Abstract

The presence of uncontrolled mechanical vibrations is typically the main precision-limiting factor of a phase-shifting interferometer. We present a method that instead of trying to insolate vibrations; it takes advantage of their presence to produce the different phase-steps. The method is based on spatial and time domain processing techniques to compute first the different unknown phase-steps and then reconstruct the phase from these tilt-shifted interferograms. In order to compensate the camera movement, it is needed to perform an affine registration process between the different interferograms. Simulated and experimental results demonstrate the effectiveness of the proposed technique without the use of any phase-shifter device.

© 2011 OSA

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References

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  1. K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).
  2. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)
  3. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
    [CrossRef]
  4. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [CrossRef] [PubMed]
  5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
    [CrossRef]
  6. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
    [CrossRef] [PubMed]
  7. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
    [CrossRef]
  8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001).
    [CrossRef]
  9. G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
    [CrossRef] [PubMed]
  10. R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).
  11. C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).
  12. J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

2008 (2)

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

2004 (1)

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

2002 (1)

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

2001 (2)

Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001).
[CrossRef]

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
[CrossRef]

2000 (1)

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[CrossRef]

1988 (1)

K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

Bokor, J.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
[CrossRef]

Chai, L.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

Chen, M.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[CrossRef]

Creath, K.

K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

Ge, Z.

Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001).
[CrossRef]

Goldberg, K. A.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
[CrossRef]

Guo, H.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[CrossRef]

Han, B.

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

Meneses-Fabian, C.

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

Rodriguez-Zurita, G.

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

Takeda, M.

Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001).
[CrossRef]

Toto-Arellano, N. I.

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

Vázquez-Castillo, J. F.

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

Wang, Z.

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

Wei, C.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[CrossRef]

Xu, J.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

Xu, Q.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

Appl. Opt. (3)

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
[CrossRef]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

Opt. Lett. (2)

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008).
[CrossRef] [PubMed]

Proc. SPIE (1)

Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001).
[CrossRef]

Prog. Opt. (1)

K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

Other (4)

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)

R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).

C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).

J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

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Figures (10)

Fig. 1
Fig. 1

Scheme of a PZT Phase-Shifter device with tilt-shift error

Fig. 2
Fig. 2

(a) phase example, (b) phase example transformed by an affine transformation corresponding to a translation of −5 and 12 pixels, a rotation of 0.5 rad and a scale change of 1.2 and 0.98 pixels in the x and y axis respectively, (c) corresponding binarized map of (a), (d) corresponding binarized map of (b), (e) phase example with the detected control points, (f) phase example transformed with the detected control points

Fig. 3
Fig. 3

Phase map described in the simulation section affected by the affine transformations shown in Table 1

Fig. 4
Fig. 4

Resultant interferogram of the phase shown in Fig. 2(b)

Fig. 5
Fig. 5

Different phases Φ 0 ( x n , y n ) shown in Fig. 2, recovered using the Fourier transform demodulation method with the detected control points

Fig. 6
Fig. 6

Difference between the phases after the affine registration process and the phase corresponding to the first interferogram shown in Fig. 2(a) that is used as reference

Fig. 7
Fig. 7

Recovered phase Φ 0 ( x , y ) after the general temporal demodulation method

Fig. 8
Fig. 8

Difference between the recovered phase and the theoretical phase

Fig. 9
Fig. 9

Results obtained by the Zygo GPI interferometer for the λ/100 high quality glass plate

Fig. 10
Fig. 10

Result obtained by the Mach-Zenhder interferometer using the proposed method for the λ/100 high quality glass plate

Tables (5)

Tables Icon

Table 1 Affine transformations introduced to the actual phase map

Tables Icon

Table 2 , Introduced tilt-shifts to generate the different interferograms in the simulation

Tables Icon

Table 3 , rms (root mean square error) of the difference between the phases after affine rectification and the phase corresponding to the first interferogram that is used as reference

Tables Icon

Table 4 , Recovered tilt-shifts in the simulation

Tables Icon

Table 5 , rms (root mean square) and pv (peak to valley) of the wave-front error measured by the Zygo interferometer and with the Mach-Zenhder interferometer for the λ/100 high quality glass plate

Equations (16)

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I n ( x , y ) = A ( x , y ) + B ( x , y ) cos [ Φ 0 ( x , y ) + d n ]
I n t ( x , y ) = A ( x , y ) + B ( x , y ) cos [ Φ 0 ( x , y ) + a n x + b n y + c n + d n ]
I n ( x n , y n , t n ) = A ( x n , y n ) + B ( x n , y n ) cos [ Φ 0 ( x n , y n ) + a ( t n ) x n + b ( t n ) y n + c ( t n ) ]
[ x n ,   y n ,   1 ] T = M ( t n ) [ x ,   y ,   1 ] T
Φ n ( x n , y n ) = Φ 0 ( x n , y n ) + a ( t n ) x n + b ( t n ) y n + c ( t n )
Φ 0 ( x n , y n ) = Φ n ( x n , y n ) P n ( x n , y n )
P n ( x n , y n ) = [ a ( t n ) + a 0 ] x n + [ b ( t n ) + b 0 ] y n + [ c ( t n ) + c 0 ]
I n ( x , y , t n ) = A ( x , y ) + B ( x , y ) cos [ Φ ^ 0 ( x , y ) + P n ( x , y ) ]
P n ( x , y ) = [ a ( t n ) + a 0 ] x + [ b ( t n ) + b 0 ] y + [ c ( t n ) + c 0 ]
I n ( x , y , t n ) = α 0 + α 1 cos [ P n ( x , y ) ] + α 2 sin [ P n ( x , y ) ]
S = n ( I n I n ) 2
A α = b
A = [ N n cos [ P n ( x , y ) ) ] n sin [ P n ( x , y ) ) ] n cos [ P n ( x , y ) ) ] n cos 2 [ P n ( x , y ) ) ] n cos [ P n ( x , y ) ) ] sin [ P n ( x , y ) ) ] n sin [ P n ( x , y ) ) ] n cos [ P n ( x , y ) ) ] sin [ P n ( x , y ) ) ] n sin 2 [ P n ( x , y ) ) ] ]
b = [ n I n n I n cos [ P n ( x , y ) ] n I n sin [ P n ( x , y ) ] ]
α = A 1 · b
Φ ^ 0 = tan 1 ( α 2 α 1 )

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