Abstract

The mean-square angle-of-arrival (AOA) difference between two counter-propagating spherical waves in atmospheric turbulence is theoretically formulated. Closed-form expressions for the path weighting functions are obtained. It is found that the diffraction and refraction effects of turbulent cells make negative and positive contributions to the mean-square AOA difference, respectively, and the turbulent cells located at the midpoint of the propagation path have no contributions to the mean-square AOA difference. If the mean-square AOA difference is separated into the refraction and diffraction parts, the refraction part always dominates the diffraction one, and the ratio of the diffraction part to the refraction one is never larger than 0.5 for any turbulence spectrum. Based on the expressions for the mean-square AOA difference, formulae for the correlation coefficient between the angles of arrival of two counter-propagating spherical waves in atmospheric turbulence are derived. Numerical calculations are carried out by considering that the turbulence spectrum has no path dependence. It is shown that the mean-square AOA difference always approximates to the variance of AOA fluctuations. It is found that the correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves ranges from 0.46 to 0.5, implying that the instantaneous angles of arrival of two counter-propagating spherical waves in atmospheric turbulence are far from being perfectly correlated even when the turbulence spectrum does not vary along the path.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (1)

2013 (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

2012 (6)

2011 (1)

2010 (1)

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[Crossref]

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

2006 (1)

Y. Baykal and H. T. Eyyuboğlu, “Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free-space optics links,” Opt. Eng. 45(5), 056001 (2006).
[Crossref]

2003 (1)

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

1999 (2)

1997 (1)

V. P. Lukin, “Comparison the efficiencies of different schemes for laser guide stars forming,” Proc. SPIE 3126, 460–466 (1997).
[Crossref]

1995 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

1994 (1)

1987 (1)

1986 (1)

1982 (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

1980 (1)

1972 (2)

1971 (2)

1968 (1)

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(1), 37–40 (1968).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Baykal, Y.

Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36(23), 4554–4556 (2011).
[Crossref] [PubMed]

Y. Baykal and H. T. Eyyuboğlu, “Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free-space optics links,” Opt. Eng. 45(5), 056001 (2006).
[Crossref]

Belen’kii, M. S.

Boeke, B. R.

Brown, J. M.

Cao, X.

Charnotskii, M.

Charnotskii, M. I.

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Churnside, J. H.

Cleis, R. A.

Consortini, A.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

Cowley, W.

Cui, L.

Dolfi, D.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Ellerbroek, B. L.

Eyyuboglu, H. T.

Y. Baykal and H. T. Eyyuboğlu, “Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free-space optics links,” Opt. Eng. 45(5), 056001 (2006).
[Crossref]

Ferraro, M.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Fried, D. L.

Fugate, R. Q.

Gerçekcioglu, H.

Giggenbach, D.

Golbraikh, E.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[Crossref]

Grant, K.

Greco, J. A.

Gurvich, A. S.

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(1), 37–40 (1968).
[Crossref]

Higgins, C. H.

Innocenti, C.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

Jelonek, M. P.

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(1), 37–40 (1968).
[Crossref]

Karis, S. J.

Kopeika, N. S.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[Crossref]

Lange, W. J.

Lataitis, R. J.

Li, Z. P.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

Lukin, V. P.

V. P. Lukin, “Some problems in the use of laser guide stars,” Proc. SPIE 3983, 90–100 (1999).
[Crossref]

V. P. Lukin, “Comparison the efficiencies of different schemes for laser guide stars forming,” Proc. SPIE 3126, 460–466 (1997).
[Crossref]

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Lutomirski, R. F.

Mahon, R.

Minet, J.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Moore, C. I.

Moroney, J. F.

Oliker, M. D.

Parenti, R. R.

Perlot, N.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Polnau, E.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Puryear, A. L.

Rabinovich, W. S.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Roth, J. M.

Ruane, R. E.

Shapiro, J. H.

Slavin, A. C.

Spinhirne, J. M.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Suite, M. R.

Sun, Y. Y.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

Swindle, D. W.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Vorontsov, M. A.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Walther, F. G.

Wandzura, S. M.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Wild, W. J.

Winick, K. A.

Winker, D. M.

Wynia, J. M.

Xue, B.

Yura, H. T.

Zhou, F.

Zilberman, A.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[Crossref]

Appl. Opt. (6)

J. Opt. (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003).
[Crossref]

Opt. Eng. (1)

Y. Baykal and H. T. Eyyuboğlu, “Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free-space optics links,” Opt. Eng. 45(5), 056001 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (5)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[Crossref]

V. P. Lukin, “Comparison the efficiencies of different schemes for laser guide stars forming,” Proc. SPIE 3126, 460–466 (1997).
[Crossref]

V. P. Lukin, “Some problems in the use of laser guide stars,” Proc. SPIE 3983, 90–100 (1999).
[Crossref]

Radiophys. Quantum Electron. (1)

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(1), 37–40 (1968).
[Crossref]

Sov. J. Quantum Electron. (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Other (5)

M. A. Vorontsov, “Conservation laws for counter-propagating optical waves in atmospheric turbulence with application to directed energy and laser communications,” in Proceedings of Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2013), paper PW3F.1.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007).

A. D. Wheelon, Electromagnetic Scintillation II: Weak Scattering (Cambridge University, 2003).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Jerusalem, 1971).

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

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Figures (4)

Fig. 1
Fig. 1 Counter-propagation of two spherical waves along a common path in atmospheric turbulence.
Fig. 2
Fig. 2 The path weighting functions in terms of ξ with various q. (a) the refraction path weighting function Wr (∙); (b) the diffraction path weighting function Wd (∙) with α = 3.1; (c) the diffraction path weighting function Wd (∙) with α = 11/3; (d) the diffraction path weighting function Wd (∙) with α = 3.9.
Fig. 3
Fig. 3 Normalized mean-square AOA difference between two counter-propagating spherical waves as a function of q and α. 3 < α < 4. The turbulence spectrum is assumed to have no path dependence.
Fig. 4
Fig. 4 Correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves as a function of q and α. The subscript v can be specified as x or y. 3 < α < 4. The turbulence spectrum is assumed to have no path dependence.

Equations (28)

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S ^ F ( r , z = L ) = k 0 L d z d v ( κ , z ) cos [ κ 2 z ( L z ) 2 k L ] exp ( i z L κ r ) ,
β F = t φ F ( r ) | r = 0 ,
β F = i 0 L d z d v ( κ , z ) κ z L cos [ κ 2 z ( L z ) 2 k L ] .
β I = i 0 L d z d v ( κ , z ) κ L z L cos [ κ 2 z ( L z ) 2 k L ] .
δ F I 2 = 2 π 0 L ( 2 z L 1 ) 2 d z d 2 κ Φ n ( κ , z ) κ 2 cos 2 [ κ 2 z ( L z ) 2 k L ] ,
δ F I 2 = 4 π 2 L 0 1 d ξ ( 1 2 ξ ) 2 0 d κ κ 3 Φ n ( κ , ξ L ) cos 2 [ κ 2 L ξ ( 1 ξ ) 2 k ] .
δ F I 2 = δ F I , r 2 δ F I , d 2
δ F I , r 2 = 4 π 2 L 0 1 d ξ ( 1 2 ξ ) 2 0 d κ κ 3 Φ n ( κ , ξ L ) ,
δ F I , d 2 = 4 π 2 L 0 1 d ξ ( 1 2 ξ ) 2 0 d κ κ 3 Φ n ( κ , ξ L ) sin 2 ( κ 2 L ξ ( 1 ξ ) 2 k ) .
δ F I , d 2 = { 0 L / k 0 δ F I , r 2 / 2 L / k .
δ F I 2 = { δ F I , r 2 L / k 0 1 2 δ F I , r 2 L / k .
Φ n ( κ , z ) = A ( α ) C ˜ n 2 ( z ) ( κ 2 + κ 0 2 ) α / 2 exp ( κ 2 / κ m 2 )
δ F I , r 2 = 2 π 2 A ( α ) Γ ( 2 α 2 ) κ m 4 α L 0 1 d ξ C ˜ n 2 ( ξ L ) W r ( ξ ) ,
δ F I , d 2 = 2 π 2 A ( α ) Γ ( 2 α 2 ) κ m 4 α L 0 1 d ξ C ˜ n 2 ( ξ L ) W d ( ξ )
W r ( ξ ) = ( 1 2 ξ ) 2 ,
W d ( ξ ) = 1 2 W r ( ξ ) [ 1 Q ( θ ) ] ,
Q ( θ ) = cos [ ( 2 α 2 ) arc tan ( θ ) ] ( 1 + θ 2 ) α / 4 1 ,
β F 2 = 4 π 2 L 0 1 d ξ ξ 2 0 d κ κ 3 Φ n ( κ , ξ L ) cos 2 [ κ 2 L ξ ( 1 ξ ) 2 k ] .
β I 2 = 4 π 2 L 0 1 d ξ ( 1 ξ ) 2 0 d κ κ 3 Φ n ( κ , ξ L ) cos 2 [ κ 2 L ξ ( 1 ξ ) 2 k ] .
δ ^ F I 2 = δ F I 2 β F 2 β I 2 = 0 1 d ξ C ˜ n 2 ( ξ L ) ( 1 2 ξ ) 2 [ 1 + Q ( θ ) ] 0 1 d ξ C ˜ n 2 ( ξ L ) ξ 2 [ 1 + Q ( θ ) ] 0 1 d ξ C ˜ n 2 ( ξ L ) ( 1 ξ ) 2 [ 1 + Q ( θ ) ] ,
δ ^ F I 2 = 0 1 d ξ ( 1 2 ξ ) 2 [ 1 + Q ( θ ) ] 0 1 d ξ ( 1 ξ ) 2 [ 1 + Q ( θ ) ] .
γ F I , v = β F , v β I , v β F , v 2 β I , v 2 = 1 2 ( β F 2 β I 2 + β I 2 β F 2 ) 1 2 δ ^ F I 2 ,
γ F I , v = 1 δ ^ F I 2 / 2.
δ F I 2 = ( β F β I ) 2 = 0 L d z 0 L d z d v ( κ , z ) d v * ( κ , z ) κ κ × { ( 2 z L 1 ) cos [ κ 2 z ( L z ) 2 k L ] } { ( 2 z L 1 ) cos [ κ 2 z ( L z ) 2 k L ] }
d v ( κ , z ) d v * ( κ , z ) = C ˜ n 2 ( z / 2 + z / 2 ) E n ( κ , | z z | ) δ ( κ κ ) d κ d κ ,
C ˜ n 2 ( z s ) 0 E n ( κ , z d ) d z d = π Φ n ( κ , z s ) ,
δ F I 2 = δ F I , x 2 + δ F I , y 2 = β F , x 2 + β I , x 2 2 β F , x β I , x + β F , y 2 + β I , y 2 2 β F , y β I , y = β F 2 + β I 2 2 β F , x β I , x 2 β F , y β I , y = β F 2 + β I 2 4 β F , v β I , v ,
γ F I , v = β F , v β I , v β F , v 2 β I , v 2 = 2 β F , v β I , v β F 2 β I 2 .

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