## Abstract

The simulation of beam propagation is used to examine the sensitivity to misalignments of a practical heterodyne lidar because of the presence of refractive turbulence. At shorter wavelengths, and under general atmospheric conditions, the performance of a realistic instrument is never well described by either of the ideal monostatic and bistatic arrangements when misalignment is taken into consideration.

©2003 Optical Society of America

## 1. Introduction

The simulation of beam propagation permits computation of the fields on the target plane [1] that determine the performance of the heterodyne lidar in turbulent atmosphere [2]. This formulation makes it possible to overcome the considerable complexity of the problem on the receiver plane, where both aerosol speckle and refractive turbulence effects define the degree of coherence of the backscattered radiation to be matched with the local oscillator [3] (see Fig. 1). The simulations used here are based on the well-known method of modeling the atmosphere by a set of two-dimensional, Gaussian, random phase screens with an appropriate phase power spectral density [4]. After formulation of the numerical limitations to the method, the range of applicability of a given simulation with a given parameter set was presented: Beam spreading, beam wander, beam coherence diameter, and variance and autocorrelation of the beam irradiance were described beyond the limits allowed by the theoretical analysis [1].

At shorter wavelengths, the effects of refractive turbulence on ground-based lidar systems are the main factors decreasing coherent signal power. Causing amplitude and phase distortions of the optical radiation propagating through the atmosphere, refractive turbulence effects are closely related and strongly dependent on near field and misalignment problems. In a companion paper [5], we addressed the near-field problem and resolved the optimal lidar telescope parameters. Relevant parameters were aperture diameter *D* and wavefront radius of curvature *F*. It was shown that, in many circumstances, the performance of the coherent lidar is not to be degraded significantly by the reduction of the aperture diameter. By simulating moderate-to-strong turbulence conditions, our analysis indicated that smaller apertures could perform at least as well as larger ones on collimated laser radar systems working at shorter wavelengths. Also under these conditions, the improvements linked to the focusing of the beam are less obvious than when longer wavelengths and weaker turbulence conditions are considered. In most practical situations collimated systems with small apertures could even exceed the performance of larger aperture systems focused into the range of interest.

By using a two-beam (transmitted and back-propagated or phase-conjugate local oscillator) approach (Section 2), the simulation technique will be extended to take account of the effects of beam misalignment (see Fig. 1) in the performance of a heterodyne lidar (Section 3). All simulations will assume uniform turbulence with range and use the von Karman turbulence spectrum with typical inner scale l_{0} of 1 cm and realistic outer scale L_{0} of the order of 5 m. It will be interesting to consider which setup would be more sensitive to misalignment in turbulent atmosphere, and how misalignments together with refractive turbulence make unnecessary to distinguish monostatic from bistatic arrangements. Their performance can not be described by either of the ideal configurations and they ought to be just considered as limit situations.

Previous analytical work considering the misalignment problem [6–8] was based on a number of simplifying approximations, whereas numerical simulations have the potential for greater realism. Differences between simulation and the most relevant and generally accepted analytical results obtained for monostatic and bistatic configurations [8, 9] were considered extensively in our earlier work [2]. We showed that although the main physical features of the problem appear in the analytical results, comparing simulated heterodyne lidar performance in the turbulent atmosphere with those that produced the analytical expressions revealed significant dissimilarities. The differences in approach and some key assumptions in the analytical theories that don’t need to be considered in the simulations explained most of the divergences.

## 2. Two-beam model for the coherent solid angle

In this study we will describe the efficiency of a practical heterodyne lidar with aperture *A*_{R}
at range *R* in terms of an effective coherent solid angle e Ω*COH* [5]:

The effective solid angle or coherent responsivity is expressed in terms of the maximum available value *A*_{R}
/*R*^{2}
and the system-antenna efficiency *η*_{S}*(R)* that describes the extent to which this value is degraded [9, 10]. The better known wideband signal-to-noise ratio (*SNR*)

differs from our coherent solid angle in just a group of parameters *C(R)* describing conversion efficiencies, various system components, and atmospheric scattering conditions. As we are mainly concerned with the effects of the refractive turbulence, those parameters are mostly irrelevant here.

By using the target-plane formulation, the equation for the coherent solid angle defines the performance of the heterodyne lidar in the overlap integral of the transmitted (T) and virtual back-propagated local oscillator (BPLO) irradiances at the target plane **p** [2]

where *λ* is the optical wavelength of the transmitted laser. The irradiances *jT* and *j*_{BPLO}
have been normalized to the laser 〈*P*_{L}
(*t*)〉 and local oscillator (LO) 〈*P*_{LO}
〉 average power, respectively. The problem of heterodyne lidar performance in the presence of atmospheric turbulence is reduced to one of computing intensity along the propagation paths. With this formulation, the random nature of the problem arises because of the randomness of the intervening atmospheric medium, which requires averaging (〈〉) over different time scales.

Now, the heterodyne solid angle Ω*COH* is easily computed by using simulation techniques: for a given propagation path, we have to estimate the overlap integral of the normalized irradiances in the two beams at the target plane in the presence of turbulence. In the absence of turbulence, the optical fields are deterministic and ensemble averages over the random media and random fields are not required, so the calculation is straightforward and not computer intensive.

Note that we can rewrite the overlap integral of the irradiances Eq. (3) as

where *C _{j}(p,R)* is the intensity fluctuations correlation of the transmitter and BPLO beams

With this two-beam model, the effects of refractive turbulence on the coherent lidar return are described by the fourth moment for wave propagation through random media. 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, the overlap integral of the transmitter and BPLO irradiances doesn’t contain any correlation term, i.e., *C _{j}(p,R)*=

*0*. For the bistatic configuration, the principal mechanism that describes the effect of atmospheric refractive turbulence is the additional expansion of both the transmitted and the BPLO beams, which always reduces performance below the free-space results. The statistically independent path result is a lower bound for the coherent solid angle.

For a monostatic configuration, where a receiving aperture co-located with the transmitter collects the backscattered light, the direct and backscattered fields travel over essentially the same atmospheric path. For perfectly matched transmitter and BPLO beams instantaneous intensity distributions coincides *j _{T}(p,R)*=

*j*and

_{BPLO}(**p**,R)*C*transforms into the beam scintillation index

_{j}(**p**,R)*${\mathrm{\sigma}}_{\mathrm{j}}^{2}$ (*. This always positive addition produces the so-called backscattering enhancement effect: Turbulence causes small bright spots on the scattering target, increasing the value of the overlap integral, and so, compared with that of the bistatic calculation, monostatic performance is enhanced. For any specific atmospheric turbulence condition, the perfectly matched beams result is an upper bound for the coherent solid angle. As a consequence of the enhancement effect, and for practical reasons, the configuration of feasible heterodyne lidars is usually monostatic.

**p**,R)≥0It results clear from Eqs. (4) and (5) that comprehending beam intensity fluctuations is fundamental to describe the performance of a heterodyne lidar. Beam wave intensity fluctuations differ from those involving plane wave or spherical wave propagation in that the lateral beam dimension plays an important role in the determination of the irradiance at the target. Initial theoretical and experimental investigations based on weak-turbulence situations (see, for example, Refs. [11, 12]) predicted that the correlation length of the irradiance fluctuations would be of the order of the first Fresnel zone √R/k. Here, *k*=*2π*/*λ* is the beam wavenumber. However, in most practical lidar systems beam paths of interest are often long enough to cause saturation effects. Notably, when the ground lidar systems working at 1–2 µm wavelengths under development [13] are considered, the effects of refractive turbulence are important for ranges as short as a few hundreds meters for any condition of turbulence. In these situations, the irradiance fluctuations are not properly described by the weak-turbulence analysis.

Theoretical calculations of beam propagation and the higher moments of the field are still difficult in this most interesting turbulence regime and, consequently, no simple analytical solutions are known, except for those obtained for simplified beam configurations and unrealistic atmospheric characterization. In any case, early works [14, 15] and -more recently- simulations [1] have shown that for this propagation regime the correlation of the intensity fluctuations contains two scales. A small-scale peak, with a width of about *r*_{0}
, the transverse-field coherence diameter of a point source, describes the generation of a diffraction speckle pattern superimposed on the beam irradiance profile. The second and longer scale is the effect of a refractive process defining a long tail that extends a distance proportional to *4R*/*kr*_{0}
. This may be interpreted as an effective beam-spreading diameter [16]. The behavior of different lidar geometries are related to the scintillation scales of the transmitted beam: The signal enhancement that occurs in the monostatic case arises because both small-scale refractive scintillation and large-scale beam wander and diffractive scintillation are common to the two beams.

Considering all these spatial scales will help us to understand how the choice of the lidar geometry and the level of misalignment may influence the efficiency of a practical heterodyne lidar in the presence of atmospheric turbulence. The simulations permit characterization of the effect on lidar performance of the analytically intractable intensity correlation *C _{j}(p,R)* in Eq. (4) that result from turbulent fluctuations for whatever lidar geometry and alignment condition. Next, simulations of beam propagation will be used to quantify these misalignment effects on the performance of a heterodyne lidar.

## 3. Lidar performance degradation due to misalignments

Even if atmospheric turbulence is not considered, when the perfect alignment of a monostatic system is disturbed, the coherent solid angle is altered by the decay of the overlap of the transmitter and BPLO beams. In this case, the lidar performance expressed in terms of the two-beams overlap integral in the target plane will depend on the single factor πD/λ |Δ**ϑ**|. Here, Δ**ϑ** is the misalignment angle and λ/πD is the diffraction limit of the lidar circular aperture. The beam divergence is directly proportional to the ratio between the wavelength λ and the aperture diameter *D* (i.e., the beam-waist diameter 2ω_{0}). To obtain a highly directional beam, a short wavelength and a large aperture should be used. For a specific misalignment angle Δ**ϑ**, if the aperture is reduced, the beam will diverges, πD/λ |Δ**ϑ**| will gets smaller and the lidar system will become less sensitivity to angular misalignment.

When more realistic lidars considering misalignment angles in the presence of refractive turbulence are considered, the differences among the performances of those simple monostatic and bistatic geometries fade away. The misalignment angle produces a spatial offset *Δ p* between the transmitted irradiance and BPLO irradiance in the target plane. For large wavelengths, or weak-turbulence conditions, it will cause a degeneration of the overlap integral of the transmitter and BPLO beams. However, when the turbulence-induced beam spreading becomes important -shorter wavelengths, higher turbulence levels, or larger ranges—this effect will be negligible. More interesting for shorter wavelengths, the small-scale scintillation structure will produces a decrease in performance with misalignment sharper and more sensitive than those observed in the free-space case: small misalignments will destroy the short-scale correlation of the transmitter irradiance and BPLO irradiance at the target and the overlap integral will become more similar to the result obtained for the bistatic situation. The enhancement in performance produced by large-scale irradiance correlation will be observed even with larger misalignment angles.

Figure 2 shows the lidar sensitivity to the misalignment angle between the transmitter and BPLO axis as a function of range. The simulation technique uses a numerical grid of 1024 1024 points with a 5-mm resolution and simulates a continuous random medium with 20 two-dimensional phase screens. Transmitted and virtual LO beams were assumed to be collimated, Gausssian, and truncated at the telescope aperture. For any aperture diameter and misalignment angle considered in the figure, the beam truncation was 1.25 (i.e., *D*=*1.25 2ω*_{0}
, where ω_{0} is the 1/e^{2} beam irradiance radius). This truncation maximizes far-field system-antenna efficiency in the ideal case of absence of turbulence. The initially considered lidar aperture *D* was 16 cm in diameter (middle figure) and its performance is compared with lidar systems with 32-cm (up) and 8-cm (down) diameter apertures. In any case, the misalignment angle |Δ**ϑ**| has values of 10 µrad, 20 µrad, 30 µrad, and 40 µrad. Because we are interested primarily in turbulent effects, and in order to simplify the graphs, we have normalized the plots so the coherent solid angle at the shortest range is 0 decibels (dB). In most practical situations, coherent solid angle decay (i.e., a wideband *SNR* decay) of about -15 dB defines the lidar maximum range. All the previously described effects associated with increasing misalignments are considered by the simulations, where the level of refractive turbulence has the typical daytime value of moderate-to-strong condition, ${C}_{n}^{\mathit{2}}$
=10^{-12} m^{-2/3}.

The misalignment angle *Δ*
**ϑ** will translate into an offset *Δ p*=

*RΔ*

**ϑ**between the transmitted irradiance and BPLO irradiance in the target plane at range

*R*, decreasing the value of the intensity fluctuations correlation

*C*in Eq. (5). Small-scale fluctuations in the two beams will make the heterodyne lidar extremely sensible to even relatively small misalignment angles. In many practical situations the two-beam offset |

_{j}(**p**,R)*Δ*| will be larger than the coherence diameter

**p***r*

_{0}characterizing small irradiance fluctuations. When larger apertures are considered (see Fig. 2, upper graph), the effects are pronounced for misalignment angles as short as 10

*µrad*. In this case, when large apertures make the lidar system most sensitive to turbulence effects, coherence diameter is less than 1 cm for ranges as short as 2000 m [17, 1]. Also at this range, turbulence spreads the angular beam diameter to near 300 µrad [16, 1], a value much larger than any considered misalignment in Fig. 2 (in the absence of turbulence, beam would spread to no more than 40 µrad). For this study, coherence diameter and beam width values were checked during the simulation process to guarantee the accuracy of our results. For smaller aperture diameters, and so larger coherence diameters in the target plane, sensitivity to misalignment slightly decreases, but still angles as small as 20–30

*µrad*make monostatic behavior almost identical to that expected for bistatic situations (see Fig. 2, middle and down graphs).

Large-scale intensity effects will be less susceptible to misalignments, and we should expect some observable effects just when the offset between transmitted and BPLO beams in the target plane is similar to the scattering disk size describing large irradiance fluctuations, i.e., |*Δ p*|

*/*

**≥**4R*kr*

_{0}. Although the residual large-scale irradiance correlation still produces an enhancement in performance, for a misalignment of just 40

*µrad*the behavior of the initially monostatic geometry closely approaches that expected for a bistatic system (see Fig. 2, up and middle graphs). Just for the smaller aperture considered in Fig. 2 (lower graph), misalignment is not able to step down the lidar performance to bistatic levels. For ranges longer -or misalignment angles slightly larger—than those shown in the figures, even this small large-scale enhancement term will eventually vanishes. Perfect angular alignment in most practical, operational lidar ground systems can not be pinned down and all the previous considerations become useful in almost any feasible experimental situation.

Figure 3 also presents the lidar responsiveness to the misalignment angle as a function of range. Simulation parameters, normalization, lidar system variables, aperture diameters, and misalignment angles are similar to those in Fig. 2 but weaker turbulence conditions are considered. As expected, the overall lidar system sensitivity to misalignment decreases with the level of turbulence, but still the monostatic configuration cannot describe properly the performance of practical geometries considering misalignment. In most realistic situations, were we should expect angles |Δ**ϑ**| even larger than those regarded in our simulations, misalignment will override almost any other consideration about the geometry of our lidar system.

At the largest misalignment angle, it is interesting to note how the decay of the overlap of the transmitter and BPLO beams at the near ranges brings the lidar performance to levels below the bistatic situation. Most important for the 8-cm aperture (Fig. 3, down), the less sensitive to atmospheric turbulence, but present to some degree for all aperture diameters considered, this effect can be interpreted with the help of the two-beam model. For short ranges and weak turbulence effects, intensity fluctuations are not large enough to produce very significant differences between the performances of monostatic and bistatic configurations. In this situation, a wide decrease of the region of overlap between the two beams at the target plane produced by large misalignment angles will become the main effect affecting the lidar performance. This geometrical, overlap situation will translate into a coherent solid angle that is less that the bistatic case. As it can be appreciated in Fig. 3, when intensity scales build up at far ranges, and turbulent beam spreading become most relevant, this effect disappears. For even weaker turbulence conditions (nighttime) than those showed in Fig. 3, we find this overlap effect to be the most relevant factor affecting the return signal on systems with angular misalignment.

## 4. Conclusions

Using numerical simulation techniques, we deal with the problem of determining the part that misalignment in the presence of refractive turbulence plays in coherent lidar systems. Our approach was based on use of the two-beam model for calculating coherent lidar signal returns, which reduces the problem to one of computing irradiance along the two paths. Comprehending intensity fluctuation scales on the propagated beams produces most useful physical insight and allows explaining in simple terms almost any significant turbulence effect linked to misalignment.

Containing the effects of atmospheric turbulence is necessary to understand the effects of angular misalignment on realistic heterodyne lidar systems. In most circumstances, the performance of the lidar is strongly degraded by inescapable lack of alignment between the transmitted and the local oscillator beams. Simulations of moderate-to-strong turbulence conditions indicate that smaller apertures could perform better than the larger ones on collimated lidar systems working at shorter wavelengths. This improvement is explained by their broader directional beams and lesser sensibility to turbulence than their large aperture counterparts. This result is complementary to those obtained in the associated study [5] dealing with the optimal telescope parameters of a practical heterodyne lidar. The near-field dependencies of turbulence effects indicated that the performance of coherent lidars working at shorter wavelengths were not to be degraded significantly by reducing the aperture diameter.

As a consequence of misalignments, our simulations also indicates that, with independency of the lidar configuration, in most practical situations its performance is described by the ideal bistatic configuration better than for the monostatic one. Misalignment tends to wipe out nearly all the enhancement associated with collocated, monostatic systems.

The author is grateful to B. J. Rye for many useful discussions that have helped to define the goals of the 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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