Abstract

The white-light interferogram in a spectrally resolved white-light interferometer is decomposed in its constituent spectral components by a spectrometer and displayed along its chromaticity axis. A piezoelectric transducer phase shifter in such an interferometer can give a desired phase shift of π/2 only at one wavelength. The phase shift varies continuously at all other wavelengths along the chromaticity axis. This situation is ideal for an experimental study of the phase error due to the phase-shift error in the phase-shifting technique, as it will be shown in this paper.

© 2007 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992), pp. 501-598.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. P. de Groot, "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window," Appl. Opt. 34, 4723-4730 (1995).
    [CrossRef]
  7. P. de Groot, "Phase-shift calibration errors in interferometers with spherical Fizeau cavities," Appl. Opt. 34, 2856-2863 (1995).
    [CrossRef]
  8. J. Schmit and K. Creath, "Window function influence on phase error in phase- shifting algorithms," Appl. Opt. 35, 5642-5649 (1996).
    [CrossRef] [PubMed]
  9. S. K. Debnath and M. P. Kothiyal, "Optical profiler based on spectrally resolved white light interferometry," Opt. Eng. 44, 013606 (2005).
    [CrossRef]
  10. S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
    [CrossRef]

2005 (1)

S. K. Debnath and M. P. Kothiyal, "Optical profiler based on spectrally resolved white light interferometry," Opt. Eng. 44, 013606 (2005).
[CrossRef]

2001 (1)

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
[CrossRef]

1996 (1)

1995 (3)

1994 (1)

1987 (1)

1983 (1)

Burow, R.

Creath, K.

de Groot, P.

Debnath, S. K.

S. K. Debnath and M. P. Kothiyal, "Optical profiler based on spectrally resolved white light interferometry," Opt. Eng. 44, 013606 (2005).
[CrossRef]

Eiju, T.

Elssner, K. E.

Grzanna, J.

Hariharan, P.

Helen, S. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
[CrossRef]

Joenathan, C.

Kothiyal, M. P.

S. K. Debnath and M. P. Kothiyal, "Optical profiler based on spectrally resolved white light interferometry," Opt. Eng. 44, 013606 (2005).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992), pp. 501-598.

Merkel, K.

Oreb, B. F.

Schmit, J.

Schwider, J.

Sirohi, R. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
[CrossRef]

Spolaczyk, R.

Appl. Opt. (7)

Opt. Eng. (2)

S. K. Debnath and M. P. Kothiyal, "Optical profiler based on spectrally resolved white light interferometry," Opt. Eng. 44, 013606 (2005).
[CrossRef]

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, "Analysis of spectrally resolved white light interferograms: use of a phase shifting technique," Opt. Eng. 40, 1329-1336 (2001).
[CrossRef]

Other (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992), pp. 501-598.

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Figures (12)

Fig. 1
Fig. 1

Function f j ( α ) for different algorithms.

Fig. 2
Fig. 2

Experimental setup of the spectrally resolved white-light interferometer.

Fig. 3
Fig. 3

Spectrally resolved white-light interferogram with a test surface.

Fig. 4
Fig. 4

Average phase shift along the chromaticity axis.

Fig. 5
Fig. 5

Difference of first and the first phase-shifted interferograms.

Fig. 6
Fig. 6

Wrapped phase map (a) using actual α values (b) assuming α = π / 2 .

Fig. 7
Fig. 7

Phase error diagrams corresponding to different algorithms (a) three-step, (b) four-step, (c) five-step, (d) six-step, (e) seven-step, and (f) eight-step.

Fig. 8
Fig. 8

Phase error for eight-step algorithms obtained by scanning along different columns in Fig. 7(f). (a) No phase-shift error, (b) 15% phase-shift error, and (c) 20% phase-shift error.

Fig. 9
Fig. 9

P–V phase error for different algorithms (curves) simulated values; (open circles) experimental points.

Fig. 10
Fig. 10

Phase error introduced in various algorithm for 20% miscalibration error. (Curves) simulated values; (open circles) experimental points.

Fig. 11
Fig. 11

Phase error as a function of the phase-shift error obtained by scanning along a row in the phase maps in Fig. 7 for different algorithms.

Fig. 12
Fig. 12

(a) Phase error diagram of the four-step type II algorithm; (b) horizontal scan from the phase error map indicating variation of phase error with phase-shift error.

Equations (25)

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I ( z , σ ) = g ( σ ) { I r + I t + 2 ( I r I t ) 1 / 2   cos [ ϕ ( z , σ ) + ϕ 0 ] } ,
ϕ ( z , σ ) = 4 π σ z ,
I i + 1 = I 0 [ 1 + V   cos ( ϕ + i α ) ] ,
tan   ϕ α j = tan ( ϕ + α ) = tan ( α / 2 ) ( I 1 I 3 I 1 + 2 I 2 I 3 ) ,
tan   ϕ α j = tan ( ϕ + 3 α / 2 ) = tan ( α / 2 ) ( I 1 I 2 + I 3 + I 4 I 1 I 2 I 3 + I 4 ) ,
tan   ϕ α j = tan ( ϕ + 2 α ) = 2   sin   α ( I 2 I 4 I 1 + 2 I 3 I 5 ) ,
tan   ϕ α j = tan ( ϕ + 5 α / 2 ) = ( 2   sin   α 1 + sin 2 α ) ( I 1 I 2 6 I 3 + 6 I 4 + I 5 I 6 4 ( I 2 I 3 I 4 + I 5 ) ) ,
tan   ϕ α j = tan ( ϕ + 3 α ) = ( 2   sin   α 1 + sin 2 α ) ( I 1 + 7 I 3 7 I 5 + I 7 4 I 2 + 8 I 4 4 I 6 ) ,
tan   ϕ α j = tan ( ϕ + 7 α / 2 ) = [ 3 + sin 2 α + 2  cos   α 3 + sin 2 α 2  cos   α   tan ( α / 2 ) ] × ( I 1 5 I 2 + 11 I 3 + 15 I 4 15 I 5 11 I 6 + 5 I 7 + I 8 I 1 5 I 2 11 I 3 + 15 I 4 + 15 I 5 11 I 6 5 I 7 + I 8 ) .
tan   ϕ α j = f j ( α ) tan   ϕ π / 2 j ,
f 3 ( α ) = tan ( α / 2 ) ,
f 4 ( α ) = tan ( α / 2 ) ,
f 5 ( α ) = sin   α ,
f 6 ( α ) = 2   sin   α 1 + sin 2 α ,
f 7 ( α ) = 2   sin   α 1 + sin 2 α ,
f 8 ( α ) = 3 + sin 2 α + 2   cos   α 3 + sin 2 α 2   cos   α   tan ( α / 2 ) .
Δ ϕ j = ϕ α j ϕ π / 2 j .
tan   Δ ϕ j = sin   2 ϕ α j x + cos   2 ϕ α j ,
tan   Δ ϕ j = sin   2 ϕ α j X m + cos   2 ϕ α j .
tan   ϕ α = tan ( ϕ + α ) = ( I 1 I 3 ) / ( I 2 I 4 ) cot   α [ ( I 1 I 3 ) / ( I 2 I 4 ) ] + cos e c α .
tan   ϕ π / 2 = I 1 I 3 I 2 I 4 .
tan   ϕ α = tan   ϕ π / 2 cot   α   tan   ϕ π / 2 + cos e c α .
tan   Δ ϕ = sin   2 ϕ α tan ( α 2 + π 4 ) ( 1 cos   2 ϕ α ) tan ( α 2 + π 4 ) [ tan ( α 2 + π 4 ) + sin   2 ϕ α ] + cos     2 ϕ α .
tan   ϕ π / 2 = I 3 I 2 I 1 I 2 .
α ( σ ) = arccos [ 1 2 ( I 5 I 1 I 4 I 2 ) ] ,

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