Abstract

A statistical analysis of the effect of measurement errors on the determination of small vector displacements of diffusely reflecting objects by holographic interferometry is reported. There are two sources of error: inaccurate measurement of fringe-order number and inaccurate measurement of illumination and observation directions. Standard deviations of errors in measured displacements due to these two sources are presented in graphical form for a variety of holographic system geometries. Absolute fringe orders should be measured whenever possible, because fringe readout by scanning leads to very large errors.

© 1978 Optical Society of America

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References

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  1. T. Matsumoto, K. Iwata, R. Nagata, Appl. Opt. 12, 961 (1973).
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    [CrossRef] [PubMed]
  4. S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1973).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. C. A. Sciammarella, T. Y. Chang, Exp. Mech. 14, 217 (1974).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. G. Golub, Numer. Math. 7, 206 (1965).
    [CrossRef]
  11. W. B. Davenport, Probability and Random Processes (McGraw-Hill, New York, 1970), pp. 345–348.
  12. W. L. Root, Univ. Mich.; private communication (1976).

1977 (1)

1974 (4)

1973 (3)

1971 (1)

1965 (1)

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Biedermann, K.

Chang, T. Y.

C. A. Sciammarella, T. Y. Chang, Exp. Mech. 14, 217 (1974).
[CrossRef]

Davenport, W. B.

W. B. Davenport, Probability and Random Processes (McGraw-Hill, New York, 1970), pp. 345–348.

Dhir, S. K.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1973).
[CrossRef]

Ek, L.

Fossati Bellani, V.

Gilbert, J. A.

Golub, G.

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Iwata, K.

King, P. W.

Matsumoto, T.

Nagata, R.

Root, W. L.

W. L. Root, Univ. Mich.; private communication (1976).

Sciammarella, C. A.

C. A. Sciammarella, T. Y. Chang, Exp. Mech. 14, 217 (1974).
[CrossRef]

C. A. Sciammarella, J. A. Gilbert, Appl. Opt. 12, 1951 (1973).
[CrossRef] [PubMed]

Shibayama, K.

Sikora, J. P.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1973).
[CrossRef]

Sona, A.

Uchiyama, H.

Appl. Opt. (7)

Exp. Mech. (2)

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1973).
[CrossRef]

C. A. Sciammarella, T. Y. Chang, Exp. Mech. 14, 217 (1974).
[CrossRef]

Numer. Math. (1)

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Other (2)

W. B. Davenport, Probability and Random Processes (McGraw-Hill, New York, 1970), pp. 345–348.

W. L. Root, Univ. Mich.; private communication (1976).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of holographic system showing object illumination source, object point whose displacement is to be measured, hologram, and coordinate system used in analysis.

Fig. 2
Fig. 2

Location of observation points on 10×13-cm holograms. All observation directions referred to in this paper are defined by lines connecting these points with the object point in Fig. 1.

Fig. 3
Fig. 3

Standard deviations of errors in dimensionless displacement L/λ when absolute fringe orders are measured from the three observation points indicated in Fig. 2(a).

Fig. 4
Fig. 4

Standard deviations of errors in dimensionless displacement L/λ when absolute fringe orders are measured from the four observation points indicated in Fig. 2(b).

Fig. 5
Fig. 5

Standard deviations of errors in dimensionless displacement L/λ when absolute fringe orders are measured from the five observation points indicated in Fig. 2(c).

Fig. 6
Fig. 6

Standard deviations of errors in dimensionless displacement L/λ when fringe order shifts are measured by scanning. Three-scan readout was along lines ab, bc, and ce in Fig. 2(c). Five-scan readout was along lines ab, bc, ce, be and ed in Fig. 2(c).

Tables (3)

Tables Icon

Table I Conversion Factors for σLZ/σN or σLz/K for Various Values of ψ

Tables Icon

Table II Standard Deviations σK of Errors in the Coefficient Matrix K when the Coordinates of the Illumination Source and Three Observation Points are Subject to Experimental Errors with 1-mm Standard Deviation

Tables Icon

Table III Standard Deviations σK of Errors in the Coefficient Matrix K when the Coordinates of the Illumination Source and Three Observation Points are Subject to Experimental Errors with 1-cm Standard Deviation

Equations (24)

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K 1 x l x + K 1 y l y + K 1 z l z = N 1 λ ,
K 2 x l x + K 2 y l y + K 2 z l z = N 2 λ ,
K 3 x l x + K 3 y l y + K 3 z l z = N 3 λ .
K 4 x l x + K 4 y l y + K 4 z l z = N 4 λ ,
K 5 x l x + K 5 y l y + K 5 z l z = N 5 λ .
K L = N ,
K L = N 0 + Δ N ,
L = K 1 N 0 + K 1 Δ N = L 0 + Δ L  .
Γ = [ σ N 1 2 0 0 0 σ N 2 2 0 0 0 σ N 3 2 ] .
Δ L = K 1 Δ N
λ = K 1  Γ K 1     T .
Γ = σ N 2 I ,
λ = σ N 2  [ K 1 K 1     T ] = σ N 2  [ KK T ] 1 .
K 1 x l x + K 1 y l y + K 1 z l z = N a b λ ,
K 2 x l x + K 2 y l y + K 2 z l z = N b c λ ,
K 3 x l x + K 3 y l y + K 3 z l z = N c e λ ,
[ K 0 + Δ K ] L = N  ,
λ = K 1    E ( Δ KLL T Δ K T ) K 1      T ,
E ( Δ KLL T Δ K T ) = ( σ K 2 i = 1 3 L i 2 ) I  ,
λ = L 2 σ K 2 K 1 K 1     T = L 2 σ K 2 [ KK T ] 1 .
σ L x / L = 0.0092 , σ L y / L = 0.0016 , and σ L z / L = 0.0051  .
σ L x / L = 0.0106 , σ L y / L = 0.0015 , and σ L z / L = 0.0047
σ L x / L = 0.0048 , σ L y / L = 0.0074 , and σ L z / L = 0.0021.
σ L x / L = 0.0073 , σ L y / L = 0.0102 , and σ L z / L = 0.0030.

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