Abstract

Multiple laser backscatter references (or guide stars) and optimal processing of the data are used to extend the useful diameter that is obtained with the use of a single laser backscatter reference for imaging at infinite conjugates. The results illustrate that the mean-square error in the estimated wave-front distortion E2 is proportional to D5/3, where D is the telescope diameter. For all the cases that are considered the results are expressible in the form E2 = (D/d0)5/3, where the quantity d0 is a measure of the useful diameter of the telescope. When nine references are used d0 is triple the value that would be achieved with a single reference when a backscatter altitude of 20 km is used. The results illustrate the impact of varying the altitude, changing the number of refrences, having references at two altitudes, and introducing reference-position uncertainty.

© 1994 Optical Society of America

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References

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  1. J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefront,” (Optical Sciences Company, Placentia, Calif., 1984).
  2. G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).
  3. R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).
  4. 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]
  5. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  6. G. A. Tyler, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston-tilt removed phase cross-covariance,” (Optical Sciences Company, Placentia, Calif., 1984).
  7. R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1405 (1992).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 22.9.3.
  9. Ref. 8, Eq. 22.3.12.
  10. Ref. 8, Eq. 15.1.1.

1992 (1)

1991 (1)

1990 (1)

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 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 22.9.3.

Belsher, J. F.

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefront,” (Optical Sciences Company, Placentia, Calif., 1984).

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Fried, D. L.

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefront,” (Optical Sciences Company, Placentia, Calif., 1984).

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

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]

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Sasiela, R. J.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 22.9.3.

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.

G. A. Tyler, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston-tilt removed phase cross-covariance,” (Optical Sciences Company, Placentia, Calif., 1984).

G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

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]

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

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

Proc. IEEE (1)

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]

Other (6)

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefront,” (Optical Sciences Company, Placentia, Calif., 1984).

G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 22.9.3.

Ref. 8, Eq. 22.3.12.

Ref. 8, Eq. 15.1.1.

G. A. Tyler, “The utility of Gegenbauer polynomials in atmospheric turbulence calculations: evaluation of piston-tilt removed phase cross-covariance,” (Optical Sciences Company, Placentia, Calif., 1984).

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Figures (7)

Fig. 1
Fig. 1

Impact of focus anisoplanatism: The normalized antenna gain (or Strehl ratio) that is achieved with one laser backscatter reference at the indicated altitude is plotted versus the aperture diameter. In all the cases a zenith angle of 45° is assumed. The short-dashed, solid, and long-dashed curves correspond to wavelengths that are equal to 0.5, 0.41, and 0.35 μm, respectively. The Submarine Laser Communications Day turbulence model was used to obtain these results.

Fig. 2
Fig. 2

Three methods of combining multiple wave fronts to reduce the impact of focus anisoplanatism: (a) butting, (b) stitching, (c) merging.

Fig. 3
Fig. 3

The modified hexagonal close-packed sample-point geometry that was chosen for the study. (a) The circular aperture of the system was decomposed into 61 equal area regions, and the position of each phase point in the computation grid was adjusted so that it was at the center of mass location within each region. (b) A limited study was performed to illustrate that d0 asymptotes to almost its exact value as the number of aperture sample points is increased to 61. The curves correspond to two cases in which the reference altitude is assumed to be equal to 20 km. NR designates the number of references that were used, and H designates the backscatter altitude. In obtaining the solution it was assumed that the wavelength λ = 0.5 μm and the zenith angle ψ = 0°.

Fig. 4
Fig. 4

Reference configurations that were studied. (a) NR = 1, (b) NR = 4, (c) NR = 9, (d) NR = 2, (e) NR = 8. In (a)–(c) the references are assumed to be at one altitude. In (d) and (e) the dots within the circles illustrate that references at a given altitude and at one half a given altitude are assumed to be at the location shown.

Fig. 5
Fig. 5

Value of d0 versus backscatter altitude for various reference configurations presented in Fig. 4. It is assumed that the piston and the tilt are removed from the reconstructed wave front and that λ = 0.5 μm and ψ = 0°. Also, the Submarine Laser Communications Day turbulence model was used. For comparison the short-dashed curve represents d0 associated with focus anisoplanatism. The lower solid curve shows that processing the wave front associated with a single reference increases d0 very little. Similarly, the long-dashed curve shows that two references, one at the given altitude and one at half the given altitude, result in only limited improvement. The top curve shows the improvement that can be obtained with four references. The lowest dot, at 20-km altitude, shows that using eight references, four at 20 km and four at 10 km, results in little improvement over the four-reference case. The highest dot, at 20 km, shows the performance when nine references are used.

Fig. 6
Fig. 6

Dependence of d0 on the number of references. Here λ = 0.5 μm, ψ = 0, H = 20 km, and the Submarine Laser Communications Day turbulence model is used. The one-, four-, and nine-reference results, indicated by filled circles, fall almost exactly on the line d0 = 0.23Nr + 0.95.

Fig. 7
Fig. 7

Impact of beacon-position uncertainty. (a) Illustrates the results for four references, and (b) illustrates the results for nine references. In both (a) and (b) the backscatter altitude pertaining to each of the curves is expressed in kilometers. In addition, it is assumed that the beacon-position uncertainty associated with each of the spherical waves is uncorrelated.

Equations (151)

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ϕ ^ ( r ) = n = 1 N R d r W ( r / R ) h n ( r , r ) ϕ n ( r ) / d r W ( r / R ) ,
W ( r ) = { 1 r 1 0 otherwise ,
E 2 = d r W ( r / R ) [ ϕ ( r ) - ϕ ^ ( r ) ] 2 / d r W ( r / R ) ,
E 2 = ( D / d 0 ) 5 / 3 ,
r = R ρ ,
r = R ρ .
ϕ ^ ( R ρ ) = 1 π n = 1 N R d ρ W ( ρ ) h ( R ρ , R ρ ) ϕ n ( R ρ ) ,
E 2 = 1 π d ρ W ( ρ ) [ ϕ ( R ρ ) - ϕ ^ ( R ρ ) ] 2 .
d r W ( r / R ) = π R 2 .
E 2 = 1 π d ρ W ( ρ ) [ ϕ 2 ( R ρ ) - 2 π n = 1 N R d ρ W ( ρ ) × h n ( R ρ , R ρ ) ϕ ( R ρ ) ϕ n ( R ρ ) + 1 π 2 n = 1 N R m = 1 N R d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) h n ( R ρ , R ρ 1 ) × h m ( R ρ , R ρ 2 ) ϕ n ( R ρ 1 ) ϕ m ( R ρ 2 ) ] .
H m ( R ρ , R ρ ) = h m ( R ρ , R ρ ) + m g m ( R ρ , R ρ ) ,
E 2 = 1 π d ρ W ( ρ ) { ϕ 2 ( R ρ ) - 2 π n = 1 N R d ρ W ( ρ ) × [ h n ( R ρ , R ρ ) + n g n ( R ρ , R ρ ) ] ϕ ( R ρ ) ϕ n ( R ρ ) + 1 π 2 n = 1 N R m = 1 N R d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) × [ h n ( R ρ , R ρ 1 ) + n g n ( R ρ , R ρ 1 ) ] × [ h m ( R ρ , R ρ 2 ) + m g m ( R ρ , R ρ 2 ) ] × ϕ n ( R ρ 1 ) ϕ m ( R ρ 2 ) } .
E 2 l = 1 π d ρ W ( ρ ) { - 2 π d ρ W ( ρ ) × g l ( R ρ , R ρ ) ϕ ( R ρ ) ϕ l ( R ρ ) + 1 π 2 n = 1 N R d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) × [ h n ( R ρ , R ρ 1 ) + n g n ( R ρ , R ρ 1 ) ] × g l ( R ρ , R ρ 2 ) ϕ n ( R ρ 1 ) ϕ l ( R ρ 2 ) + 1 π 2 m = 1 N R d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) × [ h m ( R ρ , R ρ 2 ) + m g m ( R ρ , R ρ 2 ) ] × g l ( R ρ , R ρ 1 ) ϕ l ( R ρ 1 ) ϕ m ( R ρ 2 ) } .
- 2 π 2 d ρ d ρ 1 W ( ρ ) W ( ρ 1 ) g l ( R ρ , R ρ 1 ) × [ ϕ ( R ρ ) ϕ l ( R ρ 1 ) - 1 π m = 1 N R d ρ 2 W ( ρ 2 ) × h m ( R ρ , R ρ 2 ) ϕ m ( R ρ 2 ) ϕ l ( R ρ 1 ) ] = 0.
ϕ ( R ρ ) ϕ l ( R ρ 1 ) = 1 π m = 1 N R d ρ 2 W ( ρ 2 ) h m ( R ρ , R ρ 2 ) × ϕ m ( R ρ 2 ) ϕ l ( R ρ 1 ) ,
E 2 = 1 π d ρ W ( ρ ) [ ϕ 2 ( R ρ ) - 1 π n = 1 N R d ρ W ( ρ ) × h n ( R ρ , R ρ ) ϕ ( R ρ ) ϕ n ( R ρ ) ] .
ϕ ( R ρ ) ϕ l ( R ρ 1 ) / ( D / r 0 ) 5 / 3 = 1 π m = 1 N R d ρ 2 W ( ρ 2 ) h m ( R ρ , R ρ 2 ) × ϕ m ( R ρ 2 ) ϕ l ( R ρ 1 ) / ( D / r 0 ) 5 / 3 ,
E 2 / ( D / r 0 ) 5 / 3 = 1 π d ρ W ( ρ ) [ ϕ 2 ( R ρ ) / ( D / r 0 ) 5 / 3 - 1 π n = 1 N R d ρ W ( ρ ) h n ( R ρ , R ρ ) × ϕ ( R ρ ) ϕ n ( R ρ ) / ( D / r 0 ) 5 / 3 ] .
ϕ a ( R ρ 1 ) ϕ b ( R ρ 2 ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 [ d h C n 2 ( h ) ] - 1 0 H b d h C h 2 ( h ) ( 1 - h / H a ) 5 / 3 × [ Q 5 / 3 F 1 ( ρ 1 + Ω Q ) + F 1 ( Q ρ 2 - Ω ) - ρ 1 - Q ρ 2 + Ω 5 / 3 - F 2 ( Q , Ω ) ] ,
F 1 ( Ω ) = { / ₁₁ F 2 1 ( - ¹¹ / , - ; 1 ; Ω 2 ) for Ω 1 Ω 5 / 3 F 2 1 ( - , - ; 2 ; 1 / Ω 2 ) for Ω 1 ,
F 2 ( Q , Ω ) = 2 π Q 2 0 1 + Q d ρ K 1 ( ρ , Q ) H ( ρ , Ω ) ,
K 1 ( ρ , Q ) = { π Q 2 for ρ 1 - Q cos - 1 ( t 0 ) - t 0 1 - t 0 2 + Q 2 [ cos - 1 ( s 0 / Q ) - ( s 0 / Q ) 1 - ( s 0 / Q ) 2 for 1 - Q < ρ < 1 + Q , 0 otherwise
t 0 = 1 - Q 2 2 ρ + ½ ρ ,
s 0 = - 1 - Q 2 2 ρ + ½ ρ ,
H ( p , Ω ) = { ρ 8 / 3 F 2 1 ( - , - ; 1 ; Ω 2 / ρ 2 ) for Ω ρ ρ Ω 5 / 3 F 2 1 ( - , - ; 1 ; ρ 2 / Ω 2 ) for Ω ρ ,
F 2 1 ( α , β ; γ ; z ) = Γ ( γ ) Γ ( α ) Γ ( β ) n = 0 Γ ( α + n ) Γ ( β + n ) Γ ( γ + n ) z n n ! ,
Q = ( 1 - h / H b ) / ( 1 - h / H a ) ,
Ω = ( r a / R H a - r b / R H b ) h / ( 1 - h / H a ) ,
H a H b .
ϕ ( R ρ 1 ) ϕ b ( R ρ 2 ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 [ d h C n 2 ( h ) ] - 1 0 H d h C n 2 ( h ) × [ Q 0 5 / 3 F 1 ( ρ 1 + Ω 0 Q 0 ) + F 1 ( Q 0 ρ 2 - Ω 0 ) - ρ 1 - Q 0 ρ 2 + Ω 0 5 / 3 - F 2 ( Q 0 , Ω 0 ) + 4 G 1 ( ρ 1 , Q 0 ρ 2 - Ω 0 ) - 4 G 2 ( ρ 1 , Q 0 - Ω 0 ) ] ,
G 1 ( r 1 , r 2 ) = { - / ₁₁ r 1 · r 12 F 1 ( - ¹¹ / , ; 2 ; r 2 2 ) for r 2 1 - / ₁₂ r 1 - 2 / 3 r 1 · r 22 F 1 ( - , ; 3 ; 1 / r 2 2 ) for r 2 1 ,
G 2 ( r , Q , Ω ) = 2 r · Ω π Q 2 Ω Q - Ω Q + Ω d ρ ρ 2 Ω { 1 - [ ( Ω 2 + ρ 2 - Q 2 ) / ( 2 ρ Ω ) ] 2 } 1 / 2 × G ^ 1 ( r , ρ ) ,
G ^ 1 ( r 1 , r 2 ) = { - / ₁₁ F 2 1 ( - ¹¹ / , ; 2 ; r 2 2 ) for r 2 1 - / ₁₂ r 2 - 2 / 3 F 2 1 ( - , ; 3 ; 1 / r 2 2 ) for r 2 1 ,
Q 0 = 1 - h / H b ,
Ω 0 = - ( r b / R ) ( h / H b ) .
ϕ 2 ( R ρ ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 { 6 11 [ 2 2 F 1 ( - ¹¹ / , - ; 1 ; ρ 2 ) - F 2 1 ( - ¹¹ / , - ; 2 ; 1 ) ] - 20 11 ρ 2 [ 2 2 F 1 ( - ¹¹ / , ; 2 ; ρ 2 ) - F 2 1 ( - ¹¹ / , ; 3 ; 1 ) ] } .
d 0 = 0.23 N R + 0.95
ϕ θ ( r ) = k θ · r .
ϕ θ ( r 1 ) ϕ θ ( r 2 ) = k 2 ( θ x x 1 + θ y y 1 ) ( θ x x 2 + θ y y 2 ) ,
θ x 2 = σ θ 2 ,
θ y 2 = σ θ 2 ,
θ x θ y = 0.
ϕ θ ( r 1 ) ϕ θ ( r 2 ) = k 2 σ θ 2 r 1 · r 2 .
r 1 = R ρ 1 ,
r 2 = R ρ 2 ,
k = 2 π / λ ,
ϕ θ ( R ρ 1 ) ϕ θ ( R ρ 2 ) / ( D / r 0 ) 5 / 3 = σ N 2 ρ 1 · ρ 2 ,
σ N 2 = π 2 σ θ 2 ( λ / D ) 2 ( D / r 0 ) 5 / 3 .
ϕ a ( R ρ 1 ) ϕ b ( R ρ 2 ) / ( D / r 0 ) 5 / 3 ϕ a ( R ρ 1 ) ϕ b ( R ρ 2 ) / ( D / r 0 ) 5 / 3 + σ N 2 ρ 1 · ρ 2 δ a b ,
ϕ T ( r ) = k 0 R d z n [ r ( 1 - z / R ) + r α z / R , z ] ,
ϕ ¯ = 1 π R 2 d r W ( r / R ) ϕ T ( r ) ,
θ = 4 π k R 4 d r W ( r / R ) ϕ T ( r ) r .
W ( r ) = { 1 for r < 1 0 otherwise .
ϕ ( r ) = ϕ T ( r ) - ϕ ¯ - γ k θ · r .
ϕ a ( r ) = k π R 2 0 R a d r d z W ( r / R ) { n [ r ( 1 - z R a ) + r a z R a , z ] - n [ r ( 1 - z R a ) + r a z R a , z ] - 4 γ a R 2 r · r n [ r ( 1 - z R a ) + r a z R a , z ] } ,
ϕ b ( r ) = k π R 2 0 R b d r d z W ( r / R ) { n [ r ( 1 - z R b ) + r b z R b , z ] - n [ r ( 1 - z R b ) + r b z R b , z ] - 4 γ b R 2 r · r n [ r ( 1 - z R b ) + r b z R b , z ] } .
ϕ a ( r 1 ) ϕ b ( r 2 ) = k 2 π 2 R 4 0 R b 0 R b d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) × { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - 4 γ a R 2 r 1 · r 1 n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] } × { n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] - 4 γ b R 2 r 2 · r 2 n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } .
R b R a .
[ n ( r 1 , z 1 ) - n ( r 1 , z 1 ) ] n ( r 2 , z 2 ) = - 1 2 { [ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 - [ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 } .
( r 1 · r 1 ) ( r 2 · r 2 ) n ( r 1 , z 1 ) n ( r 2 , z 2 ) = - 1 2 ( r 1 · r 1 ) ( r 2 · r 2 ) [ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2
ϕ a ( r 1 ) ϕ b ( r 2 ) = - k 2 2 π 2 R 4 0 R b 0 R b d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) [ { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 - { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 - { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 + { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R b , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 - 4 γ a R 2 r 1 · r 1 ( { n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] - n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] } 2 - { n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] - n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] } 2 ) - 4 γ b R 2 r 2 · r 2 ( { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 - { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 ) + 16 γ a γ b R 4 ( r 1 · r 1 ) ( r 2 · r 2 ) { n [ r 1 ( 1 - z 1 R a ) + r a z 1 R a , z 1 ] - n [ r 2 ( 1 - z 2 R b ) + r b z 2 R b , z 2 ] } 2 ] .
[ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 = C n 2 ( z 1 + z 2 2 ) × [ r 1 - r 2 2 + z 1 - z 2 2 ] 1 / 3
ϕ a ( r 1 ) ϕ b ( r 2 ) = - k 2 2 π 2 R 4 0 R b 0 R b d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) C n 2 ( z 1 + z 2 2 ) × [ [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 - [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 + z 1 - z 2 2 / 3 - [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 + z 1 - z 2 2 / 3 + [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 - 4 γ a R 2 r 1 · r 1 ( { [ | r 1 ( 1 - z 1 R a ) + r a z 1 R b - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 } - { [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 } ) - 4 γ b R 2 r 2 · r 2 ( { [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 } - { [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 } ) + 16 γ a γ b R 4 ( r 1 · r 1 ) ( r 2 · r 2 ) { [ | r 1 ( 1 - z 1 R a ) + r a z 1 R a - r 2 ( 1 - z 2 R b ) - r b z 2 R b | 2 + z 1 - z 2 2 ] 1 / 3 - z 1 - z 2 2 / 3 } ] .
z + = 1 2 ( z 1 + z 2 ) ,
z - = z 1 - z 2 .
d z [ ( A 2 + z 2 ) 1 / 3 - z 2 / 3 ] = κ 1 A 5 / 3 ,
κ 1 = 2 1 / 3 5 Γ 2 ( 1 / 6 ) Γ ( 1 / 3 ) 2.914381.
ϕ a ( r 1 ) ϕ b ( r 2 ) = - κ 1 k 2 2 π 2 R 4 0 R b d r 1 d r 2 d z W ( r 1 / R ) W ( r 2 / R ) C n 2 ( z ) { | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 - | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 - | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 + | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 - 4 γ a R 2 r 1 · r 1 [ | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 - | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 ] - 4 γ b R 2 r 2 · r 2 [ | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 - | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 ] - 16 γ a γ b R 4 ( r 1 · r 1 ) ( r 2 · r 2 ) × | r 1 ( 1 - z R a ) + r a z R a - r 2 ( 1 - z R b ) - r b z R b | 5 / 3 } .
r 1 = R ρ 1 ,
r 2 = R ρ 2 ,
r 1 = R ρ 1 ,
r 2 = R ρ 2 .
Q = ( 1 - z / R b ) / ( 1 - z / R a ) ,
Ω = ( r a / R R a - r b / R R b ) z / ( 1 - z / R a ) .
ϕ a ( ρ 1 R ) ϕ b ( ρ 2 R ) = κ 1 k 2 R 5 / 3 2 π 2 0 R b d ρ 1 d ρ 2 d z W ( ρ 1 ) W ( ρ 2 ) × C n 2 ( z ) ( 1 - z / R a ) 5 / 3 [ ρ 1 - Q ρ 2 + Ω 5 / 3 - ρ 1 - Q ρ 2 + Ω 5 / 3 + ρ 1 - Q ρ 2 + Ω 5 / 3 - ρ 1 - Q ρ 2 + Ω 5 / 3 + 4 γ a ρ 1 · ρ 1 ( ρ 1 - Q ρ 2 + Ω 5 / 3 - ρ 1 - Q ρ 2 + Ω 5 / 3 ) + 4 γ b ρ 2 · ρ 2 ( ρ 1 - Q ρ 2 + Ω 5 / 3 - ρ 1 - Q ρ 2 + Ω 5 / 3 ) - 16 γ a γ b ( ρ 1 · ρ 1 ) ( ρ 2 · ρ 2 ) × ρ 1 - Q ρ 2 + Ω 5 / 3 ] .
r 0 = [ κ 1 κ 2 k 2 d z C n 2 ( z ) ] - 3 / 5 ,
κ 2 = 2 [ 24 5 Γ ( 6 / 5 ) ] 5 / 6 6.883877.
1 π d ρ W ( ρ ) = 1.
ϕ a ( ρ 1 R ) ϕ b ( ρ 2 R ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 [ d z C n 2 ( z ) ] - 1 0 R b d z C n 2 ( z ) ( 1 - z / R a ) 5 / 3 × { 1 π d ρ 2 W ( ρ 2 ) ρ 1 - Q ρ 2 + Ω 5 / 3 + 1 π d ρ 1 W ( ρ 1 ) ρ 1 - Q ρ 2 + Ω 5 / 3 - ρ 1 - Q ρ 2 + Ω 5 / 3 - 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - Q ρ 2 + Ω 5 / 3 + 4 γ a [ 1 π d ρ 1 W ( ρ 1 ) ρ 1 · ρ 1 ρ 1 - Q ρ 2 + Ω 5 / 3 - 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 · ρ 1 ρ 1 - Q ρ 2 + Ω 5 / 3 ] + 4 γ b [ 1 π d ρ 2 W ( ρ 2 ) ρ 2 · ρ 2 ρ 1 - Q ρ 2 + Ω 5 / 3 - 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 2 · ρ 2 ρ 1 - Q ρ 2 + Ω 5 / 3 ] - 16 γ a γ b 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) × [ ( ρ 1 · ρ 1 ) ( ρ 2 · ρ 2 ) ρ 1 - Q ρ 2 + Ω 5 / 3 ] } .
Γ ( 11 / 5 ) = ( 6 / 5 ) Γ ( 6 / 5 ) .
F 1 ( Ω ) = 1 π d ρ W ( ρ ) ρ - Ω 5 / 3 .
F 1 ( Ω ) = 1 π 0 1 d ρ ρ 0 2 π d ϕ [ ρ 2 - 2 ρ Ω cos ( ϕ ) + Ω 2 ] 5 / 6 .
n = 0 C n ( α ) ( x ) z n = ( 1 - 2 x z + z 2 ) - α             valid for z < 1 , α 0 ,
C n ( α ) cos ( ϕ ) = m = 0 n Γ ( α + m ) Γ ( α + n - m ) m ! ( n - m ) ! [ Γ ( α ) ] 2 × cos [ ( n - 2 m ) ϕ ] .
[ ρ 2 - 2 ρ Ω cos ( ϕ ) + Ω 2 ] 5 / 6 = { ρ 5 / 3 [ 1 - 2 Ω ρ cos ( ϕ ) + Ω 2 ρ 2 ] 5 / 6 for ρ Ω Γ 5 / 3 [ 1 - 2 ρ Ω cos ( ϕ ) + ρ 2 Ω 2 ] 5 / 6 for ρ Ω .
[ ρ 2 - 2 ρ Ω cos ( ϕ ) + Ω 2 ] 5 / 6 = { ρ 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] 2 cos [ ( n - 2 m ) ϕ ] ( Ω / ρ ) n ρ Γ Ω 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] 2 cos [ ( n - 2 m ) ϕ ] ( ρ / Ω ) n Γ ρ .
0 2 π d ϕ [ ρ 2 - 2 ρ Ω cos ( ϕ ) + Ω 2 ] 5 / 6 = { 2 π ρ 5 / 3 n = 0 a n ( Ω / ρ ) 2 n ρ Ω 2 π Ω 5 / 3 n = 0 a n ( ρ / Ω ) 2 n ρ Ω ,
a n = [ Γ ( - + n ) n ! Γ ( - ) ] 2 .
F 1 ( Ω ) = { 2 n = 0 a n ( Ω 5 / 3 - 2 n 0 Ω d ρ ρ 2 n + 1 + Ω 2 n Ω 1 d ρ ρ 8 / 3 - 2 n ) Ω 1 2 n = 0 a n Ω 5 / 3 - 2 n 0 1 d ρ ρ 2 n + 1 Ω 1 .
F 1 ( Ω ) = { 2 n = 0 a n ( Ω 11 / 3 2 n + 2 + Ω 2 n 11 / 3 - 2 n - Ω 11 / 3 11 / 3 - 2 n ) Ω 1 Ω 5 / 3 n = 0 a n Ω - 2 n n + 1 Ω 1 .
F 1 ( Ω ) = { 6 11 Γ ( 1 ) Γ ( - ¹¹ / ) Γ ( - ) n = 0 Γ ( - ¹¹ / + n ) Γ ( - + n ) Γ ( n + 1 ) Ω 2 n n ! for Ω 1 Ω 5 / 3 Γ ( 2 ) Γ ( - ) Γ ( - ) n = 0 Γ ( - + n ) Γ ( - + n ) Γ ( 2 + n ) Ω - 2 n n ! for Ω 1 .
Γ ( z + 1 ) = z Γ ( z )
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) z n n ! .
F 1 ( Ω ) = { 6 11 F 2 1 ( - ¹¹ / , - ; 1 ; Ω 2 ) for Ω 1 Ω 5 / 3 F 2 1 ( - , - ; 2 ; 1 / Ω 2 ) for Ω 1 .
1 π d ρ 2 W ( ρ 2 ) ρ 1 - Q ρ 2 + Ω 5 / 3 = Q 5 / 3 F 1 ( ρ 1 + Ω Q ) ,
1 π d ρ 1 W ( ρ 1 ) ρ 1 - Q ρ 2 + Ω 5 / 3 = F 1 ( Q ρ 2 - Ω ) .
F 2 ( Q , Ω ) = 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - Q ρ 2 + Ω 5 / 3 .
ρ + = 1 2 ( ρ 1 + Q ρ 2 ) ,
ρ - = ρ 1 - Q ρ 2 .
F 2 ( Q , Ω ) = 1 π 2 Q 2 d ρ - K 1 ( ρ - , Q ) ρ - + Ω 5 / 3 ,
K 1 ( ρ , Q ) = d ρ + W ( ρ + + 1 2 ρ - ) W [ ( ρ + - 1 2 ρ - ) / Q ] .
K 1 ( ρ , Q ) = { π Q 2 ρ 1 - Q cos - 1 ( t 0 ) - t 0 1 - t 0 2 + Q 2 [ cos - 1 ( S 0 / Q ) - ( S 0 / Q ) 1 - ( S 0 / Q ) 2 ] 1 - Q < ρ < 1 + Q , 0 otherwise
t 0 = 1 - Q 2 2 ρ + 1 2 ρ ,
S 0 = - 1 - Q 2 2 ρ + 1 2 ρ .
0 < Q 1
ρ - = - ρ .
F 2 ( Q , Ω ) = 1 π 2 Q 2 d ρ K 1 ( ρ , Q ) ρ - Ω 5 / 3 .
F 2 ( Q , Ω ) = 2 π Q 2 0 1 + Q d ρ K 1 ( ρ , Q ) H ( ρ , Ω ) ,
H ( ρ , Ω ) = { ρ 8 / 3 F 2 1 ( - , - ; 1 ; Ω 2 / ρ 2 ) for Ω ρ ρ Ω 5 / 3 F 2 1 ( - , - ; 1 ; ρ 2 / Ω 2 ) for Ω ρ .
G 1 ( r 1 , r 2 ) = 1 π d ρ W ( ρ ) ρ · r 1 ρ - r 2 5 / 3 .
G 1 ( r 1 , r 2 ) = r 1 π 0 2 π d ϕ 0 1 d ρ ρ 2 cos ( ϕ - ϕ 1 ) × [ ρ 2 - 2 ρ r 2 cos ( ϕ - ϕ 2 ) + r 2 2 ) ] 5 / 6 ,
θ = ϕ - ϕ 2 .
cos ( ϕ - ϕ 1 ) = cos ( θ ) cos ( Δ ) + sin ( θ ) sin ( Δ ) ,
Δ = ϕ 1 - ϕ 2
G 1 ( r 1 , r 2 ) = r 1 · r 2 π r 2 0 1 d ρ ρ 2 0 2 π d θ cos ( θ ) × [ ρ 2 - 2 ρ r 2 cos ( θ ) + r 2 2 ] 5 / 6 .
cos ( Δ ) = r 1 · r 2 r 1 r 2 .
0 2 π d ϕ cos ( k ϕ ) cos ( l ϕ ) = π δ k l             for k 0 ,
n - 2 m = ± 1.
n = 2 p + 1 ,
m = p , p + 1 ,
p = 0 , 1 , 2 , .
0 2 π d θ cos ( θ ) [ ρ 2 2 ρ r 2 cos ( θ ) + r 2 2 ] 5 / 6 = { 2 π ρ 5 / 3 p = 0 b p ( r 2 / ρ ) 2 p + 1 ρ r 2 π r 2 5 / 3 p = 0 b p ( ρ / r 2 ) 2 p + 1 ρ r ,
b p = Γ ( - + p ) Γ ( + p ) p ! ( p + 1 ) ! [ Γ ( - ) ] 2 .
G 1 ( r 1 , r 2 ) = { - / ₁₁ r 1 · r 22 F 1 ( - ¹¹ / , ; 2 ; r 2 2 ) for r 2 1 - / ₁₂ r 2 - 1 / 3 r 1 · r 22 F 1 ( - , ; 3 ; 1 / r 2 2 ) for r 2 1 .
1 π d ρ 1 W ( ρ 1 ) ρ 1 · ρ 1 ρ 1 - Q ρ 2 + Ω 5 / 3 = G 1 ( ρ 1 , Q ρ 2 - Ω ) ,
1 π d ρ 2 W ( ρ 2 ) ρ 2 · ρ 2 ρ 1 - Q ρ 2 + Ω 5 / 3 = Q 5 / 3 G 1 ( ρ 2 , ρ 1 + Ω Q ) .
G 2 ( r , Q , Ω ) = 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) r · ρ 1 × ρ 1 - Q ρ 2 - Ω 5 / 3 .
G 2 ( r , Q , Ω ) = 1 π d ρ 2 W ( ρ 2 ) G 1 ( r , Q ρ 2 + Ω ) .
ρ = Q ρ 2 + Ω
G 1 ( r 1 , r 2 ) = r 1 · r 2 G ^ ( r 2 ) .
G 2 ( r , Q , Ω ) = 1 π Q 2 d ρ W [ ( ρ - Ω ) / Q ] r · ρ G ^ ( ρ ) .
W [ ( ρ - Ω ) / Q ] = l f l ( ρ ) exp [ i l ( ϕ - ϕ 0 ) ] ,
f l ( ρ ) = 1 2 π 0 2 π d ϕ W [ ( ρ - Ω ) / Q ] exp ( - i l ϕ ) .
f 0 ( ρ ) = { 1 for ρ Q - Ω , Q Ω Ω ( ρ ) π for Q - Ω ρ Q + Ω 0 otherwise ,
f l ( ρ ) = { 0 for ρ Q - Ω ,             l 0 sin [ l Ω ( ρ ) ] π l for Q - Ω ρ Q + Ω 0 for ρ > Q + Ω ,
Ω ( ρ ) = sin - 1 { [ 1 - ( Ω 2 + ρ 2 - Q 2 2 Ω ρ ) 2 ] 1 / 2 } .
f l ( ρ ) = f - l ( ρ ) .
G 2 ( r , Q , Γ ) = 1 π Q 2 0 2 π d ϕ d ρ ρ × { f 0 ( ρ ) + 2 l = 1 f l ( ρ ) cos [ l ( ϕ - ϕ 0 ) ] } × r ρ cos ( ϕ - ϕ ^ ) G ^ ( ρ ) ,
θ = ϕ - ϕ 0 ,
Δ = ϕ ^ - ϕ 0 .
G 2 ( r , G , Γ ) = r π Q 2 d ρ ρ 2 G ^ ( ρ ) 0 2 π d θ cos ( θ - Δ ) × [ f 0 ( ρ ) + 2 l = 1 f l ( ρ ) cos ( l θ ) ] ,
cos ( Δ ) = r · Ω r Ω
G 2 ( r , Q , Ω ) = 2 r · Ω π Q 2 Ω Q - Ω Q + Ω d ρ ρ 2 × [ 1 - ( Ω 2 + ρ 2 - Q 2 2 Ω ρ ) 2 ] 1 / 2 G ^ ( ρ ) .
1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 · ρ 1 ρ 1 - Q ρ 2 + Ω 5 / 3 = G 2 ( ρ 1 , Q , - Ω ) ,
1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 2 · ρ 2 ρ 1 - Q ρ 2 + Ω 5 / 3 = Q 5 / 3 G 2 ( ρ 2 , 1 / Q , Ω / Q ) .
G 4 ( r 1 , r 2 , Q , Ω ) = 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ( r 1 · ρ 1 ) ( r 2 · ρ 2 ) × ρ 1 - Q ρ 2 - Ω 5 / 3 .
G 4 ( r 1 , r 2 , Q , Ω ) = 1 π d ρ 2 W ( ρ ) r 2 · ρ 2 G 1 ( r 1 , Q ρ 2 + Ω ) .
G 4 ( r 1 , r 2 , Q , Ω ) = 1 π Q 3 d ρ W [ ( ρ - Ω ) / Q ] r 2 · ρ G 1 ( r 1 , ρ ) - r 2 · Γ π Q 3 d ρ W [ ( ρ - Ω ) / Q ] G 1 ( r 1 , ρ ) .
G 4 ( r 1 , r 2 , Q , Ω ) = 1 π Q 3 d ρ W [ ( ρ - Ω ) / Q ] r 2 · ρ G 1 ( r 1 , ρ ) - r 2 · Ω Q G 2 ( r 1 , Q , Ω ) .
G 4 ( r 1 , r 2 , Q , Ω ) = r 1 · r 2 Q 3 0 max ( Q - Ω , 0 ) d ρ ρ 3 G ^ ( ρ ) + r 1 · r 2 π Q 3 × Q - Ω Q + Ω d ρ ρ 3 G ^ ( ρ ) ( 1 - | Ω 2 + ρ 2 - Q 2 2 Ω ρ | ) × [ 1 - ( Ω 2 + ρ 2 - Q 2 2 Ω ρ ) 2 ] 1 / 2 + 2 ( r 1 · Ω ) ( r 2 · Ω ) π Q 3 Ω 2 × Q - Ω Q + Ω d ρ ρ 3 G ^ ( ρ ) ( | Ω 2 + ρ 2 - Q 2 2 Ω ρ | - Ω ρ ) × [ 1 - ( Ω 2 + ρ 2 - Q 2 2 Ω ρ ) 2 ] 1 / 2 .
ϕ a ( ρ 1 R ) ϕ b ( ρ 2 R ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 [ d z C n 2 ( z ) ] - 1 0 R b d z C n 2 ( z ) ( 1 - z / R a ) 5 / 3 × { Q 5 / 3 F 1 ( ρ 1 + Ω Q ) + F 1 ( Q ρ 2 - Ω ) - ρ 1 - Q ρ 2 + Ω 5 / 3 - F 2 ( Q , Ω ) + 4 γ a [ G 1 ( ρ 1 , Q ρ 2 - Ω ) - G 2 ( ρ 1 , Q , - Ω ) ] + 4 γ b [ Q 5 / 3 G 1 ( ρ 2 , ρ 1 + Ω Q ) - Q 5 / 3 G 2 ( ρ 2 , 1 / Q , Ω / Q ) ] - 16 γ a γ b G 4 ( ρ 1 , ρ 2 , Q , - Ω ) } .

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