Abstract

This paper shows how measurement errors in phase shifting interferometry (PSI) can be described to a high degree of accuracy in a linear approximation. System error sources considered here are light source instability, imperfect reference phase shifting, mechanical vibrations, nonlinearity of the detector, and quantization of the detector signal. The measurement inaccuracies resulting from these errors are calculated in linear approximation for several formulas commonly used for PSI. The results are presented in tables for easy calculation of the measurement error magnitudes for known system errors. In addition, this paper discusses the measurement error reduction which can be achieved by choosing an appropriate phase calculation formula.

© 1991 Optical Society of America

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References

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489–1497 (1985).
    [CrossRef] [PubMed]
  3. K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455–1464 (1985).
    [CrossRef]
  4. F. H. Groen, H. J. Frankena, A. Apituley, “Wavefront Measurement of Optical Waveguides by Digital Inteferometry,” ECOISA Summaries, F3, Amsterdam (1989).
  5. A. A. M. Maas, H. A. Vrooman, “In-Plane Strain Measurement by Digital Phase Shifting Speckle Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 248–256 (1989).
  6. K. Creath, “Phase-Measurement Interferometry Techniques,” Prog Optics 26, 351–398 (1988).
  7. K. Creath, “Comparisons of Phase-Measurement Algorithms,” in Proc. Soc. Photo-Opt. Instrum. Eng. 680, 19–28 (1986).
  8. C. Ai, J. C. Wyant, “Effect of Piezoelectric Transducer Nonlinearity on Phase Shift Interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]
  9. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systematic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  10. P. Hariharan, B. F. Oreb, T. Eiju, “Digitial Phase-Shifting Interferometry: a Simple Error-Compensating Phase Calculation Algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  11. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of Phase Shifting Interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  12. J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
    [CrossRef]
  13. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  14. K. A. Stetson, W. R. Brohinsky, “Electrooptic Holography and Its Application to Hologram Interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef] [PubMed]
  15. Another 3Buckets (120) phase calculation formula,8Φ=arctan3(I3-I1)I1-2I2+I3, is fully equivalent. Both formulas result in the same measurement errors for all error sources. This will not be proven here, but can be checked by performing the calculations of the measurement errors in the following sections for both formulas.
  16. The minus sign here is caused by our choice of an increasing reference phase value for subsequent buckets in Eq. (1).
  17. In fact, we made the approximation ∫αiαi+1cos[Φ+p(α′)]dα′≈∫αiαi+1cos(Φ+α′+{∫αiαi+1[p(β)-β]dβ})dα′, where Φ is the initial phase, αi and αi+1 are the ideal initial and ideal final values of the reference phase, respectively and p(α′) is given by p(α′)=α′+∊1α′+∊2(α′)2/2π+∊3(α′)3/(2π)2+….
  18. In performing the simulations for phase integration much calculation time can be saved by using the following expression: ∫αiαi+1cos[Φ+p(α′)]dα′=cos(Φ)∫αiαi+1cos[p(α′)]dα′-sin(Φ)∫αiαi+1sin[p(α′)]dα′, where the remarks of Ref. 17 apply again. Now, the integrations, which were performed numerically with Simpson’s rule, are independent of the initial phase Φ. Hence, the evaluation of the integrals has only to be carried out once for the calculation of the measurement error over a range of initial phase values.
  19. For the Carré formula, being independent of a linear δΦref, the measurement error vanishes exactly and not only in linear approximation. For the 5Buckets (90), Averaging 3 (90) and Averaging 4 (90) formulas the quadratic dependence of the measurement error amplitudes on ∊1 is calculated from simulation results and given in footnotes 2–7 of Table I. The quadratic coefficient for the 5Buckets (90) formula is about a factor of 2 smaller than for the other two formulas and its numerical value is in agreement with the result derived analytically by Hariharan.10

1989 (1)

A. A. M. Maas, H. A. Vrooman, “In-Plane Strain Measurement by Digital Phase Shifting Speckle Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 248–256 (1989).

1988 (2)

1987 (2)

1986 (1)

K. Creath, “Comparisons of Phase-Measurement Algorithms,” in Proc. Soc. Photo-Opt. Instrum. Eng. 680, 19–28 (1986).

1985 (3)

1984 (1)

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

1983 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Ai, C.

Apituley, A.

F. H. Groen, H. J. Frankena, A. Apituley, “Wavefront Measurement of Optical Waveguides by Digital Inteferometry,” ECOISA Summaries, F3, Amsterdam (1989).

Bhushan, B.

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef] [PubMed]

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Cheng, Y.-Y.

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Creath, K.

K. Creath, “Phase-Measurement Interferometry Techniques,” Prog Optics 26, 351–398 (1988).

K. Creath, “Comparisons of Phase-Measurement Algorithms,” in Proc. Soc. Photo-Opt. Instrum. Eng. 680, 19–28 (1986).

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Eiju, T.

Elssner, K.-E.

Frankena, H. J.

F. H. Groen, H. J. Frankena, A. Apituley, “Wavefront Measurement of Optical Waveguides by Digital Inteferometry,” ECOISA Summaries, F3, Amsterdam (1989).

Gallagher, J. E.

George, O. E.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Groen, F. H.

F. H. Groen, H. J. Frankena, A. Apituley, “Wavefront Measurement of Optical Waveguides by Digital Inteferometry,” ECOISA Summaries, F3, Amsterdam (1989).

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Kinnstaetter, K.

Koliopoulos, C. L.

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef] [PubMed]

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Lohmann, A. W.

Maas, A. A. M.

A. A. M. Maas, H. A. Vrooman, “In-Plane Strain Measurement by Digital Phase Shifting Speckle Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 248–256 (1989).

Merkel, K.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

Spolaczyk, R.

Stetson, K. A.

Streibl, N.

Vrooman, H. A.

A. A. M. Maas, H. A. Vrooman, “In-Plane Strain Measurement by Digital Phase Shifting Speckle Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 248–256 (1989).

White, A. D.

Wyant, J. C.

C. Ai, J. C. Wyant, “Effect of Piezoelectric Transducer Nonlinearity on Phase Shift Interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef] [PubMed]

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Appl. Opt. (7)

ASLE Trans. (1)

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, “An Optical Profilometer for Surface Characterization of Magnetic Media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Acta (1)

K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

A. A. M. Maas, H. A. Vrooman, “In-Plane Strain Measurement by Digital Phase Shifting Speckle Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 1162, 248–256 (1989).

K. Creath, “Comparisons of Phase-Measurement Algorithms,” in Proc. Soc. Photo-Opt. Instrum. Eng. 680, 19–28 (1986).

Prog Optics (1)

K. Creath, “Phase-Measurement Interferometry Techniques,” Prog Optics 26, 351–398 (1988).

Other (6)

Another 3Buckets (120) phase calculation formula,8Φ=arctan3(I3-I1)I1-2I2+I3, is fully equivalent. Both formulas result in the same measurement errors for all error sources. This will not be proven here, but can be checked by performing the calculations of the measurement errors in the following sections for both formulas.

The minus sign here is caused by our choice of an increasing reference phase value for subsequent buckets in Eq. (1).

In fact, we made the approximation ∫αiαi+1cos[Φ+p(α′)]dα′≈∫αiαi+1cos(Φ+α′+{∫αiαi+1[p(β)-β]dβ})dα′, where Φ is the initial phase, αi and αi+1 are the ideal initial and ideal final values of the reference phase, respectively and p(α′) is given by p(α′)=α′+∊1α′+∊2(α′)2/2π+∊3(α′)3/(2π)2+….

In performing the simulations for phase integration much calculation time can be saved by using the following expression: ∫αiαi+1cos[Φ+p(α′)]dα′=cos(Φ)∫αiαi+1cos[p(α′)]dα′-sin(Φ)∫αiαi+1sin[p(α′)]dα′, where the remarks of Ref. 17 apply again. Now, the integrations, which were performed numerically with Simpson’s rule, are independent of the initial phase Φ. Hence, the evaluation of the integrals has only to be carried out once for the calculation of the measurement error over a range of initial phase values.

For the Carré formula, being independent of a linear δΦref, the measurement error vanishes exactly and not only in linear approximation. For the 5Buckets (90), Averaging 3 (90) and Averaging 4 (90) formulas the quadratic dependence of the measurement error amplitudes on ∊1 is calculated from simulation results and given in footnotes 2–7 of Table I. The quadratic coefficient for the 5Buckets (90) formula is about a factor of 2 smaller than for the other two formulas and its numerical value is in agreement with the result derived analytically by Hariharan.10

F. H. Groen, H. J. Frankena, A. Apituley, “Wavefront Measurement of Optical Waveguides by Digital Inteferometry,” ECOISA Summaries, F3, Amsterdam (1989).

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Figures (4)

Fig. 1
Fig. 1

General scheme of an interferometer with a reference phase shifter to enable PSI.

Fig. 2
Fig. 2

(a) Measurement error amplitudes for the 4Buckets (90) formula in the case of phase stepping calculated by simulation for linear, quadratic and third-order δΦref (i = 1, 2, and 3, respectively). The linearly approximated amplitudes for these cases are also plotted. (b) Relative errors of the linearly approximated amplitudes shown in (a).

Fig. 3
Fig. 3

(a) Measurement error amplitudes for the 4Buckets (90) formula in the case of phase integration calculated by simulation for linear, quadratic and third-order δΦref (i = 1, 2, and 3, respectively). The linearly approximated amplitudes for these cases are also plotted. (b) Relative errors of the linearly approximated amplitudes shown in (a).

Fig. 4
Fig. 4

(a) Measurement error amplitudes for the 4Buckets (90) formula for third-power detector nonlinearity [c is defined as in Eq. (24)] in the case of phase stepping and phase integration (bias intensity and modulation depth are 0.5 and 1, respectively). The linearly approximated amplitudes for those cases are also shown. (b) Relative errors of the linearly approximated amplitudes for the same cases as (a). In this graph the relative error for phase integration is indistinguishable from that for phase stepping.

Tables (6)

Tables Icon

Table I Linearly Approximated Measurement Error Amplitudes for Linear, Quadratic, Linearly Compensated Quadratic and Third-Order δΦref

Tables Icon

Table II Measurement Errors Caused by Nonlinearity of the Detector Using a Polynomial Description for the Detector Nonlinearity [Eq. (24)]

Tables Icon

Table III Measurement Error Caused by the Nonlinearity of the Detector from Eq. (29)

Tables Icon

Table IV Contribution from Each Bucket to the Measurement Error Caused by Light Source Instability

Tables Icon

Table V Contribution from Each Bucket to the Measurement Error Caused by Deviation of the Reference Phase

Tables Icon

Table VI Correction Factors for Eq. (32) If the Phase Calculation Formula Using RBuckets does not follow from the General Buckets Formula [Eq. (6)]

Equations (42)

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I i ( x , y ) = I 0 ( x , y ) { 1 + m cos [ Φ ( x , y ) + α i ] } ,             ( i = 1 , 2 , ) ,
Φ = arctan 3 ( I 3 - I 2 ) 2 I 1 - I 2 - I 3 ,             ( α i = ( i - 1 ) 2 π 3 ) 1 , 15 ,
Φ = arctan I 3 - I 2 I 1 - I 2 ,             ( α i = ( i - 1 ) π 2 ) 12 ,
Φ = arctan I 4 - I 2 I 1 - I 3 ,             ( α i = ( i - 1 ) π 2 ) 1 ,
Φ = arctan 2 ( I 4 - I 2 ) I 1 - 2 I 3 + I 5 ,             ( α i = ( i - 1 ) π 2 ) 9 .
Φ = - arctan i = 1 R I i sin α i i = 1 R I i cos α i ,             ( α i = ( i - 1 ) 2 π R ) 16 .
Φ = 1 2 arctan I 3 - I 2 I 1 - I 2 + 1 2 arctan I 4 - I 3 I 2 - I 3 ,             ( α i = ( i - 1 ) π 2 ) 7 ,
Φ = 1 2 arctan 3 ( I 3 - I 2 ) 2 I 1 - I 2 - I 3 + 1 2 arctan 3 ( I 4 - I 3 ) 2 I 2 - I 3 - I 4 ,             ( α i = ( i - 1 ) 2 π 3 ) 8 ,
Φ = 1 2 arctan I 4 - I 2 I 1 - I 3 + 1 2 arctan I 5 - I 3 I 2 - I 4 ,             ( α i = ( i - 1 ) π 2 ) 8 .
Φ = arctan [ 3 ( I 2 - I 3 ) - ( I 1 - I 4 ) ] [ ( I 2 - I 3 ) + ( I 1 - I 4 ) ] ( I 2 + I 3 ) - ( I 1 + I 4 ) ,             ( α i = ( i - 1 ) α 0 ) .
δ α i = 1 α i + 2 α i 2 2 π + 3 α i 3 4 π 2 + ,
δ Φ = i = 1 N ( Φ I i ) ( I i α i ) δ α i ,
δ Φ = 1 4 ( δ α 1 + δ α 2 + δ α 3 + δ α 4 ) + 1 4 ( δ α 2 + δ α 4 - δ α 1 - δ α 3 ) cos ( 2 Φ ) .
δ α i = 1 ( i - 1 ) π 2 ,
δ Φ = 1 [ 3 π 4 + π 4 cos ( 2 Φ ) ] .
δ Φ = 2 [ 7 π 16 + 3 π 16 cos ( 2 Φ ) ] ,
δ Φ = 3 [ 9 π 32 + 5 π 32 cos ( 2 Φ ) ] .
δ Φ = 1 [ π + π 4 cos ( 2 Φ ) ] ,
δ Φ = 2 [ 2 π 3 + π 4 cos ( 2 Φ ) ] ,
δ Φ = 3 [ π 2 + 31 π 128 cos ( 2 Φ ) ] .
1 = - 2 .
δ Φ = - 2 [ 0.982 + 0.196 cos ( 2 Φ ) ] .
δ Φ = 2 [ 1.047 + 0.054 cos ( 2 Φ ) ] .
1 π R sin ( 2 π / R )
2 π R ( R - 2 ) sin 2 ( 2 π / R ) + 1 R 2 sin 2 ( 2 π / R )
2 π ( R 2 - 1 ) sin 2 ( 2 π / R ) + 1 R 2 sin 2 ( 2 π / R )
2 π R 2 sin 2 ( 2 π / R ) , 2 π cos ( 2 π / R ) R 2 sin 2 ( 2 π / R )             ( R > 10 )
δ I i = I i - I i = b I i 2 + c I i 3 + d I i 4 ,
δ Φ = i = 1 N ( Φ I i ) δ I i .
δ Φ = - c ( m I 0 ) 2 4 sin ( 4 Φ )
δ Φ = c ( m I 0 ) 2 12 sin ( 4 Φ )
I i = I i γ             ( 0 < γ < 1 ) .
I i = I 0 γ [ 1 + γ m cos ( Φ + α i ) + γ ( γ - 1 ) 2 m 2 cos 2 ( Φ + α i ) + γ ( γ - 1 ) ( γ - 2 ) 6 m 3 cos 3 ( Φ + α i ) + γ ( γ - 1 ) ( γ - 2 ) ( γ - 3 ) 24 m 4 cos 4 ( Φ + α i ) ] .
δ I i = β i I i ,
( δ Φ ) 2 = i = 1 R ( Φ I i ) 2 ( δ I i ) 2 .
( δ Φ ) 2 = 2 m 2 R ( δ I ) 2 ,
δ I = 1 + m 3 2 N + 1 .
δ Φ = 1 + m 3 R 2 N + 1 / 2 m .
Φ=arctan3(I3-I1)I1-2I2+I3,
αiαi+1cos[Φ+p(α)]dααiαi+1cos(Φ+α+{αiαi+1[p(β)-β]dβ})dα,
p(α)=α+1α+2(α)2/2π+3(α)3/(2π)2+.
αiαi+1cos[Φ+p(α)]dα=cos(Φ)αiαi+1cos[p(α)]dα-sin(Φ)αiαi+1sin[p(α)]dα,

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