## 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]. 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]. A standard technique, in which the atmosphere is modeled as a set of two-dimensional Gaussian random phase screens [3], was extended for computing beam propagation as described in a separate paper [4].

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, 6].

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] with typical inner scale l_{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]. 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, 9]

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 *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].

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 *P*_{on}
and the off-line beam *P*_{off}
[10]

Here, Δ*R* [*m*] is the cell resolution between ranges *R*-Δ*R*/2 and *R*+Δ*R*/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, ${{\sigma}_{P}}^{2}$
(*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 *C*_{P}
(*R*-Δ*R*/2,*R*+Δ*R*/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 ${{\sigma}_{\text{P}}}^{2}$ and covariances *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

where R_{2}-R_{1}=ΔR. The covariance is a generalization of the variance in that *C*_{P}
(R, R)=${{\mathrm{\sigma}}_{\mathrm{P}}}^{2}$(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. [11–13]). 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

The physical interpretation of the coherent return fluctuations due to turbulence is somewhat similar to that for speckle fluctuations in the return from a direct-detection lidar. In the latter situation, the average return irradiance (calculated at the receiver plane) is uniform, but at any instant it contains the bright patches known as *speckle*. Speckle increases the mean square irradiance (relative to the square of the mean irradiance) and results in the excess variance, above that due to shot noise, which is attributed to spatial fluctuations. In this case, it has been established that when the aperture contains a large number of speckles, the variance is reduced by what is known as *aperture averaging* [14]. For heterodyne detection, the overlap between *j*_{T}
and *j*_{BPLO}
determines the heterodyne return. The irradiance-related integral is evaluated in the target plane, not the receiver plane, and the integral is estimated over the beam area in this plane, which is variable and not fixed like a telescope aperture. However, we expect the same averaging principle to apply, so that a large number of bright spot scintillations caused by either the small area of each or a large beam area tend to reduce the power fluctuations.

Figures 1–2 show the normalized covariance of power fluctuations for different separations Δ*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.

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

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

When moderate turbulence levels (${{C}_{n}}^{2}$
=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]. 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.

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 (${{C}_{n}}^{2}$
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.

Received power normalized covariance presents a behavior very similar to that observed for the power variance and commented in the previous paragraphs (see Figs. 1 and 2). Besides, some specific remarks must be considered. Physically, power fluctuations are correlated along the lidar profiling path due to the propagation of beam turbulence effects. Along with the well-known transversal covariance, phase and intensity beam fluctuations show a longitudinal correlation as the turbulence perturbations on the plane *R* propagates along with the unscattered wave to the plane *R*+Δ*R*. As expected, the physical significance of the perturbation terms on the correlation of the coherent power fluctuations becomes decreasingly important as the thickness of the medium increases so that the total integrated coherent power fluctuation becomes larger. In Figs. 1 and 2 we observe how, for any of the considered turbulence levels, decreasing system range resolution by shorting the spatial intervals Δ*R* enhances the covariance.

When strong turbulence levels are considered for the 2-µm system in Fig. 1 (left), covariance shows an interesting performance at far ranges. With independence of the cell resolution Δ*R*, the power covariance trends towards the power variance value. This could be interpreted as a consequence of beam intensity saturation: For stronger turbulence and ranges larger than a few hundred meters, beam scintillation reaches a maximum and then stay there or decrease slowly [4]. Consequently, coherent power fluctuations at different ranges will be mostly correlated and the covariance for any range resolution Δ*R* will equal the power variance.

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 ${{C}_{n}}^{2}$
levels are considered (Fig. 2, left). For moderate turbulence levels and 10-µm wavelength in Fig. 2 (right), the cell resolution Δ*R* 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 Δ*R* 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 <*j*_{T}
(* p*,

*z*,

*t*)><

*j*

_{BPLO}(

*,*

**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].

Figures 3–4 are similar to the previous Figs. 1–2 but for bistatic case. Most of the previous remarks on monostatic systems apply for the bistatic situations, as can easily appreciated by comparing the two couple of figures. Certainly, as we should have expected, bistatic variance and covariance levels are slightly smaller than those estimated in the previous monostatic case. For the bistatic configuration, the principal mechanism that describes the effect of refractive turbulence is the additional expansion of both the transmitted and the BPLO beams, which always reduces performance below the free-space results. That beam spread translates into a supplemental beam averaging in the target plane and, eventually, into smaller coherent power fluctuations. The partial focusing characterizing the turbulence enhancement on monostatic systems also increases its coherent power fluctuations by defining high peaks of intensity on the target plane. The absence of this enhancement -small-scale fluctuations in the two beams are spatially uncorrelated- apparently makes bistatic systems less fit to lidar applications requiring high mean coherent power, but it certainly may turn them less sensitive to power fluctuations resulting from atmospheric turbulence. (Recently, a heuristic approximation to the heterodyne detection problem has obtained a rough estimation to the root-mean-square *SNR* valid for bistatic configurations [17]. The results of our simulations for the coherent power variance could be used to test the accuracy of the heuristic analysis.)

## 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] 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.

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