Abstract

The imaging properties of a real-time shearing interferometer are presented. The use of Savart elements, both as a beam displacer and an analyzer in a polarization phase-stepping scheme, is demonstrated in a real-time, two-camera, four-bucket shearing interferometer. A simple calculation scheme for ray propagation through uniaxial, birefringent elements is presented, and the effects on the image formation through 6-cm-long Savart elements is discussed.

© 1998 Optical Society of America

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References

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  1. A. L. Weijers, “Two-camera shearing speckle interferometer for real-time deformation measurements,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1997).
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    [CrossRef]
  3. H. Z. Hu, “Polarization heterodyne interferometry using a simple rotating analyzer. 1: Theory and error analysis,” Appl. Opt. 22, 2052–2056 (1983).
    [CrossRef] [PubMed]
  4. M. P. Kothiyal, C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
    [CrossRef] [PubMed]
  5. A. L. Weijers, H. van Brug, H. J. Frankena, “Real-time phase-stepped shearing speckle interferometer for nondestruc-tive testing,” in Fringe ’97, Automatic Processing of Fringe Patterns, Optical Metrology, W. Jüptner, W. Olsten, eds. (Wiley, New York, 1997), pp. 540–543.
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    [CrossRef] [PubMed]
  7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).
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    [CrossRef]
  9. M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Vol. 2.
  10. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  11. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. Dainty, ed. (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  12. A. J. P. van Haasteren, “Real-time phase stepped speckle interferometry,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).
  13. M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
    [CrossRef]

1996 (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

1993 (1)

1990 (1)

1985 (1)

1983 (1)

1972 (1)

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Bryngdahl, O.

Chipman, R. A.

Delisle, C.

Frankena, H. J.

A. L. Weijers, H. van Brug, H. J. Frankena, “Real-time phase-stepped shearing speckle interferometer for nondestruc-tive testing,” in Fringe ’97, Automatic Processing of Fringe Patterns, Optical Metrology, W. Jüptner, W. Olsten, eds. (Wiley, New York, 1997), pp. 540–543.

Goodman, J. W.

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

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

Hillman, L. W.

Hu, H. Z.

Kothiyal, M. P.

Lehmann, M.

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Liang, Q.-T.

McClain, S. C.

van Brug, H.

A. L. Weijers, H. van Brug, H. J. Frankena, “Real-time phase-stepped shearing speckle interferometer for nondestruc-tive testing,” in Fringe ’97, Automatic Processing of Fringe Patterns, Optical Metrology, W. Jüptner, W. Olsten, eds. (Wiley, New York, 1997), pp. 540–543.

van Haasteren, A. J. P.

A. J. P. van Haasteren, “Real-time phase stepped speckle interferometry,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

Weijers, A. L.

A. L. Weijers, “Two-camera shearing speckle interferometer for real-time deformation measurements,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1997).

A. L. Weijers, H. van Brug, H. J. Frankena, “Real-time phase-stepped shearing speckle interferometer for nondestruc-tive testing,” in Fringe ’97, Automatic Processing of Fringe Patterns, Optical Metrology, W. Jüptner, W. Olsten, eds. (Wiley, New York, 1997), pp. 540–543.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Other (7)

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Vol. 2.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

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

A. J. P. van Haasteren, “Real-time phase stepped speckle interferometry,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

A. L. Weijers, “Two-camera shearing speckle interferometer for real-time deformation measurements,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1997).

A. L. Weijers, H. van Brug, H. J. Frankena, “Real-time phase-stepped shearing speckle interferometer for nondestruc-tive testing,” in Fringe ’97, Automatic Processing of Fringe Patterns, Optical Metrology, W. Jüptner, W. Olsten, eds. (Wiley, New York, 1997), pp. 540–543.

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Figures (4)

Fig. 1
Fig. 1

Scheme of the real-time shearing setup.

Fig. 2
Fig. 2

Relative directions of the vectors in case of an extraordinary beam entering a uniaxial material.

Fig. 3
Fig. 3

Configuration as used in the simulation. For the position and orientation of the focal lines, see Fig. 4.

Fig. 4
Fig. 4

Focal lines as formed by imaging through a single piece of calcite. For the simulations, the optical axis ĉ was taken to lie in the (y, z) plane.

Equations (20)

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η ˆ × n i k ˆ i = η ˆ × n t k ˆ t ,
k ˆ t = n i k ˆ i + Γ n ˆ | n i k ˆ i + Γ η ˆ | ,
Γ = - n i k ˆ i · η ˆ ± n i 2 k ˆ i · η ˆ 2 + n t 2 - n i 2 1 / 2 .
n t ϕ = n e n o n o 2 sin 2 ϕ + n e 2 cos 2 ϕ 1 / 2 ,
ϕ = arccos k ˆ t · c ˆ .
ρ ˆ = k ˆ t + λ c ˆ | k ˆ t + λ c ˆ | ,
α = arctan n o 2 n e 2 tan ϕ .
λ = cos ϕ n e 2 n o 2 - 1 .
H ˆ = c ˆ × k ˆ t | c ˆ × k ˆ t | = c ˆ × ρ ˆ | c ˆ × ρ ˆ | .
D = H ˆ × k ˆ t
E = H ˆ × ρ ˆ .
n o 2 = 1 + 0.8559 λ 2 λ 2 - 0.0588 2 + 0.8391 λ 2 λ 2 - 0.141 2 + 0.0009 λ 2 λ 2 - 0.197 2 + 0.6845 λ 2 λ 2 - 7.005 2 ,
n e 2 = 1 + 1.0856 λ 2 λ 2 - 0.07897 2 + 0.0988 λ 2 λ 2 - 0.142 2 + 0.3170 λ 2 λ 2 - 11.486 2
ψ = arctan I 4 - I 2 I 1 - I 3 ,
| μ I i ;   I j | 2 = I i I j - I i I j I i 2 - I i 2 1 / 2 I j 2 - I j 2 1 / 2 ,
I i x ,   y = I B 1 + M   cos ψ x ,   y + Δ i with   i = 1 ,   2 ,   3 ,   4 ,
M = I 1 - I 3 2 + I 4 - I 2 2 1 / 2 2 I B .
I M = π n 1 / 2 I B ,
I M = π NA 2 1 / 2 I B .
M = I M I B     1 NA ,

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