Abstract

Previous implementations of the iterative phase shifting method, in which the phase of a test object is computed from measurements using a phase shifting interferometer with unknown positions of the reference, do not provide an accurate way of knowing when convergence has been attained. We present a new approach to this method that allows us to deterministically identify convergence. The method is tested with a home-built Fizeau interferometer that measures optical surfaces polished to λ/100 using the Hydra tool. The intrinsic quality of the measurements is better than 0.5nm. Other possible applications for this technique include fringe projection or any problem where phase shifting is involved.

© 2009 Optical Society of America

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 2005), Chap. 7.
    [CrossRef]
  2. D. Malacara, Optical Shop Testing (Wiley, 2007), Chap. 14.
    [CrossRef]
  3. Y. Morimoto and M. Fujisawa, “Fringe-pattern analysis by phase-shifting method using extraction of characteristic,” Exp. Tech. 20(4), 25-29 (1996).
    [CrossRef]
  4. A. Nava-Vega, L. Salas, E. Luna, and A. Cornejo-Rodriguez, “Correlation algorithm to recover the phase of a test surface using phase-shifting interferometry,” Opt. Express 12, 5296-5306 (2004).
    [CrossRef] [PubMed]
  5. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
  6. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671-1673 (2004).
    [CrossRef] [PubMed]
  7. J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47, 480-485(2008).
    [CrossRef] [PubMed]
  8. L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
    [CrossRef]
  9. E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).
  10. M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).
  11. M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
    [CrossRef]

2008 (1)

2004 (3)

2003 (1)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

1996 (1)

Y. Morimoto and M. Fujisawa, “Fringe-pattern analysis by phase-shifting method using extraction of characteristic,” Exp. Tech. 20(4), 25-29 (1996).
[CrossRef]

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Chai, L.

Cornejo-Rodriguez, A.

Cruz-Gonzalez, I.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

Fujisawa, M.

Y. Morimoto and M. Fujisawa, “Fringe-pattern analysis by phase-shifting method using extraction of characteristic,” Exp. Tech. 20(4), 25-29 (1996).
[CrossRef]

Garcia, V.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

Han, B.

López, E.

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Luna, E.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

A. Nava-Vega, L. Salas, E. Luna, and A. Cornejo-Rodriguez, “Correlation algorithm to recover the phase of a test surface using phase-shifting interferometry,” Opt. Express 12, 5296-5306 (2004).
[CrossRef] [PubMed]

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 2005), Chap. 7.
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, 2007), Chap. 14.
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 2005), Chap. 7.
[CrossRef]

Martinez, B.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

Morimoto, Y.

Y. Morimoto and M. Fujisawa, “Fringe-pattern analysis by phase-shifting method using extraction of characteristic,” Exp. Tech. 20(4), 25-29 (1996).
[CrossRef]

Nava, A.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

Nava-Vega, A.

Nunez, M.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Quiroz, F.

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Ruiz, E.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).

Salas, L.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

A. Nava-Vega, L. Salas, E. Luna, and A. Cornejo-Rodriguez, “Correlation algorithm to recover the phase of a test surface using phase-shifting interferometry,” Opt. Express 12, 5296-5306 (2004).
[CrossRef] [PubMed]

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Salinas, J.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Servin, M.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 2005), Chap. 7.
[CrossRef]

Sohn, E.

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Wang, Z.

Xu, J.

Xu, Q.

Opt. Commun. (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Appl. Opt. (1)

Exp. Tech. (1)

Y. Morimoto and M. Fujisawa, “Fringe-pattern analysis by phase-shifting method using extraction of characteristic,” Exp. Tech. 20(4), 25-29 (1996).
[CrossRef]

Opt. Eng. (1)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

M. Nunez, J. Salinas, E. Luna, L. Salas, E. Ruiz, E. Sohn, A. Nava, I. Cruz-Gonzalez, and B. Martinez, “Surface roughness results using a hydrodynamic polishing tool (HyDra),” Proc. SPIE 5494, 459-467 (2004).
[CrossRef]

Other (4)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 2005), Chap. 7.
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, 2007), Chap. 14.
[CrossRef]

E. Ruiz, E. Sohn, L. Salas, and E. Luna, “Hydrodynamic radial flux polishing and grinding tool for optical and semiconductor surfaces,” U.S. patent 7,169,012 (30 January 2007).

M. Nunez, E. Luna, L. Salas, E. López, F. Quiroz, and J. Salinas, “Interferometro de Fizeau para pruebas de superficies opticas,” Technical Report RT-2004-01 (Instituto de Astronomía, Universidad Nacional Autónoma de México, 2004).

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Figures (5)

Fig. 1
Fig. 1

Deviations from a perfect cosine as the iterative phase shifting algorithm converges. The top left panel shows the errors with the initial trial for the positions χ 0 ( l ) . Subsequent panels show improved fits as the iterative algorithm converges.

Fig. 2
Fig. 2

Zerodur test sample with imperfections, measured by (a) Zygo interferometer; (b) our own Fizeau interferometer and method gives a difference shown in (c). Gray scales are common for all three panels and are shown on top from 30 to 30 nm . A scan of (a) (dotted curve) and (b) (solid curve) is shown in (d).

Fig. 3
Fig. 3

Surface polished with Hydra to λ / 100 used in the experiments. The area of interest is inscribed in the central circular region.

Fig. 4
Fig. 4

Surface error as a function of the number of interferograms. The solid curve shows the theoretically expected uncertainty decreasing as N. The measured surface errors from independent experiments are marked with crosses and error bars. The surface uncertainty calculated from the dispersion in the fit to a single cosine function is shown as a dashed curve. All three results show perfect agreement.

Fig. 5
Fig. 5

Comparison of convergence criteria. Micro-roughness measurements of surface in two separate experiments, one (images on the left) with no convergence, and the other (right) with successful convergence. The top panels show the sequential convergence indicator (solid curve) always reaching convergence levels (left scale) in a few iterations (bottom scale), while the absolute convergence criteria (dashed curve) differentiates both cases according to its criteria (right scale). The middle panels reinforce this conclusion by showing the fit of the solutions to the cosine function and the bottom panels by showing the obtained phase images.

Equations (19)

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I ( x , y , l ) = A ( x , y ) + B ( x , y ) cos ( Φ o ( x , y ) + 2 π λ χ ( i ) ( l ) ) ,
I ( x , y , l ) = A ( x , y ) + C ( x , y ) cos ( 2 π λ χ ( i ) ( l ) ) + D ( x , y ) sin ( 2 π λ χ ( i ) ( l ) ) ,
Φ o ( x , y ) = tan 1 D ( x , y ) C ( x , y ) .
S = l = 1 N ( I M ( x , y , l ) I ( x , y , l ) ) 2
I ( x , y , l ) A ( x , y ) B ( x , y ) = cos ( Φ o ( x , y ) + 2 π λ χ ( i ) ( l ) ) .
d ( x , y , l ) = I M ( x , y , l ) A ( x , y ) B ( x , y ) ,
S 2 = ( x , y ) in R ( d ( x , y , l ) Y ( x , y , l ) ) 2 ,
Y ( x , y , l ) = cos ( Φ o ( x , y ) + 2 π λ χ ( i + 1 ) ( l ) ) ,
Y ( x , y , l ) = E l cos ( Φ o ( x , y ) ) + F l sin ( Φ o ( x , y ) ) ,
χ ( i + 1 ) ( l ) = tan 1 F l E l .
δ Y = 1 M N l = 1 N ( x , y ) in R M ( d ( x , y , l ) cos ( X ( x , y , l ) ) 2
X ( x , y , l ) = Φ o ( x , y ) + 2 π λ χ ( l ) .
δ Y = 2 π δ X .
δ Φ = 1 3 N 2 π λ δ χ ,
δ X = 2 π λ δ χ .
δ Φ = 1 3 N π 2 δ Y .
δ S = 1 2 3 N δ χ ,
δ S = λ 8 3 N δ Y .
e average = 2 π λ ( N 1 ) l = 2 N | ( χ i ( l ) χ i ( 1 ) ) ( χ i 1 ( l ) χ i 1 ( 1 ) ) | .

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