## 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

The approach using simulation techniques was used to look at the uncertainty inherent to the process of heterodyne optical power measurement in the presence of atmospheric refractive turbulence [1]. Fluctuations in received power owing to turbulence have the same impact as those that result from speckle and, consequently, a precise description of this effect was needed to fully characterize the performance of heterodyne lidars in the atmosphere. The analytically intractable problem of describing the coherent return variance was considered by simulations of beam propagation in a realistic way. This work preceded using of the simulation technique to take account of the effects of refractive turbulence in the performance of a heterodyne differential absorption lidar (DIAL) system, where the error associated with the measurement depends on the estimation error of the power received signal [2].

The results for the coherent power variance indicated the presence of a maximum at shorter ranges when strong turbulence levels are considered, whereas for weak turbulence the variance increased monotonically [1]. The concept of beam averaging was a simple way of interpreting this behavior. It was shown that 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: In most cases, the normalized power standard deviation σ_{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].

Averaging techniques have been devised to deal with lidar signal fluctuations. Speckle-induced fluctuations can be spatially averaged out. As a pulse propagates, it illuminates different spatial volumes of aerosols. Since every *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.

Although spatial averaging can still be used somewhat to reduce fluctuations in received signal level due to turbulence [1], temporal averaging may be more difficult to apply: Because of the increased time scale of the fluctuations, the time required to reduce the variance may be lengthened to values incompatible with lidar measuring systems requiring a large bandwidth. Roughly, the time scale of the intensity fluctuations due to turbulence is related to the wind velocity across the beam *V* and given by τ_{t}=*l*/*V* (*Taylor’s hypothesis* of ‘frozen’ atmosphere), where *l* is the spatial correlation distance of the irradiance fluctuations. In most situations of interest, the correlation distance *l* is compress between the beam transverse-field coherence diameter, *r*_{0}
and the Fresnel diameter, √*R*/*k* [3]. Here, *k*=*2π*/*λ* is the beam wavenumber and *R* range along the propagation path. For typical values of *l* and *V*, the time scale of the fluctuations due to turbulence τ_{t} is several orders of magnitude larger than that of speckle-induced fluctuations (milliseconds rather than microseconds).

It should be noted, however, that for spaceborne systems, the slew rate associated with the satellite moving respect to the ground becomes important in making temporal calculations. In most situations, ground wind speed has little effect and slew rate governs the temporal behavior of the lidar signal. Now, the time scale of fluctuations for satellite-borne systems is remarkably smaller than for their counterpart ground-based systems and, consequently, temporal averaging may result effective.

In this study we will show how this power uncertainty may become a greater problem in ground-based coherent systems owing to the long time constant associated with fluctuations. When dealing with the turbulence uncertainty added to the measurement process, this time constant will prevent to a certain extent taking advantage of the techniques (i.e., signal averaging) used regularly to wipe out speckle effects. Accordingly, refractive turbulence may become as prevalent as speckle in limiting lidar accuracy. In Section 2 we will estimate the time constant associated with the turbulence-induced fluctuations of the received power. This analysis will be based in simulations which consider the effect of the coherent signal temporal correlation. In Section 3, correlation coefficients will be used to calculate the standard deviation of the coherent power fluctuations as function of the number of pulses averaged.

The standard simulation technique, in which the atmosphere is modeled as a set of two-dimensional Gaussian random phase screens, is based on the Fresnel approximation to the wave propagation [4, 5]. In a time-independent calculation, the treatment of turbulence is built on the generation of random phase screens at each calculation step along the propagation path: at a given position they are generated anew for each successive simulation run. In this study we need to use both time and space correlated phase screens to properly describe the temporal correlation of the power received by a coherent system: The inclusion of time correlated phase screens does allow to simulate the observed temporal dependency of both the beam intensity and beam phase fluctuations. In a time-dependent calculation the random phase screens need to be moved with the wind at a given position for each successive time: Wind velocity across the beam is the only parameter required to describe properly in our simulations the temporal evolution of the atmospheric turbulence [6, 7].

All our simulations assume uniform turbulence with range, i.e., horizontal propagation paths, and use the Hill turbulence spectrum [8] – with typical inner scale l_{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] (in general, every position require the generation of statistically independent phase screens). For any given temporal resolution δt, and all 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.

Phase screens are generated via a standard number generator used to compute pseudorandom arrays of statistically independent Gaussian numbers containing zero spatial correlation. By using the Hill spectrum to filter the Fourier transform of the uncorrelated arrays and inverse transforming the result, one obtains two-dimensional spatial fields of phase fluctuations with the right spatial correlation. To create successive spatial correlated phase screens that are also correlated in time, we just use repeatedly the same array of statistically independent Gaussian numbers with no spatial correlation, shift the columns a number of positions proportional to the wind velocity Vδt/δx, and replace the columns left blank by the shifting with new independent Gaussian numbers. After the shifting, we use again the Fourier transform methods to filter this modified array. With this procedure, there are no limits to the number of successive time realizations that can be simulated.

In Fig. 1, a movie presents an example of time-evolving phase screen along with the simulated temporal effects of atmospheric turbulence on both beam phase and beam intensity. Beam phase has been considered very close to the lidar aperture plane and beam intensity is shown at the target plane, 3 km away: Phase fluctuations at short ranges translate into intensity fluctuations at ranges far away from the lidar transmitter. Beam wander, beam-shape changes, and scintillation can be clearly appreciated in the presentation.

## 2. Coherent power temporal correlation

We need to estimate the time constant associated with the turbulence-induced fluctuations of the received power. To evaluate the temporal correlation of the lidar signal returns *P*(*R*, *t*)

at range *R* and different delay times Δt=t_{2}-t_{1}, we will use the target-plane formulation [9]. In this model, the 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**

where the term *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].

Eq. (2) is especially relevant to our simulations, reducing the problem of calculating lidar returns to one of computing the transmitted and backpropagated beams along the propagation path and estimating the overlap of the two normalized irradiances at the target plane [10]. As we are mainly concerned with the effects of the refractive turbulence, term *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.

Figure 2 shows the correlation coefficients *ρ*(t_{1}, t_{2}) [temporal correlation *C*_{P}
(t_{1}, t_{2}) normalized to the variance *C*_{P}
(t, t)=${{\mathrm{\sigma}}_{\mathrm{P}}}^{2}$(t)] of power fluctuations for different ranges R as a function of the delay time Δt=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], longer correlation times corresponds to larger ranges. A simple physical explanation for this behavior follows from the fact that, as argued previously, coherent power fluctuations are a consequence of beam scintillation after averaging over the illuminated area. Actually, it is the beam diameter along with the irradiance scales, rather than scintillation alone, that define the time scales of coherent power fluctuations. For increasing ranges, the beam becomes increasingly larger and, consequently, so does the temporal correlation length.

## 3. Limitations to temporal averaging

The coherent power measurement uncertainty is characterized by its normalized standard deviation. Assuming that *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 [11–13]. Our results regarding turbulence effects complement those earlier analyses.

Simulations were carried out to measure the effect of signal averaging on turbulence-induced coherent power fluctuations. As expected, the study shows that temporal correlations alters the gain available through signal averaging, especially when high pulse repetition frequency (*PRF*) systems are considered. To obtain the variance for the average of *N* pulse returns, ${{\sigma}_{N}}^{\mathit{2}}$
, we made use of the single-pulse coherent power variance, ${{\sigma}_{P}}^{\mathit{2}}$[1], and the simulated correlation coefficients, *ρ*(see Section 2):

Here, the n-th pulse return occurs at time t_{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]. It allows us to quantify the deviation of our lidar return averaging process from the *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.)

The limitations of signal averaging due to atmospheric turbulence are illustrated in Fig. 3, which shows the values of the coherent power uncertainty σ
_{N}
as a function of the number of averaged pulses *N* when the same typical diurnal conditions of turbulence and wind that were used in Fig. 2 are considered. In Fig. 3, we also present measurement uncertainty due to speckle effects. In this case, the variance of the power fading for a single pulse is unity for any wavelength and range considered. Fluctuations in received power due to speckle are temporally uncorrelated. Therefore, we can always assume that any *N* power samples are independent and the measurement will improve its accuracy as *N*
^{-1/2}. In the figure, we use the results for ideal speckle averaging to check the efficiency of signal averaging when dealing with refractive turbulence. In most cases, the decrease of the uncertainty with *N* is significantly smaller than the *N*
^{-1/2} dependence predicted for independent measurements.

In Fig. 3, coherent power accuracy is shown for different system PRF. The values of the system PRF were chosen such that the effect of the sampling rate on the averaging process was clearly established. The value of the temporal correlation length restricts the improvement in the standard deviation that can be attain by signal averaging in spite of the number of pulses averaged during that time interval. Distinctly, any increases on the pulse repetition rate won’t necessarily reduce the coherent power measurement uncertainty.

Interestingly, when 2-µm systems and strong turbulence conditions are considered, power uncertainty always seems to be smaller than speckle uncertainty regardless of the sample size: Although sample averaging is severely affected by temporal correlation, beam averaging in the target plane limits the single-pulse level of fluctuations in the return signal [1]. However, in any other of the analyzed situations, return signal fluctuations quickly overcome speckle fading when temporal averaging is considered. Even a small sample (less than 10 pulses) makes turbulence more prevalent than speckle in limiting coherent power measurement accuracy. For 10-µm systems, where beam averaging in the target plane is less intense and single-pulse power standard deviation increases monotonically with range [1], signal averaging does noticeably worse. In any of the atmospheric conditions considered in this study, refractive turbulence makes irrelevant the improvements achievable by averaging over small samples: In the figure, just when 100 or more pulses are used to average, turbulence-induced uncertainty is consistently reduced.

## 4. Concluding remarks

Although using a system with a sufficiently high pulse repetition rate can reduce the measurement variance produced by the speckle, the turbulence-induced fluctuations cannot be easily averaged out. Therefore, we may have to increase the averaging time required to reduce the measurement uncertainty, limiting the effective bandwidth of the lidar system. Regarding this temporal averaging analysis, a 10-µm lidar seems to be in a more adverse situation that an equivalent 2-µm system: Signal averaging may result more difficult to apply due to longer correlation times associated with larger wavelengths (see Fig. 2).

Heterodyne differential absorption lidar (DIAL) systems are strongly affected by return signal fluctuations [2], so it would be worth to consider how the results of our study may impact this important sensing technique. Actually, the use of coherent DIAL systems to determine the concentration of molecular species in the atmosphere requires a differential measurement of the return power at two different wavelengths (two closely spaced longitudinal modes): Since the effect of signal averaging works also on the power differential measurements, the preceding comments on the limitations of signal-averaged measurements can be readily applied to DIAL sensing.

Still, being the ratio of the return powers that is to be averaged in DIAL measurements, the difficulties over signal averaging may be overcome if the assumption that turbulence-induced fluctuations for transmission on adjacent longitudinal modes are correlated would prove to be certain. In Fig. 4 we have tested the reach of this hypothesis for a 16-cm aperture, monostatic lidar. Systems working at both 2-µm and 10-µm wavelength are considered. It shows the results of estimating the correlation among coherent power fluctuations induced by turbulence on adjacent modes. We considered line separations ranging from 1GHz to 3000GHz (roughly, that’s equivalent to 1 µm for the 10-µm lidar and 0.1 µm for the 2-µm lidar). In order to make the estimations in a worst-case scenario, both beams were propagated 3 km through the same strong atmospheric turbulence. It shows that decorrelation terms are almost negligible for all line separations but the largest ones and, even in these cases, it is really small (in abscissa, the decorrelation term is expressed in rate per thousand).

Now, everything seems to point to the advantages of using short time intervals between the two pulses of a shot-pair: As the fluctuations of the return power in the two DIAL wavelengths will be essentially correlated and this correlation will extend over a long period of time, the ratio of simultaneous or almost simultaneous observations in the two channels could be estimated without bias and then, directly, temporally averaged within a correlation time by using a high PRF system. By getting an estimator in a short time, the differential nature of DIAL measurement would take away the effects of fluctuations that are correlated for the two wavelengths. In fact, the long time constant of the return signal fluctuation due to turbulence is relevant only if we find the mean return power at the two wavelengths before estimating the power ratio required by DIAL techniques [12].

This approach relies on the correlation of the power fluctuations for both DIAL wavelengths, as shown in Fig. 4. In most circumstances, however, the performance of the lidar may be strongly degraded by the inescapable lack of alignment between the on-line and the off-line beams: Misalignment will tend to wipe out nearly all possible correlation of turbulence-induced DIAL measurement fluctuations at different wavelengths. Accordingly, almost simultaneous shot-pair may have some limitations. A more complete discussion of these effects is beyond the scope of this study: Although the sensitivity to misalignments has already being clearly established on practical heterodyne lidars [14], some more work will be necessary to quantify shot-pair misalignment impact on the performance of coherent DIAL and clarify the possibilities of using this short-time approach to the estimation of DIAL power ratios in the presence of refractive turbulence.

This research was partially supported by the Spanish Department of Science and Technology MCYT grant No. REN 2003-09753-C02-02.

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