Abstract

A method for closed-loop stable control of two deformable mirrors for compensation of both amplitude and phase fluctuations is described. A generic implementation is described as well as an implementation that integrates the concept behind a point diffraction interferometer with a two-deformable-mirror system. The relationship of the closed-loop control algorithm to previously developed open-loop iterative algorithms is described. Simulation results are presented that indicate that the system is stable and provides superior performance over that of a single-deformable-mirror system. The impact of finite servo bandwidth on control of two deformable mirrors is evaluated by means of wave optical simulation, and it is found that to achieve a performance improvement attributable to compensation of amplitude fluctuations, the bandwidth of the two-deformable-mirror system must be at least twice the Greenwood frequency.

© 2002 Optical Society of America

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References

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  1. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  2. R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
    [CrossRef]
  3. J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
    [CrossRef]
  4. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  5. B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, R. J. Plemmons, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
    [CrossRef]
  6. D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
    [CrossRef]
  7. A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
    [CrossRef]
  8. M. C. Roggemann, D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4585 (1998).
    [CrossRef]
  9. M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bisonnette, eds., Proc. SPIE3763, 29–40 (1999).
    [CrossRef]
  10. R. A. Gonsalves, “Compensation of scintillation with a phase-only adaptive optic,” Opt. Lett. 22, 588–590 (1997).
    [CrossRef] [PubMed]
  11. J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
    [CrossRef]
  12. J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
    [CrossRef]
  13. 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]
  14. J. D. Barchers, B. L. Ellerbroek, “Increase in the compensated field of view in strong scintillation by use of two deformable mirrors,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed., available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf .
  15. H. Stark, Y. Yang, Vector Space Projections (Wiley, Inc., New York, 1998).
  16. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  17. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  18. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  19. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
    [CrossRef] [PubMed]
  20. J. D. Barchers, “Convergence rates for iterative vector space projection methods for control of two deformable mirrors for compensation of both amplitude and phase fluctuations,” Appl. Opt. 41, 2213–2218 (2002).
    [CrossRef] [PubMed]
  21. D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atm. Ocean. Opt. 11, 982–990 (1998).
  22. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
  23. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).
  24. In the case of imaging applications where the objective is to increase the compensated field of view, it is generally assumed that the optimal conjugate range to the second deformable mirror is somewhere in the upper atmosphere. However, the mathematical approach taken for full-wave compensation via iterative algorithms does not require the second deformable mirror to be conjugate to a location in the atmosphere. As the closed-loop approach for control of two deformable mirrors described here is closely related to the previously developed iterative algorithms, there is also no requirement that the second deformable mirror be conjugate to some upper-altitude turbulence layer. The important exception is the application of wide-field-of-view imaging, in which case it was found that the optimal range to the second deformable mirror was indeed conjugate to an upper-altitude turbulence layer.14
  25. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. A 64, 59–67 (1974).
    [CrossRef]
  26. C. L. Phillips, H. T. Nagle, Digital Control System Analysis and Design (Prentice-Hall, Englewood Cliffs, N.J., 1990).
  27. V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” (National Technical Information Service, Springfield, Va., 1968).
  28. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–374 (1977).
    [CrossRef]
  29. D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
    [CrossRef]
  30. J. D. Barchers, D. J. Lee, D. E. Lane, “Single-input–single-output analysis of latency and quadrant detector saturation in adaptive optical systems,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 308–320 (1999).
    [CrossRef]
  31. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  32. J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
    [CrossRef] [PubMed]
  33. J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
    [CrossRef] [PubMed]
  34. B. L. Ellerbroek, Gemini Observatory, Hilo, Hawaii 96720 (personal communication, August2001).
  35. A. Tokovinin, “The maximum separation between guide stars in atmospheric tomography,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed., available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf .
  36. D. L. Fried, P.O. Box 680, Moss Landing, California 95039 (personal communication, April2001).
  37. B. L. Ellerbroek, “Power series evaluation of covariances for turbulence-induced phase distortions including outer scale and servo-lag effects,” J. Opt. Soc. Am. A 16, 533–548 (1999).
    [CrossRef]

2002

2001

2000

1999

1998

1997

1995

1994

1992

1984

1982

1977

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–374 (1977).
[CrossRef]

1975

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

1974

H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. A 64, 59–67 (1974).
[CrossRef]

Barchers, J. D.

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]

J. D. Barchers, “Convergence rates for iterative vector space projection methods for control of two deformable mirrors for compensation of both amplitude and phase fluctuations,” Appl. Opt. 41, 2213–2218 (2002).
[CrossRef] [PubMed]

J. D. Barchers, D. L. Fried, 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, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
[CrossRef]

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

J. D. Barchers, D. J. Lee, D. E. Lane, “Single-input–single-output analysis of latency and quadrant detector saturation in adaptive optical systems,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 308–320 (1999).
[CrossRef]

Beckers, J. M.

J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

Deng, S.

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bisonnette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

Ellerbroek, B. L.

Fried, D. L.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atm. Ocean. Opt. 11, 982–990 (1998).

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

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–374 (1977).
[CrossRef]

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

D. L. Fried, P.O. Box 680, Moss Landing, California 95039 (personal communication, April2001).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

Johnston, D. C.

Kotzer, T.

Lane, D. E.

J. D. Barchers, D. J. Lee, D. E. Lane, “Single-input–single-output analysis of latency and quadrant detector saturation in adaptive optical systems,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 308–320 (1999).
[CrossRef]

Le Louarn, M.

Lee, D. J.

M. C. Roggemann, D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4585 (1998).
[CrossRef]

J. D. Barchers, D. J. Lee, D. E. Lane, “Single-input–single-output analysis of latency and quadrant detector saturation in adaptive optical systems,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 308–320 (1999).
[CrossRef]

Levi, A.

Link, D. J.

Nagle, H. T.

C. L. Phillips, H. T. Nagle, Digital Control System Analysis and Design (Prentice-Hall, Englewood Cliffs, N.J., 1990).

Phillips, C. L.

C. L. Phillips, H. T. Nagle, Digital Control System Analysis and Design (Prentice-Hall, Englewood Cliffs, N.J., 1990).

Pitsianis, N. P.

Plemmons, R. J.

Roggemann, M. C.

M. C. Roggemann, D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4585 (1998).
[CrossRef]

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bisonnette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

Rosen, J.

Sarazin, M.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).

Shamir, J.

Stark, H.

Tatarskii, V. I.

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” (National Technical Information Service, Springfield, Va., 1968).

Tokovinin, A.

Van Loan, C.

Vaughn, J. L.

Welsh, B. M.

Yang, Y.

H. Stark, Y. Yang, Vector Space Projections (Wiley, Inc., New York, 1998).

Yura, H. T.

H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. A 64, 59–67 (1974).
[CrossRef]

Appl. Opt.

Astrophys. J.

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

J. Atm. Ocean. Opt.

D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atm. Ocean. Opt. 11, 982–990 (1998).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
[CrossRef]

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]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, R. J. Plemmons, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
[CrossRef]

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. A 64, 59–67 (1974).
[CrossRef]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. A 67, 370–374 (1977).
[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

B. L. Ellerbroek, “Power series evaluation of covariances for turbulence-induced phase distortions including outer scale and servo-lag effects,” J. Opt. Soc. Am. A 16, 533–548 (1999).
[CrossRef]

Opt. Commun.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Opt. Lett.

Other

J. D. Barchers, B. L. Ellerbroek, “Increase in the compensated field of view in strong scintillation by use of two deformable mirrors,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed., available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf .

H. Stark, Y. Yang, Vector Space Projections (Wiley, Inc., New York, 1998).

J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bisonnette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

B. L. Ellerbroek, Gemini Observatory, Hilo, Hawaii 96720 (personal communication, August2001).

A. Tokovinin, “The maximum separation between guide stars in atmospheric tomography,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed., available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf .

D. L. Fried, P.O. Box 680, Moss Landing, California 95039 (personal communication, April2001).

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

J. D. Barchers, D. J. Lee, D. E. Lane, “Single-input–single-output analysis of latency and quadrant detector saturation in adaptive optical systems,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 308–320 (1999).
[CrossRef]

C. L. Phillips, H. T. Nagle, Digital Control System Analysis and Design (Prentice-Hall, Englewood Cliffs, N.J., 1990).

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” (National Technical Information Service, Springfield, Va., 1968).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

In the case of imaging applications where the objective is to increase the compensated field of view, it is generally assumed that the optimal conjugate range to the second deformable mirror is somewhere in the upper atmosphere. However, the mathematical approach taken for full-wave compensation via iterative algorithms does not require the second deformable mirror to be conjugate to a location in the atmosphere. As the closed-loop approach for control of two deformable mirrors described here is closely related to the previously developed iterative algorithms, there is also no requirement that the second deformable mirror be conjugate to some upper-altitude turbulence layer. The important exception is the application of wide-field-of-view imaging, in which case it was found that the optimal range to the second deformable mirror was indeed conjugate to an upper-altitude turbulence layer.14

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Figures (8)

Fig. 1
Fig. 1

Optical implementation of the propagation and filtering operator. A field that is conjugate to some distance z0 is brought to focus. A quadratic field lens and a filter or an amplifier is applied so that after propagation through focus and recollimation of the beam, the resultant field is conjugate to a distance z0+z and is a spatially filtered or amplified version of the input field.

Fig. 2
Fig. 2

General description of an approach for closed-loop control of two deformable mirrors (DM 1, 2) for scintillation compensation. The phase error between the incoming and outgoing beams is measured at the plane of each deformable mirror, and these error signals are used to drive the two deformable mirrors.

Fig. 3
Fig. 3

Preferred implementation for closed-loop control of two deformable mirrors for scintillation compensation. The phase error between the incoming and outgoing beams is measured at the plane of each deformable mirror by interfering the incoming beam with a reference that is generated by spatial filtering of the corrected incoming beam.

Fig. 4
Fig. 4

Analytic and observed stability margins for the CLMCAO system for Rytov numbers (a) 0.4 and (b) 0.8. The observed stability margins are in good agreement with the analytic predictions, particularly as α/l1.

Fig. 5
Fig. 5

Example performance results for (a) fG/f3 dB=0.1 and (b) fG/f3 dB=1.0, l/r0=1/2, and Rytov number equal to 0.8. The upper and lower solid curves indicate the Strehl loss due to scintillation and the Strehl, respectively, for the CLMCAO system. The upper and lower dashed curves indicate the Strehl loss due to scintillation and the Strehl, respectively, for the single-deformable-mirror system using a point diffraction interferometer. The dotted–dashed curves are the Strehl for the single-deformable-mirror system using a Hartmann sensor and a least-squares reconstructor. For small fG/f3 dB, the CLMCAO system exhibits improved performance.

Fig. 6
Fig. 6

Average steady-state Strehl as a function of normalized range to the second deformable mirror for several values of the Rytov number. Normalized range is defined as the range to the second deformable mirror divided by the total turbulence profile length. The turbulence is frozen (fG/f3 dB=0) to isolate the effect of adjusting range to the second deformable mirror.

Fig. 7
Fig. 7

Evaluation of the performance of a CLMCAO system and two single-deformable-mirror systems (SRI denotes self-referencing point diffraction interferometer, and FH denotes a Hartmann sensor in the Fried geometry) as a function of fG/f3 dB for Rytov numbers (a) 0.05 and (b) 0.8. The CLMCAO system has superior performance for sufficiently small values of fG/f3 dB.

Fig. 8
Fig. 8

Evaluation of the performance of a CLMCAO system and two single-deformable-mirror systems as a function of Rytov number for (a) fG/f3 dB=0.1 and (b) fG/f3 dB=1.0. At small values of fG/f3 dB, the CLMCAO system has improved performance as a result of compensation of scintillation; however, a performance degradation is observed at large values of fG/f3 dB.

Tables (1)

Tables Icon

Table 1 Propagation and System Parameters

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

Tz(·)=F-1[F(·)exp(iπλzκ¯2)].
Tz,α(·)=F-1F(·)expiπλzκ¯2-sgn(α)πα22κ¯2,
S=dr¯1M0(r¯1)Ub(r¯1)exp[iϕ1(r¯1)]Tz{Ur*(r¯2)exp[-iϕ2(r¯2)]}2dr¯1 Ub(r¯1)Ub*(r¯1)dr¯2 Ur(r¯2)Ur*(r¯2).
U˜1(r¯1)=D(r¯1)dr¯1Ub*(r¯1)exp[-iϕ1(r¯1)]×Tz,α*{Ur(r¯2)exp[iϕ2(r¯2)]},
U˜2(r¯2)=D(r¯2)dr¯2Ur*(r¯2)exp[-iϕ2(r¯2)]×Tz,α{Ub(r¯1)exp[iϕ1(r¯1)]},
U˜1(r¯1)=D(r¯1)dr¯1Ub*(r¯1)exp[-iϕ1(r¯1)]×D(r¯1)dr¯1Tz,α*{Ur(r¯2)exp[iϕ2(r¯2)]},
U˜2(r¯2)=D(r¯2)dr¯2Ur*(r¯2)exp[-iϕ2(r¯2)]×D(r¯2)dr¯2Tz,α{Ub(r¯1)exp[iϕ1(r¯1)]}.
ϕl(r¯l)=arg{exp[iϕl(r¯l)]exp[iμl(r¯l)]},
LS[ϕ]=(GTG)-1GTPV(Gϕ),
ϕ˘1(r¯1)=LS[ϕ1(r¯1)]+arg(exp[iϕ1(r¯1)]exp{-iLS[ϕ1(r¯1)]}),
ϕ˘2(r¯2)=LS[ϕ2(r¯2)]+arg(exp[iϕ2(r¯2)]exp{-iLS[ϕ2(r¯2)]}).
Sexp[iϕ1(r¯1)], Ub*(r¯1)Tz*{Ur(r¯2)exp[iϕ2(r¯2)]},
Sexp[iϕ2(r¯2)], Ur*(r¯2)Tz{Ub(r¯1)exp[iϕ1(r¯1)]}.
ϕ1(r¯1)=arg(Ub*(r¯1)Tz*{Ur(r¯2)exp[iϕ2(r¯2)]}).
ϕ2(r¯2)=arg(Ur*(r¯2)Tz{Ub(r¯1)exp[iϕ1(r¯1)]}).
1(r¯1)=arg(exp[-iϕ1(r¯1)]Ub*(r¯1)×Tz*{Ur(r¯2)exp[iϕ2(r¯2)]}),
2(r¯2)=arg(exp[-iϕ2(r¯2)]Ur*(r¯2)Tz{Ub(r¯1)exp[iϕ1(r¯1)]}).
ϕl(r¯l)=arg{exp[iϕl(r¯l)]exp[iμl(r¯l)]},
ϕ1,k+1(r¯, 0)=ϕ1,k(r¯, 0)+μ1,ϕ,k(r¯, 0),
ϕ2,k+1(r¯, L)=ϕ2,k(r¯, L)+μ2,ϕ,k(r¯, L).
1,k,ϕ(r¯, 0)=-ϕ(r¯, 0)-ϕ1,k(r¯, 0)+ϕ2,k(r¯, 0)*exp[-(α/r¯)2],
2,k,ϕ(r¯, L)=[ϕ(r¯, L)+ϕ1,k(r¯, L)]*×exp[-(α/r¯)2]-ϕ2,k(r¯, L),
1,k,ϕ(r¯, 0)=-ϕ(r¯, 0)-ϕ1,k(r¯, 0)+ϕ2,k(r¯, 0),
1,k,χ(r¯, 0)=-χ(r¯, 0)+χ2,k(r¯, 0),
2,k,ϕ(r¯, L)=ϕ(r¯, L)+ϕ1,k(r¯, L)-ϕ2,k(r¯, L),
2,k,χ(r¯, L)=χ(r¯, L)+χ1,k(r¯, L).
ϕˆ1,k(κ¯, L)χˆ1,k(κ¯, L)=ϕˆ1,k(κ¯, 0)cos(πκ2λL)sin(πκ2λL),
ϕˆ2,k(κ¯, 0)χˆ2,k(κ¯, 0)=ϕˆ2,k(κ¯, L)cos(πκ2λL)sin(πκ2λL).
ϕˆk(κ¯, s)χˆk(κ¯, s)=k0j=1Nnˆk(κ¯, zj)cos(πκ2λ|s-zj|)sin(πκ2λ|s-zj|).
k0n¯ˆk(κ¯)=k0[nˆk(κ¯, z1)nˆk(κ¯, zN)]T,
ϕ¯ˆk(κ¯)=[ϕˆ1,k(κ¯, 0)ϕˆ2,k(κ¯, L)]T,
¯ˆk(κ¯)=[ˆ1,k,ϕ(κ¯, 0)ˆ1,k,χ(κ¯, 0)ˆ2,k,ϕ(κ¯, L)ˆ2,k,χ(κ¯, L)]T.
ϕ¯ˆk+1(κ¯)=A(κ, L, k0, μ, α, z¯)ϕ¯ˆk(κ¯)+B(κ, L, k0, μ, α, z¯)k0n¯ˆk(κ¯),
¯ˆk(κ¯)=C(κ, L, k0, μ, α, z¯)ϕ¯ˆk(κ¯)+D(κ, L, k0, μ, α, z¯)k0n¯ˆk(κ¯),
A(κ, L, k0, μ, α, z¯)=1-μμ cos(πκ2λL)exp-πα22κ2μ cos(πκ2λL)exp-πα22κ21-μ,
D(κ, L, k0, μ, α, z¯)
=D1,1:2(κ, L, k0, μ, α, z¯)D2,1(κ, L, k0, μ, α, z¯)D2.2(κ, L, k0, μ, α, z¯),
D(κ, L, k0, μ, α, z¯)=-cos(πκ2λz1-cos(πκ2λzN)-sin(πκ2λz1)-sin(πκ2λzN)cos(πκ2λ|L-z1|)cos(πκ2λ|L-zN|)sin(πκ2λ|L-z1|)sin(πκ2λ|L-zN|),
B(κ, L, k0, μ, α, z¯)=μ100000exp[-(πα2)2κ2]0
×D(κ, L, k0, μ, α, z¯),
C(κ, L, k0, μ, α, z¯)=-1cos(πκ2λL)0sin(πκ2λL)cos(πκ2λL)-1sin(πκ2λL)0.
T¯ˆ,k0n¯ˆ(z˜, κ, L, k0, μ, α, z¯)
=C(κ, L, k0, μ, α, z¯)[z˜I-A(κ, L, k0, μ, α, z¯)]-1×B(κ, L, k0, μ, α, z¯)+D(κ, L, k0, μ, α, z¯),
det[z˜I-A(κ, L, k0, μ, α, z¯)]=z˜2+2(μ-1)z˜+(μ-1)2-μ2β2(κ, L, k0, α),
Uˆ(r¯1)=D(r¯1)dr¯Ub(r¯)exp[-iϕ(r¯)],
Δϕb(r¯1)
=D(r¯1)dr¯|Ub(r¯)|2 arg{exp[iϕb(r¯)]exp[-iϕ(r¯)]}D(r¯1)dr¯|Ub(r¯)|2.
f3dB=12πTscos-12-0.5[1+(μ-1)2]2[0.5(μ-1)+1],
SA=dr¯1|Ub(r¯1)|M0(r¯1)2dr¯1Ub(r¯1)Ub*(r¯1)dr¯2Ur(r¯2)Ur*(r¯2).
SA
=dr¯1M0(r¯1)|Ub(r¯1)||Tz{Ur*(r¯2)exp[-iϕ2(r¯2)]}|2dr¯1Ub(r¯1)Ub*(r¯1)dr¯2Ur(r¯2)Ur*(r¯2).
Sˆα1Nn=1NUb,n(r¯1)exp[iϕ1(r¯1)], T-z,α,θn{Ur,n(r¯2)exp[iϕ2(r¯2)]}.
ϕ1(r¯1)=arg1Nn=1NUb,n*(r¯1)×T-z,α,θn{Ur,n(r¯2)exp[iϕ2(r¯2)]}.
ϕ2(r¯2)=arg1Nn=1NUr,n*(r¯2)×Tz,α,θn{Ub,n(r¯1)exp[iϕ1(r¯1)]}.
1(r¯1)=arg1Nn=1NUb,n*(r¯1)exp[-iϕ1(r¯1)]×T-z,α,θn{Ur,n(r¯2)exp[iϕ2(r¯2)]},
1(r¯1)=arg1Nn=1NUr,n*(r¯2)exp[-iϕ2(r¯2)]×Tz,α,θn{Ub,n(r¯1)exp[iϕ1(r¯1)]}.

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