Abstract

A criterion is introduced for testing whether a beam retains its beam-like form after it propagates any particular distance through the turbulent atmosphere. The criterion applies to monochromatic as well as to partially coherent beams and is illustrated by examples.

© 2007 Optical Society of America

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References

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  1. J. Wu, J. Mol. Spectrosc. 37, 671 (1990).
  2. J. Wu and A. D. Boardman, J. Mol. Spectrosc. 38, 1355 (1991).
  3. G. Gbur and E. Wolf, J. Opt. Soc. Am. A 19, 1592 (2002).
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  4. J. C. Ricklin and F. M. Davidson, J. Opt. Soc. Am. A 19, 1794 (2002).
    [CrossRef]
  5. X. Ji and B. Lu, Opt. Commun. 251, 231 (2005).
    [CrossRef]
  6. E. Wolf, Proc. R. Soc. London, Ser. A 204, 533 (1951).
    [CrossRef]
  7. R. Barakat, in Progress in Optics, E.Wolf, ed. (Elsevier, 1961), Vol. 1, pp. 57-108, Sec. 3.5.
    [CrossRef]
  8. V. Mahajan, in Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Vol. 49, p. 9.
    [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
    [CrossRef]

2005 (1)

X. Ji and B. Lu, Opt. Commun. 251, 231 (2005).
[CrossRef]

2002 (2)

1991 (1)

J. Wu and A. D. Boardman, J. Mol. Spectrosc. 38, 1355 (1991).

1990 (1)

J. Wu, J. Mol. Spectrosc. 37, 671 (1990).

1951 (1)

E. Wolf, Proc. R. Soc. London, Ser. A 204, 533 (1951).
[CrossRef]

J. Mol. Spectrosc. (2)

J. Wu, J. Mol. Spectrosc. 37, 671 (1990).

J. Wu and A. D. Boardman, J. Mol. Spectrosc. 38, 1355 (1991).

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

Opt. Commun. (1)

X. Ji and B. Lu, Opt. Commun. 251, 231 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

E. Wolf, Proc. R. Soc. London, Ser. A 204, 533 (1951).
[CrossRef]

Other (4)

R. Barakat, in Progress in Optics, E.Wolf, ed. (Elsevier, 1961), Vol. 1, pp. 57-108, Sec. 3.5.
[CrossRef]

V. Mahajan, in Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Vol. 49, p. 9.
[CrossRef]

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

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

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Figures (3)

Fig. 1
Fig. 1

Illustrating the notation and the definition of fractional power p ( ρ ¯ , z , ω ) [Eqs. (1, 2)].

Fig. 2
Fig. 2

Contours of the fractional power in transverse cross-sections of a monochromatic Gaussian beam, with λ = 0.633 μ m , A = 1 , w 0 = 2 cm propagating in (a) free space, (b) atmosphere with C n 2 = 10 14 m 2 3 , l 0 = 1 mm , (c) atmosphere with C n 2 = 10 13 m 2 3 , l 0 = 1 mm .

Fig. 3
Fig. 3

Contours of the fractional power in cross-sections of a Gaussian Schell-model beam, with λ = 0.633 μ m , σ S = 1 cm propagating in free space and in the atmosphere, with C n 2 = 10 13 m 2 3 . (a)–(c) in free space, (d)–(f) in the turbulent atmosphere (Kolmogorov spectrum). (a) and (d): σ μ σ S = (completely spatially coherent source), (b) and (e): σ μ σ S = 0.1 (partially coherent source), (c) and (f): σ μ σ S 0 (incoherent source).

Tables (1)

Tables Icon

Table 1 Distances of Propagation Calculated from Eqs. (15, 16) at Which the Directionality of the Gaussian Schell-Model Beam Starts to Exceed Angle θ = 1 mrad 0 ° 3 26 for Selected Degrees of Source Coherence and Strengths of Turbulence a

Equations (16)

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p ( ρ ¯ , z , ω ) = L ( ρ ¯ , z , ω ) L ( , z , ω ) ,
L ( ρ ¯ , z , ω ) = σ S ( ρ , z , ω ) d 2 ρ .
U ( 0 ) ( ρ ) = A exp [ ρ 2 w 0 2 ] ,
S ( ρ , z ) U ( ρ , z ) 2 = A 2 w 0 2 w 2 ( z ) exp [ 2 ρ 2 w 2 ( z ) ] ,
w ( z ) = w 0 1 + ( 2 z k w 0 2 ) 2 .
S ( ρ , z ) = A 2 w 0 2 w e 2 ( z ) exp ( 2 ρ 2 w e 2 ( z ) ) ,
w e ( z ) = w ( z ) 1 + 4.37 C n 2 l 0 1 3 z 3 w 2 ( z )
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = U ( 0 ) * ( ρ 1 , ω ) U ( 0 ) ( ρ 2 , ω ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) ( ρ 1 + ρ 2 2 , ω ) μ ( 0 ) ( ρ 2 ρ 1 , ω ) ,
S ( 0 ) ( ρ , ω ) = A 2 exp ( ρ 2 2 σ S 2 ) ,
μ ( 0 ) ( ρ 2 ρ 1 , ω ) = exp [ ( ρ 2 ρ 1 ) 2 2 σ μ 2 ] .
S ( ρ , z , ω ) = A 2 Δ 2 ( z , ω ) exp [ ρ 2 2 σ S 2 Δ 2 ( z , ω ) ] ,
Δ 2 ( z , ω ) = 1 + α z 2
α = 1 ( k σ ) 2 ( 1 4 σ S 2 + 1 σ μ 2 ) .
S ( ρ , z , ω ) = A 2 Δ e 2 ( z , ω ) exp [ ρ 2 2 σ S 2 Δ e 2 ( z , ω ) ] ,
Δ e 2 = 1 + α z 2 + 0.98 ( C n 2 ) 6 5 k 2 5 σ S 2 z 16 5 .

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