Abstract

Based on the extended Huygens–Fresnel principle, the propagation law of the beam matrix in terms of second moments of the Wigner distribution function for partially coherent beams propagating through atmospheric turbulence was obtained. The general formulas for the mean-squared spatial and angular widths, as well as the beam propagation factor (M2 factor) of partially coherent beams in turbulence were also derived, which can be applied to cases of different spatial power spectra of the refractive index fluctuations of the turbulent atmosphere.

© 2009 Optical Society of America

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References

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    [CrossRef]
  2. R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
    [CrossRef]
  3. G. Gbur and E. Wolf, J. Opt. Soc. Am. A 19, 1592 (2002).
    [CrossRef]
  4. S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
    [CrossRef]
  5. A. Dogariu and S. Amarande, Opt. Lett. 28, 10 (2003).
    [CrossRef] [PubMed]
  6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
    [CrossRef]
  7. M. H. Mahdieh, Opt. Commun. 281, 3395 (2008).
    [CrossRef]
  8. Y. Dan and B. Zhang, Opt. Express 16, 15563 (2008).
    [CrossRef] [PubMed]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  10. R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
    [CrossRef]
  11. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).
  12. G. Gbur and E. Wolf, Opt. Commun. 199, 295 (2001).
    [CrossRef]

2008 (2)

2003 (1)

2002 (2)

G. Gbur and E. Wolf, J. Opt. Soc. Am. A 19, 1592 (2002).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
[CrossRef]

2001 (1)

G. Gbur and E. Wolf, Opt. Commun. 199, 295 (2001).
[CrossRef]

1995 (1)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
[CrossRef]

1991 (1)

1988 (1)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

Dan, Y.

Dogariu, A.

Gbur, G.

Greffet, J.-J.

S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
[CrossRef]

Mahdieh, M. H.

M. H. Mahdieh, Opt. Commun. 281, 3395 (2008).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
[CrossRef]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, J. Opt. Soc. Am. A 8, 1094 (1991).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
[CrossRef]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, J. Opt. Soc. Am. A 8, 1094 (1991).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[CrossRef]

Piquero, G.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
[CrossRef]

Serna, J.

Simon, R.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Wolf, E.

S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
[CrossRef]

G. Gbur and E. Wolf, J. Opt. Soc. Am. A 19, 1592 (2002).
[CrossRef]

G. Gbur and E. Wolf, Opt. Commun. 199, 295 (2001).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Zhang, B.

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

Opt. Commun. (5)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

M. H. Mahdieh, Opt. Commun. 281, 3395 (2008).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, and E. Wolf, Opt. Commun. 208, 1 (2002).
[CrossRef]

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, Opt. Commun. 115, 225 (1995).
[CrossRef]

G. Gbur and E. Wolf, Opt. Commun. 199, 295 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (3)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

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Equations (20)

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W ( ρ 1 , ρ 2 ; 0 ) = E ( ρ 1 , 0 ) E * ( ρ 2 , 0 ) m ,
W ( ρ , ρ d ; z ) = ( k 2 π z ) 2 W ( ρ , ρ d ; 0 ) exp { i k z [ ( ρ ρ ) ( ρ d ρ d ) ] H ( ρ d , ρ d ; z ) } d 2 ρ d 2 ρ d ,
H ( ρ d , ρ d ; z ) = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ ρ d ξ + ( 1 ξ ) ρ d ) ] Φ n ( κ ) κ d κ ,
h ( ρ , θ ; z ) = ( k 2 π ) 2 W ( ρ , ρ d ; z ) exp ( i k θ ρ d ) d 2 ρ d ,
x n 1 y n 2 θ x m 1 θ y m 2 1 P x n 1 y n 2 θ x m 1 θ y m 2 h ( ρ , θ ; z ) d 2 ρ d 2 θ = 1 P h ( ρ , θ ; 0 ) G ( ρ , θ ; z ) d 2 ρ d 2 θ ,
G ( ρ , θ ; z ) = ( k 2 π ) 4 1 z 2 x n 1 y n 2 θ x m 1 θ y m 2 exp { i k z [ ( ρ ρ ) ( ρ d ρ d ) ] + i k θ ρ d i k θ ρ d H ( ρ d , ρ d ; z ) } d 2 ρ d 2 θ d 2 ρ d d 2 ρ d .
δ ( n ) ( s ) = 1 2 π ( i x ) n exp ( i s x ) d x , ( n = 0 , 1 , 2 ) ,
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) , ( n = 0 , 1 , 2 ) ,
G ( ρ , θ ; z ) = z n 1 + n 2 k 2 exp [ i k z ρ ( ρ d ρ d ) + i k θ ρ d H ( ρ d , ρ d ; z ) ] δ ( n 1 ) ( x d x d ) δ ( n 2 ) ( y d y d ) δ ( m 1 ) ( x d ) δ ( m 2 ) ( y d ) d 2 ρ d d 2 ρ d .
G ( ρ , θ , z ) = z 2 k 2 exp [ i k z ρ ( ρ d ρ d ) + i k θ ρ d H ( ρ d , ρ d ; z ) ] δ ( 2 ) ( x d x d ) δ ( y d y d ) δ ( x d ) δ ( y d ) d 2 ρ d d 2 ρ d .
G ( ρ , θ ; z ) = z 2 k 2 { 2 x d 2 exp [ i k z x x d + i k θ x x d H ( 0 , x d x ̂ ; z ) ] } x d = 0 .
G ( ρ , θ ; z ) = x 2 + 2 x θ x z + θ x 2 z 2 + 2 3 π 2 z 3 0 Φ n ( κ ) κ 3 d κ .
x 2 = x 2 0 + 2 x θ x 0 z + θ x 2 0 z 2 + 2 3 π 2 z 3 0 Φ n ( κ ) κ 3 d κ ,
V 0 = [ x 2 0 x y 0 x θ x 0 x θ y 0 x y 0 y 2 0 y θ x 0 y θ y 0 x θ x 0 y θ x 0 θ x 2 0 θ x θ y 0 x θ y 0 y θ y 0 θ x θ y 0 θ y 2 0 ] ,
V = M z V 0 M z t + M T ,
M z = [ 1 0 z 0 0 1 0 z 0 0 1 0 0 0 0 1 ] , M T = T [ 2 3 z 3 0 z 2 0 0 2 3 z 3 0 z 2 z 2 0 2 z 0 0 z 2 0 2 z ] ,
T = π 2 0 Φ n ( κ ) κ 3 d κ ,
ρ 2 = ρ 2 0 + 2 ρ θ 0 z + θ 2 0 z 2 + 4 3 T z 3 .
θ 2 = θ 2 0 + 4 T z .
M 2 ( z ) = [ M 4 ( 0 ) + 4 ρ 2 0 k 2 T z + 4 ρ θ 0 k 2 T z 2 + 4 3 θ 2 0 k 2 T z 3 + 4 3 k 2 T 2 z 4 ] 1 2 .

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