Abstract

On the basis of the extended Huygens-Fresnel principle, a general expression is derived for the mutual coherence function (MCF) of a finite optical beam propagating in a weakly inhomogeneous medium. The results obtained here for the beam MCF are valid both in the near and far field of the laser transmitting aperture and for an arbitrary complex disturbance in the exit pupil of the aperture. A general expression is also derived for the propagation distance zB such that for distances much less (greater) than zB, the MCF is well approximated by the plane (spherical) wave results. An analytic expression is presented for a Gaussian beam such that a numerical error in previous results is corrected. Finally, some comments regarding higher order statistical moments of the field are given.

© 1972 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. R. E. Hufnagel, N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [CrossRef]
  3. R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 61, 482 (1971).
    [CrossRef]
  4. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).
  5. M. J. Beran, A. M. Whitman, J. Opt. Soc. Am. 61, 1044 (1971).
    [CrossRef]
  6. T. L. Ho, J. Opt. Soc. Am. 60, 667 (1970).
    [CrossRef]
  7. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
    [CrossRef]
  8. M. Beran, J. Opt. Soc. Am. 60, 518 (1970).
    [CrossRef]
  9. T. L. Ho introduces the fluctuation in refractive index as a small parameter in the wave equation and expands the field in powers of this same parameter. With this procedure, the field is expressed as a hierarchy of Born-type integrals over the vacuum fields that exist at eachpoint in space. Because of the extreme complexity of these integrals, the atmospheric calculations have been limited to those cases that have a closed-form solution for the vacuum fields, i.e., a Gaussian beam.
  10. H. T. Yura, J. Opt. Soc. Am. 62,June (1972).
    [CrossRef]
  11. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  12. H. T. Yura, Appl. Opt. 10, 2771 (1971).
    [CrossRef] [PubMed]
  13. Yu. N. Barabanenkov, Sov. Phys. JETP 27, 954 (1968).
  14. W. P. Brown, J. Opt. Soc. Am. 62, 45 (1972).
    [CrossRef]
  15. D. L. Fried, H. T. Yura, J. Opt. Soc. Am.62, in press (1972).
    [CrossRef]
  16. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  17. D. L. Fried, IEEE J. Quant. Electron. QE-3, 213 (1967).
    [CrossRef]
  18. H. T. Yura, J. Opt. Soc. Am. 59, 111 (1969).
    [CrossRef]

1972

H. T. Yura, J. Opt. Soc. Am. 62,June (1972).
[CrossRef]

W. P. Brown, J. Opt. Soc. Am. 62, 45 (1972).
[CrossRef]

1971

1970

1969

1968

Yu. N. Barabanenkov, Sov. Phys. JETP 27, 954 (1968).

1967

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

D. L. Fried, IEEE J. Quant. Electron. QE-3, 213 (1967).
[CrossRef]

1966

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).

1964

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, Sov. Phys. JETP 27, 954 (1968).

Beran, M.

Beran, M. J.

Brown, W. P.

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

D. L. Fried, IEEE J. Quant. Electron. QE-3, 213 (1967).
[CrossRef]

D. L. Fried, H. T. Yura, J. Opt. Soc. Am.62, in press (1972).
[CrossRef]

Ho, T. L.

Hufnagel, R. E.

Lutomirski, R. F.

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Whitman, A. M.

Yura, H. T.

Appl. Opt.

IEEE J. Quant. Electron.

D. L. Fried, IEEE J. Quant. Electron. QE-3, 213 (1967).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Quart. Appl. Math.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).

Sov. Phys. JETP

Yu. N. Barabanenkov, Sov. Phys. JETP 27, 954 (1968).

Other

D. L. Fried, H. T. Yura, J. Opt. Soc. Am.62, in press (1972).
[CrossRef]

T. L. Ho introduces the fluctuation in refractive index as a small parameter in the wave equation and expands the field in powers of this same parameter. With this procedure, the field is expressed as a hierarchy of Born-type integrals over the vacuum fields that exist at eachpoint in space. Because of the extreme complexity of these integrals, the atmospheric calculations have been limited to those cases that have a closed-form solution for the vacuum fields, i.e., a Gaussian beam.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (2)

Fig. 1
Fig. 1

Degree of coherence for the case of a collimated Gaussian beam for δ = 1 and various values of Ω.

Fig. 2
Fig. 2

Degree of coherence for the case of a focused Gaussian beam for δ = 1 and various values of Ω.

Equations (61)

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U ( p ) = i k 2 π G ( p , r ) U A ( r ) d 2 r ,
M ( p 1 , p 2 ) U ( p 1 ) U * ( p 2 ) ,
M ( p 1 , p 2 ) = ( k 2 π ) 2 d 2 r 1 d 2 r 2 G ( p 1 , r 1 ) G * ( p 2 , r 2 ) U A ( r 1 ) U A * ( r 2 ) .
G ( r , p ) = s 1 exp [ i k s + ψ ( r , p ) ] ,
G ( r 1 , p 1 ) G * ( r 2 , p 2 ) = { exp [ i k ( | r 1 p 1 | | r 2 p 2 | ) ] / ( | r 1 p 1 | | r 2 p 2 | ) } F ,
F = exp [ ψ ( r 1 , p 1 ) + ψ * ( r 2 , p 2 ) ] .
| r 1 p 1 | | r 2 p 2 | ( 1 / 2 z ) [ r 1 2 r 2 2 + p 1 2 p 2 2 + 2 ( r 2 · p 2 r 1 · p 1 ) ] .
F = exp [ ψ ( r 1 , p 1 ) + ψ * ( r 2 , p 2 ) ] = exp { [ | ψ 1 | 2 ψ 1 ( r 1 , p 1 ) ψ 1 * ( r 2 , p 2 ) ] } ,
| ψ 1 | 2 = 2 π k 2 z Φ n ( K ) d 2 K
U 1 ( r , p ) = k 2 2 π d 3 x n 1 ( x ) U 0 ( r , x ) exp [ i k s ( x , p ) ] s ( x , p )
U 0 ( r , x ) = exp [ i k s ( x , r ) ] / s ( x , r ) ,
ψ 1 ( r 1 , p 1 ) ψ 1 * ( r 2 , p 2 ) = ( k 2 / 2 π ) 2 [ U 0 ( r 1 , p 1 ) U 0 * ( r 2 , p 2 ) ] 1 × d 3 x 1 d 3 x 2 B n ( x 1 x 2 ) U 0 ( r 1 , x 1 ) × U 0 * ( r 2 , x 2 ) exp { i k [ s ( x 1 , p 1 ) s ( x 2 , p 2 ) ] } s ( x 1 , p 1 ) s ( x 2 , p 2 ) ,
B n ( x 1 x 2 ) = n 1 ( x 1 ) n 1 ( x 2 ) .
ψ 1 ( r 1 , p 1 ) ψ 1 * ( r 2 , p 2 ) ( z k 2 / 2 π ) 2 exp { ( i k / 2 z ) [ ( r 1 p 1 ) 2 ( r 2 p 2 ) 2 ] } × d 3 x 1 d 3 x 2 B n ( x 1 x 2 ) z 1 z 2 ( z z 1 ) ( z z 2 ) × exp { i k [ ( ϱ 1 r 1 ) 2 2 z 1 + ( ϱ 1 p 1 ) 2 2 ( z z 1 ) ( ϱ 2 r 2 ) 2 2 z 2 ( ϱ 2 p 2 ) 2 2 ( z z 2 ) ] } .
B n ( x 1 x 2 ) = d 2 K F n ( K , z 1 z 2 ) e i k · ( ρ 1 ρ 2 ) ,
ψ ( r 1 , p 1 ) ψ 1 * ( r 2 , p 2 ) = ( z k 2 / 2 π ) 2 exp { ( i k / 2 z ) [ ( r 1 p 1 ) 2 ( r 2 p 2 ) 2 ] } d 2 K Q ( K ) ,
Q ( K ) = d 3 x 1 d 3 x 2 F n ( K , z 1 z 2 ) [ z 1 z 2 ( z z 1 ) ( z z 2 ) ] 1 × exp { i k [ ( ϱ 1 r 1 ) 2 2 z 1 + ( ϱ 1 p 1 ) 2 2 ( z z 1 ) ( ϱ 2 r 2 ) 2 2 z 2 ( ϱ 2 p 2 ) 2 2 ( z z 2 ) ] i K · ( ϱ 1 ϱ 2 ) } .
exp ( a x 2 + b x ) d x = ( π a ) 1 2 exp ( b 2 / 4 a ) .
Q ( K ) = ( 2 π k z ) 2 exp { i k 2 z [ ( r 1 p 1 ) 2 ( r 2 p 2 ) 2 ] } × 0 z d z 1 0 z d z 2 F n ( K , z 1 z 2 ) exp ( i { ( z 1 z p 1 z 2 z p 2 ) · K + ( z z 1 z r 1 z z 2 z r 2 ) · K + K 2 2 k z [ z 1 ( z z 1 ) z 2 ( z z 2 ) ] } ) .
( K 2 / 2 k z ) [ z 1 ( z z 1 ) z 2 ( z z 2 ) ] = K 2 ζ [ 1 ( η / z ) ] / 2 k 1 ,
Q ( K ) 2 ( 2 π / k z ) 2 exp { ( i k / 2 z ) [ ( r 1 p 1 ) 2 ( r 2 p 2 ) 2 ] } × 0 z d η 0 d ζ F n ( K , ζ ) exp { i [ η z p · K + ( 1 η z ) r · K ] } ,
0 F n ( K , ζ ) d ζ = π Φ n ( K ) ,
Q ( K ) = 2 π ( 2 π / k z ) 2 exp { ( i k / 2 z ) [ ( r 1 p 1 ) 2 ( r 2 p 2 ) 2 ] } × d 2 K Φ n ( K ) 0 z d η exp { i [ η z p · K + ( 1 η z ) r · K ] } .
ψ 1 ( r 1 , p 1 ) ψ 1 * ( r 2 , p 2 ) = 2 π k 2 z d 2 K Φ n ( K ) × 0 1 d t exp { i [ t p · K + ( 1 t ) r · K ] } .
ψ 1 ( r 1 , p 1 ) ψ * ( r 2 , p 2 ) = 4 π 2 k 2 z 0 d K K Φ n ( K ) × 0 1 d t J 0 [ | t p + ( 1 t ) r | K ] ,
F ( r , p ) = exp [ ψ ( r 1 , p 1 ) + ψ * ( r 2 , p 2 ) ] = exp ( 2 π k 2 z 0 1 d t d 2 K Φ n ( K ) { 1 exp [ t p · K + ( 1 t ) r · K ] } ) .
F ( r , p ) = exp ( 4 π 2 k 2 z 0 1 d t 0 d K K Φ n ( K ) × { 1 J 0 [ | t p + ( 1 t ) r | K ] } ) .
exp [ ( 2.91 / 2 ) C n 2 z k 2 ρ 5 3 ] ,
F ( r , p ) = exp [ 2.91 2 C n 2 z k 2 0 1 d t | t p + ( 1 t ) r | 5 3 ] = exp { 2.91 2 C n 2 z k 2 [ ( 3 8 ) ( p 8 3 r 8 3 ) | p r | ] }
M ( p 1 , p 2 ) = ( k 2 π z ) 2 d 2 r F ( r , p ) d 2 R U A ( R + r 2 ) U A * ( R r 2 ) × exp { i k 2 [ ( R P ) · ( r p ) ] } ,
M = F ( 0 , p ) = exp { 2 π k 2 z d 2 K Φ n ( K ) 0 1 d t [ 1 exp ( t p · K ) ] } = M s ( p ) ,
d 2 R H ( r , p , R ) δ [ k z ( r p ) ] ,
M ( p ) = F ( p , p ) = exp { 2 π k 2 z d 2 K Φ n ( K ) [ 1 exp ( p · K ) ] } = M p w ( p ) ,
p 0 2 ( D / 2 ) 2 + ( 4 z 2 / k 2 ) ( ρ 0 2 + D 2 ) ,
z z B = ( k D / 4 ) [ 1 / ρ 0 2 + 1 / D 2 ] 1 2 K D ρ 0 / 4 , for ρ 0 D K D 2 / 4 , for ρ 0 D .
ρ 0 = ( 0.545 k 2 C n 2 z ) 3 5 ,
z B 0.53 D 5 8 C n 3 4 k 1 8 ;
z B k D 2 / 4 .
0.022 D 11 4 C n 3 2 k 9 4 1 ,
2.2 D 11 4 C n 3 2 k 9 4 1 ,
M ( p ) = 2 [ F ( d , p ) + cos ( k p d / 2 z ) M s ( p ) ] ,
C ( p ) = M ( p ) / [ I ( p 1 ) ] 1 2 [ I ( p 2 ) ] 1 2 .
C ( p ) = [ F ( d , p ) + cos ( k p d / 2 z ) M s ( p ) ] / [ 1 + cos ( k p d / 2 z ) M s ( p ) ]
U A ( r ) = U 0 exp [ r 2 ( w 0 2 + i k f 1 ) / 2 ] ,
F ( r , p ) = exp ( ( 1 / p 0 2 ) { [ ( r p ) 2 / 3 ] + p · ( r p ) + p 2 } ) .
C ( p , z ) | U ( p 1 ) U * ( p 2 ) | / [ I ( p 1 ) ] 1 2 [ I ( p 2 ) ] 1 2 = exp [ ( p 2 / p c 2 ) ] ,
p c = p 0 { [ 1 ( z / f ) ] 2 + Ω 2 [ 1 + ( δ 2 / 3 ) ] 1 ( 13 / 3 ) ( z / f ) + ( 11 / 3 ) ( z / f ) 2 + ( Ω 2 / 3 ) [ 1 + ( δ 2 / 4 ) ] } 1 2 ,
= p 0 { [ 1 ( z / f ) ] 2 + ( z / z B ) 2 [ 1 + ( δ 2 / 3 ) ] [ 1 / ( 1 + δ 2 ) ] 1 ( 13 / 3 ) ( z / f ) + ( 11 / 3 ) ( z / f ) 2 + ( 1 / 3 ) ( z / z B ) 2 [ 1 + ( δ 2 / 4 ) ] [ 1 / ( 1 + δ 2 ) ] } 1 2 ,
U ( p 1 ) U * ( p 2 ) U ( p 3 ) U * ( p 4 ) = d 2 r 1 d 2 r 2 d 2 r 3 d 2 r 4 U A ( r 1 ) U A * ( r 2 ) U A ( r 3 ) U A * ( r 4 ) × { exp [ s ( r 1 , p 1 ) s ( r 2 , p 2 ) ] s ( r 1 , p 1 ) s ( r 2 , p 2 ) } { exp [ s ( r 3 , p 3 ) s ( r 4 , p 4 ) ] s ( r 3 , p 3 ) s ( r 4 , p 4 ) } × H ( r 1 , r 2 , r 3 , r 4 ; p 1 , p 2 , p 3 , p 4 ) ,
H ( r 1 r 4 ; p 1 p 4 ) = exp [ ψ ( r 1 , p 1 ) + ψ * ( r 2 , p 2 ) + ψ ( r 3 , p 3 ) + ψ * ( r 4 , p 4 ) ] .
U ( p 1 ) = U 0 ( p 1 , r 1 ) exp [ ψ 1 ( p 1 , r 1 ) + ψ 2 ( p 1 , r 1 ) ] ,
U ( p 2 ) = U 0 ( p 2 , r 2 ) exp [ ψ 1 ( p 2 , r 2 ) + ψ 2 ( p 2 , r 2 ) ] .
U 0 ( p , r ) = exp [ i k s ( p , r ) ] / s ( p , r ) ,
U ( p 1 ) U * ( p 2 ) = U 0 ( p 1 , r 1 ) U 0 ( p 2 , r 2 ) F ,
F = exp [ ψ 1 ( p 1 , r 1 ) + ψ 2 ( p 1 , r 1 ) + ψ 1 * ( p 2 , r 2 ) + ψ 2 * ( p 2 , r 2 ) ] .
F = exp { ψ 2 ( p 1 , r 1 ) + ψ 2 * ( p 2 , r 2 ) + 1 2 [ ψ 1 ( p 1 , r 1 ) + ψ 1 * ( p 2 , r 2 ) ] 2 } .
ψ 2 = ϕ 2 1 2 ψ 1 2 .
ϕ 2 R e ( ϕ 2 ) = 1 2 | ψ 1 | 2 .
F = exp { 1 2 [ | ψ 1 ( p 1 , r 1 ) | 2 + | ψ 1 ( p 2 , r 2 ) | 2 2 ψ 1 ( p 1 , r 1 ) ψ 1 * ( p 2 , r 2 ) ] } .
F = exp { [ | ψ 1 | 2 ψ 1 ( p 1 , r 1 ) ψ 1 * ( p 2 , r 2 ) ] } ,
| ψ 1 | 2 = 2 π k 2 z Φ n ( K ) d 2 K ,

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