## Influence of atmospheric phase compensation on optical heterodyne power measurements

Optics Express, Vol. 16, Issue 9, pp. 6756-6767 (2008)

http://dx.doi.org/10.1364/OE.16.006756

Acrobat PDF (323 KB)

### 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

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

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]

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]

*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.

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]

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]

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

## 2. Phase-compensated heterodyne detection

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

*(*of a system with a pupil of diameter

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

*Z*as

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

_{j}*a*vary randomly with time with a zero ensemble-averaged value and a well defined covariance function [14

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

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

*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*of the series in (2) are chosen to give the best fit to Φ

_{j}*(*in the least-square sense over the aperture. (Certainly, the presence of branch points in the phase function [15

**ν**)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]

*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:

*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**)*δ*is the delta function, the values of

_{ij}*a*are determined as the expansion coefficients [18

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

*J*=3), astigmatism (

*J*=6), coma (

*J*=10), and most other high-order aberrations.

*U*and the reference (local oscillator) heterodyne field

_{S}(**ν**)*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

*U*. 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

_{S}*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:

*M*and

_{S}(**ν**_{1}**,****ν**)_{2}*M*are the collected and local-oscillator mutual coherence functions on the input (pupil) plane given by

_{LO}(**ν**_{1}**,****ν**)_{2}*M*. 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.

_{S}*U*on the pupil plane are derived from the numerical simulation of atmospheric wave beam propagation [7

_{S}**39**, 5426–5445 (2000). [CrossRef]

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

_{J}

*(*=Φ

**ν**)*(*-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

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]

_{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

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]

**39**, 5426–5445 (2000). [CrossRef]

_{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.

^{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

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

*C*, receiver optic aperture diameter

^{2}_{n}*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.

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

*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.

*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.

*A*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.

_{S}*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.

*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

## Acknowledgments

## References and links

1. | D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE |

2. | J. H. Shapiro, “Imaging and Optical Communication through Atmospheric Turbulence,” in |

3. | H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta |

4. | J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. |

5. | G. -m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A |

6. | A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. |

7. | A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. |

8. | N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. |

9. | A. W. Jelalian, |

10. | J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE |

11. | N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE |

12. | M. Born and E. Wolf, |

13. | R. K. Tyson, “Using the deformable mirror as a spatial filter: application to circular beams,” Appl. Opt. |

14. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

15. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

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. | G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A |

18. | J. Y. Wang, “Phase-compensated optical beam propagation through atmospheric turbulence,” Appl. Opt. |

19. | J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of Hartmann Sensors in Strong Scintillation,” Appl. Opt. |

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. |

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 |

22. | G. A. Tyler, “Adaptive optics compensation for propagation through deep turbulence: initial investigation of gradient descent tomography,” J. Opt. Soc. Am. A |

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. |

24. | L. C. Andrews, R. L. Phillips, and C. Y. Hopen, |

**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

- D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967). [CrossRef]
- 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.
- H. T. Yura, "Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence," Opt. Acta 26, 627-644 (1979). [CrossRef]
- J. Y. Wang and J. K. Markey, "Modal compensation of atmospheric turbulence phase distortion," J. Opt. Soc. Am. 68, 78-87 (1978). [CrossRef]
- 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]
- A. Belmonte and B. J. Rye, "Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems," Appl. Opt. 39, 2401-2411 (2000). [CrossRef]
- A. Belmonte, "Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance," Appl. Opt. 39, 5426-5445 (2000). [CrossRef]
- N. Perlot, "Turbulence-induced fading probability in coherent optical communication through the atmosphere," Appl. Opt. 46, 7218-7226 (2007). [CrossRef] [PubMed]
- A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).
- J. W. Goodman, "Some effects of Target-induced Scintillation on optical radar performance," Proc. IEEE 53, 1688-1700 (1965). [CrossRef]
- N. E. Zirkind and J. H. Shapiro, "Adaptive optics for large aperture coherent laser radars," Proc. SPIE 999, paper 13 (1988).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- R. K. Tyson, "Using the deformable mirror as a spatial filter: application to circular beams," Appl. Opt. 21, 787-793 (1982). [CrossRef] [PubMed]
- R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-212 (1976). [CrossRef]
- D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15, 2759-2768 (1998). [CrossRef]
- 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]
- 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]
- J. Y. Wang, "Phase-compensated optical beam propagation through atmospheric turbulence," Appl. Opt. 17, 2580-2590 (1978). [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

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