Abstract

Assuming that the measured coordinates of the fringes of an interferogram have random errors and that they are considered Gaussian, the system of normal equations that is obtained on application of the least-squares method is converted into a nonlinear set of equations. We present an algorithm to estimate the coefficients of the nonlinear system by applying the Newton–Raphson method and starting the iteration from the standard classic solution. This algorithm is applied to a pattern of straight and equally spaced fringes, obtaining not only the right coefficients but also the adequate election of the terms to be included in the model, to show the contrast with the results of the classic method.

© 1998 Optical Society of America

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References

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  1. D. Malacara, “Twymann Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.
  2. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
    [CrossRef]
  3. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).
  4. W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
    [CrossRef]
  5. W. H. Jefferys, “On the method of least squares II,” Astron. J. 86, 149–155 (1981).
    [CrossRef]
  6. A. W. Ross, “Regression line analysis,” Am. J. Phys. 48, 409 (1980).
    [CrossRef]
  7. D. Malacara, ed., Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

1994

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
[CrossRef]

1981

W. H. Jefferys, “On the method of least squares II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

1980

A. W. Ross, “Regression line analysis,” Am. J. Phys. 48, 409 (1980).
[CrossRef]

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

Cardona-Nuñez, O.

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
[CrossRef]

Cordero-Dávila, A.

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
[CrossRef]

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

Cornejo-Rodríguez, A.

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
[CrossRef]

Jefferys, W. H.

W. H. Jefferys, “On the method of least squares II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

Malacara, D.

D. Malacara, “Twymann Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.

Ross, A. W.

A. W. Ross, “Regression line analysis,” Am. J. Phys. 48, 409 (1980).
[CrossRef]

Rufino Díaz-Uribe,

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

Am. J. Phys.

A. W. Ross, “Regression line analysis,” Am. J. Phys. 48, 409 (1980).
[CrossRef]

Appl. Opt.

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, Rufino Díaz-Uribe, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. II: Analytical study,” Appl. Opt. 31, 7343–7349 (1994).

A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuñez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
[CrossRef]

Astron. J.

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

W. H. Jefferys, “On the method of least squares II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

Other

D. Malacara, ed., Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

D. Malacara, “Twymann Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.

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Figures (2)

Fig. 1
Fig. 1

Geometric interpretation of the estimated error, Δ i .

Fig. 2
Fig. 2

Points that belong to the straight fringe pattern.

Tables (2)

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Table 1 Values of Accepted Coefficients after Application of the Traditional Evaluation of an Interferogram to a Straight and a Parallel Fringe Pattern

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Table 2 Application of Five Least-Squares Fittingsa

Equations (31)

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W x ,   y = k = 1 N t   a k g k x ,   y ,
k = 1 N t   a k g k x ,   y - m = 0 .
R t = X 1 ,   Y 1 ,   X 2 ,   Y 2 , ,   X n - 1 ,   Y n - 1 ,   X n ,   Y n ,
X i = R 2 i - 1 ,     Y i = R 2 i .
σ = diag σ 0 ,   σ 0 , ,   σ 0 .
E t = E 1 ,   E 2 , ,   E 2 N .
r = R + E ,
a t = a 1 ,   a 2 , ,   a N T ,
Δ r ,   a = k = 1 N T   a k g k x 1 ,   y 1 - m 1 k = 1 N T   a k g k x 2 ,   y 2 - m 2 · · · k = 1 N T   a k g k x N ,   y N - m N .
W = 1 σ 0 1 h xi 2 + h yi 2 0 0 0 1 h x 2 2 + h y 2 2 0 0 0 1 h xN 2 + h yN 2 ,
h xi = k = 1 N T   a k g k x a ˆ k , x ˆ i , y ˆ i , h yi = k = 1 N T   a k g k y a ˆ k , x ˆ i , y ˆ i , g ki = g k x ˆ i ,   y ˆ i .
ϕ ˆ = k = 1 N T   a ˆ k g k 1 - h x 1 X 1 - x ˆ 1 + h y 1 Y 1 - y ˆ 1 - m 1 k = 1 N T   a ˆ k g k 2 - h x 2 X 2 - x ˆ 2 + h y 2 Y 2 - y ˆ 2 - m 2 · · · k = 1 N T   a ˆ k g kN - h xN X N - x ˆ N + h yN Y N - y ˆ N - m N .
A δ ˆ = B ,
A = Δ a ˆ t W Δ a ˆ ,     B = Δ a ˆ t W ϕ ˆ .
E ˆ final = - Δ r ˆ t W f ϕ ˆ f .
E ˆ final = - h x 1 f h x 1 f 2 + h y 1 f 2 u = 1 N t   a ˆ uf g u 1 f - h u 1 f X 1 - x ˆ 1 f + h y 1 f Y 1 - y ˆ 1 f - m 1 h y 1 f h x 1 f 2 + h y 1 f 2 u = 1 N t   a ˆ uf g u 1 f - h u 1 f X 1 - x ˆ 1 f + h y 1 f Y 1 - y ˆ 1 f - m 1 · · · h xNf h xNf 2 + h yNf 2 u = 1 N t   a ˆ uf g Nf - h uNf X N - x ˆ Nf + h yNf Y N - y ˆ Nf - m N h yNf h xNf 2 + h yNf 2 u = 1 N t   a ˆ uf g uNf - h uNf X N - x ˆ Nf + h yNf Y N - y ˆ Nf - m N .
σ ˆ 2 = E ˆ final t E ˆ final t / N p - N T ,
σ ˆ 2 = i = 1 N h xif X i - x ˆ if + h yif Y i - y ˆ if 2 h xif 2 + h yif 2 N P - N T .
σ ˆ 2 = 1 2 N - N T i = 1 N i W · Δ r i i W 2 .
F exp = S N T 2 - S N T + 1 2 S N T + 1 2 N P - N T - 1 ,
Δ R + E ,   a = 0 ,
r = R + E
S 0 = 1 2   E ˆ t σ - 1 E ˆ ,
σ - 1 E ˆ + Δ r t r ˆ ,   a ˆ μ ˆ = 0 , Δ a t r ˆ ,   a ˆ μ ˆ = 0 , Δ r ˆ ,   a ˆ = 0 ,
σ - 1 E ˆ + e ˆ + Δ r t μ ˆ = 0 , Δ a ˆ t μ ˆ = 0 , Δ ˆ + Δ r ˆ e ˆ + Δ a ˆ δ ˆ = 0 ,
Δ a ˆ t W Δ a ˆ δ ˆ = - Δ a ˆ t W ϕ ˆ ,
ϕ ˆ = Δ ˆ - Δ r ˆ E ˆ ,
W = Δ r ˆ δ Δ r ˆ t - 1 .
a ˆ new = a ˆ + δ ˆ ,
E ˆ new = - σ Δ r ˆ t W ϕ ˆ + Δ a ˆ δ ˆ .
r ˆ new = R + E ˆ new ,

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