Frequency-modulated continuous-wave (FMCW) lasers are widely employed in various fields, such as light detection and ranging (LiDAR) [1], optical coherence tomography (OCT) [2], and optical frequency domain reflectometry (OFDR) [3]. In particular, the FMCW LiDAR offers many distinct advantages over the traditional time-of-flight (ToF) LiDAR, including the ability of simultaneous distance and velocity measurement, higher measurement resolution and accuracy at long distances, and reduced sensitivity to ambient light interference. In principle, the range resolution of FMCW LiDAR is inversely proportional to the laser frequency-sweep excursion. A very high resolution can be achieved with a large frequency excursion. However, the LiDAR resolution in practical applications deteriorates with increasing distance due to the limited laser linewidth and frequency-sweep linearity. There are considerable challenges in simultaneously implementing FMCW lasers with large frequency excursion, high coherence, and high linearity. Three typical techniques have been reported. The first one is external modulation, which incorporates electro-optic modulators to modulate linear FMCW radio frequency signal onto the optical carrier. This method can offer superior linearity. But the frequency excursion is constrained by the bandwidth of the microwave source, which is usually less than a few gigahertz [4]. The second technique is internal modulation, which directly alters the temperature or drive current to tune the laser output frequency [5–8]. Large frequency excursion over 100 GHz can be achieved, but the linearity is usually very poor. A phase-locked loop (PLL) is necessary to linearize the laser frequency sweep [9–12]. The third technique is external-cavity modulation. An external-cavity laser is constructed, and the output frequency is tuned by adjusting the resonant frequency of the external cavity. Typical approaches are using chirped fiber gratings [13] or tunable micro-resonators [14–16]. Ultra-narrow laser linewidth less than 1 kHz can be achieved with the external cavity. But it is still challenging to achieve both a large frequency excursion and a high linearity.

In this paper, we demonstrate a novel concept of time-frequency self-injection locking for implementing FMCW lasers simultaneously with large frequency excursion, ultra-narrow instantaneous linewidth, and high sweep linearity. In the proposed scheme, a frequency-swept laser is self-injection locked to a delayed frequency-shifted replica of itself, resulting in significant linewidth narrowing and linearity improvement. A large frequency excursion of 100 GHz and a record low relative nonlinearity of 6.4×10−7 are demonstrated with a commercial off-the-shelf distributed feedback (DFB) laser diode. When the FMCW laser is utilized for detecting moving targets, a range resolution of 1.6 mm and a velocity accuracy of 3 mm/s are achieved at a distance up to 300 m.

The concept of time-frequency self-injection locking is illustrated in Fig. 1(a). A distributed feedback (DFB) laser is current-tuned to generate linear FMCW light. To compress the linewidth and enhance the linearity, a portion of the output light is delayed and fed back into the laser. Because the laser operates in a frequency-sweeping state, the frequency of the feedback light should be shifted to match the instantaneous frequency at the laser output. Self-injection locking can then be achieved, as depicted in Fig. 1(b).

To quantitatively analyze the mechanism of injection locking, we start with the equations of a solitary laser without feedback [17]: E˙(t)=ξ2(1+iα)(N−Nth)E(t),(1)N˙=J−ΓNN−[ΓE+ξ(N−Nth)]|E(t)|2,(2)where E is the complex cavity field; N is the carrier inversion; Nth is the threshold carrier inversion; ξ is the differential gain coefficient; α is the linewidth enhancement factor; J is the pump rate; and ΓN and ΓE are the decay rates of carriers and photons, respectively.

For an FMCW laser, Eq. (1) should be modified to account for the frequency tuning mechanism. In experiments, the laser is typically tuned by varying the current. On the one hand, the variation in current induces an increase or decrease in carrier concentration and then changes the laser frequency via the linewidth enhancement factor. On the other hand, the thermal effect also affects the light frequency by altering the material refractive index and laser cavity length. We can simply assume that the temperature T in the laser active region is proportional to the total thermal energy. The change of thermal energy depends on the rate difference between heating and cooling. The former equals the total electrical power minus the optical radiant power, and the latter is proportional to the temperature difference between the active region and the laser shell. The differential equation for the temperature change is then T˙=1CT[eJU−ΓRεν|E(t)|2−Kd(T−T0)].(3)

Here, CT is the total heat capacity of the laser active region; e is the elementary charge; U is the laser drive voltage; εν is the single photon energy; T0 is the shell temperature, usually room temperature; Kd is the heat dissipation rate in unit temperature difference; and ΓR is the ratio between output photon rate and intra-cavity photon number, which is calculated by ΓR=1−RsRsτs,(4)where Rs is the facet reflection coefficient and τs is the diode-cavity round-trip time. By taking into account the thermal effect, the equation of cavity field becomes E˙(t)=ξ2(1+iα)(N−Nth)E(t)−iΩ0(αT+βT)(T−T0)E(t),(5)where Ω0 is the laser angular frequency at T0, αT is the thermal expansion coefficient, and βT is the temperature coefficient of refractive index.

By further considering the frequency-shifted feedback and the spontaneous emission noise, the equation of cavity field is then $$\begin{gathered} E˙(t)=ξ2(1+iα)(N−Nth)E(t)−iΩ0(αT+βT)(T−T0) \\ ·E(t)+κE(t−τe)exp(−iϕ0)exp(i2πfAt)+FE, \end{gathered}$$(6)where τe is the external round-trip time, ϕ0 is the external phase shift, fA is the frequency-shift amount, κ is the feedback-coupling rate related to the feedback ratio Re by κ=1−RsτsReRs,(7)and FE is the spontaneous emission noise with ⟨FE(t)FE*(u)⟩=βNδ(t−u),(8)where β is the spontaneous emission coupling coefficient, and δ(t) is the Dirac function.

The dynamics of the self-injection locked FMCW laser can be simulated by numerically integrating Eqs. (2), (3), and (6). The parameters of the equations are listed in Table 1. We first investigate the impact of external feedback on the instantaneous laser linewidth. The drive current is tailored to generate nearly perfect frequency sweeps, as shown in Fig. 2(a), and Fig. 2(b) depicts the numerical simulation result of a free-running laser without any noise. The procedure of tailoring the current is similar to that employed in experiments [16] (see the next section for more details). Suppose the laser output field is written as E(t)=Aei[2πν0t+πγt2+ϕn(t)],(9)where ν0 is the initial frequency, γ is the chirp rate, and ϕn(t) is the phase noise. The instantaneous linewidth is then evaluated by removing the frequency chirp and calculating the power spectral density (PSD) of En(t)=eiϕn(t). The effect of external feedback for frequency-fixed lasers has been thoroughly studied before [18]. Depending on the feedback ratio and phase shifts, laser linewidth narrowing, broadening, splitting, or coherence collapse may occur [19]. We find that these behaviors can also happen to FMCW lasers with frequency-shifted feedback. Figure 2(c) presents the free-running laser PSD with an intrinsic linewidth of 5 MHz. Figs. 2(e)–2(l) display the laser PSD with different feedback ratios and phase shifts. When the feedback ratio is lower than ∼−50dB, the instantaneous linewidth may be narrowed or broadened, depending on the feedback loop phase shift. When the feedback ratio is between −50 and −30dB, the linewidth is greatly reduced regardless of the loop phase shift. At a feedback ratio exceeding −30dB, incoherent side modes significantly grow up, resulting in what is termed “coherent collapse.” The linewidth evolution with the feedback ratio is plotted in Fig. 2(d). The optimum feedback ratio should thus be between −50 and −30dB.Table 1.

Simulation Parameters

Next, we investigate the impact of external frequency-shifted feedback on the laser frequency sweep nonlinearity. The spontaneous emission noise is set to zero. A perturbation is added to the drive current, causing a sinusoidal disturbance in the instantaneous frequency of the free-running laser, which is given by νnl(t)=Anlsin(2πmfrpt),(10)where Anl is the disturbance amplitude, frp is the repetition rate of frequency sweep, and m is the harmonic order. Define the residual root-mean-square (RMS) νnl,rms as the standard deviation of νnl(t). Figures 3(a)–3(c) present the residual errors of the free-running laser with different harmonic orders of 2, 20, and 100. The residual RMSs are all around 500 kHz. Figures 3(d)–3(o) display the residual errors at different feedback ratios and harmonic orders. As the feedback ratio increases, the residual RMS is greatly reduced due to the external feedback. However, when the feedback ratio exceeds an optimum point, some frequency fluctuations gradually grow up, leading to degradation of the linearity. As plotted in Fig. 3(p), when the harmonic order is 2, the optimum feedback ratio is approximately −55dB and the optimum residual RMS reaches 28 kHz. But for harmonic order of 20 and 100, the optimum feedback ratio and residual RMS are both −40dB and 47 kHz, respectively.

It can be seen that both the instantaneous linewidth and the sweep nonlinearity can be greatly reduced by the external feedback. For convenience, we may use the single parameter of RMS frequency error to evaluate the overall frequency noise arising from either the limited laser linewidth or the sweep nonlinearity. Figure 4 shows the results when both the spontaneous emission noise and the current perturbation are considered. Here, we adjust the simulation parameters to generate results close to our experimental observation (experimental results shown in the next section). The laser frequency-sweep excursion is 100 GHz, and the repetition rate is 1 kHz. The spontaneous emission coupling coefficient β is 2×103s−1, corresponding to an intrinsic linewidth of 1 MHz, and the current perturbation is identical to Fig. 3(b). Figure 4(a) shows the residual frequency error simulated with a sampling rate of 100 MSa/s when there is no external feedback. The residual RMS is around 2.4 MHz. Figure 4(b) shows the result with external feedback. The feedback ratio is −40dB, and the delay is 65 ns. The RMS frequency error is 58 kHz, which is reduced by about 50 folds compared to the free-running laser without feedback.

The experimental setup is shown in Fig. 5. A commercial DFB laser diode (Emcore 1772) with a wavelength of 1550 nm is used to generate the FMCW light. The drive current of the DFB laser is modulated by a periodic control signal generated by an arbitrary function generator (AFG, Tektronix AFG31000). The laser diode output is frequency shifted by an acousto-optic modulation module consisting of two acousto-optic modulators (AOMs). One AOM is for frequency up-shift, and the other is for frequency down-shift. Then the optical signal is amplified by an erbium-doped fiber amplifier (EDFA), and 10% is fed back to the laser diode via a circulator. The remaining 90% is exported as an FMCW light source. It is noted that the DFB diode we have at hand has an embedded isolator with an isolation of ∼−30dB. An EDFA is thus required to increase the feedback injection ratio. If a DFB diode with no embedded isolator is used, the EDFA will not be necessary.

Injection locking is only possible when the injected light frequency is sufficiently close to that of the laser. For an FWCW laser, when the laser output is fed back after delay time τe, the instantaneous frequency of the laser diode has varied by γτe, where γ is the frequency-sweep rate. To synchronize the instantaneous frequencies of the injected light and the laser diode, the optical feedback must be frequency-shifted by γτe. Therefore, the total frequency-shift amount provided by AOMs is fA=±γτe, where the + and − signs correspond to the up-tuning and down-tuning ramps, respectively. To generate positive or negative frequency shift, the input waveforms of AOMs are frequency-shift keying (FSK) RF signals triggered by a square wave signal synchronized with the FMCW light. The two FSK signals have the equal carrier and hopping frequencies but opposite polarities, and the difference between the carrier and hopping frequencies is exactly the absolute value of the net shift frequency.

We first measure the laser frequency noise with and without self-injection locking when the laser frequency is not sweeping. The total frequency-shift amount induced by the two AOMs is set to zero. The measurement set is shown in Fig. 4(a). The optical output passes through an unbalanced Mach–Zehnder interferometer (MZI) and is then detected by a photodetector [20]. The length difference between the two arms of MZI is about 20 m. An AOM driven by a 200-MHz signal is added to one arm of the MZI. The phase noise of the detected beat note is measured by a phase noise analyzer (PNA, Rohde & Schwarz FSWP50). The relation between the beat note phase noise SΔϕ(f) and the laser frequency noise Sν(f) is given by Sν(f)=f24sin2(πfτ)SΔϕ(f),(11)where f is the frequency offset, and τ is the relative delay time between the two arms of the MZI. The laser drive current in experiment is 241 mA. The total fiber length of the external feedback loop is about 13 m, corresponding to a time delay of ∼65ns, and the feedback ratio is −40dB. The measured laser frequency noise with and without external feedback is shown in Fig. 6(b). In the high frequency range, the frequency noise level tends to be white, which is the so-called quantum noise limit and mainly caused by the spontaneous emission. The intrinsic laser linewidth Δν0 is related to the white frequency noise background Sν0 by Δν0=πSν0.(12)

The Sν0 values of the free-running and self-injection locked lasers are 2.9×104Hz2/Hz and 2.0Hz2/Hz, corresponding to the intrinsic linewidths of 91 kHz and 6.3 Hz, respectively. The latter is suppressed by four orders of magnitude compared to the former.

Compared to the white frequency noise, the frequency noise in the lower frequency range has a larger impact on the LiDAR results. So we calculate the integral linewidth Δνint to reflect the low-frequency noise, which is defined by [21] ∫Δνint∞Sν(f)f2df=1π[rad2].(13)

According to the frequency noise plotted in Fig. 6(b), the integral linewidths of the free-running and self-injection locked lasers are 260 kHz and 12 kHz, respectively. The latter is suppressed by more than 20 times compared to the former.

In addition, we increase the length difference between the two arms of the MZI to about 20 km to directly measure the laser linewidth with and without self-injection locking, as depicted in Fig. 6(a). Since this length difference far exceeds the laser coherence length, the beat signal linewidth is twice the laser linewidth. Figure 6(c) presents the spectra of the beat signals. It can be calculated that the linewidths of the free-running and self-injection locked lasers are 910 kHz and 37 kHz, respectively.

Ultra-linear FMCW light generation with frequency-shifted feedback is then tested. To guarantee stable injection locking, the external feedback is first turned off and the free-running laser frequency sweep is linearized by predistorting the drive current. We employed an iterative procedure similar to that in Ref. [16]. The response of the laser frequency to the control signal can be modeled as a low-pass filter. The iteration process is given by uk+1(t)=uk(t)−p[e˙k(t)+ωcek(t)],k=0,1,2,…,(14)where uk+1(t) and uk(t) represent the control waveforms of the kth and (k+1)th iterations; p is a constant, ek(t) is the frequency-sweep error of the kth iteration, and ωc is the characteristic angle frequency. In the experiment p=−5×10−16V/Hz2, where the ratio between drive current and control waveform is 100 mA/V, and ωc=2π×10kHz.

The configuration for laser current predistortion is illustrated in Fig. 7(a). An unbalanced MZI with ∼2-m arm length difference is used to measure the instantaneous laser frequency. The laser output is sent through the MZI and mixed in a balanced photodetector (BPD). The detected beat note is acquired by an oscilloscope (Rohde & Schwarz RTO1022). Suppose the relative delay between the two arms of MZI is τ0 and the beat signal frequency is fb(t)=|ν(t+τ0)−ν(t)|, in which ν(t) represents the laser instantaneous frequency. Since τ0 is much smaller than the frequency ramp duration, fb(t) can be approximately written as fb(t)=τ0·|ν˙(t)|. The phase of the beat signal can be extracted by ϕb(t)=arctan[HT{xb(t)}xb(t)],(15)where xb(t) is the beat signal waveform and HT represents Hilbert transform. The instantaneous laser frequency can then be calculated by ν(t)=±ϕb(t)2πτ0,(16)where the signs correspond to the up and down tuning ramps, respectively.

The initial control signal at the start of iteration is a standard triangular waveform. The repetition rate is 1 kHz, corresponding to an up/down ramp duration of 500 μs. The initial frequency sweep and residual error in down and up ramps are shown in Figs. 7(f) and 7(g), respectively. The frequency excursion is about 100 GHz. The frequency curve is clearly nonlinear, and the RMS frequency error is on the gigahertz level. For each iteration, the frequency sweep is repeatedly measured over 100 periods and averaged. Figure 7(b) plots the evolution of the RMS frequency error with the number of iterations. The RMS error decreases rapidly and reaches a stable value after 25 iterations. The final drive current waveform is shown in Fig. 7(c), and the residual frequency error is shown in Figs. 7(d) and 7(e). The residual RMS reaches 1.4 MHz and 1.6 MHz in the down ramp and up ramp, respectively. The relative nonlinearities, defined as the ratio of residual RMS over frequency excursion η=νnl,rmsΔν,(17)are 1.4×10−5 and 1.6×10−5 in the down ramp and up ramp, respectively.

Once the laser frequency sweep is preliminarily linearized by predistorting the drive current, the external feedback is turned on to further suppress the frequency error. The RF drive frequencies of AOMs are carefully tuned to match the sweep slope of the FMCW laser. Figures 8(a) and 8(b) show the spectrum of the beat signal xb(t) detected after the light passes through the MZI and measured by a spectrum analyzer. For comparison, the spectrum for the free-running laser without feedback is also shown. As can be seen, the linewidth of the beat note is significantly narrowed due to the external feedback, suggesting greatly reduced frequency errors. However, enhanced side lobes also emerge due to the feedback. The free-spectral range of the side lobes is around 14.6 MHz, which is inverse to the feedback delay. The main-lobe-to-side-lobe suppression ratio is around 28 dB. It should be noted that, in most LiDAR ranging scenarios, only one reflecting target needs to be identified in the light path; therefore the presence of the side lobes will not cause any interference since they are much weaker than the main lobe.

Figures 8(c)–8(f) show the retrieved residual frequency error. A low-pass filter with a cut-off frequency of 10 MHz is employed to suppress the side lobes in the beat signal. The residual RMS is reduced to 64 kHz, corresponding to a relative nonlinearity of 6.4×10−7. For a fair comparison, the beat signal of the free-running laser is also sent through the low-pass filter, and the sweep error is calculated. The residual RMS is around 1.2 MHz, as indicated by the blue curve in Figs. 8(c) and 8(d). Compared to the free-running laser, the frequency-sweep nonlinearity of the self-injection locked laser is suppressed by approximately 20 folds.

LiDAR ranging experiments are then performed with the FMCW laser. The experimental setup is shown in Fig. 9(a). The light is split into two parts by an optical coupler, with 99% of the power utilized for target detection and the remaining 1% used as a reference. The target detection light is sent through a length of single-mode fiber to simulate different target distances and then collimated by a collimator. The target is a planar mirror mounted on a translation stage. Here the transmitted light power is 13 dBm, while the received light power after reflection from the target is approximately −7dBm. After heterodyning the reflected light with the reference, a beat signal is generated and acquired by an oscilloscope. The beat note frequencies are detected by calculating the high-resolution spectra with chirp Z-transform and finding the peaks. The target distance R and velocity v are then given by R=c4γ(fbu+fbd),(18)v=λ4(fbu−fbd),(19)where c is the light speed; λ is the laser wavelength; fbu and fbd are the beat note frequencies generated from the up and down ramps, respectively; and v>0 means the target is moving far away from the detector, while v<0 means the target is moving towards the detector.

At short distances, the range resolution is consistent with the transform-limited resolution given by δR=c2Δν,(20)where Δν is the frequency-sweep excursion. When the target distance increases, the range resolution will degrade due to the sweep nonlinearity of the light source and can be estimated by [6] δR=c2Δν(1+2πνnl,rmsτm),(21)where τm is the relative delay time between the reference and probe lights. When 2πνnl,rmsτm≫1, the relative resolution, i.e., the ratio between the range resolution and the target distance, is δRR=2πνnl,rmsΔν,(22)where νnl,rms/Δν is the relative nonlinearity that was mentioned earlier. This is the so-called nonlinearity-limited resolution.

As the target distance further increases beyond the laser coherent length, the impact of the laser linewidth becomes apparent, and the effect of relative nonlinearity weakens because of the reduced coherence between the reference and probe lights. This results in a range resolution approaching the linewidth-limited resolution given by δR=cγΔνL,(23)where ΔνL is the laser linewidth.

The experimentally measured range resolution versus the target distance is depicted in Fig. 9(b). The theoretical transform-limited, nonlinearity-limited, and linewidth-limited resolutions are simultaneously plotted. Since the intrinsic linewidth does not account for the contribution of low-frequency noise to the beat note linewidth, and the integral linewidth does not consider the broadening effect of 1/f frequency noise on the self-heterodyne signal linewidth [22], the linewidth used for estimating long-range resolution is the one measured with ultra-long MZI (e.g., 37 kHz). The experimental results align well with the theoretical analysis. Figures 9(c)–9(e) illustrate measurement results at several typical distances. As a comparison, the ranging results of FMCW light generated by a free-running laser are also presented. It clearly shows that with self-injection locking, the deterioration of range resolution with target distance is greatly relieved due to the significantly reduced linewidth and sweep nonlinearity.

Next, detection of moving targets is performed. Both the target distance and velocity are measured simultaneously while the target is moving uniformly along the light path. The accuracy is evaluated by performing 50 measurements and calculating the sample standard deviation. The relation between target distance and ranging accuracy is depicted in Fig. 10(a). The ranging accuracy is approximately 1 mm when the target distance is less than 300 m and gradually deteriorates after 300 m due to the increasing impact of frequency-sweep nonlinearity. Figure 10(b) displays the measured velocity accuracy versus target distance. Similarly, velocity accuracy worsens with increasing target distance. At a short distance, the measured velocity value is approximately 23 mm/s, with an accuracy of about 3 mm/s, and at 1 km distance the accuracy worsens to 6 mm/s.

In conclusion, we have demonstrated an FMCW light source based on a self-injection locked laser with frequency-shifted optical feedback. It achieves a frequency excursion of 100 GHz with a repetition period of 1 ms. The relative nonlinearity is only 6.4×10−7 with about 13 m feedback loop and −40dB injection ratio. Compared to the free-running laser without feedback, the nonlinearity is reduced by 20 folds. A comparison of several typical FMCW light generation methods and their metrics is shown in Table 2 [6,7,10,11,16,23].Table 2.

Methods and Metrics of Several Typical FMCW Light Sources

When the system operates for a long term, environmental temperature variations or mechanical vibrations may lead to a deterioration in sweep linearity; however, the self-injection locking state can be maintained effectively. The experiment reveals that when the total frequency-shift amount of the AOMs deviates from the optimal value by approximately ±0.1MHz, the laser remains predominantly in the locking state. Considering the chirp rate and ramp duration, it can be estimated that the locking range of the laser is around ±0.7GHz. A commercial laser diode controller allows for the control of the laser output frequency within this range. Furthermore, the characteristic of self-injection locking is that it focuses solely on the relative frequency fluctuations within a ramp duration, but the absolute frequency variations have little impact. Thus it ensures good stability of the locking state.

The present work achieves a large frequency excursion while maintaining a record-low nonlinearity. This would be particularly useful for high-resolution ranging at long distances. In our experiments, a range resolution of 1.6 mm and a velocity accuracy of 3 mm/s are achieved when the target distance is 300 m. The proposed light source is promising for long-distance FMCW LiDAR applications.