriately configured to set the and comb lines just at the edge of the passband in order to maximize the achievable bandwidth and prevent the overlap of the adjacent comb lines during the frequency chirp duration. Before being combined at a polarization-maintaining coupler (PMC) with the selected comb line from path a, the one from path b is modulated by AOFS2. The electrical fields of the two selected comb lines could be expressed asEq. (1) and (2) and the following equations due to the limited impact.
Through the beating of the selected and comb lines expressed in Eq. (1) and (2) , frequency multiplied arbitrary signal can be obtained at the high-speed PD1
The phase term , which represents the relative phase fluctuations, is induced by the separated optic paths. It is easily affected by the environmental perturbations, and will deteriorate the phase noise of the generated signals. For the sake of detecting and eliminating the phase fluctuation of the high frequency signals, we explore and extend our DHPT method  to arbitrary signals. When the comb lines from path a and b are beating with their corresponding ones from the other 10% part of the OFCG output at a low-speed PD2, the relative phase fluctuations of the selected and comb lines are firstly transferred to two intermediate frequencies (IF) 35 MHz and 75 MHz which are in accordance with the sum and difference frequencies introduced by the two AOFSs, and then further transferred to a 40-MHz IF beat signal using the following mixer
We employed a digital phase-frequency discriminator to extract the relative phase fluctuation and control the VCXO to produce the feedback signal applied to AOFS2. Thus, the 40-MHz IF beat signal is phase-locked to a low noise reference, as shown belowEq. (3) is stabilized.
To generate linearly chirped MW/sub-THz signals, the OFCG drive signal from the RF synthesizer, which is linearly swept, could be expressed asEq. (3) that represents the generated high frequency chirped signals could be further rewritten as
Therefore, the linearly chirped signal generated by the heterodyne beating between the and comb lines is chirped from to with the chirp bandwidth , where is the temporal duration of a single chirp. The chirp bandwidth of any comb line, as explicated by the filtering process illustrated in Fig. 1, is mainly limited by the OBPF whose bandwidth should not exceed the initial comb spacing in order to avoid the unwanted beat signals due to the overlap of adjacent comb lines. Thereby, the maximum achievable bandwidth is twice the initial comb spacing, namely twice the initial chirp frequency when the and comb lines are chosen to be swept oppositely.
3. Experimental result and analysis
As a proof-of-principle experimental setup of this approach, an extra erbium-doped fiber amplifier (EDFA) and another OBPF are employed in addition to the operation structure depicted in Fig. 1 to amplify the selected comb lines and filter out wideband noise before heterodyne beating. An off-the-shelf narrow linewidth distributed feedback semiconductor laser operating at 1550nm with a measured linewidth less than 600 kHz and a commercial RF synthesizer are applied to the OFCG to generate the frequency agile OFC.
The RF synthesizer output is firstly set to 25 GHz for the long time stability evaluation of the proposed approach. The suppression ratio of the phase noise is evaluated in terms of single sideband phase noise power spectrum density (SSB PN-PSD) as shown in Fig. 2 using phase noise analyzer. When comparing the phase noises of free running and closed-loop cases, we can see our DHPT approach exhibits effective suppression of the phase noise within the bandwidth of several hundred Hz. The suppression ratio reached 70 dB, 42 dB, and 13 dB at 0.01 Hz, 1 Hz, and 100 Hz Fourier frequencies, respectively. The loop bandwidth is restricted to few kHz as the bump shows in Fig. 2 ascribing to the limited modulation bandwidth of the AOFS. As most of the optic-path-induced noises coming due to the environmental perturbations are low-frequency terms within the bandwidth of few kHz, our approach shows efficient suppression performance. The phase locking parameters are optimized to serve the purpose of minimizing the frequency error during the chirp in addition to the suppression of the separate-optical-path-induced noises, especially for the low frequency region. The spursaround the bump are mainly attributed to the artificial noises from the homemade electrical circuits. Methods such as shielding and the use of ultra-low noise power supply would further reduce these technical noises.
The optical spectra of the extracted comb line pairs are observed using an optical spectrum analyzer (OSA) at the combination of path a and b. While adjusting the center wavelength and bandwidth of the two OBPFs accordingly, we obtain the optical spectra of 0th and 1st, ± 1st, ± 2rd, and ± 5th comb line pairs as shown in Figs. 3(a)-3(d) before and Figs. 3(e)-3(h) after the amplification and filtering process. The frequency separations of the comb line pairs are 25 GHz, 50 GHz, 100 GHz, and 250 GHz, respectively. The optical signal-to-noise ratio (OSNR) realizes >55 dB for all the extracted comb lines. The clean optical spectrum and high OSNR allow the generation of background-free RF signals with low spurs and harmonics. Taking the advantages of the flat-top features of the frequency-agile OFC, the power variation between different comb lines is minimized. This permits a small power variation of the generated frequency arbitrary chirped RF signals.
In order to assess the capacity of our approach to generate high frequency chirped signals, the drive signal linearly swept from 24.3 to 26.3 GHz is applied to the OFCG. The bandwidth of drive signal is limited by the RF synthesizer and electrical power amplifiers in our lab. The MW beat signal between the 0th and 1st comb lines is obtained using the high-speed PD1 with 40-GHz bandwidth. The power is finely adjusted through the EDFA to achieve the same power level compared with the drive signal. The obtained MW beat signal and the original drive signal are analyzed in an electrical spectrum analyzer (ESA). Figure 4(a) shows the measured spectra at a fixed frequency of 25 GHz at 1-kHz resolution bandwidth (RBW),while those of linearly chirped signals are shown in Fig. 4(b) at 10-kHz RBW. The obtained MW signals exhibit almost the same SNR performance compared to the original drive signals. The little deterioration in the noise floor level is mainly attributed to the noises from the EDFA. In both conditions, we could obtain a clean background spectrum of the generated MW signals, and thus eliminate the need of electrical bandpass filter or time-domain background-subtraction methods. Note that the beat signals experience a 40-MHz frequency shift because of the AOFS.
Due to the limited bandwidth of the ESA and PD in our lab, high frequency MW/sub-THz signals generated by the beating between the high order comb lines could hardly be directly measured and evaluated. Therefore, a simple scheme consisting of optical delayed self-heterodyne and self-mixing is introduced as shown below in Fig. 5 .
In Fig. 5, the extracted comb line pairs are fed to an unbalanced Mach–Zehnder delay interferometer. An AOFS is employed in one arm to shift the beat signal to a 40-MHz IF. Considering the frequency linearly chirped stage of the symmetrical and comb lines where , from the Eq. (7), we can express the PD3 output as
For drive signal linearly swept from 24.3 to 26.3 GHz in 30 ms, the electrical spectral of PD3 outputs shown in Fig. 6(a)-6(c) correspond to the ± 1st, ± 2rd, and ± 5th comb line pairs, respectively. The left and right peaks, which are in accordance with the and terms exhibited in Eq. (8), are beat frequencies produced by delay heterodyning of the linearly chirped symmetrical comb lines whose chirp-rates are both . While the drive signal is periodically repeated, the peak at 40 MHz corresponds to the IF beat signal coming from the non-sweep duration between the chirped stages, where . With the measured beat frequency and the 280-ns delay, the chirp rate can be thus obtained from Eq. (8). Through the obtained chirp-rate and the temporal duration T of RF synthesizer, the achieved chirp-bandwidth is 4.018 GHz, 8.036 GHz and 20.089 GHz respectively, which fit well with the value that directly calculated using . The frequency deviation of the measured chirp-bandwidth is probably due to the measurement error of the fiber delay and the instrumental error of the RF synthesizer and the ESA.
In order to further investigate the chirp feature, the output of PD3 is amplified and self-mixed, the output of the mixer can be written as
A high-speed oscilloscope is employed to capture the time domain waveform as shown in Fig. 7(a) of the mixer output. The waveform corresponding to the beat between the ± 5th comb line pair is analyzed to evaluate the linearity of the generated sub-THz signal.
The phase term of the mixer output contains the chirp linearity information as explained in Eq. (9). After being extracted using Hilbert transform, which allows direct characterization of the frequency error of the waveforms regardless the frequency and bandwidth, the chirp linearity from 2.2 to 7.3 ms (limited by the recording length of the oscilloscope) is evaluated in the form of instantaneous frequency and frequency error as shown in Figs. 7(b) and 7(c), respectively. We adopt a linear fit of the instantaneous frequency curve and obtain a curve slope of 669.016 GHz/s, which is in accordance with 10 times the chirp rate of the drive signal. Figure 7(c) exhibits the frequency error by subtracting the linear fit from the obtained instantaneous frequency. The RMS frequency error is maintained below 90 kHz during the entire chirp duration, demonstrating the high linearity of the generated high frequency signals. The residual nonlinearity mainly results from the error of the frequency swept drive signal. The low frequency error enables the generation of linearly chirped signals with high stability and repeatability. The high frequency noises observed in the frequency error is probably due to the distortion of the PD output signal and harmonics of the mixer output. It could be suppressed by optimizing the LPF bandwidth, thus reducing the measurement noises in the frequency error result. In Figs. 7(d) and 7(e), a 2-ms section of the captured data and analysis result is zoomed.
Additionally, the chirp bandwidth of the generated chirped sub-THz signal corresponding to the beating between ± 5th comb lines reaches up to 20.089 GHz in the 30-ms chirp duration and thus an ultra-high TBWP is experimentally obtained.
The aforementioned experiments demonstrate the capability of the proposed approach for generating low phase noise, linear chirped MW and sub-THz signals with low deviation in chirp-linearity, large and reconfigurable RF bandwidth, and high repeatability, which are vital factors in achieving high-performance applicable microwave photonic systems. This approach exhibits good phase noise characteristics compared with the previously reported techniques based on heterodyne beating between two optical carriers. Essentially, the frequency-agile OFC behaves as an ideal frequency multiplier which scales up the arbitrary baseband electrical drive signals and minimizes the power fluctuation. The phase noise induced by the separate optical paths is suppressed utilizing the DHPT method as a frequency independent phase-frequency discriminator of the optical phase-locked loop, which permits the phase-locking at unlimited high frequencies. Although only demonstrated with linearly chirped signals in this paper due to the limited functionality of our RF synthesizer, Eqs. (3)-(7) demonstrate the ability of noiseless frequency multiplying of arbitrary electrical signals, and thus allow the straightforward generation of ultra-high frequency arbitrary MW/sub-THz and even THz signals with flexible RF bandwidth and tunable center frequency.
From the application point of view, besides the aforementioned characteristics, the capability to generate broadband signals extending over flexible temporal duration and repetition period with large TBWP is also of key importance. Taking the advantages of the multiplicative nature, the temporal duration and repetition period could be controlled independently by simply changing the corresponding parameters of the electrical drive signal respectively. Furthermore, as RF bandwidth and temporal duration could be controlled independently, waveforms with large TBWP are thus enabled. Thereby, in our demonstration, reconfigurable broadband high frequency arbitrary waveforms can be attained by simply substituting the RF synthesizer with commercial electrical arbitrary waveform generators.
However, for the sake of avoiding the overlap of the adjacent comb lines during the frequency chirp, the initial comb spacing and the necessary guard-band of comb lines and OBPFs are the main limiting factor of the generated RF bandwidth. In further work, we will focus on the comb line extraction using optical phase-locking in order to eliminate the need of OBPF, and thus achieve the generation of ultra-broadband arbitrary waveforms.
In conclusion, a photonic approach for the generation of arbitrary frequency chirped MW waveforms based on the integration of frequency agile OFC and DHPT approach is experimentally demonstrated. The capability of generating arbitrary waveforms with low phase noise characteristics, broad RF bandwidth, tunable center frequency, large TBWP, flexible temporal duration and repetition period, and high repeatability is demonstrated and further discussed. Linearly chirped MW waveforms with ultra-high TBWP are achieved. The phase noise is reduced by 42-dB at 1-Hz offset, demonstrating the effective suppression of optic-path-induced noise. It is showed that the frequency error of the linearly chirped sub-THz waveform is kept below 90 kHz during the entire chirp duration. The residual frequency error is mainly attributed to the drive signal.
Benefiting from the quasi-ideal multiplying nature, the above features have made our approach a promising applicable photonic arbitrary waveform generation method which can overcome the drawbacks of most of the previously reported heterodyne-based approaches. Further improvement employing the phase-locking of continuous-wave lasers as proposed in the discussion section will allow the generation of real arbitrary waveforms extending from baseband to ultra-high frequency regions.
The authors are indebted to Fabien Bretenaker for his constructive comments and to Mohinder Jankiraman for the fruitful discussions. This work is supported by National Program on Key Basic Research Project of China (973) under Contract 2012CB315602, National Natural Science Foundation of China (NSFC) under Contract 61225004 and Chinese Government Scholarship (CSC) under Grant 201406230161.
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