Abstract

The possibility of sensing the curvature and the slope of a distorted wave front from a single defocused star image is investigated. The suggested technique is similar to the differential curvature-sensing method of Roddier [ R&D note 87-3 ( National Optical Astronomy Observatories, Tucson, Ariz., 1987)] but uses only a single sensor at a point either before or after the focus. The signal-to-noise ratio that is achievable with such a sensor is ultimately limited by atmospheric scintillation to a value of the order of Qr02/λz0, where r0 is Fried’s correlation scale, λ is the wavelength, and z0 is the root-mean-square distance through the atmosphere, weighted by the refractive-index structure constant Cn2. At the best astronomical sites, with an optimal adaptive-optics system, a value of Q50 should be achievable. Adaptive-optics systems that use such a sensor should be capable of achieving an increase in the effective atmospheric correlation scale of a factor of Q6/5; hence a single-image curvature sensor should be practical whenever D/r0Q6/5. This condition is shown to hold at good astronomical sites even for telescopes as large as 8 m and wavelengths as short as 0.5 μm. In addition to optical and mechanical simplicity, the single-image sensor offers the advantage of reduced detector read noise and potentially higher efficiency compared with those from a differential system.

© 1994 Optical Society of America

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  1. F. Roddier, “Curvature sensing: a diffraction theory,” R&D note 87-3 (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 1–5.
  2. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  3. F. Roddier, C. Roddier, N. Roddier, “Curvature sensing: a new wavefront sensing method,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 203–209 (1988).
    [CrossRef]
  4. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  5. A. H. Mikesell, Publ. U.S. Naval Obs., Second Series 17, Part TV, 141 (1955).
  6. S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
    [CrossRef]
  7. A. T. Young, “Photometric error analysis. VIII. The temporal power spectrum of scintillation,” Appl. Opt. 8, 869–885 (1969).
    [CrossRef] [PubMed]
  8. T. Wang, J. Strohbehn, “Log-normal paradox in atmospheric scintillations,” J. Opt. Soc. Am. 64, 583–591 (1974).
    [CrossRef]
  9. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1691 (1975).
    [CrossRef]
  10. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  11. S. F. Clifford, R. J. Hill, “Relation between irradiance and log-amplitude variance for optical scintillation described by the K distribution,” J. Opt. Soc. Am. 71, 112–114 (1981).
    [CrossRef]
  12. M. Borne, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  13. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.
  14. F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
    [CrossRef]
  15. B. L. Ellerbroek, “Adaptive optics performance predictions for large telescope under good seeing conditions,” in Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 411–413.
  16. P. H. Hu, J. Stone, T. Stanley, “Application of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989).
    [CrossRef]
  17. F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
    [CrossRef]
  18. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52–61 (1964).
    [CrossRef]
  19. G. Burley, Department of Geophysics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada (personal communication, 1993).

1991 (1)

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

1989 (1)

1988 (1)

1981 (1)

1979 (1)

1975 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1691 (1975).
[CrossRef]

1974 (1)

1969 (1)

1966 (1)

1964 (1)

1963 (1)

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

1955 (1)

A. H. Mikesell, Publ. U.S. Naval Obs., Second Series 17, Part TV, 141 (1955).

Azouit, M.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Beland, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Borne, M.

M. Borne, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Burley, G.

G. Burley, Department of Geophysics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada (personal communication, 1993).

Caccia, J. L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Clifford, S. F.

Cowie, L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Cowley, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, “Adaptive optics performance predictions for large telescope under good seeing conditions,” in Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 411–413.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1691 (1975).
[CrossRef]

Fried, D. L.

Graves, J.

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Graves, J. E.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Hill, R. J.

Hill, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Hu, P. H.

Hufnagel, R. E.

Limburg, E.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

McKenna, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Mikesell, A. H.

A. H. Mikesell, Publ. U.S. Naval Obs., Second Series 17, Part TV, 141 (1955).

Northcott, M.

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Parry, G.

Pusey, P. N.

Reiger, S. H.

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Roddier, C.

F. Roddier, C. Roddier, N. Roddier, “Curvature sensing: a new wavefront sensing method,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 203–209 (1988).
[CrossRef]

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Roddier, F.

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef] [PubMed]

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

F. Roddier, C. Roddier, N. Roddier, “Curvature sensing: a new wavefront sensing method,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 203–209 (1988).
[CrossRef]

F. Roddier, “Curvature sensing: a diffraction theory,” R&D note 87-3 (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 1–5.

Roddier, N.

F. Roddier, C. Roddier, N. Roddier, “Curvature sensing: a new wavefront sensing method,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 203–209 (1988).
[CrossRef]

Salmon, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Songaila, A.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Stanley, N. R.

Stanley, T.

Stone, J.

Strohbehn, J.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.

Vernin, J.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Wang, T.

Wolf, E.

M. Borne, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Young, A. T.

Appl. Opt. (2)

Astron. J. (1)

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1691 (1975).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Publ. U.S. Naval Obs. (1)

A. H. Mikesell, Publ. U.S. Naval Obs., Second Series 17, Part TV, 141 (1955).

Other (7)

G. Burley, Department of Geophysics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada (personal communication, 1993).

F. Roddier, “Curvature sensing: a diffraction theory,” R&D note 87-3 (National Optical Astronomy Observatories, Tucson, Ariz., 1987), pp. 1–5.

F. Roddier, C. Roddier, N. Roddier, “Curvature sensing: a new wavefront sensing method,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 203–209 (1988).
[CrossRef]

M. Borne, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

B. L. Ellerbroek, “Adaptive optics performance predictions for large telescope under good seeing conditions,” in Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 411–413.

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Figures (1)

Fig. 1
Fig. 1

Geometry for wave-front curvature sensing. The large circle represents the entrance pupil, whose plane intersects the optical axis at point P. Wave-front curvature fluctuations produce spatial intensity fluctuations in defocused star images (indicated by the circles at planes A and B). The small circles indicate the circular subaperture referred to in Section 4 and its projection on the entrance pupil.

Equations (52)

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Δ I I 0 ( r ) = s k β [ 2 ϕ ( r ) δ ( r R ) ϕ ( r ) r ] ,
β = ( f s ) / f
λ s β r 0 2 R 2 ,
Q = Δ S C 2 1 / 2 Δ S A 2 1 / 2 ,
S = K A I ( r ) d 2 r ,
U ( r ) = U 0 exp [ l ( r ) + i ϕ ( r ) ] ,
I ( r ) = U ( r ) U * ( r ) = I 0 exp [ 2 l ( r ) ] ,
S A = K I 0 A exp [ 2 l ( r ) ] d 2 r .
Δ S A 2 = S A 2 S A 2 = K 2 I 0 2 A ( { exp [ 2 l ( r ) ] } × { exp [ 2 l ( r ) ] } { exp [ 2 l ( r ) ] } × { exp [ 2 l ( r ) ] } ) d 2 r d 2 r .
exp ( α ) = exp ( α 2 / 2 α 2 / 2 + α ) ,
l = C l ( 0 ) ,
Δ S A 2 = K 2 I 0 2 A { exp [ 4 C l ( r , r ) ] 1 } d 2 r d 2 r ,
C l ( r , r ) = l ( r ) l ( r ) .
C l ( r , r ) = l ( κ ) exp [ i κ · ( r r ) ] d 2 κ .
( r ) = { 1 if r 1 0 otherwise
Δ S A 2 = K 2 I 0 2 ( r / a ) [ ( x r ) / a ] × { exp [ 4 C l ( x ) ] 1 } d 2 x d 2 r = K 2 I 0 2 W ( x / a ) { exp [ 4 C l ( x ) ] 1 } d 2 x ,
w ( κ ) = π [ 2 J 1 ( κ ) / κ ] 2 ,
Δ S A 2 4 K 2 I 0 2 W ( x / a ) C l ( x ) d 2 x .
Δ S A 2 4 K 2 I 0 2 W ( x / a ) exp ( i κ · x ) d 2 x l ( κ ) d 2 κ = 8 π 2 a 2 K 2 I 0 2 0 [ 2 J 1 ( κ a ) / κ a ] 2 l ( κ ) κ d κ .
Δ S C = K s I 0 k β A 2 ϕ ( r ) d 2 r .
ϕ ( r ) = ψ ( κ ) exp ( i κ · r ) d 2 κ ,
2 ϕ ( r ) = κ 2 ψ ( κ ) exp ( i κ · r ) d 2 κ .
C C ( r , r ) = [ 2 ϕ ( r ) ] [ 2 ϕ ( r ) ] = κ 2 κ 2 ψ ( κ ) ψ ( κ ) exp [ i ( κ · r κ · r ) ] × d 2 κ d 2 κ .
ψ ( κ ) ψ ( κ ) = δ ( κ κ ) ϕ ( κ ) ,
C C ( r , r ) = C C ( x ) = κ 4 ϕ ( κ ) exp ( i κ · x ) d 2 κ .
Δ S C 2 = ( K s I 0 k β ) 2 A 2 ϕ ( r ) 2 ϕ ( r ) d 2 r d 2 r = ( K s I 0 k β ) 2 A C C ( x ) d 2 r d 2 r = ( K s I 0 k β ) 2 W ( x / a ) C C ( x ) d 2 x = ( K s I 0 k β ) 2 W ( x / a ) ϕ ( κ ) exp ( i κ · x ) κ 4 d 2 κ d 2 x = 2 π 2 a 2 ( K s I 0 k β ) 2 0 [ 2 J 1 ( κ a ) / κ a ] 2 F ϕ ( κ ) κ 5 d κ .
l ( κ ) = 2 π k 2 Φ n ( κ , z ) sin 2 ( κ 2 z 2 k ) d z , ϕ ( κ ) = 2 π k 2 Φ n ( κ , z ) cos 2 ( κ 2 z 2 k ) d z ,
Φ n ( κ ) = 1 π 2 Γ ( 8 / 3 ) sin ( π / 3 ) C n ( z ) 2 κ 11 / 3 0.033 C n 2 ( z ) κ 11 / 3 ,
Q = s 2 k β ( I C I A ) 1 / 2 ,
I A = 0 C n 2 ( z ) 0 [ 2 J 1 ( κ a ) κ a ] 2 sin 2 ( κ 2 z 2 k ) κ 8 / 3 d κ d z , I C = 0 C n 2 ( z ) 0 [ 2 J 1 ( κ a ) κ a ] 2 cos 2 ( κ 2 z 2 k ) κ 4 / 3 d κ d z .
I A ξ ( 2 k ) 2 a 7 / 3 0 C n 2 ( z ) z 2 d z , I C ξ a 7 / 3 0 C n 2 ( z ) d z ,
ξ = 0 [ 2 J 1 ( x ) x ] 2 x 4 / 3 d x 2.63 .
Q s β z 0 ,
z 0 = [ 0 C n 2 ( z ) z 2 d z / 0 C n 2 ( z ) d z ] 1 / 2 .
λ s r 0 = β a ,
Q = r 0 a λ z 0
Q r 0 D 2 λ z 0 N 1 / 2 ,
Q Q opt = 1.3 r 0 2 λ z 0 .
D = D l + D ϕ ,
D l ( r , r ) = [ l ( r ) l ( r ) ] 2 .
D l ( r , r ) = 2 [ C l ( 0 ) C l ( r , r ) ] = 2 π 0 [ 1 J 0 ( κ x ) ] l κ d κ .
D ( x ) = 6.88 ( x / r 0 ) 5 / 3 ,
Δ S C 2 = Δ S A 2 .
0 [ 2 J 1 ( κ a ) / κ a ] 2 [ ϕ ( κ ) κ 4 G l ( κ ) ] κ d κ = 0 ,
ϕ = G κ 4 l ( κ < 2 / a ) .
D AO ( x ) * ( x / a ) 2 π 2 a 2 0 [ 2 J 1 ( κ a ) / κ a ] 2 [ 1 J 0 ( κ x ) ] × [ G / κ 4 + 1 ] l ( κ ) κ d κ .
G / κ 4 > ( k a 2 β / 2 s ) 2 > ( π a / r 0 ) 2 1 ,
D AO ( x ) * ( x / a ) 0.033 π 3 a 2 G 0 [ 2 J 1 ( κ a ) / κ a ] × [ 1 J 0 ( κ x ) ] κ 8 / 3 d κ 0 C n 2 ( z ) z 2 d z .
D ( r ) = 0.033 ( 8 π 2 ) k 2 0 [ 1 J 0 ( κ r ) ] κ 8 / 3 d κ × 0 C n 2 ( z ) d z ,
D ( x ) * ( x / a ) D AO ( x ) * ( x / a ) = 8 k 2 π G z 0 2 = 2 π ( s β z 0 ) 2 = 2 π Q 2 .
r AO = r 0 3 / 5 r 0 Q 6 / 5 .
D r 0 Q 6 / 5 1.4 ( r 0 2 λ z 0 ) 0 / 5 .

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