High Precision Range Extracting Method for FMCW LiDAR Using Semiconductor Laser Based on EO-PLL and NUDFT

Abstract

Frequency tuning nonlinearities in semiconductor lasers constitute a critical factor that degrades measurement precision and spectral resolution in frequency-modulated continuous-wave (FMCW) LiDAR systems. This study systematically investigates the influence of nonlinear beat signal phase distortions on spectral peak broadening and develops a phase-fitting-based pre-correction algorithm. To further enhance system performance, an electro-optic phase-locked loop architecture combined with non-uniform discrete Fourier transform signal processing is implemented, establishing a comprehensive solution for tuning nonlinearity suppression. Experimental validation demonstrates a sub-18 µm standard deviation in absolute distance measurements at a 19 m target range. This integrated approach represents a significant advancement in coherent frequency-sweep detection methodologies, offering considerable potential for high-precision photonic radar applications.

1. Introduction

Frequency-modulated continuous wave (FMCW) LiDAR has gained widespread adoption in vehicle autonomy, 3D mapping, and medical imaging applications due to its superior ranging accuracy, enhanced anti-jamming capabilities, and simultaneous position-velocity measurement capacity [1,2,3,4]. The implementation of semiconductor lasers as light sources offers distinct advantages in FMCW LiDAR systems, particularly in terms of compact form factor, cost efficiency, and reduced weight. While semiconductor lasers enable optical frequency modulation through injection current control, practical implementations reveal significant tuning nonlinearities induced by thermal effects [5], which fundamentally limit the system’s ranging precision.

Tuning nonlinearity compensation approaches can be categorized into software-based and hardware-based methodologies. Software solutions, including resampling techniques [6] and time-frequency nonlinearity compensation [7], demonstrate implementation efficiency but remain constrained by low-order nonlinear approximations. This fundamental limitation results in accumulating errors during extended-range high-speed measurements, ultimately degrading range resolution and accuracy. Hardware approaches employ optical frequency comb (OFC) [8,9,10] and phase-locked loop (PLL) technology [11,12,13]. The OFC could serve as a frequency reference to segment and recombine the measurement signal in a common-path interferometric optical configuration. However, their operational effectiveness depends critically on maintaining high signal-to-noise ratios (SNR) and constrained modulation speeds. In contrast, electro-optic phase-locked loops (EO-PLL) technology controls the frequency modulation of semiconductor lasers through an optoelectronic negative feedback, has shown remarkable success in suppressing semiconductor laser tuning nonlinearities, leading to its extensive adoption in FMCW LiDAR systems. The EO-PLL can be implemented through either digital circuits or purely analog architectures, with the former offering enhanced design flexibility and the latter achieving superior cost-effectiveness.

Despite these advancements, tuning nonlinearities persists as a critical challenge for long-range FMCW LiDAR applications, substantially compromising measurement accuracy. To address the limitation, this study investigates the impact of higher-order terms in beat signals during non-uniform discrete Fourier transform (NUDFT) processing and proposes a novel high-precision ranging methodology combining EO-PLL with NUDFT. The implemented solution employs a two-stage compensation approach: First, a phase-fitting-based pre-correction algorithm generates compensation signals for preliminary nonlinearity correction. Subsequently, real-time linearization of laser tuning is achieved through EO-PLL control, effectively suppressing higher-order nonlinear terms in beat signals as evidenced by significantly reduced full-width-at-half-maximum (FWHM) in the obtained distance spectrum. Final distance refinement is accomplished through NUDFT processing at spectral peaks, enabling micron accuracy in non-cooperative target measurements. This dual-stage compensation architecture demonstrates exceptional ranging precision across extended measurement distances.

2. Theory

In FMCW LiDAR, the optical frequency is modulated continually, which can be expressed as,

$$f(t)=f_0+{\displaystyle {\int }_{t_0}^{t}a(t^{\prime })d}{t}^{\prime }$$

(1)

Optical frequency f(t) starts with f₀ and changes with tuning rate a(t), which varies over time when injecting a linearly varying current into a semiconductor laser.

The FMCW optical path typically consists of a Mach–Zehnder interferometer (MZI) as illustrated in Figure 1. The light emitted by a semiconductor laser is split using an optical coupler and then goes to reference and measurement arms disparately, which can be expressed as

$$\left\{\begin{array}{l}{E}_r(t)=A_r{e}^{j{\varphi }_r(t)}\\ E_m(t)=A_m{e}^{j{\varphi }_m(t)}\end{array}\right.$$

(2)

where the electric fields of reference and measurement light are E_r(t) and E_m(t), with amplitudes A_r and A_m, respectively. The phases of the reference and measurement light are ${\varphi }_r(t)$ and ${\varphi }_m(t)$, expressed as Equation (3)

$$\begin{array}{l}{\varphi }_m(t)={\varphi }_r(t-\tau )\\ \tau ={\tau }_m-{\tau }_r\end{array}$$

(3)

where ${\tau }_r$ and ${\tau }_m$ are their group delays. Perform ${\varphi }_r(t-\tau )$ Taylor expansion, as depicted in Equation (4).

$${\varphi }_m(t)={\varphi }_r(t)-2\pi \tau f(t)+\pi {\tau }^{2}a(t)-{\displaystyle \frac{1}{3}}\pi {\tau }^3{\displaystyle \frac{da(t)}{dt}}\dots$$

(4)

Subsequently, the phase of the beat signal is

$$\begin{array}{ll}{\varphi }_M(t)& ={\varphi }_m(t)-{\varphi }_r(t)\\ & =-2\pi \tau f(t)+\pi {\tau }^{2}a(t)-{\displaystyle \frac{1}{3}}\pi {\tau }^3{\displaystyle \frac{da(t)}{dt}}\dots \end{array}$$

(5)

The higher-order terms in the phase, such as πτ²a(t), introduce complex nonlinear components into the phase of the interference signal. Due to the difficulty of directly compensating for these nonlinearities through numerical computation methods, a combined approach incorporating both hardware and software solutions is necessary to achieve high-precision distance measurements.

3. Approach

3.1. Pre-Correction

To address laser tuning nonlinearities, a viable compensation strategy involves real-time laser frequency monitoring through auxiliary interferometry

$$I_{aux}(t)=A_{aux}\cos \left[{\varphi }_A(t)\right]\approx A_{aux}\cos \left[2\pi {\tau }_{A}f(t)+{\varphi }_0\right]$$

(6)

However, the inherent nonlinear tuning characteristics embedded in semiconductor lasers during manufacturing processes, additional correction measures become necessary. This study implements a phase-fitting-based pre-correction method. Specifically, a specialized driving chip is utilized to convert voltage signals into laser injection currents, and a correlation between the input voltage and the laser’s optical frequency can be established

$$f_{dfb}=f\left[u(t)\right]$$

(7)

The tuning rate can be expressed as,

$$a(t)=d{f}_{dfb}/dt=df\left[u(t)\right]/dt$$

(8)

the beat signal from the auxiliary interferometer is collected by the balance photodetector, and the frequency is considered as,

$$f_b(t)=a(t){\tau }_A={\tau }_A{\displaystyle \frac{df\left[u(t)\right]}{dt}}={\tau }_{A}K(t){\displaystyle \frac{du(t)}{dt}}$$

(9)

K(t) is used to characterize the nonlinear characteristics of the semiconductor laser. It is hoped that f_b(t) is controlled by voltage rather than voltage difference, thus, an integrator is introduced

$$\begin{array}{l}u(t)={\displaystyle {\int }_{t_0}^t{u}_d}(t^{\prime })d{t}^{\prime }\\ f_b(t)={\tau }_{A}K(t)u_d(t)\end{array}$$

(10)

u_d(t) represents the control signal for the semiconductor laser, and a voltage-controlled oscillator (VCO) is constructed. For the instantaneous frequency f_b(t), the following steps are needed.

First, acquire the beat signal I_aux(t), and obtain the complex signal by the Hilbert transformation

$$X(t)=I_{aux}(t)+j{\tilde{I}}_{aux}(t)$$

(11)

and

$${\varphi }_A(t)=unwrap\left[\arctan ({\displaystyle \frac{{\tilde{I}}_{aux}(t)}{I_{aux}(t)}})\right]$$

(12)

Second, ${\varphi }_A(t)$ undergoes filtering and is fitted by a polynomial using the least squares method, the post-fitting phase is represented as ${\varphi }_{A^{\prime }}(t)=a{t}^3+b{t}^2+ct+d$. During laser frequency modulation measurements, the beat signal exhibits instability during both the initial up-sweep and final down-sweep phases of the laser tuning process. This instability induces abrupt phase discontinuities when applying the Hilbert transform for phase extraction. Such phase jumps substantially compromise the integrity of the pre-correction signal, rendering it incapable of accurately representing the laser’s inherent tuning characteristics. Therefore, in practice, the central portion of the beat signal (as illustrated in Figure 2a) is typically selected as the region of interest (ROI) to ensure reliable data analysis.

Typically, points outside the ROI would be discarded [14,15]. However, it can result in a pre-correction signal that is difficult to accurately reflect the tuning characteristics of the laser on a point-by-point basis. Adopting the least squares fitting method can effectively overcome this adverse effect, enabling point-to-point modulation of the laser’s instantaneous tuning rate and generating a smooth pre-correction signal. Therefore, this fitting is necessary. From ${\varphi }_{A^{\prime }}(t)$ the frequency of the beat signal f_b(t) can be calculated according to the sample rate.

Third, to achieve the expected beat signal frequency $f_{\xi }$, u_d(t) needs to be updated by

$$u_{d\xi }(t)={\displaystyle \frac{f_{\xi }}{f_b(t)}}{u}_d$$

(13)

To account for measurement error, multiple repetitions are necessary. It is evident from Figure 2b,c that when utilizing an integrator driven by a square wave signal to control the laser, the phases of the beat signal during the up-sweep and down-sweep exhibit nonlinear characteristics. Specifically, the 1 − r² during the up-sweep and down-sweep are 3.43 × 10⁻² and 1.35 × 10⁻², respectively. After applying pre-correction measures, the 1 − r² of the up-sweep and down-sweep are significantly reduced to 6.17 × 10⁻⁴ and 8.15 × 10⁻⁴, respectively, as shown in Figure 2e,f.

3.2. Nonlinear Correction Based on EO-PLL and NUDFT

Although nonlinearity still exists after pre-correction, it has been controlled within an acceptable range for the EO-PLL. To further suppress the tuning nonlinearity, an EO-PLL has been designed, which employs an integer clock generator chip as a phase detector and an RC filter as the loop filter (LPF), with a loop bandwidth of 100 kHz (relative to a reference frequency of 2 MHz). The photodetector collects the beat signal from the auxiliary interferometer, which is converted into a square wave by a high-speed comparator. The square wave, together with the reference signal generated by the Direct Digital Synthesizer (DDS), serves as the input for the phase detector, as Figure 1 shows. The phase detector contains a charge pump circuit that outputs a bipolar constant current. In cooperation with the low-pass filter, the loop gain can be elevated to infinity, thereby facilitating precise measurements of phase difference. The output from the low-pass filter is merged with the pre-correction signal via an adder, as illustrated in Figure 1. Following integration by an integrator, the output signal steers laser tuning, thus aiding the lock-in process of the phase-locked loop. The transfer function for the EO-PLL is delineated as follows:

$$H(s)={\displaystyle \frac{K_{d}Z(s)K_v}{s+K_{d}Z(s)K_v}}$$

(14)

K_d is the gain of the phase detector, Z(s) represents the transform of the filter, and K_v is the gain of the VCO, which comprises an auxiliary interferometer and an integrator, and is also specified. Figure 3 presents the spectrum of the beat signal from the auxiliary interferometer. This spectrum was obtained with a triangular waveform, a pre-corrected signal, and an EO-PLL.

The results indicate that the FWHM of the beat signal spectrum after phase locking has been reduced from 260 kHz (by pre-correction) to 2 kHz, and the frequency remains stable on the whole. After phase-locking control of the laser, the frequency point corresponding to the measurement target can be obtained by performing a Fast Fourier transform (FFT) on the beat signal of the measuring interferometer.

However, since the frequency resolution obtained by the FFT is determined by the number of sampling points and sampling rate, it is difficult to achieve micrometer-level distance resolution. The Chirp Z-transform has the capability to refine the spectrum, yet it is prone to the influence of tuning nonlinearity, which may compromise its accuracy. In contrast, NUDFT offers an effective solution to compensate for tuning nonlinearity, thereby enabling high-precision target distance extraction [16].

$$F(R_M)=|{\displaystyle \sum _{n=0}^{N-1}{I}_M}(n)e^{-j{R}_M/R_A{\varphi }_A(n)}|$$

(15)

R_M and R_A represent the optical path difference (OPD) of the measurement interferometer and auxiliary interferometer, respectively. The NUDFT algorithm reconstructs the beat signal of the measurement interferometer using the phase of the auxiliary interferometer, based on the maximum likelihood principle. When R_M is closest to the real range of the target, F(R_M) can take the maximum value. While the conventional DFT algorithm exhibits O(N²) complexity, FMCW laser ranging often necessitates refining merely 200–300 frequency points near the FFT peak. NUDFT computation can be significantly accelerated via GPU/FPGA implementations.

4. Experiments and Results

A measurement system has been constructed in accordance with the configuration outlined in Figure 1. The system employs a Distributed Feedback laser (DFB, FITEL’s FRL15TCWx-D86-19610A) with a central wavelength of 1550 nm, a linewidth of 10 MHz and a modulation range of 200 GHz. Single-mode fiber (SMF) is utilized as the optical path, and it is controlled by an incubator to operate at 25 degrees Celsius. Prior to ranging a target at a long distance, calibration is necessary. Collimating lenses are employed to collimate the divergent light emanating from the fiber end face, which is then irradiated onto a corner cube prism mounted on a slider. The slider moves along a guide rail, and on the opposite side of the guide rail, a Renishaw laser interferometer (Renishaw ML10, with an ultra-high accuracy of 0.7 μm/m) is positioned to measure the relative displacement of the slider. Initially, an origin point is established, and the slider is moved to five distinct positions. The results obtained from the laser interferometer, as well as the beat signals from both the measurement optical path and the reference optical path, are recorded. These results are presented in

Table 1. OPD calibration of the auxiliary interferometer.

Position (mm)	${\mathit{\tau }}_{\mathbf{M}}/{\mathit{\tau }}_{\mathit{A}}$	${\mathit{\tau }}_{\mathit{A}}$ (ns)	${\mathit{R}}_{\mathit{A}}$ (mm)
0.0000	0.1759	-	-
198.5563	0.4060	5.7572	1725.9536
398.9355	0.6381	5.7574	1726.0380
606.7893	0.8790	5.7570	1725.9160
805.2120	1.1089	5.7573	1725.9851
1004.0833	1.3394	5.7571	1725.9463

. The OPD of the auxiliary interferometer is 1725.9678 mm by calibration.

To validate the relative displacement measurement accuracy, the corner cube prism was displaced along the linear guide rail. With the first position set as the zero reference point, the results of FMCW laser ranging were compared with those obtained from the laser interferometer. The measurement error remains within 10 μm throughout the experimental procedure, as shown in Figure 4a. The tuning nonlinearity of the semiconductor laser has been significantly suppressed by EO-PLL; however, it cannot be completely eliminated. Compared to the Chirp-Z transform, the spectrum generated by the NUDFT exhibits no discernible influence from nonlinearity, especially higher-order terms in the phase, as illustrated in Figure 4b. The effectiveness of NUDFT in compensating for tuning nonlinearity is demonstrated.

Under a fixed laser output power of 0.368 mW, three-tier reflectivity targets (50%, 5%, 2%) with Lambertian scattering characteristics were interrogated at 1 m range using a single-shot acquisition time of 1 ms. The NUDFT range spectra corresponding to the beat frequency signals of the targets are presented in Figure 5. Slight positional discrepancies introduced during the manual placement of the targets under test manifest as observable shifts in the peak positions of the range spectra.

Statistical analysis of 100 sequential measurements revealed that while the standard deviation (

Table 2. Measurements of the standard deviation of the three-tier reflectivity targets.

Reflectivity	Standard Deviation (μm)
50%	1.710
5%	2.999
2%	4.228

) scaled inversely with target reflectivity, the system consistently maintained high repeatability precision throughout the experiments.

Place the target aluminum alloy board near 19 m, as shown in Figure 6. A focusing lens is used to focus the light emitted from the fiber end-face. Considering the limitation of the experimental site size, a reflector is used to extend the measurement distance.

Select a total of 12 positions, measure each position 100 times, calculate the mean and standard deviation, and obtain the distance spectrum by NUDFT as shown in Figure 7. The measurement results are shown in

Table 3. Measurements of the target at 12 locations.

Ordinal	Measurement Value (m)	Standard Deviation (μm)
1	18.001	14.837
2	18.096	11.692
3	18.198	13.583
4	18.297	17.562
5	18.400	9.856
6	18.498	15.365
7	18.594	11.771
8	18.694	16.883
9	18.795	10.275
10	18.892	16.794
11	18.995	15.985
12	19.094	17.453

. According to the result of ranging, the standard deviation of ranging near 19 m is under 18 µm. The proposed methodology, building upon EO-PLL, achieves specified measurement accuracy in target distance determination through signal processing based on NUDFT.

5. Conclusions

In this paper, a pre-correction method based on phase least-squares fitting is introduced in detail. This method involves least squares fitting for phase correction and instantaneous frequency extraction. By calculating the desired driving signal according to the target frequency and using this signal to drive the laser, the linearity of the laser output optical frequency is significantly improved. Furthermore, an EO-PLL is integrated into the laser’s control loop, effectively suppressing the tuning nonlinearity of the semiconductor laser that broadens the spectrum of the beat signal. The NUDFT is then applied to compensate for the tuning nonlinearity and further refine the beat signal spectrum, enhancing the measurement resolution. Experimental results demonstrate a measurement standard deviation under 18 µm at a target distance of 19 m. High-precision measurements can also be achieved for targets made of plastics, gypsum, and other materials in addition to aluminum alloys. To mitigate the potential impact of vibration on measurement results, a compensation method is employed by combining the target frequencies during up-sweep and down-sweep to counteract the Doppler effect. Therefore, the proposed system holds great potential for applications in large-scale industrial surveying and mapping.