## Coherent power measurement uncertainty resulting from atmospheric turbulence

Optics Express, Vol. 12, Issue 1, pp. 168-175 (2004)

http://dx.doi.org/10.1364/OPEX.12.000168

Acrobat PDF (274 KB)

### Abstract

The simulation of beam propagation is used to examine the uncertainty inherent to the process of optical power measurement with a practical heterodyne lidar because of the presence of refractive turbulence. The approach has made possible the foremost study of the statistics of the coherent return fluctuations in the turbulent atmosphere for which there is no existing theory to be considered.

© 2004 Optical Society of America

## 1. Introduction

1. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. **39**, 2401–2411 (2000). [CrossRef]

2. B. J. Rye, “Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,” J. Opt. Soc. Am. **71**, 687–691 (1981). [CrossRef]

4. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. **39**, 5426–5445 (2000). [CrossRef]

5. A. Belmonte, “Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters,” Opt. Express **11**, 2041–2046 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-2041 [CrossRef] [PubMed]

6. A. Belmonte, “Angular misalignment contribution to practical heterodyne lidars in the turbulent atmosphere,” Opt. Express **11**, 2525–2531 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2525 [CrossRef] [PubMed]

7. L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere”, J. Mod. Opt. **39**, 1849–1853, 1992. [CrossRef]

_{0}of 1 cm and realistic outer scale L

_{0}of the order of 5 m. The effects of the bump at the high frequencies that characterize the accurate Hill spectrum affects the results of our simulations just slightly, and similar conclusions on the lidar’s basic behaviour would be obtained by using the simpler von Kármán spectrum. The simulation technique uses a numerical grid of 1024×1024 points with 5-mm resolution and simulates a continuous random medium with a minimum of 20 two-dimensional phase screens [4

4. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. **39**, 5426–5445 (2000). [CrossRef]

## 2. Heterodyne optical power statistics

*P*defines the performance of the coherent lidar in terms of the overlap integral of the transmitted (T) and virtual back-propagated local oscillator (BPLO) irradiances at the target plane

**p**[8

8. B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. **18**, 1390–1398 (1979). [CrossRef] [PubMed]

9. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. **30**, 5325–5352 (1991). [CrossRef] [PubMed]

*λ*is the optical wavelength of the transmitted laser. The irradiances

*j*

_{T}and

*j*

_{BPLO}have been normalized to the laser 〈

*P*

_{L}(t)〉 and local oscillator (LO) 〈

*P*

_{LO}〉 average power, respectively. As we are mainly concerned with the effects of the refractive turbulence, parameter

*C*is mostly irrelevant here [5

5. A. Belmonte, “Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters,” Opt. Express **11**, 2041–2046 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-2041 [CrossRef] [PubMed]

*SNR*–defined as the average signal power in Eq. (1) divided by the noise power–at the output of the detector. However, this magnitude is not in itself a true indicator of system capability in some usual heterodyne lidar implementations. Coherent differential absorption lidar (DIAL) measurements of atmospheric constituents and coherent target characterization (backscatter estimation, hard target calibration) are two important lidar applications where the accuracy of the estimate of average received power at different wavelengths is actually the critical parameter. Fluctuations in the instantaneous power level, which do not affect the signal power in Eq. (1), nonetheless degrade the ability of the system to measure average power. These fluctuations, which can be caused by different physical factors, can contribute substantially to the measurement uncertainty. Because atmospheric refractive turbulence produces signal fluctuations affecting heterodyne detection systems in different ways, they must be considered –along with the average signal power in Eq. (1)- to evaluate system performance.

*δP*of the received powers from the on-line beam

*P*

_{on}and the off-line beam

*P*

_{off}[10]

*C*

_{P}. Since the statistical properties of the signal

*P*are those corresponding to the overlap integral in Eq. (1), it is straightforward to express the normalized covariance for the coherent power as

_{2}-R

_{1}=ΔR. The covariance is a generalization of the variance in that

*C*

_{P}(R, R)=

11. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in *Laser Beam Propagation in the Atmosphere*, ed. J. W. Strohbehn (Springer Verlag, Berlin, 1978). [CrossRef]

13. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves in Random Media **4**, 243–273 (1994). [CrossRef]

## 3. Power degradation due to atmospheric turbulence

*R*as a function of range of a realistic monostatic lidar system. We use Eq. (3) to compute our estimations. Two wavelengths, 2 and 10 µm, and several levels of refractive turbulence have been considered in the figures. Transmitted and virtual LO beams were assumed to be matched, collimated, perfectly aligned, Gaussian, and truncated at a telescope aperture of diameter

*D*=16

*cm*. The beam truncation was 1.25 (i.e.,

*D*=1.25×2

*ω*

_{0}, where ω

_{0}is the 1/e

^{2}beam irradiance radius). In any situation regarded in this study, the coherent power normalized variance results are generally below 0.3 (i.e., a standard deviation of almost 3 dB around the mean values). In a most favorable situation than those considered in the figures, with ground lidar systems profiling the atmosphere along slant paths with large elevation angles, the accumulated turbulence level and its effects will be markedly smaller. Our simulation technique could be extended to consideration of those non-uniform turbulence conditions. Along with the variance and the covariance, in the figures we add the corresponding mean heterodyne power, normalized such that at the shortest range is 0 dB. It will help us to understand the results of our simulations.

1. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. **39**, 2401–2411 (2000). [CrossRef]

15. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE **63**, 790–811 (1975). [CrossRef]

16. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE **63**, 1669–1692 (1975). [CrossRef]

11. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in *Laser Beam Propagation in the Atmosphere*, ed. J. W. Strohbehn (Springer Verlag, Berlin, 1978). [CrossRef]

5. A. Belmonte, “Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters,” Opt. Express **11**, 2041–2046 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-2041 [CrossRef] [PubMed]

^{-13}m

^{-2/3}) are considered in Fig. 1 (right) the same fast increase to a maximum appears at shorter ranges. However, after a short decrease, the variance of the turbulence fluctuations increases again. The reason here is likely to be that beam spreading is less important [4

4. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. **39**, 5426–5445 (2000). [CrossRef]

^{-14}m

^{-2/3}), the variance will also show a similar behavior. By using the same physical interpretation considered for strong and moderate turbulence, an eventual smoothing of these fluctuations by averaging can be expected at ranges larger than those shown.

## 4. Conclusions

6. A. Belmonte, “Angular misalignment contribution to practical heterodyne lidars in the turbulent atmosphere,” Opt. Express **11**, 2525–2531 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2525 [CrossRef] [PubMed]

## References and links

1. | A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. |

2. | B. J. Rye, “Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,” J. Opt. Soc. Am. |

3. | J. Martin, “Simulation of wave propagation in random media: theory and applications,” in |

4. | A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. |

5. | A. Belmonte, “Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters,” Opt. Express |

6. | A. Belmonte, “Angular misalignment contribution to practical heterodyne lidars in the turbulent atmosphere,” Opt. Express |

7. | L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere”, J. Mod. Opt. |

8. | B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. |

9. | R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. |

10. | R. M. Measures, |

11. | J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in |

12. | J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. |

13. | M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves in Random Media |

14. | J. H. Churnside, Aperture averaging of optical scintillation in the turbulent atmosphere, Appl. Opt. |

15. | A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE |

16. | R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE |

17. | L. L. Gurdev and T. N. Dreischuh, “An heuristic view on the signal-to-noise ratio at coherent heterodyne detection of aerosol lidar returns formed through turbulent atmosphere”, in |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.3640) Atmospheric and oceanic optics : Lidar

(030.6600) Coherence and statistical optics : Statistical optics

(280.1910) Remote sensing and sensors : DIAL, differential absorption lidar

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 2, 2003

Revised Manuscript: December 20, 2003

Published: January 12, 2004

**Citation**

Aniceto Belmonte, "Coherent power measurement uncertainty resulting from atmospheric turbulence," Opt. Express **12**, 168-175 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-168

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### References

- A. Belmonte and B. J. Rye, �??Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,�?? Appl. Opt. 39, 2401-2411 (2000). [CrossRef]
- B. J. Rye, �??Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,�?? J. Opt. Soc. Am. 71, 687-691 (1981). [CrossRef]
- J. Martin, �??Simulation of wave propagation in random media: theory and applications,�?? in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, and V. Zavorotny, eds., SPIE, Washington (1993).
- A. Belmonte, �??Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,�?? Appl. Opt. 39, 5426-5445 (2000). [CrossRef]
- A. Belmonte, "Analyzing the efficiency of a practical heterodyne lidar in the turbulent atmosphere: telescope parameters," Opt. Express 11, 2041-2046 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-2041">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-2041</a> [CrossRef] [PubMed]
- A. Belmonte, "Angular misalignment contribution to practical heterodyne lidars in the turbulent atmosphere," Opt. Express 11, 2525-2531 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2525">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2525</a>. [CrossRef] [PubMed]
- L. C. Andrews, "An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere", J. Mod. Opt. 39, 1849-1853, 1992. [CrossRef]
- B. J. Rye, �??Antenna parameters for incoherent backscatter heterodyne lidar,�?? Appl. Opt. 18, 1390-1398 (1979). [CrossRef] [PubMed]
- R. G. Frehlich and M. J. Kavaya, �??Coherent laser radar performance for general atmospheric refractive turbulence,�?? Appl. Opt. 30, 5325-5352 (1991). [CrossRef] [PubMed]
- R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Wiley-Interscience, New York, 1984).
- J. W. Strohbehn, �??Modern theories in the propagation of optical waves in a turbulent medium,�?? in Laser Beam Propagation in the Atmosphere, ed. J. W. Strohbehn (Springer Verlag, Berlin, 1978). [CrossRef]
- J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, and F. S. Henyey, �??Solution for the fourth moment of waves propagating in random media,�?? Radio Sci. 21, 929-948 (1986). [CrossRef]
- M. I. Charnotskii, �??Asymptotic analysis of finite-beam scintillations in a turbulent medium,�?? Waves in Random Media 4, 243-273 (1994). [CrossRef]
- J. H. Churnside , Aperture averaging of optical scintillation in the turbulent atmosphere , Appl. Opt. 30, 1982-1994 ( 1991). [CrossRef] [PubMed]
- A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, �??Laser irradiance propagation in turbulent media,�?? Proc. IEEE 63, 790-811 (1975). [CrossRef]
- R. L. Fante, �??Electromagnetic beam propagation in turbulent media,�?? Proc. IEEE 63, 1669-1692 (1975). [CrossRef]
- L. L. Gurdev, T. N. Dreischuh, �??An heuristic view on the signal-to-noise ratio at coherent heterodyne detection of aerosol lidar returns formed through turbulent atmosphere�??, in 12th International School on Quantum Electronics: Laser Physics and Applications, P. A. Atanasov, A. A. Serafetinides, and I. N. Kolev, eds., Proc. SPIE 5226, 310-314 (2003).

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