Abstract

Zernike polynomials are used in the study of single- and multiple-circular-aperture optical systems that are affected by atmosphere turbulence. Expressions for mean-square wave-front distortion, correlation functions, Zernike covariance coefficients, and anisoplanatism are derived and calculated for Kolmogorov atmospheric statistics. The mathematical tools provided here, however, can readily be extended to non-Kolmogorov turbulence spectra.

© 1989 Optical Society of America

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References

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  1. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  2. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,”J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  4. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [Crossref] [PubMed]
  5. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  6. G. C. Valley, S. M. Wandzura, “Spatial correlation of phase expansion coefficients for propagation through atmospheric turbulence,”J. Opt. Soc. Am. 69, 712–717 (1979).
    [Crossref]
  7. M. T. Tavis, H. T. Yura, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
    [Crossref]
  8. G. A. Tyler, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston, tilt removed phase cross covariance,” (Optical Science Company, Placentia, Calif., 1985).
  9. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  10. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
    [Crossref]
  11. R. F. Lutomirski, H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,”J. Opt. Soc. Am. 61, 482–487 (1971).
    [Crossref]
  12. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  14. 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).
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  16. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).
  17. Bateman Manuscript Project, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vols. I and II.
  18. Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vols. I and II.
  19. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).

1985 (1)

1984 (1)

1982 (1)

1980 (1)

1979 (1)

1976 (1)

1974 (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

1971 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Born, M.

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

Fried, D. L.

Goodman, J. W.

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

Lutomirski, R. F.

Marichev, O. I.

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).

Noll, R. J.

Sasiela, R. J.

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).

Silva, D. E.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Tatarski, V. I.

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

Tavis, M. T.

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, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston, tilt removed phase cross covariance,” (Optical Science Company, Placentia, Calif., 1985).

Valley, G. C.

Wandzura, S. M.

Wang, J. Y.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).

Wolf, E.

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

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (10)

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

G. A. Tyler, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston, tilt removed phase cross covariance,” (Optical Science Company, Placentia, Calif., 1985).

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

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

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).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1948).

Bateman Manuscript Project, Table of Integral Transforms (McGraw-Hill, New York, 1954), Vols. I and II.

Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vols. I and II.

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Ellis Horwood, New York, 1983).

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Figures (8)

Fig. 1
Fig. 1

The geometry shown is for two circular apertures of identical diameter D with centers specified by ri and rj, respectively. S is, the vector from the ith to the jth center normalized by D.

Fig. 2
Fig. 2

Covariance matrix elements in units of (D/r0)5/3 are plotted as a function of S. S is the aperture-center separation in units of D. These elements are calculated by assuming that the centers lie along the x axis. We shall identify each curve by the notation ( n , m , l , n , m , l , ), where n, m, and l are axial, azimuthal, and parity mode numbers, respectively. Note that curves are normalized differently in order to display them on the same chart. (a) ( 1 , 1 , - 1 1 1 , - 1 ) (b) ( 1 , 1 , 1 1 , 1 , - 1 ) (c) ( 2 , 0 , 1 2 , 0 , 1 ) × 10 3 (d) ( 3 , 1 , 1 3 , 1 , 1 ) × 10 4.

Fig. 3
Fig. 3

Symmetry operations on a system of two apertures. From baseline configuration (a), configuration (b) is a simple exchange of indices, whereas (c) is an exchange plus 180-deg rotation. Configuration (d), with l′ = −1 (angular dependence in aperture 2 proportional to sin θ2), gives zero by symmetry.

Fig. 4
Fig. 4

Relative piston–piston variance in units of (D/r0)5/3 as a function of aperture-center separation in units of D, diameter of the circular aperture.

Fig. 5
Fig. 5

Anisoplanatism for different Zernike modes in units of (D/r0)5.3 as a function of look-ahead angle divided by D. Parameters for each curve are n, m, and l indices for the Zernike mode whose anisoplanatism is being plotted.

Fig. 6
Fig. 6

Relative piston–piston anisoplanatism in units of (D/r0)5/3 as a function of look-ahead angle divided by D for different values of S, the aperture-center separation, in units of D.

Fig. 7
Fig. 7

Geometric relation of various quantities for the Bessel-function addition theorem.

Fig. 8
Fig. 8

Geometric relation of various quantities for Eq. (C2). SijR is the spatial vector from the ith aperture center to the jth aperture center. ρj gives the coordinates of the sample point relative to the jth center. pi gives the coordinates of the same point expressed relative to the ith aperture center.

Tables (1)

Tables Icon

Table 1 Forms of A(ϕ) and B(ϕ) for Various l and l

Equations (83)

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Z q ( ρ ) = ( n + 1 ) 1 / 2 R n m ( ρ ) 2 ( 1 - δ m 0 ) / 2 cos m θ ,             l = 1 , Z q ( ρ ) = ( n + 1 ) 1 / 2 R n m ( ρ ) 2 1 / 2 sin m θ ,             l = - 1 ,
R n m ( ρ ) = s = 0 ( n - m ) / 2 ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! ρ n - 2 s .
W ( ρ ) Z q ( ρ ) Z q ( ρ ) d 2 ρ = δ q q .
W ( ρ ) = 1 / π , ρ 1 = 0 , ρ > 1 ,
W ( ρ ) Z q ( ρ ) = d 2 k Q q ( k , ϕ ) exp ( - 2 π i k · ρ ) .
Q q ( k , ϕ ) = d 2 ρ W ( ρ ) Z q ( ρ ) exp ( 2 π i k · ρ ) .
Q q ( k , ϕ ) = ( 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 , Q q ( k , ϕ ) = ( n + 1 ) 1 / 2 J n + 1 ( 2 π k ) π k ( - 1 ) ( n - m ) / 2 i m 2 1 / 2 sin m ϕ ,             l = - 1 ,
W ( ρ ) R n m ( ρ ) = 2 π ( - 1 ) ( n - m ) / 2 0 d k J n + 1 ( 2 π k ) J m ( 2 π k ρ ) .
Φ ( R ρ , θ ) = q a q Z q ( ρ ) ,
a q = d 2 ρ W ( ρ ) Φ ( R ρ , θ ) Z q ( ρ ) .
a q a q = ( 0.046 / π ) ( R / r 0 ) 5 / 3 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 ( - 1 ) n + n - 2 m ) / 2 × δ m m δ l l 0 k - 8 / 3 J n + 1 ( 2 π k ) J n + 1 ( 2 π k ) k 2 d k .
HJ 2 ( λ , n 1 , n 2 , A , B ) = 0 x - λ J n 1 ( A x ) J n 2 ( B x ) d x ,
HJ 3 ( λ , n 1 , n 2 , n 3 , A , B , C ) = 0 x - λ J n 1 ( A x ) J n 2 ( B x ) J n 3 ( C s ) d x .
a q a q = 4 C 3 ( D / r 0 ) 5 / 3 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 ( - 1 ) ( n + n - 2 m ) / 2 × δ m m π 8 / 3 HJ 2 ( λ , n + 1 , n + 1 , 1 , 1 ) ,
1 r 0 5 / 3 = C 1 C 2 sec Ψ k 0 2 c n 2 ( h ) d h , C 1 = 2 1 / 3 Γ 2 ( 1 / 6 ) 5 Γ ( 1 / 3 ) 2.9148381 , C 2 = 2 [ 24 5 Γ ( 6 / 5 ) ] ( 5 / 6 ) 6.883877 , C 3 = C 2 [ Γ ( 11 / 6 ) ] 2 / ( 2 π 11 / 3 ) 0.046 , λ = 14 / 3.
D n ( r 1 , r 2 ) = [ n ( r 1 ) - n ( r 2 ) ] 2 ,
D n ( r ) = c n 2 ( h ) r 2 / 3 ,             l 0 < r < L 0 ,
D ( x , y ) [ Φ ( x ) - Φ ( y ) ] 2 = C 2 x - y 5 / 3 / r 0 5 / 3 .
D ( x , y ) = C 3 r 0 5 / 3 g ( k ) · k - 11 / 3 · [ 1 - exp [ i 2 π k · ( x - y ) ] d 2 k .
Φ ( x ) Φ ( y ) = C 3 2 r 0 5 / 3 g ( k ) k - 11 / 3 exp [ i 2 π k · ( x - y ) ] d 2 k .
n ( x , h 1 ) n ( y , h 2 ) = C 1 C 3 2 C 2 δ ( h 1 - h 2 ) c n 2 ( h 1 ) × 0 d 2 k g ( k ) k - 11 / 3 exp [ i 2 π k · ( x - y ) ] .
Φ ( r i + R ρ i ) = k 0 0 n ( r i + R ρ i , h ) d h , Φ ( r i + R ρ i ) = k 0 0 n ( r j + R ρ j , h ) d h ,
Φ ( r i + R ρ i ) = q a q i Z q ( ρ i ) , Φ ( r j + R ρ j ) = q a q j Z q ( ρ j ) .
a q i = d 2 ρ i W ( ρ i ) Φ ( r i + R ρ i ) Z q ( ρ i ) , a q j = d 2 ρ j W ( ρ j ) Φ ( r j + R ρ j ) Z q ( ρ j ) .
a q i a q j = d 2 ρ i d 2 ρ j W ( ρ i ) W ( ρ j ) Z q ( ρ i ) Z q ( ρ j ) × Φ ( r i + ρ i R ) Φ ( r j + ρ j R ) .
a q i a q j = C 3 2 r 0 5 / 3 d 2 k exp [ i 2 π k · ( r i - r j ) ] × g ( k ) k - 11 / 3 Q q ( k R , ξ ) Q q * ( k R , ξ ) .
a q i a q j = C 3 { ( n + 1 ) [ n + 1 ] } 1 / 2 π 8 / 3 ( D r 0 ) 5 / 3 × ( - 1 ) ( n + n - m - m ) / 2 4 ( 2 ) - ( δ m 0 + δ m 0 ) / 2 × [ A ( ϕ ) HJ 3 ( λ , n + 1 , n + 1 , m + m , 1 , 1 , 2 S 12 ) × ( - 1 ) m + B ( ϕ ) × HJ 3 ( λ , n + 1 , n + 1 , m - m , 1 , 1 , 2 S 12 ) ] .
Δ = 1 N { i = 1 N i th aperture d 2 ρ i W ( ρ i ) [ Φ i ( ρ i ) - a ¯ 0 ) } 2 ,
a ¯ 0 = 1 N ( a 0 1 + a 0 2 + + a 0 N ) .
Φ i ( ρ i ) = a 0 i + q = 1 a q i Z q ( ρ i ) .
Δ = 1 N [ i = 1 N q = 1 ( a q i ) 2 + i = 1 N ( a 0 i - a ¯ 0 ) 2 ] .
i = 1 N ( a 0 i - a ¯ 0 ) 2 = 1 N i = 1 N j > i N ( a 0 i - a 0 j ) 2 ,
Δ = 1 N i = 1 N q = 1 ( a q i ) 2 + 1 N 2 i = 1 N j > i N ( a 0 i - a 0 j ) 2 ,
( a 0 i - a 0 j ) 2 = 2 ( a 0 i a 0 i - a 0 i a 0 j ) = 8 C 3 π 8 / 3 ( D r 0 ) 5 / 3 H ( S ) ,
H ( S ) 0 x - 14 / 3 J 1 2 ( x ) [ 1 - J 0 ( 2 S x ) ] d x .
H ( S ) = { Γ ( 14 / 3 ) Γ ( - 5 / 6 ) 2 14 / 3 Γ ( 17 / 6 ) Γ ( 23 / 6 ) Γ ( 17 / 6 ) - Γ ( - 5 / 6 ) 8 Γ ( 11 / 6 ) × F 3 2 [ 3 / 2 - 5 / 6 - 5 / 6 2 3 1 / S 2 ] × S 5 / 3 } = { 0.8876 × F 3 2 [ 3 / 2 - 5 / 6 - 5 / 6 2 3 1 / S 2 ] × S 5 / 3 - 0.2663 } ,
Φ p ( x ) Φ p ( y ) = q = p + 1 s q = p + 1 a q a q Z q ( ρ 1 ) Z q ( ρ 2 ) ,
C ( x , y ) = [ Φ ( x ) - q = 0 p a q Z q ( ρ 1 ) ] [ Φ ( y ) - q = 0 p a q Z q ( ρ 2 ) ] = [ Φ ( x ) - a 0 - q = 1 p a q Z q ( ρ 1 ) ] × [ Φ ( y ) - a 0 - q = 1 p a q Z q ( ρ 2 ) ] .
( a - b ) ( c - d ) = ( 1 / 2 ) [ ( a - d ) 2 + ( b - c ) 2 - ( a - c ) 2 - ( b - d ) 2 ] ,
2 C ( x , y ) = [ Φ ( x ) - a 0 - q = 1 p a q Z q ( ρ 2 ) ] 2 + [ Φ ( y ) - a 0 - q = 1 p a q Z q ( ρ 1 ) ] 2 - [ Φ ( x ) - Φ ( y ) ] 2 - [ q = 1 p a q Z q ( ρ 1 ) - q = 1 p a q Z q ( ρ 2 ) ] 2 .
[ Φ ( x ) - a 0 - q = 1 p a q Z q ( ρ 2 ) ] 2 = [ Φ 2 ( x ) - a 0 2 ] - 2 q = 0 p q = 1 a q a q Z q ( ρ 1 ) Z q ( ρ 2 ) + q = 1 p q = 1 p a q a q Z q ( ρ 2 ) Z q ( ρ 2 ) .
Φ 2 ( x ) - a 0 2 = C 3 π 8 / 3 ( D r 0 ) 5 / 3 0 g ( x ) x - 8 / 3 [ 1 + 2 J 1 2 ( x ) x 2 ] d x .
Φ 2 ( x ) - a 0 2 = 1.037 ( D r 0 ) 5 / 3
C ( x , y ) / ( D / r 0 ) 5 / 3 = 1.037 - C 2 2 8 / 3 ρ 1 - ρ 2 5 / 3 + q = 1 p q = 1 p a q a q Z q ( ρ 1 ) Z q ( ρ 2 ) / ( D / r 0 ) 5 / 3 - q = 0 p Z q ( ρ 2 ) const . A m ( l , θ 1 ) × [ HJ 2 ( 11 / 3 , n + 1 , m , 1 , ρ 1 ) - 2 δ m 0 HJ 2 ( 14 / 3 , n + 1 , 1 , 1 , 1 ) ] - q = 0 p Z q ( ρ 1 ) const . × A m ( l , θ 2 ) [ HJ 2 ( 11 / 3 , n + 1 , m , 1 , ρ 2 ) - 2 δ m 0 HJ 2 ( 14 / 3 , n + 1 , 1 , 1 , 1 ) ] ,
const . = 2 C 3 ( n + 1 ) 1 / 2 ( - 1 ) ( n - m ) / 2 π 5 / 3 , A m ( - 1 , θ ) = 2 1 / 2 sin m θ , A m ( + 1 , θ ) = 2 ( 1 - δ m 0 ) / 2 cos m θ ,
C i j ( x , y ) / ( D / r 0 ) 5 / 3 = ( a 0 j - a 0 i ) 2 / ( D / r 0 ) 5 / 3 + 1.037 - C 2 2 8 / 3 ρ i - ρ j 5 / 3 - q = 1 p q = 1 p a q i a q j Z q ( ρ i ) Z q ( ρ j ) / ( D / r 0 ) 5 / 3 - q = 0 p Z q ( ρ j ) const . A m ( l , θ i ) [ HJ 2 ( 11 / 3 , n + 1 , m , 1 , ρ i ) - 2 HJ 3 ( 14 / 3 , n + 1 , 1 , m , 1 , 1 , S i j ) ] - q = 0 p Z q ( ρ i ) const . A m ( l , θ j ) · [ HJ 2 ( 11 / 3 , n + 1 , m , 1 , ρ j ) - 2 HJ 3 ( 14 / 3 , n + 1 , 1 , m , 1 , 1 , S i j ) ] ,
const . = 2 C 3 ( n + 1 ) 1 / 2 ( - 1 ) n - m ) / 2 π 5 / 3 , A m ( - 1 , θ ) = 2 1 / 2 sin m θ , A m ( + 1 , θ ) = ( 2 ) ( 1 - δ m 0 ) / 2 cos m θ .
Φ ( y , 0 ) = k 0 n ( y , h 1 ) d h 1 , Φ ( x , θ ) = k 0 n ( x + θ h 2 , h 2 ) d h 2 ,
Φ ( x , θ ) Φ ( y , 0 ) = C 1 C 3 2 C 2 k 0 2 d h c n 2 ( h ) d 2 k k - 11 / 3 × exp [ 2 i π k · ( x + θ h - y ) ] .
a q ( θ ) a q ( 0 ) = 4 C 3 π 8 / 3 ( n + 1 ) C n ( D r 0 ) 5 / 3 0 c n 2 ( h ) f ( q , h , θ ) d h ,
f ( q , h , θ ) = [ HJ 3 ( 14 / 3 , n + 1 , n + 1 , 0 , 1 , 1 , 2 θ h / D + l ( - 1 ) m × HJ 3 ( 14 / 3 , n + 1 , n + 1 , 2 m , 1 , 1 , 2 θ h / D ) ] 2 δ m 0
C n = 0 c n 2 ( h ) d h .
[ a q ( θ ) - a q ( 0 ) ] 2 = 2 [ a q 2 ( 0 ) - a q ( θ ) a q ( 0 ) ] = 8 C 3 π 8 / 3 ( n + 1 ) C n ( D r 0 ) 5 / 3 0 c n 2 ( h ) × [ HJ 2 ( 14 / 3 , n + 1 , n + 1 , 1 , 1 ) - f ( q , h , θ ) ] d h .
{ [ a 0 i ( θ ) - a 0 j ( θ ) ] - [ a 0 i ( 0 ) - a 0 j ( 0 ) ] } 2 / 2 = [ a 0 i ( 0 ) - a 0 j ( 0 ) ] 2 - a 0 i ( θ ) a 0 i ( 0 ) - a 0 i ( θ ) a 0 j ( 0 ) - a 0 j ( θ ) a 0 j ( 0 ) - a 0 j ( θ ) a 0 i ( 0 ) .
{ [ a 0 i ( θ ) - a 0 j ( θ ) ] - [ a 0 i ( 0 ) - a 0 j ( 0 ) ] } 2 = 4 C 3 π 8 / 3 C n ( D r 0 ) 5 / 3 0 c n 2 ( h ) [ 2 H ( S ) + 2 H ( θ h / D ) - H ( S + θ h / D ) - H ( S - θ h / D ) ] d h ,
H ( S ) = Γ ( 14 / 3 ) Γ ( - 5 / 6 ) 2 14 / 3 Γ ( 17 / 6 ) Γ ( 23 / 6 ) Γ ( 17 / 6 ) - [ ( S ) 14 / 3 Γ ( - 7 / 3 ) 2 π Γ ( 10 / 3 ) F 3 2 ( 3 / 2 , 1 / 2 , - 1 / 2 10 / 3 , 10 / 3 , S 2 ) + Γ ( 14 / 3 ) Γ ( - 5 / 6 ) 2 14 / 3 Γ ( 17 / 6 ) Γ ( 23 / 6 ) Γ ( 17 / 6 ) × F 3 2 ( - 5 / 6 , - 17 / 6 , - 11 / 6 - 4 / 3 , 1 , S 2 ) ] .
J n ( z ) = 1 2 π 0 2 π exp ( - i n θ + i z sin θ ) d θ .
J n ( z ) = 2 ( n - 1 ) z J n - 1 ( z ) - J n - 2 ( z ) ,
J μ ( x ) J v ( x ) = 2 π 0 π / 2 J μ + v ( 2 x cos θ ) cos ( μ - v ) θ d θ ,             Re ( μ + v ) > - 1.
0 x - λ J μ ( a x ) J v ( b x ) d x = b v Γ ( μ + v - λ + 1 2 ) 2 λ a v - λ + 1 Γ ( v + 1 ) Γ ( μ - v + λ + 1 2 ) × F 2 1 ( μ + v - λ + 1 2 , v - μ - λ + 1 2 ; v + 1 ; b 2 a 2 ) , Re ( v + μ - λ + 1 ) > 0 ,             Re λ > 0 ,             0 < b a ,
0 x - λ J μ ( a x ) J v ( a x ) d x = a λ - 1 Γ ( λ ) Γ ( v + μ - λ + 1 2 ) 2 λ Γ ( - v + μ + λ + 1 2 ) Γ ( v + μ + λ + 1 2 ) Γ ( v - μ + λ + 1 2 ) , Re ( v + μ + 1 ) > Re λ > 0 ,             a > 0 ,
0 x - p [ 1 - J 0 ( b x ) ] d x = π b p - 1 2 p { Γ [ ( p + 1 ) / 2 ] 2 sin [ π ( p - 1 ) / 2 ] } ,             0 < p < 3 ,
0 x ρ J λ ( a x ) J μ ( a x ) J v ( 2 b x ) d x = a λ + μ b - λ - μ - ρ - 1 Γ ( λ / 2 + μ / 2 + v / 2 + ρ / 2 + 1 / 2 ) 2 λ + μ + 1 Γ ( λ + 1 ) Γ ( μ + 1 ) Γ ( 1 / 2 - λ / 2 - μ / 2 + v / 2 - ρ / 2 ) × F 4 3 ( λ + μ + 1 2 , λ + μ 2 + 1 , λ + μ + v + ρ + 1 2 , λ + μ - v + ρ + 1 2 ; λ + 1 , μ + 1 , λ + μ + 1 ; a 2 b 2 ) , Re ( λ + μ + v + ρ + 1 ) > 0 ,             0 < a b = a - ρ - 1 [ ( b / a ) - ρ Γ ( ρ / 2 + v / 2 ) 2 Γ ( - λ + μ + 1 2 ) Γ ( + λ - μ + 1 2 ) Γ ( - ρ + v + 2 2 ) × F 4 3 ( λ + μ + 1 2 , λ - μ + 1 2 , - λ - μ + 1 2 , - λ + μ + 1 2 ; - ρ + v + 2 2 , - ρ - v + 2 2 , 1 2 ; b 2 a 2 ) - Γ ( λ + μ + 2 2 ) Γ ( ρ + v - 1 2 ) ( b / a ) - ρ + 1 Γ ( - λ + μ 2 ) Γ ( λ + μ 2 ) Γ ( λ - μ 2 ) Γ ( - ρ + v + 3 2 ) × F 4 3 ( λ + μ + 2 2 , λ - μ + 2 2 , - λ - μ + 2 2 , - λ + μ + 2 2 ; - ρ + v + 3 2 , - ρ - v + 3 2 , 3 2 ; b 2 a 2 ) + 2 ρ + v Γ ( - ρ - v ) Γ ( ρ + λ + μ + v + 1 2 ) ( b / a ) v Γ ( - ρ - λ + μ - v + 1 2 ) Γ ( - ρ + λ + μ - v + 1 2 ) Γ ( - ρ + λ - μ - v + 1 2 ) Γ ( v + 1 ) × F 4 3 ( ρ + λ + μ + v + 1 2 , ρ + λ - μ + v + 1 2 , ρ - λ - μ + v + 1 2 , ρ - λ + μ + v + 1 2 ; ρ + v + 2 2 , ρ + v + 1 2 , v + 1 ; b 2 a 2 ) ] , Re ( λ + μ + v + ρ + 1 ) > 0 ,             0 < b a .
0 ( 1 - 4 J 1 2 ( x ) x 2 ) x - p d x = π Γ ( p + 2 ) 2 p { [ ( p + 3 ) / 2 ] } 2 Γ [ ( p + 5 ) / 2 ] Γ [ ( 1 + p ) / 2 ] sin [ ( π / 2 ) ( p - 1 ) ] ,             0 < p < 3 ,
0 x - λ J 1 2 ( x ) [ 1 - J 0 ( a x ) ] d x = Γ ( λ ) Γ ( 3 - λ 2 ) 2 γ Γ ( 1 + λ 2 ) Γ ( 3 + λ 2 ) Γ ( 1 + λ 2 ) - 1 8 ( α 2 ) λ - 3 × Γ [ ( 3 - λ ) / 2 ] Γ [ ( λ - 1 ) / 2 ] F 3 2 ( 3 / 2 , ( 3 - λ ) / 2 , ( 3 - λ ) / 2 2 , 3 , 4 / α 2 ) , 0 < λ < 3 ,             2 < α , 0 x - λ J 1 2 ( x ) [ 1 - J 0 ( a x ) ] d x = Γ ( λ ) Γ ( 3 - λ 2 ) 2 γ Γ ( 1 + λ 2 ) Γ ( 3 + λ 2 ) Γ ( 1 + λ 2 ) - [ ( α 2 ) λ Γ ( - λ / 2 ) 2 π Γ ( λ + 2 2 ) F 3 2 ( 3 2 , 1 2 , - 1 2 λ + 2 2 , λ + 2 2 , α 2 4 ) + Γ ( λ ) Γ ( - λ + 3 2 ) 2 γ Γ ( λ + 1 2 ) Γ ( λ + 3 2 ) Γ ( λ + 1 2 ) × F 3 2 ( - λ + 3 2 , - λ - 1 2 , - λ + 1 2 - λ + 2 2 , 1 , α 2 4 ) ] , 0 < λ < 3 ,             2 > α .
F 2 1 ( a , b ; c ; x ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) x n Γ ( c + n ) n ! , F 2 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) ,             c 0 , - 1 , - 2 , , Re ( c - a - b ) > 0.
F p q ( a 1 , a 2 , , a p ; b 1 , b 2 , , b q ; x ) F p q ( a 1 , a 2 , a p , b 1 , b 2 , a q , x ) = k = 0 ( a 1 ) k ( a 2 ) k ( a p ) k x k ( b 1 ) k ( b 2 ) k ( b q ) k k ! , ( a ) 0 = 1 ,             ( a ) n = a ( a + 1 ) ( a + 2 ) ( a + n - 1 ) , p , q integers ,             p + 1 q .
J n ( W ) ( cos sin ) n χ = k = - J n + k ( u ) J k ( v ) ( cos sin ) k α ,
W 2 = u 2 + v 2 - 2 u v cos α .
δ ( y - y ) = ( y y ) 1 / 2 0 x J v ( x y ) J v ( x y ) d x ,             ν > - 1 / 2.
HJ 2 ( λ , μ , ν , a , b ) b ν Γ ( u + ν - λ + 1 2 ) 2 λ a ν - λ + 1 Γ ( ν + 1 ) Γ ( μ - ν + λ + 1 2 ) × F 2 1 ( μ + ν - λ + 1 2 , ν - μ - λ + 1 2 ; ν + 1 ; b 2 a 2 ) , Re λ > 0 ,             0 , < b a a μ Γ ( μ + ν - λ + 1 2 ) 2 λ b u - λ + 1 Γ ( μ + 1 ) Γ ( ν - μ + λ + 1 2 ) × F 2 1 ( μ + ν - λ + 1 2 , μ - ν - λ + 1 2 ; μ + 1 ; a 2 b 2 ) , Re λ > 0 ,             0 < a b .
HJ 3 ( λ , n 1 , n 2 , n 3 , A , B , C ) = 2 ( n 1 - 1 ) A HJ 3 ( λ + 1 , n 1 - 1 , n 2 , n 3 , A , B , C ) - HJ 3 ( λ , n 1 - 2 , n 2 , n 3 , A , B , C ) .
HJ 2 ( λ , n 1 , n 2 , A , B ) = 0 g ( x ) x - λ J n 1 ( A x ) J n 2 ( B x ) d x ,
HJ 3 ( λ , n 1 , n 2 , n 3 , A , B , C ) = 0 g ( x ) x - λ J n 1 ( A x ) J n 2 ( B x ) J n 3 ( C x ) d x ,
q = 1 a q a q ( D / r 0 ) 5 / 3 Z q ( ρ ) = const . A m ( l , θ ) × [ HJ 2 ( λ , n , + 1 , m , 1 , ρ ) - 2 δ m 0 HJ 2 ( λ + 1 , n + 1 , 1 , 1 , 1 ) ] ,
A m ( l , θ ) = 2 ( 1 - δ m 0 ) / 2 cos m θ , l = 1 , A m ( l , θ ) = 2 1 / 2 sin m θ , l = - 1 , q = 0 , 1 , 2 , .
q = 1 a q i a q j ( D / r 0 ) 5 / 3 Z q ( ρ j ) = const . A m ( l , θ i ) [ HJ 2 ( λ , n + 1 , m , 1 , ρ i ) - 2 HJ 3 ( λ + 1 , n + 1 , 1 , m , 1 , 1 , S i j ) ] ,
const . A m ( l , θ i ) HJ 2 ( λ , n + 1 , m , 1 , ρ i ) = q = 0 C q q Z q ( ρ i ) ,
C q q = const . d 2 ρ j A m ( l , θ i ) HJ 2 ( λ , n + 1 , m , 1 , ρ i ) Z q ( ρ j ) W ( ρ j ) .
θ i = π + ( α + θ j ) + χ ,
C 1 ( ϕ ) J m ( ρ i x ) sin cos m χ ,
ϕ = α + θ j + π .
J m ( ρ i x ) sin cos m χ = k = - J k ( ρ j x ) J m + k ( S 12 x ) sin cos k α ,

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