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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 9 — Apr. 28, 2008
  • pp: 6756–6767
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Influence of atmospheric phase compensation on optical heterodyne power measurements

Aniceto Belmonte  »View Author Affiliations


Optics Express, Vol. 16, Issue 9, pp. 6756-6767 (2008)
http://dx.doi.org/10.1364/OE.16.006756


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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

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

9. A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).

]. 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

10. J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965). [CrossRef]

,11

11. N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

] 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

7. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). [CrossRef]

] 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.

Φ(v)=j=1ajZj(v)
(1)

where aj are the expansion coefficients. For a wavefront degraded by atmospheric turbulence, the aberration coefficients aj vary randomly with time with a zero ensemble-averaged value and a well defined covariance function [14

14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–212 (1976). [CrossRef]

]. 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

13. R. K. Tyson, “Using the deformable mirror as a spatial filter: application to circular beams,” Appl. Opt. 21, 787–793 (1982). [CrossRef] [PubMed]

]. When the first J modes are removed the correcting phase may be written as

Φc(v)=j=1JajZj(v)
(2)

ΦJ(v)=Φ(v)Φc(v)=j=JajZj(v)
(3)

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 aj of the series in (2) are chosen to give the best fit to Φ(ν) in the least-square sense over the aperture. (Certainly, the presence of branch points in the phase function [15

15. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998). [CrossRef]

] 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

16. M. C. Roggemann and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000). [CrossRef]

,17

17. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000). [CrossRef]

]. 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 W(ν), a best fit in the least-square sense implies that aj may be sought from the minimization condition of the residual phase ΦJ on the circle of diameter D corresponding to the receiver aperture:

ajW(v)[Φ(v)j=1JajZj(v)]2dv=0
(4)

W(v)Zi(v)Zj(v)dv=δij
(5)

where δij is the delta function, the values of aj are determined as the expansion coefficients [18

18. J. Y. Wang, “Phase-compensated optical beam propagation through atmospheric turbulence,” Appl. Opt. 17, 2580–2590 (1978). [PubMed]

]

aj=W(v)Zj(v)Φ(v)dvW(v)Zj2(v)dv
(6)

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 US(ν) and the reference (local oscillator) heterodyne field ULO(ν) are combined on the input plane ν=(ρ,θ) of the optical system located at the propagation axis point z=0. Therefore, the average signal heterodyne power is given as

P=[W(v)US(v)ULO*(v)dv]2
(7)

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 US. 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:

P=W(v1)W(v2)MS(v1,v2)MLO*(v1,v2)dv1dv2
(8)

where MS(ν1,ν2) and MLO(ν1,ν2) are the collected and local-oscillator mutual coherence functions on the input (pupil) plane given by

MS(v1,v2)=Us(v1)US*(v2)
MLO(v1,v2)=ULO(v1)ULO*(v2).
(9)

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, MS. 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.

US(v)=AS(v)exp[jΦ(v)]exp[jΦc(v)]=AS(v)exp[jΦJ(v)]
(10)

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

19. J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of Hartmann Sensors in Strong Scintillation,” Appl. Opt. 41, 1012–1021 (2002). [CrossRef] [PubMed]

and 20

20. J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of a Shearing Interferometer in Strong Scintillation in the Absence of Additive Measurement Moise,” Appl. Opt. 41, 3674–3684 (2002). [CrossRef] [PubMed]

]. 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

21. J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002). [CrossRef]

, 22

22. G. A. Tyler, “Adaptive optics compensation for propagation through deep turbulence: initial investigation of gradient descent tomography,” J. Opt. Soc. Am. A 23, 1914–1923 (2006). [CrossRef]

]. 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

23. L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992). [CrossRef]

] with typical inner scale l0 of 1 cm and outer scale L0 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

6. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000). [CrossRef]

,7

7. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). [CrossRef]

] and represents a serious attempt to define the simulation parameters in a realistic way. In particular, the chosen value of outer scale L0 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 C2 n. The propagation modeling parameters chosen are those corresponding to typical daytime values of strong C2 n=10-14 m-2/3 and C2 n=10-13 m-2/3 turbulence. However, the results for different C2 n parameters only vary according to the Fried’s atmospheric coherence length r0 and the Rytov variance σ1 parameters. The Fried coherence length r0 describes the coherence diameter of the distorted wavefront phase. The Rytov variance σ1 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 C2 n=10-14 m-2/3, where r0=6 cm and σ1=1; then, a situation of stronger turbulence, denoted as C2 n=10-13 m-2/3, where r0=3 cm and σ1=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

24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

].

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

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967). [CrossRef]

] 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.

Fig. 1. Heterodyne efficiency gain (in decibels) as a function of the number of modes J removed by adaptive optics systems and several receiver aperture diameters by use of the simulation of realistic beam propagation in turbulent atmosphere. The levels of turbulence considered in these plots are typical daytime values of strong scintillation. In this plots, strong turbulence means a Fried’s coherent length r0=3 cm and a Rytov variance σ1=2; it is denoted as C2 n=10-13m-2/3 (right). We define a second, weaker turbulence level, denoted as as C2 n=10-14 m-2/3, where r0=6 cm and σ1=1 (left).

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 Φc 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 r0. 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 AS 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.

Fig. 2. Heterodyne efficiency gain (in decibels) as a function of the receiver aperture diameter and several number of modes corrected by the adaptive optics system. The no-scintillation case is also shown (dased lines). Here, the compensating phases are expansions through 2rd-order (astigmatism, J=6), 5th-order (J=20), and a high-order case (J=80). The levels of turbulence considered in this plots are similar to those described in Fig. 1.

Fig. 3. Optical heterodyne power uncertainty (in decibels) as a function of the number of modes J removed by adaptive optics systems and several receiver aperture diameters. Levels of turbulence are similar to those in Fig. 1.

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.

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

Fig. 4. Optical heterodyne power uncertainty (in decibels) as a function of the receiver aperture diameter and several number of modes corrected by the adaptive optics system. Again, the compensating phases are expansions through 2rdorder (astigmatism or J=6), 5th-order (J=20), and a high-order case (J=80). The no-correction case (J=0) is also considered. The no-scintillation situation (dashed line), where irradiance fluctuations have been canceled, helps us to clearly identify the amplitude effects on the uncertainty. Levels of turbulence are similar to those described in Fig. 1.

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.

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.

References and links

1.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967). [CrossRef]

2.

J. H. Shapiro, “Imaging and Optical Communication through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., (Springer Verlag, Berlin, 1978) pp. 210–220.

3.

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979). [CrossRef]

4.

J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978). [CrossRef]

5.

G. -m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995). [CrossRef]

6.

A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000). [CrossRef]

7.

A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). [CrossRef]

8.

N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. 46, 7218–7226 (2007). [CrossRef] [PubMed]

9.

A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).

10.

J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965). [CrossRef]

11.

N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

12.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

13.

R. K. Tyson, “Using the deformable mirror as a spatial filter: application to circular beams,” Appl. Opt. 21, 787–793 (1982). [CrossRef] [PubMed]

14.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–212 (1976). [CrossRef]

15.

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998). [CrossRef]

16.

M. C. Roggemann and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000). [CrossRef]

17.

G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000). [CrossRef]

18.

J. Y. Wang, “Phase-compensated optical beam propagation through atmospheric turbulence,” Appl. Opt. 17, 2580–2590 (1978). [PubMed]

19.

J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of Hartmann Sensors in Strong Scintillation,” Appl. Opt. 41, 1012–1021 (2002). [CrossRef] [PubMed]

20.

J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of a Shearing Interferometer in Strong Scintillation in the Absence of Additive Measurement Moise,” Appl. Opt. 41, 3674–3684 (2002). [CrossRef] [PubMed]

21.

J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002). [CrossRef]

22.

G. A. Tyler, “Adaptive optics compensation for propagation through deep turbulence: initial investigation of gradient descent tomography,” J. Opt. Soc. Am. A 23, 1914–1923 (2006). [CrossRef]

23.

L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992). [CrossRef]

24.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.3640) Atmospheric and oceanic optics : Lidar
(030.6600) Coherence and statistical optics : Statistical optics
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Atmospheric and oceanic optics

History
Original Manuscript: February 20, 2008
Revised Manuscript: April 11, 2008
Manuscript Accepted: April 23, 2008
Published: April 25, 2008

Citation
Aniceto Belmonte, "Influence of atmospheric phase compensation on optical heterodyne power measurements," Opt. Express 16, 6756-6767 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6756


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References

  1. D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967). [CrossRef]
  2. J. H. Shapiro, "Imaging and Optical Communication through Atmospheric Turbulence," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., (Springer Verlag, Berlin, 1978) pp. 210-220.
  3. H. T. Yura, "Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence," Opt. Acta 26, 627-644 (1979). [CrossRef]
  4. J. Y. Wang and J. K. Markey, "Modal compensation of atmospheric turbulence phase distortion," J. Opt. Soc. Am. 68, 78-87 (1978). [CrossRef]
  5. G. -m. Dai, "Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions," J. Opt. Soc. Am. A 12, 2182-2193 (1995). [CrossRef]
  6. A. Belmonte and B. J. Rye, "Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems," Appl. Opt. 39, 2401-2411 (2000). [CrossRef]
  7. A. Belmonte, "Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance," Appl. Opt. 39, 5426-5445 (2000). [CrossRef]
  8. N. Perlot, "Turbulence-induced fading probability in coherent optical communication through the atmosphere," Appl. Opt. 46, 7218-7226 (2007). [CrossRef] [PubMed]
  9. A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).
  10. J. W. Goodman, "Some effects of Target-induced Scintillation on optical radar performance," Proc. IEEE 53, 1688-1700 (1965). [CrossRef]
  11. N. E. Zirkind and J. H. Shapiro, "Adaptive optics for large aperture coherent laser radars," Proc. SPIE 999, paper 13 (1988).
  12. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  13. R. K. Tyson, "Using the deformable mirror as a spatial filter: application to circular beams," Appl. Opt. 21, 787-793 (1982). [CrossRef] [PubMed]
  14. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-212 (1976). [CrossRef]
  15. D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15, 2759-2768 (1998). [CrossRef]
  16. M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. A 17, 53-62 (2000). [CrossRef]
  17. G. A. Tyler, "Reconstruction and assessment of the least-squares and slope discrepancy components of the phase," J. Opt. Soc. Am. A 17, 1828-1839 (2000). [CrossRef]
  18. J. Y. Wang, "Phase-compensated optical beam propagation through atmospheric turbulence," Appl. Opt. 17, 2580-2590 (1978). [PubMed]
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