Abstract

On the basis of the analytic techniques presented in the first of these two companion papers [J. Opt. Soc. Am. A 27, 2169 (2010) ] we present the complete asymptotic analysis of the axial beam scintillation index for coherent Gaussian beams on the ground-to-space propagation paths. The ratio of turbulence layer thickness to overall propagation path length contributes an additional small parameter to the analysis. We show that it is possible to use three dimensionless parameters to describe the problem and that the general arrangement of the asymptotic regions established in our earlier work [Waves Random Media 4, 243 (1994) ]) is preserved. We find that on a slant propagation path, collimated beams can experience the unusual double-scattering-dominated scintillation found originally for focused beams.

© 2010 Optical Society of America

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References

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  1. M. I. Charnotskii, “Beam scintillations for ground-to-space propagation. Part 1: path integrals and analytic techniques,” J. Opt. Soc. Am. A 27, 2169–2179 (2010).
    [CrossRef]
  2. M. I. Charnotskii, “Asymptotic analysis of finite beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  3. M. I. Charnotskii, “Beam scintillations for ground-to-space propagation,” Proc. SPIE 7463, 746304 (2009).
    [CrossRef]
  4. M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768501 (2010).
  5. G. J. Baker, “Gaussian beam weak scintillation: Low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
    [CrossRef]
  6. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
    [CrossRef]
  7. V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).
  8. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  9. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
    [CrossRef]
  10. J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
    [CrossRef]

2010

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768501 (2010).

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation. Part 1: path integrals and analytic techniques,” J. Opt. Soc. Am. A 27, 2169–2179 (2010).
[CrossRef]

2009

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation,” Proc. SPIE 7463, 746304 (2009).
[CrossRef]

2007

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
[CrossRef]

2006

G. J. Baker, “Gaussian beam weak scintillation: Low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

2001

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

1999

1994

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

1977

V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Andrews, L. C.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Baker, G. J.

Charnotskii, M. I.

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation. Part 1: path integrals and analytic techniques,” J. Opt. Soc. Am. A 27, 2169–2179 (2010).
[CrossRef]

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768501 (2010).

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation,” Proc. SPIE 7463, 746304 (2009).
[CrossRef]

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

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Klyatskin, V. I.

V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Phillips, R. L.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Recolons, J.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
[CrossRef]

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Tatarskii, V. I.

V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Zavorotny, V. U.

V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

J. Opt. Soc. Am. A

Opt. Eng.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: Focused beam case,” Opt. Eng. 46, 086002-1–086002-11 (2007).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Proc. SPIE

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation,” Proc. SPIE 7463, 746304 (2009).
[CrossRef]

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768501 (2010).

Sov. Phys. JETP

V. U. Zavorotny, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–260 (1977).

Waves Random Media

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

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

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Figures (2)

Fig. 1
Fig. 1

Map of asymptotes for the beam focus. Double line is a weak–strong scintillation boundary; solid lines, boundaries of the principal domains; dashed, boundaries between the sub-domains.

Fig. 2
Fig. 2

Map of asymptotes for the case h 1 . Double line is a weak–strong scintillation boundary; solid lines, boundaries of the principal domains; dashed, boundaries between the sub-domains.

Tables (2)

Tables Icon

Table 1 Full Set of Weak Scintillation Asymptotes

Tables Icon

Table 2 Full Set of Strong Scintillation Asymptotes

Equations (33)

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Γ 2 ( R , ρ , 0 ) = exp [ R 2 a 2 ρ 2 4 a 2 i k F ( R ρ ) ] ,
Γ 4 ( R , r 1 , r 2 , ρ , 0 ) = exp [ 2 R 2 a 2 r 1 2 + r 2 2 2 a 2 ρ 2 8 a 2 i k F ( R ρ + r 1 r 2 ) ] .
N = k a 2 L , q = k r C 2 L , d = 1 L F .
N = k a 2 L T , q = k r C 2 L T = 1.46 k 7 5 L T 1 ( 0 L T d z C ε 2 ( z ) ) 6 5 ,
h = L T ( 1 L 1 F ) = θ d ,
I ( 0 , L ) = N 2 θ 2 4 π a 2 d 2 ρ exp [ ρ 2 4 a 2 ( 1 + N 2 h 2 ) π k 2 4 0 θ L d z H ( ρ , z ) ] .
I ( 0 , L ) = N 2 θ 2 4 π a 2 d 2 ρ exp [ π k 2 4 0 θ L d z H ( ρ , z ) ] .
I ( 0 , L ) = I 0 ( 0 , L ) = N 2 θ 2 1 + N 2 h 2 .
W 1 = π k 2 0 θ L d z d 2 κ Φ ε ( κ , z ) exp ( κ 2 a 2 1 + N 2 h 2 ) [ 1 cos ( κ 2 ( a 2 N h 1 + N 2 h 2 + z k ) ) ] .
W 1 = π k 2 0 θ L d z d 2 κ Φ ε ( κ , z ) [ 1 cos ( κ 2 z k ) ] ,
W 1 = π k 2 0 θ L d z d 2 κ Φ ε ( κ , z ) [ 1 cos ( κ 2 L T k h ) ] ,
W 1 = π L T 2 2 0 θ L d z ( N 2 h + z L T ) 2 d 2 κ κ 4 Φ ε ( κ , z ) exp ( κ 2 a 2 ) .
W 2 = π 2 k 4 2 0 θ L d z 0 θ L d z d 2 κ κ 4 Φ ε ( κ , z ) d 2 κ κ 4 Φ ε ( κ , z ) × exp [ a 2 ( κ 2 + κ 2 ) ] [ exp ( a 2 ( κ κ ) ) 1 ] 2 .
M 1 = N 4 θ 4 k 2 16 π a 4 0 θ L d z d 2 κ Φ ε ( κ , z ) d 2 r d 2 ρ exp ( r 2 2 a 2 ρ 2 ( 1 + N 2 h 2 ) 8 a 2 a 2 2 ( k h L T r + κ ) 2 ) × exp ( π k 2 4 0 θ L d ζ [ H ( r + ρ 2 κ k min ( z , ζ ) , ζ ) + H ( r ρ 2 κ k min ( z , ζ ) , ζ ) ] ) × [ cos ( κ ρ 2 ) cos ( κ r κ 2 z k ) ] .
M 1 = π 2 N 4 θ 4 0 θ L z 2 d z d 2 κ κ 4 Φ ε ( κ , z ) exp [ π k 2 2 0 θ L d ζ H ( κ k min ( z , ζ ) , ζ ) ] .
M 1 = π L T 2 θ 4 2 h 6 0 θ L d z d 2 κ κ 4 Φ ε ( κ , z ) exp [ π k 2 2 0 θ L d ζ H ( κ L T k h , ζ ) ] ,
M 1 = N 4 θ 4 k 2 16 π a 4 0 θ L d z d 2 κ Φ ε ( κ , z ) × ( d 2 r ( κ r κ 2 z k ) exp [ π k 2 4 0 θ L d ζ H ( r + κ k min ( z , ζ ) , ζ ) ] ) 2 .
M 1 = N 4 θ 4 k 2 32 π a 4 0 θ L d z d 2 κ Φ ε ( κ , z ) exp ( κ 2 a 2 2 ) × [ d 2 r ( κ r ) 2 ( 1 2 N h + π k 4 0 z d ζ ( z ζ ) H r ( r , ζ ) r ) exp ( π k 2 4 0 θ L d ζ H ( r , ζ ) ) ] 2 .
M 2 = k 8 a 4 θ 4 128 L T 4 0 θ L d z 1 0 θ L d z 2 d 2 κ 1 d 2 κ 2 Φ ε ( κ 1 , z 1 ) Φ ε ( κ 2 , z 2 ) exp ( ( κ 1 2 + κ 2 2 ) a 2 2 ) × [ d 2 r ( κ 1 r ) ( κ 2 r ) exp ( π k 2 4 0 θ L d ζ H ( r , ζ ) ) ] 2 .
σ I 2 = 0.563 k 7 6 0 θ L d z C ε 2 ( z ) z 5 6 = O ( q 5 6 ) .
W 1 = 0.151 a 7 3 0 θ L d z C ε 2 ( z ) z 2 = O ( q 5 6 N 7 6 ) ,
W 2 = 0.083 k 4 a 10 3 ( 0 θ L d z C ε 2 ( z ) ) 2 = O ( q 5 3 N 5 3 ) .
σ I 2 = 1 + 0.54 k 7 15 0 θ L d z C ε 2 ( z ) z 2 [ 0 θ L d ζ C ε 2 ( ζ ) [ min ( z , ζ ) ] 5 3 ] 7 5 = 1 + O ( q 1 3 ) .
σ I 2 = 1 + O ( q 1 3 ) ,
2 M 1 I ( 0 , L ) 2 = 0.68 a 7 3 0 L T d z C ε 2 ( z ) ( 0 z d ζ C ε 2 ( ζ ) ( z ζ ) ) 2 ( 0 L T d z C ε 2 ( z ) ) 2 = O ( q 5 6 N 7 6 ) ,
2 M 2 I ( 0 , L ) 2 = 2.96 k 4 5 a 2 3 ( 0 L T d z C ε 2 ( z ) ) 2 5 = O ( q 1 3 N 1 3 ) .
σ I 2 = 0.563 k 7 6 L T 5 6 h 5 6 0 L T d z C ε 2 ( z ) = O ( q 5 6 h 5 6 ) ,
W 1 = 0.151 a 7 3 0 L T z 2 d z C ε 2 ( z ) + 0.302 k 2 L T 1 a 5 3 h 0 L T z d z C ε 2 ( z ) + 0.151 k 4 a 17 3 L T 2 h 2 0 L T d z C ε 2 ( z ) = O ( q 5 6 N 7 6 ) + O ( q 5 6 N 5 6 h ) + O ( q 5 6 N 17 6 h 2 ) .
σ I 2 = 0.151 a 7 3 0 L T d z C ε 2 ( z ) z 2 + 0.302 k 2 L T 1 a 5 3 h 0 L T d z C ε 2 ( z ) + 0.151 k 4 L T 2 a 17 3 0 L T d z C ε 2 ( z ) + 0.083 k 4 a 10 3 ( 0 L T d z C ε 2 ( z ) ) 2 = O ( q 5 6 N 7 6 ) + O ( q 5 6 N 5 6 h ) + O ( q 5 6 N 7 6 h 2 ) + O ( q 5 3 N 5 3 ) .
σ I 2 = 1 + 0.54 k 7 15 L T 1 3 h 1 3 [ 0 L T d ζ C ε 2 ( ζ ) ] 2 5 = 1 + O ( q 1 3 h 1 3 ) .
2 M 1 I ( 0 , L ) 2 = 0.68 a 7 3 0 L T d z C ε 2 ( z ) ( 0 z d ζ C ε 2 ( ζ ) ( z ζ ) ) 2 ( 0 L T d z C ε 2 ( z ) ) 2 + 0.88 k 4 5 L 2 a 5 3 ( 0 L T d z C ε 2 ( z ) ) 7 5 = O ( q 5 6 N 7 6 ) + O ( q 7 6 N 5 6 h 2 ) .
σ I 2 = 1 + 2 M 1 I ( 0 , L ) 2 + 2 M 2 I ( 0 , L ) 2 = 1 + O ( q 5 6 N 7 6 ) + O ( q 7 6 N 5 6 h 2 ) + O ( q 1 3 N 1 3 ) .
max ( 1 , q 1 ) N max ( θ 1 , q 1 θ 2 ) .

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