## Abstract

The simulation of beam propagation is used to examine the uncertainty inherent to the process of optical power measurement with a practical heterodyne receiver because of the presence of refractive turbulence. Phase-compensated heterodyne receivers offer the potential for overcoming the limitations imposed by the atmosphere by the partial correction of turbulence-induced wave-front phase aberrations. However, wave-front amplitude fluctuations can limit the compensation process and diminish the achievable heterodyne performance.

© 2008 Optical Society of America

## 1. Introduction

Heterodyne detection schemes are being used in the optical-frequency band to extend the reliability of atmospheric communication techniques. Even under clear-weather conditions, turbulence-induced optical phase perturbations destroy the spatial coherence of a laser beam as it propagates through the atmosphere and degrade the overall performance of optical heterodyne systems by restricting the received heterodyne power levels. Also, turbulent phase variations along the path are manifested in high beam divergence and power variations (scintillation) that, acting as a non-additive noise source, reduce the level of any power measurement and demeans the accuracy of heterodyne signal estimates. The possibility of using adaptive compensation of atmospheric wave-front phase distortions to improve substantially the performance and reliability of atmospheric systems has being an important part of the analysis of light through the atmosphere for many years. Phase-compensated receivers offer the potential for overcoming atmospheric limitations by adaptive tracking of the beam wave-front and consequent correction of atmospherically-induced aberrations. The modal compensation method is a correction of several modes of an expansion of the total phase distortion in a set of basics functions.

The problem of heterodyne performance in the presence of atmospheric turbulence is of considerable complexity because refractive turbulence effects define the degree of coherence of the received radiation to be matched with the local oscillator. The heterodyne power has a maximum value when the spatial field of the local oscillator on the detector is proportional to the spatial field of the received radiation. Any mismatch of the amplitude and phase of the two fields will result in a loss in signal power. In earlier works [1], it was shown that, in near-field applications—where just phase fluctuations are relevant and amplitude effects can be neglected—there was a limit to the achievable coherent power no matter how large the detector collection aperture is. The minimum aperture diameter which will almost attain this limit was revealed to be the coherent wave-front diameter describing the coherent area of the beam phase fluctuation on the target plane. The analysis on heterodyne detection systems was later extended to more general conditions under the Fresnel approximation considering backscattered light from remote atmospheric targets [2,3]. The generalization of these results to describe a system with phase compensation of the atmospherically produced phase distortion proposed simple analytical expressions to describe modal atmospheric compensation [4, 5]. This formulation, also valid in the near field, considers a modified form of the wave front with the addition of the residual phase aberration after phase compensation is considered. Thus the performance of a real optical system employing a modal phase conjugate technique or utilizing a general deformable mirror for phase compensation could be predicted through appropriate formulation.

All the relevant considerations extracted from those earlier analytical and numerical results have been extremely useful tools for developing insight into the problem of modal compensation of optical heterodyne signals through atmospheric turbulence. However, in the derivation of these semi-analytical results for modal heterodyne receivers [4,5], several assumptions and idealizations were made that need to be revisited carefully in the light of more actual approaches to the problem analysis. First, both the transmitter and local oscillators (*LO*) were assumed to have uniform intensity distributions in the source plane and the propagating medium was supposed to be described by Kolmogorov turbulence in the inertial sub-range. More importantly, these analytical techniques consider the approximation that use structure functions -second order moments of the fields propagated through random media- to describe atmospheric wave-front distortions disregarding the consequences of the variance and correlation of intensity fluctuations - higher moments of the fields - at the receiver. There is reason to believe that these approximations made in the analytical work, and especially the use of second order moments of the fields, may cause significant problems when trying to elucidate the performance of modal compensated heterodyne systems. Our results regarding modal compensation of atmospheric turbulence phase distortion is intended to complement those earlier analyses by considering a more complete, full wave description of the propagation problem and the wave front to be compensated in the receiver plane.

To investigate the propagation and heterodyne detection problems in the atmosphere under general conditions, it is necessary to include spatially random fields, refractive turbulence, transmitter configuration, and detector geometry. In general, theoretical calculations involving realistic modal compensation systems are still difficult because they would have to accomplish through consideration of the higher moments of the field. Consequently, no simple analytical solutions are known besides those previously mentioned for simplified transmitter and receiver geometries and unrealistic atmospheric characterizations. We have used here the simulation of beam propagation to overcome the lack and restrictions of analytical results in the study of heterodyne atmospheric system performance. As we have shown in previous studies [6,7], the simulation technique permits computation of the fields in the receiver that determine the received power of non-compensated, static heterodyne free-space laser system in a turbulent atmosphere. Simplifying assumptions used in analytical studies of atmospheric beam propagation were tested and treated as benchmarks for determining the accuracy of the simulations [7]. A simpler, no full-wave simulation technique, that characterize the propagation problem with just two independent phase and amplitude screens on the receiver plane, has been recently tried to produce fading probabilities in coherent optical communications [8].

Along with the optical heterodyne power, another important measure of the system performance is the relative variance of the optical power that results from turbulent fluctuations. Fluctuations in the instantaneous power level 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. The analytically intractable problem of looking at the uncertainty inherent to the process of heterodyne optical power (i.e., equivalent optical power generating the heterodyne received signal) measurement in the presence of atmospheric refractive turbulence can be considered by simulations of beam propagation in a realistic way. 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 detectors in the atmosphere. Of the most importance in our analysis, the simulation also permits characterization of the effect on heterodyne performance of the return variance that result from turbulent fluctuations when modal compensation of phase distortion is considered.

Also, being concerned with the basic problem of optical heterodyne detection, in this study we will not consider the case of back-scattered light coming from remote atmospheric targets [9]. Although in remote sensing lidar systems power fluctuations could result from a number of physical mechanisms other than refractive turbulence -mainly speckle-, these mechanisms are not the focus of this analysis and are not discussed here. We certainly understand that fluctuations induced by turbulence are not as intense as those due to speckle in optical remote sensing systems but, although their normalized variance is smaller, they still need to be considered to properly describe the performance of any practical coherent lidar. This specific situation requires some additional considerations [10,11] to be thought over carefully elsewhere.

## 2. Phase-compensated heterodyne detection

This study is for conducting simulated experiments on compensation of atmospheric turbulence phase distortion on heterodyne detection systems and comprehends the implications of considering realistic amplitude fluctuations. It was shown [7] that numerical experiments allow the maximum number of parameters to be considered to properly model phase compensation systems and to investigate any significant atmospheric and illumination characteristics. We consider the problem of numerical simulation of a wave beam as simulations are able to give us a complete numerical estimation of phase and amplitude of the wave distorted by the propagation medium. We extract this information for further use in compensation parameters.

Along with propagation simulations, we need also to model the phase compensation system realizing the possibilities set in the correction process. Here, we have implemented a numerical model for a modal compensation system, a hypothetical device whose response functions are components of some expansion basis. Different sets of functions can be used for the expansion although most often they are Zernike polynomials [12, 13], a set of orthonomal basis modes defined on a unit circle and that are related to the classical Seidel aberrations [12]. The modes are a product of angular functions and radial polynomials when polar coordinates * ν*=

*(ρ,θ)*are considered. We will assume that the modal compensation system has infinite spatial resolution in the correction of phase distortions.

To develop the model of the modal compensation system, the wave-front aberration function Φ*( ν)* of a system with a pupil of diameter

*D*is obtained directly from our simulations and can be expanded in terms of an modal basis

*Z*as

_{j}(**ν**)where *a _{j}* are the expansion coefficients. For a wavefront degraded by atmospheric turbulence, the aberration coefficients

*a*vary randomly with time with a zero ensemble-averaged value and a well defined covariance function [14]. The phase compensation on the receiver plane can be represented by the removal of these spatial modes by spatial phase conjugation. They correspond to degrees of freedom of the adaptive optics system [13]. When the first J modes are removed the correcting phase may be written as

_{j}Accordingly, when using a modal corrector, the corrected phase or residual wavefront is calculated as a difference between the initial wave-front and the truncated series of expansion functions and is given by

By simulating the removing of an increasing number *J* of modes by spatial phase conjugation, we weigh up the improvement in the coherent system performance. In general, in a real system, we need to use a number *J* of modes large enough to make the residual term (3) negligible. In our calculations, the coefficients *a _{j}* of the series in (2) are chosen to give the best fit to Φ

*(*in the least-square sense over the aperture. (Certainly, the presence of branch points in the phase function [15] may give rise to errors in a least-squares continuous phase reconstructor since the underlying assumption that a wavefront can be represented as a single-valued function is not valid [16,17]. However, the performance of the least-squares continuous wavefront reconstructor falls off over relatively intense turbulence conditions and we have not considered those situations in our study.) For an aperture weighted by a function

**ν**)*W(*, a best fit in the least-square sense implies that

**ν**)*a*may be sought from the minimization condition of the residual phase Φ

_{j}_{J}on the circle of diameter

*D*corresponding to the receiver aperture:

where, assuming that the receiver aperture is circular, we choose to use the pupil function *W* as the weighting function equal to 1 for |* ν*|=

*ρ*≤

*D*/

*2*and 0 for |

*|=*

**ν***ρ*>

*D*/

*2*. Generally, the value of

*a*is obtained by solving a system of linear equations generated from (4). The result is equivalent to using a singular value decomposition of the initial wave-front Φ on the orthogonal modal basis

_{j}*Z*. Because of orthogonality of the correcting modes

_{j}*Z*with respect to the circular aperture function

_{j}(**ν**)*W(*, i.e.,

**ν**)where *δ _{ij}* is the delta function, the values of

*a*are determined as the expansion coefficients [18]

_{j}We will evaluate the performance of the optical heterodyne detection with Zernike expansion compensation through the 80th Zernike polynomial. Without loss of generality, Zernike polynomials are chosen because of their simple analytical form and because of the correspondence of the low-order Zernike polynomials to the customary aberrations modes. A 80 Zernike polynomial expansion contains all terms through tilt (*J*=3), astigmatism (*J*=6), coma (*J*=10), and most other high-order aberrations.

In the single-mode heterodyne detection regime, the information-carrying part of the signal occurs when the received radiation field *U _{S}(ν)* and the reference (local oscillator) heterodyne field

*U*are combined on the input plane

_{LO}(**ν**)**ν**=

*(ρ,θ)*of the optical system located at the propagation axis point

*z*=

*0*. Therefore, the average signal heterodyne power is given as

The operator 〈〉 denotes an ensemble average and * complex conjugate. Implicit in Eq. (7) is an average over a time which is large compared to the reciprocal bandwidth of the signal field *U _{S}*. In our approach, an outline of wave-front matching for optical heterodyning in coherent optical systems shows that Eq. (7) is equivalent to determine the performance of the heterodyne system based on the observation of the collecting lens mutual coherence functions on the pupil plane. The equation for the optical heterodyne power

*P*expresses the performance of the heterodyne system in terms of the degree of coherence of the collected radiation and its proper match with the field of the local oscillator:

where *M _{S}(ν_{1},ν_{2})* and

*M*are the collected and local-oscillator mutual coherence functions on the input (pupil) plane given by

_{LO}(**ν**_{1}**,****ν**)_{2}$${M}_{\mathit{LO}}({\mathit{v}}_{1},{\mathit{v}}_{2})={U}_{\mathit{LO}}\left({\mathit{v}}_{1}\right){\mathit{U}}_{\mathit{LO}}^{*}\left({\mathit{v}}_{2}\right).$$

The operator 〈〉 does not apply to the deterministic LO field, which is statistically independent of the collected field and stationary. Refractive turbulence effects are considered in the mutual coherence function of the collected field, *M _{S}*. Although Eqs. (7) and (8) are equivalent, the former is easier to evaluate with the simulation technique as it makes use of simple double integrals.

Indeed, in our approach the fields *U _{S}* on the pupil plane are derived from the numerical simulation of atmospheric wave beam propagation [7] by adding the phase corrections Φ

_{c}modeling the modal compensation system (see Eq. (2)):

Now, this phase-corrected fields consider just a reduced phase Φ_{J}
*( ν)*=Φ

*(*-Φ

**ν**)_{c}

*(*expressing the residual turbulence aberration term in Eq. (3). For a given propagation path, an increasing number of compensated modes

**ν**)*J*would translate into a smaller residual phase and a more effective heterodyne reception. Unaffected by phase-corrected receivers using modal compensation as those contemplated in this analysis, aperture amplitude

*A*considers beam wander, beam spreading and, more importantly, optical scintillation effects produced by atmospheric layers close to the transmitter. Except for the case of weak turbulence [2], scintillation is described by moments of the fields higher than those accounted for in existing analytical theories. This is practical situation we want to analyze with our simulations. Amplitude fluctuations can limit the phase compensation process, diminishing the achievable heterodyne performance, and needs to be considered in any realistic description of phase compensation on optical heterodyne power measurements. Also, although it would be also possible to consider phase corrections on the transmitting systems to pre-compensate partially for scintillation, the analysis is considerably more complex and the benefits are, in principle, not so intuitive. The analysis of these spatial diversity transmitters is beyond the scope of this study.

_{S}(**ν**)## 3. Performance evaluation of phase-compensated heterodyne receivers

One of the problems confronted by any phase compensation system is the effect of the scintillation on the measurement and reconstruction of wave-fronts distorted by turbulence. This problem has been extensively considered in the context of astronomical adaptive optic systems [see, per example, Ref. 19 and 20]. Because of the effect of amplitude fluctuations on the wavefront phase measurement, scintillation needs to be considered along with phase fluctuations in the performance of the system. Recently, new optical methods for compensation of both amplitude and phase fluctuations have been described [21, 22]. Although non-uniform amplitude over the receiver aperture certainly limits the phase measurement and introduces erroneous corrections into the wave front, our analysis is not concerned with fitting errors and, therefore, we assume ideal wavefront sensing and reconstruction; systematic errors due to wavefront sensing in the presence of scintillation are not considered. With important connotations, rather than fitting errors induced by amplitude fluctuations what we intend to study is the role that scintillation plays in the loss and degradation of heterodyne efficiency in phase compensated systems. For ease of presentation, we also assume the absence of additive phase reconstruction error such as noise.

The simulations used to obtain the results illustrating our analysis below are based on the method of modeling the atmosphere by a set of two-dimensional, Gaussian random phase screens with an appropriate phase power spectral density and make use of the Fresnel approximation to the wave equation. This technique provides the tools for analyzing heterodyne systems with general refractive turbulence conditions, beam truncation at the telescope aperture, initial beam wave-front aberrations, and arbitrary transmitter and receiver configurations. Here, this simulation approach has been extended to the more-complex problem of receiving systems considering adaptive tracking of the beam phase-front distorted by turbulence. All simulations contemplated here will assume uniform turbulence with range and use the Hill turbulence spectrum [23] with typical inner scale l_{0} of 1 cm and outer scale L_{0} of the order of 5 m. The choice of both inner and outer scale with relation to the grid size have been previously discussed in great detail [6,7] and represents a serious attempt to define the simulation parameters in a realistic way. In particular, the chosen value of outer scale L_{0} produces pragmatic tip/tilt components on the atmospheric phase distortions. The simulation technique uses a numerical grid of 1,024 by 1,024 points with 5-mm resolution and simulates a continuous random medium with a minimum of 20 two-dimensional phase screens. Choice of a grid sampling interval and a grid extension appropriate to physically reasonable atmospheric scales was intended to ensure the applicability, accuracy, and realism of our simulations. In any of the scenarios considered in this study, we run over 3,000 samples to reduce to less than 3% of the corresponding mean values the statistical uncertainties of our estimations describing the heterodyne optical signal.

In this analysis, beam waves at the transmitter are propagated to different distances through turbulence aberrations with uniform refractive index structure function profile C^{2}
_{n}. The propagation modeling parameters chosen are those corresponding to typical daytime values of strong C^{2}
_{n}=10^{-14} m^{-2/3} and C^{2}
_{n}=10^{-13} m^{-2/3} turbulence. However, the results for different C^{2}
_{n} parameters only vary according to the Fried’s atmospheric coherence length r_{0} and the Rytov variance σ_{1} parameters. The Fried coherence length r_{0} describes the coherence diameter of the distorted wavefront phase. The Rytov variance σ_{1} is used as a scintillation index predicting the intensity of amplitude fluctuations. The results presented in this study consider two different propagation conditions: first, a strong turbulence situation, denoted in the figures as C^{2}
_{n}=10^{-14} m^{-2/3}, where r_{0}=6 cm and σ_{1}=1; then, a situation of stronger turbulence, denoted as C^{2}
_{n}=10^{-13} m^{-2/3}, where r_{0}=3 cm and σ_{1}=2. In both cases, the Rytov variance is large enough to be relevant but still far from its saturation regime (Rytov variances usually larger than 4). When the scintillation index reaches its level of saturation the wavefront distortion is so intense that would be unrealistic to consider any phase compensation technique. In any case, most practical atmospheric optical systems use reasonable receiver apertures (with diameters D larger than 10 cm) where turbulence is within the aperture near field and amplitude fluctuations are not saturated [24].

Figures 1 to 4 show the effect of using modal-compensated heterodyne receivers in heterodyne systems. In this paper, the calculated measure of performance of a heterodyne laser system is the heterodyne mixing efficiency as a function of level of refractive turbulence *C ^{2}_{n}*, receiver optic aperture diameter

*D*, and the number of spatial modes

*J*removed by the compensation system. The heterodyne detection efficiency, a useful measure of coherent detection operation which measures the loss in coherent power when the received field and the LO field are not perfectly matched, is defined as the ensemble averaged coherent power (7) normalized to the average local oscillator and received powers. The heterodyne efficiency describes the portion of the collected optical power effectively converted to heterodyne optical power and has a maximum value of unity when

*U*is proportional to

_{S}*U*. Since we are concerned primarily in compensation of turbulence effects, we present (Figs. 1 and 2) the efficiency gain by which the heterodyne efficiency is modified in the presence of an modal correction system; it is defined as the efficiency normalized by the case of absence of compensation system (

_{LO}*J*=

*0*). Furthermore, and of the most importance, we consider the effect on heterodyne power uncertainty (normalized standard deviation or relative error) of the return variance that result from turbulent fluctuations when modal compensation of phase distortion is applied (Figs. 3 and 4). The accuracy of the estimate of average received power is actually the critical parameter in many heterodyne systems. Any relative error in the power measurement resulting from atmospheric turbulence will translate as a relative error in the heterodyne estimations. We will usually express the measures of performance in decibels (dB) as 10log

_{10}of the estimated magnitudes.

All the same, the improvement caused by the compensation technique is very remarkable. In most situations, the effects of phase correction are important even when just a few modes are eliminated from the initial wave-front. Figure 1 shows the gain in heterodyne efficiency as a function of the number of spatial modes eliminated from the initial wave-front. It is important to note that, when the received field remains coherent over the aperture area (i.e., the coherence diameter of the field in the aperture plane [1] is larger than the receiver aperture diameter *D*), the area of the receiver optics available for collecting coherent power will be identical to the effective receiver area in the absence of turbulence. In the event that atmospheric turbulence is present, the effect of turbulence is to reduce the coherence area of the signal. This in turn reduces the heterodyne detection efficiency and, consequently, the effective mean coherent power. When in our simulated experiments the phase correction system is turned on and a set of spatial modes are compensated, for the weaker turbulence (left plot) the heterodyne efficiency grows quickly for the smallest considered aperture. Obviously, for relatively small receiving apertures the main phase aberration introduced by refractive turbulence seems to be those contain in the lower order spatial modes: If aperture diameters D are comparable to the coherent diameters characterizing the atmospheric propagation, just the lowest tip/tilt (angle-of-arrival) aberrations would need to be contemplated. For larger apertures or stronger turbulence conditions (right plot), the compensation seems to be gradually less efficient and a larger number of corrected modes seem to be necessary to reach similar levels of heterodyne efficiency gains. Certainly, larger apertures improve their performance more efficiently than smaller ones. In fact, the gain in small apertures is inclined to saturate with disregard of the phase compensation degree.

One of the most important parameters in the design of optical heterodyne systems is the diameter of the receiving aperture. In Fig. 2, the heterodyne efficiency gain is presented as a function of the receiver diameter *D*. We consider expansions of the compensation phase Φ_{c} through 2rd-order (astigmatism, *J*=6), 5th-order (*J*=20), and a high-order case (*J*=80). As expected, an increase in the number of correcting modes translates into a larger heterodyne gain with respect to the aperture diameter considered. Well-known features are clearly identifiable in the plots. Firsts, there is a limit to the achievable heterodyne efficiency gain no matter how large the detector collection aperture is: when no compensation is considered this minimum aperture attaining the maximum heterodyne power is close to the coherence length r_{0}. Utilizing phase compensation just makes this maximum gain displace towards larger apertures. It is not a surprise, as the coherent length decrease with the intensity of the atmospheric turbulence, that weaker turbulence levels (left plot) tends to produce maximum gains for apertures larger than those when stronger turbulence is considered (right plot). Certainly, as it was pointed from the results in Fig. 1, larger apertures are more sensitive to phase compensation and prone to superior improvements. Up to 12-dB gains can be expected for aperture diameters D larger than 20 cm when high-order compensations (*J*=80) are applied to the receiving wavefront.

Amplitude limitations are also apparent when the no-scintillation situation is considered in Fig. 2 (dashed lines). We eliminate any scintillation effect on our estimations by imposing in Eq. (10) an aperture amplitude *A _{S}* constant all over the receiver plane. Implementation of this correction in our simulation environment is straightforward as we have access to the full field propagated through the atmosphere: we simply eliminate amplitude scintillation while phase distortion remains unaffected. This is equivalent to having an ideal amplitude-compensation system in our heterodyne receiver and allow us to estimate which effects are associated purely with scintillation rather than those due to wavefront distortion. Scintillation effects are apparent in Fig. 2: the existence of amplitude fluctuations decrease the mixing heterodyne efficiency independent of the phase correction applied to the receiver. When amplitude fluctuations are ideally eliminated, the heterodyne gain improve. In general, the improvement is larger when the aperture area and the number of correcting modes increase. Up to 4-dB differences can be appreciated between the real and the ideal, scintillation-free cases. Surprisingly, however, scintillation effects seem to be of little relevance when the aperture diameters are smaller (D<20 cm). In these cases, Fig. 2 shows differences smaller than 1 dB between real and scintillation-free heterodyne cases in any of the phase correction situations considered and both levels of turbulence studied. This conclusion could be mistaken in heterodyne lidar systems where laser speckle may become more relevant that atmospheric turbulence scintillation.

Figure 3 details the dependence on the heterodyne power uncertainty (relative error or normalized standard deviation of heterodyne power fluctuations) with the degree of phase compensation (other simulation parameters are similar to those described previously for Fig. 1). The statistical uncertainties have been estimated from the error in the mean value from an ensemble of independent realizations. As expected, small apertures respond quicker to the phase-front compensation, even if larger apertures need no less than 10 correcting modes to show a clear improvement in their performance. For any aperture diameter considered in the figure, we observe a heterodyne power normalized error that is generally well below 2.5 (i.e., a standard deviation of nearly 2 dB around the mean values) when no compensation system (*J*=0) is applied. However, when the Zernike expansion compensation through the 100th Zernike polynomial is applied we observe normalized variances near 0.1 (a standard deviation as small as -5 dB around the mean values) under all turbulence conditions considered in this study. Under most circumstances, using modal compensation techniques potentially results in a decreasing of more than 6 dB of the uncertainty associated with the measurement of heterodyne power. Certainly, this improvement of the measurement conditions is very significant, defining the performance of any heterodyne optical system working in the atmosphere.

We consider now aperture effects on the power measurement uncertainty. Figure 4 shows the normalized standard deviation of heterodyne power measurement fluctuations as a function of aperture diameter. Because atmospheric refractive turbulence produces signal fluctuations affecting heterodyne detection systems in different ways, they must be considered to evaluate system performance: It is evident that both amplitude fluctuations and phase-front distortions are relevant in the analysis of the relative error of heterodyne power. In the figures there is a clear transition between two different regimes. For aperture diameters D less than the optimal value, where we reach a minimum in the power uncertainty, the relative error is determined largely by the amount of amplitude fluctuations. In this regime the separation of the curves for different compensation values is relatively small because phase-front distortions have a minor role to play: What we are observing is the well known fact that, when a large aperture is used to collect scintillation light, the fluctuation measured is not as large as would be observed if a small aperture were used. At larger aperture values, however, the uncertainty is determined by phase-front distortions and, consequently, we observe an increased in the uncertainty level. Now, the importance of using a high-order phase corrections is more evident for larger apertures where, as it was previously commented, improvements of several dBs in the uncertainty levels can be expected. For both turbulence levels shown in the figure, a minimum -5-dB relative error is obtained for 15-cm apertures, or around, when a high order correction is used. For larger apertures, the noise introduced by the phase-front distortion can not be completely cancelled and we see a continuous uncertainty increase.

We have added to the figure the result of canceling irradiance fluctuations in our estimations (dashed line) when no phase-front *J*=0 cancellation is used. As expected from our previous comments, the first of the regime observed in the figures tends to disappear as scintillation is not an issue anymore. Now, just phase distortion is relevant in the analysis and the uncertainty increases steadily with the aperture diameter. To simplify the plots, we do not present the no-scintillation results when phase compensation is used. Once again, the aperture averaging of scintillation will reduce the effect of scintillation from the figures, letting behind a unique regime dominated by the residual phase-front distortions.

## 4. Conclusions

The effects of atmospheric distortion of an optical wave front on the performance and reliability of an optical heterodyne detection system affected by phase-front compensation have been examined numerically. Simulation tools turn out to provide a way to study the relatively intractable problems arising when heterodyning is considered in the presence of refractive turbulence and wave-front amplitude fluctuations can limit the compensation process. In this study, extensions of the technique have allowed us to describe accurately the possible improvements on heterodyne performance when the optical system in the receiver end uses wave-front correctors to compensate for turbulence-induced phase aberrations

The ultimate performance that a compensated heterodyne system can attain is of distinct relevance. The extension of previous analytical results to situations where it is necessary to include the consequences of the intensity fluctuations at the receiver needed to be pondered. More importantly, the technique has allowed us to examine the uncertainty inherent to the process of heterodyne power measurement because of the presence of turbulence, and differentiate the effects of amplitude fluctuations and phase-front distortion in the relative error.

We could make a case about the relevance of using phase-front compensation in a heterodyne receiver. A close examination of the simulation results has revealed that the gain in heterodyne mixing efficiency is important in most of the turbulence conditions and aperture diameters considered in the study. Modest compensation levels translate into gains of several decibels, up to 12 dB for the larger apertures considered in the study (30 cm) and the stronger turbulence conditions. The effects of scintillation are not as relevant as it could have been expected. The analysis has shown that, in practical small apertures, the degradation of the heterodyne mixing due to amplitude scintillation is smaller than 1 dB. Just when large apertures with high compensation orders are considered we start to observe a more relevant contribution. In any case, amplitude fluctuations effects on heterodyne efficiency are always below the 4-dB mark.

Also, we pondered the uncertainty caused by the presence of turbulence that is inherent to the process of optical power measurement with a practical heterodyne receiver. It is now possible to consider the effect on heterodyne power uncertainty of the return variance ensuing from turbulent fluctuations when modal compensation of phase distortion is applied. From our results it is reasonable to maintain that heterodyne receivers, in applications where the accuracy of power measurements is actually the critical parameter, can benefit from phase-front compensation. An increase in the measurement uncertainty will certainly decrease the capacity of any optical communication link or the accuracy of the estimates in optical remote sensing. Simulations indicate that, for optimal aperture diameters, uncertainty when phase correction is applied can be up to 6 dB smaller than for the non-corrected case. The optimal aperture diameters, those minimizing the relative error, separate two different regimes in our simulations. The regime dominated by amplitude scintillation is defined for relatively small apertures and it is virtually unaffected by phase-front corrections. When larger apertures are considered, phase distortion is the relevant effect of turbulence, amplitude fluctuations are of little influence, and we may need high-order phase corrections to decrease the uncertainty to acceptable levels.

## Acknowledgments

The author would like to thank D. R. Gerwe and W. Buell for helpful comments on this work. The research was partially supported by the Spanish Department of Science and Technology MCYT Grant No. TEC 2006-12722, and Spanish Defense Department CIDA Technical Assistance No. 108.077.

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