# Detection Probability Analysis of True Random Coding Photon Counting Lidar

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Structure and Ranging Principle

## 3. Theoretical Model of Correct Ranging Probability

#### 3.1. Count Probability

#### 3.2. Signal Recognition Threshold

#### 3.3. Probability of Detected Codes Number Equal to Signal Recognition Threshold

_{n}) is equal to or greater than the signal recognition threshold, in other words i

_{n}≥ k. Then, if we know the relationship between the incorrect ranging probability and the correct ranging probability when the detected codes number is equal to the signal recognition threshold, the correct ranging probability can also be quantitatively described.

#### 3.4. Correct Ranging Probability Model without Considering System Jitter

#### 3.5. Correct Ranging Probability Model with Considering System Jitter

## 4. Verify the Correct Ranging Probability Theoretical Model with Monte Carlo Simulation

#### 4.1. Verify the Correct Ranging Probability Model without Considering System Jitter

#### 4.2. Verify the Correct Ranging Probability Model with Considering System Jitter

## 5. Verify the Correct Ranging Probability Theoretical Model with Experiment

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

${P}_{sn}$ | correct ranging probability of single ‘1’ code | ${\psi}_{n}\left(t\right)$ | mean noise photoelectron flux |

${P}_{n}$ | detection probability of noise in each pulse width | ${\psi}_{s}\left(t\right)$ | mean signal photoelectron flux |

${P}_{f{a}_{k}}$ | incorrect ranging probability | ${t}_{d2}$ | dead time of Gm-APD2 |

${P}_{{d}_{k}}$ | correct ranging probability | ${t}_{bin}$ | time bin width |

k | signal recognition threshold | $\Delta {t}_{Jitter}$ | the FWHM of the system jitter |

$L$ | sequence length | N | transmitted codes number in the sequence. |

$\rho $ | mean pulse count density of ‘1’ code | ${T}_{\rho}$ | average pulse interval of each ‘1’ code |

${P}_{{d}_{k}}$ | correct ranging probability when the detected signal codes number equals the signal recognition threshold | ||

${P}_{f{a}_{k}}$ | incorrect ranging probability when the detected noise codes number is equal to or greater than the signal recognition threshold | ||

${P}_{D}$ | the correct ranging probability of lidar system |

## References

- Steindorfer, M.A.; Kirchner, G.; Koidl, F.; Wang, P.; Jilete, B.; Flohrer, T. Daylight space debris laser ranging. Nat. Commun.
**2020**, 11, 1–6. [Google Scholar] [CrossRef] [PubMed] - Li, Z.; Ye, J.; Huang, X.; Jiang, P.-Y.; Cao, Y.; Hong, Y.; Yu, C.; Zhang, J.; Zhang, Q.; Peng, C.-Z.; et al. Single-photon imaging over 200 km. Optica
**2021**, 8, 344–349. [Google Scholar] [CrossRef] - Degnan, J.J. Scanning, Multibeam, Single Photon Lidars for Rapid, Large Scale, High Resolution, Topographic and Bathymetric Mapping. Remote Sens.
**2016**, 8, 958. [Google Scholar] [CrossRef] [Green Version] - Prochazka, I.; Kodet, J.; Blazej, J.; Kirchner, G.; Koidl, F. Photon counting detector for space debris laser tracking and lunar laser ranging. Adv. Space. Res.
**2014**, 54, 755–758. [Google Scholar] [CrossRef] - Markus, T.; Neumann, T.; Martino, A.; Abdalati, W.; Brunt, K.; Csatho, B.; Farrell, S.; Fricker, H.; Gardner, A.; Harding, D.; et al. The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): Science requirements, concept, and implementation. Remote Sens. Environ.
**2017**, 190, 260–273. [Google Scholar] [CrossRef] - McCarthy, A.; Collins, R.J.; Krichel, N.J.; Fernández, V.; Wallace, A.M.; Buller, G.S. Long-range time-of-flight scanning sensor based on high-speed time-correlated single-photon counting. Appl. Opt.
**2009**, 48, 6241–6251. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Petit, J.; Stottelaar, B.; Feiri, M.; Kargl, F. Remote attacks on automated vehicles sensors: Experiments on camera and lidar. Black Hat Eur.
**2015**, 11, 995. [Google Scholar] - Matthey, R.; Mitev, V. Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements. Opt. Laser Eng.
**2005**, 43, 557–571. [Google Scholar] [CrossRef] - Du, P.; Geng, D.; Wang, W.; Gong, M. Laser detection of remote targets applying chaotic pulse position modulation. Opt. Eng.
**2015**, 54, 114102. [Google Scholar] [CrossRef] - González, E.; Navarro, S.A.; Pérez Serrano, A.; Vilera, M.; Tijero, J.M.G.; Esquivias, I. Theoretical Analysis of Random-Modulation Continuous Wave LIDAR. In Proceedings of the 9 ª Reunión Española de Optoelectrónica (OPTOEL’15), Salamanca, Spain, 13–15 July 2015; pp. 1–5. [Google Scholar]
- Zhang, F.; Du, P.; Liu, Q.; Gong, M.; Fu, X. Adaptive strategy for CPPM single-photon collision avoidance LIDAR against dynamic crosstalk. Opt. Express
**2017**, 25, 12237–12250. [Google Scholar] [CrossRef] [PubMed] - Cheng, C.H.; Chen, C.Y.; Chen, J.D.; Pan, D.K.; Ting, K.T.; Lin, F.Y. 3d pulsed chaos lidar system. Opt. Express
**2018**, 26, 12230–12241. [Google Scholar] - Hao, J.; Gong, M.-L.; Du, P.-F.; Lu, B.-J.; Zhang, F.; Zhang, H.-T.; Fu, X. Ultra-low power anti-crosstalk collision avoidance light detection and ranging using chaotic pulse position modulation approach. Chin. Phys. B
**2016**, 25, 250–257. [Google Scholar] [CrossRef] - Hao, J.; Gong, M.; Du, P.; Lu, B.; Zhang, F.; Zhang, H.; Fu, X. A Novel CPPM Anti-Crosstalk Collision Avoidance Lidar with Ultra-Low Laser Power. In Proceedings of the 8th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optoelectronic Materials and Devices, Suzhou, China, 26–29 April 2016; International Society for Optics and Photonics: Bellingham, WA, USA, 2016; Volume 9686, pp. 968607–968616. [Google Scholar]
- Wang, B.; Qian, J.; Zhao, T.; Wang, Y. Anti-jamming performance of chaotic lidar. Chin. J. Lasers
**2011**, 38, 0514002. [Google Scholar] [CrossRef] - Tsai, C.; Liu, Y. Anti-Interference Single-Photon LiDAR Using Stochastic Pulse Position Modulation. Opt. Lett.
**2020**, 45, 439–442. [Google Scholar] [CrossRef] - Liu, B.; Yu, Y.; Chen, Z.; Han, W. True random coded photon counting Lidar. Opto-Electron. Adv.
**2020**, 3, 19004401–19004406. [Google Scholar] [CrossRef] - Yu, Y.; Liu, B.; Chen, Z.; Hua, K. Photon Counting LIDAR Based on True Random Coding. Sensors
**2020**, 20, 3331. [Google Scholar] [CrossRef] [PubMed] - Blaj, G. Dead-time correction for spectroscopic photon counting pixel detectors. J. Synchrotron Radiat.
**2019**, 26, 1621–1630. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Arvani, F.; Carusone, T.C. Direct Time-of-Flight TCSPC Analytical Modeling Including Dead-Time Effects. In Proceedings of the 2018 IEEE International Symposium on Circuits and Systems (ISCAS), Florence, Italy, 27–30 May 2018. [Google Scholar]
- Li, Z.; Lai, J.; Wang, C.; Yan, W.; Li, Z. Influence of dead-time on detection efficiency and range performance of photon-counting laser radar that uses a Geiger-mode avalanche photodiode. Appl. Opt.
**2017**, 56, 6680–6687. [Google Scholar] [CrossRef] [PubMed] - Hwang, I.P.; Lee, C.H. Mutual Interferences of a True-Random LiDAR with Other LiDAR Signals. IEEE Access
**2020**, 8, 124123–124133. [Google Scholar] [CrossRef] - Hwang, I.P.; Lee, C.H. A rapid LiDAR without Mutual Interferences. In Proceedings of the Optical Fiber Communication Conference, San Diego, CA, USA, 3–7 March 2019. [Google Scholar]
- Gatt, P.; Johnson, S.; Nichols, T. Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics. Appl. Opt.
**2009**, 48, 3261–3276. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**(

**a**) The ranging principle of true random coding photon counting Lidar. (1) Transmitting signal $a\left(n\right)$. (2) Echo Signal $b\left(n\right)$. (3) Correlation function $g\left(\tau \right)$. (

**b**) Normalized cross correlation.

**Figure 3.**The detected number of codes at peak position of correlation operation. (

**a**) Sequence length 100 μs; (

**b**) Sequence length 500 μs.

**Figure 4.**Correct ranging probability (${P}_{{d}_{k}}$) when the detected signal codes number equals the signal recognition threshold and incorrect ranging probability (${P}_{f{a}_{k}}$) when the detected noise codes number is equal to or greater than the signal recognition threshold under different system parameters. (

**a**) Under different sequence length; (

**b**) Under different pulse count density; (

**c**) Under different noise count level.

**Figure 6.**Monte Carlo simulation verification of correct ranging probability: (

**a**) the influence of mean echo photon number and mean pulse count density on correct ranging probability (sequence length 200 μs, pulse width 1ns, mean noise count density 1 Mcps); (

**b**) the influence of sequence length on correct ranging probability (pulse count 1 Mcps, noise count density 1 Mcps, pulse width 1 ns); (

**c**) the influence of pulse width on correct ranging probability (sequence length 200 μs, mean pulse count 1 Mcps); (

**d**) the effect of noise count density on detection probability (sequence length 200 μs, mean pulse count density 1 Mcps, pulse width 1 ns).

**Figure 7.**The influence of jitter on correct ranging probability under different echo photons number.

**Figure 9.**(

**a**) True random coding photon counting lidar experimental platform. (

**b**) IRF signal detections with Gaussian curve fitting, where estimated Gaussian RMS width is 0.464 ns, thus FWHM is 1.09 ns.

**Figure 10.**Experimental verification of correct ranging probability under different system parameters: (

**a**) the influence of sequence length on correct ranging probability (mean pulse count density 1 Mcps, pulse width 1 ns, mean noise count density 1 Mcps), (

**b**) the influence of mean pulse count on correct ranging probability (sequence length 200 μs, pulse width 1 ns, mean noise count density 1 Mcps), (

**c**) the influence of pulse width on correct ranging probability (sequence length 200 μs, mean pulse count density 1 Mcps, mean noise count density 1 Mcps).

**Table 1.**Correct ranging probability (${P}_{{d}_{k}}$) when the detected signal codes number equals the signal recognition threshold (i = k) and incorrect ranging probability (${P}_{f{a}_{k}}$) when the detected noise codes number is equal to or greater than the signal recognition threshold (i

_{n}≥ k) under different sequence lengths.

Sequence Length | |||||||||
---|---|---|---|---|---|---|---|---|---|

100 μs | 150 μs | 200 μs | 250 μs | 300 μs | 350 μs | 400 μs | 450 μs | 500 μs | |

k | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 7 |

${P}_{f{a}_{k}}$ (%) | 7.9 | 61.2 | 8.5 | 32.7 | 4.1 | 12.3 | 31.4 | 71.9 | 9.3 |

${P}_{{d}_{k}}$ (%) | 66.0 | 12.0 | 70.0 | 21.0 | 82.0 | 51.0 | 25.0 | 7.0 | 71.0 |

**Table 2.**Correct ranging probability (${P}_{{d}_{k}}$) when the detected signal codes number equals the signal recognition threshold (i = k) and incorrect ranging probability (${P}_{f{a}_{k}}$) when the detected noise codes number is equal to or greater than the signal recognition threshold (I

_{n}≥ k) under different mean pulse count density.

Mean Pulse Count Density | ||||||||
---|---|---|---|---|---|---|---|---|

0.5 Mcps | 0.6 Mcps | 0.7 Mcps | 0.8 Mcps | 0.9 Mcps | 1.0 Mcps | 1.1 Mcps | 1.2 Mcps | |

k | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 |

${P}_{f{a}_{k}}$ (%) | 50.0 | 87.0 | 1.9 | 3.2 | 5.2 | 7.9 | 11.7 | 16.6 |

${P}_{{d}_{k}}$ (%) | 0.172 | 0.05 | 0.95 | 0.83 | 0.81 | 0.66 | 59.4 | 51.0 |

Mean Pulse Count Density | ||||||||

1.3 Mcps | 1.4 Mcps | 1.5 Mcps | 1.6 Mcps | 1.7 Mcps | 1.8 Mcps | 1.9 Mcps | 2.0 Mcps | |

k | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 |

${P}_{f{a}_{k}}$ (%) | 22.9 | 30.9 | 40.8 | 52.9 | 1.9 | 2.5 | 3.3 | 4.3 |

${P}_{{d}_{k}}$ (%) | 44.0 | 29.0 | 17.0 | 11.0 | 88.4 | 87.0 | 81.6 | 78.0 |

**Table 3.**Correct ranging probability (${P}_{{d}_{k}}$) when the detected signal codes number equals the signal recognition threshold (i = k) and incorrect ranging probability (${P}_{f{a}_{k}}$) when the detected noise codes number is equal to or greater than the signal recognition threshold (i

_{n}≥ k) under different mean noise count density.

Mean Noise Count Level | ||||||||
---|---|---|---|---|---|---|---|---|

0.5 Mcps | 0.6 Mcps | 0.7 Mcps | 0.8 Mcps | 0.9 Mcps | 1.0 Mcps | 1.1 Mcps | 1.2 Mcps | |

k | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 |

${P}_{f{a}_{k}}$ (%) | 0.5104 | 0.882 | 0.019 | 0.0324 | 0.052 | 0.079 | 0.116 | 0.164 |

${P}_{{d}_{k}}$ (%) | 0.08 | 0.02 | 0.91 | 0.88 | 0.81 | 0.66 | 0.61 | 0.48 |

Mean Noise Count Level | ||||||||

1.3 Mcps | 1.4 Mcps | 1.5 Mcps | 1.6 Mcps | 1.7 Mcps | 1.8 Mcps | 1.9 Mcps | 2.0 Mcps | |

k | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 |

${P}_{f{a}_{k}}$ (%) | 22.6 | 30.4 | 40.1 | 51.9 | 66.2 | 2.4 | 3.1 | 4.1 |

${P}_{{d}_{k}}$ (%) | 38.9 | 34.8 | 29.9 | 18.1 | 11.2 | 91.4 | 88.1 | 83.0 |

Parameter | Value |
---|---|

Wavelength | 1064 nm |

Dead time | 45 ns (Gm-APD1) /25 ns (GM-APD2) |

Pulse width | 1/2/4 ns |

Sequence length | 100/200/500 μs |

Mean noise count density | 1 Mcps |

Mean echo photon number | 1:0.5:5 |

Mean pulse count density | 0.5/1/2 Mcps |

Detection efficiency (Gm-APD2) | 2% |

Time jitter (FWHM/Gm-APD2) | 300 ps |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Wang, Z.; Ma, K.; Chen, C.; Wang, X.; Xue, B.; Li, X.; Zhang, F.; Pan, X.; Zhuang, Q.;
et al. Detection Probability Analysis of True Random Coding Photon Counting Lidar. *Photonics* **2021**, *8*, 545.
https://doi.org/10.3390/photonics8120545

**AMA Style**

Yu Y, Wang Z, Ma K, Chen C, Wang X, Xue B, Li X, Zhang F, Pan X, Zhuang Q,
et al. Detection Probability Analysis of True Random Coding Photon Counting Lidar. *Photonics*. 2021; 8(12):545.
https://doi.org/10.3390/photonics8120545

**Chicago/Turabian Style**

Yu, Yang, Zhangjun Wang, Kuntai Ma, Chao Chen, Xiufen Wang, Boyang Xue, Xianxin Li, Feng Zhang, Xin Pan, Quanfeng Zhuang,
and et al. 2021. "Detection Probability Analysis of True Random Coding Photon Counting Lidar" *Photonics* 8, no. 12: 545.
https://doi.org/10.3390/photonics8120545