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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 1 — Jan. 12, 2004
  • pp: 168–175
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Coherent power measurement uncertainty resulting from atmospheric turbulence

Aniceto Belmonte  »View Author Affiliations


Optics Express, Vol. 12, Issue 1, pp. 168-175 (2004)
http://dx.doi.org/10.1364/OPEX.12.000168


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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

Using numerical simulation techniques, we tackled the problem of determining the part that refractive turbulence plays in coherent lidar systems [1

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]

]. Our approach was based on use of the two-beam (transmitted and back-propagated or phase-conjugate local oscillator) model for calculating coherent lidar signal returns, which was derived previously and reduces the problem to one of computing irradiance along the two paths [2

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

]. A standard technique, in which the atmosphere is modeled as a set of two-dimensional Gaussian random phase screens [3

3. 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).

], was extended for computing beam propagation as described in a separate paper [4

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

].

The effects of all turbulence mechanisms on the performance of the coherent lidar were considered with the simulations: Turbulent enhancement at short ranges for strong and moderate turbulence conditions is a consequence of the correlation of intensity fluctuations at the target. When the range increases, the predominant effect of atmospheric turbulence is beam spreading, which reduces the coherent power below the free-space expected values. We addressed the dependencies of refractive turbulence effects on near field, resolved the optimal lidar telescope parameters, and considered the problem of angular misalignment because of the presence of turbulence [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]

, 6

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]

].

In this study the simulation technique will be use to look at the uncertainty inherent to the process of heterodyne optical power (i.e., equivalent optical power generating the heterodyne receiver signal) measurement in the presence of atmospheric refractive turbulence. Fluctuations in received power owing to turbulence have the same consequences as those that result from speckle –degrading the accuracy of the coherent signal– and, consequently, a precise description of this turbulent effect is needed to fully characterize the performance of heterodyne lidars in the atmosphere. The analytically intractable problem of describing the coherent return higher moments (variance and covariance) can be considered by simulations of beam propagation in a realistic way.

All simulations will assume uniform turbulence with range and use the Hill turbulence spectrum [7

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]

] with typical inner scale l0 of 1 cm and realistic outer scale L0 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]

]. In any of the scenarios considered in this study, we run over 4000 samples to reduce the statistical uncertainties of our estimations to less than 2% of their corresponding mean values.

2. Heterodyne optical power statistics

By using the target-plane formulation, the equation for heterodyne optical power 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

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]

]

P(R,t)=Cexp[2Rα(R)]β(R)λ2jT(p,R,t)jBPLO(p,R)dp,
(1)

where the calibration constant C groups the conversion efficiencies and parameters that describe the various system components, β is the aerosol volume backscatter coefficient, α is the atmospheric linear extinction coefficient, and λ is the optical wavelength of the transmitted laser. The irradiances jT and jBPLO have been normalized to the laser 〈PL (t)〉 and local oscillator (LO) 〈PLO 〉 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]

].

The advantage of heterodyne systems comes about because of the higher 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.

The measurement process in coherent DIAL systems can be used to exhibit the principles of the problem we are dealing with. In DIAL measurements, two lidar pulses of slightly different wavelengths are selected such that the on-line wavelength corresponds to the absorption line of the species of interests, while the off-line wavelength is placed in a transparent region of the species line. The decay of the on-line signal and off-line signal in a defined range cell is compared to obtain the concentration of the molecule to be measured. The DIAL equation solving the measurement is proportional to the ratio δP of the received powers from the on-line beam Pon and the off-line beam Poff [10

10. R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Wiley-Interscience, New York, 1984).

]

δP(R)=Pon(RΔR2)Poff(R+ΔR2)Pon(R+ΔR2)Poff(RΔR2)
(2)

Here, ΔR [m] is the cell resolution between ranges RR/2 and RR/2. The DIAL calculation is not affected by the mean power P, as long as a return signal from both beams is received for every calculated range R. From Eq. (2) we can extract that any relative error in the power measurement resulting from atmospheric turbulence will translate as a relative error in the DIAL estimation. The propagation of errors associated to the DIAL measurement depends on the error of the four values of received signal power at the two wavelengths and two range gate positions: Uncertainty in DIAL concentration could be expressed through the sample variance of the power fluctuations, σP2 (R). Also, turbulence-induced power fluctuations at different ranges could be correlated: As the power fluctuations in Eq. (2) are not independents, the covariance terms CP (RR/2,RR/2) become non-zero and an improvement in the overall measurement accuracy may result. (In most practical situations, the correlation of turbulence-induced power fluctuations at on-line and off-line wavelengths is sufficiently small that they can be ignored.)

It results apparent from the considerations on DIAL systems–which are valid for any other coherent lidar application relying on power measurement–that an accurate knowledge of the statistics of the coherent power turbulent fluctuations as a function of range R is required if we are bound to take on the estimation of relative error in the measurements. In general, the error analysis would need to estimate the normalized power variances σP2 and covariances CP . 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

CP(R1,R2)=[jT(p,R1,t)jBPLO(p,R1)dp][jT(p,R2,t)jBPLO(p,R2)dp][jT(p,R1,t)jBPLO(p,R1)dp][jT(p,R2,t)jBPLO(p,R2)dp]1
(3)

where R2-R1=ΔR. The covariance is a generalization of the variance in that CP (R, R)=σP2(R).

Theoretical calculations of beam propagation and the higher moments of the field are still difficult and just some partial results have been obtained on the second and fourth moments for simplified beam configurations and unrealistic atmospheric characterization (see, for example, Refs. [11

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

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

]). Our problem (3) involves higher powers (fourth moment of the intensity, i.e., eighth moment of the field) and, consequently, no simple analytical solutions to the statistics of the heterodyne power have been described. The simulation permits characterization of the effect on heterodyne lidar performance of the analytically intractable coherent return fluctuations that result from turbulence.

3. Power degradation due to atmospheric turbulence

Figures 12 show the normalized covariance of power fluctuations for different separations ΔR 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=1.25×2ω 0, where ω0 is the 1/e2 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.

In Fig. 1, the 2-µm lidar power variance under typical diurnal conditions of strong-to-moderate turbulence shows a characteristic maximum. A simple physical explanation for this behavior is described below. As argued in a preceding paragraph, the power fluctuations are a consequence of beam scintillation after averaging over the illuminated area. It is apparent that the maximum of the variance occurs when a single scintillation fills the beam area on the scattering target (as expected, for the same range we observe the maximum of the mean heterodyne power characterizing the turbulence enhancement [1

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]

]). At this target plane, the optical power fluctuations should match those in the beam irradiance.

For increasing ranges, the beam resolves several scintillations, producing an averaging effect. In fact, for the larger ranges the irradiance spot size is described by the coherence length of the beam phase fluctuation on the target plane rather than for the Fresnel length [15

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

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

]. The coherence length is generally smaller than the Fresnel length, so that the number of bright spots defined on the illuminated area increases and therefore the averaging effect increases. Also at these ranges, the beam intensity fluctuations saturate, establishing a limit to the continuously increasing scintillations occurring in the weak turbulence regime [11

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]

]. This saturation, together with the beam averaging, explains the decrease in coherent power variance at far ranges for strong turbulence levels (Fig. 1, left).

Fig. 1. Statistics of the coherent power turbulent fluctuations as a function of range R and different moderate-to-strong refractive turbulence Cn2 daytime values for a 2-µm wavelength, 16-cm aperture, monostatic lidar system. Along with the coherent power variance, the covariance function is shown for different range resolutions ΔR. The dashed line and y-axis labeling on the right corresponds to the mean coherent power.

In the limit of very weak turbulence-induced beam spreading, the size of the beam on the target depends on the aperture diameter of the transmitter telescope: by using smaller apertures, diffraction induces large illuminated areas on the target that improve the smoothing of the signal fluctuations. However, for higher turbulence levels and/or large propagation paths, the size of the laser beam at the target does not depend anymore on the aperture parameters [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]

]. In fact, the spot area, and thus the beam-averaging effect, is defined by the turbulence beam spreading.

When moderate turbulence levels (Cn2 =10 -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]

]. In this regime, saturation effect and averaging over the reduced beam size are no longer able to compensate the trend of the fluctuations to increase with the range.

The same behavior can be observed in Fig. 2 where a 10-µm lidar has been considered (any other lidar system parameter and levels of refractive turbulence are similar to those in Fig. 1). For strong turbulence conditions (left) the variance of the turbulence fluctuations increases again after a short decrease in the near field. Surprisingly, variance power fluctuations are larger for the 10-µm situation, where we would expect less sensitivity to turbulence, than for the 2-µm case. To explain this result of our simulations, we must consider the fact that beam averaging is now remarkably smaller: Larger beam intensity scales in the target plane reduces the potential for smoothing power fluctuations. They have a tendency for following closely fluctuations in the beam irradiance.

Fig. 2. Similar to Fig. 1 but for a 10-µm monostatic lidar. Again, along with the coherent power variance, the covariance function is shown for different range resolutions ΔR. The dashed line and y-axis labeling on the right corresponds to the mean coherent power.

For moderate turbulence levels and 10-µm wavelength in Fig. 2 (right), the variance is understandably smaller in the near fields and increases continuously with range. At lower turbulence levels than those consider for the 2-µm system in Fig. 1, where the beam irradiance fluctuations are less intense (Cn2 levels lesser than 10-14 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.

If less intense, the same effect can be observed in the 2-µm system for moderate refractive turbulence (Fig. 1, right) and the 10-µm lidar when strong Cn2 levels are considered (Fig. 2, left). For moderate turbulence levels and 10-µm wavelength in Fig. 2 (right), the cell resolution ΔR seems to be irrelevant and, for any range R, covariance terms are mostly identical to the normalized power variance. Now, interpretation is slightly different: for weaker strength turbulence and higher wavelength, the amount of intensity fluctuations produced by a single atmospheric layer of thickness ΔR is almost immaterial and, accordingly, intensity beam fluctuations (i.e., coherent power fluctuations) will remain correlated. As before, power covariance will match power variance.

The configuration of practical heterodyne lidar is usually monostatic: a receiving aperture co-located with the transmitter collects the backscattered light, so the direct and backscattered fields travel over essentially the same atmospheric path. For a bistatic system, where the separation between the transmitting and receiving apertures is large enough to ensure that the direct and backscattered light will travel through statistically independent refractive turbulence (decorrelated paths), the argument of the overlap integral of the transmitter and BPLO irradiances at the target in Eq. (1) reduces to <jT (p,z,t)><jBPLO (p,z)>. At shorter wavelengths, and under general atmospheric conditions, this bistatic situation is far from being an unrealistic lidar arrangement: As a consequence of the inescapable lack of alignment between the transmitted and the local oscillator beams, in most practical situations the performance of heterodyne lidars is rather described by the ideal bistatic configuration than for the monostatic one [6

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]

].

Fig. 3. Variance and covariance for different range resolutions ΔR of the coherent power turbulent fluctuations as a function of range R for a 2-µm bistatic lidar system. The lidar system parameters and levels of refractive turbulence are similar to those in Fig. 1 for the monostatic system. The dashed line and y-axis labeling on the right corresponds to the mean coherent power.
Fig. 4. Similar to Fig.3 but for a 10-µm monostatic lidar. The dashed line and y-axis labeling on the right corresponds to the mean coherent power.

4. Conclusions

The approach using simulations enabled this first study of the covariance of the returns generated by refractive turbulence fluctuations, for which there is no available theory. The results indicated the presence of a maximum at shorter ranges when strong turbulence levels are considered, whereas for weak turbulence the covariance increased monotonically. The concept of beam averaging is a simple way of interpreting this rather complicated behavior.

Beam averaging also explains why a bistatic lidar configuration tends to be slightly more immune to turbulence-induced power fluctuations than monostatic arrangements. As a consequence of misalignment, our simulation pointed out [6

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]

] that in most situations the lidar performance was described by the ideal bistatic configuration better than for the monostatic one. That make those bistatic results shown in this study most relevant.

The magnitude of the normalized variance is not as large as that of speckle fading (unity), but it is significant at most levels of turbulence and is a greater problem in heterodyne lidar due to the long time constant associated with the fluctuations. Our simulations also show the importance of the covariance between the fluctuations of received powers corresponding to separate ranges. Although the covariance terms associated to speckle are negligible, they are major when refractive turbulence fluctuations are considered.

The results and considerations of this research are of crucial importance when analyzing coherent DIAL systems, where the error associated to the measurement depends on the error of the power received signal at different ranges: correlations between the turbulence-induced power fluctuations may decrease the overall measurement uncertainty. The potential for greater realism of simulations will help us to understand the limitations resulting from atmospheric turbulence to coherent DIAL measurements. The author anticipates addressing them in a following study.

This research was partially supported by the Spanish Department of Science and Technology MCYT grant No. REN 2000-1754-C02-02.

References and links

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]

3.

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).

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]

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]

10.

R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Wiley-Interscience, New York, 1984).

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]

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. 21, 929–948 (1986). [CrossRef]

13.

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

14.

J. H. Churnside, Aperture averaging of optical scintillation in the turbulent atmosphere, Appl. Opt. 30, 1982–1994 (1991). [CrossRef] [PubMed]

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]

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 12th International School on Quantum Electronics: Laser Physics and Applications,P. A. Atanasov, A. A. Serafetinides, and I. N. Kolev, eds., Proc. SPIE5226, 310–314 (2003).

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

  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]
  3. 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).
  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), <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]
  6. 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]
  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]
  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]
  10. R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Wiley-Interscience, New York, 1984).
  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]
  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. 21, 929-948 (1986). [CrossRef]
  13. M. I. Charnotskii, �??Asymptotic analysis of finite-beam scintillations in a turbulent medium,�?? Waves in Random Media 4, 243-273 (1994). [CrossRef]
  14. J. H. Churnside , Aperture averaging of optical scintillation in the turbulent atmosphere , Appl. Opt. 30, 1982-1994 ( 1991). [CrossRef] [PubMed]
  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]
  17. 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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