Abstract

Temporal phase unwrapping is applied to a two-camera polarization phase-stepped system. A simple algorithm for the phase-change calculation is given, together with simulations, to indicate its validity and strength. This method can be applied directly for detection of phase changes as a function of time. It is proposed to use this method in a shearography setup. The phase distribution in the shearogram can then be obtained, without the standard 2π ambiguities, by application of the required total shear in a number of smaller steps, provided that each step is small enough to be free from these 2π phase steps.

© 1998 Optical Society of America

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References

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  1. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  2. H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator fringe projector,” Opt. Eng. 36, 610–615 (1997).
    [CrossRef]
  3. H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
    [CrossRef] [PubMed]
  4. K. A. Stetson, “Phase-step interferometry of irregular shapes by using an edge-following algorithm,” Appl. Opt. 31, 5320–5325 (1992).
    [CrossRef] [PubMed]
  5. O. Bryngdahl, “Polarization-type interference-fringe shifter,” J. Opt. Soc. Am. 62, 462–464 (1972).
    [CrossRef]
  6. H. Z. Hu, “Polarization heterodyne interferometry using a simple rotating analyzer. 1. Theory and error analysis,” Appl. Opt. 22, 2052–2056 (1983).
    [CrossRef] [PubMed]
  7. M. P. Kothiyal, C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–4442 (1985).
    [CrossRef] [PubMed]

1997 (2)

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

1985 (1)

1983 (1)

1972 (1)

Bryngdahl, O.

Delisle, C.

Hu, H. Z.

Huntley, J. M.

Kothiyal, M. P.

Saldner, H.

Saldner, H. O.

H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef] [PubMed]

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

Stetson, K. A.

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Figures (7)

Fig. 1
Fig. 1

Numerator [n(x)] and denominator [d(x)], together with the resultant phase [p(x)], pertaining to Eq. (3).

Fig. 2
Fig. 2

Numerator [n(x)] and denominator [d(x)], together with the resultant phase [p(x)], pertaining to Eq. (8), based on the four-frame method.1,4

Fig. 3
Fig. 3

Look-up table to be used in combination with the method presented. The erroneous phase steps are automatically removed by this table.

Fig. 4
Fig. 4

Results of the simulation of tilting a curved wave front in 10 steps, each small enough not to result in 2π phase ambiguities: (a) unfiltered phase difference, (b) mask to indicate the points that require filtering, (c) the filtered phase difference. The phase differences are shown wrapped for clarity, whereas the actual results of the simulation were unwrapped.

Fig. 5
Fig. 5

Possible setup for measuring wave-front changes, based on a Michelson-type interferometer and using polarization phase stepping.

Fig. 6
Fig. 6

Possible setup for measuring wave-front changes, based on a shearing interferometer and using polarization phase stepping.

Fig. 7
Fig. 7

Possible setup for measuring surface shapes. One obtains the shape of the mirror under test by tilting a mirror in small angular steps. After each angular step the phase change is measured. A summation of all phase steps yields the shearogram without phase ambiguities.

Equations (12)

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I 1 x ,   y = I B 1 - M   cos φ ref x ,   y - φ obj x ,   y
I 2 x ,   y = I B 1 + M   sin φ ref x ,   y - φ obj x ,   y
Δ φ t ,   t + T = - π 2 - 2   arc   tan I 0 t - I π / 2 t + T I π / 2 t - I 0 t + T .
I 0 t - I π / 2 t + T I π / 2 t - I 0 t + T = - cos φ + sin φ + Δ φ sin φ + cos φ + Δ φ ,
cos Δ φ / 2 + sin Δ φ / 2 sin Δ φ / 2 - cos Δ φ / 2 ,
sin Δ φ / 2 + π / 4 - cos Δ φ / 2 + π / 4 = - tan Δ φ 2 + π 4 ,
Δ φ t 1 ,   t 1 + NT = k = 0 N - 1   Δ φ t 1 + kT ,   t 1 + k + 1 T .
Δ φ t ,   t + T = arctan Δ I 42 t + T Δ I 13 t - Δ I 13 t + T Δ I 42 t Δ I 13 t + T Δ I 13 t + Δ I 42 t + T Δ I 42 t ,
Δ I ij t = I i t - I j t ,
I i t = I B 1 + M   cos φ ref x ,   y - φ obj x ,   y + i - 1 π / 2 i = 1 ,   2 ,   3 ,   4 .
LUT i ,   j = - π 2 - arc   tan i / j ,
LUT   image i ,   j = int 127.5 LUT i ,   j + π / π ,

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