Abstract

Expressions for the correlation coefficient between light-flux fluctuations of two waves counter-propagating along a common path in weak turbulence are developed. Only the aperture and inner-scale Fresnel parameters are needed for evaluation of the correlation coefficient if the turbulence spectrum has no path dependence, and of the path weighting functions for the cross-covariance and variances of normalized light-flux fluctuations if the turbulence spectrum is dependent on path locations. Under the condition that atmospheric turbulence is statistically homogeneous over a path, although good correlation between light-flux fluctuations of two counter-propagating spherical waves may be achieved for a relatively small aperture Fresnel parameter or relatively large inner-scale Fresnel parameter, the correlation coefficient between light-flux fluctuations of two counter-propagating plane waves is always lower than 1 obviously. When the aperture Fresnel parameter becomes larger than the inner-scale Fresnel parameter, the inner scale of turbulence tends to play an unimportant role in determining the correlation coefficient.

© 2017 Optical Society of America

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References

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2016 (2)

2015 (1)

2014 (1)

2013 (3)

2012 (5)

2011 (2)

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

C. Chen, H. Yang, H. Wang, S. Tong, and Y. Lou, “Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence,” Appl. Opt. 50(3), 307–312 (2011).
[Crossref] [PubMed]

2010 (1)

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

2007 (1)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

2006 (1)

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

2005 (1)

1998 (1)

1994 (1)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

1992 (1)

1991 (1)

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[Crossref]

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]

1971 (2)

1970 (1)

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[Crossref]

Andrews, L. C.

Baker, G. J.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

Charnotskii, M.

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

Charnotskii, M. I.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[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]

Chen, C.

Cheng, J.

Cheon, Y.

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Cowley, W.

Cowley, W. G.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Davidson, F. M.

Dikmelik, Y.

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]

Duncan, B. D.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Giggenbach, D.

Goda, M. E.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Grant, K.

Greco, J. A.

Ishimaru, A.

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

Kavehrad, M.

Khatoon, A.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Kon, A. I.

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[Crossref]

Lachinova, S. L.

Leeb, W. R.

Letzepis, N.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Leung, V. C. M.

Lou, Y.

Lukin, V. P.

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.

Majumdar, A. K.

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]

Muschinski, A.

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Parenti, R. R.

Perlot, N.

Phillips, R. L.

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

Puryear, L.

Roth, J. M.

Schmidt, J. D.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Shapiro, J. H.

Song, X.

Tatarskii, V. I.

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

Tong, S.

Vorontsov, M. A.

M. A. Vorontsov, S. L. Lachinova, and A. K. Majumdar, “Target-in-the-loop remote sensing of laser beam and atmospheric turbulence characteristics,” Appl. Opt. 55(19), 5172–5179 (2016).
[Crossref] [PubMed]

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]

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.

Walther, F. G.

Wang, H.

Wang, N.

Wang, T.

Winzer, P. J.

Yang, H.

Yura, H. T.

Zhang, W.

Zhou, Z.

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. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (4)

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

Radiophys. Quantum. Electron. (1)

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[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]

Waves Random Media (2)

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

Other (4)

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

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

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

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.

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Figures (8)

Fig. 1
Fig. 1 Counter-propagation geometry for two waves along a common path in atmospheric turbulence.
Fig. 2
Fig. 2 hpl(·) and hpl(·) in terms of κ ^ = κ ( L / k ) 1 / 2 .
Fig. 3
Fig. 3 μpl in terms of q p 1 / 2 with a turbulence spectrum independent of path locations.
Fig. 4
Fig. 4 The path weighting functions gpl(ξ), gpl(ξ) and gpl(ξ) in terms of ξ with various qp. (a) qp = 0; (b) qp = 1; (c) qp = 100.
Fig. 5
Fig. 5 The spectral filter functions f sp ( κ ^ ) and f   sp ( κ ^ ) with various qR. (a) qR= 0.04; (b) qR = 1; (c) qR = 16.
Fig. 6
Fig. 6 Contours of μsp as a function of q m 1 / 2 and q R 1 / 2 with a turbulence spectrum independent of path locations. The subplot (b) shows a zoomed detail of the region with relatively small q m 1 / 2 and q R 1 / 2 .
Fig. 7
Fig. 7 The path weighting functions gsp(ξ), gsp(ξ) and gsp(ξ) in terms of ξ with various qm and qR. (a) qR = 0.01; (b) qR = 1; (c) qR = 100. The solid curves correspond to qm = 0; the dashed curves correspond to qm = 0.01; the dotted curves correspond to qm = 1.
Fig. 8
Fig. 8 Set of Asymptotes. (a) asymptotes of Eq. (51); (b) asymptotes of the correlation coefficient μpl; (c) asymptotes of the correlation coefficient μsp.

Equations (54)

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ψ 1 , pl , F ( r , z = L ) = 0 L dz d v ( κ , z ) H pl , F ( κ , z ) exp ( i κ r )
H pl , F ( κ , z ) = i k exp [ i κ 2 ( L z ) 2 k ] ,
n 1 ( r , z ) = exp ( i κ r ) d v ( κ , z ) .
ψ 1 , pl , I ( ρ , z = 0 ) = 0 L d z d v ( κ , z ) H pl , I ( κ , z ) exp ( i κ ρ )
H pl , I ( κ , z ) = i k exp ( i κ 2 z 2 k ) ,
χ 1 , pl , F ( r , z = L ) = 1 2 0 L d z d v ( κ , z ) [ H pl , F ( κ , z ) + H pl , F * ( κ , z ) ] exp ( i κ r ) ,
χ 1 , pl , I ( ρ , z = 0 ) = 1 2 0 L dz d v ( κ , z ) [ H pl , I ( κ , z ) + H pl , I * ( κ , z ) ] exp ( i κ ρ ) .
P ^ = D I ( r ) d 2 r D I 0 d 2 r = S A 1 D I ( r ) / I 0 d 2 r ,
P ^ 1 + 2 S A 1 D χ 1 , pl ( r ) d 2 r = 1 + 2 χ ^ 1 , pl ,
μ pl B pl , F I σ pl , F σ pl , I ,
χ ^ 1 , pl , F = d 2 r χ 1 , pl , F ( r , z = L ) W a ( r ) ,
χ ^ 1 , pl , I = d 2 ρ χ 1 , pl , I ( ρ , z = 0 ) W a ( ρ )
W a ( r ) = { 1 / ( π R a 2 ) | r | < R a 0 | r | R a ,
W ˜ a , pl ( κ ) = d 2 r W a ( r ) exp ( i κ r ) = jinc ( κ R a ) ,
χ ^ 1 , pl , F = 1 2 0 L d z d v ( κ , z ) [ H pl , F ( κ , z ) + H pl , F * ( κ , z ) ] W ˜ a , pl ( κ ) ,
χ ^ 1 , pl , I = 1 2 0 L d z d v ( κ , z ) [ H pl , I ( κ , z ) + H pl , I * ( κ , z ) ] W ˜ a , pl ( κ ) .
B pl , F I = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin ( κ 2 z 2 k ) sin [ κ 2 ( L z ) 2 k ] ,
σ pl , F 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin 2 [ κ 2 ( L z ) 2 k ] ,
σ pl , I 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin 2 ( κ 2 z 2 k ) ,
d v ( κ , z ) d v * ( κ , z ) = F n ( | z z | , κ ) δ ( κ κ ) d 2 κ d 2 κ
B pl , F I = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f pl ( κ L / k ) ,
σ pl , F 2 = σ pl , I 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f pl ( κ L / k )
f pl ( κ ^ ) = jinc 2 ( q R 1 / 2 κ ^ ) h pl ( κ ^ ) ,
f pl ( κ ^ ) = jinc 2 ( q R 1 / 2 κ ^ ) h pl ( κ ^ ) ,
h pl ( κ ^ ) = sinc ( κ ^ 2 / 2 ) cos ( κ ^ 2 / 2 ) ,
h pl ( κ ^ ) = 1 sinc ( κ ^ 2 ) ,
b 1 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) jinc 2 ( R a κ ) h 1 , pl ( κ L / k ) ( q p 2 + 1 4 ) 5 / 12 cos [ 5 6 arctan ( 1 2 q p ) ] C 1 q p 5 / 6 C 1 ,
b 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) jinc 2 ( R a κ ) h pl ( κ L / k / 2 1 / 2 ) 12 11 ( 1 4 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( 2 q p ) + π 12 ] C 1 q p 5 / 6 C 1 .
σ pl , F 2 6 11 ( 1 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( q p ) + π 12 ] C 1 q p 5 / 6 C 1 .
μ pl ( q p ) = { 6 11 ( 1 + q p 2 ) 11 12 sin [ 11 6 arctan ( q p ) + π 12 ] q p 5 / 6 } 1 { ( q p 2 + 1 4 ) 5 / 12 × cos [ 5 6 arctan ( 1 2 q p ) ] 12 11 ( 1 4 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( 2 q p ) + π 12 ] } ,
B pl , F I = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g pl ( ξ ) ,
σ pl , F 2 = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g pl ( ξ ) ,
σ pl , I 2 = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g pl ( ξ )
h ˜ pl ( a , b ) = 3 5 Γ ( 1 6 ) [ ( q p + i a ) 5 / 6 + ( q p i a ) 5 / 6 ( q p + i b ) 5 / 6 ( q p i b ) 5 / 6 ] .
χ ^ 1 , sp , F = 1 2 0 L d z d v ( κ , z ) [ H sp , F ( κ , z ) + H sp , F ( κ , z ) ] W ˜ a , sp ( z L κ ) ,
χ ^ 1 , sp , I = 1 2 0 L d z d v ( κ , z ) [ H sp , I ( κ , z ) + H sp , I ( κ , z ) ] W ˜ a , sp ( L z L κ ) ,
H sp , F ( κ , z ) = H sp , I ( κ , z ) = i k exp [ i κ 2 z ( L z ) 2 L k ] ,
W ˜ a , sp ( γ κ ) = d 2 r W a ( r ) exp ( i γ κ r ) = jinc ( γ κ R a ) .
B sp , F I = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc ( z L R a κ ) × jinc ( L z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] ,
σ sp , F 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] ,
σ sp , I 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( L z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] .
μ sp B sp , F I σ sp , F σ sp , I .
B sp , F I = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f sp ( κ L / k ) ,
σ sp , F 2 σ sp , I 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f   sp ( κ L / k )
f sp ( κ ^ ) = 1 2 κ ^ 4 0 1 d ξ ξ 2 ( 1 ξ ) 2 jinc [ ( 1 ξ ) q R 1 / 2 κ ^ ] jinc ( ξ q R 1 / 2 κ ^ ) sinc 2 [ κ ^ 2 2 ξ ( 1 ξ ) ] ,
f   sp ( κ ^ ) = 1 2 κ ^ 4 0 1 d ξ ξ 2 ( 1 ξ ) 2 jinc 2 ( ξ q R 1 / 2 κ ^ ) sinc 2 [ κ ^ 2 2 ξ ( 1 ξ ) ] .
B sp , F I = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g sp ( ξ ) ,
σ sp , F 2 = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g   sp ( ξ ) ,
σ sp , I 2 = 0.033 π 2 k 7 / 6 L 11 / 6 0 1 d ξ C n 2 ( ξ L ) g   sp ( ξ ) ,
h ˜ sp ( x , a , b ) = x 2 ( 1 x ) 2 0 d κ ^ κ ^ 4 / 3 exp ( q m κ ^ 2 ) jinc ( a q R 1 / 2 κ ^ ) × jinc ( b q R 1 / 2 κ ^ ) sinc 2 [ x ( 1 x ) κ ^ 2 / 2 ] .
G ( α 1 , α 2 , α 3 , α 4 ) = 0.033 × 4 π 2 k 2 L C n 2 0 1 d ξ 0 d κ κ 8 / 3 exp ( κ 2 / κ m 2 ) × jinc ( α 1 κ R a ) jinc ( α 2 κ R a ) sin [ κ 2 L α 3 / ( 2 k ) ] × sin [ κ 2 L α 4 / ( 2 k ) ] .
G ( α 1 , α 2 , α 3 , α 4 ) 0.033 × 4 π 2 C n 2 k 7 / 6 L 11 / 6 0 1 d ξ 0 d x x 8 / 3 × sin ( x 2 α 3 / 2 ) sin ( x 2 α 4 / 2 ) = O ( σ χ , sp 2 ) .
G ( α 1 , α 2 , α 3 , α 4 ) 0.033 π 2 C n 2 k 7 / 6 L 11 / 6 q R 7 / 6 0 1 d ξ α 3 α 4 0 d x x 4 / 3 × jinc ( α 1 x ) jinc ( α 2 x ) = O ( σ χ , sp 2 q R 7 / 6 ) .
G ( α 1 , α 2 , α 3 , α 4 ) 0.033 π 2 C n 2 k 7 / 6 L 11 / 6 q m 7 / 6 0 1 d ξ α 3 α 4 0 d x x 4 / 3 × exp ( x 2 ) = O ( σ χ , sp 2 q m 7 / 6 ) .

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