Abstract

The calculation of the aperture-averaged angle-of-arrival variance, observed with a telescope with a circular aperture, of a plane or spherical wave propagating through homogeneous and isotropic turbulence is one of the classical problems in the theory of wave propagation through random media. We present and discuss approximate closed-form solutions on the basis of the Rytov approximation. For both plane and spherical waves, the accuracy of the approximations is better than 0.25% for all ratios of aperture diameter and Fresnel length.

© 2007 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, 1971).
  3. A. D. Wheelon, Electromagnetic Scintillation. I. Geometrical Optics (Cambridge U. Press, 2001).
  4. A. S. Gurvich and M. A. Kallistratova, "Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations," Radiophys. Quantum Electron. 11, 37-40 (1968).
    [CrossRef]
  5. J. H. Churnside and R. J. Lataitis, "Angle-of-arrival fluctuations of a reflected beam in atmospheric turbulence," J. Opt. Soc. Am. A 4, 1264-1272 (1987).
    [CrossRef]
  6. J. H. Churnside, "Angle-of-arrival fluctuations of retroreflected light in the turbulent atmosphere," J. Opt. Soc. Am. A 6, 275-279 (1989).
    [CrossRef]
  7. D. H. Tofsted, "Outer-scale effects on beam-wander and angle-of-arrival variances," Appl. Opt. 31, 5865-5870 (1992).
    [CrossRef] [PubMed]
  8. R. Conan, J. Borgnino, A. Ziad, and F. Martin, "Analytical solution for the covariance and for the decorrelation time of the angle of arrival of a wave front corrugated by atmospheric turbulence," J. Opt. Soc. Am. A 17, 1807-1818 (2000).
    [CrossRef]
  9. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
    [CrossRef]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  11. J. Haddon and E. Vilar, "Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture," IEEE Trans. Antennas Propag. AP-34, 646-657 (1986).
    [CrossRef]
  12. A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
    [CrossRef]
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).
  14. M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1972).

2000

1992

1989

1987

1986

J. Haddon and E. Vilar, "Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture," IEEE Trans. Antennas Propag. AP-34, 646-657 (1986).
[CrossRef]

1968

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
[CrossRef]

A. S. Gurvich and M. A. Kallistratova, "Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations," Radiophys. Quantum Electron. 11, 37-40 (1968).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1972).

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Borgnino, J.

Churnside, J. H.

Conan, R.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).

Gurvich, A. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
[CrossRef]

A. S. Gurvich and M. A. Kallistratova, "Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations," Radiophys. Quantum Electron. 11, 37-40 (1968).
[CrossRef]

Haddon, J.

J. Haddon and E. Vilar, "Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture," IEEE Trans. Antennas Propag. AP-34, 646-657 (1986).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Kallistratova, M. A.

A. S. Gurvich and M. A. Kallistratova, "Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations," Radiophys. Quantum Electron. 11, 37-40 (1968).
[CrossRef]

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
[CrossRef]

Lataitis, R. J.

Martin, F.

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).

Stegun, I.

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1972).

Tatarskii, V. I.

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

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, 1971).

Time, N. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
[CrossRef]

Tofsted, D. H.

Vilar, E.

J. Haddon and E. Vilar, "Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture," IEEE Trans. Antennas Propag. AP-34, 646-657 (1986).
[CrossRef]

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation. I. Geometrical Optics (Cambridge U. Press, 2001).

Ziad, A.

Appl. Opt.

IEEE Trans. Antennas Propag.

J. Haddon and E. Vilar, "Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture," IEEE Trans. Antennas Propag. AP-34, 646-657 (1986).
[CrossRef]

J. Opt. Soc. Am. A

Radiophys. Quantum Electron.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, "Fluctuations in the parameters of a light wave from a laser during propagation in the atmosphere," Radiophys. Quantum Electron. 11, 1360-1370 (1968).
[CrossRef]

A. S. Gurvich and M. A. Kallistratova, "Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations," Radiophys. Quantum Electron. 11, 37-40 (1968).
[CrossRef]

Other

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

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

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translation, 1971).

A. D. Wheelon, Electromagnetic Scintillation. I. Geometrical Optics (Cambridge U. Press, 2001).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1972).

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Figures (6)

Fig. 1
Fig. 1

Airy function and the Gaussian function with β = 0.5216 . The solid curve is the Airy function and the dashed curve is the Gaussian function.

Fig. 2
Fig. 2

Discrepancy between the Airy function and the Gaussian function, A ( a κ ) exp [ ( β a κ ) 2 ] with β = 0.5216 .

Fig. 3
Fig. 3

Aperture-averaged AOA variance, normalized by C n 2 L D 1 3 , versus q for a plane wave. The dots are the exact (numerical) solution; the solid curve is the closed-form approximation (17), and the crosses are the asymptotic fit for q < 1 .

Fig. 4
Fig. 4

Difference (%) between the exact (numerical) solution and the closed-form approximation of the aperture-averaged AOA variances for plane and spherical waves, respectively.

Fig. 5
Fig. 5

Aperture-averaged AOA variance, normalized by C n 2 L D 1 3 , versus q for a spherical wave. The dots are the exact (numerical) solution, the solid curve is the closed-form approximation (27), and the crosses are the asymptotic fit for q < 1 .

Fig. 6
Fig. 6

Ratio between the aperture-averaged AOA variances for spherical and plane waves for the exact (numerical) solutions and the closed-form approximations, respectively.

Equations (47)

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θ ¯ 2 p = γ p ( q ) C n 2 L D 1 3 ,
θ ¯ 2 s = γ s ( q ) C n 2 L D 1 3 ,
q = D f ,
f = λ L
θ ¯ 2 = π L 0 d κ 0 1 d u κ 3 Φ n ( κ ) h ( κ , u ) ,
h p ( κ ) = [ 1 + 2 π ( κ f ) 2 sin ( κ f ) 2 2 π ] A ( a κ )
h s ( κ , u ) = [ 1 + cos ( κ f ) 2 u ( 1 u ) 2 π ] u 2 A ( a κ u )
A ( x ) = ( 2 J 1 ( x ) x ) 2
Φ n ( κ ) = Γ ( 8 3 ) sin ( π 3 ) 4 π 2 C n 2 κ 11 3 .
A ( a κ ) exp [ ( β a κ ) 2 ] ,
θ ¯ 2 g = π 2 L 0 1 d u 0 d κ κ 3 Φ n ( κ ) [ 1 + 2 π ( κ f ) 2 sin ( κ f ) 2 2 π ] exp [ ( β a κ ) 2 ] .
θ ¯ 2 GO = 3 Γ ( 8 3 ) C n 2 a 2 0 d κ κ 8 3 J 1 2 ( a κ ) = C n 2 L 1296 3 [ Γ ( 8 3 ) ] 2 Γ ( 1 6 ) 2 8 3 1375 [ Γ ( 5 6 ) ] 3 a 1 3 ,
θ ¯ 2 g = 3 4 Γ ( 8 3 ) C n 2 L 0 d κ κ 2 3 exp [ ( β a κ ) 2 ] = C n 2 L 3 8 Γ ( 8 3 ) Γ ( 1 6 ) β 1 3 a 1 3 ,
β = 5 6 11 3 [ Γ ( 5 6 ) ] 9 2 16 3 6 [ Γ ( 2 3 ) ] 3 = 0.5216 .
θ ¯ 2 p = 3 8 Γ ( 8 3 ) C n 2 L 0 d κ κ 2 3 [ 1 + 2 π ( κ f ) 2 sin ( κ f ) 2 2 π ] exp [ ( β a κ ) 2 ] .
θ ¯ 2 p = γ p ( q ) C n 2 L D 1 3 ,
γ p ( q ) = 3 16 Γ ( 1 6 ) Γ ( 8 3 ) ( β 2 ) 1 3 { 1 + 6 5 ( π 2 ) 1 6 β 1 3 q 1 3 [ 1 + π 2 4 β 4 q 4 ] 5 12 sin [ 5 6 arctan ( 2 π β 2 q 2 ) ] } .
θ ¯ 2 s = 3 8 Γ ( 8 3 ) C n 2 L 0 1 d u u 2 0 d κ κ 2 3 [ 1 + cos ( ( κ f ) 2 u ( 1 u ) 2 π ) ] exp [ ( β a κ u ) 2 ] .
I 1 = 0 1 d u u 2 0 d κ κ 2 3 exp [ ( β a κ u ) 2 ] = 3 8 1 2 Γ ( 1 6 ) β 1 3 a 1 3 .
I 2 = 0 1 d u u 2 0 d κ κ 2 3 cos ( ( κ f ) 2 u ( 1 u ) 2 π ) exp [ ( β a κ u ) 2 ] = 1 4 Γ ( 1 6 ) β 1 3 a 1 3 0 1 d u u 2 [ ( C 1 u 2 C 2 u ) 1 6 + ( C 3 u 2 + C 2 u ) 1 6 ] = I 21 + I 22 ,
I 21 = 1 4 Γ ( 1 6 ) ( β a ) 1 3 6 17 ( C 2 ) 1 6 F 1 2 ( 1 6 , 17 6 ; 23 6 ; C 1 C 2 ) ,
I 22 = 1 4 Γ ( 1 6 ) ( β a ) 1 3 6 17 C 2 1 6 F 1 2 ( 1 6 , 17 6 ; 23 6 ; C 3 C 2 ) ,
C 1 = 1 + i f 2 2 π β 2 a 2 = 1 + C 2 ,
C 2 = i f 2 2 π β 2 a 2 = 2 i π β 2 q 2 ,
C 3 = 1 i f 2 2 π β 2 a 2 = 1 C 2 ,
θ ¯ 2 s = γ s ( q ) C n 2 L D 1 3 ,
γ s ( q ) = 3 8 3 16 Γ ( 1 6 ) Γ ( 8 3 ) ( β 2 ) 1 3 [ 1 + 6 17 8 3 Re { ( 2 i π β 2 q 2 ) 1 6 F 1 2 ( 1 6 , 17 6 ; 23 6 ; 1 i π β 2 q 2 2 ) } ] .
γ s ( q ) = 3 8 3 16 Γ ( 1 6 ) Γ ( 8 3 ) ( β 2 ) 1 3 [ 1 + 6 17 8 3 ( 2 π β 2 q 2 ) 1 6 ( γ s 1 + γ s 2 ) ] ,
γ s 1 = Γ ( 23 6 ) Γ ( 5 6 ) Γ ( 22 6 ) [ cos ( π 12 ) n = 0 ( 17 6 ) 2 n ( 1 ) n ( π β 2 q 2 2 ) 2 n ( 2 n ) ! sin ( π 12 ) n = 0 ( 17 6 ) 2 n + 1 ( 1 ) n + 2 ( π β 2 q 2 2 ) 2 n + 1 ( 2 n + 1 ) ! ] ,
γ s 2 = ( π β 2 q 2 2 ) 5 6 Γ ( 23 6 ) Γ ( 5 6 ) Γ ( 1 6 ) Γ ( 17 6 ) n = 0 ( 22 6 ) 2 n + 1 ( 1 ) 2 n + 1 ( 11 6 ) 2 n + 1 ( 1 ) n + 2 ( π β 2 q 2 2 ) 2 n + 1 ( 2 n + 1 ) ! .
γ p ( 0 ) = 3 16 Γ ( 1 6 ) Γ ( 8 3 ) ( β 2 ) 1 3 = π 55 2 11 3 3 1 2 [ Γ ( 2 3 ) ] 2 [ Γ ( 5 6 ) ] 4 = 1.419 ,
γ p ( ) = 2 γ p ( 0 ) = 2.838 .
γ p ( q ) γ p ( 0 ) [ 1 + 6 5 ( π 2 ) 1 6 β 1 3 q 1 3 sin ( 5 π 12 ) ] = 1.419 ( 1 + 1.006 q 1 3 ) .
γ p ( q ) = { 1.419 + 1.4275 q 1 3 for q 1 2.838 for q > 1 ,
γ s ( 0 ) = 3 8 3 16 Γ ( 1 6 ) Γ ( 8 3 ) ( β 2 ) 1 3 = π 55 2 2 3 3 3 2 [ Γ ( 2 3 ) ] 2 [ Γ ( 5 6 ) ] 4 = 0.532 ,
F 1 2 ( 1 6 , 17 6 ; 23 6 ; 1 C 2 ) = Γ ( 23 6 ) Γ ( 16 6 ) Γ ( 17 6 ) Γ ( 22 6 ) ( 1 C 2 ) 1 6 F 1 2 ( 1 6 , 16 6 ; 10 6 ; C 2 ) + Γ ( 23 6 ) Γ ( 16 6 ) Γ ( 1 6 ) Γ ( 16 6 ) ( 1 C 2 ) 17 6 F 1 2 ( 17 6 , 0 ; 22 6 ; C 2 ) ,
γ s ( ) = 2 γ s ( 0 ) = 3 8 2.838 = 1.064 ,
γ s ( q ) γ s ( 0 ) [ 1 + 6 17 8 3 Re { ( 2 i π β 2 q 2 ) 1 6 F 1 2 ( 1 6 , 17 6 ; 23 6 ; 1 ) } ] = γ s ( 0 ) [ 1 + 1.3279 6 17 8 3 ( π 2 ) 1 6 β 1 3 q 1 3 ] = 0.532 ( 1 + 1.0847 q 1 3 ) .
γ s ( q ) = { 0.532 + 0.577 q 1 3 for q 0.8 , 1.064 for q > 0.8 .
0 x μ J ν 2 ( α x ) d x = Γ ( μ ) Γ ( 2 ν + 1 μ 2 ) 2 μ [ Γ ( μ + 1 2 ) ] 2 Γ ( 2 ν + 1 + μ 2 ) α μ 1
0 x μ e α x 2 d x = 1 2 Γ ( 1 + μ 2 ) α ( μ + 1 ) 2
0 x μ 1 e α x sin δ x d x = Γ ( μ ) ( α 2 + δ 2 ) μ + 2 sin ( μ arctan δ α )
0 1 d x x m ( C 1 x 2 C 2 x ) n = ( C 2 ) n n + m + 1 F 1 2 ( n , n + m + 1 ; n + m + 2 ; C 1 C 2 ) ,
F 1 2 ( a , b ; c ; z ) = n = 0 ( a ) n ( b ) n ( c ) n z n n ! ,
F 1 2 ( a , b ; c ; z ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) F 1 2 ( a , b ; a + b c + 1 ; 1 z ) + ( 1 z ) c a b Γ ( c ) Γ ( a + b c ) Γ ( a ) Γ ( b ) F 1 2 ( c a , c b ; c a b + 1 ; 1 z )
F 1 2 ( a , b ; c ; z ) = Γ ( c ) Γ ( b a ) Γ ( b ) Γ ( c a ) ( z ) a F 1 2 ( a , 1 c + a ; 1 b + a ; 1 z ) + Γ ( c ) Γ ( a b ) Γ ( a ) Γ ( c b ) ( z ) b F 1 2 ( b , 1 c + b ; 1 a + b ; 1 z )
F 1 2 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b )

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