Abstract

The mutual coherence function of a finite laser beam propagating through a random medium is calculated from the extended Huygens principle. Formulas are presented for the transverse coherence from an arbitrarily truncated focused Gaussian beam. Expressions are derived for a modified Kolmogorov spectrum for the refractive index fluctuations and for the case of nonuniform turbulence. The formalism is used to derive the conditions under which instantaneous reciprocity exists between a transmitter and receiver in a coherent detection system. It is also shown that a maximum useful transmitter (as well as receiver) diameter exists for computing the average normalized signal-to-noise ratio of this system. This maximum dimension is the smaller of the turbulence-induced coherence length and a Fresnel zone.

© 1973 Optical Society of America

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References

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  1. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  2. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  3. L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  4. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 370.
  6. In practice, Qv can be increased by shaping the local oscillator wavefront, or by placing focusing lenses between the transmitter and receiver; in the latter case, Qv can be increased by approximately ¼(a/b + b/a)2 when Z < kab.
  7. R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 61, 482 (1970).
    [CrossRef]
  8. It can be shown (Ref. 4) that the second-order Rytov solution yields the same result.

1972 (1)

1971 (1)

1970 (1)

1967 (1)

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

1965 (1)

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 370.

Fried, D. L.

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

Lutomirski, R. F.

Mandel, L.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 370.

Yura, H. T.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

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

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 370.

In practice, Qv can be increased by shaping the local oscillator wavefront, or by placing focusing lenses between the transmitter and receiver; in the latter case, Qv can be increased by approximately ¼(a/b + b/a)2 when Z < kab.

It can be shown (Ref. 4) that the second-order Rytov solution yields the same result.

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Figures (2)

Fig. 1
Fig. 1

Optical paths from r1 to P1 and r2 to P2.

Fig. 2
Fig. 2

Correlation function for infinite Gaussian beams, and a von Kármán turbulence spectrum.

Equations (71)

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U ( P ) = - i k 2 π G ( P , r 1 ) U A ( r 1 ) d 2 r 1 ,
G ( P , r 1 ) = [ exp ( i k P - r 1 ) / P - r 1 ] G ( P , r 1 ) ,
U ( p , z ) = - i k 2 π z exp ( i k z ) G ( p , r ) × exp [ i k ( p - r 1 ) 2 / 2 z ] U A ( r 1 ) d 2 r 1
U ( p 1 , z ) U * ( p 2 , z ) = ( k 2 π z ) 2 G ( p 1 , r 1 ) G * ( p 2 , r 2 ) × exp { ( i k / 2 z ) [ ( p 1 - r 1 ) 2 - ( p 2 - r 2 ) 2 ] } U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 .
MCF F = ( k 2 π z ) 2 H exp { ( i k / 2 z ) [ p 1 - r 1 ) 2 - ( p 2 - r 2 ) 2 ] } U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
p = p 1 - p 2 ,             q = ½ ( p 1 + p 2 ) ,             p = r 1 = r 2 ,             and             r = ½ ( r 1 + r 2 ) ,
( p 1 - r 1 ) 2 - ( p 2 - r 2 ) 2 = 2 [ p · q + r · ρ - p · r - q · ρ ] = 2 ( p - ρ ) · ( q - r ) .
MCF F ( p , q , z ) ( k 2 π z ) 2 exp [ ( i k / z ) ( p . q ) ] d 2 ρ H ( p , ρ , z ) × exp { [ - ( i k / z ) ] ( q · ρ ) } exp [ ( i k / z ) r · ( ρ - p ) ] × U A ( r + ½ ρ ) U A * ( r - ½ ρ ) d 2 r .
MCF F ( p , z ) = ( k 2 π z ) 2 d 2 ρ H ( p , ρ , z ) U A ( r + ½ ρ ) × U A * ( r - ½ ρ ) exp [ ( i k / z ) r · ( ρ - p ) ] d 2 r .
γ ( p 1 , p 2 ) = [ U ( p 1 ) U * ( p 2 ) ] / [ U ( p 1 ) 2 1 / 2 U ( p 2 ) 2 1 / 2 ] = MCF F / [ I ( p 1 ) 1 / 2 ( I p 2 ) 1 / 2 ]
C F ( p 1 , p 2 ) = γ ( p 1 , p 2 ) = U ( p 1 ) U * ( p 2 ) / [ I ( p 1 ) 1 / 2 ( I ( p 2 ) 1 / 2 ] .
V l = 2 ( μ + μ - 1 ) - 1 C ( p 1 , p 2 ) ,
I ( p 1 ) = ( k 2 π z ) 2 d 2 ρ C S ( ρ , z ) exp { [ - ( i k / z ) ] p 1 · ρ } . × U A ( r + ½ ρ ) U A * ( r - ½ ρ ) exp [ ( i k / z ) ρ · r ] d 2 r ,
C F ( p 1 , p 2 ) = MCF ( p , z ) I ( p / 2 , z ) = d 2 ρ H ( p , ρ , z ) U A ( r + ½ ρ ) U A * ( r - ½ ρ ) exp [ i k / z ) r · ( ρ - p ) ] d 2 r d 2 ρ C S ( ρ , z ) exp { [ ( i k / 2 z ) ] p · ρ } U A ( r + ½ ρ ) U A ( r - ½ ρ ) exp [ ( i k / z ) ρ · r ] d 2 r .
U A ( r i ) = exp [ - ½ r i 2 ( a - 2 + i k f - 1 ) ] r i D / 2 = 0 r i > D / 2 ,
U A ( r + ½ ρ ) U A * ( r - ½ ρ ) = exp [ - a - 2 ( r 2 + ¼ ρ 2 ) - i k f - 1 ( r · ρ ) ]
MCF F ( ρ , z ) = ( k 2 π z ) 2 d 2 ρ H ( p , ρ ) × exp ( - ρ 2 / 4 a 2 ) exp ( - r 2 / a 2 ) exp ( - i W · r ) d 2 r ,
MCF F ( p , z ) = 8 π 2 β 2 0 1 d x x exp ( - 2 δ 2 x 2 ) × 0 π d ϕ H ( p , D x , ϕ , z ) 0 cos - 1 ( x ) d θ × x / cos θ 1 d u u exp [ - δ 2 ( u 2 - 2 u x cos θ ) ] × cos { 2 β [ ( 1 - z / f ) x - p D cos θ × [ u     cos θ - x ] } cos [ 2 β p D u     sin ϕ sin θ ] ,
I ( p / 2 , z ) = 8 π β 2 0 1 x J 0 ( 2 β p D x ) × C s ( D x , z ) Γ δ , β ( x ) d x ,
Γ δ , β ( x ) = exp ( - 2 δ 2 x 2 ) 0 cos - 1 d θ x / cos θ 1 × exp [ - δ 2 ( u 2 - 2 u x cos θ ) ] × cos [ 2 β x ( u cos θ - 1 ) ] u d u , x = 1 = 0 x 1
Γ 0 , β = 0 cos - 1 ( x ) sin [ 2 β x ( cos θ - x ) ] 2 β x cos θ - 1 - cos [ 2 β x ( cos θ - x ) ] ( 2 β x cos θ ) 2 d x , x 1 , = 0 , x 1.
Γ δ , β ( x ) = ½ [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] , β < 2.
2 π 0 d r r exp ( - r 2 / a 2 ) J 0 ( W r ) = π a 2 exp ( - W 2 a 2 / 4 )
MCF F ( p , z ) = k 2 a 2 4 π z 2 exp ( k 2 a 2 p 2 4 z 2 ) d 2 ρ H ( p , ρ , z ) × exp { - ρ 2 4 a 2 [ 1 + k 2 a 4 z 2 ( 1 - z / f ) 2 ] } × exp k 2 a 2 ( 2 z 2 ) ( 1 - z / f ) p · ρ · exp k 2 a 2 ( 2 z 2 ) ( 1 - z / f ) p · ρ .
M f ( p , z ) = 2 σ 2 π exp ( - σ 2 p 2 a 2 ) 0 d x x × exp { - x 2 [ ¼ + σ 2 ( 1 - z / f ) 2 ] } 0 π d ϕ × exp [ 2 σ 2 a 2 ( 1 - z / f ) p x cos ϕ ] H ( p , a x , cos ϕ , z ) .
I ( p 1 , z ) = 2 σ 2 0 exp { - x 2 [ ¼ + σ 2 ( 1 - z / f ) 2 ] } × J 0 ( p 1 x 2 a ) C S ( a x , z ) x d x
ρ 0 = [ 0.5 k 2 C n 2 z ] - 3 / 5             ( homogeneous turbulence ) , = [ 1.5 k 2 0 z C n 2 ( s ) ( s / z ) 5 / 3 d s ] - 3 / 5             ( inhomogeneous turbulence ) .
z 0 = ( k D 2 / 4 ) / 1 + ( D / ρ 0 ) 2 ] 1 / 2 .
S = ½ ( η A R g ) 2 R U ( P ) F ( p ) d 2 p 2 ,
S = ½ ( k η A R g 2 π ) 2 R U A ( r ) G ( r , p , z ) F ( p ) d 2 r d 2 p 2 ,
G ( r , P ) = G ( P , r )             or             G ( r , p , z ) = G ( p , r , z ) .
S = ½ ( k η A R g 2 π ) 2 R U A ( r 1 ) U A * ( r 2 ) F ( p 1 ) F * ( p 2 ) × exp { i k 2 z [ ( p 1 - r 1 ) 2 - ( p 2 - r 2 ) 2 ] } × H ( p 1 - p 2 , r 1 - r 2 , z ) d 2 r 1 d 2 r 2 d 2 p 1 d 2 p 2 .
N = e g 2 η ( π / 4 ) D R 2 A R 2 R ,
W = U A ( r ) 2 d 2 r = B D T 2 ,
U A ( r 1 ) U A * ( r 2 ) = ( π D T 2 / 4 W ) U A ( r 1 ) U A * ( r 2 ) .
Q = S N W = 2 π 2 ( η e ) k 2 D T 2 D R 2 z 2 × U A ( r 1 ) U A * ( r 2 ) F ( p 1 ) F * ( p 2 ) exp { i k 2 z [ ( p 1 - r 1 ) 2 × - ( p 2 - r 2 ) 2 ] } H ( p 1 - p 2 , r 1 - r 2 , z ) d 2 r 1 d 2 r 2 d 2 p 1 d 2 p 2 .
S v = C 1 z 2 U A ( r ) exp [ i k 2 z ( p - r ) 2 ] F ( p ) d 2 r d 2 p 2
U A ( r ) = U 0 exp ( - r 2 / 2 a 2 ) ,
F ( p ) = exp ( - p 2 / 2 b 2 ) ,
S v = ( C 1 U 0 2 / z 2 ) ( 16 π 4 a 4 b 4 / { 1 + [ k ( a 2 + b 2 ) / z ] 2 } ) .
Q v = S v / N W = ( C 3 / z 2 ) ( a 2 b 2 { 1 + [ k ( a 2 + b 2 / z ) ] 2 } ) ,
( Q v ) max C 3 / k 2 ,
H = exp [ - ( p 2 + ρ 2 + p · ρ ) / ρ 0 2 ] .
Q = ( C 3 / z 2 ) a 2 b 2 / [ 1 + k 2 ( a 2 + b 2 ) 2 z 2 + 4 ( a 2 + b 2 ) ρ 0 2 ( 1 + 3 k 2 a 2 b 2 z 2 ) + 12 a 2 b 2 ρ 0 4 . ]
Q f C 3 z 2 · ρ 0 4 4 [ ( ρ 0 / a ) 2 + ( ρ 0 / b ) 2 ] + 12 .
Q n C 3 z 2 · a 2 b 2 / { 12 k 2 ( a 2 + b 2 ) 2 z 2 [ 1 ( ρ 0 / a ) 2 + ( ρ 0 / b ) 2 + ( z k ρ 0 2 ) 2 ( a b a + b ) 2 ] } .
G ( P i , r i ) = exp [ i k n 1 ( R i ) d l i ] ,
d l i = d z i { 1 + [ ( p i - r i ) 2 / z 2 ] } 1 / 2 d z i .
S 1 = k 0 z n 1 ( R 1 ) d z 1 ,             S 2 = k 0 z n 1 ( R 2 ) d z 2
H = exp [ i ( S 1 - S 2 ) ] = exp [ - ½ D S ] ,
B S = k 2 0 z d z 1 0 z d z 2 B n ( R 1 , R 2 ) ,
R 1 - R 2 = [ w 2 + ( z 2 - z 1 ) 2 ] 1 / 2
w 2 - w 1 = w = ρ + ( z 2 / z ) ( p - ρ )             and             ρ = r 2 - r 1 , p = p 2 - p 1 .
B n ( w 2 - w 1 , z 2 - z 1 = Φ n ( K , K z ) × exp ( - i K · w ) exp [ - i K z ( z 2 - z 1 ) ] d 3 K .
B n ( w , z 2 - z 1 ) = 2 π 0 d K K Φ n ( K , K z ) J 0 ( K w ) × - exp [ - i K z ( z 1 - z 2 ) ] d K z ,
B S = 2 π k 2 0 z d z 1 0 z d z 2 0 d K K J 0 ( K w ) × - d K z Φ n ( K , K z ) exp [ - i K z ( z 1 - z 2 ) ]
B S = 2 π k 2 0 z d z 2 0 d K K J 0 ( K w ) × - d K z Φ n ( K , K z ) z 2 - z z 2 exp ( - i K 2 u ) d u .
- exp ( - i K z u ) d u = 2 π δ ( K z )
B S ( p , ρ , z ) = 4 π 2 k 2 0 z d z 2 0 d K K J 0 ( K w ) Φ n ( K , 0 ) .
D S ( p , ρ , z ) = 8 π 2 k 2 z 0 1 d t 0 d K K Φ n ( K , 0 ) × { 1 - J 0 [ K ρ ( 1 - t ) + p t ] }
H ( p , ρ , z ) = exp [ - ½ D S ( p , ρ , z ) ] .
C p ( p , z ) = H ( p , p , z )
C s ( p , z ) = H ( p , o , z ) .
Φ n ( K ) = 0.033 C n 2 exp [ - ( K l 0 ) 2 ] / ( K 2 + L 0 - 2 ) 11 / 6 ,
z c [ 0.4 k 2 C n 2 L 0 5 / 3 ] - 1 .
H ( p , ρ , z ) = H ( p , ρ , cos ϕ , z ) = exp [ - 2 z z c ( 1 - 5 3 0 1 d t × 0 y ( 1 + y 2 ) 11 / 6 J 0 { y [ t 2 ( p L 0 ) 2 + ( 1 - t ) 2 ( ρ L 0 ) 2 + 2 t ( 1 - t ) p ρ L 0 2 cos ϕ ] 1 / 2 } d y ) ] .
D S ( p , ρ , z ) = 8 π 2 k 2 0 z d z 0 d K K Φ n ( K , 0 , z ) × { 1 - J 0 [ K ρ ( 1 - z z ) + p ( z z ) | }
Φ n ( K , z ) = 0.033 C n 2 ( z ) K - 11 / 3
D S ( p , ρ , z ) = 2.91 k 2 0 z C n 2 ( z ) ρ ( - z z ) + p ( z z ) | 5 / 3 d z .
C p ( p , z ) = exp ( - 2 z z c { 1 - 5 3 0 J 0 [ ( p / L 0 ) y ] ( 1 + y 2 ) 11 / 6 d y } ) ,
C S ( p , z ) = exp ( - 2 z z c { 1 - 5 3 0 1 d t 0 J 0 [ ( p / L 0 ) y ] ( 1 + y 2 ) 11 / 6 d y } ) ,

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