Abstract

The recently formulated stochastic geometrical theory of diffraction is here applied to reflection, transmission, and diffraction of the high-frequency two-point coherence function when interfaces or scatterers are embedded in a randomly fluctuating medium with inhomogeneous background. This extends the previous solutions for a homogeneous background.

© 1987 Optical Society of America

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References

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  1. R. Mazar, L. B. Felsen, Opt. Lett. 12, 4 (1987).
    [CrossRef] [PubMed]
  2. R. Mazar, L. B. Felsen, Opt. Lett. 12, 146 (1987).
    [CrossRef] [PubMed]
  3. R. Mazar, L. B. Felsen, “High-frequency coherence functions propagated along ray paths in the inhomogeneous background of a weakly random medium. I—Formulation and evaluation of the second moment,” J. Acoust. Soc. Am. (to be published).
    [PubMed]
  4. V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems (Nauka, Moscow, 1972).
  5. R. J. Hill, J. Acoust. Soc. Am. 77, 1742 (1985).
    [CrossRef]
  6. V. I. Tatartskii, V. U. Zavorotnyi, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980).
  7. V. I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomogeneous Media (Nauka, Moscow, 1980).
  8. V. I. Gelfgat, Sov. Phys. Acoust. 22, 65 (1976).
  9. V. P. Aksenov, V. A. Banakh, V. L. Mironov, J. Opt. Soc. Am. A 1, 263 (1984).
    [CrossRef]

1987 (2)

1985 (1)

R. J. Hill, J. Acoust. Soc. Am. 77, 1742 (1985).
[CrossRef]

1984 (1)

1976 (1)

V. I. Gelfgat, Sov. Phys. Acoust. 22, 65 (1976).

Aksenov, V. P.

Babich, V. M.

V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems (Nauka, Moscow, 1972).

Banakh, V. A.

Buldyrev, V. S.

V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems (Nauka, Moscow, 1972).

Felsen, L. B.

R. Mazar, L. B. Felsen, Opt. Lett. 12, 4 (1987).
[CrossRef] [PubMed]

R. Mazar, L. B. Felsen, Opt. Lett. 12, 146 (1987).
[CrossRef] [PubMed]

R. Mazar, L. B. Felsen, “High-frequency coherence functions propagated along ray paths in the inhomogeneous background of a weakly random medium. I—Formulation and evaluation of the second moment,” J. Acoust. Soc. Am. (to be published).
[PubMed]

Gelfgat, V. I.

V. I. Gelfgat, Sov. Phys. Acoust. 22, 65 (1976).

Hill, R. J.

R. J. Hill, J. Acoust. Soc. Am. 77, 1742 (1985).
[CrossRef]

Klyatskin, V. I.

V. I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomogeneous Media (Nauka, Moscow, 1980).

Mazar, R.

R. Mazar, L. B. Felsen, Opt. Lett. 12, 146 (1987).
[CrossRef] [PubMed]

R. Mazar, L. B. Felsen, Opt. Lett. 12, 4 (1987).
[CrossRef] [PubMed]

R. Mazar, L. B. Felsen, “High-frequency coherence functions propagated along ray paths in the inhomogeneous background of a weakly random medium. I—Formulation and evaluation of the second moment,” J. Acoust. Soc. Am. (to be published).
[PubMed]

Mironov, V. L.

Tatartskii, V. I.

V. I. Tatartskii, V. U. Zavorotnyi, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980).

Zavorotnyi, V. U.

V. I. Tatartskii, V. U. Zavorotnyi, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980).

J. Acoust. Soc. Am. (1)

R. J. Hill, J. Acoust. Soc. Am. 77, 1742 (1985).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Sov. Phys. Acoust. (1)

V. I. Gelfgat, Sov. Phys. Acoust. 22, 65 (1976).

Other (4)

V. I. Tatartskii, V. U. Zavorotnyi, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980).

V. I. Klyatskin, Stochastic Equations and Waves in Randomly Inhomogeneous Media (Nauka, Moscow, 1980).

R. Mazar, L. B. Felsen, “High-frequency coherence functions propagated along ray paths in the inhomogeneous background of a weakly random medium. I—Formulation and evaluation of the second moment,” J. Acoust. Soc. Am. (to be published).
[PubMed]

V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems (Nauka, Moscow, 1972).

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Figures (1)

Fig. 1
Fig. 1

Ray reflection–transmission diagram. Solid lines, incident rays; dotted–dashed lines, transmitted rays; dashed lines, reflected rays.

Equations (16)

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Γ σ ̅ = i n ̅ 0 ( σ ̅ ) { [ h ̅ 2 ( p ̅ , σ ̅ ) + 1 3 s ̅ 2 4 R ̅ 2 ( σ ̅ ) ] 2 Γ p ̅ s ̅ + h ̅ ( p ̅ , σ ̅ ) R ̅ ( σ ̅ ) ( 1 3 s ̅ 4 2 Γ p ̅ 2 + s ̅ 2 Γ s ̅ 2 ) + 1 R ̅ ( σ ̅ ) [ h ̅ ( p ̅ , σ ̅ ) Γ s ̅ + 1 3 s ̅ 4 R ̅ ( σ ̅ ) Γ p ̅ ] } i 1 3 / 2 [ μ ( r ̅ + , σ ̅ ) μ ( r ̅ , σ ̅ ) ] Γ + [ H ̅ ( r ̅ + , σ ̅ ) + H ̅ ( r ̅ , σ ̅ ) ] Γ + i 2 1 n ̅ 0 ( σ ̅ ) × [ h ̅ 2 ( r ̅ + , σ ̅ ) n ̅ 2 ( r ̅ + , σ ̅ ) n ˜ ( p ̅ + 2 s ̅ 2 , σ ̅ ) h ̅ 2 ( r ̅ , σ ̅ ) n ̅ 2 ( r ̅ , σ ̅ ) n ˜ ( p ̅ 2 s ̅ 2 , σ ̅ ) ] Γ ,
Γ ( p ̅ , s ̅ , σ ̅ ) = d υ ̅ s d q ̅ s Γ s ( υ ̅ s , q ̅ s ) G ( p ̅ , s ̅ , σ ̅ | υ ̅ s , q ̅ s 0 )
= 1 2 π d υ ̅ s d q ̅ s d ρ ̅ s Γ ˜ s ( υ ̅ s , ρ ̅ s ) × G ( p ̅ , s ̅ , σ ̅ | υ ̅ s , q ̅ s , 0 ) exp ( i ρ ̅ s q ̅ s ) ,
G ( p ̅ 2 , s ̅ 2 , σ ̅ 2 | p ̅ 1 , s ̅ 1 , σ ̅ 1 ) = 1 ( 2 π ) 2 d ρ ̅ 1 η ̅ × n ̅ 0 ( σ ̅ 1 ) n ̅ 0 ( σ ̅ 2 ) h ̅ ( υ ̅ , σ ̅ 2 ) h ̅ ( p ̅ 1 , σ ̅ 1 ) exp [ i ( ρ ̅ s ̅ 2 ρ ̅ 1 s ̅ 1 ) ] × exp [ i η ̅ ( p ̅ υ ̅ ) ] C ( η ̅ , p ̅ 1 , ρ ̅ 1 , σ ̅ 1 , σ ̅ 2 ) × exp ( i 2 1 σ ̅ 1 σ ̅ 2 d ζ h ̅ 2 ( υ ̅ , ζ ) n ̅ 2 ( υ ̅ , ζ ) n ̅ 0 ( ζ ) × { n ˜ [ υ ̅ ( ζ ) + 2 η ̅ s ̅ ( ζ ) 2 , ζ ] n ˜ [ υ ̅ ( ζ ) + 2 η ̅ s ̅ ( ζ ) 2 , ζ ] } ) .
Γ b ( p ̅ b , s ̅ b , σ ̅ i ) Γ i ( p ̅ b cos θ i , s ̅ b cos θ i , σ ̅ i ) × exp ( i n b s ̅ b sin θ i ) ,
Γ ̂ b ( p ̅ b , s ̅ b , σ ̅ i ) Γ b ( p ̅ b , s ̅ b , σ ̅ i ) exp ( i k l n n b ψ ( p ̅ b , s ̅ b ) ,
ψ ( p ̅ b , s ̅ b ) = κ ( p ̅ b + 2 s ̅ b 2 ) κ ( p ̅ b 2 s ̅ b 2 ) .
Γ t s ( p ̅ t , s ̅ t ) = Γ ̂ b ( p ̅ b , s ̅ b , σ ̅ i ) K t ( p ̅ b + 2 s ̅ b 2 , θ i ) × K t * ( p ̅ b 2 s ̅ b 2 , θ i ) × exp ( i k l n n b + ψ ( p ̅ b , s ̅ b ) i n b + s ̅ t tan θ i ] ,
Γ r s ( p ̅ r , s ̅ r ) Γ i ( p ̅ r , s ̅ r , σ ̅ i ) K r [ ( p ̅ r + 2 s ̅ r 2 ) / × cos θ i , θ i ] K r * [ ( p ̅ r 2 s ̅ r 2 ) / cos θ i , θ i ] × exp [ 2 i k l n n b ψ ( p ̅ r / cos θ i , s ̅ r / cos θ i ) ] .
Γ r s ( p ̅ , s ̅ , σ ̅ b ) = χ ( p ̅ , s ̅ , σ ̅ b ) Γ i ( p ̅ , s ̅ , σ ̅ b ) ,
χ ( p ̅ , s ̅ , σ ̅ b ) = K r ( p ̅ + 2 s ̅ 2 ) K r * ( p ̅ 2 s ̅ 2 ) × exp { i k l n [ ψ r ( p ̅ + 2 s ̅ 2 ) ψ r ( p ̅ 2 s ̅ 2 ) ] } ,
ψ r ( r ̅ ) = 2 σ ̅ b σ ̅ b + κ ( r ̅ ) d t n ( r ̅ , t ) [ 1 + n ˜ ( r ̅ , t ) ] .
I ( r ̅ 0 , σ ̅ 0 ) = d υ ̅ s d q ̅ s d p ̅ d s ̅ Γ s ( υ ̅ s , q s ) × χ ( p ̅ , s ̅ , σ ̅ b ) G ( p ̅ , s ̅ , σ ̅ b | υ ̅ s , q ̅ s , 0 ) × G * ( r ̅ 0 , 0 , σ ̅ 0 | p ̅ , s ̅ , σ ̅ b ) ,
G ˜ r s ( υ ̅ s , ρ ̅ s ) = I inc k l n δ ( υ ̅ s ) | D 0 ( ρ ̅ s , θ i ; k ) | 2 ,
D 0 ( ρ ̅ s , θ i ; k ) D ( θ d + ϕ , θ i ; k ) , ϕ = tan 1 [ ρ ̅ s n ̅ 0 ( 0 ) ] ,
I ( r ̅ 0 , σ ̅ 0 ) = I s k 2 2 π l n 2 d q ̅ s d ρ ̅ s | D 0 ( ρ ̅ s , θ i ; k ) | 2 × exp ( i ρ ̅ s q ̅ s ) G ( 0 , 0 , σ ̅ i | 0 , 0 , 0 ) × G * ( r ̅ 0 , 0 , σ ̅ 0 | 0 , q ̅ s , σ ̅ i )

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