Abstract

The speckle pattern produced in laser light that has been backscattered by atmospheric aerosols is considered. The spatiotemporal correlation properties both of the optical field amplitudes and of the optical intensities are described. Analytical expressions are presented for the special cases of illumination by a short Gaussian pulse and by a cw source.

© 1983 Optical Society of America

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References

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  1. R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
    [CrossRef]
  2. M. J. Post, F. F. Hall, R. A. Richter, T. R. Lawrence, Appl. Opt. 21, 2442 (1982).
    [CrossRef] [PubMed]
  3. T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981).
    [CrossRef]
  4. J. H. Churnside, H. T. Yura, Appl. Opt. 20, 3539 (1981).
    [CrossRef] [PubMed]
  5. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  8. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  10. J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley, New York, 1964).
  11. J. A. Dutton, D. G. Deavan, “Some Observed Properties of Atmospheric Turbulence,” in Statistical Models and Turbulence, M. Rosenblatt, C. Van Atta, Eds. (Springer, New York, 1972).
    [CrossRef]
  12. F. Pasquill, Atmospheric Diffusion (Wiley, New York, 1974).
  13. J. A. Dutton, J. Hojstcup, “A Model for the Probability Structure of Atmospheric Turbulence,” in Proceedings, Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting, Portland, Ore., 19–21 June 1979
  14. J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer, New York, 1975).
  15. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
    [CrossRef]
  16. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  17. J. H. Churnside, H. T. Yura, Appl. Opt. 21, 845 (1982).
    [CrossRef] [PubMed]
  18. J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 19, 3292 (1981).
    [CrossRef]
  19. L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

1982

1981

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 19, 3292 (1981).
[CrossRef]

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981).
[CrossRef]

J. H. Churnside, H. T. Yura, Appl. Opt. 20, 3539 (1981).
[CrossRef] [PubMed]

1976

1966

Asakura, T.

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981).
[CrossRef]

Capron, B. A.

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 19, 3292 (1981).
[CrossRef]

Churnside, J. H.

Deavan, D. G.

J. A. Dutton, D. G. Deavan, “Some Observed Properties of Atmospheric Turbulence,” in Statistical Models and Turbulence, M. Rosenblatt, C. Van Atta, Eds. (Springer, New York, 1972).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Drain, L. E.

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

Dutton, J. A.

J. A. Dutton, D. G. Deavan, “Some Observed Properties of Atmospheric Turbulence,” in Statistical Models and Turbulence, M. Rosenblatt, C. Van Atta, Eds. (Springer, New York, 1972).
[CrossRef]

J. A. Dutton, J. Hojstcup, “A Model for the Probability Structure of Atmospheric Turbulence,” in Proceedings, Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting, Portland, Ore., 19–21 June 1979

Goodman, J. W.

J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Hall, F. F.

Hardesty, R. M.

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

Harney, R. C.

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 19, 3292 (1981).
[CrossRef]

Hojstcup, J.

J. A. Dutton, J. Hojstcup, “A Model for the Probability Structure of Atmospheric Turbulence,” in Proceedings, Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting, Portland, Ore., 19–21 June 1979

Keeler, R. J.

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

Kogelnik, H.

Lawrence, T. R.

Li, T.

Lumley, J. L.

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley, New York, 1964).

Panofsky, H. A.

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley, New York, 1964).

Pasquill, F.

F. Pasquill, Atmospheric Diffusion (Wiley, New York, 1974).

Post, M. J.

M. J. Post, F. F. Hall, R. A. Richter, T. R. Lawrence, Appl. Opt. 21, 2442 (1982).
[CrossRef] [PubMed]

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

Richter, R. A.

M. J. Post, F. F. Hall, R. A. Richter, T. R. Lawrence, Appl. Opt. 21, 2442 (1982).
[CrossRef] [PubMed]

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 19, 3292 (1981).
[CrossRef]

Takai, N.

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Yura, H. T.

Appl. Opt.

Appl. Phys.

T. Asakura, N. Takai, Appl. Phys. 25, 179 (1981).
[CrossRef]

J. Opt. Soc. Am.

Opt. Appl.

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, Opt. Appl. 20, 3763 (1981).
[CrossRef]

Other

L. E. Drain, The Laser Doppler Technique (Wiley, New York, 1980).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

J. L. Lumley, H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley, New York, 1964).

J. A. Dutton, D. G. Deavan, “Some Observed Properties of Atmospheric Turbulence,” in Statistical Models and Turbulence, M. Rosenblatt, C. Van Atta, Eds. (Springer, New York, 1972).
[CrossRef]

F. Pasquill, Atmospheric Diffusion (Wiley, New York, 1974).

J. A. Dutton, J. Hojstcup, “A Model for the Probability Structure of Atmospheric Turbulence,” in Proceedings, Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting, Portland, Ore., 19–21 June 1979

J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer, New York, 1975).

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Equations (41)

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U T ( r , t ) = [ 2 P T ( t ) π a 2 ] 1 / 2 exp [ ( 1 a 2 + i k 2 f ) r 2 ] ,
U I ( ρ , t ) = k e ikz 2 π iz d 2 r U T ( r , t z c ) exp [ i k 2 z ( ρ r ) 2 ] ,
U I ( ρ , t ) = [ P T t z / c ] 1 / 2 k a i z 1 i k a 2 2 z ( 1 z f ) exp [ ikz k 2 a 2 4 z f i k 2 z 1 i k a 2 2 z ( 1 z f ) ρ 2 ] .
U S = S U I exp ( ikL ) L ,
U S ( q , t ) = [ P T ( t 2 z / c ) 2 π ] 1 / 2 Ska i z 2 1 i k a 2 2 z ( 1 z f ) × exp [ 2 ikz k 2 a 2 4 z f i k 2 z 1 i k a 2 2 z ( 1 z f ) ρ 2 + i k 2 z ( q ρ ) 2 ] .
U S ( q , t + τ ) = [ P T ( t + τ 2 z / c ) 2 π ] 1 / 2 Ska i z 2 1 i k a 2 2 z ( 1 z f ) × exp [ 2 i k ( z + υ z τ ) + i k 2 z ( q ρ v T τ ) 2 k 2 a 2 4 z f i k 2 z 1 i k a 2 2 z ( 1 z f ) ( ρ + v T τ ) 2 ] .
U S ( q 1 , t ) U S * ( q 2 , t + τ ) = | S | 2 k 2 a 2 [ P T ( t 2 z c ) P T ( t + τ 2 z c ) ] 1 / 2 2 π z 4 [ 1 + ( k a 2 2 z ) 2 ( 1 z f ) 2 ] × exp [ 2 i k υ z τ + i k 2 z ( q 1 ρ ) 2 i k 2 z ( q 2 ρ v T τ ) 2 ( k a 2 z ) 2 i k 2 z [ 1 k 2 a 4 4 z f ( 1 z f ) ] 1 + ( k a 2 2 z ) 2 ( 1 z f ) 2 ρ 2 ( k a 2 z ) 2 + i k 2 z [ 1 k 2 a 4 4 z f ( 1 z f ) ] 1 + ( k a 2 2 z ) 2 ( 1 z f ) 2 ( ρ + v T τ ) 2 ] .
p ( υ z ) = 1 2 π σ z exp [ 1 2 σ z 2 ( υ z υ ̅ z ) 2 ] , p ( v T ) = 1 2 π σ t 2 exp [ 1 2 σ T 2 2 ( v T v ̅ t ) 2 ] ,
p ( ρ , z ) = 1 / V for ( ρ , z ) V = 0 elsewhere .
U S ( q 1 , t ) U S * ( q 2 , t + τ ) = d υ z d 2 v T v × d z D 2 ρ U S ( q 1 , t ) U S * ( q 2 , t + τ ) × p ( υ z ) p ( v T ) 1 V .
U s ( q 1 , t ) U s * ( q 2 , t + τ ) = o d z β ( z ) [ P T ( t 2 z c ) P T ( t + τ 2 z c ) ] 1 / 2 K z 2 × exp ( 2 k 2 σ z 2 τ 2 2 i k υ ̅ z τ 1 2 K a 2 ( q 2 v ̅ T τ ) 2 1 2 K ( k a 2 z ) 2 [ ( 1 z f ) q ( 1 2 z f ) v ̅ T τ ] 2 i k K z Q · ( q v ̅ T τ ) σ T 2 τ 2 { 1 2 K ( k 2 z ) 2 ( 4 Q 2 + q 2 ) + i k K z [ 2 a 2 ( 1 2 z f ) ( k a 2 z ) 2 z f ] Q · q } )
K = 1 + σ T 2 τ 2 [ 4 a 2 + ( 1 2 z f ) 2 ( k a 2 z ) 2 ] ,
U S ( q 1 , t ) U S * ( q 2 , t + τ ) = d z β ( z ) [ P T ( t 2 z c ) P T ( t + τ 2 z c ) ] 1 / 2 z 2 × exp { 2 k 2 σ 2 τ 2 2 i k υ ̅ z τ 1 2 a 2 ( q 2 v ̅ T τ ) 2 1 2 ( k a 2 z ) 2 [ ( 1 z f ) q ( 1 2 z f ) v ̅ T τ ] 2 i k z Q · ( q v ̅ T τ ) ] .
R I ( Q , q , τ ) = I ( q 1 , t ) I ( q 2 , t + τ ) I ( q 1 , t ) I ( q 2 , t + τ ) [ I 2 ( q 1 , t ) I ( q 1 , t ) 2 ] 1 / 2 [ I 2 ( q 2 , t + τ ) I ( q 2 , t + τ ) 2 ] 1 / 2 = | Γ ( Q , q , τ ) | 2 [ Γ ( Q + ½ q , 0 , 0 ) Γ ( Q ½ q , 0 , 0 ) ] 1 / 2 .
P T ( t ) = E π τ p exp [ t 2 / τ p 2 ] ,
Γ ( Q , q , τ ) = β c E 2 z 0 2 exp { τ 2 4 τ p 2 2 k 2 σ 2 τ 2 2 i k υ ̅ z τ 1 2 a 2 ( q 2 v ̅ T τ ) 2 1 2 ( k a 2 z 0 ) 2 [ ( 1 z 0 f ) q ( 1 2 z 0 f ) v ̅ T τ ] 2 i k z 0 Q · ( q v ̅ T τ ) } .
R I ( Q , q , τ ) = exp [ τ 2 2 τ p 2 4 k 2 σ 2 τ 2 1 a 2 ( q 2 v ̅ T τ ) 2 ( k a 2 z 0 ) 2 [ ( 1 z 0 f ) q ( 1 2 z 0 f ) v ̅ T τ ] 2 } .
Γ ( Q , q , τ ) = 2 π β P T k a | q v ̅ τ | exp [ 2 k 2 σ 2 τ 2 2 i k υ ̅ z τ 1 2 a 2 ( q 2 v ̅ T τ ) 2 2 Q 2 a 2 i k f Q · ( q 2 v ̅ T τ ) ] { erf [ g ( z 1 ) ] erf [ g ( z 2 ) ] } ,
g ( z ) = k a 8 z | q v ̅ T τ | k a 8 f ( q v ̅ T τ ) | q v ̅ T τ | · ( q 2 v ̅ T τ ) 2 i Q a .
Γ ( Q , q , τ ) = β P T z 2 z 1 z 1 z 2 exp [ 2 k 2 σ 2 τ 2 2 i k υ ̅ z τ 1 2 a 2 ( q 2 v ̅ T τ 2 1 2 ( k a 2 f ) 2 v ̅ T 2 τ 2 i k f Q · ( q v ̅ T τ ) ] .
R I ( Q , q , τ ) = 2 π z 1 2 z 2 2 k 2 a 2 ( q v ̅ T τ ) 2 ( z 2 z 1 ) 2 × exp [ 4 k 2 σ 2 τ 2 1 a 2 ( q 2 v ̅ T τ ) 2 4 Q 2 a 2 ] × | erf [ g ( z 1 ) ] erf [ g ( z 2 ] | 2 .
R I ( Q , q , τ ) = exp [ 4 k 2 σ 2 τ 2 1 a 2 ( q 2 v ̅ T τ ) 2 ( k a 2 f ) 2 υ ̅ T 2 τ 2 ] .
k a 8 | q v ̅ T τ |
Γ I ( Q , q , τ ) = exp [ 4 k 2 σ 2 τ 2 ] Γ I 0 ( Q , q , τ ) ,
σ 2 ( Δ z L 0 ) 2 / 3 σ F 2 , Δ z L o σ F 2 , Δ z > L o
| Δ Γ | N ,
Δ Γ Γ ˙ Δ τ .
Δ τ ( N / Γ ˙ ) .
t 0 = [ 2 k σ ] 1
Δ τ t o .
( SNR ) S t o Γ ˙ .
( SNR ) pulsed 2 k σ exp ( q 2 / 2 a 2 ) ( a 2 q υ t ) [ 1 + 1 2 ( k a 2 2 z o ) 2 × ( 1 z o f ) ( 1 2 z o f ) ] 1 ,
( SNR ) CW 2 k σ ( a 2 q υ T ) exp ( q 2 / 2 a 2 ) ,
( SNR ) k a ( σ υ T ) ,
S ( ω D ) S N ,
S N = N / B ,
B = 2 k Δ υ .
S ( ω ) = Γ exp [ i ( ω ω p ) τ ] d τ .
S ( ω ) = S ( ω D ) exp [ ( ω ω D ) 2 t o 2 ] .
S = S ( ω ) d ω = π S ( ω D ) t o .
SNR σ / Δ υ

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