Abstract

New methods that can be used to determine phase in phase-stepping interferometry are presented. It is shown that a combination of some of these methods can be used to reduce the error introduced by phase-stepper miscalibration and nonlinearity. Moreover these new algorithms can also be used to detect the presence of miscalibration or phase-shifter nonlinearity. A simplified approach to understanding the error introduced by miscalibration and nonlinearity of the phase stepper and its reduction in phase-shifting interferometry is also presented.

© 1994 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598, and references therein.
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 26, pp. 349–393.
    [CrossRef]
  3. J. R. 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]
  4. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  5. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensation phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  6. C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shifting interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]
  7. K. Frieschald, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  8. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement error in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]

1991 (1)

1990 (1)

1987 (2)

1985 (1)

1983 (1)

Ai, C.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598, and references therein.

Burow, R.

Cheng, Y. Y.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 26, pp. 349–393.
[CrossRef]

Eiju, T.

Elssner, K. E.

Frankena, H. J.

Frieschald, K.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598, and references therein.

Grzanna, J.

Hariharan, P.

Koliopoulos, C. L.

Merkel, K.

Oreb, B. F.

Schwider, J. R.

Smorenburg, C.

Spolaczyk, R.

van Wingerden, J.

Wyant, J. C.

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Figures (12)

Fig. 1
Fig. 1

Phase-measurement error against the ideal phase in radians for different 4-bucket algorithms. The algorithms in which frames 1, 3, 4, 5 and 1, 2, 3, 5 are used exhibit higher sensitivity to phase-step miscalibration.

Fig. 2
Fig. 2

Phase-measurement error against the ideal phase in radians for different 3-bucket algorithms. The algorithms in which frames 1, 2, 4 and 1, 3, 4 are used exhibit higher sensitivity to phase-step miscalibration.

Fig. 3
Fig. 3

Phase-measurement error in the average 4&4, average 4&4 + fraction (1245), and 5-bucket algorithms. The error fluctuation in the average 4&4 and 1, 2, 4, 5 is almost 180 deg out of phase. However, their amplitudes are not the same. The choice of a fraction value of 0.15 was determined when the phase fluctuations were the minimum. This value gives the minimum peak-to-peak phase error.

Fig. 4
Fig. 4

Phase error obtained by subtracting the phase obtained from the 2-, 3-, and 4-bucket algorithms. In the average 3&3 and 4&4 methods the phases are added, whereas in this case we subtract them. The peak-to-peak phase cycle is used for determining the phase-step calibration error.

Fig. 5
Fig. 5

Error in phase after removal of the first-order term: 4-bucket (0.1) is the plot with α = 0.1 rad, 4-bucket (0.15) is the plot with α = 0.15708 rad, and 4-bucket (0.2) is the plot with α = 0.2 rad. Note that in the 4-bucket (0.2) plot the phase ripple changes sign since the value of a is greater than the optimum value. A 10% miscalibration in the phase stepper was used.

Fig. 6
Fig. 6

Error in the measured phase against the ideal phase. A 10% calibration error in the phase step was used. 4-bucket (1) is the phase error obtained with the 10% phase-step miscalibration 4-bucket (2) is the phase error after the removal of the first-order term, and 4-bucket (3) is the phase-error obtained with the modified algorithm.

Fig. 7
Fig. 7

Errors for the 3-bucket methods: 3-bucket (1) is the phase-error plot obtained with Eq. (6), 3-bucket (1) is the phase-error plot after removal of the first-order error in the phase, and 3-bucket (3) is the phase error obtained after removal of the higher-order terms. The error was eliminated in the 3-bucket (3) plot.

Fig. 8
Fig. 8

Phase-measurement error against the ideal phase in radians for different 4-bucket algorithms when 10% calibration and 1% nonlinearity were introduced.

Fig. 9
Fig. 9

Phase-measurement error in the average 4&4, average 4&4 + fraction (1245), and 5-bucket algorithms: (a) 10% calibration and 1% nonlinearity error was introduced and (b) 10% calibration and 5% nonlinearity error was introduced.

Fig. 10
Fig. 10

Phase error obtained by subtracting the phase obtained from the two 3-bucket algorithms when 10% phase-step calibration and 5% nonlinearity error are present. The peak-to-peak phase cycle is used for determining phase-step miscalibration. However, it is observed that this value is slightly higher but close to the value of phase-step miscalibration.

Fig. 11
Fig. 11

Errors in the measured phase with 10% calibration error, with 10% calibration and 2% nonlinearity error, and after removal of the calibration and nonlinearity errors by use of the modified algorithm.

Fig. 12
Fig. 12

Error in the measured phase when the value of nonlinearity γ is varied from 0.02 to 0.04 rad. A 10% nonlinearity was introduced in the phase evaluation algorithm. The error was reduced to its minimum if not eliminated when γ was 0.0314 rad.

Equations (45)

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I 1 ( frame 1 ) = I 0 { 1 + V cos ( ϕ ) } , I 2 ( frame 2 ) = I 0 { 1 + V cos ( ϕ + β ) } , I 3 ( frame 3 ) = I 0 { 1 + V cos ( ϕ + 2 β ) } , I 4 ( frame 4 ) = I 0 { 1 + V cos ( ϕ + 3 β ) } , I 5 ( frame 5 ) = I 0 { 1 + V cos ( ϕ + 4 β ) } ,
ϕ ( 1 , 2 , 3 , 4 ) = tan - 1 { I 4 - I 2 I 1 - I 3 } .
ϕ ( 2 , 3 , 4 , 5 ) = tan - 1 { I 4 - I 2 I 5 - I 3 } .
ϕ ( 1 , 3 , 4 , 5 ) = tan - 1 { 2 I 4 - 2 I 5 + I 1 - I 3 I 1 - I 3 } ,
ϕ ( 1 , 2 , 4 , 5 ) = tan - 1 { I 5 - I 2 I 1 - I 4 } - π / 4 ,
ϕ ( 1 , 2 , 3 , 5 ) = tan - 1 { 2 I 5 - 2 I 2 + I 3 - I 1 I 1 - I 3 } .
ϕ ( 1 , 2 , 3 ) = tan - 1 { I 3 - I 2 I 1 - I 2 } + π 4 ,
ϕ ( 1 , 2 , 3 ) = tan - 1 { I 3 - 2 I 2 + I 1 I 1 - I 3 } .
ϕ ( 2 , 3 , 4 ) = tan - 1 { I 4 - I 3 I 2 - I 3 } - π 4 .
ϕ ( 2 , 3 , 4 ) = tan - 1 { I 4 - I 2 I 4 - 2 I 3 + I 2 } .
ϕ ( 1 , 3 , 4 ) = tan - 1 { I 4 - I 3 I 1 - I 4 } - π 4
ϕ ( 1 , 3 , 4 ) = tan - 1 { 2 I 4 - I 1 - I 3 I 1 - I 3 } ,
ϕ ( 1 , 2 , 4 ) = tan - 1 { I 1 - I 2 I 1 - I 4 } - π 4
ϕ ( 1 , 2 , 4 ) = tan - 1 { I 4 - I 2 2 I 1 - I 4 - I 2 } .
2 ϕ = tan - 1 { I 3 - I 2 I 1 - I 2 } + tan - 1 { I 4 - I 3 I 2 - I 3 } .
ϕ = tan - 1 ( { [ 3 ( I 2 - I 3 ) - ( I 1 - I 4 ) ] [ ( I 2 - I 3 ) + ( I 1 - I 4 ) ] } 1 / 2 ( I 2 - I 1 ) - ( I 4 - I 3 ) ) .
ϕ = tan - 1 { 2 ( I 2 - I 4 ) 2 I 3 - I 5 - I 1 } .
tan ϕ m ( 1 , 2 , 3 , 4 ) = tan ( ϕ + α ) cos α + sin α ,
tan ϕ m ( 2 , 3 , 4 , 5 ) = tan ( ϕ + 3 α ) cos α - sin α ,
tan ϕ m ( 1 , 3 , 4 , 5 ) = tan ( ϕ + α ) { cos 2 α cos α + sin 3 α cos α } + 2 sin α - 4 sin 2 α ,
tan ϕ m ( 1 , 2 , 4 , 5 ) = tan ( ϕ + 3 α / 2 - 3 π / 4 ) × cos α - sin α ,
tan ϕ m ( 1 , 2 , 3 , 5 ) = - tan ( ϕ + α ) { sin 3 α + 1 } / cos α - 4 sin 2 α .
tan ϕ m ( 1 , 2 , 3 ) = tan ( ϕ - π / 4 + α / 2 ) cos α + sin α .
tan ϕ m ( 2 , 3 , 4 ) = tan ( ϕ + π / 4 + 3 α / 2 ) cos α - sin α .
tan ϕ m ( 1 , 3 , 4 ) = [ cot ( ϕ + 3 α / 2 - π / 4 ) cos α - sin α ] × cos ( π / 4 - α ) sin ( π / 4 - 3 α / 2 ) ,
tan ϕ m ( 1 , 2 , 4 ) = [ cot ( ϕ + 3 α / 2 - π / 4 ) cos α + sin α ] × cos ( π / 4 - α ) sin ( π / 4 - 3 / 2 α ) .
2 ϕ m ( 3 & 3 ) = tan - 1 { tan ( ϕ - π / 4 + α / 2 ) cos α + sin α } + tan - 1 { tan ( ϕ + π / 4 + 3 α / 2 ) × cos α - sin α } ,
2 ϕ m ( 4 & 4 ) = tan - 1 { tan ( ϕ + α ) cos α + sin α } + tan - 1 { tan ( ϕ + 3 α ) cos α - sin α } .
tan ϕ m = tan ( ϕ + 2 α ) / cos α .
tan ϕ m ( 1 , 2 , 3 , 4 ) - tan ϕ m ( 2 , 3 , 4 , 5 ) = [ tan ( ϕ + α ) - tan ( ϕ + 3 α ) ] cos α + 2 sin α .
tan ϕ m ( 1 , 2 , 3 ) - tan ϕ m ( 2 , 3 , 4 ) = [ tan ( ϕ - π / 4 + α / 2 ) - tan ( ϕ + π / 4 + α / 2 ) ] × cos α + 2 sin α .
ϕ ( 1 , 2 , 3 , 4 ) = tan - 1 { [ ( I 4 - I 2 I 1 - I 3 ) - sin α ] / cos α } .
ϕ ( 1 , 2 , 3 ) = tan - 1 { [ ( I 3 - I 2 I 1 - I 2 ) - sin α ] / cos α } .
tan ϕ m ( 1 , 2 , 3 , 4 ) = { tan ( ϕ + α + 2 γ ) cos ( α + 3 γ ) - sin ( α + 3 γ ) } × { cos 2 γ - tan ( α + 2 γ ) sin 2 γ } ,
tan ϕ m ( 2 , 3 , 4 , 5 ) = { tan ( ϕ + 3 α + 10 γ ) cos ( α + 5 γ ) - sin ( α + 5 γ ) } × { cos 3 γ + tan ( α + 7 γ ) sin 3 γ } ,
tan ϕ m ( 1 , 2 , 3 , ) = { tan ( ϕ + α / 2 + γ / 2 - π / 4 ) × cos ( α + 2 γ ) - sin ( α + 2 γ ) } × { cos γ + tan ( π / 4 - α - γ ) sin γ } ,
tan ϕ m ( 2 , 3 , 4 ) = { tan ( π / 4 - ϕ - 3 α / 2 - 5 γ / 2 ) × cos ( α + 4 γ ) - sin ( α + 4 γ ) } × { tan ( π / 4 - α / 2 - 3 γ / 2 ) sin γ + cos γ } .
tan ϕ m = tan ( ϕ + γ / 2 ) .
tan ϕ m = tan ϕ + ( 1 - tan ϕ ) sin γ / 2 - ( 1 + 1 2 tan ϕ ) sin 2 γ / 2 + 1 / 2 tan ϕ sin 3 γ / 2 ,
tan ϕ m = tan ϕ + ( tan ϕ - 1 ) sin γ / 2 - ( 1 + 1 / 2 tan ϕ ) sin 2 γ / 2 - 1 / 2 tan ϕ sin 3 γ / 2 ,
tan ϕ m = 1 / 2 ( tan ϕ + tan ϕ ) + 1 / 2 ( tan ϕ - tan ϕ ) × sin γ / 2 + [ 1 + 1 4 ( tan ϕ + tan ϕ ) ] sin 2 γ / 2 + 1 / 4 ( tan ϕ - tan ϕ ) sin 3 γ / 2.
tan ϕ m ( 1 , 2 , 3 , 4 ) = { tan ( ϕ + α ) cos α + sin α } × [ cos γ + tan ( α + γ / 2 ) sin γ ] .
tan ϕ m ( 1 , 2 , 3 ) = { tan ( ϕ - π / 4 + α / 2 ) cos α + sin α } × [ cos γ + tan ( α / 2 + γ / 2 - π / 4 ) sin γ ] .
ϕ ( 1 , 2 , 3 , 4 ) = tan - 1 { [ ( { I 4 - I 2 I 1 - I 3 } / { cos γ + tan ( γ + α / 2 ) sin α } ) - sin α ] / cos α } .
ϕ ( 1 , 2 , 3 ) = tan - 1 { [ ( { I 3 - I 2 I 1 - I 2 } / { cos γ + tan ( γ / 2 + α / 2 - π / 4 ) sin α } ) - sin α ] / cos α } .

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