Abstract

An improved moiré deflectometry phase-shifting technique is presented. A squared grating is used to multiplex the information of the deflections in two orthogonal directions in one image. This procedure avoids the need to rotate the gratings to obtain complete deflection information. However, the use of these gratings makes impossible the application of standard phase-shifting algorithms. Specifically, the problems associated with the nonsinusoidal profile of the moiré fringes and the low-modulation areas produced by the square gratings are solved. A modified moiré deflectometry phase-shifting method is designed to deal with these problems. In addition, a method to obtain the zero order of the prismatic effect is developed. The technique configures a complete and automatic method of mapping ray deflections. From them the refractive power maps can be derived. Experimental results obtained with a progressive-addition lens are shown.

© 1998 Optical Society of America

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References

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  1. O. Kafri, I. Glat, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
    [CrossRef]
  2. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
  3. M. Servin, R. Rodriguez-Vera, M. Carpio, A. Morales, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
    [CrossRef] [PubMed]
  4. H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moiré deflectometry,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).
  5. J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).
  6. T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).
  7. J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
    [CrossRef]
  8. J. Stricker, “Diffraction effects and special advantages in electronic heterodyne moiré deflectometry,” Appl. Opt. 25, 895–902 (1986).
    [CrossRef] [PubMed]
  9. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  10. J. A. Quiroga, A. González-Cano, “Phase-measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
    [CrossRef]
  11. Y. Hotta, A. Asano, S. Yokozeki, “Automated Talbot interferometer using moiré fringe scanning and ray tracing,” in Proceedings of the Seventeenth General Congress of the International Commission for Optics, Proc. SPIE2778, 1114–1115 (1996).

1997

J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
[CrossRef]

J. A. Quiroga, A. González-Cano, “Phase-measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
[CrossRef]

1995

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

1990

1987

J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).

1986

1985

O. Kafri, I. Glat, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

Alonso, J.

J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
[CrossRef]

Asano, A.

Y. Hotta, A. Asano, S. Yokozeki, “Automated Talbot interferometer using moiré fringe scanning and ray tracing,” in Proceedings of the Seventeenth General Congress of the International Commission for Optics, Proc. SPIE2778, 1114–1115 (1996).

Bernabeu, E.

J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
[CrossRef]

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moiré deflectometry,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).

Canabal, H.

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moiré deflectometry,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).

Carpio, M.

Evertz, J.

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

Farrant, D. I.

Glat, I.

O. Kafri, I. Glat, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

Glatt, I.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Gómez-Pedrero, J. A.

J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
[CrossRef]

González-Cano, A.

Hibino, K.

Hotta, Y.

Y. Hotta, A. Asano, S. Yokozeki, “Automated Talbot interferometer using moiré fringe scanning and ray tracing,” in Proceedings of the Seventeenth General Congress of the International Commission for Optics, Proc. SPIE2778, 1114–1115 (1996).

Kafri, O.

O. Kafri, I. Glat, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Larkin, K. G.

Morales, A.

Oreb, B. F.

Pfeifer, T.

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

Quiroga, J. A.

J. A. Quiroga, A. González-Cano, “Phase-measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
[CrossRef]

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moiré deflectometry,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).

Rodriguez-Vera, R.

Servin, M.

Stricker, J.

Striker, J.

J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).

Tutsch, R.

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

Wang, B.

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

Yokozeki, S.

Y. Hotta, A. Asano, S. Yokozeki, “Automated Talbot interferometer using moiré fringe scanning and ray tracing,” in Proceedings of the Seventeenth General Congress of the International Commission for Optics, Proc. SPIE2778, 1114–1115 (1996).

Appl. Opt.

J. Opt. Soc. Am. A

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).

Opt. Eng.

O. Kafri, I. Glat, “Moiré deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 944–960 (1985).
[CrossRef]

Opthal. Physiol Opt.

J. Alonso, J. A. Gómez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Opthal. Physiol Opt. 17, 522–529 (1997).
[CrossRef]

Optik (Stuttgart)

T. Pfeifer, B. Wang, J. Evertz, R. Tutsch, “Phase-shifting moiré deflectometry,” Optik (Stuttgart) 98, 158–162 (1995).

Other

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moiré deflectometry,” in Fringe ’97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Y. Hotta, A. Asano, S. Yokozeki, “Automated Talbot interferometer using moiré fringe scanning and ray tracing,” in Proceedings of the Seventeenth General Congress of the International Commission for Optics, Proc. SPIE2778, 1114–1115 (1996).

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Figures (8)

Fig. 1
Fig. 1

Experimental setup of the phase-shifting moiré deflectometer. Lenses L1 and L2 expand and collimate, respectively, the laser beam. Gratings G1 and G2 are at a distance Z. The screen S is at a distance Zs from the tested object.

Fig. 2
Fig. 2

(a) Reference moiré deflectogram in a finite-fringe configuration with Ronchi gratings. (b) Fast Fourier transform of (a) with a 2× zoom. Harmonics up to the fourth order are clearly discernible.

Fig. 3
Fig. 3

(a) Derivative of the phase map of a progressive-addition lens obtained with the four-step algorithm. Information loss is due to the harmonics errors. (b) Derivative of the phase map of the lens of (a) obtained with the 11-step algorithm. The harmonics errors are reduced.

Fig. 4
Fig. 4

Moiré deflectogram of the progressive-addition lens in an infinite-fringe configuration with SRG’s. Two black points used for position reference can be observed.

Fig. 5
Fig. 5

(a) Phase map associated with the x-deflection component obtained with Eq. (7). The low-modulation areas corrupt the information. (b) Phase map associated with the x-deflection component obtained with Eq. (10). The low-modulation areas were suppressed. Each 2π jump in the phase map corresponds to a variation of 0.363° in the ray deflection. (c) Phase map associated with the y-deflection component with low-modulation areas suppressed.

Fig. 6
Fig. 6

Phase derivatives representing (a) the power along the x direction ϕ xx , (b) torsion ϕ xy , and (c) the power along the y direction ϕ yy . The scale is expressed in diopters (Dp).

Fig. 7
Fig. 7

Absolute ray deflections: (a) x deflection and (b) y deflection in prismatic diopters (Δ). A measure of 1Δ corresponds to a deflection of 1 cm in 1 mts. The solid lines delineate the zones of zero x and y deflections, respectively.

Fig. 8
Fig. 8

Profile along the principal meridian (centered vertical line) of the y deflection represented in Fig. 7(b). The solid curve corresponds to the values obtained by moiré deflectometry. The cross points represent the values measured with a commercial focimeter.

Equations (12)

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I x ,   y = 1 4 + 2 π 2 n = 0 cos π 2 n + 1 2 Z * 2 n + 1 2 × cos 2 π 2 n + 1 χ p + y θ p ,
I x ,   y = 1 4 + 2 π 2 n = 0 cos π 2 n + 1 2 Z * 2 n + 1 2 × cos 2 n + 1 2 π χ p + y θ p + Z φ x ,   y p .
ϕ n x ,   y = 2 n + 1 2 π Z p   φ x ,   y .
ϕ = arctan i = 1 11   a i I i i = 1 11   b i I i , a i = 3 0   - 1   - 4   - 7   - 6   0   6   7   4   1   0 , b i = - 2   - 5   - 6   - 1   8   12   8   - 1   - 6   - 5   - 2 ,
I x ,   y = I 0 + M   cos 2 π   Z φ y x ,   y p + α × cos 2 π   Z φ x x ,   y p + β ,
A 1 = i = 1 11   a i I 1 i ,     B 1 = i = 1 11   b i I 1 i ,
ϕ 1 x = arctan A 1 / B 1 ,
m 1 x ,   y = A 1 2 + B 1 2 1 / 2 = M   cos 2 π   Z φ y x ,   y p + β 1 .
m 2 x ,   y = M   sin 2 π   Z φ y x ,   y p + β 1 .
ϕ x = arctan m 1 A 1 + m 2 A 2 m 1 B 1 + m 2 B 2 .
ϕ I , II x ,   y = 2 π p   Z I , II φ x ,   y mod   2 π .
| Z I - Z II | < p 2   φ max ,

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