Ranging and velocimetry measurements by phase-based MTCW lidar

: We propose a complementary phase detection algorithm to enhance the capabilities of the multi-tone continuous wave (MTCW) lidar for single-shot simultaneous ranging and velocimetry measurements. We show that the phase of the Doppler-shifted RF tones and the amount of the induced Doppler frequency shift can be used to extract the phase and velocity information, simultaneously. A numerical case study and experimental work have been performed for the proof of concept. We show that the velocity resolutions are limited by frequency resolution and the ranging resolution is determined by the temporal resolution. Experimentally, we obtain 8.08 ± 0.8cm/s velocity measurement and 111.9cm range measurements with ± 0.75cm resolution in a 6-tone MTCW lidar system.


Introduction
The light detection and ranging (lidar) market is rapidly growing over the past years [1] and the lidar industry is always hungry for new developments in various fields such as autonomous vehicles [2], airborne lidars for terrestrial applications [3], spacecraft [4], and precision measurements [5]. The pulse time-of-flight (PToF) is the most common technique that provides a robust ranging methodology by taking advantage of high peak power pulsed light sources and conventional direct detection methods [6]. However, simultaneous ranging and velocimetry of a moving target are not achievable via PToF lidars without generating and comparing multiple frames of the environment with respect to time [7][8][9], or without renewing the PToF methodology with enhanced configurations [10]. On the other hand, lidars operating in the continuous wave (CW) mode can perform coherent light detection to extract high-resolution single-shot range and velocity information of dynamic targets [11][12][13][14]. In particular, frequency-modulated continuous-wave (FMCW) lidars operate by linearly sweeping the frequency of a CW laser and using the interference of the delayed backscattering signal with the local oscillator to produce a beating note, which corresponds to the target distance [15]. By employing triangular waveform frequency modulation, FMCW lidars can yield the Doppler shifts induced by target velocity, simultaneously [10]. Even though the FMCW lidars can perform single shot ranging and velocimetry, the performance of the measurement highly depends on the scan duration, linearity of the frequency sweep, and frequency span that the source scans [14][15][16]. Eventually, the type of application and the required measurements, cost of the system, and resolution requirements dictate the type of the lidar suitable for the given application.
As an alternative to the current lidar technologies, we have previously introduced the multitone continuous wave (MTCW) lidar that operates in CW mode and eliminates the need for frequency sweeping [17][18][19][20][21]. The MTCW lidar utilizes the different phase shifts acquired by static radiofrequency (RF) tones imposed on the CW carrier. In particular, the coherent detection at the receiver end converts these frequency-dependent phase shifts into amplitude variations at RF tones; hence, allows us to predict the range of the target from these amplitude variations in a single shot measurement [18,19]. The ranging resolution of the system is determined by the span of the static RF tones and signal-to-noise ratio (SNR) at the receiver. Since there is no need for frequency, phase, or amplitude sweeping, it is suitable for conventional off-the-shelf light sources.
We have already demonstrated the static target ranging via the MTCW lidar [17,18], as well as the ranging of slow-moving targets [19]. However, the simultaneous ranging and velocimetry measurements are affected by the speed of the target in the MTCW lidar, because the desired interference diminishes and new RF beating tones appear if the resultant Doppler shifts are bigger than the optical carrier linewidth. Since the coherence length of the light is determined by the linewidth of the laser, and narrow linewidth lasers are compulsory for long-range applications, the MTCW methodology should be enhanced to address the Doppler shifts caused by the moving targets with arbitrary velocities in practice [19]. Also, the amplitude variations are sensitive to the relative signal amplitudes at the reference arm and the collection arm.
In this manuscript, we propose a complementary phase detection algorithm to enhance the capabilities of the MTCW lidar for single-shot simultaneous ranging and velocimetry measurements. As described in the original MTCW approach [17][18][19], the range information of the target is stored in the phases of the individual RF tones. Here, we show that instead of focusing on the amplitude variations, the phase of the Doppler-shifted RF tones and the amount of the induced Doppler frequency shift can be used to extract the range and velocity information, simultaneously. Specifically, we present the distribution of tones, their phases, and the amplitude information, and how we can utilize these to enhance the single-shot measurements. Combined with quasi-CW signals that facilitate coarse PToF measurements, the proposed technique can give high resolution ranging limited by the maximum tone frequency and temporal resolution of detection electronics irrespective of the target distance. The resolution can be further enhanced by using prediction algorithms [22]. Moreover, the proposed approach has the potential to mitigate the requirement for a narrow linewidth laser for coherent detection, since we use the relative phase changes of RF tones instead of absolute phase and frequency measurements as a means to determine the target range. Furthermore, this technique eliminates the power-balance requirements in between the local oscillator and the echo signal, which forced the system to have an integrated monitoring photodetector and a variable attenuator to realize the power balance [19]. To prove the concept, a simple case study on simultaneous ranging and velocimetry of a fast dynamic target is performed via numerical simulations. We show that a <±1cm resolution in the ranging, limited by the temporal resolution of the detection system, and a 0.4cm/s speed resolution that is limited by the linewidth of the laser and frequency resolution of the detection system are achievable. Furthermore, experimental results are presented to demonstrate the capability of the proposed methodology.

Theoretical modelling
The schematic of the MTCW lidar for fast target ranging and velocimetry is presented in Fig. 1. A narrow linewidth CW laser with an output electric field of E 1 is modulated via a balanced Mach-Zehnder modulator (MZM) under push-pull configuration. Multiple RF tones, f i , with the same initial phases are fed to the MZM that yields optical field, E 2 , at the output facet of the collimator (CL). The modulator is configured to have a linear modulation with a low modulation depth of m<<1 [18] and the corresponding E 2 is shown in Eq. (1).
(1) Here, A 0 , ω 0, and ϕ 0 represent the electric field amplitude, angular optical carrier frequency, and the initial phase of the CW laser, respectively. Similarly, ω i indicates the angular frequency of i th RF modulation tone, and ϕ i is the initial phase of the corresponding tone.
The laser beam is then split into two via a beamsplitter (BS), where one arm is kept as the local oscillator and the other as the measurement branch to realize coherent detection on the photodetector (PD). The local signal is transmitted to the reference mirror that is separated from the BS by a distance L ref . The back-reflected signal from the reference mirror accumulates a The electric field in the measurement branch is represented by E m , where the target speed, v, alters the echo signal by inducing Doppler shift, ω d , to the optical carrier frequency by ω d = (2v/c)ω 0 after the laser beam travels a distance of L m [23,24]. Similarly, each modulation frequency realizes a Doppler shift of ω i d , as well. The returned signal electric field equation after the completion of the round trip is shown in Eq. (3).
The forward propagating and backscattered light have different phases due to change in the carrier and modulation frequencies. Since Unless the laser linewidth is in the order of kHz or below and the target is moving at extreme velocities, this assumption is always true for most practical applications. The Doppler shift realized by individual modulation frequencies will be in the < kHz levels even for very fast targets, while the optical carrier will realize MHz level shifts. On the other hand, to further simplify Eqs. (2) and (3), we assume all carriers and sidetones are in phase, thus ϕ 0 = ϕ i = 0.
After the beams in both arms propagate back to the PD from the reference mirror and the target, the corresponding electric fields will be converted into the detector photocurrent as , respectively. Moreover, selecting tone frequencies in a manner that prevents frequency overlap between desired beating tones and weak cross beating tones would improve the crosstalk and spur-free dynamic range of the measurement. For simplicity, the weak intermodulation terms between individual tone frequencies are neglected in Eq. (4). The expected spectral peaks in the frequency domain are stationed at ω d , ω i , 2ω i , ω i +ω d , ω i -ω d , 2ω i +ω d and 2ω i -ω d , and at their negatives if a dual side-band modulation is used. The phases of ω i and 2ω i terms are highly dependent on the reference field and have a very small contribution from the measurement arm for a highly unbalanced system. However, in our previous MTCW experiments, we had demonstrated how to utilize those tones for range measurements by comparing the relative amplitude variations [18,19].
)︂ for each tone are given in Table 1. Hence, their amplitudes and phases reveal the range information as indicated in Eq. (4) and Table 1. In particular, we focus on the phases of the measurable tones for the range information. Moreover, as shown in these definitions, for a system with N RF tones at the transmitter we have 4N frequency tones for data analysis for dynamic targets and we have 2N tones for static targets to extract the range information only, which is instrumental to increases the robustness and accuracy of the system. Here, we show an algorithm for single-shot range and velocity measurements by utilizing the phases rather than the tone amplitudes. For illustration purposes, we use the phase accumulations of tones at ω i + ω d and ω i − ω d only.
One of the challenges in the proposed technique is the modulo 2π cyclic pattern of the phase accumulation. In other words, ϕ meas ω i ±ω d represents the measured phase of the indicated frequency term, where 0 ≤ ϕ meas ω i ±ω d ≤ 2π, and yields the same phase result for every L m such that inteL m = L ω i ±ω d 0 + 2πc ω i ±ω d n i , where n i is an integer related with the i th frequency and L ω i ±ω d 0 is the measured length in the first cycle of the i th frequency when n i = 0. Therefore, we propose to use multiple tones to facilitate triangulation algorithms. In particular, if we define the integer , where λ i−RF is the RF tone wavelength, then we can define the possible measurement distance L m for a given phase measurement as in Eq. (5).
Hence, for a given maximum measurable distance L m−max that is determined by the system parameters, such as laser power, laser linewidth, SNR of the system, etc., we will have multiple solutions for the same target. While higher tone frequencies are desired for high resolution ranging, they are handicapped due to increasing n i value. Lower frequency tones produce a lower number of solutions with coarser resolutions, whereas the rapidly varying phases on the higher frequency tones generate multiple solutions with higher resolutions. The actual ranging solution is a triangulation of all tone frequencies. One method of converging to a single solution after triangulation is selecting the lowest frequency RF tone such that λ 1−RF ≥ L m−max . However, this will impose additional constraints on the detection electronics and the length of the time window that is utilized in the desired application. Similar to constraints in FMCW, if there is extensive scanning involved, using a longer time window will limit the number of scans that can be performed per second. Therefore, the number of RF tones and their frequency ranges should be determined based on the desired resolution and maximum ranging distance L m−max . However, implementation of a pseudo pulsation or quasi-CW operation that uses long pulses with multi-tone RF modulations imposed on them can further enhance this approach by eliminating the limits of n i described above and provide a higher SNR solution due to high peak power excitation. Similar to FMCW lidars, the frequency variations due to Doppler shift are used to identify the velocity information [25]. We have up to 2N degrees of freedom to estimate the velocity information. The precision of the velocity measurement is determined by the time window used to capture the ranging. For instance, a 1ms time window will yield a 1kHz spectral resolution that corresponds to 1mm/s or 1.5mm/s resolutions in velocity measurements by using a 1µm laser or by using a standard telecom laser at 1.55µm, respectively. The variations in Doppler shifts at different RF tones are negligibly small in most applications. For practical purposes, using tones with higher powers would yield high SNR velocity measurements. The value of ω d can be extracted from the photocurrent spectrum by comparing the ω i or 2ω i tones and its corresponding Doppler-shifted ω i ± ω d or 2ω i ± ω d tones, respectively, or by evaluating the Doppler peak near the baseband.

Numerical verification
A numerical verification is performed by mimicking simultaneous ranging and velocimetry of a fast-moving target to demonstrate the system's capability. To configure the MTCW lidar, a narrow linewidth CW laser is set to operate at 1µm central wavelength. The laser beam enters the MZM with a selected modulation depth of m = 0.01 to maintain the linearity of the modulator. Four modulation frequencies are fed to the MZM at 75MHz, 500MHz, 1900MHz, and 2450MHz. These tones are carefully selected in a fashion to forestall any form of frequency overlapping over a ω i ± ω d frequency span. MZM output is followed by a beam splitter. At the detector, we assume that the reference signal amplitude is α ref = 1, while the signal from the target has an attenuation of α m = 0.01 pointing out a 20dB loss due to scattering.
The receiver is assumed to be an InGaAs PIN photodetector with a 5GHz bandwidth and 0.9A/W responsivity. The load impedance of the detector is set to 50Ω. For simplicity and proof of concept purposes, detector noises such as the shot noise and the thermal noise are neglected, therefore the noise seen in Fig. 2 arises from the phase and amplitude noises of the source used in our modeling. In the simulation, we use a 256µs time window with about 61 ps temporal resolution (i.e. 2 22 samples), which corresponds to a total of a 16.4GHz frequency window and ∼4kHz frequency resolution. The L ref is preselected in the system as 10cm, and the target is set to be L m = 50m away from the lidar with a movement speed of 30m/s (108km/h) in the direction of the laser beam propagation, and hence induces a 60MHz Doppler shift to the optical carrier at 1µm central wavelength [23]. Equations (2) and (3) are used by setting ϕ o = ϕ i = 0 to acquire the resultant I PD . In practice, the initial phases of RF tones can be set by synchronization of RF generators via master-slave operation. The detected time-domain signal by the PD is converted into the RF spectrum via Fast Fourier Transform (FFT) as shown in Fig. 2. The significant frequencies at ω d , ω i , 2ω i , ω i ±ω d , and 2ω i ±ω d , are marked on the spectrum, while the resultant intermodulation tones that we don't utilize in our calculations remain unlabeled. There are in total of 8 frequency spikes that are applicable for Eq. (5). These peaks are found via a peak finding algorithm and by using the known ω i and measured ω d values as reference points. It is possible to acquire the value of ω d by analyzing the spike near the baseband or the peaks near the known modulation tones in Fig. 2. The measured Doppler shift on the RF spectrum is 60MHz and the Doppler peak is indicated in Fig. 2 as ω d . The resolution of the velocimetry is depending on the frequency resolution of the spectrum, δω. Therefore, the velocity resolution can be formalized by ∆v = (±δω/ω 0 )c. In this particular simulation, the ∆v of the MTCW system is 0.4cm/s due to the long time window. In the actual practice, there is an interplay between time window, i.e., velocimetry resolution, and the number of scans one can perform per second in most of the CW lidar systems, in particular FMCW lidar systems, which should be taken into account while configuring the lidar depending on the desired application.
The desired phases are extracted from the output voltage, V out , or the generated photocurrent I PD . After performing FFT, the resultant complex V out yields the phase observed at that particular tone in between the interval of -π,π. This process can be further improved by using Bessel filters to generate the phases of individual tones. At this point, all the variables are found to compute L m except for the value of n i in Eq. (5). It is not possible to measure the exact number of complete cycles of a modulation tone via the MTCW methodology by looking at the results in a single tone. Hence a triangulation approach is used to generate the exact distance of the target using the individual ω i ±ω d phases. It is possible to further enhance the sensitivity of the methodology by employing 2ω i ±ω d tones, however, these tones will have lower powers compared to ω i ±ω d .
To find the measurement length, the possible n i values are swept for each ω i ±ω d . The highest n i value belongs to the highest frequency tone with the smallest RF wavelength. The lowest frequency tone will have the lowest value of n i . In an actual application, it is desired to have a minimum value of n i = 1 within the maximum measurement range by selecting an appropriate tone frequency, or it is desired to use the time of arrival information of quasi-CW pulses to estimate a coarse range value. The calculated L m results for the given n i at the corresponding frequencies are shown in Fig. 3(a). Each frequency has a different repetition length of 2πc/(ω i ±ω d ) as the following, 54cm and L ω 4 +ω d = 11.94cm. To estimate the actual L m , first, we generate a data matrix M k,l such that M k, is computed by using the equations in Table 1, k = 1, 2, . . . , n i−max , and l = 2N. For illustration purposes, it is assumed that the measurement range we would like to resolve, L m−max , is within n i−max = 500 for the highest frequency tone. Since, for other lower frequencies, n i−max will be lower for the same target range, we don't need to fill values of the matrix for the estimated length L i k = k(2πc/ω i ± ω d ) + L 0 >L m−max . Since ω 4 +ω d will yield a better resolution due to its smaller repetition length, the last column of M k,l is set to the estimated L m at each n i , which will be used as the finest length resolution. The rest of the columns are filled in the same manner. However, range estimation values presented in new columns are selected in a manner to closely match the estimated L m values in the last column. As the tone frequencies decrease, 2πc/(ω i ±ω d ) will repeat itself less often, hence we obtain repetitive terms in the previous columns, as illustrated in a sample matrix in Fig. 3(b).
After establishing the M k,l , the standard deviation of each row is calculated and stored in the array σ k as σ k = Here, M k stands for the mean of the k th row. Then the last column of M k,l is matched with σ k to find the standard deviation at the corresponding L m . The length where the minimum σ k is found will yield the closest target range. Figure 4 illustrates the σ values for a target at 50m away. The minimum σ k corresponds to a target at L m = 50.0091m that deviates 0.91cm from the actual. Here the error range is dictated by the time resolution δt of the detected signal as ∆L = (±δt × c)/2.
Detecting targets that are further than the λ RF−max should be evaluated by considering the fact that the minimum standard deviation point repeats itself for every distance of L rep = 2πc/ω gcd , where ω gcd is the greatest common divisor of ω i ±ω d frequencies. In particular, for applications like aerial imaging or remote sensing through satellites or flying devices, a quasi-CW approach should be utilized to generate long pulses with high peak power and RF modulations on top of them to mitigate the cyclic behavior. In such a method, the quasi-CW pulses offer the time of arrival measurements to capture coarse range measurements, while RF modulations on top of the long pulse facilitate more precise range and velocity measurements. Therefore, the span of n i will be limited due to the time gating of the modulated pulses. At each interval, the selected n i range will yield a single solution based on the measured tone frequencies and phases. In our theoretical model, L rep is found to be ∼60m, which corresponds to the greatest common divisor of all ω i ± ω d terms that is 5MHz. As a result, the minimum standard deviation repeats itself at 50m, 110m, 170m, . . . etc. Hence, selecting tone frequencies in a way that the greatest common divisor of all ω i ±ω d terms is as small as possible will increase the L rep. Moreover, the quasi-CW pulsation and the considerate selection of the tone frequencies will prevent the potential false ranging measurements by eliminating additional close-to-minimum standard deviation points, which may occur due to the additional phase noise introduced by the lidar system. However, these calculations exclude the phase noise that might arise from the optical sources, detectors, RF generators, channel turbulences, and potential phase noise caused by the target surface roughness on the echo signal. Although the impact of the phase noise is left out for future study, in order to average the impact of phase noise, selection of tone frequencies can be engineered more carefully, and additional modulation tones can be added, if it is necessary, as a general principle.

Experimental verification
To prove the concept in the experimental domain, the test bench in [19] is used, i.e. the target in Fig. 1 is replaced by a reflector stationed on a translational motorized stage. The target is set to move with ∼8cm/s and the 40cm long stage is placed 90cm away from the MTCW lidar. A region is selected on the stage for data acquisition that corresponds to a distance of 110-115cm from the detector since the actual distance of the target cannot be measured due to the movement of the target with an integrated PToF lidar for comparison and verification. The data is acquired while the target is moving through this predetermined location on the translational stage with a constant speed. A narrow linewidth 1550nm laser is modulated by 6 RF modulation tones are used at 79, 391, 971, 1657, 2159, and 2623MHz. An 8GHz bandwidth oscilloscope with a 20GSa/s sampling rate is used for data acquisition with a time window of 100µs and 50ps time resolution, and 10 kHz frequency resolution. The L ref is measured to be 3cm and the initial tone phases, ϕ i , are acquired from the E ref spectrum to normalize the resultant ω i ±ω d phases. The triangulation algorithm is applied to the measured ω i ±ω d phases to compute the target distance. The velocity of the target is found using the RF peak near the baseband.
The final measured RF spectrum is presented in Fig. 5. Each ω i and its corresponding ω i ±ω d are labeled on the spectrum as well as the Doppler frequency near the baseband. The measured Doppler spike is at 105kHz that is equal to 8.08cm/s at 1550nm with a ±0.8cm/s accuracy due to the frequency resolution of the measurement setup. For ranging, data matrix M k,l is generated by setting l to 12 and k to 30. The phase of ω 6 +ω d is used as the reference point to maximize resolution. The final σ k and the minimum standard deviation point are shown in Fig. 6. The L m is measured as 111.9cm that is within the predetermined measurement range. The expected ranging resolution, ∆L, is ±0.75cm based on the temporal resolution. The minimum σ k is 0.15, where the minimum standard deviation should be zero in an ideal noiseless system. It is expected to have a nonzero standard deviation due to the noises. In particular, the laser phase noise, linewidth of the RF generators, and FFT leakage to side peaks caused broadening of ω 6 as shown in Fig. 6(b), which induces additional phase noise to the side peaks stationed at ω 6 ±ω d . Therefore, the measured value of the minimum σ k is higher than the ideal case. However, since the same information is engraved into 12 different tones, averaging will mitigate the impact of the noise. As stated in the previous section a further study should be performed to quantify the impact of phase error on measurements more accurately.

Conclusion
In this work, the enhancement of the MTCW lidar system to perform simultaneous ranging and velocimetry of moving targets is theoretically developed and demonstrated via numerical analysis and experimental results. Since the proposed approach uses a comparative analysis of narrowly spaced RF tones, it has the potential for eliminating the limits of the coherence length of lasers in lidar applications. For a numerical verification and explanation of the concept, a dummy target is assumed with 108km/h speed and 50m distance during the simulations. The target velocity is measured by evaluating the side peaks near the frequency tones. The acquired Doppler shift corresponds to the target's speed. Then by using the phase and measurement length relation at the individual ω i ±ω d , the potential values are determined. A triangulation algorithm is built and used to extract the range of a target by utilizing the possible solutions of L m at each frequency. The resolution of the velocimetry depends on the frequency resolution of the spectrum, while the ranging resolution is defined by the temporal resolution of detection electronics. In the presented simulations, the velocimetry and ranging resolutions are found to be ±0.4cm/s and ±1cm, respectively. Furthermore, the proposed concept is demonstrated in the experimental domain by using a dynamic target placed to a motorized translational stage. The target is measured to be at 111.9cm with 8.08cm/s speed by using the triangulation algorithm. By engineering the selected tone powers, detection electronics, and estimation algorithms, range and velocity resolution can be further improved for the proposed measurement technique. Disclosures. The authors declare that there are no conflicts of interest related to this article.