## Temporal averaging of turbulence-induced uncertainties on coherent power measurements

Optics Express, Vol. 12, Issue 16, pp. 3770-3777 (2004)

http://dx.doi.org/10.1364/OPEX.12.003770

Acrobat PDF (284 KB)

### Abstract

The process of optical power measurement with a heterodyne lidar carry an inherent statistic uncertainty because of the presence of refractive turbulence. Although these uncertainties are usually reduced by taking average values of different measurements, our analysis shows that temporal correlation of the laser-beam fluctuations restricts the effectiveness of the signal averaging in practical systems such as a coherent DIAL.

© 2004 Optical Society of America

## 1. Introduction

1. A. Belmonte, “Coherent power measurement uncertainty resulting from atmospheric turbulence,” Opt. Express **12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

2. A. Belmonte, “Coherent DIAL profiling in turbulent atmosphere,” Opt. Express **12**, 1249–1257 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1249 [CrossRef] [PubMed]

1. A. Belmonte, “Coherent power measurement uncertainty resulting from atmospheric turbulence,” Opt. Express **12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

_{P}is below -3 dB around the mean power values. This makes the turbulence-induced equivalent standard deviation at least four times (6 dB) smaller than that of speckle fading. Still it is significant at most levels of turbulence in ground coherent DIAL systems now on use or under development that work at wavelengths that range from 1 to 10 µm [2

2. A. Belmonte, “Coherent DIAL profiling in turbulent atmosphere,” Opt. Express **12**, 1249–1257 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1249 [CrossRef] [PubMed]

*τ*

_{P}seconds, where

*τ*

_{P}is pulse duration, a totally new volume is traversed, returns spaced at this interval are independent. By making

*τ*

_{P}much less than

*T*, where

*T*=

*2ΔR*/

*c*is the temporal equivalent of a range gate

*ΔR*, one reduces the variance due to speckle by T/τ

_{P}. Also, for a system with a sufficient high pulse repetition rate, target speckle effects can be reduced by temporal averaging by taking mean values of different measurements.

8. 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 – to describe the spatial correlation of the phase screens. Although simulations could as readily be extended to consideration of nonuniform situations, uniform atmospheric winds V are also considered along the propagation path. The simulation technique uses a numerical grid of 1024×1024 points with 5-mm spatial resolution δx, and simulates a continuous random medium with a minimum of 20 – and as much as 50- two-dimensional phase screens [5

5. A. Belmonte “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt.39, 5426–5445 (2000). J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of High energy laser beams through the atmosphere, [CrossRef]

## 2. Coherent power temporal correlation

*P*(

*R*,

*t*)

*R*and different delay times Δt=t

_{2}-t

_{1}, we will use the target-plane formulation [9

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

*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**

*C*(

*R*) groups the conversion efficiencies and parameters that describe the various system components and the atmospheric scattering conditions.

*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. With this formulation, the random power fluctuations arise because of the randomness of the intervening atmospheric medium over different time scales. It should be noted that, as the coherent return is evaluated in the target plane as an overlap integral, when a large number of bright spot scintillation are found in that plane, we should expect a spatial averaging principle to apply over power fluctuations [1

1. A. Belmonte, “Coherent power measurement uncertainty resulting from atmospheric turbulence,” Opt. Express **12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

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

*C*(

*R*) is mostly irrelevant here: The statistical properties of the signal

*P*are those corresponding to the overlap integral in Eq. (2). As our problem involves higher powers [fourth moment of the intensity in Eq. (1)], no simple analytical solutions to the temporal statistics of the heterodyne power have been described. Simulation is the only approach permitting the characterization of the coherent return fluctuations that result from turbulence.

*ρ*(t

_{1}, t

_{2}) [temporal correlation

*C*

_{P}(t

_{1}, t

_{2}) normalized to the variance

*C*

_{P}(t, t)=

_{2}-t

_{1}. We used simulations along with Eqs. (1) and (2) to compute our estimations. 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*×

*2ω0*, where ω

_{0}is the 1/e

^{2}beam irradiance radius). Two wavelengths, 2 and 10 µm, and several levels of refractive turbulence were considered. Although several wind velocities

*V*ranging from 5 m/s to 20 m/s were simulated, in these plots we show those results corresponding to

*V*

_{0}=10 m/s. In any case, for a specific turbulence level, the graphs are all inclusive: For any other wind velocity

*V*, the temporal (abscissa) axis just needs to be escalated by the factor

*V*

_{0}/

*V*to read the right coherent power autocorrelation. In the figure, temporal correlation length (measured at 1/e

^{2}, as indicated in the graphics) expands between 3 and 9 milliseconds. It should be noted that, although we may expect correlation times decreasing with range as the spatial scale of beam intensity fluctuations does [5

5. A. Belmonte “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt.39, 5426–5445 (2000). J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of High energy laser beams through the atmosphere, [CrossRef]

## 3. Limitations to temporal averaging

*N*independent power samples can be averaged, the measurement will improve its accuracy as

*N*

^{-1/2}. However, attending to the results in the previous Section 2 showing the probable lack of independency among consecutives lidar signal samples, we should expect a reduction in the effectiveness of signal averaging relative to

*N*

^{-1/2}. The temporal correlation extending over several milliseconds of the lidar signal due to the slow movement of atmospheric refractive turbulence necessarily modifies the random nature of successive lidar returns and, consequently, the statistical behavior of the signal-averaging process. Many relevant considerations about the limitations to lidar signal averaging have been shown elsewhere [11

11. N. Menyuk, D. K. Killinger, and C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. **21**, 3377–3383 (1982). [CrossRef] [PubMed]

13. G. M. Ancellet and R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A **4**, 367–373 (1987). [CrossRef]

*PRF*) systems are considered. To obtain the variance for the average of

*N*pulse returns,

**12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

*ρ*(see Section 2):

_{n}=nΔt, with

*Δt*=

*1*/

*PRF*the time interval between pulses. When dealing with independent pulses, i.e., null correlation coefficients, Eq. (3) predict the expected

*N*

^{-1/2}accuracy. Eq. (3) is independent of the probability-distribution function of the signals and, consequently, it is applicable to any signal averaging phenomena [11

11. N. Menyuk, D. K. Killinger, and C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. **21**, 3377–3383 (1982). [CrossRef] [PubMed]

*N*

^{-1/2}behavior. (It should be noted that exists another possible approach to the problem which don’t require estimating correlation coefficients or considering Eq. (3). In this approach,

*N*simulated pulses are directly averaged to establish the statistics σ

_{N}of the averaging process. This path of course – although demands a bigger computational effort – yields essentially the same result than the previous approach. They are not reported in this paper.)

**12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

**12**, 168–175 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168 [CrossRef] [PubMed]

## 4. Concluding remarks

2. A. Belmonte, “Coherent DIAL profiling in turbulent atmosphere,” Opt. Express **12**, 1249–1257 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1249 [CrossRef] [PubMed]

14. 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, “Coherent power measurement uncertainty resulting from atmospheric turbulence,” Opt. Express |

2. | A. Belmonte, “Coherent DIAL profiling in turbulent atmosphere,” Opt. Express |

3. | L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A |

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

5. | A. Belmonte “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt.39, 5426–5445 (2000). J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of High energy laser beams through the atmosphere, [CrossRef] |

6. | J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. |

7. | K. J. Gamble, A. R. Weeks, H. R. Myler, and W. A. Rabadi, “Results of two-dimensional time-evolved phase screen computer simulations,” in |

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

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

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

11. | N. Menyuk, D. K. Killinger, and C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. |

12. | B. J. Rye, “Power ratio estimation in incoherent backscatter lidar: Heterodyne receiver with square law detection,” Journal of Applied Meteorology |

13. | G. M. Ancellet and R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A |

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

**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

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 12, 2004

Revised Manuscript: July 26, 2004

Published: August 9, 2004

**Citation**

Aniceto Belmonte, "Temporal averaging of turbulence-induced uncertainties on coherent power measurements," Opt. Express **12**, 3770-3777 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3770

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### References

- A. Belmonte, "Coherent power measurement uncertainty resulting from atmospheric turbulence," Opt. Express 12, 168-175 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-168</a> [CrossRef] [PubMed]
- A. Belmonte, "Coherent DIAL profiling in turbulent atmosphere," Opt. Express 12, 1249-1257 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1249">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1249</a> [CrossRef] [PubMed]
- L. C. Andrews, W. B. Miller, and J. C. Ricklin, ???Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,??? J. Opt. Soc. Am. A 11, 1653-1660 (1994) [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]
- J. A. Fleck, J. R. Morris, and M. D. Feit, ???Time-dependent propagation of high energy laser beams through the atmosphere,??? Appl. Phys. 10, 129-160 (1976) [CrossRef]
- K. J. Gamble, A. R. Weeks, H. R. Myler, W. A. Rabadi, ???Results of two-dimensional time-evolved phase screen computer simulations,??? in Atmospheric Propagation and Remote Sensing IV, C. Dainty, ed., Proc. SPIE 2471, 170-180 (1995)
- 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, ???Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,??? J. Opt. Soc. Am. 71, 687-691 (1981) [CrossRef]
- A. Belmonte and B. J. Rye, ???Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,??? Appl. Opt. 39, 2401-2411 (2000) [CrossRef]
- N. Menyuk, D. K. Killinger,, and C. R. Menyuk, ???Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,???Appl. Opt. 21, 3377-3383 (1982) [CrossRef] [PubMed]
- B. J. Rye, ???Power ratio estimation in incoherent backscatter lidar: Heterodyne receiver with square law detection,??? Journal of Applied Meteorology 22, 1899-1913 (1983) [CrossRef]
- G. M. Ancellet and R. T. Menzies, ???Atmospheric correlation-time measurements and effects on coherent Doppler lidar,??? J. Opt. Soc. Am. A 4, 367-373 (1987) [CrossRef]
- 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]

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