Abstract

A method is described for 2-D control of spatial coherence by use of multiple, crossing ultrasonic waves. If all the ultrasonic waves differ in frequency from one another, the coherence modification factor results as a product of the individual modification factor formed by each ultrasonic wave itself. Two arrangements are discussed in detail together with experimental results: one with the ultrasonic waves propagating in orthogonal directions, the other with the waves propagating in three symmetric directions. The coherence modification factor obtained from either arrangement varies periodically in space but is almost uniform in the azimuthal direction if the two-point separation concerned is taken in a linear dimension a little larger than half of one ultrasonic wavelength.

© 1983 Optical Society of America

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References

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  1. B. J. Thompson, Prog. Opt. 7, 169 (1969).
    [CrossRef]
  2. G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).
  3. D. Nyyssonen, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 34 (1979).
  4. D. Nyyssonen, J. Opt. Soc. Am. 72, 1425 (1982).
    [CrossRef]
  5. See, for example, T. S. McKechnie, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9, pp. 123–170.
    [CrossRef]
  6. Y. Ohtsuka, Y. Imai, J. Opt. Soc. Am. 69, 684 (1979).
    [CrossRef]
  7. C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. Sect. A 2, 406 (1935).
  8. Y. Imai, M. Imai, Y. Ohtsuka, Appl. Opt. 19, 3541 (1980).
    [CrossRef] [PubMed]
  9. Y. Imai, Y. Ohtsuka, Optik 58, 377 (1981).
  10. Y. Imai, Y. Ohtsuka, J. Opt. Soc. Am. 71, 1427 (1981).
    [CrossRef]
  11. Y. Ohtsuka, Y. Imai, Opt. Acta 29, 781 (1982).
    [CrossRef]
  12. W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 505–508.

1982

D. Nyyssonen, J. Opt. Soc. Am. 72, 1425 (1982).
[CrossRef]

Y. Ohtsuka, Y. Imai, Opt. Acta 29, 781 (1982).
[CrossRef]

1981

Y. Imai, Y. Ohtsuka, Optik 58, 377 (1981).

Y. Imai, Y. Ohtsuka, J. Opt. Soc. Am. 71, 1427 (1981).
[CrossRef]

1980

1979

Y. Ohtsuka, Y. Imai, J. Opt. Soc. Am. 69, 684 (1979).
[CrossRef]

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

D. Nyyssonen, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 34 (1979).

1969

B. J. Thompson, Prog. Opt. 7, 169 (1969).
[CrossRef]

1967

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

1935

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. Sect. A 2, 406 (1935).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 505–508.

Cook, B. D.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

Imai, M.

Imai, Y.

Klein, W. R.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

McKechnie, T. S.

See, for example, T. S. McKechnie, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9, pp. 123–170.
[CrossRef]

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. Sect. A 2, 406 (1935).

Nyyssonen, D.

D. Nyyssonen, J. Opt. Soc. Am. 72, 1425 (1982).
[CrossRef]

D. Nyyssonen, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 34 (1979).

Ohtsuka, Y.

Raman, C. V.

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. Sect. A 2, 406 (1935).

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

Thompson, B. J.

B. J. Thompson, Prog. Opt. 7, 169 (1969).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 505–508.

Appl. Opt.

IEEE Trans. Sonics Ultrason.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

Y. Ohtsuka, Y. Imai, Opt. Acta 29, 781 (1982).
[CrossRef]

Optik

Y. Imai, Y. Ohtsuka, Optik 58, 377 (1981).

Proc. Indian Acad. Sci. Sect. A

C. V. Raman, N. S. N. Nath, Proc. Indian Acad. Sci. Sect. A 2, 406 (1935).

Proc. Soc. Photo-Opt. Instrum. Eng.

G. O. Reynolds, J. B. DeVelis, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 2 (1979).

D. Nyyssonen, Proc. Soc. Photo-Opt. Instrum. Eng. 194, 34 (1979).

Prog. Opt.

B. J. Thompson, Prog. Opt. 7, 169 (1969).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 505–508.

See, for example, T. S. McKechnie, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Heidelberg, 1975), Vol. 9, pp. 123–170.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Coordinate system for acoustooptic interactions.

Fig. 2
Fig. 2

Overview of coherence modification factors obtained with two arrangements of ultrasonic waves.

Fig. 3
Fig. 3

Dependence of coherence modification factor on the azimuthal angle for the arrangement of orthogonal ultrasonic waves.

Fig. 4
Fig. 4

Dependence of coherence modification factor on the azimuthal angle for the arrangement of three crossing ultrasonic waves.

Fig. 5
Fig. 5

Measured degree of spatial coherence for the light from a He–Ne laser.

Fig. 6
Fig. 6

Degree of spatial coherence obtained with the arrangement of orthogonal ultrasonic waves. Measured and calculated results are denoted, respectively, by three kinds of plotted mark and three solid lines.

Fig. 7
Fig. 7

Degree of spatial coherence obtained with the arrangement of three crossing ultrasonic waves. Measured and calculated results are denoted, respectively, by three kinds of plotted mark and three solid lines.

Fig. 8
Fig. 8

Coherence diagram for explaining symmetric properties of coherence: (a) for orthogonal ultrasonic waves, (b) for three crossing ultrasonic waves.

Equations (23)

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μ ( r , t ) = μ 0 + m = 1 N μ m sin ( Ω m t K m r ϕ m ) ,
2 V ( r , t ) = ( 1 c ) 2 2 t 2 [ ε ( r , t ) V ( r , t ) ] ,
V ( r , t ) = V ( r , t ) [ exp ( i K μ 0 L ) ] × p 1 p 2 p N = J p 1 ( υ 1 ) J p 2 ( υ 2 ) J p N ( υ N ) × exp { i [ p 1 ( Ω 1 t K 1 r ϕ 1 ) + p 2 ( Ω 2 t K 2 r ϕ 2 ) + + p N ( Ω N t K N r ϕ N ) ] } ,
V ( r , t ) = V ( r , t ) exp { i [ k μ 0 L + m = 1 N υ m × sin ( Ω m t K m r ϕ m ) ] } = V ( r , t ) [ exp ( i k μ 0 L ) ] m = 1 N × exp [ i υ m sin ( Ω m t K m r ϕ m ) ] ,
p = J p ( υ ) exp ( i p α ) = exp ( i υ sin α ) .
J ( P 1 , P 2 ) = V ( r 1 , t ) V * ( r 2 , t ) = J ( P 1 , P 2 ) M N ( P 1 , P 2 ) ,
J ( P 1 , P 2 ) = V ( r 1 , t ) V * ( r 2 , t )
M N ( P 1 , P 2 ) = p 1 p 2 p N = q 1 q 2 q N = J p 1 ( υ 1 ) × J p 2 ( υ 2 ) J p N ( υ N ) J q 1 ( υ 1 ) J q 2 ( υ 2 ) J q N ( υ N ) × { exp i [ q 1 ( K 1 r 1 + ϕ 1 ) p 1 ( K 1 r 2 + ϕ 1 ) + q 2 ( K 2 r 1 + ϕ 2 ) p 2 ( K 2 r 2 + ϕ 2 ) + + q N ( K N r 1 + ϕ N ) p N ( K N r 2 + ϕ N ) ] } × exp i [ ( p 1 q 1 ) Ω 1 t + ( p 2 q 2 ) Ω 2 t + + ( p N q N ) Ω N t ] .
= δ ( p 1 q 1 ) δ ( p 2 q 2 ) δ ( p N q N ) = m = 1 N δ ( p m q m ) ,
δ ( p m q m ) = { 1 when p m = q m , 0 when p m q m .
M N ( P 1 , P 2 ) = m = 1 N p m = J p m 2 ( υ m ) exp [ i p m K m ( r 1 r 2 ) ] = m = 1 N J 0 [ 2 υ m sin { ( 1 / 2 ) K m ( r 1 r 2 ) } ] ,
γ 12 ( 0 ) = J ( P 1 , P 2 ) / I ( P 1 ) I ( P 2 ) = J ( P 1 , P 2 ) M N ( P 1 , P 2 ) / I ( P 1 ) I ( P 2 ) ,
γ 12 ( 0 ) = J ( P 1 , P 2 ) / I ( P 1 ) I ( P 2 ) ,
γ 12 ( 0 ) = γ 12 ( 0 ) M N ( P 1 , P 2 ) .
| M N ( P 1 , P 2 ) | 1 M N ( P 1 , P 1 ) = M N ( P 2 , P 2 ) = 1 } ,
M N ( P 1 , P 2 ) = 1 when υ m = 0 for all m .
γ 12 ( 0 ) = M N ( P 1 , P 2 ) .
M 2 ( P 1 , P 2 ) = J 0 [ 2 υ 1 sin ( K 1 r / 2 ) ] × J 0 [ 2 υ 2 sin ( K 2 r / 2 ) ] ,
M 2 ( P 1 , P 2 ) = J 0 [ 2 υ 1 sin ( π r cos θ / Λ 1 ) ] × J 0 [ 2 υ 2 sin ( π r sin θ / Λ 2 ) ] ,
M 3 ( P 1 , P 2 ) = J 0 [ 2 υ 1 sin ( K 1 r / 2 ) ] J 0 [ 2 υ 2 sin ( K 2 r / 2 ) ] × J 0 [ 2 υ 3 sin ( K 3 r / 2 ) ] .
M 3 ( P 1 , P 2 ) = J 0 [ 2 υ 1 sin ( π r cos θ / Λ 1 ) ] × J 0 { 2 υ 2 sin [ π r ( cos θ + 3 sin θ ) / 2 Λ 2 ] } × J 0 { 2 υ 3 sin [ π r ( cos θ 3 sin θ ) / 2 Λ 3 ] } .
| γ 12 ( 0 ) | = I max I min I max + I min I 1 + I 2 2 ( I 1 I 2 ) 1 / 2 ,
M 1 ( P 1 , P 2 ) = J 0 { 2 υ 1 sin [ π ( x 1 x 2 ) / Λ 1 ] } ,

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