Abstract

The averaged intensity distribution of the diffraction spectrum of light transmitted by a superimposed ultrasonic wave and diffuser is investigated in the far field. The diffuser plate makes each spectral component broaden owing to its random nature. The intensity at an observation plane is represented as a summation of each component of the diffraction spectrum. A measure to distinguish the zeroth-order spectral component from the first-order component is given under the assumption that the diffuser obeys Gaussian statistics. It is suggested theoretically and experimentally that the roughness parameter of the diffuser can be obtained from the measurement of the ratio between the peak intensities of the zeroth- and first-order spectra. The advantage of using an ultrasonic wave is that its wavelength can be utilized as a marker for the determination of the roughness parameter.

© 1979 Optical Society of America

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References

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  1. See, for example, R. W. Damon, W. T. Maloney, and D. H. McMahon, “Interaction of Light with Ultrasound: Phenomena and Applications,” in Physical Acoustics Principles and Methods Vol. 7, edited by M. P. Mason and R. N. Thurston (Academic, New York, 1970).
  2. C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I–IV”, 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).
  3. See, for example, P. Beckmann, “Scattering of Light by Rough Surfaces,” in Progress in Optics, Vol. 6, edited by E. Wolf (North-Holland, Amsterdam, 1967).
    [Crossref]
  4. Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975).
  5. J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 15, edited by E. Wolf (North-Holland, Amsterdam, 1976).
  6. See, for example, H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image contrast,” J. Opt. Soc. Am. 66, 1217–1222 (1976).H. Fujii and T. Asakura, “Development of laser speckle and its application to surface inspection,” Appl. Opt. 15, 180–183 (1977).
    [Crossref]
  7. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 507.
  8. Y. Ohtsuka, “Modulation effects of a sound wave on the mutual coherence function of light,” Optics Commun. 17, 234–237 (1976).
    [Crossref]
  9. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 518.
  10. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 182.
  11. N. Takai, “Statistics of dynamic speckle produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
    [Crossref]

1976 (2)

1974 (1)

N. Takai, “Statistics of dynamic speckle produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

1935 (1)

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I–IV”, 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).

Asakura, T.

Beckmann, P.

See, for example, P. Beckmann, “Scattering of Light by Rough Surfaces,” in Progress in Optics, Vol. 6, edited by E. Wolf (North-Holland, Amsterdam, 1967).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 507.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 518.

Dainty, J. C.

J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 15, edited by E. Wolf (North-Holland, Amsterdam, 1976).

Damon, R. W.

See, for example, R. W. Damon, W. T. Maloney, and D. H. McMahon, “Interaction of Light with Ultrasound: Phenomena and Applications,” in Physical Acoustics Principles and Methods Vol. 7, edited by M. P. Mason and R. N. Thurston (Academic, New York, 1970).

Fujii, H.

Maloney, W. T.

See, for example, R. W. Damon, W. T. Maloney, and D. H. McMahon, “Interaction of Light with Ultrasound: Phenomena and Applications,” in Physical Acoustics Principles and Methods Vol. 7, edited by M. P. Mason and R. N. Thurston (Academic, New York, 1970).

McMahon, D. H.

See, for example, R. W. Damon, W. T. Maloney, and D. H. McMahon, “Interaction of Light with Ultrasound: Phenomena and Applications,” in Physical Acoustics Principles and Methods Vol. 7, edited by M. P. Mason and R. N. Thurston (Academic, New York, 1970).

Nath, N. S. N.

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I–IV”, 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 a sound wave on the mutual coherence function of light,” Optics Commun. 17, 234–237 (1976).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 182.

Raman, C. V.

C. V. Raman and N. S. N. Nath, “The diffraction of light by high frequency sound waves: part I–IV”, 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).

Shindo, Y.

Takai, N.

N. Takai, “Statistics of dynamic speckle produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 518.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 507.

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

N. Takai, “Statistics of dynamic speckle produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[Crossref]

Optics Commun. (1)

Y. Ohtsuka, “Modulation effects of a sound wave on the mutual coherence function of light,” Optics Commun. 17, 234–237 (1976).
[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–IV”, 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 (7)

See, for example, P. Beckmann, “Scattering of Light by Rough Surfaces,” in Progress in Optics, Vol. 6, edited by E. Wolf (North-Holland, Amsterdam, 1967).
[Crossref]

Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Heidelberg, 1975).

J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 15, edited by E. Wolf (North-Holland, Amsterdam, 1976).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 518.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 182.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 507.

See, for example, R. W. Damon, W. T. Maloney, and D. H. McMahon, “Interaction of Light with Ultrasound: Phenomena and Applications,” in Physical Acoustics Principles and Methods Vol. 7, edited by M. P. Mason and R. N. Thurston (Academic, New York, 1970).

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Figures (4)

Fig. 1
Fig. 1

Optical system under consideration.

Fig. 2
Fig. 2

Average intensity distribution in the far field. Each diffraction spectrum is denoted by a dotted line.

Fig. 3
Fig. 3

Photographs of diffraction patterns. Zeṙoth and ± first order spectra are indicated by 0 and ± 1, respectively.

Fig. 4
Fig. 4

Traced intensity distributions of the diffraction patterns. Each diffraction order is denoted by number under the traced curves.

Equations (30)

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μ ( x , t ) = μ 0 + Δ μ sin ( Ω t K x δ ) ,
V ( x , t ) = A exp [ ikL μ ( x , t ) ] ,
J ( P 1 , P 2 ) = lim T ( 1 / 2 T ) T T V ( x 1 , t ) V * ( x 2 , t ) d t = | A | 2 lim T ( 1 / 2 T ) T T exp { i υ [ sin ( Ω t K x 2 δ ) sin ( Ω t K x 1 δ ) ] } d t ,
exp ( i υ sin B ) = n = + J n ( υ ) exp ( i n υ B ) ,
J ( P 1 , P 2 ) = | A | 2 p = + q = + J p ( υ ) J q ( υ ) lim T ( 1 / 2 T ) × T T exp { i υ [ Ω t ( p q ) p ( K x 2 + δ ) + q ( K x 1 + δ ) ] } d t .
lim t ( 1 / 2 T ) T T exp [ i Ω t ( p q ) ] d t = { 0 for p q 1 for p = q ,
J ( P 1 , P 2 ) = | A | 2 p = + J p 2 ( υ ) exp [ ipK ( x 1 x 2 ) ] .
J ( Q 1 , Q 2 ) = | A | 2 p = + J p 2 ( υ ) exp [ i p ( K / m ) ( y 1 y 2 ) ]
O ( y ) = exp [ i θ ( y ) ] ,
I ( z ) = + J ( Q 1 , Q 2 ) O ( y 1 ) O * ( y 2 ) × exp [ i ( k / f ) ( y 1 y 2 ) z ] d y 1 d y 2 .
I ( z ) = | A | 2 + ( p = + J p 2 ( υ ) exp [ i p ( K / m ) ( y 1 y 2 ) ] ) × [ exp i { θ ( y 1 ) θ ( y 2 ) } ] × exp { i ( k / f ) ( y 1 y 2 ) z } d y 1 d y 2 ,
I ( z ) = | A | 2 + ( p = + J p 2 ( υ ) exp [ p ( K / m ) y ] ) C ( y ) × exp { i ( k / f ) y z } d y ,
C ( y ) = + exp i [ θ ( y 1 ) θ ( y 1 y ) ] d y 1 .
I ( z ) = | A | 2 p = + J p 2 ( υ ) + { exp [ i p ( K / m ˙ ) y ] } × C ( z ) exp [ i ( k / f ) y z ] d y ,
C ( y ) = + exp i [ θ ( y 1 ) θ ( y 1 y ) ] d y 1 ,
ρ = exp ( | y 1 y 2 | 2 / α 2 ) ,
C ( y ) = exp { θ 2 [ 1 ρ ( y ) ] } .
C ( y ) exp [ ( θ 2 / α 2 ) y 2 ] ,
θ 2 > π .
θ 2 = ( 2 π Δ n σ ) / λ ,
σ > ( λ / 2 Δ n ) .
I ( z ) = | A | 2 p = + J p 2 ( υ ) × + { exp [ i p ( K / m ) y ] } { exp [ ( θ 2 / α 2 ) y 2 ] } × exp [ i ( k / f ) z y ] d y .
I ( z ) = | A | 2 p = + J p 2 ( υ ) × + { exp [ i p ( K / m ) y ] } exp [ i ( k / f ) z y ] d y + { exp [ ( θ 2 / α 2 ) y 2 ] } exp { i ( k / f ) z y } d y ,
I ( z ) = ( π / θ 2 ) 1 / 2 | α | | A | 2 p = + J p 2 ( υ ) × exp [ ( α 2 / 4 θ 2 ) ( k / f ) 2 ( z z p ) 2 ] ,
z p = p ( K f / m k ) .
[ I 0 ( 0 ) + I 1 ( 0 ) ] ( 1 / 2 ) > [ I 0 ( z 1 / 2 ) + I 1 ( z 1 / 2 ) ] .
( 2 m Λ / π ) ln 2 < | α | / θ 2 .
R = I 1 ( z 1 ) + I 0 ( z 1 ) + I 2 ( z 1 ) I 0 ( 0 ) + I 1 ( 0 ) + I 1 ( 0 ) = J 1 2 ( υ ) + J 0 2 ( υ ) B + J 2 2 ( υ ) B J 0 2 ( υ ) + [ J 1 2 ( υ ) + J 1 2 ( υ ) ] B ,
B = exp { ( α 2 / θ 2 ) ( π / m Λ ) 2 } .
| α | θ 2 = m Λ π ( ln [ 1 + [ J 2 2 ( υ ) / J 0 2 ( υ ) ] R { [ J 1 2 ( υ ) / J 0 2 ( υ ) ] + [ J 1 2 ( υ ) / J 0 2 ( υ ) ] } R [ J 1 2 ( υ ) / J 0 2 ( υ ) ] ] ) 1 / 2 .