Abstract

The application of the speckle phenomenon to the analysis of in-plane translations and oscillations is first reviewed briefly. Then a practical method of investigating out-of-plane rotations (tilts) even in the presence of in-plane movements is studied. This goal can be achieved by recording the speckle patterns in the Fourier transform plane before and after the tilt or as a time-average exposure for an oscillating object. Young’s fringes related to the tilts are observed in the Fraunhofer diffraction pattern when the developed photographic plate is illuminated with coherent light. This leads to a very simple engineering tool for the analysis of movements. Theoretical and experimental results will be shown.

© 1972 Optical Society of America

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References

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  1. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  2. H. J. Tiziani, thesis, London University (1967).
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  4. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  5. J. A. Leendertz, J. Phys. E. 3, 214 (1970).
    [CrossRef]
  6. J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).
  7. S. Debrus, M. Françon, M. May, in Proceedings of ICO Conference, Reading (1969).
  8. J. N. Butters, J. A. Leendertz, J. Phys. E. 4, 277 (1971).
    [CrossRef]
  9. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
    [CrossRef]
  10. H. J. Tiziani, Opt. Acta 18, 891 (1971).
    [CrossRef]
  11. H. J. Tiziani, Optik 34, 442 (1972).
  12. U. Köpf, Optik 33, 517 (1971).
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    [CrossRef]
  14. E. Eliasson, F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
    [CrossRef]
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  18. H. J. Tiziani, Opt. Commun., July1972.

1972 (2)

H. J. Tiziani, Optik 34, 442 (1972).

H. J. Tiziani, Opt. Commun., July1972.

1971 (5)

E. Eliasson, F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
[CrossRef]

N. Fernelius, C. Tome, J. Opt. Soc. Am. 61, 556 (1971).
[CrossRef]

U. Köpf, Optik 33, 517 (1971).

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

J. N. Butters, J. A. Leendertz, J. Phys. E. 4, 277 (1971).
[CrossRef]

1970 (3)

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

J. A. Leendertz, J. Phys. E. 3, 214 (1970).
[CrossRef]

1969 (1)

E. Archbold et al., Nature 222, 263 (1969).
[CrossRef]

1968 (1)

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

1966 (1)

1965 (2)

Archbold, E.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

E. Archbold et al., Nature 222, 263 (1969).
[CrossRef]

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Butters, J. N.

J. N. Butters, J. A. Leendertz, J. Phys. E. 4, 277 (1971).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Debrus, S.

S. Debrus, M. Françon, M. May, in Proceedings of ICO Conference, Reading (1969).

Eliasson, E.

Ennos, A. E.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

Fernelius, N.

N. Fernelius, C. Tome, J. Opt. Soc. Am. 61, 556 (1971).
[CrossRef]

Françon, M.

S. Debrus, M. Françon, M. May, in Proceedings of ICO Conference, Reading (1969).

Goldfischer, L. I.

Hopkins, H. H.

H. H. Hopkins, H. J. Tiziani, in Proceedings of the International Symposium on Holography, Besançon (1970).

Köpf, U.

U. Köpf, Optik 33, 517 (1971).

Kozma, A.

Leendertz, J. A.

J. N. Butters, J. A. Leendertz, J. Phys. E. 4, 277 (1971).
[CrossRef]

J. A. Leendertz, J. Phys. E. 3, 214 (1970).
[CrossRef]

May, M.

S. Debrus, M. Françon, M. May, in Proceedings of ICO Conference, Reading (1969).

Mottier, F. M.

Powel, R. L.

Stetson, K. A.

Tiziani, H. J.

H. J. Tiziani, Optik 34, 442 (1972).

H. J. Tiziani, Opt. Commun., July1972.

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

H. H. Hopkins, H. J. Tiziani, in Proceedings of the International Symposium on Holography, Besançon (1970).

H. J. Tiziani, thesis, London University (1967).

Tokarski, J. M. J.

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Tome, C.

N. Fernelius, C. Tome, J. Opt. Soc. Am. 61, 556 (1971).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Phys. E. (2)

J. A. Leendertz, J. Phys. E. 3, 214 (1970).
[CrossRef]

J. N. Butters, J. A. Leendertz, J. Phys. E. 4, 277 (1971).
[CrossRef]

Nature (1)

E. Archbold et al., Nature 222, 263 (1969).
[CrossRef]

Opt. Acta (4)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Opt. Commun. (1)

H. J. Tiziani, Opt. Commun., July1972.

Optik (2)

H. J. Tiziani, Optik 34, 442 (1972).

U. Köpf, Optik 33, 517 (1971).

Other (3)

S. Debrus, M. Françon, M. May, in Proceedings of ICO Conference, Reading (1969).

H. J. Tiziani, thesis, London University (1967).

H. H. Hopkins, H. J. Tiziani, in Proceedings of the International Symposium on Holography, Besançon (1970).

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Figures (10)

Fig. 1
Fig. 1

Photograph of the tuning fork.

Fig. 2
Fig. 2

Photographs of the fringes obtained in the FDP of a time-average exposure of a speckle pattern of the tuning fork recorded in the image plane (a) at the center of the symmetrical tuning fork where ρ01 = 9.5 μm, (b) at position 2 where ρ02 = 5.4 μm.

Fig. 3
Fig. 3

Fringes displayed in the FDP when the central portion of the image with speckle patterns was illuminated. The image was recorded in the time average before and after a lateral displacement of 5 μm (ρ01 = 9.5 μm).

Fig. 4
Fig. 4

Arrangement to record the speckle patterns in the Fourier transform plane.

Fig. 5
Fig. 5

Photographs of the fringes obtained in the FDP from the speckle patterns recorded in the Fourier transform plane of an Al plate, illuminated by a plane wave (β = 45°). (a) Doubly exposed speckle patterns with a tilt γξ = 60 see of arc introduced between the two exposures (γξγ in Fig. 4), (b) two doubly exposed speckle patterns with a tilt γξ = 60 see of arc between the first and second exposure and a tilt γη = 60 see of arc perpendicular to γξ between the second and third exposure, (c) microdensitometer trace of (a) slightly below the center.

Fig. 6
Fig. 6

The inverse of the fringe spacings obtained in the FDP is plotted against the tilt angle for tilts γξ(γξγ in Fig. 4) and tilts γη perpendicular to γξ marked by 0. The results for tilts γξ with an additional translation of 40 μm are marked by ⊗. (β = 45°).

Fig. 7
Fig. 7

Photographs of the fringes displayed in the FDP for the time-average speckle patterns recorded in the Fourier transform plan of region 2 in Fig. 1. The resonant frequency was about 1050 Hz, and the voltages applied to the coils were U = 1.0 V, 1.5 V, and 2.5 V for Fig. 7(a), (b), and (c) yielding tilts in the azimuth perpendicular to the fringes; (d) and (e) show fringes obtained in the FDP of the doubly exposed speckle patterns (stationary and time-average) for U = 2.5 V and 3.5 V.

Fig. 8
Fig. 8

Amplitudes of the mechanical oscillations as function of the voltages applied for (a) in-plane movements of the central portion 1 and (b) tilts γ0 of position 2.

Fig. 9
Fig. 9

Fringes in the FDP of the speckling recorded in the image plane for two incident light beams (position 2 in Fig. 1) for U = 2.5 V (β1 ≈ −β2).

Fig. 10
Fig. 10

Example of the fringes in the FDP obtained from a time-average speckle pattern recorded in the Fourier transform plane, of an oscillating quartz.

Equations (13)

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Δ ξ = λ f 1 / M p ,             ξ 0 = 0.76 ( λ f 1 / M p 1 ) ,
A 0 ( ξ ) = A ( ξ ) exp [ i ( 2 π / λ ) ξ sin β ] .
[ Q P 2 ] = 2 ( ξ - ξ 0 ) sin ( γ / 2 ) cos [ β - ( γ / 2 ) ] ( ξ - ξ 0 ) γ cos β
A ( ξ ) exp [ i 2 π ξ ( sin β / λ ) ] exp [ i 2 π ( γ / λ ) ξ ] × exp [ i 2 π ( ξ - ξ 0 ) ( γ / λ ) cos β ] .
a [ x - ( sin β / λ ) - ( γ / λ ) ( 1 + cos β ) ] exp [ i 2 ( π ξ 0 / λ ) γ cos β ] ,
i 1 ( x ) = a x - ( sin β / λ ) 2 + a x - ( sin β / λ ) - ( γ / λ ) ( 1 + cos β ) ] 2 .
i 2 ( x ) t = a [ x - ( sin β / λ ) - ( 1 + cos β ) ( γ 0 / λ ) cos 2 π ν t ] 2 t ,
A ( u ) = T ( x ) ( exp i 2 π u x ) d x ,
I 1 ( u ) = 2 A ( u ) A ( - u ) 2 × { 1 + cos [ 2 π u ( γ / λ ) ( 1 + cos β ) ] } ,
I 2 ( u ) = A ( u ) A * ( - u ) 2 J 0 2 [ 2 π ( γ 0 / λ ) u ( 1 + cos β ) ] .
γ = λ f 1 / f ( 1 + cos β ) Δ ξ
γ 0 = 0.76 λ f 1 / f ( 1 + cos β ) a 1 .
I 3 ( u ) = A ( u ) A * ( - u ) 2 × K + J 0 [ 2 π ( γ 0 / λ ) u ( 1 + cos β ) ] 2 ,

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