Abstract

The mutual coherence function of a light wave modulated by a progressive ultrasonic wave having harmonics is formulated on the basis of Raman-Nath’s phase lattice theory. The resultant degree of spatial coherence derived under the assumption that only the fundamental of the ultrasonic wave exists is in good agreement with the results obtained from Young’s interference experiments for rather low ultrasonic pressures. It is evidently shown that the partial coherence condition can be controlled electronically by ultrasonic pressures. Laser speckle reduction is demonstrated as one of its applications.

© 1979 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).
  2. See, for example, T. Asakura, “Resolution of two unequally bright points with partially coherent light,” Nouv. Rev. Opt.169–177 (1974).
    [Crossref]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. Y. Ichioka, K. Yamamoto, and T. Suzuki, “Image of a sinusoidal complex object in a partially coherent optical system,” J. Opt. Soc. Am. 65, 892–902(1975).
    [Crossref]
  5. See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975).
    [Crossref]
  6. H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453(1970).
    [Crossref]
  7. Y. Ohtsuka, “Diffraction and interference of partially coherent light traversing two superposed sound fields,” Opt. Acta 20, 263–270 (1973).
    [Crossref]
  8. Y. Ohtsuka, “Modulation effects of sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
    [Crossref]
  9. Y. Ohtsuka, “Effects of sound-light interaction on partial coherence in image-forming optical system,” Opt. Commun. 17, 238–241(1976).
    [Crossref]
  10. C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Ind. Acad. Sci. 2A, 406–412(1935);Proc. Ind. Acad. Sci. 2A, 413–420(1935);Proc. Ind. Acad. Sci. 3A, 75–84(1936);Proc. Ind. Acad. Sci. 3A, 119–125 (1936).
  11. M. V. Berry, The diffraction of light by ultrasound (Academic, New York, 1966).
  12. R. Extermann and G. Wannier, “Thèorie de la difraction de la lumière par les ultrasons,” Helv. Phys. Acta 9, 520–532 (1936).
  13. E. Schröder, “Elimination of granulation in laser beam projection by means of moving diffusers,” Opt. Commun. 3, 68–72(1971).
    [Crossref]
  14. S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971).
    [Crossref]
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., 1966), pp. 350–359.

1976 (2)

Y. Ohtsuka, “Modulation effects of sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
[Crossref]

Y. Ohtsuka, “Effects of sound-light interaction on partial coherence in image-forming optical system,” Opt. Commun. 17, 238–241(1976).
[Crossref]

1975 (1)

1974 (1)

See, for example, T. Asakura, “Resolution of two unequally bright points with partially coherent light,” Nouv. Rev. Opt.169–177 (1974).
[Crossref]

1973 (1)

Y. Ohtsuka, “Diffraction and interference of partially coherent light traversing two superposed sound fields,” Opt. Acta 20, 263–270 (1973).
[Crossref]

1971 (2)

E. Schröder, “Elimination of granulation in laser beam projection by means of moving diffusers,” Opt. Commun. 3, 68–72(1971).
[Crossref]

S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971).
[Crossref]

1970 (1)

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453(1970).
[Crossref]

1936 (1)

R. Extermann and G. Wannier, “Thèorie de la difraction de la lumière par les ultrasons,” Helv. Phys. Acta 9, 520–532 (1936).

1935 (1)

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Ind. Acad. Sci. 2A, 406–412(1935);Proc. Ind. Acad. Sci. 2A, 413–420(1935);Proc. Ind. Acad. Sci. 3A, 75–84(1936);Proc. Ind. Acad. Sci. 3A, 119–125 (1936).

Arsenault, H.

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453(1970).
[Crossref]

Asakura, T.

See, for example, T. Asakura, “Resolution of two unequally bright points with partially coherent light,” Nouv. Rev. Opt.169–177 (1974).
[Crossref]

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, 2nd ed. (Pergamon, New York, 1964).

Extermann, R.

R. Extermann and G. Wannier, “Thèorie de la difraction de la lumière par les ultrasons,” Helv. Phys. Acta 9, 520–532 (1936).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Ichioka, Y.

Joyeux, D.

Lowenthal, S.

S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971).
[Crossref]

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453(1970).
[Crossref]

McKechnie, T. S.

See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975).
[Crossref]

Nath, N. S. N.

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Ind. Acad. Sci. 2A, 406–412(1935);Proc. Ind. Acad. Sci. 2A, 413–420(1935);Proc. Ind. Acad. Sci. 3A, 75–84(1936);Proc. Ind. Acad. Sci. 3A, 119–125 (1936).

Ohtsuka, Y.

Y. Ohtsuka, “Modulation effects of sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
[Crossref]

Y. Ohtsuka, “Effects of sound-light interaction on partial coherence in image-forming optical system,” Opt. Commun. 17, 238–241(1976).
[Crossref]

Y. Ohtsuka, “Diffraction and interference of partially coherent light traversing two superposed sound fields,” Opt. Acta 20, 263–270 (1973).
[Crossref]

Raman, C. V.

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Ind. Acad. Sci. 2A, 406–412(1935);Proc. Ind. Acad. Sci. 2A, 413–420(1935);Proc. Ind. Acad. Sci. 3A, 75–84(1936);Proc. Ind. Acad. Sci. 3A, 119–125 (1936).

Schröder, E.

E. Schröder, “Elimination of granulation in laser beam projection by means of moving diffusers,” Opt. Commun. 3, 68–72(1971).
[Crossref]

Suzuki, T.

Wannier, G.

R. Extermann and G. Wannier, “Thèorie de la difraction de la lumière par les ultrasons,” Helv. Phys. Acta 9, 520–532 (1936).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., 1966), pp. 350–359.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Yamamoto, K.

Helv. Phys. Acta (1)

R. Extermann and G. Wannier, “Thèorie de la difraction de la lumière par les ultrasons,” Helv. Phys. Acta 9, 520–532 (1936).

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. (1)

See, for example, T. Asakura, “Resolution of two unequally bright points with partially coherent light,” Nouv. Rev. Opt.169–177 (1974).
[Crossref]

Opt. Acta (1)

Y. Ohtsuka, “Diffraction and interference of partially coherent light traversing two superposed sound fields,” Opt. Acta 20, 263–270 (1973).
[Crossref]

Opt. Commun. (4)

Y. Ohtsuka, “Modulation effects of sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
[Crossref]

Y. Ohtsuka, “Effects of sound-light interaction on partial coherence in image-forming optical system,” Opt. Commun. 17, 238–241(1976).
[Crossref]

E. Schröder, “Elimination of granulation in laser beam projection by means of moving diffusers,” Opt. Commun. 3, 68–72(1971).
[Crossref]

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453(1970).
[Crossref]

Proc. Ind. Acad. Sci. (1)

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I,” Proc. Ind. Acad. Sci. 2A, 406–412(1935);Proc. Ind. Acad. Sci. 2A, 413–420(1935);Proc. Ind. Acad. Sci. 3A, 75–84(1936);Proc. Ind. Acad. Sci. 3A, 119–125 (1936).

Other (5)

M. V. Berry, The diffraction of light by ultrasound (Academic, New York, 1966).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., 1966), pp. 350–359.

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Figures (6)

FIG. 1
FIG. 1

Coordinate system for light-sound interaction.

FIG. 2
FIG. 2

Schematic representation of partial coherence measurements.

FIG. 3
FIG. 3

A wedge type of double slits S1 and S2 combined with a fixed single slit S3. The double slits are movable along the y axis. The x axis denotes the propagation direction of the ultrasonic wave. The resultant double square pinholes are shown in (b).

FIG. 4
FIG. 4

Examples of interference fringes obtained and their intensity distributions. Pictures (a) and (b) are taken without and with the ultrasonic wave, respectively.

FIG. 5
FIG. 5

Degree of coherence |μ12| dependent on Raman-Nath parameter v. The solid lines and dotted points indicate the theoretical and measured results, respectively. The top figure shows the results obtained with no ultrasonic wave.

FIG. 6
FIG. 6

Demonstration of laser speckle reduction. The picture (a) is taken with no ultrasonic wave. The ultrasonic power in (b) is the same as in (c) but the picture (c) is taken by making the laser beam cross the ultrasonic column three times.

Equations (22)

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V ( r , t ) = V ( r , t ) exp [ iLk μ ( r , t ) ] ,
μ ( r , t ) = μ 0 + m = 1 Δ μ a m sin m ( Ω t K r δ ) ,
Γ ( P 1 , P 2 ; t 1 , t 2 ) = Γ ( P 1 , P 2 ; t 1 , t 2 ) G ( P 1 , P 2 ; t 1 , t 2 ; Δ μ ) ,
G ( P 1 , P 2 ; t 1 , t 2 ; Δ μ ) = exp { iLk [ μ ( r 1 , t 1 ) μ ( r 2 , t 2 ) ] }
G = m = 1 n = 1 exp ( i υ ) [ a n sin ( n F 2 ) a m sin ( m F 1 ) ] ,
υ = k Δ μ L , F 1 = Ω t 1 K r 1 δ , F 2 = Ω t 2 K r 2 δ .
G ( P 1 , P 2 ; τ ; υ ) = m = 1 J 0 { 2 a m υ | sin ( m / 2 ) [ K ( r 1 r 2 ) + Ω τ ] | } ,
G ( P 1 , P 2 ; τ ; υ ) = J 0 { 2 υ | sin ( 1 / 2 ) [ K ( r 1 r 2 ) + Ω τ ] | } ,
G ( p 1 , P 1 ; υ ) = G ( P 2 , P 2 ; υ ) = 1 ,
G ( P 1 , P 2 ; υ = 0 ) = 1 .
μ 12 ( υ ) = μ 12 ( υ = 0 ) G ( P 1 , P 2 ; υ ) .
| μ 12 ( υ ) | = I max I min I max + I min I 1 + I 2 2 ( I 1 I 2 ) 1 / 2 ,
| μ 12 ( υ ) | = | μ 12 ( υ = 0 ) m = 1 J 0 [ 2 a m υ | sin ( m π x / Λ ) | ] | ,
D = ( π λ L ) / ( μ 0 Λ 2 ) < 1 and D υ / 2 < 1 ,
exp ( i a j sin z j ) = q = + J q ( a j ) exp ( i q z j )
j = 1 exp ( i a j sin z j ) = q 1 q 2 = J q 1 ( a 1 ) J q 2 ( a 2 ) J q j ( a j ) × exp i ( q 1 z 1 + q 2 z 2 + + q j z j + ) ,
G = q 1 q 2 = p 1 p 2 = × J q 1 ( a 1 υ ) J q m ( a m υ ) J p 1 ( a 1 υ ) J p n ( a n υ ) × exp ( i ) m = 1 n = 1 ( m q m F 1 n p n F 2 ) .
F 1 = Ω t K r 1 δ , F 2 = Ω ( t + τ ) K r 2 δ
exp ( ist ) = { 0 for s 0 , 1 for s = 0 ,
G = q 1 q 2 = J q 1 2 ( a 1 υ ) J q m 2 ( a m υ ) exp i { m = 1 m q m [ K ( r 1 r 2 ) + Ω τ ] } .
s = J s 2 ( W ) exp ( isZ ) = J 0 [ 2 W | sin ( Z / 2 ) | ]
G = m = 1 J 0 { 2 a m υ | sin ( m / 2 ) [ K ( r 1 r 2 ) + Ω τ ] | } .