Abstract

We consider an ideal Twyman–Green interferogram with equally spaced straight fringes parallel to the x axis and fringe coordinates that are affected by Gaussian errors. We adjust the data points by polynomial fitting to the interferograms. We use a statistical analysis to obtain analytical formulas for the expected values of the aberration coefficients. The result of the analysis shows that the expected coefficients are zero, except for tilt about x and for the comatic term, and that such deviation increases with the noise level and decreases with the number of fringes. Formulas are also obtained for the expected values of the sum of squares of the residuals. We show that the problem of choosing the wrong polynomial order is a consequence of erroneous adjustment of the data points.

© 1994 Optical Society of America

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References

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  1. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nunez, “Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. I: Computer simulations,” Appl. Opt. 33, 7339–7342 (1994).
    [CrossRef] [PubMed]
  2. E. Chavez, C. Menchaca, D. Malacara, “Ajuste bidimensional de datos con mínimos cuadrados,” Reporte Técnico No. 7 (Centro Investigaciones Opticas, León, Guanajuato, Mexico, 1982).
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9.
  4. M. P. Rimmer, “A method for evaluating lateral shearing interferograms,” Itek Internal Rep. 72-5802-1 (Itek Corporation, Lexington, Mass., 1972).
  5. J. S. Loomis, “Analysis of interferometric data for the multiple mirror telescope optics,” J. Opt. Soc. Am. 66, 1116–1117 (1976).
  6. J. S. Loomis, “A computer program for analysis of interferogramic data,” in Optical Interferometric Reductions and Interpretations, A. H. Guenther, D. H. Liebenberg, eds., (American Society for Testing and Materials, Philadelphia, Pa., 1978).
    [CrossRef]
  7. J. Y. Wang, D. E. Silva, “Wavefront interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  8. D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  9. G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
    [CrossRef]
  10. R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. 27, 94–99 (1925–1926).
  11. V. S. Koroliuk, Manual de la Teoria de Probabilidades y Estadística Matemática (Mir, Moscow, 1981), p. 120.

1994

1990

D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

1980

1976

J. S. Loomis, “Analysis of interferometric data for the multiple mirror telescope optics,” J. Opt. Soc. Am. 66, 1116–1117 (1976).

1957

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9.

Cardona-Nunez, O.

Carpio-Valadez, J. M.

D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Chavez, E.

E. Chavez, C. Menchaca, D. Malacara, “Ajuste bidimensional de datos con mínimos cuadrados,” Reporte Técnico No. 7 (Centro Investigaciones Opticas, León, Guanajuato, Mexico, 1982).

Cordero-Dávila, A.

Cornejo-Rodríguez, A.

Forsythe, G. E.

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Kingslake, R.

R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. 27, 94–99 (1925–1926).

Koroliuk, V. S.

V. S. Koroliuk, Manual de la Teoria de Probabilidades y Estadística Matemática (Mir, Moscow, 1981), p. 120.

Loomis, J. S.

J. S. Loomis, “Analysis of interferometric data for the multiple mirror telescope optics,” J. Opt. Soc. Am. 66, 1116–1117 (1976).

J. S. Loomis, “A computer program for analysis of interferogramic data,” in Optical Interferometric Reductions and Interpretations, A. H. Guenther, D. H. Liebenberg, eds., (American Society for Testing and Materials, Philadelphia, Pa., 1978).
[CrossRef]

Malacara, D.

D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

E. Chavez, C. Menchaca, D. Malacara, “Ajuste bidimensional de datos con mínimos cuadrados,” Reporte Técnico No. 7 (Centro Investigaciones Opticas, León, Guanajuato, Mexico, 1982).

Menchaca, C.

E. Chavez, C. Menchaca, D. Malacara, “Ajuste bidimensional de datos con mínimos cuadrados,” Reporte Técnico No. 7 (Centro Investigaciones Opticas, León, Guanajuato, Mexico, 1982).

Rimmer, M. P.

M. P. Rimmer, “A method for evaluating lateral shearing interferograms,” Itek Internal Rep. 72-5802-1 (Itek Corporation, Lexington, Mass., 1972).

Sánchez Mondragón, J. J.

D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Silva, D. E.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9.

Appl. Opt.

J. Opt. Soc. Am.

J. S. Loomis, “Analysis of interferometric data for the multiple mirror telescope optics,” J. Opt. Soc. Am. 66, 1116–1117 (1976).

J. Soc. Ind. Appl. Math.

G. E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting on a digital computer,” J. Soc. Ind. Appl. Math. 5, 74–80 (1957).
[CrossRef]

Opt. Eng.

D. Malacara, J. M. Carpio-Valadez, J. J. Sánchez Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Trans. Opt. Soc.

R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. 27, 94–99 (1925–1926).

Other

V. S. Koroliuk, Manual de la Teoria de Probabilidades y Estadística Matemática (Mir, Moscow, 1981), p. 120.

J. S. Loomis, “A computer program for analysis of interferogramic data,” in Optical Interferometric Reductions and Interpretations, A. H. Guenther, D. H. Liebenberg, eds., (American Society for Testing and Materials, Philadelphia, Pa., 1978).
[CrossRef]

E. Chavez, C. Menchaca, D. Malacara, “Ajuste bidimensional de datos con mínimos cuadrados,” Reporte Técnico No. 7 (Centro Investigaciones Opticas, León, Guanajuato, Mexico, 1982).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 9.

M. P. Rimmer, “A method for evaluating lateral shearing interferograms,” Itek Internal Rep. 72-5802-1 (Itek Corporation, Lexington, Mass., 1972).

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Figures (9)

Fig. 1
Fig. 1

Expected deviations for tilt (t x ) and coma (c) coefficients (●) without and (+) with Gaussian noise for three fringes in the interference pattern.

Fig. 2
Fig. 2

Same as Fig. 1 but for 11 fringes in the interference pattern.

Fig. 3
Fig. 3

Same as Fig. 1 but for 21 fringes in the interference pattern.

Fig. 4
Fig. 4

Comparison of the results obtained for the expected values of t x using the analytical analysis (●) and computation simulations (+) for the three fringes in the interference pattern.

Fig. 5
Fig. 5

Same as Fig. 4 but for 11 fringes in the interference pattern.

Fig. 6
Fig. 6

Same as Fig. 4 but for 21 fringes in the interference pattern.

Fig. 7
Fig. 7

T% is defined in Eq. (45) to define the rms variations with N ϕ the number of scanned lines and three fringes in the interference pattern.

Fig. 8
Fig. 8

Same as Fig. 7 but for 11 fringes in the interference pattern.

Fig. 9
Fig. 9

Same as Fig. 7 but for 21 fringes in the interference pattern.

Equations (45)

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x = 2 k S 0 - 1 ,
y = 2 m N f + 1 ,
X i = x i ,
Y i = y i + i ,
i ~ N 0 ( 0 , σ 2 ) ,
W = k = 1 T a k g k ( x , y ) ,
S 2 = i = 1 N [ k = 1 T a k g k ( X i , Y i ) - m i ] 2 .
F ( a ^ 1 a ^ 2 a ^ T ) = B ,
f ln = i = 1 N g l ( X i , Y i ) g n ( X i , Y i ) ,
b l = i = 1 N m i g i ( X i , Y i ) ,
W 2 = c t + t y x + t x y + d ( x 2 + y 2 ) + a ( x 2 + 3 y 2 ) + c y ( x 2 + y 2 ) + S ( x 2 + y 2 ) 2 .
E p q = E [ i = 1 N X i p Y i q ] ,
E p q = i = 1 N { x i p [ y i q + q y i q - 1 E ( i ) + q ( q - 1 ) 2 ! × y i q - 2 E ( i 2 ) + + q y i E ( i q - 1 ) + 0 ( i q ) ] } .
E ( i γ ) = { 0 if γ is odd 1 · 3 · 5 ( γ - 1 ) σ γ if γ is even .
E p q = i = 1 N { [ x i p Y i q ] + q ( q - 1 ) 2 ! σ 2 [ x i p y i q - 2 ] + + [ 1 · 3 · 5 ( q - 2 ) q σ q - 2 [ x i p y i ] 1 · 3 · 5 ( q - 1 ) σ q x i p ] } q odd q even .
S γ δ = i = 1 N x i γ y i δ ,
S γ δ * = [ 1 + ( - 1 ) γ + ( - 1 ) γ + δ + ( - 1 ) δ ] x γ y δ .
( F 11 0 0 F 14 F 15 0 F 17 0 F 22 0 0 0 0 0 0 0 F 33 0 0 F 36 0 F 14 0 0 F 44 F 45 0 F 47 F 15 0 0 F 45 F 55 0 F 57 0 0 F 36 0 0 F 66 0 F 17 0 0 F 47 F 57 0 F 77 ) ( c ^ t t ^ y t ^ x d ^ a ^ c ^ s ^ ) = ( 0 0 B 3 0 0 B 6 0 ) ,
( F 11 0 F 15 0 0 F 17 0 F 22 0 0 0 0 F 14 0 F 44 F 45 0 F 47 F 15 0 F 45 F 55 0 F 57 F 17 0 F 47 F 57 0 F 77 ) ( c ^ t t ^ y d ^ a ^ s ^ ) = ( 0 0 0 0 0 ) ,
( F 33 F 36 F 36 F 66 ) ( t ^ x c ^ ) = ( B 3 B 6 ) .
f 33 = i = 1 N Y i 2 ,
f 36 = i = 1 N Y i 2 ( X i 2 + Y i 2 ) ,
f 66 = i = 1 N Y i 2 ( X i 2 + Y i 2 ) 2 ,
F 33 = i = 1 N y i 2 + N p σ 2 ,
F 36 = i = 1 N y i 2 ( x i 2 + y i 2 ) + i = 1 N ( x i 2 + 6 y i 2 ) σ 2 + 3 N σ 4 ,
F 66 = i = 1 N y i 2 ( x i 2 + y i 2 ) 2 + i = 1 N ( x i 4 + 12 x i 2 y i 2 + 15 y i 4 ) σ 2 + i = 1 N 3 ( 2 x i 2 + 15 y i 2 ) σ 4 + 15 N σ 6 .
B 3 = i = 1 N m i y i ,
B 6 = i = 1 N m i y i ( x i 2 + y i 2 ) + 3 i = 1 N m i y i σ 2 .
W t = c t + t y x + t x y + d f ( x 2 + y 2 ) .
( F 11 0 0 F 14 0 F 22 0 0 0 0 F 33 0 F 14 0 0 F 44 ) ( c ^ t t ^ y t ^ x d ^ f ) = ( 0 0 B 3 0 ) .
t ^ x = i = 1 N m i y i i = 1 N y i 2 + N σ 2 .
W 1 = t x y
W 1 = t x 1 y ,
W 2 = t x 2 y + c 2 y ( x 2 + y 2 ) .
S 1 2 = i = 1 N ( m i - t ^ x 1 Y i ) 2 ,
S 2 2 = i = 1 N [ m i - t ^ x 2 Y i - c ^ 2 Y i ( X i 2 + Y i 2 ) ] .
E ( S 1 2 ) = i = 1 N E ( m i 2 + t ^ x 1 2 Y i 2 - 2 t ^ x 1 m i Y i ) 2 ,
E ( S 2 2 ) = i = 1 N E [ m i 2 + t ^ x 2 2 Y i 2 + c ^ 2 2 Y i 2 ( X i 2 + Y i 2 ) 2 - 2 t ^ x 2 m i Y i - 2 c ^ 2 m i Y i ( X i 2 + Y i 2 ) + 2 t ^ x 2 c ^ 2 Y i 2 ( X i 2 + Y i 2 ) ] .
E ( S i 2 ) = i = 1 N m i 2 + t ^ x 1 2 F 33 - 2 t ^ x 1 B 3 ,
E ( S 2 2 ) = i = 1 N m i 2 + t ^ x 2 2 F 33 + c ^ 02 2 F 66 - 2 t ^ x 2 B 3 - 2 c ^ 2 B 6 + 2 t ^ x 2 c ^ 2 F 36 ,
E ( S 1 2 ) = i = 1 N ( m i 2 ) - B 3 2 F 33 ,
E ( S 2 2 ) = i = 1 N ( m i 2 ) + ( - F 66 B 3 2 + 2 F 36 B 3 B 6 - F 33 B 6 2 ) ( F 33 F 66 - F 36 2 ) .
rms 1 = [ i = 1 N ( m i 2 ) - B 3 2 F 33 N - 1 ] 1 / 2 ,
rms 2 = [ i = 1 N m i 2 + ( - F 66 B 3 2 + 2 F 36 B 3 B 6 - F 33 B 6 2 ) ( F 33 F 66 - F 36 2 ) N - 2 ] 1 / 2 ,
T % = ( rms 1 - rms 2 rms 1 ) 100 ,

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