Abstract

When a rough surface illuminated by coherent light is displaced perpendicularly to the optical axis of an imaging optical system the speckle pattern in the conjugate plane is transversally displaced too. This displacement has two components. The first one is proportional to the object displacement, and the second one depends on wave-front aberrations and, consequently, is strongly related to the optical system that is used. Usually, well-corrected photographic objectives are used for the measurement of transverse displacements by double-exposure laser speckle photography. Since in well-corrected objectives aberrations tend to compensate one another, it seems that the complementary displacement of the speckle pattern, caused by aberrations, is near zero and does not affect the accuracy of the measurement. Here it is analytically shown that the compensation of spherical aberrations does not guarantee a negligible complementary displacement. From the results obtained it follows that well-corrected objectives for laser speckle photography can be regarded as a particular class of photographic objectives, since they not only yield high-quality images but also minimize complementary displacement.

© 2000 Optical Society of America

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References

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  1. T. Asakura, N. Takai, “Dynamic speckle and their application to velocity measurement of diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]
  2. Y. Fainman, J. Shamir, E. Lenz, “Static and dynamic behavior of speckle patterns described by operator algebra,” Appl. Opt. 20, 3526–3538 (1981).
    [CrossRef] [PubMed]
  3. Y. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image field for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  4. D. A. Gregory, “Basic physical principles of defocused laser speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
    [CrossRef]
  5. L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).
  6. K. J. Stetson, “The vulnerability of speckle-photography to lens aberration,” J. Opt. Soc. Am. 67, 1587–1590 (1977).
    [CrossRef]
  7. Yu. I. Rakushin, “Analysis of perturbation in the recording of speckle-photograph of a strained object,” in Applied Questions of Holography, Materials of the XIV All-Union School on Holography (Institute of Nuclear Physics of Leningrad, Leningrad, 1982), pp. 141–145 (in Russian).
  8. Yu. I. Rakushin, “Spatially local variation of accuracy in speckle photography,” Opt. Spectrosc. 6, 823–825 (1987).
  9. D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle-photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
    [CrossRef]
  10. L. Martı́-López, “Effect of primary aberrations on transverse displacement of laser speckle patterns,” Opt. Laser Technol. 28, 15–19 (1996).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  13. L. Martı́-López, “Effect of the scattering on speckle interferograms,” Zh. Tekh. Fiz. 52, 2032–2035 (1982) (in Russian).

1996 (1)

L. Martı́-López, “Effect of primary aberrations on transverse displacement of laser speckle patterns,” Opt. Laser Technol. 28, 15–19 (1996).
[CrossRef]

1987 (1)

Yu. I. Rakushin, “Spatially local variation of accuracy in speckle photography,” Opt. Spectrosc. 6, 823–825 (1987).

1986 (1)

1985 (1)

L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).

1982 (1)

L. Martı́-López, “Effect of the scattering on speckle interferograms,” Zh. Tekh. Fiz. 52, 2032–2035 (1982) (in Russian).

1981 (3)

Y. Fainman, J. Shamir, E. Lenz, “Static and dynamic behavior of speckle patterns described by operator algebra,” Appl. Opt. 20, 3526–3538 (1981).
[CrossRef] [PubMed]

T. Asakura, N. Takai, “Dynamic speckle and their application to velocity measurement of diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Y. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image field for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1977 (1)

1976 (1)

D. A. Gregory, “Basic physical principles of defocused laser speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

Asakura, T.

T. Asakura, N. Takai, “Dynamic speckle and their application to velocity measurement of diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Chiang, F. P.

Fainman, Y.

Goodman, J. W.

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

Gregory, D. A.

D. A. Gregory, “Basic physical principles of defocused laser speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

Lenz, E.

Li, D. W.

Marti´-López, L.

L. Martı́-López, “Effect of primary aberrations on transverse displacement of laser speckle patterns,” Opt. Laser Technol. 28, 15–19 (1996).
[CrossRef]

L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).

L. Martı́-López, “Effect of the scattering on speckle interferograms,” Zh. Tekh. Fiz. 52, 2032–2035 (1982) (in Russian).

Ostrovsky, Yu. I.

L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).

Rakushin, Yu. I.

Yu. I. Rakushin, “Spatially local variation of accuracy in speckle photography,” Opt. Spectrosc. 6, 823–825 (1987).

Yu. I. Rakushin, “Analysis of perturbation in the recording of speckle-photograph of a strained object,” in Applied Questions of Holography, Materials of the XIV All-Union School on Holography (Institute of Nuclear Physics of Leningrad, Leningrad, 1982), pp. 141–145 (in Russian).

Serra-Toledo, R.

L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).

Shamir, J.

Stetson, K. J.

Takai, N.

T. Asakura, N. Takai, “Dynamic speckle and their application to velocity measurement of diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Yamaguchi, Y.

Y. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image field for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

T. Asakura, N. Takai, “Dynamic speckle and their application to velocity measurement of diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

Y. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image field for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Laser Technol. (2)

D. A. Gregory, “Basic physical principles of defocused laser speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

L. Martı́-López, “Effect of primary aberrations on transverse displacement of laser speckle patterns,” Opt. Laser Technol. 28, 15–19 (1996).
[CrossRef]

Opt. Spectrosc. (1)

Yu. I. Rakushin, “Spatially local variation of accuracy in speckle photography,” Opt. Spectrosc. 6, 823–825 (1987).

Zh. Tekh. Fiz. (2)

L. Martı́-López, “Effect of the scattering on speckle interferograms,” Zh. Tekh. Fiz. 52, 2032–2035 (1982) (in Russian).

L. Martı́-López, Yu. I. Ostrovsky, R. Serra-Toledo, “About the transverse displacement of speckle-structures in two-lens optical systems,” Zh. Tekh. Fiz. 55, 929–931 (1985) (in Russian).

Other (3)

Yu. I. Rakushin, “Analysis of perturbation in the recording of speckle-photograph of a strained object,” in Applied Questions of Holography, Materials of the XIV All-Union School on Holography (Institute of Nuclear Physics of Leningrad, Leningrad, 1982), pp. 141–145 (in Russian).

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (3)

Fig. 1
Fig. 1

Optical arrangement: W, point source of coherent light; T, transparent object; L, optical system; P, real-image plane; OO, optical axis; r1r3, coordinates; p1,p2,p3, positions.

Fig. 2
Fig. 2

AB.

Fig. 3
Fig. 3

AB.

Equations (34)

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U1(r1, d0)=t(r1-d0)expj Kr122p1,
U2(r3, d0)=J1[(Kwr3)/p3](Kwr3)/p3 expj Kr322m2p2U1r3m, 0 δ(r3-md0)  F{exp[jΦa(r1+d0,r2+d1)]},
d2=md0+dc,
ARRD=|dc||md0|.
ARRD0.005.
01πw20w02π[Φa(r1, r2)-Φmean(r1)]2r2dr2dθπ249,
Φmean(r1)=(1/πw2)0w02πΦa(r1, r2)r2dr2dθ,
01πw20w02π[Φa(r1+d0,r2+d1)-Φmean(r1+d0)]2r2d r2dθπ249,
Φa(r1, r2)=Φa(r2)=α0f 2 r22+α1f 4 r24+α2f 6 r26+α3f 8 r28=q=03αqf 2(q+1) r22(q+1),
0112α02+445 g4α12+9112 g8α22+16225 g12α32+16 g2α0α1+320 g4α0α2+215 g6α0α3+16 g6α1α2+320 g10α2α3+16105 g8α1α3π249g4,
Φa(r2+d1)=Φa(r2)+P(r2, d1)+Q(d1)r2  d1,
P(r2, d1)
=2r22q=02k=0q(2+q)×αq+1f 2(q+2) (r22+d12)q-kd12k×r2d1+22α1f 4+3 α2f 6 (r22+d12)+6 α3f 8 (r22+d12)2×(r2d1)2+23α2f 6+4 α3f 8 (r22+d12)(r2d1)3+24α3f 8 (r2d1)4+d12q=03k=0qαqf 2(q+1) (r22+d12)q-kr22k,
Q(d1)
=2 α0f 2+4 α1f 4 d12+6 α2f 6 d14+8 α3f 8 d16=2k=03(k+1) αkf 2(k+1) d12k.
U2(r3, d0)=J1(Kwr3/p3)Kwr3/p3 expj Kr322m2p2U1r3m, 0 F{exp[jΦa(r2)+jP(r2, d1)]} δ(r3-md0-dc),
dc=λp32π Q(d1)d1dc=λp3πfk=03(k+1) αkf 2k+1 d12k+1
ARRD=λp32π|Q(d1)d1||md0|=λπf 2p2p2p1+1×k=03αk(k+1)p2p1+12kd0f2k.
-0.005 πf 2λp2p2p1+1-1
k=03αk(k+1)p2p1+12kd0f2k0.005 πf 2λp2p2p1+1-1.
k=03αk(k+1)p2p1+12kd0f2k
=-0.005 πf 2λp2p2p1+1-1,
k=03αk(k+1)p2p1+12kd0f2k
=0.005 πf 2λp2p2p1+1-1.
-12π7 g-2α012π7 g-2.
ARRD=λπf 2p2p2p1+1α012λ7f 2 g-2p2p2p1+1.
α0=πλΔp3 f 2p32.
Δp3p3127λp3f 2 g-2.
ARRD=p2p3p2p1+1Δp3p3.
-(2k+3)π7k+2k+1 g-2(k+1)
αk(2k+3)π7k+2k+1 g-2(k+1),
ARRDλ7w2  (2k+3)(k+2)p2
×p2p1+12k+1d0w2k.
0α02+1615 g4α12+2g2α0α112π249g4.

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