Abstract

An analysis of holographic interferometry using two reference beams to record and reconstruct the two interfering light fields independently is carried out. The multiplicity of reconstructions is discussed. Guidelines for the recording setup are derived to separate the wanted from the unwanted reconstructions. The influence of misalignment of hologram and reference waves on the fringe pattern in the image is evaluated in detail. It is shown that by proper choice of the experimental arrangement, two-reference-beam holography can safely be used for quantitative measurements of surface displacements.

© 1976 Optical Society of America

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  1. M. Dé and L. Sévigny, Appl. Phys. Lett. 10, 98 (1967).
    [CrossRef]
  2. M. Dé and L. Sévigny, Appl. Opt. 10, 1665 (1967).
    [CrossRef]
  3. A. W. Lohmann, Appl. Opt. 4, 1667 (1965).
    [CrossRef]
  4. O. Bryngdahl, J. Opt. Soc. Am. 57, 545 (1967).
    [CrossRef] [PubMed]
  5. M. E. Fourney, A. P. Waggoner, and K. V. Mate, J. Opt. Soc. Am. 58, 701 (1968).
    [CrossRef]
  6. G. S. Ballard, J. Appl. Phys. 39, 8846 (1968).
    [CrossRef]
  7. J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
    [CrossRef]
  8. L. Kersch, Mater. Evaluation 29, 125 (1971).
  9. J. Surget, Schéma d’holographie à deux sources de référence, Euromech 55, Bochum (Germany) March, 1974 (unpublished).
  10. R. Crane, Appl. Opt. 8, 538 (1969).
  11. G. S. Ballard, The Quantitative Measurement of Small Phase Differences in Holographically Reconstructed Optical Fields, Conference on Holography and Optical Filtering, Huntsville (Alabama), May, 1971 (unpublished).
  12. R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
    [CrossRef]
  13. R. Dändliker, B. Ineichen, and F. M. Mottier, Proceedings of International Optics Computing Conference, Zurich (Switzerland) April, 1974 (IEEE, New York, 1974), p. 69.
  14. R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.
  15. B. Ineichen, U. Kogelschatz, and R. Dändliker, Appl. Opt. 12, 2554 (1973).
    [CrossRef] [PubMed]
  16. E. D. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  17. R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
    [CrossRef]
  18. S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [CrossRef]

1973 (2)

R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

B. Ineichen, U. Kogelschatz, and R. Dändliker, Appl. Opt. 12, 2554 (1973).
[CrossRef] [PubMed]

1972 (1)

R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
[CrossRef]

1971 (2)

J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
[CrossRef]

L. Kersch, Mater. Evaluation 29, 125 (1971).

1970 (1)

1969 (1)

R. Crane, Appl. Opt. 8, 538 (1969).

1968 (2)

1967 (4)

O. Bryngdahl, J. Opt. Soc. Am. 57, 545 (1967).
[CrossRef] [PubMed]

M. Dé and L. Sévigny, Appl. Phys. Lett. 10, 98 (1967).
[CrossRef]

M. Dé and L. Sévigny, Appl. Opt. 10, 1665 (1967).
[CrossRef]

E. D. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[CrossRef]

1965 (1)

Arsenault, H.

Ballard, G. S.

G. S. Ballard, J. Appl. Phys. 39, 8846 (1968).
[CrossRef]

G. S. Ballard, The Quantitative Measurement of Small Phase Differences in Holographically Reconstructed Optical Fields, Conference on Holography and Optical Filtering, Huntsville (Alabama), May, 1971 (unpublished).

Ben-Uri, J.

J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
[CrossRef]

Bryngdahl, O.

Champagne, E. D.

Crane, R.

R. Crane, Appl. Opt. 8, 538 (1969).

Dändliker, R.

B. Ineichen, U. Kogelschatz, and R. Dändliker, Appl. Opt. 12, 2554 (1973).
[CrossRef] [PubMed]

R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
[CrossRef]

R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.

R. Dändliker, B. Ineichen, and F. M. Mottier, Proceedings of International Optics Computing Conference, Zurich (Switzerland) April, 1974 (IEEE, New York, 1974), p. 69.

Dé, M.

M. Dé and L. Sévigny, Appl. Phys. Lett. 10, 98 (1967).
[CrossRef]

M. Dé and L. Sévigny, Appl. Opt. 10, 1665 (1967).
[CrossRef]

Eliasson, B.

R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.

Fourney, M. E.

Ineichen, B.

B. Ineichen, U. Kogelschatz, and R. Dändliker, Appl. Opt. 12, 2554 (1973).
[CrossRef] [PubMed]

R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.

R. Dändliker, B. Ineichen, and F. M. Mottier, Proceedings of International Optics Computing Conference, Zurich (Switzerland) April, 1974 (IEEE, New York, 1974), p. 69.

Kersch, L.

L. Kersch, Mater. Evaluation 29, 125 (1971).

Kogelschatz, U.

Lohmann, A. W.

Lowenthal, S.

Marom, E.

R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
[CrossRef]

Mate, K. V.

Mottier, F. M.

R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
[CrossRef]

R. Dändliker, B. Ineichen, and F. M. Mottier, Proceedings of International Optics Computing Conference, Zurich (Switzerland) April, 1974 (IEEE, New York, 1974), p. 69.

R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.

Politch, J.

J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
[CrossRef]

Sévigny, L.

M. Dé and L. Sévigny, Appl. Phys. Lett. 10, 98 (1967).
[CrossRef]

M. Dé and L. Sévigny, Appl. Opt. 10, 1665 (1967).
[CrossRef]

Shamir, J.

J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
[CrossRef]

Surget, J.

J. Surget, Schéma d’holographie à deux sources de référence, Euromech 55, Bochum (Germany) March, 1974 (unpublished).

Waggoner, A. P.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

M. Dé and L. Sévigny, Appl. Phys. Lett. 10, 98 (1967).
[CrossRef]

J. Appl. Phys. (1)

G. S. Ballard, J. Appl. Phys. 39, 8846 (1968).
[CrossRef]

J. Opt. Soc. Am. (4)

Mater. Evaluation (1)

L. Kersch, Mater. Evaluation 29, 125 (1971).

Opt. Commun. (2)

R. Dändliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, E. Marom, and F. M. Mottier, Opt. Commun. 6, 368 (1972).
[CrossRef]

Opt. Laser Technol. (1)

J. Politch, J. Shamir, and J. Ben-Uri, Opt. Laser Technol. 3, 226 (1971).
[CrossRef]

Other (4)

J. Surget, Schéma d’holographie à deux sources de référence, Euromech 55, Bochum (Germany) March, 1974 (unpublished).

R. Dändliker, B. Ineichen, and F. M. Mottier, Proceedings of International Optics Computing Conference, Zurich (Switzerland) April, 1974 (IEEE, New York, 1974), p. 69.

R. Dändliker, B. Eliasson, B. Ineichen, and F. M. Mottier, Proc. of Symposium on the Engineering Uses of Coherent Optics, Glasgow (Scotland), April, 1975 (Cambridge U. P., Cambridge, England, 1975) p. 125.

G. S. Ballard, The Quantitative Measurement of Small Phase Differences in Holographically Reconstructed Optical Fields, Conference on Holography and Optical Filtering, Huntsville (Alabama), May, 1971 (unpublished).

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Figures (6)

FIG. 1
FIG. 1

Setup for recording double exposure holograms with two reference beams. R1 and O1 are used for the first exposure, while R2 and O2 are used for the second one.

FIG. 2
FIG. 2

Propagation directions associated with various image terms are marked as points on the surface of the k sphere. The wave vectors of the object and the reference waves are k 0 , k 1, and k 2, respectively.

FIG. 3
FIG. 3

Reconstructed images from a two-reference-beam holographic interferometry setup. Reconstruction with (A) R1 only, (B) R1 and R2 simultaneously, and (C) R2 only.

FIG. 4
FIG. 4

Schematic view of hologram and beam orientations. (A) Initial orientation of hologram and reference beams used in the recording setup. (B) View of the reconstruction setup and the reorientation of hologram plate to reduce fringes in its plane.

FIG. 5
FIG. 5

Schematic arrangement for evaluating misalignment fringe effects on the interference pattern. (A) Setup for inspecting the interference of O1 and O2 in the image plane I. The image is formed by the lens L. O1 and O2 are reconstructed from the hologram H by their respective references waves R1 and R2. (B) Equivalent setup to (A), where the misalignment of R1 is substituted by a wedge D1 in the hologram plane H acting on the image O1 only but not on the image O2.

FIG. 6
FIG. 6

Influence of misalignment of the hologram on the interference pattern. The distance between object and hologram is a = 1.3 m and the focal length of the imaging lens is f = 360 mm for all pictures. /π is the number of misalignment fringes across the aperture of the lens.

Equations (29)

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τ = τ 0 - β ( R 1 * O 1 + R 1 O 1 * + R 2 * O 2 + R 2 O 2 * ) ,
1 / ρ I = 1 / ρ 0 + ( 1 / ρ l - 1 / ρ k )
1 / ρ I * = - 1 / ρ 0 + ( 1 / ρ l + 1 / ρ k ) ,
ψ = e i ( k 1 r - k 1 r ) + e i ( k 2 r - k 2 r ) ,
k 1 = ( k x , k y , k z ) ,             k 2 = ( k x , - k y , k z ) ,
r - r = Δ r ( Δ x , Δ y , Δ z ) .
ψ = e i k 1 Δ r + e i k 2 Δ r = 2 e i ( k x Δ x + k z Δ z ) cos ( k y Δ y ) ,
ψ 2 = 4 cos 2 ( k y Δ y ) .
ψ = e i ( - k y z Δ α + k z y Δ α ) + e i ( k y z Δ α + k z y Δ α ) = 2 e i k z y Δ α cos ( k y z Δ α ) ,
ψ 2 = 4 cos 2 ( k y z Δ α ) .
ψ 2 = 4 cos 2 k y y ( Δ α ) 2 .
ψ = e i ( k x z Δ β - k z x Δ β ) + e i ( k x z Δ β - k z x Δ β ) ,
ψ 2 = 4.
ψ = e i ( - k x y Δ γ + k y x Δ γ ) + e i ( - k x y Δ γ - k y x Δ γ ) ,
ψ 2 = 4 cos 2 ( k y x Δ γ ) .
k 1 - k 1 = Δ k 1 ,             k 2 - k 2 = Δ k 2 .
ψ = e i ( Δ k 1 · r ) + e i ( Δ k 2 · r ) ,
ψ 2 = 4 cos 2 [ 1 2 ( Δ k 1 - Δ k 2 ) · r ] .
( Δ k 1 - Δ k 2 ) · r = ( k 1 - k 2 ) · r - ( k 1 - k 2 ) · r = 2 k ( sin δ - sin δ ) y ,
cos 2 δ = ( k 1 · k 2 ) and δ = δ + Δ δ .
ψ 2 = 4 cos 2 ( k Δ δ y cos δ ) .
e i k y Δ y = e i ( π / 2 ) ( 2 n + 1 ) = ( i ) 2 n + 1 ,             n = 0 , 1 , 2 , ,
e - i k y Δ y = ( - i ) 2 n + 1 = - ( i ) 2 n + 1 ,             n = 0 , 1 , 2 , .
O 1 e i k y Δ y + O 2 e - i k y Δ y 2 = O 1 2 + O 2 2 + 2 Re ( O 1 O 2 * e i 2 k y Δ y ) ,
V 1 ( x 4 ) = d x 1 d x 2 d x 3 O 1 ( x 1 ) × e i ( k / 2 a ) ( x 1 - x 2 ) 2 e i k α x 2 e i ( k / 2 b ) ( x 2 - x 3 ) 2 × P ( x 3 ) e - i ( k / 2 f ) x 3 2 e i ( k / 2 c ) ( x 4 - x 3 ) 2 ,
V 1 ( x 4 ) = d x 1 d x 3 O 1 ( x 1 ) × e [ k α b / ( a + b ) ] x 1 e i [ k α a / ( a + b ) ] x 3 e i [ k / 2 ( a + b ) ] ( x 1 - x 3 ) 2 × P ( x 3 ) e - i ( k / 2 f ) x 3 2 e i ( k / 2 c ) ( x 4 - x 3 ) 2 .
V 1 ( x 4 ) = d x 1 O 1 ( x 1 ) e i [ k α b / ( a + b ) ] x 1 × P ˆ [ 1 λ ( a + b ) ( α a - x 1 - x 4 a + b c ) ] × e i ( k / 2 ) [ x 1 2 / ( a + b ) + x 4 2 / c ] ,
V 2 ( x 4 ) = d x 1 O 2 ( x 1 ) P ˆ [ 1 λ ( a + b ) ( - x 1 - x 4 a + b c ) ] × e i ( k / 2 ) [ x 1 2 / ( a + b ) + x 4 2 / c ] .
V 1 V 2 * = d x 1 Ō 1 ( x 1 ) Ō 2 * ( x 1 ) e i [ k α b / ( a + b ) ] x 1 × P ˆ [ 1 λ ( a + b ) ( α a - x 1 - x 4 a + b c ) ] × P ˆ * [ 1 λ ( a + b ) ( - x 1 - x 4 a + b c ) ] ,