Long Range Sensitivity of Coherent LiDAR

Two coherent LiDAR modulation formats are compared by the minimum required number of photons per pixel. Phase-coded and linear frequency chirp modulations are found to result in similar sensitivity of a few photons per pixel.

(Image credit: iStock/Pobytov)

There are many different LiDARs on the market today. They have a variety of specifications, and standards are yet to be developed. On a very fundamental level LiDARs can be classified by how they transmit (modulation format) and how they receive (detection method) light. Most commercial LiDARs use high power amplitude-modulated lasers and direct detection.

In coherent detection LiDAR, a low power frequency modulated continuous wave (FMCW) laser source is commonly used. Such LiDARs are gaining industry attention due to their ability to measure radial velocity directly, quantum-limited sensitivity and immunity to interference. The two most popular modulation formats of coherent LiDAR are linear frequency modulation (LFM) and phase shift keying (PSK, phase-coded LiDAR).

While coherent LiDAR has been around for a while, its automotive application is new and is at the cutting edge of R&D with probably a dozen commercial projects around the world. So when I recently asked myself a non-trivial question, I couldn’t quickly find an answer:

“Is phase-coded LiDAR (PSK) more or less sensitive than linearly chirped (LFM)?”

This question is asked for an ideal case, considering the detection mechanisms and quantum noise only.

Fortunately, the answer for LFM exists [1]. The probability of false return (PFR, probability of incorrectly ranging a target) drops below 50% when the number of detected photons exceeds roughly ln(M), where M is the number of range slots. In other words, if there are 328 range slots (e.g. 200 m and 0.6 m resolution), one needs to detect at least ln(328), or 6 photons per pixel to record 50% of pixels at the correct range. The curve is steep, and PFR dips under 10 % for 10 photons per pixel. It is worth mentioning that in LFM coherent detection, the signal-to-noise ratio (SNR) is equal to the number of received photons per pixel.

Probability that estimator correctly finds the target location for 328 range bins as a function of the mean signal photoelectron level. The open circles are experimental values calculated from the 8000 trials for each of several signal levels. Adapted from [1].

A phase-coded LiDAR is implemented by a few coherent LiDAR companies and internal projects. As in optical communications, the phase-coded LiDAR laser beam is modulated to transmit a bitstream of data, where each bit is represented by a phase flip of the optical waveform. Each pixel can be represented by a pre-determined sequence of bits, for example 1000 bits long. When the optical wave reflects from a target and returns to LiDAR, it is attenuated and delayed. Detecting PSK data with an extremely weak optical signal leads to bit errors, which can be quantified by the bit error rate (BER). One then performs cross-correlations between the returned data word and the delayed version of the original one. When delay equals the roundtrip time to target, the cross-correlation shows a peak. When the returned code word is corrupted, the peak weakens and noise peaks can be recorded as false returns.

How much of the code word can get corrupted before we stop seeing the target in the correct range bin? A good question is half the answer, and it’s better to ask “how does probability of false return depend on the number of photons per pixel (per code word)?” The dependence of BER on the number of photons per bit in coherent detection of PSK modulation is [3] BER=erfc(sqrt(Ns))/2. In optical communications, a sufficiently high number of photons per bit (high SNR) is required for low BER. For example, 9 photons per bit for BER of 1/10⁹. In order to conceptually evaluate the phase-coded LiDAR for long range operation we have to look at the opposite side of the BER curve. Here, only a few photons are available per code word and BER is quite high. From BER we can numerically estimate the probability of false return (PFR):

Left: bit error rate as a function of average signal power in a general two-detector BPSK coherent receiver configuration [3]. Right: probability of recording target range incorrectly as a function of photons per pixel (per code word). PFR was estimated by counting incorrect ranging over 1000 trials for each power level.

It looks like, again, one needs roughly ln(M) photons per code word for 50% PFR with phase-coded LiDAR. Here M is the code word length and the number of range slots as well.

The sensitivities of LFM and phase-coded coherent detection are similar.

Interestingly, special code sequences like Golay pairs and MPSL sequences, which are good for improved side lobe suppression, are no different from random codes in terms of PFR in the photon-starved regime.

Pulse compression with amplitude modulation and coherent detection is also possible [2], and sensitivities around 800 photons were achieved. Other permutations of modulation and detection formats may be possible. For SPAD-LiDAR with amplitude modulation and direct detection, a single photon might be in principle sufficient for ranging in complete darkness. In practice, SPAD have non-ideal detection efficiency, after-pulsing, dead time, dark count rate, etc. Importantly, ambient light leads to histogram “pile-up” effect [4] which requires many more photons for accurate ranging. Interference caused by sunlight and light from other LiDARs is generally a problem [5] for direct time-of-flight SPAD and other direct detection sensors.

Ultimately, the difference between the LFM and phase-coded modulation coherent LiDARs lies in the implementation specifics. For phase-coded, the Doppler effect is detrimental [6] to code integrity and has to be addressed with a more complex data processing and detection setup compared to LFM hardware. These difficulties may explain inconsistent velocity color mapping in some published point cloud examples. Also, phase coding requires modulators which are lossy if integrated photonics is used, leaving less light for sensing.

The main challenge of the LFM approach is to make a laser source with fast and sufficiently linear frequency sweeps and sufficiently narrow laser line width. On the other hand, velocity can be directly obtained from Doppler shifts of the FFT spectrum peaks. The signal is present for targets located well beyond the laser coherence length. In addition, target location can be more accurately determined by interpolation. A low cost and simple solution may find many applications.

References:

[1] Zeb W. Barber, Jason R. Dahl, Tia L. Sharpe, and Baris I. Erkmen, “Shot noise statistics and information theory of sensitivity limits in frequency-modulated continuous-wave ladar,” J. Opt. Soc. Am. A 30, 1335–1341 (2013).

[2] Keren Shemer, Gil Bashan, H. Hagai Diamandi, Yosef London, Tzur Raanan, Yochai Israelashvili, Alon Charny, Itzik Cohen, Arik Bergman, Nadav Levanon, and Avi Zadok, “Sequence-coded coherent laser ranging with high detection sensitivity,” OSA Continuum 3, 1274–1282 (2020).

[3] P. Gallion and F. J. Mendieta, “Minimum energy per bit in high bit rate optical communications and quantum communications,” Proc. SPIE 8065, SPIE Eco-Photonics 2011, 80650F (20 April 2011).

[4] Anant Gupta, Atul Ingle, Andreas Velten, Mohit Gupta, “Photon-Flooded Single-Photon 3D Cameras,” Proc. CVPR 2019

[5] Padmanabhan, P.; Zhang, C.; Charbon, E. “Modeling and Analysis of a Direct Time-of-Flight Sensor Architecture for LiDAR Applications.” Sensors 2019, 19, 5464.

[6] Foucras, M., Julien, O., Macabiau, C., Ekambi, B., “Detailed Analysis of the Impact of the Code Doppler on the Acquisition Performance of New GNSS Signals,” Proceedings of the 2014 International Technical Meeting of The Institute of Navigation, San Diego, California, January 2014, pp. 513–524.