Abstract

To estimate the modulating wavefront of an interferogram in Phase Shifting Interferometry (PSI) one frequently uses a Phase Shifting Algorithm (PSA). All PSAs take as input N phase-shifted interferometric measures, and give an estimation of their modulating phase. The first and best known PSA designed explicitly to reduce a systematic error source (detuning) was the 5-steps, Schwider-Hariharan (SH-PSA) PSA. Since then, dozens of PSAs have been published, designed to reduce specific data error sources on the demodulated phase. In Electrical Engineering the Frequency Transfer Function (FTF) of their linear filters is their standard design tool. Recently the FTF is also being used to design PSAs. In this paper we propose a technique for designing PSAs by fine-tuning the few spectral zeroes of a PSA to approximate a template FTF spectrum. The PSA’s spectral zeroes are moved (tuned) while gauging the plot changes on the resulting FTF’s magnitude.

© 2011 OSA

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References

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  1. 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(21), 3421–3432 (1983).
    [CrossRef] [PubMed]
  2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [CrossRef] [PubMed]
  3. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).
  4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [CrossRef]
  5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
    [CrossRef]
  6. J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
    [CrossRef]
  7. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
    [CrossRef] [PubMed]
  8. J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35(28), 5642–5649 (1996).
    [CrossRef] [PubMed]
  9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [CrossRef] [PubMed]
  10. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th ed. (Prentice Hall, 2007).
    [PubMed]
  11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).
  12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [CrossRef] [PubMed]

2009

1996

1995

1993

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

1992

1990

1987

1983

Burow, R.

Creath, K.

Eiju, T.

Elssner, K. E.

Estrada, J. C.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Freischlad, K.

Groot, P.

Grzanna, J.

Hariharan, P.

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

Oreb, B. F.

Quiroga, J. A.

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

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(21), 3421–3432 (1983).
[CrossRef] [PubMed]

Servin, M.

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Surrel, Y.

Zöller, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Opt. Express

Other

J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th ed. (Prentice Hall, 2007).
[PubMed]

J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).

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Figures (5)

Fig. 1
Fig. 1

Magnitude of HSH (ω) Eq. (11), and its CP-diagram. The PSA’ spectral plot |HSH (ω)| has two first-order zeroes at 0, and π, and a second-order one at -π/2.

Fig. 2
Fig. 2

Magnitude of H 5(ω), and its CP-diagram. The PSA spectrum, has been considerably flattened around π/2 with respect to HSH (ω) for the same measured interferograms. From the CP-diagram alone, the spectral shape outside the 4 zeroes shown is absent. In contrast the plot of |H 5(ω)| shows it clearly. One must be aware of the small ripples within the stop-band.

Fig. 3
Fig. 3

Simulated speckle-like interferograms applied to our modified PSA (Eq. (13)). The 5 interferograms used in Eq. (13) may also be used for the SH-PSA. However the PSA detuning robustness is higher in our modified PSA (Eq. (13)) than in the SH-PSA (Eq. (11)).

Fig. 4
Fig. 4

Magnitude of P 9(e ) and its CP-diagram. The resulting spectral rejection band is wide (6th order) around ω 0, and 2nd order around the origin.

Fig. 5
Fig. 5

CP diagram, and magnitude of H 9(ω). The resulting PSA’ spectral shape has been further flattened, with respect to P 9(e ) around ω 0, and around of the origin. . From the CP-diagram alone, one may only wonder the spectral amplitude outside the zeroes shown, making almost impossible the fine-tuning task performed herein.

Equations (15)

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I ( x , y , t ) = k = 0 N 1 { a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ω 0 k ] } δ ( t k ) .
I ( x , y , ω ) = a δ ( ω ) + b 2 exp [ i φ ] δ ( ω + ω 0 ) + b 2 exp [ i φ ] δ ( ω ω 0 ) .
tan [ φ ( x , y ) ] = k = 0 N 1 a k sin ( ω 0 t ) I ( k ) k = 0 N 1 a k cos ( ω 0 t ) I ( k ) .
h ( t ) = {     k = 0 N 1 a k δ ( t k ) } e i ω 0 t ,       H ( ω ) = F [ h ( t ) ] = k = 0 N 1 a k e i k ( ω ω 0 ) .
S = [ I ( t ) h ( t ) ] t = N 1 = k = 0 N 1 a k e i ω 0 k I ( k ) .
| H ( ω ) | = | F [ h ( t ) ] | = R e ( ω ) 2 + I m ( ω ) 2 .
P ( x ) = k = 0 N 1 a k e i ω 0 k x k = k = 0 N 2 ( x d k ) .
H ( ω ) = e i ( N 1 ) ω P ( e i ω ) .
h ( t ) = [ δ ( t ) δ ( t 1 ) ] e i ω 0 t .
H ( ω ) = F [ h ( t ) ] = 1 e i ( ω + ω 0 ) .
H S H ( ω ) = ( 1 e i ω ) [ 1 e i ( ω + π / 2 ) ] 2 [ 1 e i ( ω + π ) ] .
H 5 ( ω ) = ( 1 e i ω ) [ 1 e i ( ω + 0.45 ω 0 ) ] [ 1 e i ( ω + ω 0 ) ] [ 1 e i ( ω + 1.4 ω 0 ) ] .
tan [ ϕ ] = 2.4 I ( π / 2 ) + 2.9 I ( π ) + 0.57 I ( 3 π / 2 ) I ( 2 π ) I ( 0 ) 1.2 I ( π / 2 ) 2.3 I ( π ) + 2.7 I ( 3 π / 2 ) 0.23 I ( 2 π ) .
H 9 ( ω ) = [ 1 e i ( ω 0.3 ω 0 ) ] [ 1 e i ω ] [ 1 e i ( ω + 0.3 ω 0 ) ] [ 1 e i ( ω + 0.6 ω 0 ) ] [ 1 e i ( ω + ω 0 ) ]                               [ 1 e i ( ω + 1.4 ω 0 ) ] [ 1 e i ( ω + 1.7 ω 0 ) ] [ 1 e i ( ω + 2 ω 0 ) ] .
S / N ( ω 0 ) = | H 9 ( ω 0 ) | 2 1 2 π π π | H 9 ( ω ) | 2 d ω = 5.3 ,     S / N ( 1.16 ω 0 ) = | H 9 ( 1.16 ω 0 ) | 2 1 2 π π π | H 9 ( ω ) | 2 d ω = 6.6.

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