1. Introduction

Linearly chirped microwave waveforms (LCMWs) are of prime importance in scientific and technological areas, such as precise measurement, radar system and frequency-hopping communications [1-3]. For a pulse compression radar system, an LCMW with a large time-bandwidth product is essential to simultaneously achieve high-precision location and long-range detection [4,5]. The general method of producing LCMWs is using a voltage-controlled oscillator (VCO) [6] or a direct digital synthesizer (DDS) [7], which, however, suffers from limited frequency and bandwidth (generally below several gigahertz) due to the well-known electronic bottleneck. In order to conquer this limitation confronted by electronic approaches, various methods based on microwave photonics technique have been proposed in recent years [8-15], including direct space-to-time pulse shaping [10], spectral shaping cascaded by wavelength-to-time mapping [11], temporal pulse shaping [12], wavelength-to-time mapping cascaded by heterodyne detection [13], optical injection [14] and self-heterodyne technique [15]. The approaches based on ultra-short optical pulses in [10-13] can generate LCMWs with large bandwidths (beyond tens of gigahertz). However, the pulse duration is short (in the order of nanosecond). Hence, the time-bandwidth product is generally below 1000. The short pulse duration is not favorable for realizing a long-range pulse compression radar due to the limited pulse energy after power amplification. The methods based on direct-modulated distributed feedback laser diode in [14,15] can generate LCMWs with large time-bandwidth products (tens of thousands) and large durations (generally beyond tens of microseconds, even up to the order of a second). Nevertheless, the frequency sweep linearity is generally poor, which is not beneficial to achieve a high range resolution [16]. Therefore, it is still a hot topic to find a way to generate broadband LCMWs for a practical radar application, which should be simultaneously with a large time-bandwidth product (>10000), a large duration (beyond tens of microseconds), an excellent frequency sweep linearity (<1%), and a good coherence.

In 2018, T. F. Hao et al., demonstrated an approach to generating LCMWs based on a Fourier domain mode locking (FDML) optoelectronic oscillator (OEO) [17]. Different from a conventional tunable OEO, the frequency sweep period in an FDML OEO must be set to be equal to or a faction of the round-trip delay time in the OEO cavity. In such a case, the eigenmodes at different frequencies can oscillate with a continuous phase in a time-division multiplexing mode to form a periodic LCMW. In an FDML OEO, a microwave photonic filter (MPF) with a fast tuning speed is essential to guarantee FDML. To date, several bandpass MPF schemes have been used to realize FDML OEOs [17-19]. In [17], a narrowband MPF is achieved based on phase-modulation-to-intensity-modulation (PM-IM) conversion via a phase-shifted fiber Bragg granting (PS-FBG), where fast tuning is realized by the chirp effect in a laser diode driven by a sawtooth current. In the experiment, an LCMW ranging from 6.25 GHz to 13.75 GHz and with a duration of 22.22 μs has been generated, which corresponds to a time-bandwidth product of 166650. The main problem of this scheme lies in that the frequency sweep linearity is not good due to the nonlinear chirp of the laser diode, which may degrade the range resolution in a pulse compression radar [16]. In addition, the frequency consistency between adjacent waveforms is poor due to the frequency drift between the optical carrier and the notch point of the PS-FBG. In [18], an equivalent MPF is achieved based on optical injection, where fast tuning is realized through varying the injected optical power into the slave laser diode via an intensity modulator driven by a sawtooth current after the master laser diode. Through using this approach, an LCMW ranging from 15.4 GHz to 31 GHz and with a duration of 887.12 ns has been generated, which corresponds to a time-bandwidth product of 13839.1. Nevertheless, the frequency sweep linearity and the frequency stability are still unsatisfactory due to the nonlinear relationship between the output microwave frequency and the injected optical power, and the instability of the Period-one state. In [19], a narrowband MPF is achieved based on PM-IM conversion via stimulated Brillouin scattering (SBS) effect by using two independent laser sources, where fast tuning is realized by varying the probe light wavelength through driving the probe laser diode via a sawtooth current. In this scheme, an LCMW with a time-bandwidth product of 22446 and a chirp rate up to 0.84 GHz/µs is generated. The main drawback of this method is that the relatively large wavelength fluctuation of two separate laser sources will deteriorate the frequency stability of the generated LCMWs. In addition, it is difficult to precisely set the operation frequency range of the generated LCMWs due to the rough tunability of the two laser sources.

In this paper, we propose and experimentally demonstrate an SBS-based FDML OEO with an excellent frequency sweep linearity, where its output frequency range can be precisely set. The MPF in this scheme is realized based on PM-IM conversion via SBS by using a single laser source, where both the probe light and the pump light are generated through electro-optic frequency shifting. FDML is guaranteed by setting the frequency sweep period of the probe light to be equal to the round-trip delay time in the OEO cavity, and the frequency range of the generated LCMWs is accurately tuned by varying the frequency shift of the pump light. In the proof-of-concept experiment, LCMWs with flexibly-tunable center frequency and bandwidth are generated, where the maximum time-bandwidth product and chirp rate are 82000 and 0.195 GHz/µs, respectively. In addition, the generated LCMW has an excellent frequency sweep linearity and a small frequency deviation, where the frequency sweep linearity is smaller than 1.7%, and the frequency deviation is less than 1 MHz.

2. Operation principle

Figure 1 shows the schematic diagram of the proposed SBS-based FDML OEO. In order to maintain the frequency stability of the generated LCMWs, a single laser source is employed to generate the pump light and the probe light. The operation principle is introduced in detail as follows. Continuous-wave (CW) light at ${f_c}$ from a laser diode is divided into two branches by a 3-dB optical coupler. In the upper branch (i.e., the pump light branch), the frequency of the CW light is down-shifted to ${f_c} - {f_{RF}}$ via an electro-optic frequency shifter composed of a dual-parallel Mach-Zehnder modulator (DPMZM) and an electronic 90° hybrid, where the DPMZM works at carrier-suppressed single-sideband modulation (CS-SSB) mode through biasing the parent Mach-Zehnder modulator (MZM) at its quadrature point and the two sub MZMs at their minimum transmission points. After boosting by an erbium-doped fiber amplifier (EDFA), the pump light at ${f_c} - {f_{RF}}$ enters the high nonlinear fiber (HNLF) via an optical circulator, where ${f_{RF}}$ is the frequency of the single-tone radio-frequency (RF) signal applied to the DPMZM (i.e., DPMZM1 in Fig. 1). In the lower branch (i.e., the probe light branch), the CW light propagates through another electro-optic frequency shifter which is also composed of a DPMZM and an electronic 90° hybrid. The frequency of the output CW light from the DPMZM is up-shifted to ${f_c} + {f_{LFM}}$, where ${f_{LFM}}$ is the frequency of the microwave signal applied to the DPMZM (i.e., DPMZM2 in Fig. 1). After boosting by an EDFA and passing through an electro-optic phase modulator (PM), the phase-modulated probe light counter-propagates with the pump light in the HNLF. The phase modulation sideband of the probe light falling into the Brillouin gain spectrum at the vicinity of ${f_c} - {f_{RF}} - {f_B}$ (${f_B}$ is the Brillouin frequency shift in the HNLF) is amplified, which breaks the amplitude balance of the phase-modulated sidebands and achieves frequency-selective PM-IM conversion. After photoelectric conversion in a photodetector, an MPF centered at ${f_{LFM}} + {f_{RF}} + {f_B}$ is realized, which is used to achieve mode selection in the OEO loop.

 

Fig. 1. Schematic diagram of the proposed SBS-based FDML OEO. LD: laser diode; OC: optical coupler; PC: polarization controller; DPMZM: dual-parallel Mach-Zehnder modulator; EDFA: erbium-doped fiber amplifier; PM: phase modulator; ISO: isolator; HNLF: high nonlinear fiber; NZ-DSF: none-zero dispersion-shifted fiber; PD: photodetector; LNA: low noise amplifier; EC: electrical coupler; FDML: Fourier domain mode locking; OEO: optoelectronic oscillator; MPF: microwave photonic filter. ${f_{cn}}$ and ${f_n}$ (n=1,2,⋯) denote the optical carrier frequency of the probe light and the central frequency of the bandpass MPF, respectively.

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In order to build up FDML, a fast tuning MPF is constructed by setting the frequency sweep period of the probe light to be equal to the round-trip delay time in the OEO cavity, which is realized by applying a periodic fast frequency sweep electrical signal to the electro-optic frequency shifter in the probe light branch. Therefore, a specific mode selector will return to the same position in the next scanning period to produce a quasi-stationary oscillation, which guarantees that all the eigenmodes matching with the mode selector in a scanning period can be activated in a time-division multiplexing mode to form an LCMW with a period identical to the round-trip delay time in the OEO cavity. The frequency sweep linearity of the generated LCMWs depends on the sweep linearity of ${f_{LFM}}$, which is excellent since the sweep linearity of the frequency sweep electrical signal applied to DPMZM2 in Fig. 2 can be guaranteed by a commercially-available electronic device. In addition, the frequency range of the generated LCMW can be precisely tuned by varying the frequency of the single-tone RF signal (${f_{RF}}$) applied to DPMZM1 in Fig. 1.

 

Fig. 2. Open-loop response of the SBS-based OEO by varying (a) ${f_{RF}}$ from 1.000 GHz to 3.000 GHz and (b) ${f_{LFM}}$ from 4.000 GHz to 8.000 GHz.

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3. Experimental results and discussion

A proof-of-concept experiment is carried out to demonstrate the proposed scheme. In the experiment, a narrow-linewidth CW light at 1550.04 nm and with a power of 9.7 dBm is generated by a tunable laser source (TeraXion PS-TNL), which is then divided into two branches by a 3-dB optical coupler. Two properly-biased 40-Gb/s DPMZMs (COVEGA Mach40086) with assistance of two electronic 90° hybrids (ABACUS MICROWAVE 9-010180, 1–18 GHz) are used to construct the electro-optic frequency shifters in the two branches. In the probe light branch, a periodic linear frequency-modulated signal with a fixed center frequency of 6 GHz and a software-definable bandwidth from a home-made DDS is applied to the electro-optic frequency shifter to obtain a periodic frequency-scanning optical carrier. After boosting by an EDFA (Amonics AEDFA-IL-23-B-FA), the frequency-scanning optical carrier passes through a 40-Gb/s electro-optic PM (Thorlabs PM-1550-40-PFA-PFA), and then enters a spool of HNLF (OFS HNLF Standard) with a length of 1 km and a Brillouin frequency shift of 9.644 GHz. In the pump light branch, a tunable single-tone RF signal generated by a microwave source (R&S SMB100A, 100 kHz-12.75 GHz) is applied to the electro-optic frequency shifter to generate the frequency-controllable pump light. The power of the pump light launched into the HNLF is adjusted by an EDFA (Amonics AEDFA-IL-23-B-FA). The phase-modulated probe light after frequency-selective PM-IM conversion propagates through a spool of none-zero dispersion-shifted fiber (NZ-DSF) with a length of 3 km, and achieves photoelectric conversion in a high-speed photodetector (Discovery Semiconductors DSC-10H) with a bandwidth larger than 40 GHz and a responsivity of 0.6 A/W. The signal loss in the OEO loop is compensated by two cascaded electronic amplifiers, where the first one (Qotana) has a power gain of 27 dB in the frequency range of 2 GHz to 20 GHz, and the second one (Mini-Circuits ZX60-183-S+) has a power gain of 10 dB in the frequency range of 5 GHz to 18 GHz. A 6-dB resistance power divider (GTPD-COMB50G, 0-50 GHz) is employed to close the OEO loop, where one port is used to output the generated LCMWs, and the other port is connected to the RF port of the electro-optic PM to maintain the oscillation in the OEO loop. In the experiment, the spectra and the waveforms of the generated LCMWs are measured by an electrical spectrum analyzer (ESA, R&S FSU50, 20 Hz-50 GHz) and a real-time high-speed oscilloscope (OSC, Tektronix DPO75002SX, 100 GS/s, 33 GHz), respectively.

The open-loop response of the OEO is measured by a vector network analyzer (VNA, Agilent N5235A) to examine the frequency sweep accuracy. Figure 2(a) shows the measured open-loop response when ${f_{LFM}}$ is set to be 6.000 GHz, and ${f_{RF}}$ is tuned from 1.000 GHz to 3.000 GHz with a step of 0.500 GHz. The measured center frequency of the MPF varies from 16.644 GHz to 18.644 GHz, which fits in with the theoretical value. Figure 2(b) presents the measured open-loop response when ${f_{RF}}$ is set to be 0.000 GHz, and ${f_{LFM}}$ is tuned from 4.000 GHz to 8.000 GHz with a step of 1.000 GHz. The measured center frequency of the MPF varies from 13.644 GHz to 17.644 GHz, which also fits in with the theoretical value. Therefore, it can be seen from the measurement results that the frequency deviation of the open-loop frequency response in the proposed FDML OEO is smaller than 1 MHz.

The performance of the FDML OEO is tested by closing the loop and applying a linear frequency-modulated signal with a period of 20.5 µs to DPMZM2. The period of the applied linear frequency-modulated signal is equal to the round-trip delay time of the OEO cavity, which is calculated from the measured free spectral range (i.e., 48.78 kHz) of the OEO under single-frequency oscillation condition. Figure 3 exhibits the measured spectrum of the generated LCMW when ${f_{RF}}$ is set to be 0 GHz and ${f_{LFM}}$ is sweeping from 5 GHz to 7 GHz. It can be seen from Fig. 3 that the expected LCMW with a center frequency of 15.644 GHz and a bandwidth of 2 GHz is generated in the FDML OEO. Nevertheless, some spurious signals are also generated, which are attributed to the nonideal CS-SSB modulation and the limited isolation of the optical circulator in the experiment. The broadband spurious signals in the low frequency range (i.e., 5-7 GHz) are generated by the beating between the residual carrier and the 1st-order modulation sidebands from DPMZM2 in Fig. 1, which are then amplified in the OEO cavity. The most effective way to suppress these spurious components is increasing the carrier-suppression ratio in the electro-optic frequency shifting. In general, the carrier-suppression ratio can be larger than 30 dB in a commercial LiNbO₃-based DPMZM through finely controlling the bias voltages. Therefore, these spurious signals can be greatly suppressed compared with those in Fig. 3 if bias optimization is implemented. The spurious signals around 9.644 GHz are generated by the beating between the backward spontaneous Brillouin scattering of the pump light in the HNLF and the leaked pump light from port one to port three of the optical circulator, which are also amplified in the OEO cavity. These spurious signals can be suppressed by using an optical circulator with a high isolation, and properly decreasing either the power of the pump light or the HNLF length. It can also be seen in Fig. 3 that the signal-to-noise ratio (SNR) of the generated LCMW is not very good, which is mainly attributed to the poor SNR of the linear frequency-modulated signal from the home-made DDS (lower than 40 dB). There are two methods to improve the SNR of the generated LCMW. The first one is using a frequency-swept microwave source with a better SNR to replace the home-made DDS. The second method is employing optical bandpass filters after EDFAs in Fig. 1 to filter out the out-of-band spontaneous emission noise. Figure 4(a) shows the spectra of the generated LCMWs when ${f_{RF}}$ is set to be 0 GHz, and the bandwidth of the applied linear frequency-modulated signal is tuned from 1 GHz to 4 GHz with a step of 1 GHz. Figure 4(b) presents the spectra of the generated LCMWs when the bandwidth of the applied linear frequency-modulated signal is set to be 2 GHz, and ${f_{RF}}$ is tuned from 1 GHz to 2.5 GHz with a step of 0.5 GHz. It can be seen from Fig. 4 that a linear frequency-modulated signal with a low center frequency (6 GHz in the experiment) is up-converted to an LCMW in the high frequency range by using the proposed FDML OEO scheme, where the center frequency shift is equal to ${f_B}\textrm{ + }{f_{RF}}$. In addition, the frequency sweep range of the generated LCMW is software definable, where the frequency deviation depends on the stability of${f_B}$. Two main factors influence${f_B}$ in the experiment setup – the variation of the pump light wavelength and the ambient temperature of the HNLF. The relationship between ${f_B}$ and the pump wavelength ${\lambda _p}$ is${f_B} = {{2n{v_A}} \mathord{\left/ {\vphantom {{2n{v_A}} {{\lambda_p}}}} \right.} {{\lambda _p}}}$, where n and ${v_A}$ are the effective refractive index of the pump light and the speed of the acoustic wave in the HNLF, respectively. For the pump light with a wavelength at the vicinity of 1550.04 nm, the maximum variation of ${f_B}$ is below 3 MHz even if the electro-optic frequency shift (${f_{RF}}$) reaches 40 GHz. In the experiment, ${f_{RF}}$ is smaller than 4 GHz. Therefore, the frequency deviation induced by the variation of the pump light wavelength is below 0.3 MHz. Secondly, the temperature sensitivity of ${f_B}$ in a silica fiber is generally in the order of 1 MHz/^○C (e.g., 1.06 MHz/^○C in a G.652 fiber). Hence, the temperature-induced frequency deviation can be smaller than 1 MHz through maintaining the ambient temperature fluctuation of the HNLF below ±0.5 ^○C. Besides, frequency deviation may also be introduced by the mode hopping effect. The intrinsic Brillouin gain spectrum has a Lorentz shape, and it can be seen from Fig. 2 that the 3-dB bandwidth of the MPF passband in the OEO loop is smaller than 34 MHz. Therefore, mode hopping only happens among the several modes near the MPF passband center since these modes dominate in the mode competition process. This may introduce a frequency drift below hundreds of kHz (the mode spacing is ∼49 kHz in the OEO).

 

Fig. 3. Measured spectrum of the generated LCMW with a center frequency of 15.644 GHz and a bandwidth of 2 GHz.

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Fig. 4. Superimposed spectra of the generated LCMWs with different (a) bandwidth and (b) center frequency.

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Figure 5(a) exhibits the temporal waveform of the generated LCMW with a bandwidth of 4 GHz and a center frequency of 16.644 GHz. Figure 5(b) presents its instantaneous frequency distribution obtained by short-time Fourier transform. It can be seen from Fig. 5(b) that the generated LCMW has a duration time of 20.5 µs and a bandwidth of 4 GHz, which corresponds to a time-bandwidth product of 82000 and a chirp rate of 0.195 GHz/µs. In Fig. 5(a), it can be seen that there is an amplitude variation in each waveform period, which is mainly attributed to the limited operation bandwidth of the electronic amplifier (Mini-Circuits ZX60-183-S+, 5-18 GHz) used in the experiment. Figure 6 presents the measured frequency response of this electronic amplifier by using a VNA. It can be seen that the gain of this electronic amplifier severely decreases beyond 17 GHz. Therefore, the amplitude of the generated LCMW decreases as the frequency increases as shown in Figs. 5(a) and 5(b). This amplitude variation can be effectively suppressed by employing devices (i.e., electronic amplifiers, PM, PD) with larger operation bandwidths or adding an RF equalizer after the electronic amplifiers in the OEO cavity. The frequency sweep linearity ($\delta$) of the generated LCMW can be calculated according to the definition in [20] as

 

Fig. 5. Generated LCMW with a bandwidth of 4 GHz and a center frequency of 16.644 GHz. (a) Measured temporal waveform, (b) instantaneous frequency distribution and (c) the compression result by autocorrelation.

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Fig. 6. Measured frequency response of the electronic amplifier (Mini-Circuits ZX60-183-S+).

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Fig. 7. Cross-correlation result of the generated LCMW.

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The prominent advantage of an OEO is that it can generate a microwave signal with an ultra-low phase noise [21,22]. Since direct measurement of a LCMW is difficult, the phase noise of a single-tone microwave signal from the proposed SBS-based OEO is measured as an alternative. Figure 8 shows the measured phase noise of the generated single-tone microwave signal at 15.644 GHz by using a phase noise analyzer (R&S FSWP50), where a single-tone signal at 6 GHz from the DDS is applied to DPMZM2, and ${f_{RF}}$ is set to be 0 GHz. In Fig. 8, the phase noise of the single-tone signal at 6 GHz from the DDS is also presented. It should be pointed out that the phase noise of the single-tone signal at 6 GHz is not good since a crystal oscillator with a poor phase noise is employed in the home-made DDS. It can be seen from Fig. 8 that the phase noise of the generated single-tone microwave signal at 15.644 GHz is measured to be -113.98 dBc/Hz@10kHz, which is 35 dB lower than the value of -78.29 dBc/Hz@10kHz for the single-tone signal at 6 GHz from the DDS. This phase noise improvement characteristic is favorable when the DDS is replaced by any fast frequency sweep microwave source with a broader sweep range to enhance the bandwidth of the generated LCMW, even though the employed microwave source is with a bad phase noise (e.g., a VCO driven by a properly-designed sawtooth voltage, or a microwave source through frequency multiplication of a frequency sweep signal from either a DDS or a VCO). In fact, LCMW generation by an FDML OEO is a quasi-stationary process. Compared with the stationary single-tone signal generation process in the OEO, the phase noise of the generated LCMW may degrade due to the dynamic variation of the OEO loop. For example, the near end phase noise may deteriorate due to the passband center deviation from period to period, and the round-trip delay deviation in the scanning process brought by the weak optical nonlinear effect induced by the fast-scanning probe light.

 

Fig. 8. Phase noise of the generated single-tone microwave signal from the OEO and the single-tone signal from the DDS.

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4. Conclusion

In summary, we have proposed and experimentally demonstrated an SBS-based FDML OEO for LCMW generation with an excellent frequency sweep linearity, where the frequency sweep range is precisely controllable. The frequency-definable SBS-based bandpass MPF, which is the key component in the proposed scheme, is realized by PM-IM conversion through employing a single laser source to generate both the pump light and the probe light via electro-optic frequency shift. FDML is guaranteed by generating a frequency-swept probe light with its period equal to the round-trip time of the OEO cavity. The bandwidth of the generated LCMW is equal to the bandwidth of the electrical linear frequency-modulated signal used to achieve electro-optic frequency shift of the probe light. The center frequency of the generated LCMW can be subtly tuned through varying the frequency of the single-tone RF signal used to achieve electro-optic frequency shift of the pump light. In the proof-of-concept experiment, frequency-definable LCMWs with bandwidths from 1 GHz to 4 GHz and center frequencies from 15.644 GHz to 17.144 GHz were generated, where the frequency deviation was lower than 1 MHz. Besides, an LCMW with a maximal time-bandwidth product of 82000 and a chirp rate of 0.195 GHz/µs was generated, whose frequency sweep linearity was smaller than 1.7%. The pulse compression ratio of this LCMW through autocorrelation and cross-correlation calculation was 78244 and 65495, respectively, which indicated that the generated LCMW had a good coherence. Hence, the proposed SBS-based FDML OEO is a potential solution to generate a frequency-definable LCMW in various applications such as pulse compression radar system.