1. Introduction

Highly coherent monochromatic continuous-wave (CW) lasers capable of broadband phase-continuous linear frequency sweep (LFS) while exhibiting low phase noise, high spectral purity, precision, and stability are key elements in a variety of applications. Often regarded as frequency-agile or swept-frequency lasers, they have played indispensable roles in many scenarios such as atomic physics [1], optical spectrum analyzer [2], spectroscopy [3], optical frequency-modulated continuous-wave (OFMCW) interferometry [4,5], distributed fiber-optic sensing [6], and optical coherence tomography [7], where highly coherent broadband optical LFS is essential since the performance is largely harnessed by the sweep properties. Particularly from the viewpoint of OFMCW interferometry, the important metrics including the spatial resolution and range window are limited by the sweep linearity, bandwidth, as well as the dynamic coherence i.e., the phase noise. Intrinsically, lasers capable of broadband sweep, such as intracavity modulated semiconductor lasers (SCLs), since being loosely confined usually exhibit poor dynamic coherence and sweep linearity manifested as large phase noises with limited reproducibility. On the opposite, the sweeping bandwidth would be compromised for those of high coherence and narrow linewidth, e.g. lasers with longer cavities such as fiber lasers.

Efforts have long been made to fill this gap. Concerning narrow linewidth laser, external frequency actuation using different types of modulations allows generating LFS at the sidebands. Though the coherence features can be directly inherited, the sweep range is, nevertheless, strictly limited according to the bandwidth of the modulators and driving electronics [8].

On top of this, one popular approach relies on sweep linearization and coherence enhancement for direct modulated lasers. To fully exploit the capability of broadband sweeping while to cope with the phase noise and nonlinearities, a variety of techniques have been studied. By actively referenced against an optical etalon or locked to an unbalanced Mach-Zehnder interferometer (UMZI) [9] using servo control or optical phase-locked loop (OPLL) [10], both rectified LFS as well as coherence enhancement can be simultaneously achieved [11] with versatility for either frequency fixed or swept operations. However, inevitable trade-offs between loop bandwidth and discrimination gain has limited the locking thus the sweep performance. Another important technology explores well-established optical frequency combs [12] as precise frequency grids. This enables nonlinear calibration for swept CW lasers at fixed intervals [3,13], however, without acting on the coherence of the CW lasers. To reduce the phase noise, phase-locking the beat note between the CW laser and individual comb mode allows for active referencing. In general, this is often regarded as the single-frequency optical synthesizer [14,15]. Nevertheless, though attentions have been paid [16–18], the confounded phase still makes it problematic to preserve the phase-continuity when swithcing the locking point over adjacent modes, thus hindering the realization of broadband LFS [19]. Recently, serrodyne frequency shifting [20] has been proposed [21] with applicable potential for optical LFS generation. Its sweeping ability relying on external comb-carrier-frequency shift [22] has been verified for broadband spectroscopy [23].

To deal with these challenges and to avoid the phase ambiguities in comb-based LFS generations, in this work, we report broadband optical LFS generation using multi-loop composite OPLL and a specially designed mode-spacing swept agile opto-electronic frequency comb (AFC) whose mode-spacing is capable of frequency sweep with arbitrary waveform in a phase-continuous manner. This way, the high-order mode of the AFC acts as a swept reference with a multiplied sweep range and sweep rate and high mutual coherence in a dynamic fashion rather than fixed frequency grid. Broadband optical LFS can be obtained by phase-locking a CW laser to any specified mode using tightened multi-loop composite OPLL, which realizes highly coherent extraction and amplification for individual high-order mode. Practical implementation has been demonstrated by locking a distributed-feedback (DFB) SCL to the high-order mode of a AFC that is seeded by a narrow linewidth fiber laser. Application oriented assessment in terms of OFMCW interferometry in fiber-optic links has been conducted and the performance metrics such as coherence, precision, and agility have also been discussed.

2. Principle of operation

The operational principle relies on the coherent transfer of the broadband optical LFS from the high-order mode of the AFC to a CW laser via multi-loop composite OPLL. The AFC is basically comprised by a specially designed opto-electronic frequency comb where one intensity modulator (IM) and one phase modulator (PM) are placed in tandem with precisely balanced optical and electrical transmission delays. It exploits the fact that the mode-spacing is determined by the frequency of driving signal. In general, to achieve a stable and uniform power and phase distribution per mode for a certain fixed mode-spacing, which is manifested as a flat spectrum, precise temporal phase matching between the driving signals at the two modulators is required [24]. It should be noted that such phase alignment is valid only for one certain value of the mode-spacing. According to the phase-frequency relation, any slight frequency change inevitably leads to a severe phase mismatch, and thus to drastic fluctuations in both phase and power distribution per mode, as well as impairing the spectrum flatness. Therefore, to maintain the flat-top spectrum, the phase matching has to be adjusted even for slight changes of the mode-spacing. This is particularly detrimental for phase-stable frequency sweep of the mode-spacing and, hence hinders the generation of LFS at high-order modes because the phase and power distribution per mode are necessarily to be stable and uniform during the sweeps. In allusion to this, precise delay matching for the absolute optical and electrical paths between the two modulators that could achieve broadband phase matching regardless of the frequency of the mode-spacing is necessary. With an accurate measure for the difference between the propagation delays [25], precise compensation for the delay difference can be achieved using electrical tunable delay line. This way, the phase and power distribution per mode can be steadily maintained and become insensitive to any change of the mode-spacing, even for the desired broadband arbitrary frequency sweeps, i.e. LFS. As a consequence, the mode-spacing of the AFC can be swept with a high agility in an elastic manner while preserving the phase and power stability.

This is of unique significance because it permits the high-order modes to accurately track the original arbitrary driving waveform while inheriting the precision of the comb in a fully coherent manner thanks to the mutual coherence. In the meantime, the sweep range and sweep rate are inherently multiplied at high-order modes owing to the fact that the frequency of the mode-spacing scales in proportion to the mode order. It is also worth noting that the flat spectrum, i.e. uniform power and phase distribution among all the modes, is also critical to minimize the amplitude to phase noise conversion, thus improving the signal-to-noise ratio (SNR) when acting as a sweep reference for further coherent extraction.

Exploiting this feature, the implementation is mainly consist of an AFC acting as the sweeping reference in connection with a multi-loop composite OPLL as conceptually sketched in Fig. 1. The AFC is seeded by a narrow linewidth laser. In this context, highly coherent broadband LFS can be readily obtained at the high-order mode by driving with an intermediate frequency (IF) LFS signal. The OPLL is utilized for the coherent extraction and amplification of an arbitrary individual mode. By phase-locking a tunable CW laser to any specified mode, usually a high-order mode, the broadband LFS together with the coherence of the seed laser is transferred to the CW laser. In order to improve the efficiency for the coherence transfer while maintain the phase-locking in case of fast sweep, a highly tightened OPLL exhibiting large loop bandwidth and high loop gain is demanded. This is accomplished by a multi-loop composite OPLL which combines several different kinds of actuators and will be introduced in the following. It is also worth mentioning that on top of the LFS waveform, arbitrary driving waveforms can be also imprinted and multiplied with such AFC. Such kind of frequency agility is, therefore, quite useful for arbitrary frequency-modulated optical waveform generation.

 

Fig. 1. Conception for the generation of LFS relying on coherent transfer from the high-order mode of the AFC to a CW laser via optical phase-locking. The original sweep range $\Delta f$ and sweep rate are extended by $N$ at the $N^{\textrm{th}}$ high-order mode.

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3. Experimental setup

Proof-of-concept experiment was carried out based on the configuration shown in Fig. 2. The implemented AFC is seeded with a commercial narrow linewidth CW fiber laser (Orbits Lightwave, Ethernal laser) operating around 1551 nm with a measured 3-dB linewidth of about a few kHz. The laser output is injected into two cascaded commercial modulators, a PM (EOSpace, PM-0S5-20) followed by a Mach-Zehnder IM (EOSpace, AX-0MSS-20). The driving signal is provided by a homemade fractional-N RF synthesizer in conjunction with a wideband voltage-controlled oscillator and has been properly amplified before being fed to the two modulators, respectively. One portion of the AFC output containing the mode of interest is optically filtered with optical band-pass filter (OBPF) (Yenista, XTM-50) and coupled to a balanced photodetector (BPD) after mixing with the output from a commercial DFB laser (Box Optronics, BFLD-1550) which serves as the CW laser with $\sim$800 kHz measured 3-dB linewidth. The beat note is compared with a reference at phase frequency discriminator (PFD) using an off-the-shelf mixer, producing an error signal. As shown in the lower part of Fig. 2, the composite OPLL consists of several distinct loops with their respective characteristics. This architecture allows achieving a large loop bandwidth and a high loop gain, simultaneously. Hence, it allows fully exploiting the tunability of the DFB laser while ensuring a efficient coherence transfer.

 

Fig. 2. Experimental setup. PM: phase modulator; IM: intensity modulator; ETDL: electrical tunable delay line; OBPF: optical band-pass filter; BPD: balanced photodetector; EDFA: erbium-doped fiber amplifier; AOFS: acousto-optic frequency shifter; A: electrical amplifier; BPF: electrical band-pass filter; PFD: phase frequency discriminator; T, I: temperature and current controllers, respectively.

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To achieve fast broadband optical LFS, direct SCL current control is employed, enabling a loop bandwidth up to few MHz limited by the inherent DFB thermal response. In addition, a temperature control loop is further incorporated for stable operation. In light of this, the combination of both loops is basically sufficient to provide phase locking within a large spectral range for scenarios where fast and broadband LFS is demanded. Actually, a even broader LFS range can be realized using temperature tuning with, however, a limited bandwidth due to its Hz-level response time. This could be interesting when a broad optical LFS range is required with a compromised sweep rate.

To further suppress the phase noise lying at higher frequencies, as well as to optimize the loop gain in low frequency region, external modulations based composite loop comprising two external actuators is introduced. An acousto-optic frequency shifter (AOFS) (BrimRose, IMF-500) operating at 500 MHz with $\sim$50 MHz shifting bandwidth is placed at the output of the DFB laser. Such high operation frequency could allow for sufficient frequency response thus capable of enlarging the loop gain. Its traveling acoustic path and interaction zone have been particularly optimized, permitting to provide a fast response time of about 65 ns, however, at the expense of a slightly decreased efficiency ($\sim$6 dB insertion loss for first-order diffraction). Moreover, further noise suppression is performed using an additional PM cascaded with the AOFS whose rise-time is measured to be $\sim$6 ns. It is worth noting that a simple current loop is enough to implement a phase-locking with, however, limited loop gain. This way, the adoption of the AOFS and PM loops could allow for a tighter phase-locking thus a more efficient coherence transfer. A small portion of the output after the PM and the AOFS, is used to generate the beat note with the specified AFC mode. All the loops are constructed using homemade loop filters with optimized circuitries. Cares have been taken to minimize the noise and to extend the bandwidth while maintaining stable operation. The fiber pigtails of the DFB laser, AOFS, and PM, including other passive components, are customized to minimize the overall propagation delay, permitting a large loop bandwidth up to more than ten MHz range. This ensures a tight phase-locking for efficient coherence transfer.

It should be pointed out that the DFB laser is actually offset locked with respect to any specified AFC mode due to the AOFS induced frequency shift $f_{\textrm{AOFS}}$. The phase of the output when phase-locked to $N^{\textrm{th}}$ order AFC mode can be expressed as

Concerning the total phase noise, homemade RF synthesizer is referenced to an atomic clock, which in general outperforms commercial narrow linewidth fiber lasers in terms of phase noise. Hence, thanks to the high mutual coherence, even after the comb assisted photonic multiplication, the eventual phase noise, i.e. the coherence property of the high-order mode, is still dominated by the seed laser. Thus, starting from a stabilized narrow linewidth seed, it permits a high degree of coherence on all the modes that can be coherently transferred using OPLL.

4. Results and discussions

4.1 Performance of AFC and OPLL

The flat-top optical spectrum of the AFC together with the closed-loop DFB laser locking to the $9^{\textrm{th}}$ mode is captured before the BPD as illustrated in Fig. 3(a). Due to the limited resolution bandwidth (RBW), the linewidth for the DFB, comb modes, and AOFS induced frequency offset, are hardly visible. The optical power per mode is about −16 dBm while that for the generated LFS is more than 6 dBm and can be conveniently boosted with the in-loop EDFA. The achievable sweep range is restricted both by the tuning range of the DFB laser and by that of the high-order mode. In this demonstration, the current and temperature tuning range of the DFB laser is $\sim$60 GHz and $\sim$220 GHz, respectively, within its nominal operation range. On top of that, the bandwidth of the driving RF amplifiers applied to the PM and IM for the AFC generation is from 8.5 to 19.0 GHz. The implemented AFC contains 19 flat-top modes concerning a better uniform frequency response, so the largest multiplication factor is 9. Taking into account these technical issues, here the maximum sweep range is limited to about 60 GHz for fast and broadband LFS generation using DFB current tuning in this implementation. Further improvement with higher power or wider bandwidth amplifier could allow for broader sweep range. With the help of temperature tuning, a broader LFS with a relative slow sweep rate can also be realized. In open-loop condition, by sweeping the current and temperature control signal, the DFB output spectrum is obtained as presented in Fig. 3(b). The tuning range of the adopted DFB laser could fully cover the sweep range of the higher-order mode. It is worth noting that the sweep rate is mainly limited by tuning response of the DFB laser not by the proposed method. Turning off the temperature sweeping can improve the sweep speed.

 

Fig. 3. Captured optical spectra when the DFB is phase-locked to the $9^{\textrm{th}}$ mode: (a) The AFC together with the DFB; (b) Total 220 GHz sweeping range of the DFB laser.

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The effect of phase-locking is assessed by an extra out-of-loop UMZI with a 20 km fiber delay (not shown in the figure). The delayed self-heterodyne (DSH) spectra while the DFB laser is locked to the $9^{\textrm{th}}$ mode at a fixed 19.0 GHz mode-spacing are reproduced in Fig. 4(a) with 1 kHz RBW. In free-running case, the obtained DSH spectrum is simply a measure for the phase noise, which is in a fair agreement with the 800 kHz 3-dB nominal linewidth announced by the manufacturer. When only the AOFS loop is applied in combination with the temperature and current control loops, the resulting DSH spectrum exhibits a clear linewidth reduction. A narrow peak with a RBW-limited linewidth is observed together with a broad pedestal due to the residual amplitude and phase noises. A bandwidth equal to four times the RBW around the central peak contains 95.9% of the total power presented in the 40 MHz span, implying an efficient coherent enhancement. The loop bandwidth is manifested as the two narrow bumps located at approximately 2 MHz frequency offset, which agrees well with the predictions from loop stability theory when the AOFS response time is taken into account.

 

Fig. 4. Beat note spectrum obtained using an out-of-loop 20-km DSH when the DFB is free-running, phase-locked via AOFS only, or via both AOFS and PM, respectively, to the $9^{\textrm{th}}$ mode of the AFC at (a) 1 kHz and (b) 1 Hz RBW. (c) SSB PN-PSD of the in-loop beat note after the BPD.

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By further closing the PM loop as seen in Fig. 4(a), the noise sidebands in the DSH spectrum are reduced in a bandwidth of the order of 11 MHz, which is in accordance with the loop bandwidth expected from the PM response time. This is also manifested as the bumps in the spectrum, indicating that the loop bandwidth have been pushed towards higher frequency region. The fractional power concentrated in the RBW-limited central peak is remarkably high, amounting to 98.6% of the total power within the 40 MHz span. Both closed-loop spectra exhibit a few discrete lines at about $\pm$1 MHz with respect to the central peak. These spurs are probably resulting from mechanical resonances of the circuits and from the power supplies.

High resolution DSH spectra are also recorded in Fig. 4(b) with 1 Hz RBW. Again, when both AOFS and PM loops are closed, this shows a RBW-limited coherent $\delta$-function peak at the offset frequency with more than 75 dB carrier-to-noise ratio, testifying the reliable coherence transfer from the comb to the DFB laser.

It is worth noting that although the 20 km DSH could be insufficient for a precise measurement for the linewidth in phase-locked cases, it still serves as an effective characterization for the efficiency of the OPLL based coherence transfer. It should be also pointed out that in some scenarios the phase noise at high frequencies may have significant contribution, but here, the power contained in the noise pedestal of the beat note is nearly negligible compared with that in the center peak. This way, the observed 3 dB linewidth in connection with the DSH spectra still constitutes a reliable coherence metric.

Additionally, the locking efficiency is independently confirmed by measuring the single-sideband (SSB) phase noise power spectral density (PN-PSD) of the in-loop beat note obtained after the BPD as shown in Fig. 4(c). Clearly, as indicated by the bumps in the SSB phase noise PSD, the loop bandwidths in the two situations are in accordance with the observation in the DSH spectra. One may also notice the asymmetry of the in-band noise spectra in both the DSH spectra. This imbalance is mainly attributed to the AOFS and the PM induced parasitic amplitude modulations. Besides, since the loop bandwidth of the current tuning loop is quite close to that of the AOFS loop, they are thus hardly distinguishable in the spectra.

4.2 Linear frequency sweep characteristics

In order to evaluate the fidelity of the generated LFS, as well as the frequency agility, we assess the generated LFS when the DFB laser is locked to the $+9^{\textrm{th}}$ mode of the AFC. In all situations, the AOFS and PM loops are always activated. Fast and broadband LFS of 60 GHz in a 10 ms duration is carried out employing the current control loop, corresponding to 6 THz/s slope. The sweep rate is mainly limited by the frequency ramping speed of the fractional-N RF synthesizer. The instantaneous frequency is assessed by another short DSH with 63 m fiber delay and subsequently extracted via Hilbert transform as illustrated in Fig. 5. The measured sweep rate and the RMS frequency error are extracted by using a linear fit to be $\sim$6.000 THz/s and 160.829 kHz, respectively. Via monitoring the lock detect signal in connection with the error signal at the PFD, no glitch over $2\pi$ occurs during the entire sweep. Thus, the absence of the cycle slip indicates the excellent phase-locking maintained in the obtained LFS, implying the efficient coherence transfer of the broadband phase-continuous frequency agility from the AFC to the DFB CW laser. The overall residual frequency error is probably limited by the nonlinearity and precision of the fractional-N RF synthesizer and the phase noise of the AFC seed laser according to the operational principle.

The accuracy, precision, and stability of the generated LFS are bounded by the quality of the AFC seed source, RF driving signal, and locking scheme. The dominant SSB phase noise of commercial fractional-N RF synthesizer remains steadily below −60 dBc/Hz at Fourier frequencies of 10 kHz. The nominal resolution is higher than 0.01 Hz. Taking into account an overestimated 6 dB average noise figure of the wideband RF amplifier and also the multiplication factor as stated in Eq. (1), the contribution of the RF synthesizer on the overall frequency error stays almost below 1 Hz. Meanwhile, the frequency noise of the seed laser is about $55\,\textrm {Hz}/\sqrt {\textrm {Hz}}$ at 100 Hz which raises to above $\sim 100\,\textrm {Hz}/\sqrt {\textrm {Hz}}$ in frequency region below 20 Hz according to its specification. The contribution from the former usually remain below the frequency noise of the AFC seed thus constituting a lower bound for the precision in this demonstration. However, in case of frequency sweep, the sweep nonlinearity and the additional frequency error from the RF synthesizer should be accounted for. Therefore, the overall frequency noise is dictated by these above noise contributions. This also has been cross-validated by the conclusions drawn from Fig. 4. Therefore, in connection with a cavity or a femtosecond comb referenced or stabilized narrow linewidth seed source, the low inherent uncertainty of the seed laser source can be transferred to the AFC, then the high-order modes and the phase-locked CW laser subsequently. This is quite promising for metrology and spectroscopy applications where highly coherent rapid frequency sweeps across a broad spectral range with a high precision and stability is essential. Moreover, the center wavelength of the AFC can be readily shifted by simply tuning that of the seed laser source, leading to the straightforward extension of the spectral coverage, especially attractive for spectroscopy. Besides, the sweep range is limited by the electrical devices for AFC in the current implementation, but not by the proposed approach.

 

Fig. 5. Frequency error (left-hand axis) and instantenous frequency (right-hand axis) for the generated fast and broadband LFS when locked to the $9^{\textrm{th}}$ mode.

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4.3 Optical frequency domain reflectometry based on OFMCW interferometry

In order to verify the sweep properties and coherence characteristics, from an application standpoint, field demonstrations of OFMCW interferometry in the form of optical frequency domain reflectometry (OFDR) has been carried based on the current implementation as schematically displayed in Fig. 6(a). The generated optical LFS when the DFB laser is phase-locked to the $+9^{\textrm{th}}$ AFC mode is coupled to and split by a polarization maintaining fiber (PMF) coupler with one portion acting as the interrogation probing signal by sending to the fiber-under-test (FUT) through a circulator and the other portion being the reference. A fiber-coupled AOFS is inserted into the reference path, yielding a frequency shift for eliminating low frequency noise and internal multi-interference. The back scattering signals from the FUT are mixed with the frequency-shifted reference. The polarization alignment is accomplished using a polarization controller. The resulting photo-current from the BPD is filtered, amplified, and then digitized by the data-acquisition card for further Fourier domain post-processing. The FUT used here is consisted of several spools of standard single-mode fibers as depicted in the inset of Fig. 6 (h) , which are connected using FC/APC connectors.

 

Fig. 6. (a) Experiment configuration for the OFDR based on the proposed LFS generation. DAQ: Data acquisition; Reflection peaks at three different distances when the DFB is frequency-swept in case of free-running by direct current ramping (b), (c), (d) and in case of phase-locked to the +9th AFC mode (e), (f), (g), respectively; (h) The resulting OFDR trace with insets showing the detailed reflection peaks and the structure of the FUT.

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The OFDR trace for the implemented FUT obtained using the generated LFS with $\sim$60 GHz sweep range in 10 ms is presented in Fig. 6(h). As aforementioned, the performance of OFDR strongly relies on the characteristics of the applied LFS of the laser source. It suffers from a severe trade-off between sweep range which limits the spatial resolution and the dynamic coherent as well as the sweep linearity, both of which determines the valid measurement range. Therefore, Fourier transform-limited spatial resolution of $\sim$1.67 mm at all the 3 APC connectors at different lengths along the entire FUT link has been readily achieved as can be inferred from Fig. 6(e)–6(f), which is in an excellent accordance with the sweep range. Compared with the free-running case, improvements of $\sim 10^4$ times in spatial resolution are observed at all connectors. The $\sim$0.5 m patchcord near the second connector can clearly identified as shown in the inset of Fig. 6(h). Note that this is achieved without any post-processing for neither phase noise compensation nor nonlinear error correction, testifying the capability for on-line requirements. Since the dynamic coherent follows that of the AFC seed laser source, the $\sim$3 km measurement range in optical fiber is right within the penalty-free measurement range [26] for FMCW lidar without scarifying the spatial resolution. Nevertheless, this remains far beyond (more than 80 times) the intrinsic coherent length of the DFB laser, which is about several tens of meters according to its measured linewidth, confirming the efficient coherence transfer in a fully dynamic manner. Slight degeneration in SNR at the reflection peaks can, however, be observed along with the increase of the measurement range. For instance, the SNR for the reflection at $\sim$1.87 km is degraded by more than 5 dB for that at $\sim$3.23 km. This can be attributed to the deteriorated coherence at longer distances while the inherent sweep nonlinearity of the RF driving signal may also become a limiting factor. The defect of the APC connector can also be impairing. Moreover, according to the SNR margin revealed at the last reflection, the measurement range can be further extended. From an application standpoint, combining with practical phase error compensation techniques, it would allow for further improvements.

5. Conclusion

In conclusion, we report a generation of optical LFS relying on coherent transfer of the LFS from the high-order mode of a specically designed AFC to a CW DFB laser using tightened multi-loop composite OPLL. Thanks to the phase-stable operation during the LFS of the mode-spacing of the AFC, sweep range and sweep rate multiplication are realized at the high-order modes while preserving the mutual coherence. Meanwhile the composite OPLL could offer a large loop bandwidth, allowing for the efficient transfer of the LFS of the AFC mode to the CW laser in a highly coherent manner. The concept, functionality, and performance of the key techniques, AFC and multi-loop composite OPLL, have been carefully discussed and evaluated.

Demonstration of the proposed system has been realized using a narrow linewidth fiber laser and a DFB laser as the AFC seed laser and the swept CW laser, respectively, achieving highly coherent fast, broadband optical LFS of 60 GHz in 10 ms with about 160.8 kHz frequency error. From a practical point of view, such implementation has been verified in OFMCW interferometry experiments, exhibiting Fourier transform-limited spatial resolution at the distance of more than 80 times of the intrinsic coherence length of the DFB laser, corresponding to more than 4 orders of magnitude spatial resolution improvement than DFB free running case. The unique phase-stable frequency swept nature, the high coherence property, and the highly tightened coherent transfer with arbitrary waveforms are of particular significance and beneficial for numerous potential applications.