Abstract

In a previous article we have shown that the two speckle patterns produced from the same rough surface illuminated by two coherent plane waves under two different angles of incidence are correlated. The correlation depends on the surface roughness. In this paper a method is described where the rough surface is illuminated simultaneously by the two plane waves. The ensemble-averaged coherence function, that is, the correlation function, of the scattered field is measured by using a two-waves interferometer. This affords a real-time measurement of the surface roughness in the range of large roughness (σ > λ). The theoretical calculations have been performed for a normally distributed surface. The experimental results are in good agreement with theory. We describe the optical arrangement of an instrument based on this principle.

© 1976 Optical Society of America

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References

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  1. J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15, 101–111 (1968).
  2. G. Tribillon, “Correlation entre deux speckles obtenus avec deux longueurs d’onde. Application à la mesure de la rugositè moyenne,” Opt. Commun. 11, 172–174 (1974).
    [CrossRef]
  3. J. A. Mendez and M. L. Roblin, “Utilisation des franges d’interférence en lumière diffuse pour l’étude de l’état de surface d’un diffuseur,” Opt. Commun. 13, 142–147 (1975).
    [CrossRef]
  4. D. Léger, E. Mathieu, and J. C. Perrin, “Optical surface roughness determination using speckle correlation technique,” Appl. Opt. 14, 872–877 (1975).
    [CrossRef] [PubMed]
  5. J. N. Butters, “Electronic speckle pattern interferometry,” in Proceedings of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April 1975, p. 403–411.
  6. P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces” (Pergamon, New York, 1963).
  7. K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
    [CrossRef]
  8. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  9. D. Léger, “Deux méthodes de mesure de rugosité par corrélation de speckle,” Thesis, University of Orsay, France (1976).

1975 (2)

J. A. Mendez and M. L. Roblin, “Utilisation des franges d’interférence en lumière diffuse pour l’étude de l’état de surface d’un diffuseur,” Opt. Commun. 13, 142–147 (1975).
[CrossRef]

D. Léger, E. Mathieu, and J. C. Perrin, “Optical surface roughness determination using speckle correlation technique,” Appl. Opt. 14, 872–877 (1975).
[CrossRef] [PubMed]

1974 (1)

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longueurs d’onde. Application à la mesure de la rugositè moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

1970 (1)

K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
[CrossRef]

1968 (1)

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15, 101–111 (1968).

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Beckmann, P.

P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces” (Pergamon, New York, 1963).

Burch, J. M.

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15, 101–111 (1968).

Butters, J. N.

J. N. Butters, “Electronic speckle pattern interferometry,” in Proceedings of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April 1975, p. 403–411.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Léger, D.

D. Léger, E. Mathieu, and J. C. Perrin, “Optical surface roughness determination using speckle correlation technique,” Appl. Opt. 14, 872–877 (1975).
[CrossRef] [PubMed]

D. Léger, “Deux méthodes de mesure de rugosité par corrélation de speckle,” Thesis, University of Orsay, France (1976).

Mathieu, E.

Mendez, J. A.

J. A. Mendez and M. L. Roblin, “Utilisation des franges d’interférence en lumière diffuse pour l’étude de l’état de surface d’un diffuseur,” Opt. Commun. 13, 142–147 (1975).
[CrossRef]

Nagata, K.

K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
[CrossRef]

Nishiwaki, J.

K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
[CrossRef]

Perrin, J. C.

Roblin, M. L.

J. A. Mendez and M. L. Roblin, “Utilisation des franges d’interférence en lumière diffuse pour l’étude de l’état de surface d’un diffuseur,” Opt. Commun. 13, 142–147 (1975).
[CrossRef]

Spizzichino, A.

P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces” (Pergamon, New York, 1963).

Tokarski, J. M. J.

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15, 101–111 (1968).

Tribillon, G.

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longueurs d’onde. Application à la mesure de la rugositè moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

Yoshida, N.

K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
[CrossRef]

Appl. Opt. (1)

Jpn. J. Appl. Phys. (1)

K. Nagata, N. Yoshida, and J. Nishiwaki, “Ensemble-averaged coherence function of light reflected from rough surface. Determination of its correlation length,” Jpn. J. Appl. Phys. 9, 505–515 (1970).
[CrossRef]

Opt. Acta (1)

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15, 101–111 (1968).

Opt. Commun. (2)

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longueurs d’onde. Application à la mesure de la rugositè moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

J. A. Mendez and M. L. Roblin, “Utilisation des franges d’interférence en lumière diffuse pour l’étude de l’état de surface d’un diffuseur,” Opt. Commun. 13, 142–147 (1975).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Other (3)

D. Léger, “Deux méthodes de mesure de rugosité par corrélation de speckle,” Thesis, University of Orsay, France (1976).

J. N. Butters, “Electronic speckle pattern interferometry,” in Proceedings of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April 1975, p. 403–411.

P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces” (Pergamon, New York, 1963).

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Figures (10)

FIG. 1
FIG. 1

A rotation δθ1 of the incident plane wave A induces a rotation δθ2 = (cosθ1/cosθ2)δθ1 of the speckle pattern scattered by the rough surface in the direction θ2.

FIG. 2
FIG. 2

The interferometer I1 divides the illuminating plane wave. The interferometer I2 superimposes the two speckle patterns corresponding with the two incident waves.

FIG. 3
FIG. 3

Plotting of the theoretical visibility of the fringes as a function of δθ2 for given θ1, θ2, and δθ1.

FIG. 4
FIG. 4

Experimental setup.

FIG. 5
FIG. 5

Equal inclination fringes observed in the plane of the detector. Their visibility decreases when the angle is increasing [with δθ2 = (cosθ1/cosθ2)δθ1].

FIG. 6
FIG. 6

Experimental results and theoretical curves for the decrease of the visibility V by increasing the angle for different values of σ. —— Theoretical curves: V = 1 2 exp { - 2 [ 2 π / λ ) sin θ 1 σ δ θ 1 ] 2 }; experimental points: □, σ = 32 μm; +, σ = 16 μm; ○, σ = 8 μm.

FIG. 7
FIG. 7

Arrangement for obtaining an autoadjusted interferometer I2: the mirrors M1 and M2 are on the same mounting.

FIG. 8
FIG. 8

Variation of δθ2 in the field of view.

FIG. 9
FIG. 9

Introducing prisms P and P′ into the interferometer I2, we obtain a field-compensated interferometer.

FIG. 10
FIG. 10

Experimental results showing the action of the compensation for δθ2 = 1° 50′.

Equations (39)

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δ θ 2 = ( cos θ 1 / cos θ 2 ) δ θ 1 .
V = exp { - [ ( 2 π / λ ) σ sin θ 1 δ θ 1 ] 2 } .
E ( θ 2 ) = A ( θ 2 ) + A ( θ 2 ) e - i φ + A ( θ 2 + δ θ 2 ) + A ( θ 2 + δ θ 2 ) e - i φ 2 ,
E ( θ 2 ) = A ( θ 2 ) 2 + A ( θ 2 ) 2 + A ( θ 2 + δ θ 2 ) 2 + A ( θ 2 + δ θ 2 ) 2 + Re [ A ( θ 2 ) A * ( θ 2 + δ θ 2 ) + A ( θ 2 ) A * ( θ 2 + δ θ 2 ) ] + Re { [ A ( θ 2 ) A * ( θ 2 ) + A ( θ 2 ) A * ( θ 2 + δ θ 2 ) + A ( θ 2 + δ θ 2 ) A * ( θ 2 ) + A ( θ 2 + δ θ 2 ) A * ( θ 2 + δ θ 2 ) ] e i φ } .
A ( θ 2 ) 2 = A ( θ 2 ) 2 = A ( θ 2 + δ θ 2 ) 2 = A ( θ 2 + δ θ 2 ) 2 = I 0 .
A ( θ 2 ) = ( A 0 F 2 / 2 L ) - L + L exp { i [ V x x + V z ζ ( x ) ] } d x ,
V x = ( 2 π / λ ) ( sin θ 1 - sin θ 2 ) , V 2 = ( 2 π / λ ) ( cos θ 1 - cos θ 2 ) .
A ( θ 2 ) A * ( θ 2 + δ θ 2 ) = ( A 0 F 2 2 L ) 2 - L + L e i V z ζ ( x 1 ) e i V x x 1 d x 1 × - L + L e - i V z ζ ( x 2 ) e - i V x x 2 d x 2 ,
A ( θ 2 ) A * ( θ 2 + δ θ 2 ) = ( A 0 F 2 2 L ) 2 - L + L e i [ V z ζ ( x 1 ) - V z ζ ( x 2 ) ] e i ( V x x 1 - V x x 2 ) × d x 1 d x 2 .
A ( θ 2 ) A * ( θ 2 + δ θ 2 ) = ( A 0 F 2 2 L ) 2 - L + L exp ( i Δ V x x ) d x × - L + L χ ( V z , - V z ) exp ( i V x τ ) d τ .
Y = - L + L χ ( V z , - V z ) exp ( i V x τ ) d τ .
χ ( V z , - V z ) = exp { - ( σ 2 / 2 ) [ V z 2 + V z 2 - 2 C ( τ ) V z V z ] } .
Y = exp { - [ ( σ 2 / 2 ) Δ V z 2 ] } × - L + L exp { - σ 2 V z V z [ 1 - C ( τ ) ] } exp ( i V x τ ) d τ .
Y = exp { - [ ( σ 2 / 2 ) Δ V z 2 ] } × - exp [ - σ 2 V z V z ( τ 2 / T 2 ) ] exp ( i V x τ ) d τ .
- exp ( - a t 2 ) exp ( i b t ) d t = π / a exp ( - b 2 / 4 a )             ( a > 0 )
A ( θ 2 ) A * ( θ 2 + δ θ 2 ) = ( A 0 F 2 2 L ) 2 2 L sinc ( Δ V x L ) × exp ( - σ 2 2 Δ V z 2 ) T σ ( π V z V z ) 1 / 2 exp [ - V x 2 T 4 V z V z σ 2 ] ,
Δ V x = ( 2 π / λ ) [ cos ( θ 1 ) δ θ 2 - cos ( θ 2 ) δ θ 2 ] ,
Δ V z = ( 2 π / λ ) [ sin ( θ 1 ) δ θ 1 + sin ( θ 2 ) δ θ 2 ] ;
I 0 = ( A 0 F 2 2 L ) 2 2 L T σ ( π V z 2 ) 1 / 2 exp ( - V x 2 T 4 V z 2 ) ,
A ( θ 2 ) A * ( θ 2 + δ θ 2 ) = I 0 sinc ( Δ V x L ) exp [ ( - σ 2 / 2 ) Δ V z 2 ] .
δ θ 2 = ( cos θ 1 / cos θ 2 ) δ θ 1 .
E ( θ 2 ) = 4 I 0 + 2 A ( θ 2 ) A * ( θ 2 + δ θ 2 ) cos φ .
V = A ( θ 2 ) A * ( θ 2 + δ θ 2 ) 2 I 0 .
V = 1 2 exp [ - 1 2 ( 2 π λ sin ( θ 1 + θ 2 ) cos θ 2 σ δ θ 1 ) 2 ] .
δ θ 2 + Δ ( δ θ 2 ) = [ cos θ 1 / cos ( θ 2 + Δ θ 2 ) ] δ θ 1
Δ ( δ θ 2 ) / δ θ 2 = tan ( θ 2 ) Δ θ 2 .
V = V max exp { - 2 [ ( 2 π / λ ) sin θ σ δ θ ] 2 } ,
Δ i = Δ i + D i Δ i .
Δ θ 2 = Δ θ 2 + D i Δ θ 2 .
Δ θ 2 = Δ θ 2 .
Δ θ 2 = Δ θ 2 + D i Δ θ 2 .
D = D + D i δ θ 2 2 .
Δ θ 2 = Δ θ 2 + ( D i + 2 D i 2 δ θ 2 2 ) Δ θ 2 .
Δ θ 2 = Δ θ 2 ( 1 + 2 D i 2 δ θ 2 2 ) .
δ θ 2 + Δ θ 2 = δ θ 2 + Δ θ 2 .
δ θ 2 - δ θ 2 = 2 D i 2 δ θ 2 2 Δ θ 2 .
2 D i 2 δ θ 2 2 Δ θ 2 = tan θ 2 δ θ 2 Δ θ 2 .
2 D i 2 = 2 tan θ 2
tan θ 2 = sin A cos 2 A ( 1 - n 2 sin 2 A ) 1 / 2 ( n 2 - 1 n ) ,