Abstract

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.

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References

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2006 (2)

1995 (1)

1994 (1)

1984 (1)

1981 (3)

1978 (2)

1977 (1)

1976 (1)

1974 (1)

1966 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), App. VII, p. 910.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Sec. 9.2.1.

Butts, R. R.

Dai, G.-m.

Fried, D. L.

Hardy, J. W.

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

Hogge, C. B.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Lum, B. K. C.

Mahajan, V. N.

Markey, J. K.

Noll, R. J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

Wang, J. Y.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Sec. 9.2.1.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), App. VII, p. 910.

Appl. Opt. (4)

J. Opt. Soc. Am. (7)

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

Opt. Lett. (1)

Other (8)

G.-m. Dai, 'Theoretical studies and computer simulations of post-detection atmospheric turbulence compensation,' Ph.D. thesis (Lund University, 1995).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), App. VII, p. 910.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Sec. 9.2.1.

V. N. Mahajan, Optical Imaging and Aberrations. Part II. Wave Diffraction Optics, 2nd printing (SPIE Press, 2004).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

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Figures (13)

Fig. 1
Fig. 1

Long-exposure MTF for different values of obscuration ratio ϵ and several values of the normalized diameter D r 0 : (a) D r 0 = 0 ; (b) D r 0 = 1 ; (c) D r 0 = 10 . ν is a spatial frequency normalized by the cutoff frequency 1 λ F of an optical imaging system.

Fig. 2
Fig. 2

Long-exposure Strehl ratio S : (a) as a function of D r 0 for several values of ϵ; (b) as a function of obscuration ratio ϵ for D r 0 = 1 , 3, and 10.

Fig. 3
Fig. 3

Long-exposure Strehl ratio η: (a) as a function of D r 0 for several values of ϵ; (b) as a function of obscuration ratio ϵ for D r 0 = 0.4 , 1, and 10.

Fig. 4
Fig. 4

Mean square value of Δ J of the residual phase error in units of ( D r 0 ) 5 3 as a function of obscuration ratio ϵ for J = 1 , 3, 6, and 11.

Fig. 5
Fig. 5

Variance of aberration coefficients in units of ( D r 0 ) 5 3 as a function of obscuration ratio ϵ.

Fig. 6
Fig. 6

Covariance of aberration coefficients in units of ( D r 0 ) 5 3 as a function of obscuration ratio ϵ.

Fig. 7
Fig. 7

Residual phase structure function D J ( ν ; ϵ ) for modal correction of 1, 3, 4, and 6 modes.

Fig. 8
Fig. 8

Residual MTF after modal correction of 1, 3, 6, 11, and modes for ϵ = 0 and ϵ = 0.25 when D r 0 = 5 .

Fig. 9
Fig. 9

Strehl ratio S after modal correction of 1, 3, 6, 11, and modes as a function of normalized diameter ( D r 0 ) for ϵ = 0 and ϵ = 0.25 .

Fig. 10
Fig. 10

Strehl ratio η after modal correction of 3, 6, 11, and modes as a function of normalized diameter ( D r 0 ) for ϵ = 0 and ϵ = 0.25 .

Fig. 11
Fig. 11

Short-exposure MTF for different values of obscuration ratio and normalized diameter D r 0 .

Fig. 12
Fig. 12

Short-exposure Strehl ratio S for different values of obscuration ratio as a function of the normalized diameter D r 0 .

Fig. 13
Fig. 13

Short-exposure Strehl ratio η for different values of obscuration ratio as a function of the normalized diameter D r 0 .

Tables (7)

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Table 1 Expansion Coefficients g n ( n , m ; ϵ ) for a Radial Annular Polynomial R n m ( ρ ; ϵ ) for n 6

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Table 2 Expansion Coefficients h n ( n ; ϵ ) for a Polynomial R n m ( ϵ ρ ) for n 6

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Table 3 Radial Part T n m ( κ ; ϵ ) of the Fourier Transform of a Zernike Annular Polynomial for n 6

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Table 4 Variances of Zernike Annular Coefficients in Units of ( D r 0 ) 5 3 for Various Values of Obscuration Ratio ϵ

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Table 5 Covariances of Zernike Annular Coefficients in Units of ( D r 0 ) 5 3 for Various Values of Obscuration Ratio ϵ

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Table 6 Mean Square Value Δ J of the Residual Phase Error in Units of ( D r 0 ) 5 3 When the First J Modes of Aberrations Are Corrected, for Several Values of ϵ

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Table 7 Mode Shape Functions ( d j ) and Cross-Mode Shape Functions ( c j , j ) up to n 6 for Zernike Circle Polynomials

Equations (62)

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Φ ( ρ , θ ; ϵ ) = j a j ( ϵ ) Z j ( ρ , θ ; ϵ ) ,
Z j ( ρ , θ ; ϵ ) = Z n m ( ρ , θ ; ϵ ) = n + 1 R n m ( ρ ; ϵ ) Θ m ( θ ) ,
Θ m ( θ ) = { 2 cos m θ m 0 even j 2 sin m θ m 0 odd j 1 m = 0 } ,
1 π ( 1 ϵ 2 ) ϵ 1 0 2 π Z j ( ρ , θ ; ϵ ) Z j ( ρ , θ ; ϵ ) ρ d ρ d θ = δ j j ,
ϵ 1 R n m ( ρ ; ϵ ) R n m ( ρ ; ϵ ) ρ d ρ = 1 ϵ 2 2 ( n + 1 ) δ n n .
R n m ( ρ ; ϵ ) = N n m [ R n m ( ρ ) j = 1 ( n m ) 2 R n 2 j m ( ρ ; ϵ ) 2 ( n 2 j + 1 ) 1 ϵ 2 ϵ 1 R n m ( ρ ) R n 2 j m ( ρ ; ϵ ) ρ d ρ ] ,
a j ( ϵ ) = 1 π ( 1 ϵ 2 ) ϵ 1 0 2 π Φ ( ρ , θ ; ϵ ) Z j ( ρ , θ ; ϵ ) ρ d ρ d θ .
R n m ( ρ ; ϵ ) = n = 0 n g n ( n , m ; ϵ ) R n m ( ρ ) ,
g n ( n , m ; ϵ ) = N n m ( ϵ ) 2 1 ϵ 2 ϵ 1 R n m ( ρ ) R n m ( ρ ; ϵ ) ρ d ρ .
ρ n = [ ( n + m ) 2 ] ! [ ( n m ) 2 ] ! s = 0 ( n m ) 2 ( n + 1 2 s ) R n 2 s m ( ρ ) s ! ( n + 1 s ) ! ,
R n m ( ϵ ρ ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! ( ϵ ρ ) n 2 s s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! = n = 0 n h n ( n ; ϵ ) R n m ( ρ ) ,
h n ( n ; ϵ ) = s = 0 ( n m ) 2 s = 0 [ ( n m ) 2 ] s ( 1 ) s ϵ n 2 s ( n s ) ! ( n + 1 2 s 2 s ) s ! s ! ( n + 1 2 s s ) ! .
S j ( κ , ϕ ; ϵ ) = ϵ 1 0 2 π Z j ( ρ , θ ; ϵ ) exp ( i 2 π ρ κ ) d ρ = n + 1 ϵ 1 R n m ( ρ ; ϵ ) ρ d ρ 0 2 π exp [ i 2 π κ ρ cos ( θ ϕ ) ] Θ m ( θ ) d θ = 2 π ( 1 ) 3 m 2 n + 1 Θ m ( ϕ ) ϵ 1 R n m ( ρ ; ϵ ) J m ( 2 π κ ρ ) ρ d ρ = 2 π ( 1 ) 3 m 2 n + 1 Θ m ( ϕ ) n = 0 n g n ϵ 1 R n m ( ρ ) J m ( 2 π κ ρ ) ρ d ρ ,
0 1 R n m ( ρ ) J m ( κ ρ ) ρ d ρ = ( 1 ) ( n m ) 2 J n + 1 ( κ ) κ ,
ϵ 1 R n m ( ρ ) J m ( 2 π κ ρ ) ρ d ρ = 0 1 R n m ( ρ ) J m ( 2 π κ ρ ) ρ d ρ 0 ϵ R n m ( ρ ) J m ( 2 π κ ρ ) ρ d ρ = 0 1 R n m ( ρ ) J m ( 2 π κ ρ ) ρ d ρ ϵ 2 0 1 R n m ( ϵ ρ ) J m ( 2 π κ ϵ ρ ) ρ d ρ = ( 1 ) ( n m ) 2 J n + 1 ( 2 π κ ) 2 π κ ϵ 2 n = 0 n h n 0 1 R n m ( ρ ) J m ( 2 π κ ϵ ρ ) ρ d ρ = 1 2 π κ ( 1 ) ( n m ) 2 [ J n + 1 ( 2 π κ ) ϵ n = 0 n ( 1 ) ( n n ) 2 h n J n + 1 ( 2 π ϵ κ ) ] .
S j ( κ , ϕ ; ϵ ) = n + 1 T n m ( κ ; ϵ ) Θ m ( ϕ ) ,
T n m ( κ ; ϵ ) = 1 κ n = 0 n g n ( n , m ; ϵ ) ( 1 ) ( n 2 ) + m [ J n + 1 ( 2 π κ ) ϵ n = 0 n ( 1 ) ( n n ) 2 h n ( n ; ϵ ) J n + 1 ( 2 π ϵ κ ) ] .
D ( ρ , ρ ) = 6.88 2 5 3 ( D r 0 ) 5 3 ρ ρ 5 3 ,
D Φ ( ρ , ρ ) = Φ ( ρ ) Φ ( ρ ) 2 = 2 [ R Φ ( 0 ) R Φ ( ρ ρ ) ] ,
R Φ ( ρ ρ ) = Φ ( ρ ) Φ ( ρ )
Q Φ ( κ ) = R Φ ( ρ ) exp ( i 2 π ρ κ ) d ρ = 2 π R Φ ( ρ ) J 0 ( 2 π κ ρ ) ρ d ρ ,
R Φ ( ρ ) = Q Φ ( κ ) J 0 ( 2 π κ ρ ) d κ = 2 π Q Φ ( κ ) J 0 ( 2 π κ ρ ) κ d κ .
Q Φ ( κ ) = 0.023 r 0 5 3 κ 11 3 ,
τ ( ν ; ϵ ) = 1 1 ϵ 2 [ τ ( ν ) τ 12 ( ν ; ϵ ) + ϵ 2 τ ( ν ϵ ) ] ,
τ ( ν ) = 2 π [ cos 1 ( ν ) ν 1 ν 2 ] ( 0 ν 1 )
τ 1 , 2 ( ν ; ϵ ) = { 2 ϵ 2 0 ν ( 1 ϵ ) 2 2 π ( θ 1 + ϵ 2 θ 2 2 ν sin θ 1 ) ( 1 ϵ ) 2 ν ( 1 + ϵ ) 2 0 otherwise } .
cos θ 1 = 4 ν 2 + 1 ϵ 2 4 ν ,
cos θ 2 = 4 ν 2 1 + ϵ 2 4 ϵ ν .
τ ( ν ; ϵ ; D r 0 ) = τ ( ν ; ϵ ) exp [ 1 2 D ( ν ) ] = τ ( ν ; ϵ ) M ( ν ) ,
S ( ϵ ; D r 0 ) = 8 1 ϵ 2 0 1 τ ( ν ; ϵ ; D r 0 ) ν d ν ,
η ( ϵ ; D r 0 ) = ( 1 ϵ 2 ) ( D r 0 ) 2 S ( ϵ ; D r 0 ) .
a j * ( ϵ ) a j ( ϵ ) = 1 π 2 ( 1 ϵ 2 ) 2 ϵ 1 0 2 π R Φ ( ρ ρ ) Z j * ( ρ ; ϵ ) Z j ( ρ ; ϵ ) ρ d ρ d θ .
a j * ( ϵ ) a j ( ϵ ) = 1 π 2 ( 1 ϵ 2 ) 2 S j * ( κ ; ϵ ) S j ( κ ; ϵ ) d κ d κ × exp [ i 2 π ( κ κ ) ρ ] R Φ ( ρ ρ ) exp [ i 2 π κ ( ρ ρ ) ] d ρ d ρ = 1 π 2 ( 1 ϵ 2 ) 2 0 0 2 π S j * ( κ , ϕ ; ϵ ) Q ( 2 κ D ) S j ( κ , ϕ ; ϵ ) κ d κ d ϕ = 0.023 2 2 3 π ( 1 ϵ 2 ) 2 ( n + 1 ) ( n + 1 ) ( D r 0 ) 5 3 0 [ T n m ( κ ; ϵ ) ] * T n m ( κ ; ϵ ) κ 8 3 d κ = 0.023 2 2 3 π ( 1 ϵ 2 ) 2 ( D r 0 ) 5 3 I ( n , n ; ϵ ) ,
Φ 2 = R Φ ( 0 ) = 0.023 π 2 2 3 ( D r 0 ) 5 3 0 κ 8 3 d κ ,
a 1 2 ( ϵ ) = 0.023 2 2 3 π ( D r 0 ) 5 3 0 [ T 0 0 ( κ ; ϵ ) ] 2 κ 8 3 d κ .
Δ 1 ( ϵ ) = Φ 2 a 1 2 ( ϵ ) = 0.046 π ( D 2 r 0 ) 5 3 0 κ 8 3 d κ 0.046 π ( 1 ϵ 2 ) 2 0 ( D 2 r 0 ) 5 3 1 κ 2 [ J 1 ( 2 π κ ) ϵ J 1 ( 2 π ϵ κ ) ] 2 κ 8 3 d κ = 0.046 π 8 3 ( D r 0 ) 5 3 0 { 1 1 + ϵ 17 3 ( 1 ϵ 2 ) 2 [ 2 J 1 ( x ) x ] 2 + 8 ϵ ( 1 ϵ 2 ) 2 [ J 1 ( x ) J 1 ( ϵ x ) x 2 ] } x 8 3 d x = 0.046 π 11 3 2 5 3 Γ ( 17 6 ) Γ ( 11 6 ) ( 1 ϵ 2 ) 2 [ Γ ( 14 3 ) ( 1 + ϵ 17 3 ) Γ ( 17 6 ) Γ ( 23 6 ) 2 ϵ 2 F 1 ( 5 6 , 11 6 ; 2 ; ϵ 2 ) ] ( D r 0 ) 5 3 ,
Δ 1 ( ϵ ) = 3 ( 6.88 ) 11 π ( 1 ϵ 2 ) 2 [ 8 ( 4 3 ) ! ( 1 2 ) ! ( 17 6 ) ! ( 1 + ϵ 17 3 ) 2 2 3 π ϵ 2 ( 1 + ϵ ) 5 3 F 1 ( 5 6 , 3 2 ; 3 ; 4 ϵ ( 1 + ϵ ) 2 ) ] ( D r 0 ) 5 3 .
a 2 2 ( ϵ ) = a 3 2 ( ϵ ) = 0.023 Γ ( 1 6 ) π 8 3 2 2 3 ( 1 + ϵ 2 ) ( 1 ϵ 2 ) 2 Γ ( 17 6 ) ( D r 0 ) 5 3 [ ( 1 + ϵ 23 3 ) Γ ( 14 3 ) Γ ( 17 6 ) Γ ( 29 6 ) ϵ 4 F 1 ( 1 6 , 11 6 ; 3 ; ϵ 2 ) ] .
a n n ( ϵ ) 2 = 1 π 2 ( 1 ϵ 2 ) 2 0 [ S n n ( κ , ϕ ; ϵ ) ] * Q ( 2 κ D ) S n n ( κ , ϕ ; ϵ ) κ d κ = 0.023 ( D 2 r 0 ) 5 3 2 π ( 1 ϵ 2 ) 2 0 [ n + 1 κ 1 + ϵ 2 + + ϵ 2 n ] 2 [ J n + 1 ( 2 π κ ) ϵ n + 1 J n + 1 ( 2 π ϵ κ ) ] 2 κ 8 3 d κ = 0.023 ( n + 1 ) Γ ( n 5 6 ) π 8 3 2 5 3 Γ ( 17 6 ) ( 1 ϵ 2 ) [ 1 ϵ 2 ( n + 1 ) ] ( D r 0 ) 5 3 [ ( 1 + ϵ 2 n + 17 3 ) Γ ( 14 3 ) Γ ( 17 6 ) Γ ( n + 23 6 ) 2 ϵ 2 ( n + 1 ) ( n + 1 ) ! F 1 ( n 5 6 , 11 6 ; n + 2 ; ϵ 2 ) ] .
a n n 2 = 0.023 ( n + 1 ) Γ ( 14 3 ) Γ ( n 5 6 ) π 8 3 2 5 3 [ Γ ( 17 6 ) ] 2 Γ ( n + 23 6 ) ( D r 0 ) 5 3 .
Φ c ( ρ ; ϵ ) = j = 1 J a j ( ϵ ) Z j ( ρ ; ϵ ) .
Φ J ( ρ ; ϵ ) = Φ ( ρ ) Φ c ( ρ ; ϵ ) = j = J + 1 a j ( ϵ ) Z j ( ρ ; ϵ ) .
Δ J ( ϵ ) = Δ 1 ( ϵ ) j = 2 J a j 2 ( ϵ ) .
D J ( ρ , ρ ; ϵ ) = [ Φ J ( ρ ) Φ J ( ρ ) ] 2 = { [ Φ ( ρ ) Φ ( ρ ) ] [ Φ c ( ρ ) Φ c ( ρ ) ] } 2 = [ Φ ( ρ ) Φ ( ρ ) ] 2 + [ Φ c ( ρ ) Φ c ( ρ ) ] 2 2 [ Φ ( ρ ) Φ ( ρ ) ] [ Φ c ( ρ ) Φ c ( ρ ) ] = D Φ ( ρ , ρ ) [ Φ c ( ρ ) Φ c ( ρ ) ] 2 2 [ Φ c ( ρ ) Φ c ( ρ ) ] [ Φ J ( ρ ) Φ J ( ρ ) ] .
D J ( ν ; ϵ ) = D Φ ( ν ; ϵ ) j = 2 J a j 2 ( ϵ ) d j ( ν ; ϵ ) 2 j = 2 J j = J + 1 a j ( ϵ ) a j ( ϵ ) c j , j ( ν ; ϵ ) ,
d j ( ν ; ϵ ) = w ( ν ) [ Z j ( ρ ; ϵ ) Z j ( ρ ; ϵ ) ] 2 d 3 ν ,
c k , l ( ν ; ϵ ) = w ( ν ) [ Z k ( ρ ; ϵ ) Z k ( ρ ; ϵ ) ] [ Z l ( ρ ; ϵ ) Z l ( ρ ; ϵ ) ] d 3 ν .
w ( ν ) = 1 2 π 2 ( 1 ν ) 2 .
c j , j ( ν ; ϵ ) = w ( ν ) k = 1 j g k ( j ; ϵ ) [ Z k ( ρ ) Z k ( ρ ) ] l = 1 j g l ( j ; ϵ ) [ Z l ( ρ ) Z l ( ρ ) ] d 3 ν = k = 1 j g k ( j ; ϵ ) l = 1 j g l ( j ; ϵ ) { w ( ν ) [ Z k ( ρ ) Z k ( ρ ) ] [ Z l ( ρ ) Z l ( ρ ) ] d 3 ν } = k = 1 j g k ( j ; ϵ ) l = 1 j g l ( j ; ϵ ) c k , l ( ν ) .
d 4 ( ν ; ϵ ) = [ g 2 ( 2 , 0 ; ϵ ) ] 2 d 4 ( ν ) = 48 ν 2 ( 1 ν ) 2 ( 1 ϵ 2 ) 2 ,
c 2 , 8 ( ν ; ϵ ) = g 1 ( 1 , 1 ; ϵ ) g 1 ( 3 , 1 ; ϵ ) c 2 , 8 ( ν ) + [ g 1 ( 1 , 1 ; ϵ ) ] 2 d 2 ( ν ) = 8 2 ν 2 [ 6 ( 1 + ϵ 2 ) ν 2 6 ( 1 + ϵ 2 ) ν + 1 + ϵ 2 2 ϵ 4 ] ( 1 ϵ 2 ) 1 + 6 ϵ 2 + 10 ϵ 4 + 6 ϵ 6 + ϵ 8 .
M J ( ν ) = exp { i [ Φ J ( ρ ) Φ J ( ρ ) ] 2 } = exp { 1 2 D J ( ν ; ϵ ) } ,
τ ( ν ; ϵ ; D r 0 ) = τ ( ν ; ϵ ) exp [ 1 2 D J ( ν ; ϵ ) ] .
D 3 ( ν ; ϵ ) = D Φ ( ν ) 2 a 2 2 ( ϵ ) d 2 ( ν ; ϵ ) 4 a 2 ( ϵ ) a 8 ( ϵ ) c 2 , 8 ( ν ; ϵ ) = 6.8839 ν 5 3 ( D r 0 ) 5 3 16 ν 2 1 + ϵ 2 a 2 2 ( ϵ ) 32 2 ν 2 [ 6 ( 1 + ϵ 2 ) ν 2 6 ( 1 + ϵ 2 ) ν + 1 + ϵ 2 2 ϵ 4 ] ( 1 ϵ 2 ) 1 + 6 ϵ 2 + 10 ϵ 4 + 6 ϵ 6 + ϵ 8 a 2 ( ϵ ) a 8 ( ϵ ) .
D 3 ( ν ; 0 ) = D Φ ( ν ) 16 ν 2 a 2 2 ( 0 ) 32 2 ν 2 ( 6 ν 2 6 ν + 1 ) a 2 ( 0 ) a 8 ( 0 )
= 6.8839 ν 5 3 ( 1 0.9503 ν 1 3 0.5585 ν 4 3 + 0.5585 ν 7 3 ) ( D r 0 ) 5 3 .
τ ( ν ; ϵ ; D r 0 ) SE = τ ( ν ; ϵ ) exp [ 1 2 D 3 ( ν ; ϵ ) ] .
I ( n , n ; ϵ ) ( n + 1 ) ( n + 1 ) 0 [ T n m ( κ ; ϵ ) ] * T n m ( κ ; ϵ ) κ 8 3 d κ = ( n + 1 ) ( n + 1 ) 0 { n 1 = 0 n g n 1 ( n , m ; ϵ ) ( 1 ) ( n 1 2 ) + m [ J n 1 + 1 ( 2 π κ ) ϵ n 2 = 0 n 1 ( 1 ) ( n 2 n 1 ) 2 h n 2 ( n 1 ; ϵ ) J n 2 + 1 ( 2 π ϵ κ ) ] } * { n 3 = 0 n g n 3 ( n , m ; ϵ ) ( 1 ) ( n 3 2 ) + m [ J n 3 + 1 ( 2 π κ ) ϵ n 4 = 0 n 3 ( 1 ) ( n 4 n 3 ) 2 h n 4 ( n 3 ; ϵ ) J n 4 + 1 ( 2 π ϵ κ ) ] } κ 14 3 d κ = ( 1 ) n ( n + 1 ) ( n + 1 ) n 1 = 0 n n 3 = 0 n g n 1 ( n , m ; ϵ ) g n 3 ( n , m ; ϵ ) [ ( 1 ) ( n 1 + n 3 ) 2 I 1 ϵ n 4 = 0 n 3 ( 1 ) ( n 1 + n 4 ) 2 h n 4 ( n 3 ; ϵ ) I 2 ϵ n 2 = 0 n 1 ( 1 ) ( n 2 + n 3 ) 2 h n 2 ( n 1 ; ϵ ) I 3 + ϵ 2 h 2 = 0 n 1 n 4 = 0 n 3 ( 1 ) ( n 2 + n 4 ) 2 h n 2 ( n 1 ; ϵ ) h n 4 ( n 3 ; ϵ ) I 4 ] δ m m ,
I 1 = 0 J n 1 + 1 ( 2 π κ ) J n 3 + 1 ( 2 π κ ) κ 14 3 d κ = Γ ( 14 3 ) Γ [ ( n 1 + n 3 ) 2 5 6 ] π 11 3 2 Γ [ ( n 1 n 3 ) 2 + 17 6 ] Γ [ ( n 3 n 1 ) 2 + 17 6 ] Γ [ ( n 1 + n 3 ) 2 + 23 6 ] ,
I 2 = 0 J n 1 + 1 ( 2 π κ ) J n 4 + 1 ( 2 π ϵ κ ) κ 14 3 d κ = Γ [ ( n 1 + n 4 ) 2 5 6 ] π 11 3 ϵ n 4 + 1 2 ( n 4 + 1 ) ! Γ [ ( n 1 n 4 ) 2 + 17 6 ] × F 1 [ n 1 + n 4 2 5 6 , n 4 n 1 2 11 6 ; n 4 + 2 ; ϵ 2 ] ,
I 3 = 0 J n 2 + 1 ( 2 π ϵ κ ) J n 3 + 1 ( 2 π κ ) κ 14 3 d κ = Γ [ ( n 2 + n 3 ) 2 5 6 ] π 11 3 ϵ n 2 + 1 2 ( n 2 + 1 ) ! Γ [ ( n 3 n 2 ) 2 + 17 6 ] × F 1 [ n 2 + n 3 2 5 6 , n 2 n 3 2 11 6 ; n 2 + 2 ; ϵ 2 ] ,
I 4 = 0 J n 2 + 1 ( 2 π ϵ κ ) J n 4 + 1 ( 2 π ϵ κ ) κ 14 3 d κ = Γ ( 14 3 ) Γ [ ( n 2 + n 4 ) 2 5 6 ] ( ϵ π ) 11 3 2 Γ [ ( n 2 n 4 ) 2 + 17 6 ] Γ [ ( n 4 n 2 ) 2 + 17 6 ] Γ [ ( n 2 + n 4 ) 2 + 23 6 ] .

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