Abstract

Time-varying laser speckle in the image plane is used to measure the shape and rotation rate of rotationally symmetric objects. The analysis is based on the space–time correlation function of the intensity in the image plane, which is derived from a geometrical model for the correlation function of the optical field in the object plane. The geometrical model is developed using the method of stationary phase. Experimental results obtained using a linear detector array located in the image plane are given.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. George, “Speckle from rough, moving objects,” J. Opt. Soc. Am. 66, 1182–1193 (1976).
    [Crossref]
  2. J. C. Erdmann, R. I. Gellert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser spot,”J. Opt. Soc. Am. 66, 1194–1203 (1976).
    [Crossref]
  3. 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]
  4. A. Hayashi, Y. Kitagawa, “High-resolution rotation-angle measurement of a cylinder using speckle displacement detection,” Appl. Opt. 22, 3520–3525 (1983).
    [Crossref] [PubMed]
  5. J. C. Leader, “An analysis of the frequency spectrum of laser light scattered from moving rough objects,” J. Opt. Soc. Am. 67, 1091–1098 (1977).
    [Crossref]
  6. J. Van Bladel, “Electromagnetic fields in the presence of rotating bodies,” Proc. IEEE 64, 301–318 (1976).
    [Crossref]
  7. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  8. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  9. R. D. Kodis, “A note on the theory of scattering from an irregular surface,” IEEE Trans. Antennas Propag. AP-14, 77–82 (1966).
    [Crossref]
  10. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).
  11. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
    [Crossref]
  12. E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
    [Crossref]
  13. J. W. Goodman, Introduction to Fourier Objects (McGraw-Hill, New York, 1968).
  14. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. G. Dainty, ed (Springer-Verlag, Berlin, 1975).
    [Crossref]

1983 (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]

1977 (1)

1976 (3)

1975 (1)

E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

1968 (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[Crossref]

1966 (1)

R. D. Kodis, “A note on the theory of scattering from an irregular surface,” IEEE Trans. Antennas Propag. AP-14, 77–82 (1966).
[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]

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[Crossref]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Erdmann, J. C.

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

Gellert, R. I.

George, N.

Goodman, J. W.

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. G. Dainty, ed (Springer-Verlag, Berlin, 1975).
[Crossref]

Hayashi, A.

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]

Jakeman, E.

E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

Kitagawa, Y.

Kodis, R. D.

R. D. Kodis, “A note on the theory of scattering from an irregular surface,” IEEE Trans. Antennas Propag. AP-14, 77–82 (1966).
[Crossref]

Leader, J. C.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

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]

Van Bladel, J.

J. Van Bladel, “Electromagnetic fields in the presence of rotating bodies,” Proc. IEEE 64, 301–318 (1976).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Appl. Opt. (1)

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]

IEEE Trans. Antennas Propag. (2)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[Crossref]

R. D. Kodis, “A note on the theory of scattering from an irregular surface,” IEEE Trans. Antennas Propag. AP-14, 77–82 (1966).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. A (1)

E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centers in uniform motion,” J. Phys. A 8, L23–L28 (1975).
[Crossref]

Proc. IEEE (1)

J. Van Bladel, “Electromagnetic fields in the presence of rotating bodies,” Proc. IEEE 64, 301–318 (1976).
[Crossref]

Other (5)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, New York, 1979).

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. G. Dainty, ed (Springer-Verlag, Berlin, 1975).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Coordinate system with object rotating about the l axis. The optical axis of the imaging system is the z axis, and the object plane, or reference plane, is the ξ, η plane.

Fig. 2
Fig. 2

Cross section of an object with exaggerated roughness. The angle θ1 designates a particular point on the surface, and the angle θ2 is measured between the surface normal and the radius vector. The distance D(l) is also shown.

Fig. 3
Fig. 3

Two-lens imaging system. A reflected ray leaves the object at an angle α with respect to the optical axis.

Fig. 4
Fig. 4

An incident ray is reflected to the point ξ1. The distance P is given in Eq. (22).

Fig 5
Fig 5

Position and phase shift of a ray. Before rotation the ray contributes to the point ξ1. After a rotation of Δθ1, the ray contributes to the point ξ2 The phase and the direction of the ray also change.

Fig. 6
Fig. 6

The central lobes of the correlation coefficient as a function of detector separation. The magnitude of this function is plotted for various values of Δθ1. The envelope function (dashed line) is [1 − 2F#Ω(t2t1)].

Fig. 7
Fig. 7

Experimental measurement of the magnitude of the correlation coefficient as a function of detector separation for various values of Δθ1. Measured values are shown as dots, and the dashed lines are interpolated results. The solid lines are the theoretical predictions based on Eq. (40). Experimental parameters were r = 2.78 cm, F# = 50, M = 1.90, λ = 0.5145 μm, D = 0.0, and θ1′ = 0.0.

Fig. 8
Fig. 8

Experimental measurement of the detector separation for peak correlation coefficient as a function of Δθ1. For D = 0.0 mm measured values are shown as dots and theoretical prediction is the solid line. Experimental parameters were the same as for Fig. 7. For D = 5.0 mm measured values are denoted by crosses, and the experimental parameters were the same as in Fig. 7.

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

RATE DIFF = 2 N s Ω d / λ ,
RATE IM = N s Ω d / 4 λ F # .
Δ θ 1 = Ω ( t 2 t 1 ) .
U ( ξ , η ) = 1 4 π S { u υ n υ u n } d S .
u n = i k exp ( i k q ) cos ( q , n ) ,
υ n = exp ( i k s ) s ( i k 1 / s ) cos ( s , n ) .
U ( ξ , η ) = 1 4 π S exp [ i k ( q + s ) ] s × [ ( i k 1 / s ) cos ( s , n ) i k cos ( q , n ) ] d S .
U ( ξ , η ) = S exp [ i k f ( θ 1 , l ) ] g ( θ 1 , l ) d θ 1 d l .
f ( θ 1 , l ) θ 1 | θ 1 n , l n = f ( θ 1 , l ) l | θ 1 n , l n = 0 ,
U ( ξ , η ) = n = 0 N a n
= 1 k n = 0 N exp [ i k f ( θ 1 n , l n ) ] g ( θ 1 n , l n ) S n ,
I ( ξ , η ) = n = 0 N a n m = 0 N a m * ,
a n a m * = 1 k 2 | g ( θ 1 n , l n ) S n | 2 δ n m ,
I ( ξ , η ) = N c | a | 2 ,
| tan 2 ( θ 1 + θ 2 ) | < 1 / ( 2 F # ) ,
| 2 ( θ 1 + θ 2 ) | < 1 / ( 2 F # ) .
| Δ θ 1 | < 1 / ( 2 F # ) .
J u ( ξ 1 , η 1 , ξ 2 , η 2 ; t 1 , t 2 ) = U ( ξ 1 , η 1 ; t 1 ) U * ( ξ 2 , η 2 ; t 2 ) = 1 k 2 δ [ ξ 2 ξ 1 κ Ω ( t 2 t 1 ) ] δ ( η 2 η 1 ) × n = 0 N exp [ i k f ( θ 1 n , l n ) ] g ( θ 1 n , l n ) S n × n = 0 N exp [ i k f ( θ 1 n , l n ) + i ϕ ] g * ( θ 1 n , l n ) S n * .
J u ( ξ 1 , η 1 , ξ 2 , η 2 ; t 1 , t 2 ) = I ( ξ , η ) e i ϕ × δ [ ξ 2 ξ κ Ω ( t 2 t 1 ) ] δ ( η 2 η 1 ) ,
ξ 1 = [ r ( l ) + h ( θ 1 , l ) ] sin θ 1 + P tan 2 ( θ 1 + θ 2 )
η 1 = l + P tan 2 θ 3 ,
P = D ( l ) + r ( l ) [ r ( l ) + h ( θ 1 , l ) ] cos θ 1 .
Δ ξ = ξ 2 ξ 1 = ξ 1 θ 1 Δ θ 1
Δ η = η 2 η 1 = η 1 θ 1 Δ θ 1 ,
ξ 1 θ 1 = [ r ( l ) + h ( θ 1 , l ) ] cos θ 1 + 2 P + 2 ( θ 1 + θ 2 ) P θ 1
η 1 θ 1 = 2 θ 3 P θ 1 .
ξ 2 ξ 1 = [ 2 D ( l ) + r ( l ) ( 2 cos θ 1 ) ] Δ θ 1
η 2 η 1 = 0 .
| h ( θ 1 , l ) cos θ 1 | 4 λ F # 2
| r ( l ) sin θ 1 | 8 λ F # 3 ,
θ 1 1 / ( 4 F # ) < θ 1 < θ 1 + 1 / ( 4 F # ) ,
| r ( l ) sin θ 1 | 16 λ F # 3 .
ξ 2 ξ 1 = κ Ω ( t 2 t 1 ) = [ 2 D ( l ) + r ( l ) ( 2 cos θ 1 ) ] Ω ( t 2 t 1 ) .
ϕ = k [ ( P + Δ P ) [ 1 + 1 / ( cos 2 Δ θ 1 ) ] 2 P ] ;
ϕ = k 2 Δ P
| D ( l ) + r ( l ) ( 1 cos θ 1 ) ] < λ 2 F # 2 .
ϕ = k 2 r ( l ) sin θ 1 Δ θ 1 = k 2 Ω ( t 2 t 1 ) ξ 1 .
U ( x , y ; t ) = U ( ξ , η ; t ) Z ( x , y ; ξ , η ) d ξ d η ,
J u ( r 1 , r 2 ; t 1 , t 2 ) = U ( r 1 ; t 1 ) U * ( r 2 ; t 2 ) = J u ( ρ 1 , ρ 2 ; t 1 , t 2 ) Z ( r 1 ; ρ 1 ) Z * ( r 2 ; ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
Z ( x , y ; ξ , η ) = sinc 1 λ F # ( x M + ξ ) sinc 1 λ F # ( y M + η ) ,
| μ u ( r 1 , r 2 ; t 1 , t 2 ) | = | J u ( r 1 , r 2 ; t 1 , t 2 | ( | J u ( r 1 , r 1 ; t 1 , t 1 ) | | J u ( r 2 , r 2 ; t 2 , t 2 ) | ) 1 / 2 = | [ 1 2 F # Ω ( t 2 t 1 ) ] sinc { [ 1 2 F # Ω ( t 2 t 1 ) ] λ F # × [ x 2 x 1 M + κ Ω ( t 2 t 1 ) ] } × sinc [ ( y 2 y 1 ) M λ F # ] | ,
κ = 2 D ( l ) + r ( l ) ( 2 cos θ 1 ) .
| J u ( r 1 , r 2 ; t 1 , t 2 ) | 2 = I ( r 1 ; t 1 ) I ( r 2 ; t 2 ) I ( r 1 ; t 1 ) I ( r 2 ; t 2 ) ,
I ( r 1 ; t 1 ) = | U ( r 1 ; t 1 ) | 2 .
( x 1 x 2 ) peak = M [ 2 D ( l ) + r ( l ) ( 2 cos θ 1 ) ] Ω ( t 2 t 1 ) ,
| μ u ( r 1 , r 2 ; t 1 , t 2 ) | peak = [ 1 2 F # Ω ( t 2 t 1 ) ] ,
I = + δ ( ξ 2 ξ 1 κ Δ θ 1 ) exp ( i k 2 Δ θ 1 ξ 1 ) × sinc [ 1 λ F # ( x 1 M + ξ 1 ) ] sinc [ 1 λ F # ( x 2 M + ξ 2 ) ] d ξ 1 , d ξ 2 .
I = exp ( i k 2 Δ θ 1 ξ 1 ) sinc [ 1 λ F # ( x 1 M + ξ 1 ) ] × sinc [ 1 λ F # ( x 2 M + ξ 1 + κ Δ θ 1 ) ] d ξ 1 .
sinc [ 1 λ F # ( x 2 M + ξ 1 + κ Δ θ 1 ) ] = rect ( f x ) exp [ i 2 π f x λ F # ( x 2 M + ξ 1 + κ Δ θ 1 ) ] d f x ,
rect ( f x ) = { 1 if ½ f x ½ 0 otherwise .
I = rect ( f x ) rect ( f x 2 F # Δ θ 1 ) × exp [ i 2 π λ F # f x ( x 2 x 1 M + κ Δ θ 1 ) ] × exp ( i 2 π x 1 2 Δ θ 1 λ M ) λ F # d f x .
| I | = | ( 1 2 Δ θ 1 F # ) sinc [ 1 λ F # ( 1 2 Δ θ 1 F # ) × ( x 2 x 1 M + κ Δ θ 1 ) ] | .

Metrics