Abstract

The theory of short-term average modulation transfer function developed by Fried is extended by considering the correlation between the turbulence-induced phase distortion and the residual phase aberrations after adaptive phase compensations. Calculation of the optical resolution shows that the original approximation is valid for the normalized lens diameter D/r0 less than 4.0; conversely, for D/r0 greater than 4.0 the optical resolution becomes arbitrarily large because the MTF is unrealistically overcorrected at high spatial frequencies. The extended formulation is then applied to the cases with higher-order phase compensations such as focus and astigmatism. The peak optical resolution occurs at larger values of D/r0 when the higher-order corrections are included.

© 1977 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
    [CrossRef]
  3. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
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  4. H. T. Yura, Appl. Opt. 10, 2771 (1971).
    [CrossRef] [PubMed]
  5. D. A. deWolf, “Effects of Turbulence Instabilities on Laser Propagation,” RADC-TR-72-119 (April1972) (unpublished).
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  7. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  8. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  9. G. R. Heidbreder, IEEE Trans. Antennas Propag. AP-15, 90 (1967).
    [CrossRef]
  10. H. T. Yura, J. Opt. Soc. Am. 63, 567 (1973).
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  11. T. Chiba, Appl. Opt. 10, 2456 (1971).
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  12. J. A. Dowling and P. M. Livingston, J. Opt. Soc. Am. 63, 846 (1973).
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  13. J. R. Dunphy and J. R. Kerr, J. Opt. Soc. Am. 64, 1015 (1974).
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  14. W. T. Cathey, C. L. Hayes, W. C. Davis, and V. F. Pizzurro, Appl. Opt. 9, 701 (1970).
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  15. J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
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  16. A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
    [CrossRef]
  17. R. V. Wick and R. W. Goranson, “Infrared Position Sensing Detector: A Photopot for CO2 Lasers,” in Laser Digest, AFWL-TR-75-229 (October1975) (unpublished).
  18. J. C. Wyant, Appl. Opt. 14, 2622 (1975).
    [CrossRef] [PubMed]
  19. J. Feinleib and J. W. Hardy, “Wideband Adaptive Optics for Imaging,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 103.
    [CrossRef]
  20. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  21. J. Herrmann, “Properties of Phase-Conjugate COAT,” in Proceedings of the Optical Society of America Annual Meeting, Boston, October 1975, see J. Opt. Soc. Am. 65, 1212A (1975).
  22. D. R. Dean and L. T. James, “Adaptive Laser Optics Techniques (ALOT),” in Proceedings of the High Energy Laser Conference, San Diego, Calif., October 1975 (unpublished).
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 465.
  24. C. B. Hogge and R. R. Butts, IEEE Trans. Antennas Propag. AP-24, 144 (1976).
    [CrossRef]
  25. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]

1976 (3)

1975 (1)

1974 (1)

1973 (2)

1971 (5)

1970 (1)

1967 (2)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

G. R. Heidbreder, IEEE Trans. Antennas Propag. AP-15, 90 (1967).
[CrossRef]

1966 (1)

1965 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 465.

Bridges, W. B.

Brown, W. P.

Buffington, A.

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

Butts, R. R.

C. B. Hogge and R. R. Butts, IEEE Trans. Antennas Propag. AP-24, 144 (1976).
[CrossRef]

Cathey, W. T.

Chiba, T.

Crawford, F. S.

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

Davis, W. C.

Dean, D. R.

D. R. Dean and L. T. James, “Adaptive Laser Optics Techniques (ALOT),” in Proceedings of the High Energy Laser Conference, San Diego, Calif., October 1975 (unpublished).

deWolf, D. A.

D. A. deWolf, “Effects of Turbulence Instabilities on Laser Propagation,” RADC-TR-72-119 (April1972) (unpublished).

Dowling, J. A.

Dunphy, J. R.

Feinleib, J.

J. Feinleib and J. W. Hardy, “Wideband Adaptive Optics for Imaging,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 103.
[CrossRef]

Fried, D. L.

Goranson, R. W.

R. V. Wick and R. W. Goranson, “Infrared Position Sensing Detector: A Photopot for CO2 Lasers,” in Laser Digest, AFWL-TR-75-229 (October1975) (unpublished).

Hansen, S.

Hardy, J. W.

J. Feinleib and J. W. Hardy, “Wideband Adaptive Optics for Imaging,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 103.
[CrossRef]

Hayes, C. L.

Heidbreder, G. R.

G. R. Heidbreder, IEEE Trans. Antennas Propag. AP-15, 90 (1967).
[CrossRef]

Herrmann, J.

J. Herrmann, “Properties of Phase-Conjugate COAT,” in Proceedings of the Optical Society of America Annual Meeting, Boston, October 1975, see J. Opt. Soc. Am. 65, 1212A (1975).

Hogge, C. B.

C. B. Hogge and R. R. Butts, IEEE Trans. Antennas Propag. AP-24, 144 (1976).
[CrossRef]

James, L. T.

D. R. Dean and L. T. James, “Adaptive Laser Optics Techniques (ALOT),” in Proceedings of the High Energy Laser Conference, San Diego, Calif., October 1975 (unpublished).

Kerr, J. R.

Livingston, P. M.

Lutomirski, R. F.

Muller, R. A.

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

Noll, R. J.

Nussmeier, T. A.

Pearson, J. E.

Pedinoff, M. E.

Pizzurro, V. F.

Sancer, M. I.

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[CrossRef]

Schwemiu, A. J.

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

Smith, R. G.

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

Tatarski, V. I.

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

Varvatsis, A. D.

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[CrossRef]

Wick, R. V.

R. V. Wick and R. W. Goranson, “Infrared Position Sensing Detector: A Photopot for CO2 Lasers,” in Laser Digest, AFWL-TR-75-229 (October1975) (unpublished).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 465.

Wyant, J. C.

Yura, H. T.

Appl. Opt. (6)

Can. J. Phys. (1)

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. R. Heidbreder, IEEE Trans. Antennas Propag. AP-15, 90 (1967).
[CrossRef]

C. B. Hogge and R. R. Butts, IEEE Trans. Antennas Propag. AP-24, 144 (1976).
[CrossRef]

J. Opt. Soc. Am. (7)

Proc. IEEE (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Other (8)

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

D. A. deWolf, “Effects of Turbulence Instabilities on Laser Propagation,” RADC-TR-72-119 (April1972) (unpublished).

A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemiu, and R. G. Smith, “Active Image Restoration with a Flexible Mirror,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 90.
[CrossRef]

R. V. Wick and R. W. Goranson, “Infrared Position Sensing Detector: A Photopot for CO2 Lasers,” in Laser Digest, AFWL-TR-75-229 (October1975) (unpublished).

J. Herrmann, “Properties of Phase-Conjugate COAT,” in Proceedings of the Optical Society of America Annual Meeting, Boston, October 1975, see J. Opt. Soc. Am. 65, 1212A (1975).

D. R. Dean and L. T. James, “Adaptive Laser Optics Techniques (ALOT),” in Proceedings of the High Energy Laser Conference, San Diego, Calif., October 1975 (unpublished).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 465.

J. Feinleib and J. W. Hardy, “Wideband Adaptive Optics for Imaging,” in Proceedings of the SPIE, Imaging through the Atmosphere, Vol. 75,March1976, p. 103.
[CrossRef]

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Figures (5)

FIG. 1
FIG. 1

Short-term average normalized resolution. Curve A, from Ref. 6; curve B, from Eq. (29); curve C, from Eq. (16).

FIG. 2
FIG. 2

Modulation transfer function with adaptive phase compensations for D/r0 = 2.0. Curve A, no correction; curve B, tilt correction; curve C, tilt plus focus corrections; curve D, tilt plus focus plus astigmatism corrections; and curve E, ideal phase compensations and diffraction-limited performance. Dashed curves, from Eq. (43).

FIG. 3
FIG. 3

Same as Fig. 2 except that D/r0 = 7.0.

FIG. 4
FIG. 4

Dependence of normalized resolution on normalized lens diameter for D ≫ (λz)1/2. Curve descriptions are the same as in Fig. 2.

FIG. 5
FIG. 5

Dependence of normalized resolution on normalized lens diameter for D ≪ (λz)1/2. Curve A, no correction; curve B, tilt correction; curve C, tilt plus focus corrections; curve D, tilt plus focus plus astigmatism corrections; curve E, ideal phase compensations; and curve F, diffraction-limited performance.

Equations (74)

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I ( P ) = ( k 2 π z ) 2 × exp [ i k ( s 1 - s 2 ) ] G ( r 1 ) G * ( r 2 ) U ( r 1 ) U * ( r 2 ) d r 1 d r 2 ,
I ( P ) = ( k 2 π z ) 3 d ρ exp ( - i k z ρ · P ) G ( r + 1 2 ρ ) G * ( r - 1 2 ρ ) × U ( r + 1 2 ρ ) U * ( r - 1 2 ρ ) exp ( i k z ρ · r ) d r ,
τ ( ρ ) = G ( r + 1 2 ρ ) G * ( r - 1 2 ρ ) U ( r + 1 2 ρ ) × U * ( r - 1 2 ρ ) exp [ ( i k / z ) ρ · r ] d r ,
G ( r + 1 2 ρ ) G * ( r - 1 2 ρ ) = exp { [ l ( r + 1 2 ρ ) + i φ ( r + 1 2 ρ ) ] + [ l ( r - 1 2 ρ ) - i φ ( r - 1 2 ρ ) ] } = exp [ - D ( ρ ) / 2 ] ,
D ( ρ ) = 2.91 C n 2 z k 2 ρ 5 / 3 ,
r 0 = 1.68 ( C n 2 z k 2 ) - 3 / 5 ;
D ( ρ ) = 6.88 ( ρ / r 0 ) 5 / 3 .
G ( r + 1 2 ρ ) G * ( r - 1 2 ρ ) = exp { l ( r + 1 2 ρ ) + i [ φ ( r + 1 2 ρ ) - Φ ( r + 1 2 ρ ) ] + l ( r - 1 2 ρ ) - i [ φ ( r - 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] }
W ( r ) = { 1 if r D / 2 , 0 if r > D / 2 ,
τ ( ρ ) = G ( r + 1 2 ρ ) G * ( r - 1 2 ρ ) W ( r + 1 2 ρ ) W ( r - 1 2 ρ ) d r .
Φ ( r ) = a 2 F 2 ( r ) + a 3 F 3 ( r ) ,
F 2 ( r ) = ( 64 / π D 4 ) 1 / 2 x ,
F 3 ( r ) = ( 64 / π D 4 ) 1 / 2 y ,
τ ( ρ ) SE = exp [ - 1 2 D ( ρ ) ] d r W ( r + 1 2 ρ ) W ( r - 1 2 ρ ) × exp { - 1 2 [ Φ ( r + 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] 2 + [ φ ( r + 1 2 ρ ) - φ ( r - 1 2 ρ ] [ Φ ( r + 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] } .
τ ( ρ ) SE = exp [ - 1 2 D ( ρ ) ] d r W ( r + 1 2 ρ ) W ( r - 1 2 ρ ) × exp { 1 2 [ Φ ( r + 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] 2 } .
τ ( ρ ) SE = D 2 exp [ - 1 2 D ( ρ ) + 16 π D 2 ( ρ D ) 2 a 2 2 + a 3 2 ] Γ ( ρ D ) ,
Γ ( ρ D ) = 1 2 { cos - 1 ( ρ D ) - ( ρ D ) [ 1 - ( ρ D ) 2 ] 1 / 2 } ,
a 2 2 + a 3 2 = 0 1 [ F C ( x ) - F L ( x ) ] D φ ( x ) x d x ,
F C ( x ) = D 2 [ 2 cos - 1 x - 2 x ( 1 - x 2 ) 1 / 2 ] ,
F L ( x ) = D 2 [ 6 cos - 1 x - ( 14 x - 8 x 3 ) ( 1 - x 2 ) 1 / 2 ] .
16 π D 2 ( ρ D ) 2 a 2 2 + a 3 2 = 1 2 D ( ρ )
a i = d x W ( x ) φ ( x ) F i ( x ) ,
C ( ρ ) [ φ ( r + 1 2 ρ ) - φ ( r - 1 2 ρ ) ] [ Φ ( r + 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] = i [ F i ( r + 1 2 ρ ) - F i ( r - 1 2 ρ ) ] × d x W ( x ) F i ( x ) { [ φ ( x ) φ ( r + 1 2 ρ ) - φ ( x ) φ ( r - 1 2 ρ ) ] } .
d x W ( x ) F i ( x ) = 0 ,
d x W ( x ) [ - 1 2 φ ( r + 1 2 ρ ) 2 + 1 2 φ ( r - 1 2 ρ ) 2 - 1 2 φ ( x ) 2 + 1 2 φ ( x ) 2 ] F i ( x ) = 0.
D φ ( ρ ) = [ φ ( r + 1 2 ρ ) - φ ( r - 1 2 ρ ) ] 2 ,
C ( ρ ) = 1 2 i [ F i ( r + 1 2 ρ ) - F i ( r - 1 2 ρ ) ] d x W ( x ) F i ( x ) × [ - D φ ( r + 1 2 ρ - x ) + D φ ( r - 1 2 ρ - x ) ] .
P i ± ( ρ ; r ) = 1 2 [ F i ( r + 1 2 ρ ) - F i ( r - 1 2 ρ ) ] × d x W ( x ) F i ( x ) D φ ( r ± 1 2 ρ - x ) ,
τ ( ρ ) SE = 4 D 2 exp [ - 1 2 D ( ρ ) - 16 π D 2 ( ρ D ) 2 a 2 2 + a 3 2 ] × ϕ ( π / 2 ) + ϕ d θ 0 L ( θ ) u d u × exp [ i P i + ( ρ ; u , θ ) - i P i - ( ρ ; u , θ ) ] ,
L ( θ ) = - 1 2 ( ρ D ) cos ( θ - ϕ ) + 1 2 ( ρ D ) [ ( ρ D ) - 2 - sin 2 ( θ - ϕ ) ] 1 / 2 ,
R = d ρ τ ( ρ ) .
R R max = 16 π ( D r 0 ) 2 0 1 τ ( ρ ) ρ d ρ ,
Φ ( r ) = i = 1 a i F i ( r ) ,
F 4 ( r ) = ( 768 / π D 6 ) 1 / 2 ( x 2 + y 2 - D 2 / 8 )
F 5 ( r ) = ( 384 / π D 6 ) 1 / 2 ( x 2 - y 2 ) ,
F 6 ( r ) = ( 1536 / π D 6 ) 1 / 2 x y
τ ( ρ ) APC = 4 D 2 exp ( - D ( ρ ) 2 ) ϕ ( π / 2 ) + ϕ d θ 0 L ( θ ) u d u × exp ( - Q ( u , θ ) 2 - i = 2 6 P i + ( ρ ; u , θ ) + i = 2 6 P i - ( ρ ; u , θ ) ) ,
Q ( u , θ ) = [ Φ ( r + 1 2 ρ ) - Φ ( r - 1 2 ρ ) ] 2 = 32 π D 2 ( ρ D ) 2 [ a 2 2 + a 3 2 + 24 u 2 ( a 4 2 + a 4 2 + a 5 2 + a 6 2 ) + 48 u 2 a 4 2 cos ( 2 θ - 2 ϕ ) ] .
a 4 2 = 0 1 [ F L ( x ) - F S ( x ) ] D φ ( x ) x d x
a 4 2 + a 5 2 + a 6 2 = 0 1 [ F L ( x ) - F Q ( x ) ] D φ ( x ) x d x .
F S ( x ) = D 2 [ 8 cos - 1 x - ( 24 x - 80 3 x 3 + 32 3 x 5 ) ( 1 - x 2 ) 1 / 2 ] ,
F Q ( x ) = D 2 [ 12 cos - 1 x - ( 44 x - 64 x 3 - 32 x 5 ) ( 1 - x 2 ) 1 / 2 ] .
τ ( ρ ) APC = 4 D 2 exp ( - D ( ρ ) 2 ) × ϕ ( π / 2 ) + ϕ d θ 0 L ( θ ) u d u exp ( Q ( u , θ ) 2 ) .
a i = d x W ( x ) F i ( x ) φ ( x ) .
a i a j = d x d x W ( x ) W ( x ) F i ( x ) F j ( x ) φ ( x ) φ ( x ) .
d x W ( x ) F i ( x ) = 0 ,
- 1 2 d x d x W ( x ) W ( x ) F i ( x ) F j ( x ) [ φ ( x ) 2 + φ ( x ) 2 ] = 0.
a i a j = - 1 2 d x d x W ( x ) W ( x ) F i ( x ) F j ( x ) D φ ( x - x ) ,
x - x = r ,
1 2 ( x + x ) = r ,
a i a j = - 1 2 d r d r W ( r + 1 2 r ) W ( r - 1 2 r ) F i ( r + 1 2 r ) × F j ( r - 1 2 r ) D φ ( r ) .
F i j ( r ) = d r W ( r + 1 2 r ) W ( r - 1 2 r ) F i ( r + 1 2 r ) F j ( r - 1 2 r ) ,
a i a j = - 1 2 d r F i j ( r ) D φ ( r ) .
F 2 ( r + 1 2 r ) F 5 ( r - 1 2 r ) = 64 ( 6 ) 1 / 2 π D 5 [ ( p + r 2 ) cos θ + q sin θ ] × [ ( p 2 - p r + r 2 4 - q 2 ) cos 2 θ + ( p q - q r 2 ) sin 2 θ ] .
0 2 π sin m θ sin n θ d θ = 0 ,             m n
F 5 ( r + 1 2 r ) F 6 ( r - 1 2 r ) + F 5 ( r - 1 2 r ) F 6 ( r + 1 2 r ) = ( 768 / π D 6 ) [ ( - 1 2 p q r 2 + 2 p 3 q - 2 p q 3 ) cos 2 2 θ + ( 1 4 p q r 2 + p q 3 - p 3 q ) sin 2 2 θ + g ( sin 4 θ ) ] ,
P i ± ( ρ ; r ) = 1 2 [ F i ( r + 1 2 ρ ) - F i ( r - 1 2 ρ ) ] × d x W ( x ) F i ( x ) D φ ( r ± 1 2 ρ - x ) .
G i ± = d x W ( x ) F i ( x ) D φ ( r ± 1 2 ρ - x ) .
y ± = - ( r ± 1 2 ρ - x ) ,
G i ± = d y W ( y ) F i ( y ± + r ± 1 2 ρ ) D φ ( y ± ) .
r = r cos θ î + r sin θ ĵ ,
ρ = ρ cos ϕ î + ρ sin ϕ ĵ ,
y = ξ cos δ î + ξ sin δ ĵ ,
G i ± = 0 2 π d δ 0 η ± F i ( ξ , δ ) D φ ( ξ ) ξ d ξ ,
η ± = - Δ ± cos ( δ - δ 0 ± ) + [ 1 4 - ( Δ ± ) 2 sin 2 ( δ - δ 0 ± ) ] 1 / 2 ,
Δ ± = [ r 2 + 1 4 ρ 2 ± ρ r cos ( θ - ϕ ) ] 1 / 2 ,
δ 0 ± = tan - 1 ( r sin θ ± 1 2 ρ sin ϕ r cos θ ± 1 2 ρ cos ϕ ) .
P 2 ± = 32 A π cos ϕ 0 2 π d δ ( + H 2 ± cos δ + r D H 1 ± cos θ ± 1 2 ρ D H 1 ± cos ϕ ) ,
P 3 ± = 32 A π sin ϕ 0 2 π d δ ( + H 2 ± sin δ + r D H 1 ± sin θ ± 1 2 ρ D H 1 ± sin ϕ ) ,
P 4 ± = 768 A π r D cos ( θ - ϕ ) × 0 2 π d δ { H 3 ± + 2 r D H 2 ± cos ( δ - θ ) ± ρ D H 2 ± cos ( δ - ϕ ) + [ ( r D ) 2 + 1 4 ( ρ D ) 2 - 1 8 ] H 1 ± ± ρ D r D H 1 ± cos ( θ - ϕ ) } ,
P 5 ± = 384 A π r D cos ( θ + ϕ ) × 0 2 π d δ { H 3 ± cos 2 δ + 2 r D H 2 ± cos ( δ + θ ) ± ρ D H 2 ± cos ( δ + ϕ ) + [ ( r D ) 2 cos 2 θ + 1 4 ( ρ D ) 2 cos 2 ϕ ] H 1 ± ± ρ D r D H 1 ± cos ( θ + ϕ ) } ,
P 6 ± = 768 A π r D sin ( θ + ϕ ) × 0 2 π d δ { 1 2 H 3 ± sin 2 δ + r D H 2 ± sin ( δ + θ ) ± 1 2 ρ D H 2 ± sin ( δ + ϕ ) + [ 1 2 ( r D ) 2 sin 2 θ + 1 8 ( ρ D ) 2 sin 2 ϕ ] H 1 ± ± 1 2 ρ D r D H 1 ± sin ( θ + ϕ ) } ,
A = 6.88 ( D r 0 ) 5 / 3 ρ D ,
H n ± = ( η ± ) 8 / 3 + n 8 / 3 + n ,             n = 1 , 2 , 3.