Abstract

It is shown that, in the case of a bounded sound column, scattering calculations are much simplified by the introduction of Laplace transforms.

© 1981 Optical Society of America

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References

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  1. A. Korpel, J. Opt. Soc. Am. 69, 678 (1979).
    [CrossRef]
  2. A. Korpel, T. C. Poon, J. Opt. soc. Am. 70, 817 (1980).
    [CrossRef]
  3. T. C. Poon, A. Korpel, J. Opt. Soc. Am. 71, 1202 (1981).
    [CrossRef]
  4. R. D. Mattuck, A Guide to Feynman Diagrams in the Many Body Problems (McGraw-Hill, New York, 1976).
  5. V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464, U.S. Department of Commerce, (U.S. Government Printing Office, Washington, D.C., 1961).
  6. A. Korpel, “Acousto-optics” in Applied Solid State Science, Vol. 3, R. Wolfe, ed. (Academic, New York, 1972).
  7. A. Goldman, Laplace Transform Theory and Electrical Transients (Dover, New York, 1966).

1981

1980

1979

Goldman, A.

A. Goldman, Laplace Transform Theory and Electrical Transients (Dover, New York, 1966).

Korpel, A.

Mattuck, R. D.

R. D. Mattuck, A Guide to Feynman Diagrams in the Many Body Problems (McGraw-Hill, New York, 1976).

Poon, T. C.

Tatarskii, V.

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464, U.S. Department of Commerce, (U.S. Government Printing Office, Washington, D.C., 1961).

J. Opt. soc. Am.

Other

R. D. Mattuck, A Guide to Feynman Diagrams in the Many Body Problems (McGraw-Hill, New York, 1976).

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, TT-68-50464, U.S. Department of Commerce, (U.S. Government Printing Office, Washington, D.C., 1961).

A. Korpel, “Acousto-optics” in Applied Solid State Science, Vol. 3, R. Wolfe, ed. (Academic, New York, 1972).

A. Goldman, Laplace Transform Theory and Electrical Transients (Dover, New York, 1966).

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Figures (5)

Fig. 1
Fig. 1

General configuration for sound–light interaction showing how diffracted orders n + 1 and n − 1 contribute to n through sound fields S n + 1 and S n 1 +.

Fig. 2
Fig. 2

Feynman diagrams illustrating some possible scattering paths from inc to 3

Fig. 3
Fig. 3

Interaction configuration for scattering with arbitrary angle of incidence.

Fig. 4
Fig. 4

Feynman diagram used in illustrative example of scattering calculation.

Fig. 5
Fig. 5

Feynman diagrams showing some possible scattering paths for +1-order generation in the exact Bragg case.

Equations (24)

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n = j a n 1 z S n 1 + n 1 d z j a n + 1 l z S n + 1 n + 1 d z .
0 = inc j a 1 z S 1 + 1 d z j a 1 z S 1 1 d z ,
( j ) 5 a 2 a 1 a 2 a 1 a 0 z S 2 + d z 1 z 1 S 1 + d z 2 × z 2 S 2 d z 3 z 3 S 1 + d z 4 z 4 S 0 + inc d z 5 .
n = all paths ( j ) n a n 1 a 0 z S n 1 ± d z × z m 1 S 0 ± inc d z m .
S n + = S [ z , z ( ϕ n + ϕ B ) ] = S { z , z [ ϕ inc + ( 2 n + 1 ) ϕ B ] } ,
S n = S * [ z , z ( ϕ n ϕ B ) ] = S { z , z [ ϕ inc + ( 2 n 1 ) ϕ B ] } ,
S n + = A exp { j K [ ϕ inc + ( 2 n + 1 ) ϕ B ] z } = A exp { j 1 / 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ξ }
S n = A exp { + j K [ ϕ inc + ( 2 n 1 ) ϕ B ] z } = A exp { + j 1 / 2 [ ϕ inc ϕ B + ( 2 n 1 ) ] Q ξ } ,
a n , n + 1 = 1 / 2 [ ϕ inc ϕ B + ( 2 n + 1 ) ] Q ,
a n , n 1 = 1 / 2 [ ϕ inc ϕ B + ( 2 n 1 ) ] Q ,
S n + = A exp ( j a n , n + 1 ξ ) ,
S n = A exp ( j a n , n 1 ξ ) .
( j α ̂ 2 ) 5 0 1 exp ( j a 23 ξ 1 ) d ξ 1 0 ξ 1 exp ( j a 12 ξ 2 ) d ξ 2 × 0 ξ 2 exp ( j a 21 ξ 3 ) d ξ 3 0 ξ 3 exp ( j a 12 ξ 4 ) d ξ 4 × 0 ξ 4 exp ( j a 01 ξ 5 ) inc d ξ 5 ,
g new ( ξ ) = 0 ξ g old ( ξ ) exp ( j a ξ ) d ξ ,
G new ( s ) = 1 s G old ( s + j a ) ,
G new ( s ) = G 1 ( 1 ) ( s ) = 1 s 1 s + j a 01 inc ,
G 2 ( 1 ) ( s ) = 1 s G 1 ( 1 ) ( s + j a 12 ) inc .
G 2 ( 1 ) ( s ) = inc × 1 s ( 1 s × 1 s + j a 01 ) s = s + j a 12 = 1 s × 1 s + j a 12 × 1 s + j a 01 + j a 12 × inc .
G 1 ( 2 ) ( s ) = 1 s G 2 ( 1 ) ( s + j a 21 ) inc .
G 1 ( 2 ) ( s ) = 1 s × 1 s + j a 21 × 1 s + j a 12 + j a 21 × 1 s + j a 01 + j a 12 + j a 21 inc .
G 3 ( 1 ) ( s ) = 1 s × 1 s + j a 23 × 1 s + j a 12 + j a 23 × 1 s + j a 21 + j a 12 + j a 23 × 1 s + j a 12 + j a 21 + j a 12 + j a 23 × 1 s + j a 01 + j a 12 + j a 21 + j a 12 + j a 23 × inc .
inc ( j α ̂ 2 ) 5 L 1 { G 3 ( 1 ) ( s ) } .
n = inc L 1 { n + 2 l ( j α ̂ 2 ) n 1 s × 1 s + a n ± 1 , n × × 1 s + a n ± 1 , n + + a 0 , ± 1 } ,
1 = L 1 [ l = 0 ( j α ̂ 2 ) 2 l + 1 1 s 1 s 2 l + 1 ] inc = L 1 [ ( j α ̂ 2 ) 1 s 2 l = 0 ( 1 ) l ( α ̂ 2 ) 2 l 1 s 2 l ] inc = L 1 [ ( j α ̂ 2 ) 1 s 2 1 1 + ( α ̂ 2 1 s ) 2 ] inc = L 1 [ ( j α ̂ 2 ) 1 s 2 + ( α ̂ 2 ) 2 ] inc = ( j ) sin ( α ̂ 2 ξ ) inc ,

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