Abstract

A novel phase-step calibration technique is presented on the basis of a two-run-times-two-frame phase-shift method. First the symmetry factor M is defined to describe the distribution property of the distorted phase due to phase-shifter miscalibration; then the phase-step calibration technique, in which two sets of two interferograms with a straight fringe pattern are recorded and the phase step is obtained by calculating M of the wrapped phase map, is developed. With this technique, a good mirror is required, but no uniform illumination is needed and no complex mathematical operation is involved. This technique can be carried out in situ and is applicable to any phase shifter, whether linear or nonlinear.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, 1988), Vol. 26, pp. 350-393.
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital beam measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]
  3. P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
    [CrossRef]
  4. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]
  5. Y. Y. Cheng and J. C. Wyant, "Phase-shifter calibration in phase-shifting interferometry," Appl. Opt. 24, 3049-3052 (1985).
    [CrossRef] [PubMed]
  6. K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
    [CrossRef]
  7. N. A. Ochoa and J. M. Huntley, "Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry," Opt. Eng. 37, 2501-2505 (1998).
    [CrossRef]
  8. H. van Brug, "Phase-step calibration for phase-stepped interferometry," Appl. Opt. 38, 3549-3555 (1999).
    [CrossRef]
  9. X. Chen, M. Gramaglia, and J. A. Yeazell, "Phase-shift calibration algorithm for phase-shifting interferometry," J. Opt. Soc. Am. A 17, 2061-2066 (2000).
    [CrossRef]
  10. K. A. Goldberg and J. Bokor, "Fourier-transform method of phase-shift determination," Appl. Opt. 40, 2886-2894 (2001).
    [CrossRef]
  11. C. Guo, Z. Rong, J. He, H. Wang, L. Cai, and Y. Wang, "Determination of global phase shifts between interferograms by use of an energy-minimum algorithm," Appl. Opt. 42, 6514-6519 (2003).
    [CrossRef] [PubMed]
  12. X. Zhong, "A four-frame phase shift method insensitive to phase shifter nonlinearity," J. Opt. 8, 300-303 (2006).
    [CrossRef]
  13. X. Zhong, "Methods for removal of spurious reflection effect on phase-shifting interferometry," J. Opt. 8, 617-624 (2006).
    [CrossRef]

2006 (2)

X. Zhong, "A four-frame phase shift method insensitive to phase shifter nonlinearity," J. Opt. 8, 300-303 (2006).
[CrossRef]

X. Zhong, "Methods for removal of spurious reflection effect on phase-shifting interferometry," J. Opt. 8, 617-624 (2006).
[CrossRef]

2003 (1)

2001 (1)

2000 (1)

1999 (1)

1998 (1)

N. A. Ochoa and J. M. Huntley, "Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry," Opt. Eng. 37, 2501-2505 (1998).
[CrossRef]

1995 (1)

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

1985 (1)

1983 (1)

1974 (1)

1966 (1)

P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

Bokor, J.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Cai, L.

Carré, P.

P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

Chen, X.

Cheng, Y. Y.

Creath, K.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, 1988), Vol. 26, pp. 350-393.

Dobbins, B. N.

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

Elssner, K. E.

Gallagher, J. E.

Goldberg, K. A.

Gramaglia, M.

Grzanna, J.

Guo, C.

He, J.

He, S. P.

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

Herriott, D. R.

Huntley, J. M.

N. A. Ochoa and J. M. Huntley, "Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry," Opt. Eng. 37, 2501-2505 (1998).
[CrossRef]

Jambunathan, K.

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

Merkel, K.

Ochoa, N. A.

N. A. Ochoa and J. M. Huntley, "Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry," Opt. Eng. 37, 2501-2505 (1998).
[CrossRef]

Rong, Z.

Rosenfeld, D. P.

Schwider, J.

Spolaczyk, R.

van Brug, H.

Wang, H.

Wang, L. S.

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

Wang, Y.

White, A. D.

Wyant, J. C.

Yeazell, J. A.

Zhong, X.

X. Zhong, "Methods for removal of spurious reflection effect on phase-shifting interferometry," J. Opt. 8, 617-624 (2006).
[CrossRef]

X. Zhong, "A four-frame phase shift method insensitive to phase shifter nonlinearity," J. Opt. 8, 300-303 (2006).
[CrossRef]

Appl. Opt. (6)

J. Opt. (2)

X. Zhong, "A four-frame phase shift method insensitive to phase shifter nonlinearity," J. Opt. 8, 300-303 (2006).
[CrossRef]

X. Zhong, "Methods for removal of spurious reflection effect on phase-shifting interferometry," J. Opt. 8, 617-624 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Metrologia (1)

P. Carré, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

Opt. Eng. (1)

N. A. Ochoa and J. M. Huntley, "Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry," Opt. Eng. 37, 2501-2505 (1998).
[CrossRef]

Opt. Laser Technol. (1)

K. Jambunathan, L. S. Wang, B. N. Dobbins, and S. P. He, "Semi-automatic phase shift calibration using digital speckle pattern interferometry," Opt. Laser Technol. 27, 145-151 (1995).
[CrossRef]

Other (1)

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, 1988), Vol. 26, pp. 350-393.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Response plots of a phase shifter in two runs of phase shifting, under the same operating condition: (a) with stability, irreproducibility, and nonlinearity; (b) with instability, reproducibility, and nonlinearity; (c) with stability, reproducibility, and linearity.

Fig. 2
Fig. 2

(a) Retrieved phase plots over the interval [ β π β ] . The solid curve is for β = π / 2   rad ; the dashed curve is for β = 1.2   rad , and the dotted curve is for β = 0.9   rad . (b) Ratio of the areas of the two curvilinear triangles AOC and BOD depends on the phase step β.

Fig. 3
Fig. 3

Computed plot of M versus β.

Fig. 4
Fig. 4

Plot of the computational error of M versus β in the case of P = 1000 .

Fig. 5
Fig. 5

(a) Horizontal π block. (b) Slightly tilted π block. (c) Dimensions of a general π block.

Fig. 6
Fig. 6

Schematics of the optical path difference between a spherical wavefront and a planar wavefront.

Fig. 7
Fig. 7

Distorted phase maps due to phase-step miscalibration ( β = 1.4   rad ) . (a) Horizontal π blocks without noise. (b) Slightly tilted π blocks with random noise.

Tables (2)

Tables Icon

Table 2 Phase-Step Calibration Results

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

I i ( x , y ) = A ( x , y ) + B ( x , y ) cos [ ϕ ( x , y ) + α i ] ,
I 1 = A + B cos ( ϕ α / 2 ) ,
I 2 = A + B cos ( ϕ + β α / 2 ) ,
I 3 = A + B cos ( ϕ + α / 2 ) ,
I 4 = A + B cos ( ϕ + β + α / 2 ) ,
tan ϕ = ( I 3 I 1 ) sin β ( I 4 I 2 ) ( I 3 I 1 ) cos β .
tan ϕ = I 3 I 1 I 4 I 2 .
ϕ = f ( ϕ ) = arctan ( sin ϕ sin ( ϕ + β ) ) .
S = β 0 ϕ d ϕ ,   S + = 0 π β ϕ d ϕ .
S = π P ( π 4 + ϕ ) ,   S + = π P ( π 4 + ϕ + ) ,
M = ( π 4 + ϕ L ) / ( π 4 + ϕ + L ) ,   ( | S | S + ) .
M = ( π 4 + ϕ h ) / ( π 4 + ϕ + h ) ,   ( | S | S + ) .
X * X ¯ = t r 2 / 2 R ,
X * X ¯ = t r θ + r 2 / 2 R .
C * D ¯ A * B * ¯ 2 / 8 R A B ¯ 2 / 8 R .

Metrics