Abstract

We propose a general approach to eliminating some error source effects in phase-calculation algorithms for phase-shifting interferometry. We express the actual phase shift in a convenient form that takes the errors into account and develop in series the detected phase from a generic algorithm. Setting to zero the terms of the series that involve unwanted errors leads to a set of linear equations for the algorithm coefficients, which can thus be found. By using this approach, one could develop an algorithm series for an individual interferometer based on relevant concerns about the main error sources in it and eliminate the error source effects to any desired order. Two examples of algorithm series, to eliminate distorted phase shifts caused by the geometric effect in an interferometer with a spherical Fizeau cavity and to eliminate vibration effects, are discussed.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measurement interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  3. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  4. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  5. 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]
  6. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  7. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
    [CrossRef]
  8. P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
    [CrossRef]
  9. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]

2000

P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

1997

1996

1995

1987

1974

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

de Groot, P.

Eiju, T.

Farrant, D. I.

Gallagher, J. E.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Larkin, K. G.

Oreb, B. F.

Phillion, D. W.

Rosenfeld, D. P.

Schmit, J.

Surrel, Y.

White, A. D.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Phase-calculation errors that are due to the numerical apertures for Eqs. (17), (19), (21), and (23) in the interferometric test of a spherical surface [Eq. (17) is a five-frame, Eq. (19) is a seven-frame, Eq. (21) is a nine-frame, and Eq. (23) is an eleven-frame formula].

Fig. 2
Fig. 2

Comparison of seven-frame Eqs. (31)–(33). [We designed Eq. (31) by assigning v = 0; Eq. (32), by assigning v = 1; and Eq. (33) by assigning v = 0.667.)

Fig. 3
Fig. 3

Comparison of predicted rms errors for Eqs. (42) and (23). [Equation (42) is the eleven-frame algorithm for eliminating low-frequency vibration and 8.75-Hz vibration; Eq. (23), the eleven-frame algorithm with insensitivity to low-frequency vibration.]

Equations (42)

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

gj=Q1+V cosθ+ϕj, j=-n,  , -3, -2, -1, 0, 1, 2, 3,  , n,
ϕj=jπ/2.
ϕj=j π2+jεPZT-jπ sin2α/2,
gj=Q1+V cosθ+j π2+jεPZT-jπ sin2α/2.
ϕj=j π2+jεPZT+ka sinjνπ4
ν=4fvib/fCCD.
gj=Q1+V cosθ+j π2+jεPZT+ka sinjνπ4.
θ=tan-1×j=02j+1n-1/2 A2j+1g2j+1-g-2j+1j=12jn-1/2 A2jg2j+g-2j-2 j=12jn-1/2 A2jg0.
εsphe=-2Δ sin2α/2,
gj=Q1+V cosθ+j π2+jε,
θ=tan-1×A1g1-g-1+A3g3-g-3+A5g5-g-5A2g2+g-2+A4g4+g-4-2A2+A4g0.
θtan-1×-A1+A3-A5+A1-9A3+25A5ε2/2+-A1+81A3-625A5ε4/24+A1-729A3+15625A5ε6/720+-A1+6561A3-390625A5ε8/40320-2A2+4A2-16A4ε2/2+-16A2+256A4ε4/24+64A2-4096A4ε6/720+-256A2+65536A4ε8/40320×tan θ.
A1-A3+A5=2A2
A1-9A3+25A5=4A2-16A4,
A1-81A3+625A5=16A2-256A4,
A1-729A3+15625A5=64A2-4096A4,
θ=tan-12g1-g-1g2+g-2-2g0.
Δθfive=ε2/4sin2θ.
θ=tan-17g1-g-1-g3-g-34g2+g-2-8g0.
Δθseven=ε4/16sin2θ.
θ=tan-126g1-g-1-6g3-g-316g2+g-2-g4+g-4-30g0.
Δθnine=ε6/64sin2θ.
θ=tan-1×98g1-g-1-29g3-g-3+g5-g-564g2+g-2-8g4+g-4-112g0.
θ=tan-1×-2+2A3-2A5+1-9A3+25A5εPZT2+sin2νπ4-A3 sin23νπ4+A5 sin25νπ4k2a2-4A2+4A2-16A4εPZT2+A2 sin2νπ2-A4 sinνπk2a2×tan θ.
1-A3+A5=2A2
1-9A3+25A5=4A2-16A4,
sin2νπ4-A3 sin23νπ4+A5 sin25νπ4=A2 sin2νπ2-A4 sin2νπ,
sin2ν2π4-A3 sin23ν2π4+A5 sin25ν2π4=A2 sin2ν2π2-A4 sin2ν2π.
A2=cosνπ2-cos3νπ21+cosνπ-2 cos3νπ2,
A3=1-2A2.
θ=tan-17g1-g-1-g3-g-34g2+g-2-8g0,
θ=tan-13g1-g-1-g3-g-32g2+g-2-4g0,
θ=tan-15g1-g-1-g3-g-33g2+g-2-6g0,
A2=-4+8 cosνπ2-8 cos3νπ2+4 cos2νπ1+8 cosνπ-16 cos3νπ2+7 cos2νπ,
A3=-2A2+1,
A4=4-7A2/8.
θ=tan-1×g1-g-1+A3g3-g-3A2g2+g-2+A4g4+g-4-2A2+A4g0.
θ=tan-112g1-g-1-4g3-g-38g2+g-2-g4+g-4-14g0.
4-8 cosνπ2+8 cos3νπ2-4 cos2νπ+1+8 cosνπ-16 cos3νπ2+7 cos2νπ×A2+-8+8 cos3νπ2+8 cos2νπ-8 cos5νπ2A5=0,
A3=-2A2+1+A5,
A4=4-7A2-8A5/8.
θ=tan-1g1-g-1-0.64444g3-g-3+0.06488g5-g-50.85466g2+g-2-0.3127g4+g-4-1.08392g0.

Metrics