Frequency-modulated continuous-wave (FMCW) light detection and ranging (LiDAR) is an advanced imaging and sensing technology that offers several advantages over traditional direct time-of-flight (dToF) LiDAR [1–3] and amplitude-modulated continuous-wave (AMCW) LiDAR [4–6]. Due to its coherent detection mechanism, FMCW LiDAR provides enhanced sensitivity, along with improved resistance to environmental interference [7]. Furthermore, FMCW LiDAR can simultaneously measure both distance and velocity through the Doppler effect at a high axial resolution and signal-to-noise ratio (SNR), enabling the generation of 4D imaging point clouds [8]. This capability makes it highly valuable for various applications in fields, such as robotics, autonomous driving [9], remote sensing [10], and atmospheric spectroscopy [11].

For FMCW LiDAR systems, achieving high lateral resolution and fast scanning rates are crucial criteria for most of the applications. This places greater demands on tunable laser sources, which serve as the core components in the systems. Vertical-cavity surface-emitting lasers (VCSELs) have emerged as a cost-effective alternative due to their small footprint, wafer-scale testing capabilities, and high energy efficiency [12]. By integrating a micro-electro-mechanical-systems (MEMS)-actuated mirror on top of a VCSEL [13], the wavelength can be easily tuned by applying a single control voltage to a compact, monolithic device. This integration facilitates low-cost manufacturing and low power consumption [14], while offering continuous, mode-hop-free, and deterministic tuning characteristics [15]. In addition, tunable MEMS-VCSELs have demonstrated a wavelength tuning range and a sweep rate exceeding 150 nm [16] and 500 kHz [17,18], respectively. As a result, fast and widely tunable MEMS-VCSELs have been developed rapidly in recent years, establishing themselves as promising candidates for FMCW LiDAR applications.

When a tunable MEMS-VCSEL operates at fast sweep rates with wide tuning ranges, the tuning rate, a specific parameter for tunable lasers determining the potential system performance expressed in GHz/μs, becomes significantly large. Consequently, the beat frequency for a distant target after coherent detection rises substantially, thereby heavily increasing the burden on the detector bandwidth and the data acquisition sampling rates, as dictated by the Nyquist–Shannon sampling theorem [19]. This additional load results in substantial system costs and challenges for real-time FMCW LiDAR applications. Therefore, down-shifting the beat frequency through system optimization becomes essential for alleviating the requirement on high detector bandwidth and sampling rates, hence effectively reducing acquired data volume and thereby enhancing data efficiency and overall system performance.

Various approaches have been proposed for beat frequency down-shifting in FMCW LiDAR systems. Acousto-optic modulators (AOMs) [20–22] and acousto-optic deflectors (AODs) [23] introduce optical frequency shifts of the laser passing through the devices via acoustic frequency, with AODs additionally serving as solid-state beam scanners in free space for altering the deflection angle of collimated light. While AOMs and AODs can offer adjustability based on measurable distance and velocity ranges, the frequency shifts induced by acoustic waves are typically in limited range, which is inadequate for addressing beat frequencies over several hundreds of MHz. Additionally, AOMs and AODs contribute to increased system size and cost.

An alternative method involves using an electrical frequency mixer, where the beat frequency is combined with a swept optical frequency over time to identify a moment at which the two frequencies become the same for beat frequency extraction [24]. However, this method is primarily effective in scenarios where a specific frequency range is desired and ignores nonlinear laser frequency sweep. Moreover, the trade-off between measurable frequency and axial resolution, along with insertion or conversion of electrical loss induced by the mixer, restricts the practical implementation. Another method leverages optical parametric-assisted frequency modulation [25,26], where a four-wave mixing process in a highly nonlinear medium enables frequency down-shifting by doubling the tuning rate of the idler light. Nevertheless, the additional laser components increase system complexity, and the interference from the unsuppressed pump and signal may induce additional noise and thus degrade the SNR of the beat signal. High optical power from adjacent pump frequencies may also significantly impact the idler-derived beat signal. A further approach involves applying a programmable voltage signal to an electro-optic modulator to produce discrete optical frequency tuning through a stepped-frequency-swept laser [27], but with a deteriorated axial resolution.

In this study, we introduce a low-cost, fiber-based Mach–Zehnder interferometer (MZI) integrated into the FMCW LiDAR system to address the challenges of (1) an excessively high beat frequency that hinders real-time imaging, (2) the limited beat frequency down-shifting range, and (3) the increased system complexity and cost associated with existing approaches. The MZI provides arbitrarily adjustable frequency shifts by altering the fiber delay length, effectively down-shifting the beat frequency while enabling the tunable MEMS-VCSEL to operate at an exceptionally high tuning rate of 146.88 GHz/μs. Additionally, an interpolated modulation voltage and a second MZI are employed to correct nonlinear laser frequency sweep and compensate for laser phase noise. By incorporating a semi-solid-state 2D beam scanning structure, the dual-MZI-based FMCW LiDAR system achieves high-resolution, ultra-fast real-time 3D imaging, as well as 4D imaging. This study demonstrates the effectiveness of the MZI-based beat frequency down-shifting structure in significantly alleviating the burden on the detector bandwidth and data acquisition, thereby improving data efficiency and advancing real-time 4D FMCW LiDAR with minimal system complexity.

Figure 1 illustrates the coherent detection principle of FMCW LiDAR, demonstrating the variation in optical frequency over time for both the local oscillator and received signals. The time delay τ between these signals corresponds to the time-of-flight of light in free space, which is related to the target distance d by τ=2d/c, where c is the speed of light. The interference between the local oscillator and received signals occurs at the photodetector, producing a beat signal with a frequency of fb=γτ, where γ denotes the tuning rate. Consequently, the measurement of τ is the determination of the beat frequency [23].

For a stationary target, the beat frequency remains consistent across both up and down ramps as fb, enabling the calculation of the target distance using d=cfb/(2γ). While the target is moving, a Doppler frequency shift fd is introduced, resulting in distinct beat frequencies, fb,up and fb,down, for the up and down ramps, respectively, expressed as {fb,up=fb−fdfb,down=fb+fd.(1)

In this scenario, the beat frequency over a triangular sweep period is analyzed to simultaneously derive the target distance d and velocity v as d=c2γfb=c2γfb,up+fb,down2,(2)v=λ2fd=−λ2fb,up−fb,down2,(3)where λ is the central wavelength of the laser frequency sweep. Notably, the direction of v can be determined from the sign of Eq. (3). Furthermore, as the minimum frequency resolution δf of a Fourier transform is inversely proportional to the chosen time duration T [28], i.e., δf=1/T, the theoretical resolutions for d and v are given by δd=c2γδfb=c2γ·1T=c2Δf,(4)δv=λ2δfd=λ2T,(5)where Δf represents the laser frequency sweep range.

To down-shift the beat frequency, the swept laser is routed through a 50/50 fiber coupler to a fiber-based MZI. The principle of beat frequency down-shifting and the laser frequency sweeps for different fiber paths is demonstrated in Fig. 2(a). In the MZI, the up-ramp optical signals in paths 1 and 2 are expressed as E1(t)=η1Aei[2πf0(t−τ1)+πγ(t−τ1)2],(6)E2(t)=η2Aei[2πf0(t−τ2)+πγ(t−τ2)2],(7)where A is the amplitude of the laser, f0 is the initial optical frequency, η1 and η2 are the fiber attenuation coefficients, and τ1 and τ2 are the fiber delay times for paths 1 and 2, respectively.

The optical signals from paths 1 and 2 are coupled and transmitted through a 95/5 fiber coupler for coherent detection, where 95% of the laser is routed via a circulator for ranging. Subsequently, the reflected laser is collected and transmitted through path 3, denoted as $$\begin{gathered} E3(t)=η1η3Aei[2π(f0+fd)(t−τ1−τ3)+πγ(t−τ1−τ3)2] \\ +η2η3Aei[2π(f0+fd)(t−τ2−τ3)+πγ(t−τ2−τ3)2], \end{gathered}$$(8)where η3 and τ3 are the fiber and free-space attenuation coefficient and the fiber delay time in path 3, respectively. Concurrently, 5% of the laser serves as the local oscillator signal, routed through path 4 with a fiber delay time of τ4 and given by $$\begin{gathered} E4(t)=η4[E1(t−τ4)+E2(t−τ4)] \\ =η1η4Aei[2πf0(t−τ1−τ4)+πγ(t−τ1−τ4)2] \\ +η2η4Aei[2πf0(t−τ2−τ4)+πγ(t−τ2−τ4)2], \end{gathered}$$(9)where η4 is the fiber attenuation coefficient of path 4. The fiber delay in path 4 is employed to balance the delay in path 3 and thereby calibrate the zero-distance point for ranging. The optical signals from these two paths are coupled at a 50/50 coupler and interfered at a balanced photodetector (BPD) to generate the beat signal. Ignoring the high order and constant terms, the detected beat signal for up ramp Bup(t) is represented as $$\begin{gathered} Bup(t)=[E3(t)+E4(t)]2 \\ =η1η2(η32+η42)A2cos(2πγτMZIt) \\ +(η12+η22)η3η4A2cos[2π(γτ−fd)t] \\ +η1η2η3η4A2cos[2π(γτ+γτMZI−fd)t] \\ +η1η2η3η4A2cos[2π(γτ−γτMZI−fd)t], \end{gathered}$$(10)where τMZI is the fiber time difference of the MZI (expressed as τMZI=τ2−τ1=nLMZI/c), τ2>τ1, LMZI is the fiber length difference of the MZI, n is the fiber refractive index, and τ=τ3−τ4 with τ3>τ4. Similarly, the detected beat signal for down ramp Bdown(t) is written as $$\begin{gathered} Bdown(t)=η1η2(η32+η42)A2cos(2πγτMZIt) \\ +(η12+η22)η3η4A2cos[2π(γτ+fd)t] \\ +η1η2η3η4A2cos[2π(γτ+γτMZI+fd)t] \\ +η1η2η3η4A2cos[2π(γτ−γτMZI+fd)t]. \end{gathered}$$(11)

In Eqs. (10) and (11), the beat signal primarily comprises four distinct frequency components for a stationary target, among which γτ−γτMZI is the desired down-shifted beat frequency. Since BPD generally incorporates a low-pass filter (LPF), the components γτ+γτMZI and γτ are suppressed as high-frequency signals and become undetectable. To address potential misleading arising from γτMZI during beat spectral peak identification and to ensure that γτ−γτMZI remains positive, as detectors are insensitive to frequency signs, the relationship between τMZI and τ must satisfy the condition γτ−γτMZI<γτMZI<γτ, i.e., τ/2<τMZI<τ. Consequently, LMZI is adjusted to fulfill d/n<LMZI<2d/n for different target distances. For instance, considering a target distance of 1 m and n=1.468, LMZI is adjusted between 0.68 m and 1.36 m. Assuming γ=120MHz/μs and setting LMZI=1m, the simulated beat spectra with and without a 300-kHz cutoff-frequency Butterworth LPF in BPD are plotted in Fig. 2(b). The results demonstrated the relative magnitude of the four distinct frequency components. With the LPF applied, the component corresponding to γτ−γτMZI is preserved, while the other three frequency components are effectively filtered out. Additionally, for a moving target, fd is constrained by γτ−γτMZI rather than γτ, i.e., fd<γ(2d−nLMZI)/c. Hence, the measurable velocity is restricted by the target distance and the fiber length difference of the MZI, as given by v<γλ(d−nLMZI/2)/c.

The schematic diagram of the proposed dual-MZI-based FMCW LiDAR system is shown in Fig. 3(a). The tunable MEMS-VCSEL (BW10-1550-T-PSFA, Bandwidth10) employed in our work utilizes a high-contrast grating (HCG) mirror as the top mirror instead of a conventional DBR-based movable mirror [14]. Due to the thinner structure of the HCG, the footprint of the 1550-nm tunable MEMS-VCSEL is compact enough to reduce parasitic capacitance, achieving a sweep rate exceeding 10 kHz and a maximum tuning range of 8 nm with a single modulation voltage corresponding to a typical tuning rate of 19.96 GHz/μs. An arbitrary function generator (AFG, AFG3011C, Tektronix) provides modulation voltages ranging from 1 V to 19 V with a 10-μs period, corresponding to a sweep rate of 100 kHz. This results in a wavelength sweep from 1557.7510 nm to 1551.8290 nm, as measured by an optical spectrum analyzer (AQ6370D, Yokogawa). Consequently, the tuning range is 5.9220 nm, and the tuning rate is 146.88 GHz/μs.

In an ideal scenario, the tunable MEMS-VCSEL exhibits a perfectly linear laser frequency sweep, as depicted in Fig. 1. However, due to the intrinsic nonlinearity between the output wavelength and the modulation voltage, as shown in Fig. 3(b), a linear modulation voltage leads to a nonlinear frequency sweep [29]. This nonlinearity is exacerbated at high tuning rates, causing spectral energy diffusion after Fourier transform and adversely affecting the accurate determination of the beat frequency.

To address this repetitive nonlinearity, a proper pre-distorted modulation voltage is applied. As the equal modulation voltage intervals correspond to the nonlinear optical frequency, the pre-distorted voltage is derived by interpolating the characterized relationship between the optical frequency and the modulation voltage with equal optical frequency intervals. Figure 3(c) presents the modulation voltage and the corresponding corrected optical frequency, demonstrating a reduction in relative nonlinearity from 14.32% and 11.62% to 4.37% and 5.86% for the up and down ramps, respectively. Further reduction of relative nonlinearity is constrained by the trade-off with the tuning range, as the pre-distorted modulation voltage required for achieving minimal relative nonlinearity would inherently limit the overall tuning range of the MEMS-VCSEL at high sweep rates, such as 100 kHz [30].

Subsequently, 95% of the linearized swept laser is transmitted to MZI 1 for beat frequency down-shifting and amplified using an erbium-doped fiber amplifier (EDFA, EDFA-D-O-23, Fiber-Photonics). In the coherent detection path, 95% of the laser is routed via a circulator and collimated by a collimator into a transmission grating, dispersing the swept laser along the fast vertical axis (y-axis). A rotating galvo mirror, driven by a sine-wave voltage, reflects the dispersed laser onto the target along the slow horizontal axis (x-axis), enabling 2D beam scanning and 4D imaging after retrieving d and v along the z-axis. The 2D beam scanning pattern is shown in Fig. 3(d), illustrating the spatial correspondence between the wavelengths and the target coordinates. The tuning range Δλ from λ1 to λ2 determines the field of view (FOV) along the y-axis (FOVy), while the sine-wave voltage from −ugm to ugm determines the x-axis FOV (FOVx). The fast-axis scanning rate and frame rate depend on the sweep rate and sine-wave frequency. The 2D beam scanning structure adopts a co-axial detection scheme, preventing adjacent laser beam interference.

In addition to systematic nonlinearity, which is mitigated through interpolated modulation voltage, laser phase noise impacts the imaging performance as well. Laser phase noise, also characterized by the linewidth of the laser, arises primarily from spontaneous emission within the active regions [31] and mechanical fluctuations of the MEMS structure induced by Brownian motion [12]. These factors cause random fluctuations in optical and thus beat frequencies after coherent detection [32], degrading the SNR of the beat signal with increasing target distance, SNR=10log10[Tτc·1e2ττc−(1+2ττc)],(12)where τc is the coherence time determined by the laser linewidth [33]. Notably, a swept laser exhibits greater phase noise compared to when its operation is under a constant voltage, which is inevitable, particularly at higher tuning rates [34].

Due to the non-repetitive character, laser phase noise cannot be corrected through pre-distortion. Instead, an additional MZI 2 with a fixed fiber time difference of τMZI2 is introduced. Here, 5% of the tunable MEMS-VCSEL laser is routed to MZI 2, following a long fiber delay for alignment with the fiber delay before the other 95/5 fiber coupler.

Since laser phase noise is inherently stochastic, the trajectory shown in Fig. 4(a) represents a random realization rather than a deterministic waveform. The coherence time τc illustrated in Fig. 4(a) denotes the average length of time over which the optical frequency or phase remains statistically correlated. Within this duration, the optical frequency sweep can be considered quasi-linear, despite underlying noise contributions [35].

Since τ<τc and spectral windows are applied to retrieve pixel depths along the dispersion-scanned y-axis, the optical frequency sweeps associated with different paths in Fig. 4(b) remain approximately linear within τc. These sweeps exhibit nearly parallel trajectories over the duration, characterized by an effective tuning rate γ′, which incorporates the influence of the laser phase noise. Consequently, the relationship of (fBPD1,up+fBPD1,down)/[2(τ−τMZI1)]=fBPD2/τMZI2 is established, where fBPD1,up and fBPD1,down are the beat frequencies of up and down ramps detected by BPD 1 (PDB570C, Thorlabs), and fBPD2 are the beat frequencies detected by BPD 2 (PDB570C, Thorlabs). The compensated distance d is thereby derived as $$\begin{gathered} d=cτMZI22fBPD2fBPD1,up+fBPD1,down2+cτMZI12 \\ =nLMZI2(fBPD1,up+fBPD1,down)4fBPD2+nLMZI12. \end{gathered}$$(13)

It is worth noting that the laser phase noise influences only the beat frequency associated with the time-of-flight and does not affect the Doppler shift. Thus, velocity calculations remain consistent with Eq. (3).

Finally, the beat signals from BPD 1 and BPD 2 are sampled using a high-speed oscilloscope (OSC, MSO73304DX, Tektronix) and transferred to a PC for digital signal processing.

To validate the system’s high-resolution imaging capabilities, three 3D-printed epoxy resin objects and a flat panel covered with highly reflective sheets were employed as targets, as illustrated in Fig. 5(a). Among the objects, a large cube with a 4.5-cm edge length and a smaller cube with a 3-cm edge length were imaged, focusing on the front plane (positioned 80 cm from the system), slopes, and edge. A cylinder with a 4.5-cm diameter and height was also imaged for surface details. All objects were axially separated by 10 cm in front of the flat panel, which served as the background. The front face of the large cube, the plane of the smaller cube featuring face and space diagonals, and the background were aligned within the x–y plane.

With LMZI1=1.00m and LMZI2=10.60cm, the measured and reference beat signals detected by BPD 1 and BPD 2 were sampled and plotted in Figs. 5(b) and 5(c). Instead of performing a Fourier transform on the entire beat signal for a single ramp, an overlapped short-time Fourier transform (STFT) was applied to extract the target depths [36]. The number of pixels along the fast vertical axis correlates with the spectral window length of the STFT, further determining the trade-off between the lateral resolution along the y-axis and the axial resolution. According to Eq. (4), a larger spectral window length results in greater Δf, providing superior axial resolution but reduced lateral resolution. Here, the optimal spectral window length was determined based on the desired axial resolution. In Figs. 5(b) and 5(c), adjacent spectral windows overlapped by half the length to enhance axial resolution while maintaining consistent pixel numbers. To mitigate the impact of unsatisfactory mixing at the ramp ends, which leads to a suddenly changed beat frequency, only the middle 80% of each ramp was used for the STFT. Utilizing a 2D beam scanning structure, a field of view FOVx×FOVy=22cm×6cm was achieved. Consequently, 39 spectral windows were applied per ramp, resulting in 39 pixels along the y-axis, while 20 non-overlapping windows were segmented within 4 μs. This configuration provided lateral and axial resolutions of 1.54 mm and 5.10 mm, respectively.

Subsequently, the beat signals in each spectral window were zero-padded and Fourier-transformed to derive the beat spectra, shown in Figs. 5(d) and 5(e). These spectra demonstrate the effectiveness of the system in down-shifting beat frequencies from over 700 MHz to below the BPD bandwidth limit. Zero-padding improved the frequency resolution of the beat spectrum, allowing for precise spectral peak localization. Benefitting from the MZI-based beat frequency down-shifting structure, the 100-kHz sweep rate of the tunable MEMS-VCSEL and thus the 200-kHz fast-axis scanning rate were implemented. Moreover, the structure enabled the OSC sampling rate to be reduced to 1.25 GS/s.

Limited by the recording length of the OSC, the sine-wave frequency applied to the galvo mirror was set to 50 Hz, enabling one complete frame to be acquired in a single data acquisition. This resulted in 1999 pixels along the x-axis and a high lateral resolution of 0.11 mm. Figure 5(f) presents the high-resolution 3D imaging result, where scattering pixels below 1 cm along the y-axis were due to the low reflectivity of the platform supporting the objects. Detailed depth deviations for the background, plane, slopes, and surface are analyzed in Figs. 5(g)–5(j), revealing standard deviations of 3.19 mm, 2.70 mm, 2.61 mm, and 2.62 mm, respectively. The larger deviation observed for the background was attributed to the degraded SNR caused by the diminished reflected laser power at farther distances. The mean depth error between the front face of the large cube and the background was 0.27 mm, surpassing the theoretical axial resolution. Depth profiles along the red dashed line in Fig. 5(a) are illustrated in Fig. 5(k), highlighting the surface and contour characteristics of the targets and demonstrating the superior high-resolution 3D imaging capabilities of the system.

Apart from high-resolution imaging, the proposed system also demonstrates ultra-fast imaging capabilities for moving targets, achieving a frame rate exceeding the kHz level. Three acrylic plates, with the letters “T”, “H”, “U” carved into them and covered with highly reflective sheets, were considered as targets, as depicted in Fig. 6(a). These plates were mounted on a rotator with different intervals for spinning and placed at a distance of 130 cm in the x–y plane, ensuring an FOVy coverage exceeding 10 cm. Nevertheless, at this target distance, the beat frequency reached 1.27 GHz, over three times the bandwidth limit of BPD. To address this, LMZI1 was adjusted to 1.60 m. The sine-wave frequency driving the galvo mirror was increased to 1 kHz, while other parameters remained unchanged. This enabled 3D imaging with 39×99 pixels along the y- and x-axes and a frame rate of 2 kHz. Although lateral resolution along the x-axis was sacrificed to achieve this ultra-high frame rate, the axial resolution of the imaging system is preserved.

Setting the rotator spinning at a maximum speed of 1350 r/min, Fig. 6(b) presents 10 frames of 3D imaging results of the spinning letters spanning 11cm×10cm at different times. These results clearly capture the motion, contours, and depth variations of the letters. To further verify the ultra-fast 3D imaging capability with high axial resolution, the target was switched to a highly reflective fan with four tilted blades, as shown in Fig. 6(c). Under the same spinning speed, the shape of the fan at different times is illustrated in Fig. 6(d), with the color bar indicating the depth variations across the blades. Moreover, two 10-second real-time 3D imaging videos corresponding to Figs. 6(b) and 6(d) are provided (see Visualization 1 and Visualization 2), showcasing 100 frames at 0.005×playback speed to emphasize the ultra-fast imaging capabilities of the proposed system.

The proposed system also enables 4D imaging, capturing both depth and velocity information simultaneously. This was demonstrated using a rotator with a stationary flat panel in the background, as depicted in Fig. 7(a). Both surfaces were covered with highly reflective sheets. The rotator, with a radius of r=16cm, was positioned 0.75 m behind the galvo mirror and spun at 1350 r/min, corresponding to a tangential velocity of vt=11.31m/s. A 50-Hz sine-wave voltage was applied to the galvo mirror, enabling scanning along the x-axis with an FOVx coverage of 27.20 cm. The center of the rotator was approximately aligned with the scanning origin of the galvo mirror. The angles α and β represented the orientation between these points and each laser beam.

As the laser scanned the tangential direction of the rotator surface, the maximum angular values were αmax=5.56° and βmax=84.44°. Within the range −5.56°<α<5.56°, the beat spectra at five different angles are shown in Fig. 7(b). At α=0°, the spectral peaks of the up and down ramps overlapped due to the absence of fd and v along the laser propagation axis. As |α| increased, spectral peaks began to split, indicating the presence of fd, although with widened spectral widths caused by the jitter from high-speed rotation.

The 4D imaging results are presented in Fig. 7(c), with a color bar indicating the velocity. By utilizing the beat frequencies for both up and down ramps within a single frequency sweep period, the pixel count along the x-axis was halved to 999 while maintaining a high lateral resolution of 0.27 mm. The high axial resolution effectively captured the contour characteristics of the rotator surface, while the velocity of the background remained at 0 m/s, denoting the stationary state.

To further evaluate velocity measurement performance during one frame of 4D imaging, the velocity distribution of the upper pixels along the x-axis was analyzed, as shown in Fig. 7(d). The theoretical velocity along the laser propagation axis, represented by the black dashed line, was calculated as vtsin(α+β), where β=arcsin[(d/r)sin(α)] [37]. The measured velocity closely matched the theoretical distribution from positive to negative, except at large |α|, where the broadened beat spectra led to deviations, as indicated in Fig. 7(b). To mitigate uneven scanning steps at both sides from the 50-Hz sine-wave voltage, the middle 100 pixels in Fig. 7(d) were segmented for error and deviation analysis, as shown in Fig. 7(e). The results revealed a maximum velocity error within 0.50 m/s and a standard deviation of 0.18 m/s.

In practical FMCW LiDAR applications, the adjustment of LMZI is essential to ensure valid down-shifted beat frequency generation over varying target distances. As described in Section 2, the condition d/n<LMZI<2d/n must be satisfied. This requirement inherently constrains the permissible variation range of d for a fixed LMZI, specifically to a span of nLMZI/2. Introducing an additional tunable fiber length delay ΔLMZI extends this operating range to n(LMZI/2 +ΔLMZI). Accordingly, the choice of different adjustment strategies of LMZI depends on the required variation range of d and the system integration level.

For fine adjustments over millimeter-scale ranges, piezoelectric fiber stretchers offer sub-picosecond to several-picosecond delay control. For instance, a tapered optical nanofiber stretcher actuated by a piezoelectric element has been reported to provide up to 19 ps of optical delay [38], corresponding to a physical fiber delay change of approximately 3.88 mm. Similarly, optical delay lines based on cascaded chirped fiber Bragg gratings (CFBGs) can yield comparable millimeter-scale optical delay through micrometer-strain-induced stretching, though these are typically constrained to applications requiring compact delay variation, such as optical coherence tomography (OCT) [39].

For longer-range adjustments spanning tens of centimeters to meters, mechanical free-space optical delay lines incorporating fiber collimators and movable mirrors provide significant flexibility. Commercial solutions provide tunable optical delays of up to 1.2 m with sub-femtosecond-level repeatable delay shifts and excellent stability. However, these solutions are relatively bulky and slower in response, restricting the suitability for compact or mobile applications.

In our implementation, we adopt a pragmatic and cost-effective strategy by manually configuring LMZI through interchangeable fiber segments of predefined lengths. This modular approach enables discrete yet robust reconfiguration of LMZI, well suited for static or semi-static target distance ranges without introducing mechanical complexity or stability concerns. Such scalability facilitates deployment across diverse platforms, from benchtop prototypes to compact embedded systems, and provides a flexible foundation for future architectural extensions. In prospective on-chip integrated implementations, the manual switching can be replaced by optical switches selecting among predefined waveguide segments, allowing automated and reconfigurable LMZI control while preserving the modular design [40].

To highlight the advantages of the proposed fiber-based MZI approach, a comparative analysis against conventional beat frequency down-shifting techniques, including AOMs, AODs, and electrical mixers, is presented in Table 1. The fiber-based MZI approach offers a compact size, low implementation cost, and a significantly larger frequency shift. By alleviating the burden on the photodetector bandwidth and the data acquisition sampling rate, the proposed method improves overall data efficiency and system scalability. This allows the adoption of integrated low-sampling-rate signal acquisition circuits and facilitates implementation on compact FPGA-based processing platforms. Such integration-oriented compatibility further reduces system complexity and cost, making the approach well suited for data-efficient, real-time 4D imaging FMCW LiDAR systems targeting embedded applications.Table 1.

Comparative Summary of Beat Frequency Down-Shifting Methods in FMCW LiDAR Systems

The proposed dual-MZI-based FMCW LiDAR system demonstrates superior performance, particularly in terms of high tuning rate, resolution, and capability for real-time imaging of dynamic scenes with ultra-high fast-axis scanning rate and frame rate. Table 2 provides a comparative analysis of our system with recent advancements in FMCW LiDAR, focusing on key imaging performance parameters.Table 2.

Comparison of the Recent FMCW LiDAR Systems^a

Compared to conventional systems using dual-axis galvo mirrors, our system achieves a significantly improved axial resolution of 5.10 mm and a higher number of depth pixels of 39×1999 (at 100-Hz frame rate), even when applying spectral windows for depth retrieval. This excels compared to other systems that rely on the entire beat signal within a single ramp [8,22,24,27,37]. While the 100-kHz sweep rate of our system aligns with the flutter-wavelength-swept laser [23] operating at a center wavelength of 1535 nm, our system outperforms them across all key metrics in both static and dynamic imaging scenarios.

Our method achieves a tuning rate of 146.88 GHz/μs, comparable to the state-of-the-art akinetic all-semiconductor swept laser [36], while delivering significantly higher fast-axis scanning rates and frame rates with the same scanning structure. These improvements are attributed to the center wavelength of 1316 nm, which inherently offers a broader tuning range compared to lasers centered at 1550 nm, thereby contributing to enhanced axial resolution and an increased number of depth pixels along the fast axis. Nevertheless, our system achieves a higher overall depth pixel acquisition rate of 7.80 MHz, compared to 7.60 MHz in the system described in Ref. [36].

Table 2 also presents a comparative evaluation of velocity measurement performance. The velocity standard deviation of our system is primarily attributed to unintended vibrations of the rotating target and its inherent ellipticity, which introduce fluctuations in the beat frequency and thereby impair Doppler estimation precision. Furthermore, the theoretical velocity resolution δv, as defined by Eq. (5), plays a crucial role in determining precision. For instance, systems such as in Ref. [22], which exhibit a finer δv in the mm/s level, achieve notably lower standard deviation and velocity error. In contrast, both our system and Ref.  [23] feature larger δv values due to higher sweep rates, leading to increased velocity uncertainty.

The maximum velocity error observed in our system is mainly attributed to two sources, which are the relatively large δv and residual nonlinearity in the laser frequency sweep. Nonlinear frequency sweep broadens the beat spectrum, reducing the accuracy of the spectral peak and thus the Doppler shift estimation. Notably, while fast sweep rates are essential for enabling high axial resolution and ultra-fast real-time imaging, they inherently reduce δv, as governed by Eq. (5), presenting a fundamental trade-off between the velocity resolution and the sweep rate. It is important to clarify that the total laser frequency sweep range does not directly affect the velocity error. For example, despite their limited sweep ranges, systems such as in Refs. [8,22,37] achieve high velocity accuracy owing to their low sweep rates, which lead to narrower δv values.

To further improve velocity accuracy and precision, several strategies can be considered. First, optimized pre-distorted voltage waveforms may be employed to reduce MEMS-VCSEL sweep nonlinearity, thereby narrowing the beat spectra. Second, reducing the sweep rate can effectively increase δv, thereby improving Doppler estimation fidelity. Although lowering the sweep rate may reduce the tuning rate and the fast-axis scanning rate, it simultaneously extends the laser frequency sweep range within each ramp. This trade-off does not compromise the overall frame rate. Instead, under a fixed axial resolution, it results in a redistribution of depth pixels per frame, specifically, increasing the pixel density along the fast vertical axis (y-axis) while decreasing the pixel count along the slow horizontal axis (x-axis). Furthermore, reduced sweep rates inherently lead to lower residual nonlinearity after correction, which, in turn, helps reduce both velocity error and standard deviation. These optimization directions will be explored in future implementations to strike a better balance between precise velocity measurement and high-performance 3D imaging.

In conclusion, we present a dual-MZI-based FMCW LiDAR system to address the critical challenges of high beat frequencies posed by tunable MEMS-VCSELs operating at fast sweep rates and wide tuning ranges. The proposed system enables precise beat frequency down-shifting through adjustable fiber delay lengths of the fiber-based MZI, allowing the tunable MEMS-VCSEL to achieve an exceptional tuning rate of 146.88 GHz/μs. Key contributions include the integration of an interpolated modulation voltage and an additional MZI to correct nonlinear frequency sweeps and compensate for laser phase noise, ensuring high imaging performance. The system achieves state-of-the-art 3D imaging performance metrics, including lateral and axial resolutions of 0.11 mm and 5.10 mm, respectively, at a target distance of 80 cm. Additional features include sub-millimeter-level accuracy and millimeter-level precision, along with a 200-kHz fast-axis scanning rate and a 2-kHz frame rate over a 39×99 pixel frame. Furthermore, it has the capability to produce 4D imaging point clouds with a velocity error below 0.50 m/s and a standard deviation of 0.18 m/s. This cost-effective dual-MZI-based system paves the way for developing data-efficient and real-time 4D imaging FMCW LiDAR, with promising applications in industrial automation and robotics. With continued advances toward photonic integration, the architecture also holds long-term potential for enabling compact high-performance LiDAR systems in AR and VR scenarios, where precise scene reconstruction, high-accuracy motion tracking, and real-time 4D perception are critical.