Abstract

On the basis of the extended Huygens–Fresnel principle, a general expression is derived for the short-term average optical-beam spread, as measured with respect to the instantaneous center of energy of the beam, of an initially coherent optical-beam wave propagating in a weakly inhomogeneous medium. The present analysis applies to the near field of the effective coherent transmitting aperture, where the beam wanders (dances) as a whole and does not break up into multiple patches or blobs. Central to the analysis is the short-term average mutual coherence function (MCF) of a spherical wave. This quantity is obtained from the corresponding long-term MCF by removing the random tilt of the wave front. Analytic expressions for the short-term beam spread are presented for the case of a Kolmogorov spectrum and the short-term average MCF derived by Fried. As expected, the short-term, turbulence-induced beam spread is always less than the corresponding long-term beam spread. Analytic and numerical results are given for the short-term average irradiance at focal range f, which is always greater than the corresponding long-term average irradiance.

© 1973 Optical Society of America

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References

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  1. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
    [Crossref]
  2. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  4. A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).
  5. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  7. D. L. Fried and H. T. Yura, J. Opt. Soc. Am. 62, 600 (1972).
    [Crossref]
  8. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  9. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
    [Crossref]
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    [Crossref]

1972 (1)

1971 (4)

1970 (2)

A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).

R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
[Crossref]

1966 (1)

1964 (1)

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Figures (5)

Fig. 1
Fig. 1

Ratio of short-term to long-term MCF of a spherical wave for a Kolmogorov spectrum as a function of ρ/D for various values of ρ0LT/D.

Fig. 2
Fig. 2

Ratio of short-term to long-term, turbulence-induced beam spread as a function of ρ0LT/D.

Fig. 3
Fig. 3

Ratio of short-term to long-term, on-axis irradiance as a function of ρ0LT/D.

Fig. 4
Fig. 4

Short-term reduction in on-axis irradiance compared with its value in the absence of turbulence as a function of ρ0LT/D.

Fig. 5
Fig. 5

Normalized irradiance profile as a function of α for ρ0LT/D = 1 and β = 1 for a Kolmogorov spectrum and an initially gaussian wave function. (Curves A, B, and C refer to long-term average, short-term average, and the absence of turbulence, respectively.)

Equations (34)

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I ( p ) = ( k 2 π z ) 2 exp [ i k ( s 1 s 2 ) ] × exp [ ψ ( r 1 ) + ψ * ( r 2 ) ] U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
I ( p ) LT = ( k 2 π z ) 2 exp [ i k ( s 1 s 2 ) ] × M LT ( r 1 , r 2 ; z ) U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
I ( p ) ST = ( k 2 π z ) 2 exp [ i k ( s 1 s 2 ) ] M ST ( r 1 , r 2 ; z ) × U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
M ST pw ( ρ , z ) = M LT pw ( ρ , z ) exp { 1 2 ρ 2 η } ,
η = 64 D 2 0 1 d x x [ F C ( x ) F L ( x ) ] D ϕ ( D x , z ) ,
F C ( x ) = ( π ) 1 { 2 cos 1 x 2 x ( 1 x 2 ) 1 2 } ,
F L ( x ) = ( π ) 1 { 6 cos 1 x [ 14 x 8 x 2 ] ( 1 x 2 ) 1 2 } .
D ϕ ( ρ ) = g ( ρ ) D ( ρ ) ,
M LT pw = exp { 2.91 2 C n 2 k 2 z ρ 5 / 3 } ,
η ( 1 2 ) [ 0.7 A δ ρ 2 D 1 2 ] ,
A = 2.91 k 2 C n 2 z
δ = 1 for ρ ( λ z ) 1 2 = 1 2 for ρ ( λ z ) 1 2
M ST ( ρ , z ) = exp { ( ρ ρ 0 LT ) 5 / 3 [ 1 0.62 δ ( ρ / D ) 1 2 ] } ,
ρ 0 LT [ 0.545 C n 2 k 2 z ] 3 / 5
I ( p ) ST = ( k 2 π z ) 2 d 2 ϱ M ST ( ϱ , z ) exp [ ( i k / z ) p · ϱ ] × d 2 R U A ( R + ϱ 2 ) U A * ( R ϱ 2 ) × exp [ ( i k / z ) ϱ · R ] ,
p 1 2 = p 0 2 + p T 2 ,
p T 2 z k ρ 0 .
p 0 2 = ω 0 2 [ 1 ( z / f ) ] 2 + z 2 / k 2 ω 0 2 .
ρ 0 ST ρ 0 LT [ 1 + 0.37 ( ρ 0 LT / D ) 1 2 ] .
p T ST p T LT [ 1 0.37 ( ρ 0 LT / D ) 1 2 ] ,
p T LT = 2 z k ρ 0 LT
R ST 1 1 + [ D / ρ 0 ST ] 2 ,
R LT 1 1 + [ D / ρ 0 LT ] 2 ,
I ( 0 ) ST I ( 0 ) LT = 1 + [ D / ρ 0 LT ] 2 1 + [ D / ρ 0 ST ] 2
[ ρ 0 ST ρ 0 LT ] 2 for ρ 0 LT D .
U A ( r ) = U 0 exp ( r 2 2 a 2 ) ,
I ( p ) ST = ( k 2 π z ) | U 0 | 2 d 2 ρ M ST ( ρ ) × exp [ i k z p · ϱ ] { π a 2 exp [ k 2 a 2 ρ 2 4 z 2 ] }
= 2 β 2 | U 0 | 2 0 d x x M ST ( D x ) J 0 ( 2 α x ) × exp [ x 2 ( 1 + β 2 ) ] ,
β = k D 2 4 z , α = k D p 2 z , D = 2 a ,
M ST ( D x ) exp { x 5 / 3 ( D ρ 0 LT ) 5 / 3 [ 1 0.62 x 1 2 ] } .
I ( α ) ST = 2 β 2 | U 0 | 2 0 d x x J 0 ( 2 α x ) exp [ ( D x / ρ 0 LT ) 5 / 3 ] × exp { x 2 [ 1 + β 2 ( 1 0.62 β 0 2 ( ρ 0 LT / D ) 7 / 3 ) ] } ,
β 0 = k ( ρ 0 LT ) 2 4 z .
I ( α ) LT = 2 β 2 | U 0 | 2 0 d x x J 0 ( 2 α x ) exp [ ( D x ρ 0 LT ) 5 / 3 ] × exp [ x 2 ( 1 + β 2 ) ] ,
I 0 ( α ) = 2 β 2 | U 0 | 2 0 d x x J 0 ( 2 α x ) exp [ x 2 ( 1 + β 2 ) ] = β 2 1 + β 2 | U 0 | 2 exp [ α 2 1 + β ] .