Abstract

The concept that the focus anisoplanatism effect, which limits the useful diameter of an adaptive-optics system that relies on an artificial guide star [(AGS), a laser atmospheric backscatter spot] as a reference source for determining the turbulence-induced wave-front distortion, can be eliminated (or greatly reduced) by use of a multiplicity of AGS spots is evaluated. The case of an infinite density of such spots with an infinite density of wave-front sensor subapertures (each infinitely small) is analyzed assuming that performance is limited only by the fact that turbulence is distributed along the propagation path rather than being contained in a single plane. It is found that even in this case focus anisoplanatism limits performance. Relative to what can be achieved with a single AGS spot, it is found that at most approximately a factor-of-2.5 increase in the useful aperture diameter can be obtained by use of infinitely many AGS spots and that this increase is available only for a laser backscatter altitude as high as the 90-km mesospheric sodium layer.

© 1995 Optical Society of America

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References

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  1. J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  2. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  3. D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
    [CrossRef]
  4. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
    [CrossRef]
  5. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
    [CrossRef]
  6. G. A. Tyler, “Rapid evaluation of d0: the effective diameter of a laser-guide-star adaptive-optics system,” J. Opt. Soc. Am. A 11, 325–338 (1994).
    [CrossRef]
  7. The term focus anisoplanatism was coined by Tom O’Meara (Hughes Research Laboratories, Malibu, Calif.) in approximately 1980 and has been in general use in the interested community since then, although the strict suitability of the phrase might be questioned. As a more suitable terminology we might use the words range-dependent anisoplanatism, but use of the phrase focus anisoplanatism has become so customary that we will not undertake to advocate its replacement. (In retrospect, it seems that the nomenclature applied to almost all the various anisoplanatic effects in adaptive optics has been less than optimal.)
  8. R. Benedict, J. B. Breckinridge, D. L. Fried, eds., Feature issue, “Atmospheric-Compensation Technology,” J. Opt. Soc. Am. A11, 255–451, 779–945 (1994).This double-issue Feature (January and February 1994) contains 30 papers on the subject of adaptive optics, with a substantial number being concerned with matters related to the use of an artificial guide star.
    [CrossRef]
  9. R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).
  10. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [CrossRef]
  11. D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,” J. Opt. Soc. Am. 58, 961–969 (1968), App. B.
    [CrossRef]
  12. D. L. Fried, “Turbulence-induced variation in the apparent separation of a pair of point sources,” Rep. TR-721R (Optical Sciences Company, Placentia, Calif., 1989), App. A.
  13. D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications,” in Proceedings of the NATO Conference on Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer, Dordrecht, The Netherlands, 1994), pp. 25–57.
    [CrossRef]
  14. D. L. Fried, J. L. Vaughn, “M-method performance for M = 3,” Rep. TR-995R (Optical Sciences Company, Placentia, Calif., 1989).

1994

1991

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

1990

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

1985

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

1982

1978

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1968

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Belsher, J. F.

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Fried, D. L.

D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

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

D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,” J. Opt. Soc. Am. 58, 961–969 (1968), App. B.
[CrossRef]

D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications,” in Proceedings of the NATO Conference on Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer, Dordrecht, The Netherlands, 1994), pp. 25–57.
[CrossRef]

D. L. Fried, J. L. Vaughn, “M-method performance for M = 3,” Rep. TR-995R (Optical Sciences Company, Placentia, Calif., 1989).

D. L. Fried, “Turbulence-induced variation in the apparent separation of a pair of point sources,” Rep. TR-721R (Optical Sciences Company, Placentia, Calif., 1989), App. A.

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Hardy, J.

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Tyler, G. A.

Vaughn, J. L.

D. L. Fried, J. L. Vaughn, “M-method performance for M = 3,” Rep. TR-995R (Optical Sciences Company, Placentia, Calif., 1989).

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Astron. Astrophys.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Proc. IEEE

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

J. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other

D. L. Fried, “Turbulence-induced variation in the apparent separation of a pair of point sources,” Rep. TR-721R (Optical Sciences Company, Placentia, Calif., 1989), App. A.

D. L. Fried, “Atmospheric turbulence optical effects: understanding the adaptive-optics implications,” in Proceedings of the NATO Conference on Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer, Dordrecht, The Netherlands, 1994), pp. 25–57.
[CrossRef]

D. L. Fried, J. L. Vaughn, “M-method performance for M = 3,” Rep. TR-995R (Optical Sciences Company, Placentia, Calif., 1989).

The term focus anisoplanatism was coined by Tom O’Meara (Hughes Research Laboratories, Malibu, Calif.) in approximately 1980 and has been in general use in the interested community since then, although the strict suitability of the phrase might be questioned. As a more suitable terminology we might use the words range-dependent anisoplanatism, but use of the phrase focus anisoplanatism has become so customary that we will not undertake to advocate its replacement. (In retrospect, it seems that the nomenclature applied to almost all the various anisoplanatic effects in adaptive optics has been less than optimal.)

R. Benedict, J. B. Breckinridge, D. L. Fried, eds., Feature issue, “Atmospheric-Compensation Technology,” J. Opt. Soc. Am. A11, 255–451, 779–945 (1994).This double-issue Feature (January and February 1994) contains 30 papers on the subject of adaptive optics, with a substantial number being concerned with matters related to the use of an artificial guide star.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Q(z) presentation. The function Q(z) is defined by Eq. (44). If plotted directly in log–log form, it appears very nearly to be a straight line with a −2 slope. In fact, it is very close to having a −5/3 slope for small values of z but is close to having a −2 slope for large values of z. By plotting the function [Q(z) − 1]z11/6 as is done here, we bring out the details of Q(z).

Fig. 2
Fig. 2

F(X) presentation. The function F(X) is defined by Eq. (49). The function is very nearly equal to unity for small values of X and goes to zero very rapidly for X much greater than approximately 0.3. It is this rapid transition from its near-unity value to a nearly zero value that establishes the role that F(X) plays in Eq. (48) in setting the value of G(x). What really matters are that the function falls off rapidly and just where the transition point is. By plotting 1 − F(X) [rather than F(X) itself] we place emphasis on these features and distance our attention from irregularities in the shape of the curve in its rapid falloff Also shown here is the approximation to F(X) defined by Eq. (53). It is this approximation that was used in calculating values for G(x).

Fig. 3
Fig. 3

G(x) presentation. The function G(x) is defined by Eq. (48). The straight line is a simple x5/3 power law, which is shown as being asymptotic to G(x). The values of G(x) are calculated by use of the approximation for F(X) given by Eq. (53).

Fig. 4
Fig. 4

Presentation of g↓1(x) and g↓2(x). The two functions are defined by Eqs. (52). Each is shown after multiplication by 2/β. The lower straight line represents g↓1(x) [or rather 2 g↓1(x)/β]. It has a slope of −2. The curve represents g↓2(x) [or rather G(x) = 2g↓2(x)/β]. The upper (nearly) straight line represents the sum of the two curves, with a slope of −5/3. It is interesting to note how the comparative significance of the two effects associated g↓1(x) with and with g↓2(x) [going back to the distinction made in Eq. (12) between ↓1 and ↓2] changes at a range approximately equal to one sixth of the backscatter range.

Fig. 5
Fig. 5

Ξ (x) presentation. The function Ξ (x) is defined by Eq. (55) for the case analyzed here, in which an essentially infinite density of AGS reference spots is assumed, whereas Ξ (x) is defined by Eq. (57), as obtained from Refs. 3 and 11, and applies when there is only a single AGS reference spot. The Ξ (x) curve for the single AGS reference spot is the upper curve of the two. For small values of x this curve is approximately a factor of 6 larger than the Ξ (x) curve for the infinity of AGS reference spots. The quantity x should be understood as corresponding to the normalized range, i.e., the range s over the backscatter range S. The quantity Ξ (x) determines the mean-square phase error σ2, as given by Eq. (54), and determines the effective aperture diameter d0, as given by Eq. (59).

Fig. 6
Fig. 6

Refractive-index structure constant C N 2. The results shown here are for the so-called Hufnagel–Valley5/7 turbulence model.

Fig. 7
Fig. 7

Sample results for d0. Results are calculated for a propagation path (from the ground) going straight up through the atmosphere, with a zenith angle of zero. The wavelength is λ = 1.0 μm. The results shown here are calculated with Eq. (59). The upper of the two curves corresponds to the use of an infinitely dense array of AGS reference spots, whereas the lower curve corresponds to the use of only a single AGS reference spot. Even at the highest altitude (100 km) the separation between the two d0 values corresponds to a factor of only approximately 2.5. If shown on a log–log plot the two curves would be seen to be nearly parallel at the higher altitudes, implying that this factor of 2.5 would not be improved even if we could somehow work with a higher AGS laser backscatter altitude. Still, the difference between 6 and 15 m is not really trivial. (Analysis that is not presented here suggests that most of this factor-of-2.5 advantage could be obtained by use of just six, rather than an infinity of, AGS laser reference spots.)

Equations (59)

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[ n ( r , s ) n ( r , s ) ] 2 = C N 2 ( s ¯ ) [ | r r | 1 + ( s s ) 2 ] 1 / 3 ,
ϕ ( r ) = k 0 d s n ( r , s ) ,
W ( x ) = { 1 if | x | ½ 0 if otherwise .
( r ) = [ ϕ ( r ) φ ( r ) ] ( 1 4 π D 2 ) 1 d r W ( r ) × [ 1 + ( 4 / D ) 2 ( r · r ) ] [ ϕ ( r ) φ ( r ) ] = ( 1 4 π D 2 ) 1 d r W ( r ) [ 1 + ( 4 / D ) 2 ( r · r ) ] × { [ ϕ ( r ) φ ( r ) ] [ ϕ ( r ) φ ( r ) ] } .
σ 2 = ( 1 4 π D 2 ) 1 d r W ( r / D ) | ( r ) | 2 .
μ ( x ) = { 1 x if 0 x 1 0 if otherwise .
n ( r , s ) = r n ( r , s ) .
ν ( x ) = { x if 0 x 1 0 if otherwise .
n [ μ ( s / S ) r + ν ( s / S ) r , s ] = [ 1 / ν ( s / S ) ] r n [ μ ( s / S ) r + ν ( s / S ) r , s ] .
r φ ( r ) = k d s μ ( s / S ) r n ( r , s ) k ( 1 4 π D 2 ) 1 × d r W ( r / D ) d s [ μ ( s / S ) / ν ( s / S ) ] × r n [ μ ( s / S ) r + ν ( s / S ) r , s ] = r { k d s μ ( s / S ) n ( r , s ) k ( 1 4 π D 2 ) 1 × d r W ( r / D ) d s [ μ ( s / S ) / ν ( s / S ) ] × n [ μ ( s / S ) r + ν ( s / S ) r , s ] } .
φ ( r ) = k d s μ ( s / S ) n ( r , s ) k ( 1 4 π D 2 ) 1 × d r W ( r / D ) d s [ μ ( s / S ) / ν ( s / S ) ] × n [ μ ( s / S ) r + ν ( s / S ) r , s ] .
( r ) = 1 ( r ) + 2 ( r ) + ( r ) .
( r ) = ( 1 4 π D 2 ) 1 d r W ( r / D ) [ 1 + ( 4 / D ) 2 ( r · r ) ] × k S d s [ n ( r , s ) n ( r , s ) ] .
1 ( r ) = ( 1 4 π D 2 ) 1 d r W ( r / D ) [ 1 + ( 4 / D ) 2 ( r · r ) ] × { [ k d s n ( r , s ) k d s μ ( s / S ) n ( r , s ) ] [ k d s n ( r , s ) k d s μ ( s / S ) n ( r , s ) ] } = ( 1 4 π D 2 ) 1 k d r W ( r / D ) [ 1 + ( 4 / D ) 2 ( r · r ) ] × d s ν ( s / S ) [ n ( r , s ) = n ( r , s ) ] ,
2 ( r ) = ( 1 4 π D 2 ) 1 d r W ( r / D ) [ 1 + ( 4 / D ) 2 ( r · r ) ] × ( 1 4 π D 2 ) 1 k d r W ( r / D ) × { d s [ μ ( s / S ) / ν ( s / S ) ] n [ μ ( s / S ) r + ν ( s / S ) r , s ] d s [ μ ( s / S ) / ν ( s / S ) ] n [ μ ( s / S ) r + ν ( s / S ) r , s ] } = ( 1 4 π D 2 ) 2 d r d r W ( r / D ) W ( r / D ) × [ 1 + ( 4 / D ) 2 ( r · r ) ] d s [ μ ( s / S ) / ν ( s / S ) ] × { n [ μ ( s / S ) r + ν ( s / S ) r , s ] n [ μ ( s / S ) r + ν ( s / S ) r , s ] } .
σ 2 = ( 1 4 π D 2 ) 1 d r W ( r / D ) [ | ( r ) | 2 + | 1 ( r ) | 2 + | 2 ( r ) | 2 .
σ 2 = σ 2 + σ 1 2 + σ 2 2 ,
σ 2 = ( 1 4 π D 2 ) 1 d r W ( r / D ) | ( r ) | 2 = k 2 ( 1 4 π D 2 ) 3 S S d s d s d r d r 1 d r 2 × W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) [ 1 + ( 4 / D ) 2 ( r · r 1 ) ] × [ 1 + ( 4 / D ) 2 ( r · r 2 ) ] [ n ( r , s ) n ( r 1 , s ) ] × [ n ( r , s ) n ( r 2 , s ) ] ,
σ 1 2 = ( 1 4 π D 2 ) 1 d r W ( r / D ) | 1 ( r ) | 2 = k 2 ( 1 4 π D 2 ) 3 d s d s d r d r 1 d r 2 × W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) [ 1 + ( 4 / D ) 2 ( r · r 1 ) ] × [ 1 + ( 4 / D ) 2 ( r · r 2 ) ] { ν ( s / S ) [ n ( r , s ) n ( r 1 , s ) ] } × { ν ( s / S ) [ n ( r , s ) n ( r 2 , s ) ] } ,
σ 2 2 = ( 1 4 π D 2 ) 1 d r W ( r / D ) | 2 ( r ) | 2 = k 2 ( 1 4 π D 2 ) 5 d s d s d r d r 1 d r 2 d r 3 d r 4 × W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) W ( r 3 / D ) W ( r 4 / D ) × [ 1 + ( 4 / D ) 2 ( r · r 2 ) ] [ 1 + ( 4 / D ) 2 ( r · r 4 ) ] × ( [ μ ( s / S ) / ν ( s / S ) ] { n [ μ ( s / S ) r 1 + ν ( s / S ) r , s ] n [ μ ( s / S ) r 1 + ν ( s / S ) r 2 , s ] } ) ( [ μ ( s / S ) / ν ( s / S ) ] × { n [ μ ( s / S ) r 3 + ν ( s / S ) r , s ] n [ μ ( s / S ) r 3 + ν ( s / S ) r 4 , s ] } ) .
d x [ ( 1 + x 2 ) α ( x 2 ) α ] = 2 α 2 α + 1 Γ ( ½ ) Γ ( ½ α ) Γ ( 1 α ) , ( α ) < ½
σ 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 3 S d s ¯ C N 2 ( s ¯ ) × d r d r 1 d r 2 W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) × [ 1 + ( 4 / D ) 2 ( r · r 1 ) + ( 4 / D ) 2 ( r · r 2 ) + ( 4 / D ) 4 ( r · r 1 ) ( r · r 2 ) ] ( | r r 2 | 5 / 3 | r r 2 | 5 / 3 + | r 1 r 2 | 5 / 3 )
σ 1 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 3 d s ¯ [ ν ( s ¯ / S ) ] 2 C N 2 ( s ¯ ) × d r d r 1 d r 2 W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) × [ 1 + ( 4 / D ) 2 ( r · r 1 ) + ( 4 / D ) 2 ( r · r 2 ) + ( 4 / D ) 4 ( r · r 1 ) ( r · r 2 ) ] ( | r r 2 | 5 / 3 | r r 1 | 5 / 3 + | r 1 r 2 | 5 / 3 ) .
σ 2 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 5 × d s ¯ [ μ ( s ¯ / S ) ] 2 [ ν ( s ¯ / S ) ] 2 C N 2 ( s ¯ ) × d r d r 1 d r 2 d r 3 d r 4 W ( r / D ) W ( r 1 / D ) W ( r 2 / D ) × W ( r 3 / D ) W ( r 4 / D ) [ 1 + ( 4 / D ) 2 ( r · r 2 ) + ( 4 / D ) 2 ( r · r 4 ) + ( 4 / D ) 4 ( r · r 2 ) ( r · r 4 ) ] { [ μ ( s ¯ / S ) 5 / 3 | r 1 r 3 | 5 / 3 | μ ( s ¯ / S ) r 1 + ν ( s ¯ / S ) r μ ( s ¯ / S ) r 3 ν ( s ¯ / S ) r 4 | 5 / 3 | μ ( s ¯ / S ) r 1 + ν ( s ¯ / S ) r 1 μ ( s ¯ / S ) r 3 ν ( s ¯ / S ) r | 5 / 3 | μ ( s ¯ / S ) r 1 + ν ( s ¯ / S ) r 2 μ ( s ¯ / S ) r 3 ν ( s ¯ / S ) r 4 | 5 / 3 } .
σ 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 2 S d s C N 2 ( s ) × d r 1 d r 2 W ( r 1 / D ) W ( r 2 / D ) × [ 1 + ( 4 / D ) 2 ( r 1 · r 2 ) ] | r 1 r 2 | 5 / 3 = ( 1 2 ) 2.9144 k 2 D 5 / 3 S d s C N 2 ( s ) ( 1 4 π ) 2 × d r 1 d r 2 W ( r 1 ) W ( r 2 ) × [ 1 + 16 ( r 1 · r 2 ) ] | r 1 r 2 | 5 / 3
σ 1 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 2 0 S d s [ ν ( s / S ) ] 2 C N 2 ( s ) × d r 1 d r 2 W ( r 1 / D ) W ( r 2 / D ) × [ 1 + ( 4 / D ) 2 ( r 1 · r 2 ) ] | r 1 r 2 | 5 / 3 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ ν ( s / S ) ] 2 C N 2 ( s ) ( 1 4 π ) 2 × d r 1 d r 2 W ( r 1 ) W ( r 2 ) × [ 1 + 16 ( r 1 · r 2 ) ] | r 1 r 2 | 5 / 3 .
σ 2 2 = ( 1 2 ) 2.9144 k 2 ( 1 4 π D 2 ) 2 0 S d s [ μ ( s / S ) ] 11 / 3 × [ ν ( s / S ) ] 2 C N 2 ( s ) d r 1 d r 3 W ( r 1 / D ) W ( r 3 / D ) × { | r 1 r 3 | 5 / 3 + ( 1 4 π D 2 ) 2 d r 2 d r 4 W ( r 2 / D ) × W ( r 4 / D ) [ 1 + ( 4 / D ) 2 ( r 2 · r 4 ) ] × | r 1 r 3 + η ( s / S ) ( r 2 r 4 ) | 5 / 3 } = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ μ ( s / S ) ] 11 / 3 × [ ν ( s / S ) ] 2 C N 2 ( s ) ( 1 4 π ) 2 d r 1 d r 3 W ( r 1 ) W ( r 3 ) × { | r 1 r 3 | 5 / 3 + ( 1 4 π ) 2 d r 2 d r 4 W ( r 2 ) × W ( r 4 ) [ 1 + 16 ( r 2 · r 4 ) ] × | r 1 r 3 + η ( s / S ) ( r 2 r 4 ) | 5 / 3 } ,
η ( s / S ) = ν ( s / S ) / μ ( s / S ) .
( 1 4 π ) 1 d x W ( x + 1 2 X ) W ( x 1 2 X ) ( x 2 1 4 X 2 ) = 1 8 ( 2 / π ) [ cos 1 ( X ) X ( 3 2 X 2 ) 1 X 2 ]
( 1 4 π ) 1 d x W ( x + 1 2 X ) W ( x 1 2 X ) = ( 2 / π ) [ cos 1 ( X ) X 1 X 2 ] ,
σ 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 S d s C N 2 ( s ) ( 16 / π ) × 0 1 d X X 8 / 3 [ 3 cos 1 ( X ) X ( 7 4 X 2 ) 1 X 2 ] = ( 1 2 ) 2.9144 k 2 D 5 / 3 S d s C N 2 ( s ) β ,
σ 1 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ ν ( s / S ) ] 2 C N 2 ( s ) ( 16 / π ) × 0 1 d X X 8 / 3 [ 3 cos 1 ( X ) X ( 7 4 X 2 ) 1 X 2 ] = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ ν ( s / S ) ] 2 C N 2 ( s ) β ,
β = ( 1 4 π ) 2 d r 2 d r 4 W ( r 2 ) W ( r 4 ) ( 1 + 16 r 2 · r 4 ) × | r 2 r 4 | 5 / 3 = ( 16 / π ) 0 1 d X X 8 / 3 [ 3 cos 1 ( X ) X ( 7 4 X 2 ) 1 X 2 ] = 0.0391243738 .
f ( X , z ) = ( 1 4 π ) 2 d r 2 d r 4 W ( r 2 ) W ( r 4 ) [ 1 + 16 ( r 2 · r 4 ) ] × | 1 X + η ( z ) ( r 2 r 4 ) | 5 / 3 ,
σ 2 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ μ ( s / S ) ] 11 / 3 [ ν ( s / S ) ] 2 × C N 2 ( s ) ( 1 4 π ) 2 d r 1 d r 3 W ( r 1 ) W ( r 3 ) × [ | r 1 r 3 | 5 / 3 + f ( | r 1 r 3 | , s / S ) ] .
f ( X , z ) = ( 1 4 π ) 2 d x d x W ( x + 1 2 x ) W ( x 1 2 x ) × [ 1 + 16 ( x 2 1 4 x 2 ) ] | 1 X + η ( z ) x | 5 / 3 = ( 1 4 π ) 1 d x | 1 X + η ( z ) x | 5 / 3 × ( 2 / π ) [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 ρ 2 ] .
( 2 π ) 1 0 2 π d θ ( a 2 + 2 a b cos θ + b 2 ) 5 / 6 = I 0 ( a , b ) ,
I 0 ( a , b ) = ( a b ) 5 / 6 { J 0 ( a / b ) if a < b J 0 ( b / a ) if a b ,
J 0 ( α ) = α 5 / 6 n = 0 [ Γ ( n 5 / 6 ) Γ ( n + 1 ) Γ ( 5 / 6 ) ] 2 α 2 n for α 1 .
f ( X , z ) = ( 16 / π ) 0 1 x d x I 0 [ X , η ( z ) x ] × [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] .
σ 2 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ μ ( s / S ) ] 11 / 3 × [ ν ( s / S ) ] 2 C N 2 ( s ) ( 16 / π ) 0 1 X d X × [ cos 1 ( X ) X 1 X 2 ] [ f ( X , s / S ) X 5 / 3 ] .
( 16 / π ) 0 1 x d x [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] = 1 ,
σ 2 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ μ ( s / S ) ] 11 / 3 [ ν ( s / S ) ] 2 × C N 2 ( s ) β ( 16 / π ) 0 1 X d X [ cos 1 ( X ) X 1 X 2 ] × { ( ( 16 / π ) 0 1 x d x { I 0 [ X , η ( s / S ) x ] X 5 / 3 } [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] ( 16 / π ) 0 1 x d x x 5 / 3 [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] ) } .
Q ( x ) = { x 5 / 3 n = 0 [ Γ ( n 5 / 6 ) Γ ( n + 1 ) Γ ( 5 / 6 ) ] 2 ( x 2 ) n x < 1 n = 0 [ Γ ( n 5 / 6 ) Γ ( n + 1 ) Γ ( 5 / 6 ) ] 2 ( x 2 ) n x 1 ,
I 0 ( a , b ) = a 5 / 3 Q ( a / b ) .
σ 2 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s [ μ ( s / S ) ] 11 / 3 [ ν ( s / S ) ] 2 × C N 2 ( s ) β ( 16 / π ) 0 1 X d X X 5 / 3 [ cos 1 ( X ) X 1 X 2 ] × { ( ( 16 / π ) 0 1 x d x { Q [ X η ( s / S ) x ] 1 } [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] ( 16 / π ) 0 1 x d x x 5 / 3 [ 3 cos 1 ( x ) x ( 7 4 x 2 ) 1 x 2 ] ) } .
σ 2 2 = ( 1 2 ) 2.9144 k 2 D 5 / 3 0 S d s G ( s / S ) C N 2 ( s ) β ,
G ( x ) = ( 16 / π ) [ ( 1 x ) 2 x 1 / 3 ] 0 1 d t t F [ t ( 1 x ) x ] × [ cos 1 ( x ) x 1 x 2 ] ,
F ( X ) = ( 16 / π ) 0 1 z d z [ Q ( X / z ) 1 ] [ 3 cos 1 ( z ) z ( 7 4 z 2 ) 1 z 2 ] ( 16 / π ) 0 1 z d z ( z / X ) 5 / 3 [ 3 cos 1 ( z ) z ( 7 4 z 2 ) 1 z 2 ] .
σ 2 = 2.9144 k 2 D 5 / 3 S d s g ( s / S ) C N 2 ( s ) ,
σ ( 1 or 2 ) 2 = 2.9144 k 2 D 5 / 3 0 S d s g ( 1 or 2 ) ( s / S ) C N 2 ( s ) ,
g ( x ) = ½ β , g 1 ( x ) = ½ β x 2 , g 2 ( x ) = ½ β G ( x ) .
F ( X ) 1 [ 1 + ( a X p ) 5 / 4 ] 4 / 5 ,
σ 2 = k 2 D 5 / 3 0 d s Ξ ( s / S ) C N 2 ( s ) .
Ξ ( x ) = 2.9144 [ g ( x ) + g 1 ( x ) + g 2 ( x ) ] .
Ξ ( x ) = { Ξ low ( x ) if x 1 Ξ high ( x ) if x > 1 ,
Ξ low ( x ) = A [ 1 + ( 1 x ) 5 / 3 ] B [ 6 / 11 F 1 2 [ 11 / 6 , ; 2 ; ( 1 x ) 2 ] 10 / 11 ( 1 x ) F 1 2 [ 11 / 6 , ; 3 ; ( 1 x ) 2 ] 6 / 11 x 5 / 3 } ,
Ξ high ( x ) = A ,
d 0 = k 6 / 5 [ 0 d s Ξ ( s / S ) C N 2 ( s ) ] 3 / 5 .

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