Abstract

A novel, to our knowledge, method for the measurement of angular displacement for arbitrarily shaped objects is presented in which the angular displacement is perpendicular to the optical axis. The method is based on Fourier-transforming the scattered field from a single laser beam that illuminates the target. The angular distribution of the light field at the target is linearly mapped on a linear image sensor placed in the Fourier plane. Measuring this displacement facilitates the determination of the angular displacement of the target. It is demonstrated both theoretically and experimentally that the angular-displacement sensor is insensitive to object shape and target distance if the linear image sensor is placed in the Fourier plane. A straightforward procedure for positioning the image sensor in the Fourier plane is presented. Any transverse or longitudinal movement of the target will give rise to partial speckle decorrelation, but it will not affect the angular measurement. Furthermore, any change in the illuminating wavelength will not affect the angular measurements. Theoretically and experimentally it is shown that the method has a resolution of 0.3 mdeg (≈5 μrad) for small angular displacements, and methods for further improvement in resolution is discussed. No special surface treatment is required for surfaces giving rise to fully developed speckle. The effect of partially developed speckle is considered both theoretically and experimentally.

© 1998 Optical Society of America

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References

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  1. I. Yamaguchi, T. Fujita, “Laser speckle rotary encoder,” Appl. Opt. 28, 4401–4406 (1989).
    [CrossRef] [PubMed]
  2. N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
    [CrossRef]
  3. J. C. Marron, K. Schroeder, “Speckle from rough rotating objects,” Appl. Opt. 27, 4279–4287 (1988), where numerous references to previous research are cited therein.
    [CrossRef] [PubMed]
  4. X. Dai, O. Sasaki, J. E. Greivenkamp, T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380–6388 (1995).
    [CrossRef] [PubMed]
  5. H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
    [CrossRef]
  6. H. T. Yura, S. G. Hanson, “Optical beam propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.
  8. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [CrossRef]
  9. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef] [PubMed]
  10. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986).
    [CrossRef]
  11. M. S. Beck, A. Pla̧skowski, Cross Correlation Flowmeters—Their Design and Application (Adam Hilger, Bristol, UK, 1987).
  12. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “Effects of target structure on the performance of laser time-of-flight velocimeter systems,” Appl. Opt. 36, 518–533 (1997).
    [CrossRef] [PubMed]

1997 (1)

1995 (1)

1994 (1)

1993 (1)

1989 (1)

1988 (1)

1987 (1)

1986 (1)

1981 (1)

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

1972 (1)

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Asakura, T.

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

Beck, M. S.

M. S. Beck, A. Pla̧skowski, Cross Correlation Flowmeters—Their Design and Application (Adam Hilger, Bristol, UK, 1987).

Dai, X.

Fujita, T.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

Greivenkamp, J. E.

Grum, T. P.

Hanson, S. G.

Imam, H.

Iwai, T.

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

Marron, J. C.

Pla¸skowski, A.

M. S. Beck, A. Pla̧skowski, Cross Correlation Flowmeters—Their Design and Application (Adam Hilger, Bristol, UK, 1987).

Rose, B.

Sasaki, O.

Schroeder, K.

Sjödahl, M.

Suzuki, T.

Takai, N.

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Yamaguchi, I.

Yoshimura, T.

Yura, H. T.

Appl. Opt. (5)

Appl. Phys. B (1)

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B 26, 185–192 (1981).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Other (2)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

M. S. Beck, A. Pla̧skowski, Cross Correlation Flowmeters—Their Design and Application (Adam Hilger, Bristol, UK, 1987).

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Figures (15)

Fig. 1
Fig. 1

Fourier transform system (receiver part of an angular-displacement sensor) with a Gaussian shaped lens aperture.

Fig. 2
Fig. 2

Optical diagram for the angular-displacement sensor setup. B.S., beam splitter; f col., collimating lens.

Fig. 3
Fig. 3

Normalized intensity distribution at the linear image sensor for (a) specularly, (b) diffusely, and (c) combined specularly and diffusely scattered light, respectively.

Fig. 4
Fig. 4

Measured angular displacements and Gaussian widths of the resulting cross covariances versus the applied angular displacement for a matte aluminum shaft (fully developed speckle field).

Fig. 5
Fig. 5

Measured versus applied angular displacements in steps of 0.43 mdeg on a matte silicon surface (fully developed speckle field).

Fig. 6
Fig. 6

Total angular displacement of 6000 mdeg measured in steps of 300 mdeg.

Fig. 7
Fig. 7

Measured versus applied angular displacement for varying image-sensor positions. The target is placed in the Fourier plane (z 1 = 0 mm). The symbols represent measurements, and the solid curves represent theory. The radius of curvature is 15 mm.

Fig. 8
Fig. 8

Measured versus applied angular displacement for varying image-sensor positions. The target is displaced from the Fourier plane (z 1 = 20 mm). The symbols represent measurements, and the solid curves represent theory. The radius of curvature is 15 mm.

Fig. 9
Fig. 9

Measured versus applied angular displacement for varying image-sensor positions. The target is placed in the Fourier plane (z 1 = 0 mm). The symbols represent measurements, and the solid curves represent theory. The radius of curvature is 47.5 mm.

Fig. 10
Fig. 10

Measured versus applied angular displacement for varying image-sensor positions. The target is displaced from the Fourier plane (z 1 = 20 mm). The symbols represent measurements, and the solid curves represent theory. The radius of curvature is 47.5 mm.

Fig. 11
Fig. 11

Measured versus applied angular displacement for varying image-sensor positions. The target is displaced from the Fourier plane (z 1 = -15 mm). The symbols represent measurements, and the solid curves represent theory. The radius of curvature is 47.5 mm.

Fig. 12
Fig. 12

Measured angular displacement versus the linear target displacement on a matte aluminum shaft. The symbols represent measurements, and the solid curves represent the applied angular displacement. Rot., rotation.

Fig. 13
Fig. 13

Measured angular displacements and Gaussian widths of the resulting cross covariances versus the applied angular displacement for a highly reflective aluminum shaft (R = 5 mm), giving rise mainly to reflection. The symbols represent measurements, and the solid straight line represents theory.

Fig. 14
Fig. 14

Measured angular displacements and Gaussian widths of the resulting cross covariances versus the applied angular displacement for a stainless steel shaft (R = 0.1 mm), giving rise to partially developed speckle. The symbols represent measurements, and the solid straight line represents theory.

Fig. 15
Fig. 15

Measured angular displacements and Gaussian widths of the resulting cross covariances versus the applied angular displacement for the inside of a rough aluminum tube wall (partially developed speckle field). The symbols represent measurements, and the solid straight line represents theory.

Equations (46)

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C k = j = k N   i j i j - k - i j i j - k ,
C i Δ p x = i p x i p x + Δ p x - i p x i p x + Δ p x ,
i p x = α     d p W pix p - p x e x I p ,
α = q η h ν ,
i p x α A pix I p ,
I p =   d r U 0 r G r ,   p 2 ,
G r ,   p = ik 2 π B x B y × exp - ik 2 B x D x p x 2 - 2 xp x + A x x 2 × exp - ik 2 B y D y p y 2 - 2 yp y + A y y 2 ,
U 0 r = U i r ψ r ,
I i r = | U i r | 2 = 2 P 0 π r s 2 exp - 2 r 2 r s 2 ,
B ψ r 1 ,   r 2 = 4 k 2 r c 2 1 - exp - σ ϕ 2 × exp - r 1 - r 2 2 r c 2 + exp - σ ϕ 2 ,
ψ r = ψ r - R θ e x .
U 0 r 1 U 0 * r 2 U 0 * r 2 U 0 r 1 ,
U 0 r 1 U 0 * r 2 U 0 * r 2 U 0 r 1 = U 0 r 1 U 0 * r 2 U 0 * r 2 U 0 r 1 + U 0 r 1 U 0 * r 2 U 0 * r 2 U 0 r 1 .
C i Δ p x =   d p 1 W 2 p 1 K p 1 ,   p 1 + Δ p x e x a 2 A pix 2 K 0 ,   Δ p x e x ,
K 0 ,   Δ p x e x =   d r 1   d r 2     dr 1     d r 2 × G r 1 ,   0 G * r 2 ,   0 G r 1 ,   Δ p x e x × G * r 2 ,   Δ p x e x U 0 r 1 U 0 * r 2 × U 0 * r 2 U 0 r 1 .
K 0 ,   Δ p x e x =     d r 1 U i r 1 G r 1 ,   0     d r 2 U i * r 2 × G * r 2 ,   Δ p x e x B ψ r 1 ,   r 2 2 ,
r 2 = r 2 + R θ e x .
C i Δ p x = α P 0 A pix π B x 2 F 2 1 - exp - σ ϕ 2 2 exp - 2 Δ p x 2 ω x 2 × exp - R θ 2 r s 2 F exp - 1 ρ 0 2 F Δ p x - A x R θ 2 ,
ρ 0 = λ B x π r s ,
F = 1 + r c 2 2 r s 2 1 + A x 2 k 2 r s 4 4 B x 2 ,
ω x 2 = A x 2 r s 2 + ρ 0 2 1 + 2   r s 2 r c 2 .
1 ρ ˜ 0 2 = 1 ρ 0 2 F 1 + r c 2 r s 2 .
A x = 2 f 2 - Rz 2 - 2 z 1 z 2 fR ,
B x = f - z 1 z 2 f ,
D x = - z 1 f ,
p x 0 = A x R θ = 2 θ f - z 2 f R 2 + z 1
Δ p x 0 = 4 ln   2 1 / 2 F ρ 0 ,
ρ 0 F = λ π r s f - z 1 z 2 f F .
ρ 0 σ 2 = 4 f 2 k 2 r s 2 1 + 2 f k σ 2 2 + 4 f 2 k 2 σ 2 .
D Sp θ = exp - R θ 2 r s 2 F
D Sp θ = exp - R θ 2 1 r s 2 F + 8 f 2 R 2 σ 2 ,
α spec = Δ p x Δ x target = - z 2 f ,
var τ = 0.038 TB 3 1 ρ Corr 2 - 1
var p x 0 = 0.038 ρ 0 3 L 1 ρ Corr 2 - 1 ,
σ pix = 0.038 ρ 0 3 Δ p pix 2 L 1 ρ Corr 2 - 1 1 / 2 ,
σ θ = 0.0175 f λ 3 Lr s 3 1 ρ Corr 2 - 1 1 / 2 .
ψ r = | ψ r | exp i ϕ r
ϕ r = k 1 + cos   β h r ,
σ ϕ 2 = k 1 + cos   β 2 σ h 2 .
B ψ r 1 ,   r 2 = exp i ϕ r 1 - ϕ r 2 = exp - σ ϕ 2 1 - b h r ¯ 1 ,   r ¯ 2 ,
b h r = exp - r r h 2 ,
B ψ r = exp ( - σ ϕ 2 1 - exp - r / r h 2 ) .
B ψ r 1 - exp - σ ϕ 2 exp r 2 r c 2 + exp - σ ϕ 2 ,
c σ ϕ σ ϕ 2 1 - exp - σ ϕ 2 ,
r c = r h c σ ϕ 1 / 2 ,
B ψ r 1 ,   r 2 = 4 π k 2 1 π r c 2 1 - exp - σ ϕ 2 × exp - r 1   -   r 2 2 r c 2 + exp - σ ϕ 2 .

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