Abstract

Modal decomposition of the phase variance between the wave front that is sampled to make a correction and the desired wave front allows inappropriate piston and tilt terms to be excluded from statistical estimates. Mathematical techniques that were developed for calculating Zernike-mode covariances for multiaperture optical systems lead to statistical expressions for angular and focal anisoplanatism and combinations thereof. The 5/3-law dependence that characterizes the angular dependence and also appears in connection with the standard Greenwood frequency is shown to be an artifact of infinite outer scale that is entirely removable by exclusion of the piston. Large piston and/or tilt components significantly affect critical angle and frequency estimates under conditions of practical interest. A finite inner scale or the removal of high-frequency components also produces a characteristic deviation from the 5/3 power law, including a substantial quadratic region for small angles.

© 1994 Optical Society of America

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References

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  1. 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]
  2. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  3. R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1402 (1992).
    [Crossref]
  4. D. L. Fried, “Analysis of focal anisoplanatism: the fundamental limitation in performance of artificial guide star adaptive optics system (AGS/AOS),” in Laser Guide Star Adaptive Optics Workshop: Proceedings (Phillips Laboratory, Kirtland Air Force Base, N.M., 1992), Vol. 1, pp. 37–80.
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.
  7. This is equivalent to Eq. (3) of Ref. 4, when expressed in terms of real coordinates (as opposed to Noll’s aperture-scaled coordinate).
  8. We can easily imagine using more general aperture-size versus distance functions to treat, to various degrees of approximation, other types of solutions, but these are beyond the scope of the present paper.
  9. Rather simple modifications with the Rytoff approximation can be used to account approximately for some diffractive effects. [For a discussion see, e.g., V. I. Tatarski, Wave Propagation in a Thrbulent Medium (McGraw-Hill, New York, 1961), p. 122.]
  10. J. Stone, P. H. Hu, “Single and multiaperture Zernike correlations in the atmospheric turbulence problem,” in Proceedings of the International Conference on Lasers ’89, D. G. Harris, T. M. Shay, eds. (STS, McLean, Va., 1990), pp. 709–716.
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 363.
  12. Evaluating single or multiparameter integrals with Mellin transforms to generate multiple series solutions is discussed in R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).
  13. G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” (Optical Sciences Company, Placentia, Calif., 1988).
  14. Only in the case of pure angular anisoplanatism, the expression ℒ(η1=0)(r) is independent of the position within the aperture, so the aperture average could be omitted.
  15. See Ref. 1. HJ1(8/3,0,1) itself is Γ(−5/6)/[28/3Γ(11/6)] (or −π/{25/3[Γ(11/6)]2}) = −1.11833 (see Ref. 11, p. 486).
  16. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensative systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
    [Crossref]
  17. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,”J. Opt. Soc. Am. 67, 390–392 (1977).
    [Crossref]
  18. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 386.
  19. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).
  20. Ref. 11, pp. 256–258.
  21. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
    [Crossref]
  22. Ref. 11, p. 70.
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.
  24. Here, in the orthogonality relation, and in Eq. (15) the factors in area Aarise from 1/πin the window function and from factors of Rmthat convert the d2ρfor the dimensionless variable ρbetween 0 and 1 to the equivalent form (1/R2)d2r.

1992 (1)

1989 (1)

1984 (1)

1982 (1)

1977 (1)

1976 (2)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 363.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

Fried, D. L.

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

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensative systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

D. L. Fried, “Analysis of focal anisoplanatism: the fundamental limitation in performance of artificial guide star adaptive optics system (AGS/AOS),” in Laser Guide Star Adaptive Optics Workshop: Proceedings (Phillips Laboratory, Kirtland Air Force Base, N.M., 1992), Vol. 1, pp. 37–80.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 386.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.

Greenwood, D. P.

Hu, P. H.

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]

J. Stone, P. H. Hu, “Single and multiaperture Zernike correlations in the atmospheric turbulence problem,” in Proceedings of the International Conference on Lasers ’89, D. G. Harris, T. M. Shay, eds. (STS, McLean, Va., 1990), pp. 709–716.

Noll, R. J.

Sasiela, R. J.

R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1402 (1992).
[Crossref]

Evaluating single or multiparameter integrals with Mellin transforms to generate multiple series solutions is discussed in R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

Stanley, T.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 363.

Stone, J.

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]

J. Stone, P. H. Hu, “Single and multiaperture Zernike correlations in the atmospheric turbulence problem,” in Proceedings of the International Conference on Lasers ’89, D. G. Harris, T. M. Shay, eds. (STS, McLean, Va., 1990), pp. 709–716.

Tatarski, V. I.

Rather simple modifications with the Rytoff approximation can be used to account approximately for some diffractive effects. [For a discussion see, e.g., V. I. Tatarski, Wave Propagation in a Thrbulent Medium (McGraw-Hill, New York, 1961), p. 122.]

Tyler, G. A.

G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
[Crossref]

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” (Optical Sciences Company, Placentia, Calif., 1988).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Other (17)

Ref. 11, p. 70.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.

Here, in the orthogonality relation, and in Eq. (15) the factors in area Aarise from 1/πin the window function and from factors of Rmthat convert the d2ρfor the dimensionless variable ρbetween 0 and 1 to the equivalent form (1/R2)d2r.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 386.

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

Ref. 11, pp. 256–258.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

This is equivalent to Eq. (3) of Ref. 4, when expressed in terms of real coordinates (as opposed to Noll’s aperture-scaled coordinate).

We can easily imagine using more general aperture-size versus distance functions to treat, to various degrees of approximation, other types of solutions, but these are beyond the scope of the present paper.

Rather simple modifications with the Rytoff approximation can be used to account approximately for some diffractive effects. [For a discussion see, e.g., V. I. Tatarski, Wave Propagation in a Thrbulent Medium (McGraw-Hill, New York, 1961), p. 122.]

J. Stone, P. H. Hu, “Single and multiaperture Zernike correlations in the atmospheric turbulence problem,” in Proceedings of the International Conference on Lasers ’89, D. G. Harris, T. M. Shay, eds. (STS, McLean, Va., 1990), pp. 709–716.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 363.

Evaluating single or multiparameter integrals with Mellin transforms to generate multiple series solutions is discussed in R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” (Optical Sciences Company, Placentia, Calif., 1988).

Only in the case of pure angular anisoplanatism, the expression ℒ(η1=0)(r) is independent of the position within the aperture, so the aperture average could be omitted.

See Ref. 1. HJ1(8/3,0,1) itself is Γ(−5/6)/[28/3Γ(11/6)] (or −π/{25/3[Γ(11/6)]2}) = −1.11833 (see Ref. 11, p. 486).

D. L. Fried, “Analysis of focal anisoplanatism: the fundamental limitation in performance of artificial guide star adaptive optics system (AGS/AOS),” in Laser Guide Star Adaptive Optics Workshop: Proceedings (Phillips Laboratory, Kirtland Air Force Base, N.M., 1992), Vol. 1, pp. 37–80.

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Figures (2)

Fig. 1
Fig. 1

Anisoplanatic degradation (mean-square wave-front error) as a function of angle for various aperture diameters for full wave front without removal of the piston or the tilt [(θ/θ0)5/3, top curve in each figure], with only the piston removed (middle curves), and with both the piston and the tilt (x and y) removed (bottom curves): (a) 5.0-m diameter, (b) 1.0-m diameter, (c) 0.5-m diameter.

Fig. 2
Fig. 2

Anisoplanatism for finite sum of Zernike modes, excluding the piston and the tilt and including (a) 18 individual modes (all the modes for n = 2 through n = 5, inclusive), (b) 42 individual modes (all the modes for n = 2 through n = 8, inclusive). The dashed curve is the quadratic approximation.

Equations (66)

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[ ϕ ( 0 ) - ϕ ( θ ) ] 2 = ( θ / θ 0 ) 5 / 3 .
L ( r ) = [ ϕ l ( r ) - η 0 a 0 ( I ) - η 1 a 1 ( I ) Z 1 ( r R m ) - η 2 a 2 ( I ) Z 2 ( r R m ) - ϕ I I ( r ) + η 0 a 0 ( I I ) + η 1 a 1 ( I I ) Z 1 ( r R m ) + η 2 a 2 ( I I ) Z 2 ( r R m ) ] 2 .
Z 1 ( r R ) = 2 r R cos ( θ ) = 2 x R ,
Z 2 ( r R ) = 2 r R sin ( θ ) = 2 y R ,
ϕ I ( r ) = q = 0 a q ( I ) Z q ( r R m ) ,
ϕ I I ( r ) = q = 0 a q ( I I ) Z q ( r R m ) .
r < R Z q ( r R ) Z q ( r R ) d 2 r = π R 2 δ q q = A δ q q .
σ 2 ¯ 1 A A d 2 r L ( r ) = { 1 A A d 2 r [ ϕ I ( r ) - ϕ I I ( r ) ] 2 } + ( η 0 2 - 2 η 0 ) [ a 0 ( I ) - a 0 ( I I ) ] 2 + ( η 1 2 - 2 η 1 ) [ a 1 ( I ) - a 1 ( I I ) ] 2 + ( η 2 2 - 2 η 2 ) [ a 2 ( I ) - a 2 ( I I ) ] 2
σ 2 ¯ = { 1 A A d 2 r [ ϕ I ( r ) - ϕ I I ( r ) ] 2 } - [ a 0 ( I ) - a 0 ( I I ) ] 2 - [ a 1 ( I ) - a 1 ( I I ) ] 2 - [ a 2 ( I ) - a 2 ( I I ) ] 2 .
{ 1 A A d 2 r [ ϕ I ( r ) - ϕ I I ( r ) ] 2 } = q = 0 [ a q ( I ) - a q ( I I ) ] 2 .
f I ( r , S ) = r R 1 ( S ) R m ,
f I I ( r , S ) = r R 2 ( S ) R m + S θ ,
ϕ J ( r ) ϕ K ( r ) = k 0 2 0 L 0 L n [ f J ( r , s 1 ) , s 1 ] n [ f K ( r , s 2 ) , s 2 ] d s 1 ds 2 .
n ( f 1 , s 1 ) n ( f 2 , s 2 ) = C A δ ( s 1 - s 2 ) C n 2 [ h ( s 1 ) ] × d 2 κ g ( κ ) κ - 11 / 3 exp [ ( i 2 π κ · ( f 2 - f 1 ) ] ,
C A 5 36 2 1 / 3 1 π 5 / 3 Γ ( 1 / 3 ) = 0.0096932 C 1 C 3 / ( 2 C 2 ) .
a q ( J ) = 1 A r < R Z q ( r R m ) ϕ J ( r ) d 2 r .
[ a 0 ( I ) - a 0 ( I I ) ] 2 = 4 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R 1 ( S ) ] 5 / 3 × { H J 2 ( 14 / 3 , 1 , 1 , 1 , 1 ) + [ R 2 ( S ) R 1 ( S ) ] 2 × H J 2 [ 14 / 3 , 1 , 1 , R 2 ( S ) R 1 ( S ) , R 2 ( S ) R 1 ( S ) ] - 2 R 1 ( S ) R 2 ( S ) H J 3 [ 14 / 3 , 1 , 1 , 0 , 1 , R 2 ( S ) R 1 ( S ) , S θ R 1 ( S ) ] } ,
q = 1 2 [ a q ( I ) - a q ( I I ) ] 2 = 16 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R 1 ( S ) ] 5 / 3 × { H J 2 ( 14 / 3 , 2 , 2 , 1 , 1 ) + [ R 1 ( S ) R 2 ( S ) ] 2 × H J 2 [ 14 / 3 , 2 , 2 , R 2 ( S ) R 1 ( S ) , R 2 ( S ) R 1 ( S ) ] - 2 R 1 ( S ) R 2 ( S ) H J 3 [ 14 / 3 , 2 , 2 , 0 , 1 , R 2 ( S ) R 1 ( S ) , S θ R 1 ( S ) ] } ,
H J n ( ρ , k 1 , k 2 , , k n , a 1 , a 2 , , a n ) 0 x - ρ J k 1 ( a 1 x ) J k 2 ( a 2 x ) J k n ( a n x ) d x .
σ 2 ¯ = 4 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R 1 ( S ) ] 5 / 3 × [ 1 2 0 ( 1 - 2 J 1 { u [ 1 - R 2 ( S ) R 1 ( S ) ] } u [ 1 - R 2 ( S ) R 1 ( S ) ] J 0 [ S θ u R 1 ( S ) ] ) × u - 8 / 3 d u - n = 0 1 ( n + 1 ) 2 H J 2 ( 14 / 3 , n + 1 , n + 1 , 1 , 1 ) + [ R 1 ( S ) R 2 ( S ) ] 2 H J 2 [ 14 / 3 , n + 1 , n + 1 , R 2 ( S ) R 1 ( S ) , R 2 ( S ) R 1 ( S ) ] - 2 R 1 ( S ) R 2 ( S ) H J 3 [ 14 / 3 , n + 1 , n + 1 , 0 , 1 , R 2 ( S ) R 1 ( S ) , S θ R 1 ( S ) ] ] .
I A 1 2 0 ( 1 - 2 J 1 { u [ 1 - R 2 ( S ) R 1 ( S ) ] } u [ 1 - R 2 ( S ) R 1 ( S ) ] J 0 [ S θ u R 1 ( S ) ] ) u - 8 / 3 d u
I A = - R 1 ( S ) R 1 ( S ) - R 2 ( S ) H J 2 [ 11 / 3 , 1 , 0 , 1 - R 2 ( S ) R 1 ( S ) , S θ R 1 ( S ) ] .
J 1 ( x - y ) x - y = 2 k = 0 ( k + 1 ) 2 J k + 1 ( x ) x J k + 1 ( y ) y
I A = R 1 ( S ) R 2 ( S ) - R 1 ( S ) H J 2 [ 11 / 3 , 1 , 0 , 1 - R 2 ( S ) R 1 ( S ) , 0 ] = R 1 ( S ) R 1 ( S ) - R 2 ( S ) H J 1 [ 11 / 3 , 1 , R 1 ( S ) - R 2 ( S ) R 1 ( S ) ] ,             θ = 0.
I A = - lim R 2 ( S ) R 1 ( S ) R 1 ( S ) R 1 ( S ) - R 2 ( S ) × H J 2 [ 11 / 3 , 1 , 0 , R 1 ( S ) - R 2 ( S ) R 1 ( S ) , S θ R 1 ( S ) ] = - 1 2 H J 1 [ 8 / 3 , 0 , S θ R 1 ( S ) ] ,             R 1 ( S ) = R 2 ( S ) .
H J 1 [ 8 / 3 , 0 , S θ R 1 ( S ) ] = [ S θ R 1 ( S ) ] 5 / 3 H J 1 ( 8 / 3 , 0 , 1 ) ,
σ 2 ¯ ( η i = 0 ) = θ 5 / 3 × { 2 ( 2 π ) 8 / 3 C A H J 1 ( 8 / 3 , 0 , 1 ) k 0 2 0 L d S C n 2 [ h ( s ) ] S 5 / 3 } .
θ 0 { 2 ( 2 π ) 8 / 3 C A H J 1 ( 8 / 3 , 0 , 1 ) k 0 2 0 L d S C n 2 [ h ( S ) ] S 5 / 3 } - 3 / 5 = { C 1 k 0 2 0 L d S C n 2 [ h ( S ) ] S 5 / 3 } - 3 / 5 ,
σ 2 ¯ = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R 1 ( S ) ] 5 / 3 × ( [ S θ R 1 ( S ) ] 5 / 3 | H J 1 ( 8 3 , 0 , 1 ) | - 4 n = 0 1 ( n + 1 ) 2 { H J 2 ( 14 3 , n + 1 , n + 1 , 1 , 1 ) - H J 3 [ 14 3 , n + 1 , n + 1 , 0 , 1 , 1 , S θ R 1 ( S ) ] } )
σ 2 ¯ = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × 0 d u u - 8 / 3 { 1 - J 0 [ S θ u R ( S ) ] } × { 1 - 4 [ J 1 ( u ) u ] 2 - 16 [ J 2 ( u ) u ] 2 } .
σ 2 ¯ = 2 ( 2 π ) 8 / 3 C A k 0 2 0 d S C n 2 [ h ( S ) ] [ R 1 ( S ) ] 5 / 3 Y ( S ) ,
Y ( S ) = [ S θ R 1 ( S ) ] 5 / 3 | H J 1 ( 8 3 , 0 , 1 ) | - A 0 [ S θ 2 R 1 ( S ) ] - A 1 [ S θ 2 R 1 ( S ) ] ,
A η - 1 ( β ) = η 2 2 - 5 / 3 Γ ( ¹⁴ / ) Γ ( η - ¹¹ / ) [ Γ ( ¹⁷ / ) ] 2 Γ ( ¹⁷ / + η ) - η 2 β - 2 ( η - 11 / 6 ) Γ ( η - ¹¹ / ) 2 2 ( η - 1 ) [ Γ ( η + 1 ) ] 2 Γ ( ¹⁷ / - η ) × F 3 2 ( η + 1 2 , η - 11 6 , η - 11 6 ; η + 1 , 2 η + 1 ; 1 β 2 ) , β > 1 ,
A η - 1 ( β ) = - η 2 2 - 5 / 3 Γ ( ¹⁴ / ) Γ ( η - ¹¹ / ) [ Γ ( ¹⁷ / ] 2 Γ ( ¹⁷ / + η ) × F ¯ 3 2 ( η - 11 6 , - η - 11 6 , - 11 6 ; - 4 3 , 1 ; β 2 ) - 4 η 2 β 14 / 3 Γ ( - / ) π Γ ( ¹⁰ / ) F 3 2 ( η + 1 2 , - η + 1 2 , 1 2 ; 10 3 , 10 3 ; β 2 ) , β < 1 ,
F ¯ 3 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; χ ) k = 1 ( a 1 ) k ( a 2 ) k ( a 3 ) k χ k ( b 1 ) k ( b 2 ) k k ! ,
( a ) n j = 0 n - 1 ( a + j ) .
Y ( S ) = [ S θ 2 R 1 ( S ) ] 5 / 3 Γ ( - ) Γ ( ¹¹ / ) × F ¯ 3 2 { 3 2 , - 5 6 , - 5 6 ; 2 , 3 ; [ 2 R 1 ( S ) s θ ] 2 } - Γ ( - ) [ Γ ( ¹⁷ / ) ] 2 Γ ( ²³ / ) - A 1 [ S θ 2 R 1 ( S ) ] ,
σ r 2 = 0 d f Φ ( f ) ( f / f 3 dB ) 2 1 + ( f / f 3 dB ) 2 ,
σ 2 ¯ = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × 0 d u u - 8 / 3 { 1 - J 0 [ ν t u R ( S ) ] } × { 1 - 4 η 0 [ J 1 ( u ) u ] 2 - 16 η 1 [ J 2 ( u ) u ] 2 } ,
Φ ( f ) = 2 ( 2 π ) 8 / 3 C A k 0 2 0 d t exp ( - 2 π i f t ) × 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 0 d u u - 8 / 3 J 0 [ 2 π t u τ ( S ) ] × { 1 - 4 η 0 [ J 1 ( u ) u ] 2 - 16 η 1 [ J 2 ( u ) u ] 2 } ,
- d t J 0 [ 2 π t u τ ( S ) ] exp ( - 2 τ i f t ) d t = { τ ( S ) π u { 1 - [ f τ ( S ) u ] 2 } 1 / 2 u > f τ ( S ) 0 , u < f τ ( S ) ,
τ ( S ) 2 π R ( S ) ν ( S ) .
Φ ( f ) = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × f - 8 / 3 [ τ ( S ) ] - 5 / 3 π 0 1 d w w 5 / 3 ( 1 - w 2 ) 1 / 2 × { 1 - 4 η 0 [ J 1 ( f τ / w ) f τ / w ] 2 - 16 η 1 [ J 2 ( f τ / w f τ / w ] 2 } .
σ r 2 = 2 ( 2 π ) 8 / 3 C A k 0 2 ( f 3 dB ) 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 [ τ ( S ) ] - 5 / 3 π × 0 1 d w w 5 / 3 ( 1 - w 2 ) 1 / 2 0 d f f - 2 / 3 1 + ( f / f 3 dB ) 2 × { 1 - 4 η 0 [ J 1 ( f τ / w ) f τ / w ] 2 - 16 η 1 [ J 2 ( f τ / w ) f / w ] 2 } .
σ r 2 = 2 ( 2 π ) 8 / 3 C A k 0 2 ( f 3 dB ) 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 [ τ ( S ) ] - 5 / 3 π × ( [ 0 1 d w w 5 / 3 ( 1 - w 2 ) 1 / 2 ] [ f 3 dB 1 / 3 0 d y y - 2 / 3 1 + y 2 ] - { 0 1 d w w 5 / 3 ( 1 - w 2 ) 1 / 2 [ w τ ( S ) ] 1 / 3 } × { 0 d y y - 8 / 3 4 η 0 J 1 2 ( y ) + 16 η 1 J 2 2 ( y ) ] } ) .
σ r 2 = k 0 2 ( f 3 dB ) 5 / 3 0 L d S C n 2 [ h ( S ) ] [ ν ( S ) ] 5 / 3 × { C G - C P T [ ν ( S ) ] 1 / 3 f 3 dB 1 / 3 R ( S ) 1 / 3 } ,
C G = 2 1 / 3 Γ ( 1 / 6 ) 18 π 7 / 6 = 0.102478 , C P T = 50 594 π 1 / 2 [ Γ ( 11 / 6 ) ] 3 ( η 0 + 4 17 η 1 ) = { 0.05704 η 0 = 1 , η 1 = 0 0.07048 η 0 = η 1 = 1 .
σ 2 ¯ FR = n = 2 n max m , l [ a n m l ( I ) - a n m l ( I I ) ] 2 .
σ 2 ¯ FR = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × n = 2 n max A n [ S θ 2 R ( S ) ]
σ 2 ¯ FR = - 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × η = 3 n max + 1 ( η 2 2 - 5 / 3 Γ ( ¹⁴ / ) Γ ( η - ¹¹ / ) [ Γ ( ¹⁷ / ) ] 2 Γ ( ¹⁷ / + η ) F ¯ 3 2 × { η - 11 6 , - η - 11 6 , - 11 6 ; - 4 3 , 1 ; [ S θ 2 R ( S ) ] 2 } + 4 η 2 [ / 2 R ( S ) S θ ] 14 / 3 Γ ( - / ) π Γ ( ¹⁰ / ) F ¯ 3 2 × { η + 1 2 , - η + 1 2 , 1 2 ; 10 3 , 10 3 ; [ S θ 2 R ( S ) ] 2 } ) .
σ 2 ¯ FR θ 2 [ 11 8 π 8 / 3 C A k 0 2 ] Γ ( ¹⁴ / ) [ Γ ( ¹⁷ / ) ] 2 × 0 L d S C n 2 [ h ( S ) ] S 2 [ R ( S ) ] - 1 / 3 F ( n max ) ,
F ( N ) η = 3 N + 1 η 2 ( η + ¹¹ / ) ( η - ¹¹ / ) Γ ( η - ¹¹ / ) Γ ( ¹⁷ / + η ) .
f ( κ 2 ) = C A C n 2 exp ( - κ 2 / κ max 2 ) ( κ 2 + κ min 2 ) 11 / 6 ,
σ 2 ¯ = 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] 5 / 3 × 0 d u u - 8 / 3 exp { - [ u k max R ( S ) ] 2 } × { 1 - J 0 [ S θ u R ( S ) ] } × { 1 - 4 [ J 1 ( u ) u ] 2 - 16 [ J 2 ( u ) u ] 2 } ,
σ 2 ¯ θ 2 [ 1 2 ( 2 π ) 8 / 3 C A ] k 0 2 0 L d S C n 2 [ h ( S ) ] [ R ( S ) ] - 1 / 3 S 2 × 0 d u u - 2 / 3 exp { - [ u k max R ( S ) ] 2 } × { 1 - 4 [ J 1 ( u ) u ] 2 - 16 [ J 2 ( u ) u ] 2 } .
Q q ( k ) = ( n + 1 ) 1 / 2 J n + 1 ( 2 π k ) π k ( - 1 ) ( n - m ) / 2 i m 2 ( 1 - δ m 0 ) / 2 × { cos ( m ϕ ) l = 1 sin ( m ϕ ) l = - 1 , n = 0 , 1 , ,             m = n , n - 2 , { 0 1 ,
Q q ( k ) 1 A r < R m Z q ( r R m ) exp ( 2 π i k · r R m ) d 2 r ,
[ ϕ I ( r ) - ϕ I I ( r ) ] 2 = 2 [ ϕ I ( r ) 2 - ϕ I I ( r ) ϕ I ( r ) ] = 2 C A k 0 2 0 L d S C n 2 [ h ( S ) ] d 2 κ g ( κ ) κ - 11 / 3 × [ 1 - exp ( i 2 π κ · { r R m [ R 2 ( S ) - R 1 ( S ) ] + S θ } ) ] .
{ 1 A A d 2 r [ ϕ I ( r ) - ϕ I I ( r ) ] 2 } = 2 C A k 0 2 0 L d S C n 2 [ h ( S ) ] d 2 κ g ( κ ) κ - 11 / 3 × ( 1 - J 1 { 2 π κ [ R 2 ( S ) - R 1 ( S ) ] } π κ [ R 2 ( S ) - R 1 ( S ) ] exp [ i 2 π κ · [ ( S θ ) ] ) .
J q ( z ) = ( - i ) q 2 π 0 2 π cos ( q ϕ ) exp [ i z cos ( ϕ ) ] d ϕ ,
{ 1 A A d 2 r [ ϕ I ( r ) - ϕ I I ( r ) ] 2 } = 4 π k 0 2 C A 0 L d S C n 2 [ h ( S ) ] × 0 d κ κ - 8 / 3 ( 1 - J 1 { 2 π κ [ R 2 ( S ) - R 1 ( S ) ] } π κ [ R 2 ( S ) - R 1 ( S ) ] J 0 ( 2 π κ S θ ) .
[ a q ( I ) - a q ( I I ) ] 2 = [ 1 A r < R Z q ( r R m ) ϕ I ( r ) d 2 r - 1 A r < R Z q ( r R m ) ϕ I I ( r ) d 2 r ] × [ 1 A r < R Z q ( r R m ) ϕ I ( r ) d 2 r - 1 A r < R Z q ( r R m ) ϕ I I ( r ) d 2 r ] .
[ a q ( I ) - a q ( I I ) ] 2 = 1 A 2 r < R d 2 r Z q ( r R m ) r < R d 2 r Z q ( r R m ) × [ ϕ I ( r ) ϕ I ( r ) + ϕ I I ( r ) ϕ I I ( r ) - 2 ϕ I ( r ) ϕ I I ( r ) ] ,
[ ϕ I ( r ) ϕ I ( r ) + ϕ I I ( r ) ϕ I I ( r ) - 2 ϕ I ( r ) ϕ I I ( r ) ] = C A k 0 2 0 L d S C n 2 [ h ( S ) ] d 2 κ g ( κ ) κ - 11 / 3 × ( exp { i 2 π κ · [ r - r R m R 1 ( S ) ] } + exp { i 2 π κ · [ r - r R m R 2 ( S ) ] } - 2 exp { i 2 π κ · [ r R m R 2 ( S ) - r R m R 1 ( S ) + S θ ] } ) .
[ a q ( I ) - a q ( I I ) 2 = ( n + 1 ) ( - 1 ) ( n - m ) 2 ( 1 - δ m 0 ) C A k 0 2 0 L d S C n 2 [ h ( S ) ] × 1 2 0 d κ κ - 8 / 3 g ( κ ) [ ( { J n + 1 [ 2 π κ R 1 ( S ) ] π κ R 1 ( S ) } 2 + { J n + 1 [ 2 π κ R 2 ( S ) ] π κ R 2 ( S ) } 2 ) 0 2 π d ϕ [ 1 + l cos ( 2 m ϕ ) ] - 2 { J n + 1 [ 2 π κ R 1 ( S ) ] π κ R 1 ( S ) } { J n + 1 [ 2 π κ R 2 ( S ) ] π κ R 2 ( S ) } × 0 2 π d ϕ exp [ i 2 π κ S θ cos ( ϕ - α ) ] [ 1 + l cos ( 2 m ϕ ) ] ] .
m , l [ a n m l ( I ) - a n m l ( I I ) ] 2 = 4 ( n + 1 ) 2 ( 2 π ) 8 / 3 C A k 0 2 0 L d S C n 2 [ h ( s ) ] [ R 1 ( S ) ] 5 / 3 × 0 d u u - 8 / 3 ( [ J n + 1 ( u ) u ] 2 + { J n + 1 [ u R 2 ( S ) / R 1 ( S ) ] u R 2 ( S ) / R 1 ( S ) } 2 - 2 J n + 1 ( u ) u J n + 1 [ u R 2 ( S ) / R 1 ( S ) ] u R 2 ( S ) / R 1 ( S ) J 0 [ u S θ R 1 ( S ) ] ) .

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