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Abstract
We report on an enhanced photonic generation of frequency-modulated continuous-wave (FMCW) signals by injection-locking a semiconductor laser operating in period-one (P1) nonlinear dynamic with an intensity modulated electro-optic frequency comb. When the cavity mode is injection-locked with respect to any of the comb modes, through linearly sweeping the frequency of the injected comb mode while synchronously modulating the injected intensity, the center wavelength of the cavity mode can be tuned following the injected comb mode. This way, it allows maintaining the phase-locking between the cavity mode and comb mode even if beyond the original locking bandwidth of the cavity mode, since it is tuned accordingly. It thus leads to the generation of FMCW signal with efficient phase noise suppression and improved achievable sweep range compared with the limited original injection-locking bandwidth. Such injection enhanced phase-locking is investigated and a demonstration with the injection of −4th order comb mode has realized photonic FMCW generation with enhanced sweep range and suppressed phase noise. Thanks to the flexibility in sweep parameters, this method can also be readily applied for the generation of arbitrary waveforms.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Frequency-modulated continuous-wave (FMCW) technique, based on the measurement using linearly frequency swept (LFS) waveforms, has been found appealing in various applications such as displacement measurement [1], radar [2], imaging [3], and sensing [4]. Conceptually, broadband frequency sweep with high linearity, tunable center frequency, flexible sweep rate and large time-bandwidth product is essentially demanded to achieve a long measurement range, high spatial resolution and high precision. Conventional electronic approaches, due to the so-called “electronic bottleneck”, has encountered limitations and difficulties in the pursuit of high quality FMCW signals in terms of large bandwidth and high center frequency. Thanks to the ultra-short operation wavelength, photonic approaches that exhibits natural advantages such as large bandwidth, high speed, and electromagnetic interference immunity, have attracted a wide range of research interests in recent decades.
Amongst a variety of reported methods, one common way is to constitute a heterodyne beating between two optical carriers with different frequency sweeping features, where the sweeps can be produced by such as direct wavelength tuning of the lasers [5,6], Fourier domain mode-locking (FDML) [7–10], and external modulation [11]. In addition, space- or frequency-to-time mapping relying on temporal or spectral optical pulse shaping [12–15] that allows yielding LFS waveforms with limited flexibility in bandwidth and center frequency are also widely studied. In last few years, under proper optical injection, nonlinear dynamics of semiconductor laser (SCL) has been put forward for the generation of FMCW signals [7–9,16,17] with, however, inevitably large phase noise and poor stability that not only deteriorates seriously the signal quality but also hinders further applications [17]. To deal with this issue, different kinds of stabilization methods have been introduced. By applying feedback loops [7–9], either optical or electrical, it leads to the suppression of the phase noise while the sweep period is, nevertheless, required to precisely match the delay of feedback loop, bringing about inconveniences in the flexibility of the sweep period. Simultaneous stabilization and phase noise suppression can be achieved by injection-locking (IL) the nonlinear dynamics of SCL with respect to an optical frequency comb [17–20]. The resulting phase-locked cavity mode exhibits an improved coherence for the yielded FMCW signal in addition to the flexibility in sweep properties. Nonetheless, the achieved sweep range is strictly limited due to the intrinsic locking bandwidth of the cavity mode.
To this end, in this paper, we propose an enhanced photonic generation of FMCW signals by IL a semiconductor laser operating in period-one (P1) nonlinear dynamic with an intensity modulated frequency comb. The phase-locking between the cavity mode and the injected comb mode is maintained by synchronously modulating the intensity of the injected comb with respect to the frequency sweep of the injection-locked comb mode. It thus allows generating FMCW signals within a broader frequency range without limited by the original locking bandwidth in addition to the efficient phase noise suppression. Characterizations and demonstrations are both carried out experimentally, testifying the performance of the approach including the flexibility in center frequency, sweep range, time-bandwidth product, and sweep fidelity, as well as the discussion regarding the potential in arbitrary waveform generation.
2. Concept and experimental setup
In conventional optical injection schemes where usually a single carrier with a detuning of fd from a free-running SCL (usually denoted as slave laser, SL) acts as the master laser (ML), due to the undamped relaxation resonance, the emission from the SCL under P1 nonlinear dynamics [16,21] consists of a regeneration mode, a cavity mode and several harmonics uniformly spaced on both sides of the formers. The beat note frequency f1 that coincides with the separation between the adjacent modes as illustrated in Fig. 1(a-ii), is usually positively correlated to the injection ratio (IR) between the power of injected carrier (Pinj) and SCL (P0) in most circumstances, i.e. the cavity mode red shifts when IR increases. Due to the intrinsic noise of the ML, the linewidth of the cavity mode is usually deteriorated compared to that in free-running case. Accordingly, similar behaviors for the cavity mode can be found when injected with an optical frequency comb consisting of a series of evenly spaced modes. This way, by varying the IR, if the cavity mode could be pushed into the locking range of any of the other injected comb modes, it will be captured as IL between this comb mode to the pushed cavity mode as depicted in Fig. 1(b-ii), hence giving rise to the emergence of the injection-locked cavity mode according to the mode competition during the injection process. The modes from the injection-locked SL are phase-correlated thanks to the mutual coherence of the comb, leading to the phase-locking for the P1 dynamics. This way, by tuning the mode spacing of the injected comb, the phase-locked cavity mode will also be tuned in the same manner while preserving the phase coherence with respect to the injected comb and thus the regeneration mode as well. The tuning range is, nevertheless, strictly limited within the IL bandwidth of the cavity mode.
Fig. 1. The schematic diagram for optical injection under different conditions. (a-i) The laser injection scheme when the SL is injected with a single optical carrier. (a-ii) The output from SL when the P1 dynamic status is activated. (b-i) The laser injection scheme when the SL is injected with optical comb modes. (b-ii) The output from SL when the cavity mode is injection-locked by one of the sidebands. (c) The locking bandwidth extension effect when the IR is tuned.
To further deal with such constrain, the intensity-frequency conversion that allows tuning the cavity mode thus its associated locking bandwidth will be exploited here. By synchronically changing the IR during the frequency sweep of the mode-spacing, it allows to maintain a stable phase-locked state for the cavity mode even when beyond the initial locking bandwidth as sketched in Fig. 1(c). This is achieved because the cavity mode is shifted along with the tuning of the IR, thus the center frequency of the associated locking bandwidth accordingly. In this manner, such injection-locked comb mode remains within the locking bandwidth of the cavity mode even it is swept over a much broader frequency range. To take full advantage of this property, by IL the shifted cavity mode with a high-order mode from an electro-optic frequency comb (EOFC) [22], a low phase noise microwave signal with a high center frequency and broad tuning range can be attained from such phase-locked P1 dynamic state, thus enabling the generation of low phase noise FMCW signals with a flexible center frequency, a large sweep range, and high fidelity by driving the comb with a linearly swept RF signal.
As a matter of fact, the underlying IL laser dynamics could be intuitively described by an extension of the dual-beam injection [23,24] in particular when the power of the injected comb modes is significantly lower than that of the injected carrier so that the carrier nevertheless induces a much stronger impact. With moderate modifications taking into account all the comb modes, a similar analysis that would allow for an insight of the complicated nonlinear dynamics of such bandwidth enhancement, which will be further investigated in our future study.
The demonstrative experimental setup for the proposed system is schematically shown in Fig. 2, where a low phase noise fiber laser with a few kHz nominal linewidth is used as the ML. A commercial distributed feedback (DFB) SCL with output power P0 = ∼6.6 dBm when biased above the threshold plays the role of the SL, whose linewidth is measured to be ∼400 kHz. The ML output is modulated by two intensity modulators (IM) in tandem. The injection power Pinj that determines the IR is controlled through tuning the DC-bias of the former with a high precision programmable power supply while the latter is adopted to yield an EOFC with a linearly frequency swept RF driving signal fm from an analog signal generator (ASG, Keysight) with a proper modulation bias. Then this comb is injected through a circulator to stimulate the nonlinear states in the SCL, from which the output is simultaneously analyzed by an optical spectra analyzer (OSA, Apex) and an oscilloscope (OSC, Agilent) after being detected using a high-speed photodetector (PD).
Fig. 2. The schematic of the proposed system. ML: master laser; IM: intensity modulator; DC: programmable DC power supply; ASG: analog signal generator; CIR: circulator; DFB-SCL: distributed feedback semiconductor laser; FC: fiber coupler; OSA: optical spectra analyzer; PD: photodetector; AMP: amplifier; OSC: oscilloscope.
The effect of frequency tuning in comb injection is firstly verified, where the detuning fd between the ML (1550.024 nm) and SL (1550.081 nm) is set ∼7 GHz. The RF power is adjusted to generate a pair of ${\pm} $ 1st order modes with a power ratio of r = ∼−36 dB with respect to the carrier. The total injection power is tuned to be −20, −10, −5, and 0 dBm, respectively, while the driving frequency fm is correspondingly adjusted to attain IL for the −1st order comb mode to the cavity mode. It can be observed that compared to the case when only the carrier is injected, for any of the injection power, the optical spectrum of the cavity mode has been sharpened after the IL of the comb mode (see Fig. 3, left column), inferring a clear coherence enhancement for the cavity mode. This is also confirmed by their corresponding RF beat note spectra calculated using a 1-µs-long acquisition data as depicted in Fig. 3. As a matter of fact, when without IL, the RF beat note exhibits a poor stability manifested as obvious spectral drifts that hinders the assessment of the noise property. For a quantitative comparison, the full width at half maximum (FWHM) linewidth is extracted with a Lorentzian fitting [25]. The results that range from several MHz to a few tens of MHz agree well with previously reported work [18], implying a rather poor stability in this case. Meanwhile, the same assessment in phase-locked state presents a resolution bandwidth (RBW) limited FWHM linewidth of 15 Hz which is only limited by the RBW of the spectrum analyzer in our lab, verifying a significant phase noise suppression in this state. It should be noted that in both cases the varied frequency shifts at different injection powers is attributed to the IR induced cavity mode shift. This way, the linewidth reduction effect can be sustained at different nonlinear dynamics output frequencies through adopting proper IR and RF driving signal.
Fig. 3. (a-i), (b-i), (c-i), (d-i) are the optical spectra and (a-ii), (b-ii), (c-ii), (d-ii) are the calculated beat note spectra of the SL output when the injection power is at −20, −10, −5 and 0 dBm, respectively. The blue lines represent the circumstances of single carrier injection, and the red lines represent those of comb injection.
3. Feature of comb assisted injection-locking
The variation for the locking characteristics at different IRs (precisely defined as Pinj/P0, where Pinj now refers to as the total injection power of the EOFC, which is different from the definition in case of single carrier injection) are plotted in Fig. 4(a) when r remains at ∼−36 dB. Here, we regard the situation in which the carrier to noise ratio (CNR) of the beat signal is no less than 30 dB as an effective “phase-locked state”, corresponding to efficient noise suppression, while the “locking bandwidth” refers to the achievable frequency ranges of the beat signal under the aforementioned condition. The black dots represent the beat note frequency f1 between the regeneration mode and cavity mode in case of single carrier injection. While the colored regions hold for the corresponding locking bandwidth. A zoom-in view for the absolute locking bandwidth with respect to f1 is additionally shown in Fig. 4(b). Above all, it is directly seen that when the IR is within the range from about −23.6 to −17.6 dB, the SL would operate in chaos state, in which IL thus phase locking is not able to be achieved.
Fig. 4. When fd = 7 GHz, (a) the locking bandwidth under different IRs when injection-locked by the −1st order harmonic with r = ∼−36 dB; (b) the locking bandwidth with respect to f1; (c) the locking bandwidth of the cavity mode under different power ratio r when the IR is set to −6.6 dB.
In principle, for conventional single carrier IL, the locking bandwidth is strongly dependent on not only the IR but also the spectral width of SL [26], where a larger linewidth and a higher IR will lead to a wider locking bandwidth. When operating in nonlinear dynamic state, the injection induced linewidth degradation of the cavity mode will further facilitate the broadening of locking bandwidth. Similar phenomena can also be found in case of comb injection, which indeed contributes a lot in the enlarged locking bandwidth at higher IRs as can be directly verified in Fig. 4(b), in particular thanks to the linewidth deterioration of the cavity mode attributed to all the injected comb modes. Such influence due to the injection of the comb mode is further studied in terms of the power ratio r between the comb mode and the main carrier as shown in Fig. 4(c). While the IR is kept at −6.6 dB, when r is tuned from −39.5 to −22.5 dB by finely changing the driving RF power, the locking bandwidth is consequently extended by almost 9 times from 0.42 to 3.72 GHz. Hence, resulting from the combination of these effects, a significant extension in the achievable locking bandwidth from 1.65 GHz (marked in Fig. 4(b)) to 13.29 GHz can be directly verified by the total excursion range (orange regions) as indicated in Fig. 4(a).
Based on above analysis, continuous frequency sweep of the RF beat signal within a broader frequency range with a maintained IL for the injected comb mode to cavity mode can be achieved by synchronously tuning the IR in line with the frequency sweep of the IL comb mode. This allows the injected comb mode stays well within the locking bandwidth of the cavity mode (colored regions in Fig. 4(a)) during the sweep because the center wavelength of such locking bandwidth is tuned accordingly and synchronously. In addition, by employing high-order mode yielded from an EOFC, the difficulties in the electrical generation of high frequency and large sweep range can be effectively dealt with thanks to the multiplication nature of such comb. This way, it therefore, leads to IL assisted phase-locked frequency sweep with high center frequency, broad sweep range, versatile sweep waveform and variable sweep rate, as well as flexibility in these properties that can be readily enabled by proper adjustment of the driving RF signal.
With a similar experimental setup but for the IL of high-order modes, the optical spectra for the injected comb and free-running SL with detuning fd = 38.5 GHz are shown in Fig. 5(a). Through carefully tuning the DC-bias and the driving RF signal, an EOFC with 10 GHz mode-space is injected, whose −4th order mode is adjusted to be ∼55 dB lower than the carrier, namely r = −55 dB. The IL assisted phase-locking is manifested as the obvious linewidth compression in addition to the more than 45 dB CNR for the cavity mode as indicated by Fig. 5(b).
Fig. 5. Optical spectra for (a) the injected comb (red) and free-running SL (blue) with 38.5 GHz detuning; (b) the IL assisted phase-locked SL in P1 dynamic status (blue) and the injection-locked comb (red), where a zoom-in view for the phase-locked cavity mode is shown in the inset. (c) Phase-locked bandwidth under different IRs when the cavity mode is locked by the −4th order comb mode.
Besides the 30 dB constraint in the CNR, another condition is added to the well-defined phase-locked state that the power of the cavity mode should be no less than −3 dB compared with the maximum power that can be achieved in the phase-locked case under the same IR, which may occur when the IR is relatively high. Though may limits the maximum achievable sweep range, it helps guaranteeing the potential power fluctuation during the frequency sweeps. It is worth mentioning again that the injection power is the overall power of all the combs.
It is worth noting that in the generation of FMCW signal, the actual sweep range is dictated by the total excursion range of the cavity mode as verified above and by the sweep range of the comb mode. Concerning the limited bandwidth for the swept RF driving signal generated by the arbitrary waveform generator (AWG) in our lab, so far, a high-order, precisely the −4th order comb mode is chosen in this demonstration. As a result, when the mode-spacing sweeps from 9.99 to 11.7 GHz, a stable phase-locked state can be sustained by correspondingly tuning the IR from −5 to 1.6 dB. With the IL of the −4th comb mode, it enables the generation of a swept microwave signal from 39.96 to 46.8 GHz, verifying a clear enhancement compared with the original locking bandwidth limited sweep range. The sweep parameters can be readily adjusted by changing the experimental settings.
4. FMCW generation
4.1 Experimental setup
The experimental setup for the FMCW generation is depicted in Fig. 6(a). To realize frequency sweep while maintaining the phase-locking, the signals applied to IM-1 and IM-2 are precisely synchronized by using an arbitrary waveform generator (AWG) with the same time length T, where one channel generates the sawtooth signal to control the bias of the IM-1 and the other yields the swept RF signal with a sweep rate of γ to drive the EOFC through IM-2. The RF signal is linearly swept from 10 to 11.5 GHz within a duration of T = 1 ms, and the cavity mode is phase-locked by IL to the −4th order comb mode of the obtained EOFC.
Fig. 6. Schematic of the experimental setup. ML: master laser; IM: intensity modulator; AWG: arbitrary waveform generator; AMP: amplifier; CIR: circulator; DFB-SL: distributed feedback semiconductor laser; FC: fiber coupler; OSA: optical spectrum analyzer; MPOF: multi-passband optical filter; AOFS: acousto-optic frequency shifter; SG: signal generator; BPD: balanced photodetector; DIV: power divider; MIX: mixer; BPF: bandpass filter.
In our demonstration, the phase-locked P1 dynamic oscillation reaches over 40 GHz, which is way beyond the bandwidth of the photodetector in our lab. For a feasible evaluation, a simple alternative based on optical delayed self-heterodyne followed by self-mixing [11] is utilized as depicted in Fig. 6(b). After being filtered by the multi-passband optical filter (MPOF), the regeneration mode and phase-locked cavity mode, which is equivalent to an optically-carried FMCW signal, is guided to an unbalanced Mach-Zehnder interferometer (UMZI) in which one arm is fiber-delayed by τ and an AOFS is inserted in the other arm to shift the frequency by fIF. The coupled optical signals are combined and received by a balanced photodetector (BPD). The detected signal is divided, self-mixed, and band-pass filtered before being captured by the oscilloscope.
Given that fm is linearly swept with a slope γ, the electric fields of the regeneration mode Er and the phase-locked cavity mode Es can be respectively expressed as
Worth noting that the phase noises could be somewhat cancelled out owing to the high mutual coherence inherited from the comb. This way, the output of mixer can be then written as
It is seen that the phase terms in Eq. (4) contains the information regarding the sweep slope. Provided the delay of the UMZI is known and remained stable while the mutual coherence built by the phase-locked IL, a linear approximation can be applied without loss of information, leaving only the slope related phase term. Therefore, with the help of Hilbert transform, the instantaneous frequency can be extracted regardless of the actual frequency or bandwidth of the generated microwave beat signal, allowing for the direct assessment for the sweep linearity in terms of the frequency error with respect to an ideal linear sweep.
4.2 Experimental results
The frequency shift provided by the AOFS is fIF = −40 MHz while the UMZI delay is set τ = 240 ns. The spectrum of the BPD output recorded by a real-time spectra analyzer (RSA) is shown in Fig. 7(a). The peaks correspond to the phase-locked cavity mode and the regeneration mode can be found in the beat notes spectrum, respectively, at ∼38.567 MHz and 40 MHz.
Fig. 7. (a) Measured electrical spectrum of the BPD output signals. (b) The output temporal waveform. (c) Relative instantaneous frequency of the generated linearly swept signal (blue) and the frequency error (orange).
The captured temporal waveform for a single period is recorded in Fig. 7(b). According to Eq. (4), after employing Hilbert transforming, the instantaneous frequency of the generated FMCW signal can be directly extracted. The chirp rate of the generated FMCW signal is supposed to be 4 times of that of the RF drive signal for the EOFC, with which the instantaneous frequency can be figured out as described in Fig. 7(c). We also adopt a linear fit to extract the frequency error which is well below 120 kHz, implying a high fidelity in addition to the stability and repeatability. The residual nonlinearity probably stems from the inherent error of the RF drive signal and the nonlinear response of components in the system.
This leads to a clear extension for the sweep range compared to the original IL bandwidth, achieving a large time-bandwidth product up to ∼$6 \times {10^6}$. In addition, by simply changing the swept RF driving signal and the intensity modulation synchronously, it is feasible to change the sweep parameters even including the sweep waveform, testifying the flexibility of the system.
5. Conclusion
In conclusion, by exploiting the intensity-frequency conversion in P1 nonlinear dynamics under proper optical injection, we have proposed and demonstrated photonic generation of FMCW signals with enhanced sweep range exceeding the limit induced by the original IL bandwidth as well as the flexibility in sweep parameters and time-bandwidth product. This is achieved by synchronously modulating the intensity of the injected EOFC while whose comb mode-spacing is linearly swept through driving with a linearly swept RF signal. This way, the phase-locking between the cavity mode and the injected comb mode can be maintained even when beyond the original locking bandwidth since the cavity mode is tuned following the injected comb mode, leading the generation of FMCW signal with low phase noise and flexible sweep parameters in addition to the enlarged sweep range. In our demonstrative verification, the actual achievable locking bandwidth has been extended from 1.65 to 13.29 GHz while the linewidth of the output RF signal is significantly suppressed to less than 15 Hz thanks to the highly mutual coherence established between the regeneration mode and the phase-locked cavity mode. The above features allow for photonic generation of FMCW signals with low phase noise and improved sweep range compared with the original limited locking bandwidth. As a result, FMCW signal of 6 GHz sweep range and 1 ms duration has been realized with RMS frequency error below 120 kHz and ∼$6 \times {10^6}$ time-bandwidth product at a high center frequency over 40 GHz by IL of the −4th order comb mode of the EOFC. And the realized sweep range is actually limited by the RF driving signal but not the proposed method. The flexibility in the sweep parameters has been discussed and verified, testifying the potential for arbitrary waveform generation.
Funding
National Natural Science Foundation of China (61827807, 61805014); National Key Research and Development Program of China (2019YFB2204004).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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