Abstract

Visible holographic interferometry is generally too sensitive for the measurement of large deformations. We present a holographic method that permits an increase in the range of measurable deformations. It requires the use of two different wavelengths, λ1 and λ2, and two holograms in series. We develop the theoretical basis of a method that permits the obtention of an interferogram as if a longer equivalent wavelength, λeq = λ1λ2/|λ1 − λ2|, were used. The method is experimentally tested by use of a setup that can be easily converted into a classical single-wavelength holographic interferometer, permitting comparison of the interferograms of the same deformation produced with both methods. Significant results are presented.

© 1995 Optical Society of America

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References

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  1. See, for example, C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  2. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [CrossRef] [PubMed]
  3. B. P. Hildebrand, K. A. Haines, “Multiple-wavelength and multiple-source holography applied to contour generation,” J. Opt. Soc. Am. 57, 155–162 (1967).
    [CrossRef]
  4. J. C. Wyant, B. F. Oreb, P. Hariharan, “Testing aspherics using two-wavelength holography: use of digital electronic techniques,” Appl. Opt. 23, 4020–4023 (1984).
    [CrossRef] [PubMed]
  5. C. W. Chen, “Real-time two-wavelength holography for coarse aspherical surface contouring,” in Contemporary Methods of Optical Manufacturing and Testing, G. M. Sanger, ed., Proc. Soc. Photo-Opt. Instrum. Eng.433, 158–164 (1983).
  6. P. Lam, J. D. Gaskill, J. C. Wyant, “Two-wavelength holorgaphic interferometer,” Appl. Opt. 23, 3079–3081 (1984).
    [CrossRef] [PubMed]
  7. W. B. Ribbens, “Surface roughness measurement by two-wavelength holographic interferometry,” Appl. Opt. 13, 1085–1088 (1974).
    [CrossRef] [PubMed]

1984

1974

1971

1967

Chen, C. W.

C. W. Chen, “Real-time two-wavelength holography for coarse aspherical surface contouring,” in Contemporary Methods of Optical Manufacturing and Testing, G. M. Sanger, ed., Proc. Soc. Photo-Opt. Instrum. Eng.433, 158–164 (1983).

Gaskill, J. D.

Haines, K. A.

Hariharan, P.

Hildebrand, B. P.

Lam, P.

Oreb, B. F.

Ribbens, W. B.

Wyant, J. C.

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Figures (7)

Fig. 1
Fig. 1

First step of the method: recording of the interference of the object and reference beams consecutively at two different wavelengths, λ1 and λ2, on the same holographic plate, H.

Fig. 2
Fig. 2

Second step of the method: readout of H by use of only the deformed object wave, O′, at λ1. Angle α is such that sin α = λ12 sin θ2. The beam emerging in direction θ1 is recorded on a second holographic plate, H′, with the reference beam at incidence angle θ3.

Fig. 3
Fig. 3

Third step of the method: readout of H by use of the deformed object wave, O′, at λ2. The beam diffracted in direction θ2 is incident to hologram H′. At the same time, H′ is read out by reference wave R′ at λ2 with incidence angle θ4, complying to condition (14). The interferogram results from the superimposition of the beam diffracted by H′ and the beam diffracted by H propagating through H′; their directions are collinear.

Fig. 4
Fig. 4

Configuration used to produce the interferogram with single-wavelength holographic interferometry (e.g., λ1) to be compared with the interferogram produced with the two-wavelength method: the object is illuminated with both deformed object beam O′ and readout wave R′.

Fig. 5
Fig. 5

Scheme of the overall setup: Sh’s, shutters; M1, M3–M6, plane mirrors; M2, spherical mirror, first part of the object; L1, spherical lens, second part of the object; BS1–BS3, beam splitters; M7, retrievable plane mirror; L2, L3, lenses for imaging on the CCD camera.

Fig. 6
Fig. 6

Interferogram of the difference between two spherical wave fronts observed with classical single-wavelength holographic interferometry at 633 nm. Both wave fronts have their centers of curvature displaced from each other along the optical axis, leading to circular fringes. The square edge (which is visible in the bottom right of the interferogram) is a sharp-cut blade placed at the surface of the spherical mirror in order to ensure good focusing of the CCD camera imaging system.

Fig. 7
Fig. 7

Interferogram corresponding to the same wave-front difference as in Fig. 6 but using our two-wavelength holographic interferometry method at an equivalent wavelength of 3819 nm. The number of fringes is six times less than in fig. 6, as expected.

Equations (17)

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m cos [ 2 π ( W - W ) / λ ] ,
m cos [ 2 π W ( 1 / λ 1 - 1 / λ 2 ) ] or m cos ( 2 π W / λ eq ) ,
cos [ 2 π ( W - W ) / λ eq ] .
O ( λ 1 ) = O 0 ( λ 1 ) exp ( i 2 π W / λ 1 ) ,
R ( λ 1 ) = R 0 ( λ 1 ) exp ( i 2 π x sin θ 1 / λ 1 ) .
I = [ O ( λ 1 ) + R ( λ ) ] [ O ( λ 1 ) + R ( λ 1 ) ] * = 2 + 2 cos [ 2 π ( W - x     sin θ 1 ) / λ 1 ] ,
t H I 1 + I 2 = 4 + exp [ i 2 π ( W / λ 1 - x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( - W / λ 1 + x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( W / λ 2 - x sin θ 2 / λ 2 ) ] + exp [ i 2 π ( - W / λ 2 + x sin θ 2 / λ 2 ) ]
O ( λ 1 ) = O 0 ( λ 1 ) exp ( i 2 π W / λ 1 ) .
A H ( λ 1 ) = O ( λ 1 ) t H = 4 exp [ i 2 π ( W / λ 1 ) ] + exp [ i 2 π ( W / λ 1 + W / λ 1 - x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( W / λ 1 - W / λ 1 + x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( W / λ 1 + W / λ 2 - x sin θ 2 / λ 2 ) ] + exp [ i 2 π ( W / λ 1 - W / λ 2 + x sin θ 2 / λ 2 ) ] .
t H 2 + exp [ i 2 π ( W / λ 1 - W / λ 1 + x sin θ 1 / λ 1 - x sin θ 3 / λ 1 ) ] + exp [ i 2 π ( - W / λ 1 + W / λ 1 - x     sin θ 1 / λ 1 + x sin θ 3 / λ 1 ) ]
A H ( λ 2 ) = O ( λ 2 ) t H = 4 exp [ i 2 π ( W / λ 2 ) ] + exp [ i 2 π ( W / λ 2 + W / λ 1 - x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( W / λ 2 - W / λ 1 + x sin θ 1 / λ 1 ) ] + exp [ i 2 π ( W / λ 2 + W / λ 2 - x sin     θ 2 / λ 2 ) ] + exp [ i 2 π ( W / λ 2 - W / λ 2 + x sin θ 2 / λ 2 ) ] .
( W - W ) / λ 2 + x sin θ 2 / λ 2 .
A H ( λ 2 ) = R ( λ 2 ) t H = 2 exp ( i 2 π x sin θ 4 / λ 2 ) + exp [ i 2 π ( W / λ 1 - W / λ 1 + x sin θ 1 / λ 1 - x sin θ 3 / λ 1 + x sin θ 4 / λ 2 ) ] + exp [ i 2 π ( - W / λ 1 + W / λ 1 - x sin θ 1 / λ 1 + x sin θ 3 / λ 1 + x sin θ 4 / λ 2 ) ] .
( W - W ) / λ 1 + x sin θ 1 / λ 1 - x sin θ 3 / λ 1 + x sin θ 4 / λ 2 .
sin θ 4 = sin θ 2 - ( λ 2 / λ 1 ) ( sin θ 1 - sin θ 3 ) .
I = 2 + cos { 2 π [ ( W - W ) / λ 2 - ( W - W ) / λ 1 ] } = 2 + cos [ 2 π ( W - W ) / λ eq ] .
m cos [ 2 π W ( 1 / λ 1 - 1 / λ 2 ) + Δ / λ 2 ] .

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