Abstract

The scattering of light by sound has been examined in the past mostly in terms of a single incident plane wave and approximations have often been used that are overly restrictive for many practical cases. To gain a better understanding of the basic diffraction process associated with realistically bounded light sources, we employ a rigorous approach to examine the irradiance from a line source or scatterer embedded in a medium traversed by an acoustic disturbance. As usual, we assume that the acoustic wave modulates the dielectric properties of the medium and thus produces a periodic stratification that serves to scatter the radiant energy. After deriving a formal solution for the scattered radiant flux produced by an arbitrary periodic modulation, we show that the principal diffraction aspects can be found by examining the simpler case of sinusoidal stratification. Since this particular case may be regarded as a prototype for the more general problem, we restrict our detailed quantitative analysis to the basic sinusoidal acoustic modulation. We find that the resulting illumination exhibits regions characterized by different diffraction effects that are intimately related to the Bragg directions prescribed by the acoustic wave fronts. These directions serve to classify the various diffraction regions into three basic varieties: (1) a set of wide angular domains wherein the radiant flux density is explainable by simple geometric-optical arguments, (2) another set of wide angular domains wherein the predominant diffraction effect is described in terms of a Bragg wave mechanism, and (3) a set of narrow angular domains that serve as transition regions between the other diffraction domains. Each transition region contains a caustic that accounts for multiple peaks of radiant energy density in the neighborhood of the Bragg directions. Physical interpretations and quantitative estimates for each diffraction domain are presented by employing asymptotic results that are very accurate for the sound intensities that are usually available.

© 1970 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Ch. 52, p. 579.
  2. T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).
  3. M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965).
    [Crossref]
  4. T. Tamir and H. C. Wang, Can. J. Phys. 44, 2703 (1966).
  5. T. Tamir, Can. J. Phys. 44, 2461 (1966).
    [Crossref]
  6. M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).
  7. W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).
  8. R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).
  9. L. Bergstein and D. Kermisch, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1968; Wiley, New York, distr.), p. 655.
  10. H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
    [Crossref]
  11. D. H. McMahon, IEEE Trans. SU-16, 41 (1969).
  12. B. Singer and T. Tamir, Alta Frequenza 38(special issue), 70 (1969).
  13. K. F. Casey, Can. J. Phys. 46, 2543 (1968).
    [Crossref]
  14. B. Singer, Ph.D. dissertation, Dept. of Electrophysics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1969.
  15. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947).
  16. T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).
  17. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 19.2, p. 245.
  18. H. Bremmer, Comm. Pure Appl. Math. 4, 105 (1951).
    [Crossref]
  19. G. Tyras, Radiation and Propagation of Electromagnetic Waves, (Academic, New York, 1969), Ch. 4, p. 64.
  20. Reference 17, p. 483.
  21. L. B. Felsen, Radio Sci. 69D, 155 (1965).

1970 (1)

R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).

1969 (3)

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
[Crossref]

D. H. McMahon, IEEE Trans. SU-16, 41 (1969).

B. Singer and T. Tamir, Alta Frequenza 38(special issue), 70 (1969).

1968 (1)

K. F. Casey, Can. J. Phys. 46, 2543 (1968).
[Crossref]

1967 (1)

W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).

1966 (2)

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2703 (1966).

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

1965 (2)

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965).
[Crossref]

L. B. Felsen, Radio Sci. 69D, 155 (1965).

1964 (1)

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).

1963 (1)

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

1951 (1)

H. Bremmer, Comm. Pure Appl. Math. 4, 105 (1951).
[Crossref]

Bergstein, L.

L. Bergstein and D. Kermisch, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1968; Wiley, New York, distr.), p. 655.

Berry, M. V.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Ch. 52, p. 579.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 19.2, p. 245.

Bremmer, H.

H. Bremmer, Comm. Pure Appl. Math. 4, 105 (1951).
[Crossref]

Casey, K. F.

K. F. Casey, Can. J. Phys. 46, 2543 (1968).
[Crossref]

Chu, R. S.

R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).

Cohen, M. G.

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965).
[Crossref]

Cook, B. D.

W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).

Felsen, L. B.

L. B. Felsen, Radio Sci. 69D, 155 (1965).

Gordon, E. I.

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965).
[Crossref]

Kermisch, D.

L. Bergstein and D. Kermisch, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1968; Wiley, New York, distr.), p. 655.

Klein, W. R.

W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
[Crossref]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947).

McMahon, D. H.

D. H. McMahon, IEEE Trans. SU-16, 41 (1969).

Oliner, A. A.

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

Singer, B.

B. Singer and T. Tamir, Alta Frequenza 38(special issue), 70 (1969).

B. Singer, Ph.D. dissertation, Dept. of Electrophysics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1969.

Tamir, T.

R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).

B. Singer and T. Tamir, Alta Frequenza 38(special issue), 70 (1969).

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2703 (1966).

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

Tyras, G.

G. Tyras, Radiation and Propagation of Electromagnetic Waves, (Academic, New York, 1969), Ch. 4, p. 64.

Wang, H. C.

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2703 (1966).

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Ch. 52, p. 579.

Alta Frequenza (1)

B. Singer and T. Tamir, Alta Frequenza 38(special issue), 70 (1969).

Bell System Tech. J. (2)

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
[Crossref]

M. G. Cohen and E. I. Gordon, Bell System Tech. J. 44, 693 (1965).
[Crossref]

Can. J. Phys. (3)

T. Tamir and H. C. Wang, Can. J. Phys. 44, 2703 (1966).

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[Crossref]

K. F. Casey, Can. J. Phys. 46, 2543 (1968).
[Crossref]

Comm. Pure Appl. Math. (1)

H. Bremmer, Comm. Pure Appl. Math. 4, 105 (1951).
[Crossref]

IEEE Trans (1)

T. Tamir, H. C. Wang, and A. A. Oliner, IEEE Trans MTT-12, 323 (1964).

IEEE Trans. (3)

W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).

R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).

D. H. McMahon, IEEE Trans. SU-16, 41 (1969).

Proc. IEE (London) (1)

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

Radio Sci. (1)

L. B. Felsen, Radio Sci. 69D, 155 (1965).

Other (8)

G. Tyras, Radiation and Propagation of Electromagnetic Waves, (Academic, New York, 1969), Ch. 4, p. 64.

Reference 17, p. 483.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Sec. 19.2, p. 245.

B. Singer, Ph.D. dissertation, Dept. of Electrophysics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1969.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947).

L. Bergstein and D. Kermisch, in Proceedings of the Symposium on Modern Optics, edited by J. Fox (Polytechnic Press, Brooklyn, 1968; Wiley, New York, distr.), p. 655.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Ch. 52, p. 579.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the periodically stratified medium. The filamentary source is located normal to the plane of the diagram at Z=Z0.

Fig. 2
Fig. 2

The Mathieu stability diagram.

Fig. 3
Fig. 3

Branch-cut singularities in the uppermost sheet of the four-sheeted complex Riemann plane kx=kxr+ikxi.

Fig. 4
Fig. 4

Typical saddle-point solutions sj vs the angular variable θ. To indicate the labeling of sj(j= 1,2,3), the three real saddle points at an arbitrary angle θ=θ2>θc are displayed. The two points s2 and s3 would coalesce at θ=θc and their value would be complex for θ <θc.

Fig. 5
Fig. 5

Paths of integration for observation points in region 1.

Fig. 6
Fig. 6

Paths of integration for observation points in regions 2 and 3.

Fig. 7
Fig. 7

Graphical description of the Bragg field in region 2.

Fig. 8
Fig. 8

The radiant-power density P vs θ for a typical case, with M=10−3, k =2, and θB=π/3. To delineate the transition region, σ0 was chosen equal to 5 in all the cases shown.

Fig. 9
Fig. 9

Wavenumber diagram when only a first-order (m=1) Bragg condition is satisfied.

Fig. 10
Fig. 10

Diffraction effects when two Bragg directions θ1, and θ2 occur. (a) Wavenumber diagram. (b) Diffraction domains; for clarity the Bragg flux along θ1 and θ2 was omitted and only that along πθ1 and πθ2 is shown.

Equations (58)

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[ 2 + K 2 / 0 ] E = i ω μ 0 I δ ( X ) δ ( Z - Z 0 ) ,
K = ω ( μ 0 0 ) 1 2 = 2 π / λ ,
[ 2 + K 2 0 - 1 Z Z ] H = i ω M δ ( X ) δ ( Z - Z 0 ) ,
x = π X / L ,             z = π Z / L ,             and             r = π R / L ,
k = K L / π = 2 L / λ .
G = G ( x , z ; z 0 ) = 1 2 π - g ( k x ; x ) exp ( i k x x ) d k x ,
d 2 g / d z 2 + [ k z 2 + p ( z ) ] g = - δ ( z - z 0 ) ,
G = i E ω μ 0 I ,
p ( z ) = k 2 [ ( z ) 0 - 1 ] .
G = i H / ω [ ( z ) ( z 0 ) ] 1 2
p ( z ) = k 2 [ ( z ) 0 - 1 ] - 3 4 [ 1 ( z ) ( z ) z ] 2 + 1 2 ( z ) 2 ( z ) z 2 .
k z 2 = k 2 - k x 2 ,
g h ( k x ; z ) = e ± i κ z P ( z ) ,
g ( k x ; z ) = [ P ( z > ) P ( - z < ) / W ( κ ) ] exp ( i κ z - z 0 ) ,
W ( κ ) = 2 P ( z ) [ i κ P ( z ) + P ( z ) ] ,
P ( z ) = n = - a n e 2 i n z ,
G = n = - r = - I n r exp [ 2 i ( n z > - r z < ) ] ,
I n r = I n r ( x , z ; z 0 ) = - a n ( κ ) a r ( κ ) 2 π W ( κ ) exp [ i ( k x x + κ z - z 0 ) ] d k x .
( z ) = 0 ( 1 - M cos 2 z ) ,
d 2 g / d z 2 + ( k z 2 - 2 q cos 2 z ) g = - δ ( z - z 0 ) ,
q = 1 2 k 2 M
k cos θ B m = ( 2 L / λ ) cos θ B m = m .
κ 2 = 1 ± [ ( k z 2 - 1 ) 2 - q 2 ] 1 2 .
a 0 ( κ ) = 1 ,
α ± 1 ( κ ) = q / [ k z 2 - ( κ ± 2 ) 2 ] ,
b 1 2 2 = k 2 - 1 ± q .
lim q 0 κ 2 = k z 2
b 3 = [ k 2 + ( 1 + q ) 1 2 - 1 ] 1 2 k [ 1 + ( q / 2 k ) 2 ] .
κ = 1 + U ,
I n r = I n r ( x , z ; z 0 ) = - F n r ( k x ) exp [ i ( k x x + κ z - z 0 ) ] d k x ,
F n r ( k x ) = a n ( κ ) a r ( κ ) / 2 π W ( κ ) .
x + z - z 0 d η / d k x = 0.
tan θ = x z - z 0 = - d κ d k x = k z 2 - 1 η 2 - 1 k x κ .
k sin θ c = ( k 2 - 1 ) 1 2 { 1 + 3 [ q / 2 k ( k 2 - 1 ) ] 2 3 } 1 2 .
region 1 : 0 < θ < θ c - δ , region 2 : θ c + δ < θ < π / 2 , region 3 : θ c - δ < θ < θ c + δ ,
s 1 ( 1 ) = { 1 - [ ( 3 k 2 cos 2 θ - 1 ) / ( k 2 cos 2 θ - 1 ) 2 ] × ( q / 2 k ) 2 } k sin θ ,
G ( 1 ) ~ exp i ( k r - π / 4 ) 2 i ( 2 π k r ) 1 2 × [ 1 - q 4 ( e - 2 i z > + e 2 i z < 1 - k cos θ + e 2 i z > + e - 2 i z < 1 + k cos θ ) ] .
k x 2 = k 2 - 1 q ( 1 - ( k x / κ ) cot θ ) - 1 2 .
s 1 2 ( 2 ) = [ k 2 - 1 q sin θ / ( 1 - k 2 cos 2 θ ) 1 2 ] 1 2 ,
s 3 ( 2 ) = s 1 ( 1 ) .
G ( 2 ) = G 1 ( 2 ) + G 2 ( 2 ) + G 3 ( 2 ) ,
G 3 ( 2 ) = G ( 1 ) .
G 1 2 ( 2 ) ~ 1 8 i ( q π i ) 1 2 sin θ + ( k 2 - 1 ) 1 2 cos θ ( k 2 - 1 ) 1 2 ( 1 - k 2 cos 2 θ ) 3 4 × exp [ i ( k 2 - 1 ) 1 2 x i π / 4 ] · [ e - i z < ± A ( θ ) e i z < ] [ e i z > ± A ( θ ) e - i z > ] ,
A ( θ ) = a - 1 ( s 1 ) = - a - 1 ( s 2 ) = [ sin ( θ - θ B ) / sin ( θ + θ B ) ] 1 2 .
G B = G 1 ( 2 ) + G 2 ( 2 ) ~ 1 4 i k [ q sin θ π k r sin θ B sin ( θ - θ B ) sin ( θ + θ B ) ] 1 2 · { [ A ( θ ) ] - 1 2 exp [ i k r cos ( θ - θ B ) - i φ ] - i [ A ( θ ) ] 1 2 exp [ - i k r cos ( θ + θ B ) + i φ ] } ,
tan φ = A ( θ ) .
s c = k sin θ c ( k 2 - 1 ) 1 2 + [ 3 / 2 ( k 2 - 1 ) 1 / 6 ] ( q / 2 k ) 2 3 ,
s 1 ( 3 ) = ( k 2 - 1 ) 1 2 - [ 1 / 2 ( k 2 - 1 ) 1 / 6 ] ( q / C k ) 2 3 ,
G 1 ( 3 ) ~ exp [ i k r cos ( θ - θ c ) - π / 4 ] 2 i [ 2 π k r ( 1 + 2 C 2 ) ] 1 2 × [ 1 + 1 2 ( C 2 k 2 q k 2 - 1 ) 1 3 ( e - 2 i z > + e 2 i z < ) ] .
G c ~ 1 2 i 2 1 / 9 3 1 3 ( q / k ) 2 / 9 ( k r ) 1 3 ( k 2 - 1 ) 1 / 18 A i ( σ ) × exp [ i k r cos ( θ c - θ ) ] ,
σ = ( 2 1 / 9 / 3 1 3 ) [ ( k r ) 2 3 sin ( θ c - θ ) / ( k 2 - 1 ) 1 / 18 ] ( q / k ) 2 / 9 .
δ = ( 3 1 3 / 2 1 / 9 ) [ ( k 2 - 1 ) 1 / 18 / ( k r ) 2 3 ] ( k / q ) 2 / 9 σ 0 ,
D = k r q 1 3 1.
Power θ - δ θ + δ G c 2 r d θ 2 2 / 9 3 2 3 ( q / k ) 4 / 9 ( k r ) 2 3 ( k 2 - 1 ) 1 / 9 A m 2 r δ = B q 4 / 9 r 1 3 ,
tan θ = - ( d κ / d k x ) k x = s j .
a n - 1 + D n a n + a n + 1 = 0 ,             ( n = 0 , 1 , 2 ) ,
D n = [ ( κ + 2 n ) 2 - k z 2 ] / q .
Δ 3 - 8 Δ 2 + 2 ( 8 - 8 k z 2 - q 2 ) Δ - 8 q 2 = 0 ,