## Abstract

The simulation of beam propagation permits examination of the signal degradation in a heterodyne receiver caused by refractive turbulence under general atmospheric conditions and at arbitrary transmitter and receiver configurations. At shorter wavelengths, an understanding of turbulence effects is essential for deciding the optimal telescope parameters, i.e., focal length and aperture diameter, of a practical heterodyne lidar.

©2003 Optical Society of America

## 1. Introduction

For a given heterodyne lidar system, optimal detection sensitivity occurs when the incoherent backscatter field accepted by the receiver mixes effectively with the local- oscillator (LO) field. That is possible only when all avenues for the loss of coherence are negligible. Simulations of beam propagation in random media were used in the study of reduction in coherent signal power as a result of refractive turbulence [1]. It was shown that, along with the common factors that affect the return signal whatever the lidar wavelength (i.e., low aerosol concentration, increased atmospheric absorption), one has to consider properly the effects of refractive turbulence on ground-based systems working at shorter wavelengths.

The simulation technique provides the tools with which to analyze laser radar with general atmospheric conditions and arbitrary transmitter and receiver configurations (see Section 2). In this analysis we shall discuss the relevant telescope (antenna) parameters, i.e., collected area and focal length, and the conditions of turbulence under which these parameters may maximize the performance of a practical monostatic lidar system (Section 3). All simulations will assume uniform turbulence with range and will use the von Kármán turbulence spectrum with typical inner scale *l*_{0}
of 1 cm and outer scale *L*_{0}
of the order of 5 m. Outer-scale effects of turbulence are less pronounced than inner scale effects in our analysis because monostatic systems have immunity to beam wander or tilt [2]. Although the effects of the bump at the high frequencies that characterize the more-accurate Hill spectrum could affect the results of our simulations slightly, much of the lidar’s basic behavior is retained when the simpler von Kármán spectrum is used. Also, because alignment of a diffraction-limited system is critical, we assume throughout this paper that any angle between the axis of the transmitted and the LO beams is negligible.

Calculations for coherent laser radar (CLR) will be made in the target plane under different turbulent conditions. The simulation technique was described in a previous paper [3]. Briefly, we use a numerical grid of 1024×1024 points with 5-mm resolution and simulate a continuous random medium with 20 two-dimensional phase screens. We run a minimum of 250 samples to reduce the statistical uncertainties of our estimations to less than 1% of their corresponding mean values.

## 2. Coherent solid angle

The performance of a *CLR* is often determined by the signal-to-noise ratio (SNR). Although in atmospheric backscatter lidar the speckle patterns in the plane of the receiver aperture fluctuate in both space and time, only the spatial dependency is relevant to the telescope-related problems discussed here. Once we have assumed temporal stationarity, we consider time averages, so temporal fluctuations will be irrelevant to our estimations. In the discussion below, we consider the wideband SNR---also referred as carrier-to-noise ratio—defined as the ratio of the time-averaged mean signal power to the mean LO shot-noise power, assuming that the latter is the dominant noise.

By analogy with the well-known lidar equation for incoherent lidar, the SNR for a shot-noise-limited heterodyne lidar can be expressed as a function of solid angle Ω=*A*_{R}
/*R*^{2}
presented by a lidar telescope with aperture area *A*_{R}
at range *R*:

*C(R)* groups the conversion efficiencies and parameters that describe the various system components and the atmospheric scattering conditions: The SNR is maximized through high pulse energy, narrow receiver bandwidth, high transmission optics, high backscatter coefficient, and low extinction. One’s first thought would be also to make the telescope aperture as large as possible. After propagating back through the system optics, the backscattered energy must mix effectively with the LO beam at the signal detector: *η*_{S}*(R)* is the system-antenna efficiency [4], a parameter that describes the portion of return optical signal converted to effective coherent signal by the CLR. As in this study we are concerned mainly with the effects of the telescope parameters on the performance of a heterodyne lidar in the turbulent atmosphere, we can describe the efficiency of a practical CLR in terms of an effective coherent solid angle Ω*COH*:

The conversion efficiencies and parameters described by the term *C(R)* in Eq. (1) are irrelevant. The coherent solid angle is expressed in terms of the maximum available value *A*_{R}
/*R*^{2}
and the efficiency parameter *η*_{S}*(R)* that describes the extent to which this value is degraded. In the ideal case of deterministic transmitted optical fields, optimal beam truncation, far-field conditions, and absence of turbulence, the system-antenna efficiency can reach a maximum of approximately 0.4 [5]. Overcoming the several limitations of previously proposed analytical methods [2, 6, 7], simulations make possible the estimation of Ω*COH* (i.e., antenna efficiency) under any realistic conditions. The coherent solid angle is equal in every respect to the so-called coherent responsivity [7].

Lidars intended for profiling over long ranges are generally operated with collimated transmitting beams. As was discussed previously [8], for free-space propagation, for which the only relevant beam propagation effect is beam expansion, the physics of the heterodyne efficiency depends on range, wavelength *λ*, and aperture diameter *D* through the same dimensionless Fresnel number *N*_{F}
=*kD*^{2}
/*4R* that defines the beam’s far-field conditions, *N*_{F}
<*1*. Here, *k*=*2π*/*λ* is the beam wave number.

In any turbulence situation, small-scale (refractive) and large-scale (diffractive) intensity fluctuations [3, 9] on the target plane will modify the gain of coherence of the initially incoherent light propagating back to the receiver plane from the target plane. Diffractive scattering spreads the wave as it propagates to the target plane, causing a loss of spatial coherence on the receiver plane. Furthermore, a finite-diameter beam propagating through refractive large-scale inhomogeneities will experience beam wander, painting out a larger long-term spot size and also decreasing the effective receiver area. However, the same large turbulent cells act as refractive lenses, and they can effectively focus the light that illuminates the target plane-essentially characterizing the short-term spot size there-and increase the coherence of the wave coming into the receiver. All these factors that delimit the available coherent solid angle are described in the general expression (2) through modified system antenna efficiency. In this case, Fresnel number *N*_{F}
is not enough to describe the heterodyne lidar returns.

The dimensionless parameter *N*_{T}
=${\mathit{\text{kr}}}_{\mathit{0}}^{\mathit{2}}$
/*4R* has been used to describe both the magnitude of the irradiance fluctuations and the beam expansion associated with arbitrary turbulence conditions [10, 11]. Here, for a given wave number *k* and level of refractive turbulence ${C}_{n}^{\mathit{2}}$
, the parameter *r*_{0}
=*(0.42 k*^{2}${RC}_{n}^{\mathit{2}}$*)*
^{-3/5} is the transverse-field coherence diameter on the receiver plane of a point source located at the target at range *R*. It is appealing to appreciate the similarity between this parameter *N*_{T}
, which describes the refractive-turbulence effects on the propagated beam, and the previous Fresnel number *N*_{F}
, which describes the free-space propagation: Now coherence diameter *r*_{0}
assumes the role of aperture diameter *D*.

Although in previous research [8] parameters that resemble *N*_{T}
were used to describe the performance of CLR, the results referred only to a lidar that employed well-separated receiver and transmitting optics, i.e., a bistatic system, under restrictive turbulence conditions. It was shown from simulation of beam propagation [12] that just the combination of both numbers *N*_{F}
and *N*_{T}
permits the contribution of refractive turbulence to incoherent backscatter heterodyne lidar returns to be made in a comprehensive way. Any two monostatic lidars with the same *N*_{F}
and *N*_{T}
should exhibit the same turbulent effects in their performance. Distinctive interdependencies of the parameters that define the coherent system configuration and the atmospheric condition appear easily from a study of the similarity between parameters *N*_{F}
and *N*_{T}
[12]. Those results will help us to understand how the choice of telescope parameter may affect the performance of a practical heterodyne lidar in the presence of atmospheric turbulence.

## 3. Analysis of relevant telescope-related parameters

It is of engineering interest to be able to predict the dependency, in the antenna and beam geometry, of refractive-turbulence contributions to heterodyne lidar returns. In this paper the relevant parameters are wavelength, beam waist radius or aperture diameter, and wave-front radius of curvature *F*.

Even if atmospheric turbulence is not considered, when the aperture size is modified the coherent solid angle Ω*COH* pattern is altered in several ways. Indeed, a reduction factor *r*=*D*_{2}
/*D*_{1}
of the aperture diameter would degrade Ω*COH* by 1/*r*^{2}
(a factor of 2 would translate into a Ω*COH* loss of 6 dB). With respect only to this mostly obvious consideration, it would erroneously seem necessary to increase the size of the system aperture to improve the lidar performance. Otherwise, reducing the aperture size by a factor of *r* will yield the far-field condition *N*_{F}
<*1*, so the maximum antenna efficiency *η*_{S}
will be reached for ranges *r*^{2}
times smaller (for example, reducing *D* by a factor of 2 will produce the optimum 0.4 described in Section 2 for ranges that are four times closer to the transmitter). That would improve Ω*COH* in the short ranges.

Refractive turbulence makes our analysis less simple. It has been shown [12] that coherent solid angle Ω*COH* is less sensitive to turbulence when lidar aperture decreases. Parameterization of the effects of turbulence on heterodyne systems shows that reducing the aperture diameter by a factor of *r* is equivalent to attenuating the refractive-turbulence effects to situations in which the turbulence intensity level ${C}_{n}^{\mathit{2}}$
is a factor of *r*^{11/3}
smaller (*r*=*2* corresponds to a reduction in turbulence level of more than 1 order of magnitude). It is worth noting that this payoff is useful in the far ranges, where the principal mechanism that describes the effect of atmospheric refractive turbulence is the additional expansion of the beam, which always reduces the overall signal coherence over the receiver plane. Equivalent weak turbulence levels may eventually translate into less-intense beam spreading

Figure 1 compares the expected heterodyne solid angle Ω*COH* of an operating lidar system with that which results from considering different aperture diameters several times smaller. Transmitted and virtual LO beams were assumed to be matched, collimated, perfectly aligned, Gausssian, and truncated at the telescope aperture. For any aperture diameter shown in Fig. 1, 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 [5]. Because we are interested primarily in turbulent effects, and to simplify the figures, we have normalized the plots such that the coherent solid angle for the initial aperture *D*=16 cm at the shortest range is 0 dB. In most practical situations, a coherent solid angle decay (i.e., a wideband SNR decay) of -15 dB defines the lidar’s maximum range. All the effects described previously that were associated with an aperture reduction are included in the simulations, for which two different realistic turbulent levels ${C}_{n}^{\mathit{2}}$
have been contemplated. In both cases, the performance of the coherent lidar seems not to have been degraded significantly by the reduction of the aperture diameter: In fact, at shorter ranges Ω*COH* is improved as a consequence of improved system-antenna efficiency (see above), and, at larger ranges, the effect of the beam size is mostly compensated for by the decrease of the refractive-turbulence effects (see the previous paragraph). Certainly, in many circumstances, smaller initial beams could perform at least as well as their larger counterparts.

The loss of mixing efficiency from the diffraction at the illuminated target is eliminated when the transmitter is much larger than the receiver [2, 8]. A compact illuminated image is produced at the target that approximates a point source and produces a coherent wave over the dimensions of the LO at the receiver. However, the condition for eliminating the diffraction component of the coherent losses is not a feasible geometry because the transmitter must be much larger than the receiver. At larger wavelengths, practical heterodyne systems overcome this limitation by using focused beams: The loss of antenna efficiency from the diffraction component of the illuminated target is partially eliminated if one considers a transmitting lens with a focal length similar to the range of interest, *F*_{T}
=*R*. A more-compact target is illuminated, resulting in an almost backscattering point source. When a focused system is considered, Fresnel number *N*_{F}
that characterizes the free-space antenna efficiency must be modified as *N*_{F}*(1*-*R*/*F*_{T}*)* to encompass focal length *F*_{T}
or the radius of curvature of the transmitted beam. Now the far-field condition-small Fresnel number-occurs at ranges *R* close to *F*_{T}
, and the angular size of the illuminated area at this range is the collimated beam divergence once again. Using focusing techniques to improve the performance at ranges of interest was an extensive practice in earlier CO_{2} lidar systems working at wavelengths in the far infrared (9–11 µm). At these wavelengths, turbulence effects are less remarkable.

The enhancement associated with the transmitter focusing can be less efficient when shorter wavelengths are considered. Parameterization of the refractive-turbulence effects in heterodyne systems also shows [12] that reducing the wavelength by a factor of *r* is equivalent to increasing the refractive-turbulence effects such that turbulence intensity level ${C}_{n}^{\mathit{2}}$
is a factor of *r*^{3}
larger (for *r*=*5*, i.e., when a *2-µm* system is compared with a *10-µm* lidar, that means an increase in the turbulence level of more than two 2 orders of magnitude).. Consequently, the same turbulence that reduces the coherence of the received beam may destroy the initial phase curvature introduced by the transmitter, decreasing its ability to focus the transmitting beam’s energy in a small area at the target plane.

Figure 2 compares the expected coherent solid angle of the collimated system with those that correspond to focusing the same lidar at different ranges (transmitter focal lengths of 1, 2, and 3 km). Normalization, lidar system parameters, and levels of refractive turbulence are similar to those for Fig. 1. For *F*_{T}* 2 km*, under most conditions the beam wave is so distorted by the turbulence that the initial phase that describes the focusing is barely sufficient, and just some weak enhancement can be appreciated. Shorter focal lengths (1 km or less) are only slightly more effective. In any case, for longer ranges the initially focused lidar exhibits a performance similar to that observed in the simpler collimated system. The problem is characterized mainly by the turbulence conditions, independently of the initial beam conditions.

The goal of this study could have been condensed to answer the ultimate question of whether a smaller-aperture telescope focused at infinity can outperform a larger-aperture telescope that is focused in the near field. As can be seen from Figs. 2 and 3, it happens that, at shorter wavelengths, in many situations small apertures can even exceed the performance of larger telescopes with focal lengths that resemble the range of interest. These considerations are of the most practical importance in ground lidar systems now under development that work at wavelengths that range from 1 to 2 µm [13].

## 4. Conclusions

Simulating tools turn out to be extremely useful for insight into problems as complex as the propagation of coherent lidar signals through atmospheric turbulence. We have described here some results of the simulations and the relations that appear from dimensionless *N*_{F}
and *N*_{T}
parameter dependencies.

Comprehending turbulence effects is fundamental for resolving the optimal lidar telescope parameters. In many circumstances, the performance of the coherent lidar is not degraded significantly by the reduction of the aperture diameter. Even better, by simulating moderate-to-strong turbulence conditions, our analysis seems to indicate that smaller apertures could perform at least as well as larger ones in collimated laser radar systems working at shorter wavelengths. Also under these conditions, the improvements linked to the focusing of the beam are less apparent than when longer wavelengths and weaker turbulence conditions are considered. Only at very near ranges short focal distances may result in decreasing the diffraction component of the coherent losses to some degree.

The simulation technique could as readily be extended to take account of aberrations in the optics, nonuniform turbulence, and the effects of beam misalignment in the performance of a heterodyne lidar. With respect to misalignments, some important considerations that complement those considered in this analysis still need to be addressed: It will be interesting to think about which setup would be more sensitive to misalignment in a turbulent atmosphere, a smaller diameter focused at infinity or a larger diameter focused in the near field. Also, although the configuration of practical heterodyne lidars is usually monostatic, misalignments together with refractive turbulence may make unnecessary the distinction from bistatic arrangements: Their performance may never be well described by either of the ideal configurations, and maybe they ought to be considered simply limit situations. We have avoided considering these important points in detail here. The author intends to address them in a later study.

The author is greatly indebted to B. J. Rye of the NOAA Environmental Technology Laboratory for his insight and many constructive comments that have helped to clarify the contents of the manuscript. This research was partially supported by Spanish Department of Science and Technology MCYT grant REN- 2000-1754-C02-02.

## References and links

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**12. **A. Belmonte, B. J. Rye, W. A. Brewer, and R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., June28–July 2, 1999.

**13. **See papers on device technology presented at the Twelfth Biennial Coherent Laser Radar Technology and Applications Conference, Bar Harbor, Me., June 15–20, 2003.