Abstract

Displacement measurements by optical interferometry depend on the induced phase difference and on the interferometer’s sensitivity vector; the latter depends in turn on the illuminating sources and on the geometry of the optical arrangement. We have performed an uncertainty analysis of the in-plane displacements measured by electronic speckle-pattern interferometry with spherical incident wave fronts. We induced the displacements by applying a uniaxial tensile load on a nominally flat elastic sample. We approached the displacement uncertainty by propagating the uncertainties that we considered reasonable to assign to the measured phase difference and to the characteristic parameters of the interferometer’s sensitivity vector. Special attention was paid to evaluating contributions to the displacement uncertainty. Moreover, we observed that the uncertainty decreases if the angles of incidence and the source–target distances are increased.

© 2005 Optical Society of America

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References

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  1. A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
    [CrossRef]
  2. C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
    [CrossRef] [PubMed]
  3. R. R. Cordero, I. Lira, “Uncertainty analysis of displacements measured by phase shifting moiré interferometry,” Opt. Commun. 237, 25–36 (2004).
    [CrossRef]
  4. A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
    [CrossRef]
  5. R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
    [CrossRef]
  6. H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
    [CrossRef]
  7. W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
    [CrossRef]
  8. International Organization for Standardization, (International Organization for Standardization, Geneva, 1993).
  9. I. Lira, Evaluating the Uncertainty of Measurement: Fundamentals and Practical Guidance (Institute of Physics Publishing, Bristol, UK, 2002), Chap. 3, pp. 45–102.
  10. R. R. Cordero, P. Roth, “Assigning probability density functions in a context of information shortage,” Metrologia 41(4), L22–L25 (2004)
    [CrossRef]
  11. T. Kreis, “Speckle metrology,” in Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie Verlag, New York, 1996), Chap. 3, pp. 71–74.
  12. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, Singapore, 1970), Chap. 1, pp. 1–14.
  13. Ref. 11, Chap. 4, pp. 125–149.

2004 (3)

R. R. Cordero, I. Lira, “Uncertainty analysis of displacements measured by phase shifting moiré interferometry,” Opt. Commun. 237, 25–36 (2004).
[CrossRef]

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

R. R. Cordero, P. Roth, “Assigning probability density functions in a context of information shortage,” Metrologia 41(4), L22–L25 (2004)
[CrossRef]

2003 (2)

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

2002 (1)

H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
[CrossRef]

2001 (1)

W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
[CrossRef]

1994 (1)

Cordero, R. R.

R. R. Cordero, I. Lira, “Uncertainty analysis of displacements measured by phase shifting moiré interferometry,” Opt. Commun. 237, 25–36 (2004).
[CrossRef]

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

R. R. Cordero, P. Roth, “Assigning probability density functions in a context of information shortage,” Metrologia 41(4), L22–L25 (2004)
[CrossRef]

Goodier, J. N.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, Singapore, 1970), Chap. 1, pp. 1–14.

Joenathan, C.

Kreis, T.

T. Kreis, “Speckle metrology,” in Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie Verlag, New York, 1996), Chap. 3, pp. 71–74.

Lira, I.

R. R. Cordero, I. Lira, “Uncertainty analysis of displacements measured by phase shifting moiré interferometry,” Opt. Commun. 237, 25–36 (2004).
[CrossRef]

I. Lira, Evaluating the Uncertainty of Measurement: Fundamentals and Practical Guidance (Institute of Physics Publishing, Bristol, UK, 2002), Chap. 3, pp. 45–102.

Martínez, A.

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
[CrossRef]

Petzing, J. N.

W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
[CrossRef]

Puga, H. J.

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
[CrossRef]

Rayas, J. A.

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

Rodríguez-Vera, R.

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
[CrossRef]

Roth, P.

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

R. R. Cordero, P. Roth, “Assigning probability density functions in a context of information shortage,” Metrologia 41(4), L22–L25 (2004)
[CrossRef]

Timoshenko, S. P.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, Singapore, 1970), Chap. 1, pp. 1–14.

Tyrer, J. R.

W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
[CrossRef]

Wan Abdullan, W. S.

W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

W. S. Wan Abdullan, J. N. Petzing, J. R. Tyrer, “Wavefront divergence: a source of error in quantified speckle shearing data,” J. Mod. Opt. 48, 757–772 (2001).
[CrossRef]

Metrologia (1)

R. R. Cordero, P. Roth, “Assigning probability density functions in a context of information shortage,” Metrologia 41(4), L22–L25 (2004)
[CrossRef]

Opt. Commun. (3)

R. R. Cordero, I. Lira, “Uncertainty analysis of displacements measured by phase shifting moiré interferometry,” Opt. Commun. 237, 25–36 (2004).
[CrossRef]

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Error in the measurement due to the divergence of the object illumination wavefront for in-plane interferometers,” Opt. Commun. 223, 239–246 (2003).
[CrossRef]

R. R. Cordero, A. Martínez, R. Rodríguez-Vera, P. Roth, “Uncertainty evaluation of displacements measured by electronic speckle-pattern interferometry,” Opt. Commun. 241, 279–292 (2004).
[CrossRef]

Opt. Laser Technol. (1)

H. J. Puga, R. Rodríguez-Vera, A. Martínez, “General model to predict and correct errors in phase map interpretation and measurement for out-of-plane ESPI interferometers,” Opt. Laser Technol. 34, 81–92 (2002).
[CrossRef]

Opt. Lasers Eng. (1)

A. Martínez, R. Rodríguez-Vera, J. A. Rayas, H. J. Puga, “Fracture detection by grating moiré and in-plane ESPI techniques,” Opt. Lasers Eng. 39, 525–536 (2003).
[CrossRef]

Other (5)

International Organization for Standardization, (International Organization for Standardization, Geneva, 1993).

I. Lira, Evaluating the Uncertainty of Measurement: Fundamentals and Practical Guidance (Institute of Physics Publishing, Bristol, UK, 2002), Chap. 3, pp. 45–102.

T. Kreis, “Speckle metrology,” in Holographic Interferometry, W. Jüptner, W. Osten, eds. (Akademie Verlag, New York, 1996), Chap. 3, pp. 71–74.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, Singapore, 1970), Chap. 1, pp. 1–14.

Ref. 11, Chap. 4, pp. 125–149.

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Figures (11)

Fig. 1
Fig. 1

Diagram to define sensitivity vector e of a dual-beam interferometer.

Fig. 2
Fig. 2

(a) Photograph of the optical system used. (b) Schematic of a dual-beam ESPI optical setup with sensitivity mostly along x, as shown in (a). PZTs, piezoelectric transducers.

Fig. 3
Fig. 3

Estimates of the sensitivity vector components evaluated by Eqs. (3): (a) ex, (b) ey, (c) ez.

Fig. 4
Fig. 4

(a) Wrapped phase, (b) relative displacement induced along x.

Fig. 5
Fig. 5

Standard uncertainties associated with the estimates of the sensitivity vector components: (a) u(ex), (b) u(ey), (c) u(ez).

Fig. 6
Fig. 6

Mutual uncertainties of the sensitivity vector components: (a) u(ex, ey), (b) u(ex, ez), (c) u(ey, ez).

Fig. 7
Fig. 7

Relative standard uncertainty of the sensitivity vector component along x. The plot was built up by use of the values depicted in Figs. 5(a) and 3(a). The angles of incidence measured with respect to axis z were both equal to θ = 6°, and the source–target distance was r = 166.4 cm.

Fig. 8
Fig. 8

Contributions to the square of u(ex) along x = 5 cm. C(x1) = (∂ex/∂x1)2u2(x1), C(y1) = (∂ex/∂y1)2u2(y1), C(z1) = (∂ex/∂z1)2u2(z1), C(x2) = (∂ex/∂x2)2u2(x2), C(y2) = (∂ex/∂y2)2u2(y2), C(z2) = (∂ex/∂z2)2u2(z2), C(α) = (∂ex/∂α)2u2(α), C(β) = (∂ex/∂β)2u2(β).

Fig. 9
Fig. 9

Relative standard uncertainty of the sensitivity vector component along x. (a). Angle of incidence with respect to sample normal θ = 6°, source distance with respect to sample center r = 166.4 cm, and 20 × 20 cm illuminated area. (b) Angle of incidence θ = 50°, source–target distance r = 166.4 cm, and 10 × 10 cm illuminated area. (c) θ = 6°, r = 100 cm, and 10 × 10 cm illuminated area.

Fig. 10
Fig. 10

Standard uncertainties associated with the estimates of the displacement component along x.

Fig. 11
Fig. 11

Contributions to the square of u(dx) along x = 5 cm. C(ex) = (∂dx/∂ex)2u2(ex), C(ey) = (∂dx/∂ey)2u2(ey), C(ez) = (∂dx/∂ez)2u2(ez), C(v) = (∂dx/∂v)2u2(v)

Equations (27)

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Δ ϕ = d · e ,
e = 2 π λ [ n ^ 1 - n ^ 2 ] ,
e x = 2 π λ ( x - x 1 ) [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + ( z - z 1 ) 2 ] 1 / 2 - ( x - x 2 ) [ ( x - x 2 ) 2 + ( y - y 2 ) 2 + ( z - z 2 ) 2 ] 1 / 2 ,
e y = 2 π λ ( y - y 1 ) [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + ( z - z 1 ) 2 ] 1 / 2 - ( y - y 2 ) [ ( x - x 2 ) 2 + ( y - y 2 ) 2 + ( z - z 2 ) 2 ] 1 / 2 ,
e z = 2 π λ ( z - z 1 ) [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + ( z - z 1 ) 2 ] 1 / 2 - ( z - z 2 ) [ ( x - x 2 ) 2 + ( y - y 2 ) 2 + ( z - z 2 ) 2 ] 1 / 2 ,
z = α x + β y ,
ɛ x = d x x ,
ɛ y = d y y ,
ɛ z = d z z .
ɛ x = d x x p ,
ɛ y = d y y p .
ɛ z = d W W ,
ɛ y = ɛ z = - ν ɛ x ,
d x = x Δ ϕ x e x - ν e y y - ν e z W .
d x = Δ ϕ / e x .
u 2 ( p ) = [ u 2 ( p 1 ) u ( p 1 , p n ) u ( p 1 , p n ) u 2 ( p n ) ] ,
u 2 ( q ) = S u 2 ( p ) S T ,
S = - [ S ( q ) ] - 1 S ( p ) ,
S ( q ) = [ M 1 q 1 M 1 q m M m q 1 M m q m ] ,
S ( p ) = [ M 1 p 1 M 1 p n M m p 1 M m p n ] .
u 2 ( p ) = [ u 2 ( x ) u ( x , y ) u ( x , y ) u 2 ( y ) ] .
u 2 ( z ) = ( f / x ) 2 u 2 ( x ) + ( f / y ) 2 u 2 ( y ) + 2 ( f / x ) ( f / y ) u ( x , y ) .
u ( X ) = δ X / 12 .
u ( α ) = u ( β ) = 2 ( 3 ° ) 12 ( π 180 ) .
u ( x 1 ) = 2 ( 1 ) 12 [ mm ] .
u 2 ( e ) = [ u 2 ( e x ) u ( e x , e y ) u ( e x , e z ) u ( e x , e y ) u 2 ( e y ) u ( e y , e z ) u ( e x , e z ) u ( e y , e z ) u 2 ( e z ) ] .
u ( ν ) = 0.05 / 12 .

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