Abstract

In a previous paper, we derived expressions for the irradiance distribution of a finite beam. Here we derive expressions for the full coherence function. We find an asymptotic analytic expression for the coherence function for very large propagation distances. In this region, the on-axis angular spectrum of the finite beam is one fourth that of an initial plane wave and three fourths that of an initial spherical wave. When the scattering dominates diffraction, we find overlapping asymptotic representations for the coherence function for all distances. We present some explicit results for an initially coherent gaussian wave.

© 1971 Optical Society of America

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References

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  1. A. Whitman and M. Beran, J. Opt. Soc. Am. 60, 1595 (1970).
    [CrossRef]
  2. M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964).
  3. M. Beran, J. Opt. Soc. Am. 60, 518 (1970).
    [CrossRef]
  4. In Refs. 1 and 3 it was stated that the derivation of this equation was subject to the condition a≫lM. In Appendix A, we point out that this restriction is not generally required.
  5. A. Erdélyi, A symptotic Expansions (Dover, New York, 1956).
  6. Note that here the ratio of spherical and plane-wave angular spectra is 1/3 and not 3/8 as in the Kolmogorov range (lm≪ρ).
  7. T. L. Ho, J. Opt. Soc. Am. 60, 667 (1970).
    [CrossRef]
  8. G. Watson, Bessel Functions (Cambridge U. P., New York, 1966).

1970 (3)

Beran, M.

Erdélyi, A.

A. Erdélyi, A symptotic Expansions (Dover, New York, 1956).

Ho, T. L.

Van Dyke, M.

M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964).

Watson, G.

G. Watson, Bessel Functions (Cambridge U. P., New York, 1966).

Whitman, A.

J. Opt. Soc. Am. (3)

Other (5)

G. Watson, Bessel Functions (Cambridge U. P., New York, 1966).

M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964).

In Refs. 1 and 3 it was stated that the derivation of this equation was subject to the condition a≫lM. In Appendix A, we point out that this restriction is not generally required.

A. Erdélyi, A symptotic Expansions (Dover, New York, 1956).

Note that here the ratio of spherical and plane-wave angular spectra is 1/3 and not 3/8 as in the Kolmogorov range (lm≪ρ).

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Figures (4)

Fig. 1
Fig. 1

Geometric definitions.

Fig. 2
Fig. 2

Domains of validity of the asymptotic expansions. // region of validity of Eq. (5). \\ region of validity of Eq. (11). × × overlapregion.

Fig. 3
Fig. 3

The radial distribution of the on-axis coherence function for various values of α, with β = 4 and δ = 103. The dotted curves represent a Kolmogorov correlation function, whereas the solid curves represent a gaussian correlation function.

Fig. 4
Fig. 4

The radial distribution of the on-axis coherence function for various values of α, with β = 4 and δ = 31.62. The dotted curve represents a Kolmogorov correlation function, whereas the solid curves represent a gaussian correlation function.

Equations (61)

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{ Γ ˆ } z = k 2 f ( s ) { Γ ˆ } + i 2 k ( x 2 - 2 x · s ) { Γ ˆ } .
{ Γ ˆ } = Γ ˆ * ( R , s , γ , 0 ) ,             when z = 0.
f ( s ) = σ ¯ ( s ) - σ ¯ ( 0 ) ,
σ ¯ ( s ) = 1 4 - σ [ ( s 2 + z 2 ) 1 2 ] d z .
f ( s ) = Λ n = 1 A n ( s l n ) 2 n = Λ A 1 3 g ( s l n ) ,
{ Γ ˆ } z = 1 l F ( n = 1 3 A n ρ 2 n A 1 ) { Γ ˆ } + i l D { 1 2 β [ 1 R ˆ R ˆ ( R ˆ R ˆ ) + 1 R ˆ 2 2 γ 2 ] - cos γ ( 2 ρ R ˆ - 1 ρ R ˆ 2 γ 2 ) + sin γ γ ( 1 R ˆ ρ + 1 ρ R ˆ ) } { Γ ˆ } .
ρ = s l n ,             R ˆ = R a ,             β = a l n ,             l F = 3 k 2 A 1 Λ ,             l D = k a l n .
δ = l D / l F = k 3 a A 1 Λ l n / 3 ,
{ Γ ˆ } / α = g ( ρ ) { Γ ˆ } + ( i / δ ) L ˆ Γ ˆ .
{ Γ ˆ } = j = 0 { Γ ˆ j } δ - j .
{ Γ ˆ o } = Γ ˆ * exp [ g ( ρ ) α ] .
{ Γ ˆ 1 } α = g ( ρ ) { Γ ˆ 1 } + i [ L ˆ Γ ˆ * - g ( ρ ) α ( cos γ Γ * R ˆ - sin γ R ˆ Γ * γ ) ] exp [ g ( ρ ) α ] , { Γ ˆ 1 } = 0             at             α = 0.
{ Γ ˆ 1 } = i { Γ ˆ 0 } Γ ˆ * [ L ˆ Γ ˆ * α - ( cos γ R ˆ - sin γ R ˆ γ ) Γ * g ( ρ ) α 2 2 ] .
{ Γ ˆ } = { Γ ˆ o } [ 1 + O ( α 2 δ - 1 ) ] .
{ Γ ˆ 2 } α = g ( ρ ) { Γ ˆ 2 } - { L ˆ 2 Γ ˆ * α - ( Ŝ , L ˆ ) + Γ * g ( ρ ) α 2 2 + [ g ( ρ ) α 2 2 + g 2 ( ρ ) α 3 2 ] Ŝ 2 Γ * } exp [ g ( ρ ) α ] .
{ Γ ˆ 2 } = - { Γ ˆ 0 } Γ ˆ * [ L ˆ 2 Γ ˆ * α 2 2 - ( Ŝ , L ˆ ) + Γ * g ( ρ ) α 3 6 + Ŝ 2 Γ * ( g 2 α 4 8 + g α 3 6 ) ] .
ζ = α / δ 2 3 ,             ρ ˜ = ρ δ 1 3 .
{ Γ ˆ } ζ = A ρ ˜ 2 { Γ ˆ } - i [ cos γ ( 2 ρ ˜ R ˆ - 1 ρ R ˆ 2 γ 2 ) - sin γ γ ( 1 R ˆ ρ ˜ + 1 ρ ˜ R ˆ ) ] { Γ ˆ } + 1 δ 1 3 { i 2 β [ 1 R ˆ R ˆ ( R ˆ R ˆ ) + 1 R ˆ 2 2 γ 2 ] + n = 2 A n ρ ˜ 2 n A 1 δ ( 2 n - 3 ) } { Γ ˆ } .
{ Γ ˆ } = j = 0 { Ĝ j } δ - j / 3 .
{ Ĝ 0 } ζ = A ρ ˜ 2 { Ĝ 0 } - i [ cos γ ( 2 ρ ˜ R ˆ - 1 R ˆ ρ ˜ 2 γ 2 ) - sin γ ( γ 1 R ˆ ρ ˜ + 1 ρ ˜ R ˆ ) ] { Ĝ 0 } ,
{ Ĝ 0 } = 0 { G ˆ 0 * ( u ) } exp [ A ρ ˜ 2 ζ + A 3 ζ 3 u 2 ] J 0 ( D u ) u d u .
G ˆ 0 * ( u ) = 0 Ĝ 0 * ( R ˆ ) J 0 ( R ˆ u ) R ˆ d R ˆ ,
D = ( R ˆ 2 - 2 i A ζ 2 ρ ˜ R cos γ - A 2 ζ 4 ρ ˜ 2 ) 1 2 .
{ Γ ˆ o } = Γ ˆ * ( R ˆ , ρ ˜ / δ 1 3 , γ , 0 ) exp [ g ( ρ ˜ / δ 1 3 ) ζ δ - 2 3 ] { Γ ˆ o } = Ĝ 0 * exp ( A ρ ˜ 2 ζ ) + O ( δ - 1 3 ) .
{ Ĝ 0 } = exp ( A ρ 2 α ) 0 G 0 * exp ( A α 3 u / 3 δ 2 ) × J 0 { [ R ˆ 2 - 1 δ ( 2 i A α 2 ρ R ˆ cos γ + A 2 α 4 ρ 2 ) ] u } u d u .
{ Ĝ 0 } = exp ( A ρ 2 α ) 0 Ĝ 0 * J 0 ( R ˆ u ) u d u + O ( δ - 1 ) ,
{ Ĝ 0 } = exp ( A ρ 2 α ) Ĝ 0 * + O ( δ - 1 ) .
{ Ĝ 0 } = - Ĝ 0 * ( 0 , 0 ) 4 3 A ζ 3 exp { 1 4 A ρ ˜ 2 ζ - 3 2 i ρ ˜ R ζ cos γ + 3 4 A R ˆ 2 ζ 3 } × [ 1 + O ( ζ - 1 ) ] .
Γ * = I c exp [ - R ˆ 2 - ( R ˆ ρ / β ) cos γ - ( ρ 2 / 2 β ) ] .
{ Γ ˆ } = Γ * exp [ g ( ρ ) α ] { 1 - [ ( ρ β + 2 R ˆ cos γ ) × ( ( 1 - 2 β ) 2 β 2 ρ α + g ( ρ ) 2 α 2 ) ] i δ + O ( δ - 2 ) } .
{ Γ ˆ } = - I c κ exp { 1 κ [ ( - A ζ + 1 3 A 2 ζ 4 ) ρ ˜ 2 - 2 i A ζ 2 ρ ˜ R ˆ cos γ + R ˆ 2 ] } ,
κ = - 1 + 4 3 A ζ 3 .
σ ¯ ( s ) - σ ¯ ( 0 ) = π 1 2 { 2 } l n 4 [ exp ( - s 2 / l n 2 ) - 1 ]
g ( ρ ) = 3 [ F 1 1 ( 1 ; 1 ; - ρ 2 ) - 1 ] 3 [ exp ( - ρ 2 ) - 1 ] ,
σ ¯ ( s ) - σ ¯ ( 0 ) = 0.132 π 2 C n 2 l m 5 / 3 × 0 [ J 0 ( s x l m ) - 1 ] { exp [ - x 2 / ( 5.91 ) 2 ] } x - 8 / 3 d x
g ( ρ ) = - 0.132 π 2 Γ ( 1 6 ) 0.9 ( 5.91 ) 5 / 3 { F 1 1 [ - 5 / 6 ; 1 ; - ( 2.96 ) 2 ρ 2 ] - 1 } .
{ Γ ˆ } / z ˆ = δ g ( ρ ) { Γ ˆ } + i L ˆ { Γ ˆ } .
{ Γ ˆ } = j = 0 { Γ ˆ j } δ j .
{ Γ ˆ 0 } / z ˆ = i L ˆ { Γ ˆ 0 } z ˆ = 0 ;             Γ 0 = Γ * ,
{ Γ ˆ 1 } / z ˆ = i L ˆ { Γ ˆ 1 } + g ( ρ ) { Γ ˆ 0 } z ˆ = 0 ;             { Γ ˆ 1 } = 0.
R ˜ = R ˆ δ .
{ Γ ˆ } α = g ( ρ ) { Γ ˆ } - i [ cos γ ( 2 ρ R ˜ - 1 ρ R ˜ 2 γ 2 ) - sin γ γ ( 1 R ˜ ρ + 1 ρ R ˜ ) ] { Γ ˆ } + δ i 2 β [ 1 R ˜ R ˜ ( R ˜ R ˜ ) + 1 R ˜ 2 2 γ 2 ] { Γ ˆ } .
{ Γ ˆ } = 1 ( 2 π ) 0 0 2 π w exp [ - i μ R ˆ cos ( β - ψ ) ] μ d μ d β
w = w * [ cos β ρ sin ( β - θ ) , ρ cos ( β - θ ) - μ α δ , μ ] × exp ( - i μ 2 α 2 β - δ μ { Q [ ρ cos ( β - θ ) - μ α δ , ρ 2 sin 2 ( β - θ ) ] - Q [ ρ cos ( β - θ ) , ρ 2 sin 2 ( β - θ ) ] } ) .
Q [ ( a , b ) ] = 0 a g [ ( q 2 + b ) 1 2 ] d q .
Q [ ρ cos ( β - θ ) - μ α / δ , ρ 2 sin 2 ( β - θ ) ]
Q [ ρ cos ( β - θ ) - μ α δ , ρ 2 sin 2 ( β - θ ) ] - Q [ ρ cos ( β - θ ) , ρ 2 sin 2 ( β - θ ) ] = - g ( ρ ) μ α δ + g ρ cos ( β - θ ) μ 2 α 2 δ 2 - 1 3 μ 3 α 3 δ 3 × [ g + 2 g ρ 2 cos 2 ( β - θ ) ] + .
w = w * exp { - i μ 2 α 2 β + g α - g ρ cos ( β - θ ) α 2 μ δ + 1 3 [ g + 2 g ρ 2 cos 2 ( β - θ ) ] α 3 μ 2 δ 2 + } .
{ Γ ˆ } ~ 1 2 π 0 0 2 π w * ( 0 , 0 , 0 ) × exp [ A ρ 2 α - A ρ α 2 δ cos ( β - θ ) μ + A 3 α 3 δ 2 μ 2 - i R ˆ cos ( β - ψ ) μ ] μ d μ d β .
{ Γ ˆ } ~ Ĝ 0 * ( 0 ) 0 exp [ A ρ 2 α + A 3 α 3 δ 2 μ 2 ] J 0 ( D μ ) μ d μ .
α 2 ρ = ζ 2 ρ ˜ δ .
{ Γ ˆ } ~ - G 0 * ( 0 , 0 ) δ 2 4 3 A α 3 × exp ( A 4 ρ 2 α - 3 2 i ρ R ˆ δ α cos γ + 3 R ˜ 2 δ 2 4 A α 3 ) .
l F = l n 1 2 / [ ( k 2 l n 2 ) ( 0.54 ) C n 2 ] δ = k 3 l n 3 [ a ( 0.54 C n 2 ) / l n 1 3 ] .
{ Γ ˆ } = G ( z ) exp [ - 1.6 4 ( s 2 l n 2 ) ( k 2 l n 2 ) ( C n 2 z l n 1 3 ) - 3 2 i s R k z - R 2 / ( 0.54 C n 2 z 3 / l n 1 3 ) ] ,
0 [ I ˆ ( R , 0 ) ] R d R = 0 [ I ˆ ( R , z ) ] R d R .
{ Γ ˆ } = Γ ˆ * ( s ) exp { - k 2 z [ σ ¯ ( 0 ) - σ ¯ ( s ) ] } , σ ¯ ( s ) - σ ¯ ( 0 ) = 1.32 π 2 C n 2 l n 5 / 3 × 0 [ J o ( s x l n ) - 1 ] exp [ - x 2 / ( 5.91 ) 2 ] x 8 / 3 d x , [ α = o ( δ 2 3 ) ] .
{ Γ ˆ } = { I ˆ } exp [ - 1.6 s 2 l n 2 ( k 2 l n 2 ) ( C n 2 z l n 1 3 ) ] ,
{ Γ ˆ } z = k 2 f ( s ) { Γ ˆ } + i 2 k ( x 2 - 2 x · s ) { Γ ˆ } ,
[ Γ ˆ ( z + Δ z ) ] - [ Γ ˆ ( z ) ] = k 2 f ( s ) [ Γ ˆ ( z ) ] Δ z .
Δ z l M Δ z k l m 2
( l M / l m ) ( 1 / k l m ) 1 ,