Abstract

Plane-wave scintillation is shown to impose multiconjugate adaptive optics (MCAO) correctability limitations that are independent of wavefront sensing and reconstruction. Residual phase and log-amplitude variances induced by scintillation in weak turbulence are derived using linear (diffraction-based) diffractive MCAO spatial filters or (diffraction-ignorant) geometric MCAO proportional gains as open-loop control parameters. In the case of Kolmogorov turbulence, expressions involving the Rytov variance and/or weighted Cn2 integrals apply. Differences in performance between diffractive MCAO and geometric MCAO resemble chromatic errors. Optimal corrections based on least squares imply irreducible performance limits that are validated by wave-optic simulations.

© 2006 Optical Society of America

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  1. M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
    [CrossRef]
  2. B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).
  3. E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
    [CrossRef]
  4. A. Tokovinin, M. Le Louarn, and M. Sarazin, "Isoplanatism in a multiconjugate adaptive optics system," J. Opt. Soc. Am. A 17, 1819-1827 (2000).
    [CrossRef]
  5. A. Tokovinin and E. Viard, "Limiting precision of tomographic phase estimation," J. Opt. Soc. Am. A 18, 873-882 (2001).
    [CrossRef]
  6. J. D. Barchers and 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]
  7. J. D. Barchers, "Noise gain and misregistration calculations for a two-deformable mirror-field conjugation system," in Laser Weapons Technology III, W. E. Thompson and P. H. Merritt, eds., Proc. SPIE 4724, 1-16 (2002).
    [CrossRef]
  8. J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations," J. Opt. Soc. Am. A 19, 926-945 (2002).
    [CrossRef]
  9. J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations: errata," J. Opt. Soc. Am. A 19, 1831-1832 (2003).
    [CrossRef]
  10. M. C. Roggemann and 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]
  11. A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
    [CrossRef]
  12. R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
    [CrossRef]
  13. J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
    [CrossRef]
  14. R. C. Flicker, "Sequence of phase correction in multiconjugate adaptive optics," Opt. Lett. 26, 1743-1745 (2001).
    [CrossRef]
  15. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. L. Lee, G. Baker, and R. Benson, "Wavefront control limitations due to scintillation in multiconjugate adaptive optics," in Proceedings of the 2005 AMOS Technical Conference (Maui Economic Development Board, 2005), pp. 678-687.
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  20. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  21. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).
  22. R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
    [CrossRef]
  23. P. H. Hu, J. Stone, and T. Stanley, "Application of Zernike polynomials to atmospheric propagation problems," J. Opt. Soc. Am. A 6, 1595-1608 (1989).
    [CrossRef]
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    [CrossRef]

2005 (1)

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

2004 (1)

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

2003 (3)

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations: errata," J. Opt. Soc. Am. A 19, 1831-1832 (2003).
[CrossRef]

2002 (3)

2001 (3)

2000 (1)

1998 (1)

1992 (1)

1989 (1)

1983 (1)

1975 (1)

A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
[CrossRef]

1970 (1)

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

1969 (1)

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Apsell, S.

A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
[CrossRef]

Baker, G.

L. Lee, G. Baker, and R. Benson, "Wavefront control limitations due to scintillation in multiconjugate adaptive optics," in Proceedings of the 2005 AMOS Technical Conference (Maui Economic Development Board, 2005), pp. 678-687.

Barchers, J. D.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations: errata," J. Opt. Soc. Am. A 19, 1831-1832 (2003).
[CrossRef]

J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations," J. Opt. Soc. Am. A 19, 926-945 (2002).
[CrossRef]

J. D. Barchers, "Noise gain and misregistration calculations for a two-deformable mirror-field conjugation system," in Laser Weapons Technology III, W. E. Thompson and P. H. Merritt, eds., Proc. SPIE 4724, 1-16 (2002).
[CrossRef]

J. D. Barchers and 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]

Benson, R.

L. Lee, G. Baker, and R. Benson, "Wavefront control limitations due to scintillation in multiconjugate adaptive optics," in Proceedings of the 2005 AMOS Technical Conference (Maui Economic Development Board, 2005), pp. 678-687.

Boyer, C.

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

Brennan, T. J.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Britton, M.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Buchroeder, R. A.

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

Dekany, R.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Devaney, A. J.

A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
[CrossRef]

Ellerbroek, B.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Ellerbroek, B. L.

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

B. L. Ellerbroek, "Efficient computation of minimum-variance wave-front reconstructors with sparse-matrix techniques," J. Opt. Soc. Am. A 19, 1803-1816 (2002).
[CrossRef]

J. D. Barchers and 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]

Flicker, R. C.

Fried, D. L.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Fugate, R. Q.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Gavel, D.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

Harp, J. C.

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Hegwer, S.

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

Herriot, G.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Hu, P. H.

Hunten, M. R.

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

James, E.

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

Langlois, M.

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

Le Louarn, M.

Lee, D. J.

Lee, L.

L. Lee, G. Baker, and R. Benson, "Wavefront control limitations due to scintillation in multiconjugate adaptive optics," in Proceedings of the 2005 AMOS Technical Conference (Maui Economic Development Board, 2005), pp. 678-687.

Lee, R. W.

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Liff, H. J.

A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
[CrossRef]

Link, D. J.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Macintosh, B.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Moretti, W.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Moretto, G.

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

Richards, K.

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

Rimmele, T. R.

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

Roggemann, M. C.

Sarazin, M.

Sasiela, R. J.

R. J. Sasiela, "Strehl ratios with various types of anisoplanatism," J. Opt. Soc. Am. A 9, 1398-1405 (1992).
[CrossRef]

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
[CrossRef]

Stanley, T.

Stoesz, J.

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

Stone, J.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tokovinin, A.

Tyler, G. A.

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

Viard, E.

Wallner, E. P.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (1)

A. J. Devaney, H. J. Liff, and S. Apsell, "Spectral representations for free space propagation of complex phase perturbations of optical fields," Opt. Commun. 15, 1-5 (1975).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Proc. SPIE (5)

J. D. Barchers, D. L. Fried, D. J. Link, G. A. Tyler, W. Moretti, T. J. Brennan, and R. Q. Fugate, "The performance of wavefront sensors in strong scintillation," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 217-227 (2003).
[CrossRef]

M. Langlois, G. Moretto, K. Richards, S. Hegwer, and T. R. Rimmele, "Solar multiconjugate adaptive optics at the Dunn Solar Telescope: preliminary results," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 59-66 (2004).
[CrossRef]

B. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B. Macintosh, and J. Stoesz, "Adaptive optics for the thirty meter telescope," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE 5903, 20-31 (2005).

E. James, C. Boyer, R. A. Buchroeder, B. L. Ellerbroek, and M. R. Hunten, "Design considerations of the AO module for the Gemini South multiconjugate adaptive optics system," in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE 4839, 67-80 (2003).
[CrossRef]

J. D. Barchers, "Noise gain and misregistration calculations for a two-deformable mirror-field conjugation system," in Laser Weapons Technology III, W. E. Thompson and P. H. Merritt, eds., Proc. SPIE 4724, 1-16 (2002).
[CrossRef]

Other (6)

L. Lee, G. Baker, and R. Benson, "Wavefront control limitations due to scintillation in multiconjugate adaptive optics," in Proceedings of the 2005 AMOS Technical Conference (Maui Economic Development Board, 2005), pp. 678-687.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (10)

Fig. 1
Fig. 1

Abstract propagation model. An initially planar-wave U propagates from an entry plane at z = 0 to an exit plane at z = L . Positions denoted by z = z 0 are possible locations for a turbulent layer that applies a phase aberration ϕ 0 . Correctors (or their optical conjugates) located at z = z 1 , , z M apply corresponding phase corrections ϕ 1 , , ϕ M .

Fig. 2
Fig. 2

Linearized open-loop control system block diagram.

Fig. 3
Fig. 3

Normalized minimum error variance versus wavelength ratio λ λ for diffractive SCAO with chromatic error.

Fig. 4
Fig. 4

Minimum-variance correction gains versus aberration location with two corrections.

Fig. 5
Fig. 5

Minimum-variance error sensitivity versus aberration location with one or two corrections.

Fig. 6
Fig. 6

Normalized minimum error variance versus locations of two corrections with uniform C n 2 Kolmogorov turbulence between z = 0 and z = L .

Fig. 7
Fig. 7

Normalized minimum error variance with many uniformly spaced corrections and uniform C n 2 Kolmogorov turbulence.

Fig. 8
Fig. 8

Simulated compensation architectures: (a) transmitter, (b) receiver, (c) embedded.

Fig. 9
Fig. 9

Average Strehl ratio versus scintillation index for geometric SCAO conjugated to suboptimal ( z = 0 or z = L ) or optimal ( z = 0.5 L ) locations.

Fig. 10
Fig. 10

Average Strehl ratio versus scintillation index for geometric MCAO conjugated to suboptimal ( z = 0 and z = L ) or optimal ( z = 0.2 L and z = 0.8 L ) locations.

Tables (1)

Tables Icon

Table 1 Normalized Minimum Error Variance versus Locations of One or Two Corrections with Uniform C n 2 Kolmogorov Turbulence between z = 0 and z = L

Equations (68)

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

U ( f ; z + ζ ) = exp ( j π λ ζ f 2 ) U ( f ; z ) ,
U ( x ; z ) = exp [ χ ( x ; z ) + j ϕ ( x ; z ) ] U ref ( x ; z ) ,
exp [ χ ( x ; z ) + j ϕ ( x ; z ) ] 1 + χ ( x ; z ) + j ϕ ( x ; z ) ,
[ χ ( f ; z + ζ ) ϕ ( f ; z + ζ ) ] [ cos ( π λ ζ f 2 ) sin ( π λ ζ f 2 ) sin ( π λ ζ f 2 ) cos ( π λ ζ f 2 ) ] [ ζ ( f ; z ) ϕ ( f ; z ) ] .
Ω ( f ; z 0 ) = 2 π Γ ( 8 3 ) sin ( π 3 ) ( 2 π λ ) 2 C n 2 ( z 0 ) Δ z ( 2 π f ) 11 3 .
σ χ 2 ( z 0 ; z 1 ) = 0.563 ( 2 π λ ) 7 6 z 1 z 0 5 6 C n 2 ( z 0 ) Δ z ,
σ χ 2 = 0.563 ( 2 π λ ) 7 6 0 L L z 5 6 C n 2 ( z ) d z ,
σ χ 2 = 0.563 ( 2 π λ ) 7 6 6 11 L 11 6 C n 2 .
Γ ( 8 3 ) sin ( π 3 ) 2 11 6 3 π 1 2 Γ ( 7 12 ) 5 Γ ( 11 12 ) 0.563 .
j ψ ( f ; z 0 ) = χ ( f ; L ) + j ϕ ( f ; L ) ,
ψ ( f ; z 0 ) = exp [ j π λ ( L z 0 ) f 2 ] ϕ 0 ( f ; z 0 ) + m = 1 M exp [ j π λ ( L z m ) f 2 ] ϕ m ( f ; z 0 ) .
ϕ m ( f ; z 0 ) = g m ( f ; z 0 ) ϕ 0 ( f ; z 0 ) .
g m ( f ; z 0 ) = c m ( z 0 ) ,
m = 1 M c m ( z 0 ) = 1 .
W ψ ( f ; z 0 ) = ψ ( f ; z 0 ) 2 ,
σ ψ 2 ( z 0 ) = + + W ψ ( f ; z 0 ) d f ,
σ ψ 2 = 0 L σ ψ 2 ( z 0 ) Δ z d z .
W ψ ( f ; z 0 , R ) = Im [ ψ ( f ; z 0 ) ] 2 + ( 1 2 J 1 ( 2 π R f ) 2 π R f 2 ) Re [ ψ ( f ; z 0 ) ] 2 ,
S ( R ) = x < Re U * ( x ; L ) U ref ( x ; L ) d x 2 [ x < R U ( x ; L ) 2 d x ] [ x < R U ref ( x ; L ) 2 d x ] ,
S ( R ) exp ( σ ψ 2 ) ,
ψ ( f ; z 0 ) = exp [ j π λ ( L z 0 ) f 2 ] G ( f ; z 0 ) ϕ 0 ( f ; z 0 ) ,
W ψ ( f ; z 0 ) = G ( f ; z 0 ) 2 W ϕ ( f ; z 0 ) ,
G ( f ; z 0 ) = 1 m = 1 M exp [ j π λ ( z 0 z m ) f 2 ] g m ( f ; z 0 ) ,
G ( f ; z 0 ) 2 = 1 2 m = 1 M cos [ π λ ( z 0 z m ) f 2 ] g m ( f ; z 0 ) + m , n = 1 M cos [ π λ ( z n z m ) f 2 ] g m ( f ; z 0 ) g n ( f ; z 0 )
G ( f ; z 0 ) 2 = 1 2 a T ( f ; z 0 ) g ( f ; z 0 ) + g T ( f ; z 0 ) A ( f ) g ( f ; z 0 ) ,
g ° ( f ; z 0 ) = A # ( f ) a ( f ; z 0 ) ,
min g G ( f ; z 0 ) 2 = 1 a T ( f ; z 0 ) A # ( f ) a ( f ; z 0 )
G ( f ; z 0 ) 2 min g G ( f ; z 0 ) 2 = [ g ( f ; z 0 ) g ° ( f ; z 0 ) ] T A ( f ) [ g ( f ; z 0 ) g ° ( f ; z 0 ) ] .
D ( z ; z 0 ) = + + 2 [ 1 cos ( π λ z f 2 ) ] W ϕ ( f ; z 0 ) d f ,
σ ψ 2 ( z 0 ) = m = 1 M D ( z 0 z m ; z 0 ) c m ( z 0 ) 1 2 m , n = 1 M D ( z n z m ; z 0 ) c m ( z 0 ) c n ( z 0 ) .
σ ψ 2 ( z 0 ) = d T ( z 0 ) c ( z 0 ) 1 2 c T ( z 0 ) D ( z 0 ) c ( z 0 ) ,
c ° ( z 0 ) = D 1 ( z 0 ) d ( z 0 ) + 1 1 T D 1 ( z 0 ) d ( z 0 ) 1 T D 1 ( z 0 ) 1 D 1 ( z 0 ) 1 ,
min c σ ψ 2 ( z 0 ) = 1 2 d T ( z 0 ) D 1 ( z 0 ) d ( z 0 ) 1 2 [ 1 1 T D 1 ( z 0 ) d ( z 0 ) ] 2 1 T D 1 ( z 0 ) 1 .
D ( z ; z 0 ) = 0.563 ( 2 π λ ) 7 6 2 7 6 z 5 6 C n 2 ( z 0 ) Δ z
σ ψ 2 ( z 0 ) = 0.563 ( 2 π λ ) 7 6 F ( z 0 ) C n 2 ( z 0 ) Δ z ,
σ ψ 2 = 0.563 ( 2 π λ ) 7 6 0 L F ( z ) C n 2 ( z ) d z ,
F ( z ) = m = 1 M 2 7 6 z z m 5 6 c m ( z ) 1 2 m , n = 1 M 2 7 6 z n z m 5 6 c m ( z ) c n ( z ) .
F ( z ) = e T ( z ) c ( z ) 1 2 c T ( z ) E ( z ) c ( z ) ,
c ° ( z 0 ) = E 1 ( z 0 ) e ( z 0 ) + 1 1 T E 1 ( z 0 ) e ( z 0 ) 1 T E 1 ( z 0 ) 1 E 1 ( z 0 ) 1 ,
min c F ( z ) = 1 2 e T ( z ) E 1 ( z ) e ( z ) 1 2 [ 1 1 T E 1 ( z ) e ( z ) ] 2 1 T E 1 ( z ) 1 .
g ° ( f ; z 0 ) = cos [ π λ ( z 0 z 1 ) f 2 ] ,
min g G ( f ; z 0 ) 2 = sin 2 [ π λ ( z 0 z 1 ) f 2 ] ,
min g σ ψ 2 ( z 0 ) = σ χ 2 ( z 0 ; z 1 ) .
G ( f ; z 0 ) 2 = 2 { 1 cos [ π λ ( z 0 z 1 ) f 2 ] } ,
σ ψ 2 ( z 0 ) = D ( z 0 z 1 ; z 0 ) .
σ ψ 2 ( z 0 ) = 2 7 6 σ χ 2 ( z 0 ; z 1 )
g ( f ; z 0 ) = cos [ π λ ( z 0 z 1 ) f 2 ] .
G ( f ; z 0 ) 2 = sin 2 [ π λ ( z 0 z 1 ) f 2 ] + { cos [ π λ ( z 0 z 1 ) f 2 ] cos [ π λ ( z 0 z 1 ) f 2 ] } 2 .
σ ψ 2 ( z 0 ) σ χ 2 ( z 0 ; z 1 ) = 1 + 2 1 6 λ λ λ 5 6 [ 1 + 5 72 λ λ λ + λ 7 6 F 2 3 ( 7 12 , 1 , 13 12 ; 3 2 , 2 ; λ λ λ + λ 2 ) ]
g ° ( f ; z 0 ) = 1 sin [ π λ ( z 2 z 1 ) f 2 ] [ sin [ π λ ( z 2 z 0 ) f 2 ] sin [ π λ ( z 0 z 1 ) f 2 ] ] ,
min g G ( f ; z 0 ) 2 = 0 ,
g ° ( f ; z 0 ) = 1 4 [ cos [ π λ ( z 0 z 1 ) f 2 ] cos [ π λ ( z 0 z 2 ) f 2 ] ] + ( 1 ) N 1 4 [ cos [ π λ ( z 0 z 2 ) f 2 ] cos [ π λ ( z 0 z 1 ) f 2 ] ] ,
min g G ( f ; z 0 ) 2 = sin 2 [ π λ ( z 0 z 1 ) f 2 ] ,
lim f 0 g ° ( f ; z 0 ) = 1 z 2 z 1 [ z 2 z 0 z 0 z 1 ] .
c ° ( z 0 ) = [ 1 2 1 2 ] + z 0 z 2 5 6 z 0 z 1 5 6 z 1 z 2 5 6 [ 1 2 1 2 ] ,
min c F ( z 0 ) 2 7 6 z 1 z 2 5 6 = z 0 z 1 z 1 z 2 5 6 z 0 z 2 z 1 z 2 5 6 1 4 ( 1 z 0 z 1 z 1 z 2 5 6 z 0 z 2 z 1 z 2 5 6 ) 2 .
g ( k ; z 0 ) = 1 sin [ λ ( z 2 z 1 ) k 2 4 π ] [ sin [ λ ( z 2 z 0 ) k 2 4 π ] sin [ λ ( z 0 z 1 ) k 2 4 π ] ]
G ( f ; z 0 ) 2 = ( { sin [ π λ ( z 0 z 1 ) f 2 ] sin [ π λ ( z 0 z 1 ) f 2 ] } sin [ π λ ( z 0 z 1 ) f 2 ] sin [ π λ ( z 2 z 1 ) f 2 ] { sin [ π λ ( z 2 z 1 ) f 2 ] sin [ π λ ( z 2 z 1 ) f 2 ] } ) 2 + ( { cos [ π λ ( z 0 z 1 ) f 2 ] cos [ π λ ( z 0 z 1 ) f 2 ] } sin [ π λ ( z 0 z 1 ) f 2 ] sin [ π λ ( z 2 z 1 ) f 2 ] { cos [ π λ ( z 2 z 1 ) f 2 ] cos [ π λ ( z 2 z 1 ) f 2 ] } ) 2 .
Ξ ( z ; z 0 ) = + + cos ( π λ z f 2 ) W ϕ ( f ; z 0 ) d f ,
σ ψ 2 ( z 0 ) = Ξ ( 0 ; z 0 ) 2 m = 1 M Ξ ( z 0 z m ; z 0 ) c m ( z 0 ) + m , n = 1 M Ξ ( z n z m ; z 0 ) c m ( z 0 ) c n ( z 0 ) .
σ ψ 2 ( z 0 ) = [ 1 m = 1 M c m ( z 0 ) ] 2 Ξ ( 0 ; z 0 ) + m = 1 M D ( z 0 z m ; z 0 ) c m ( z 0 ) 1 2 m , n = 1 M D ( z n z m ; z 0 ) c m ( z 0 ) c n ( z 0 ) ,
D ( z ; z 0 ) = ( 2 π ) 5 3 Γ ( 8 3 ) sin ( π 3 ) C n 2 ( z 0 ) Δ z ( 2 π λ ) 2 0 2 [ 1 cos ( π λ z f 2 ) ] f 8 3 d f .
D ( z ; z 0 ) = Γ ( 8 3 ) sin ( π 3 ) C n 2 ( z 0 ) Δ z ( 2 π λ ) 7 6 z 5 6 0 2 [ 1 cos ( θ ) ] ( 2 θ ) 11 6 d θ .
D ( z ; z 0 ) = Γ ( 8 3 ) sin ( π 3 ) C n 2 ( z 0 ) Δ z ( 2 π λ ) 7 6 z 5 6 [ 3 π 1 2 2 2 3 Γ ( 7 12 ) 5 Γ ( 11 12 ) ] ,
G ( f ; z 0 ) 2 = sin 2 [ π λ ( z 0 z 1 ) f 2 ] + 4 sin 2 [ π ( 1 a ) λ ( z 0 z 1 ) f 2 ] sin 2 [ π a λ ( z 0 z 1 ) f 2 ] ,
σ ψ 2 ( z 0 ) = ( 2 π ) 5 3 Γ ( 8 3 ) sin ( π 3 ) C n 2 ( z 0 ) Δ z ( 2 π λ ) 2 × 0 { sin 2 [ π λ ( z 0 z 1 ) f 2 ] + 4 sin 2 [ π ( 1 a ) λ ( z 0 z 1 ) f 2 ] sin 2 [ π a λ ( z 0 z 1 ) f 2 ] } f 8 3 d f .
σ ψ 2 ( z 0 ) = Γ ( 8 3 ) sin ( π 3 ) C n 2 ( z 0 ) Δ z ( 2 π λ ) 7 6 z 0 z 1 5 6 × 0 { sin 2 ( θ ) + 4 sin 2 [ ( 1 a ) θ ] sin 2 ( a θ ) } ( 2 θ ) 11 6 d θ .
σ ψ 2 ( z 0 ) σ χ 2 ( z 0 ; z 1 ) = 1 + 2 a 5 6 [ 1 + 5 72 a 1 a 7 6 F 2 3 ( 7 12 , 1 , 13 12 ; 3 2 , 2 ; a 1 a 2 ) ] ,

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