Abstract

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. The algorithm is obtained from a generalized Fourier transform of a symmetrical quadrature filter. This formalism allows us to represent the detuning phase shift error and bias modulation as geometrical conditions. Therefore, the design of the filter becomes a set of solvable linear equations. Hence, to prove our method, several general tunable filters, like three and four frame algorithms, are obtained. Finally, from our results we reproduce particular symmetrical four frame algorithms reported in literature.

© 2009 OSA

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References

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  1. K. Freischland and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [CrossRef]
  2. Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997).
    [CrossRef] [PubMed]
  3. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 31(36), 8098–8115 (1997).
    [CrossRef]
  4. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [CrossRef] [PubMed]
  5. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
    [CrossRef] [PubMed]
  6. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).
  7. M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).
  8. D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).
  9. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009).
    [CrossRef] [PubMed]

2009

1997

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 31(36), 8098–8115 (1997).
[CrossRef]

Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997).
[CrossRef] [PubMed]

1996

1995

1990

Creath, K.

Estrada, J. C.

Freischland, K.

Koliopoulos, C. L.

Mosiño, J. F.

Phillion, D. W.

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 31(36), 8098–8115 (1997).
[CrossRef]

Quiroga, J. A.

Schmit, J.

Servin, M.

Surrel, Y.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Other

H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).

M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).

D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).

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Figures (1)

Fig. 1
Fig. 1

Graphical representation of a quadrature filter

Tables (1)

Tables Icon

Table 1 Several particular four frame TPS algorithms class A, B and C

Equations (33)

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I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ω0t]   .
I(ω)=2aπδ(ω)+bπexp(iφ)δ(ω+ω0)+bπexp(iφ)δ(ωω0).
G(ω)=2aπH(ω)δ(ω)+bπH(ω)exp(iφ)δ(ω+ω0)+bπH(ω)exp(iφ)δ(ωω0).
G(ω0)=bπH(ω0)exp(iφ).
G(ω0)=bπH(ω0)exp(iφ).
tan(φ)=k=1NbkIkk=1NakIk.
h(t)=k=1Nak[δ(tp)]+ik=1Nbk[δ(tp)].
H(ω)=k=1N/2akcos(pω)k=1N/2bksin(pω);  For even N.
H(ω)=a(N+1)/22+k=1(N1)/2akcos(pω)k=1(N1)/2bksin(pω);  For odd N
H(0)=0;  H(α)=0.
H(α)=0; H(α)=0; H(α)=0 ...  Hn(α)=0.
H(0)=0; H(0)=0; H(0)=0 ...  Hm(0)=0.
tan(φ)=b1I1b1I3a1I1+a2I2+a1I3.
H(ω)=1+a1cos(ω)b1sin(ω).
H(0)=1+a1=0.
H(α)=1+a1cos(α)+b1sin(α)=0.
b1=[1cos(α)]/sin(α).
tan(φ)=[1cos(α)sin(α)]I1I3I12I2+I3=tan(α/2)(I1I3I12I2+I3).
tan(φ)=b1I1+b2I2b2I3b1I4a1I1+a2I2+a2I3+a1I4.
H(ω)=a1cos(3ω/2)+cos(ω/2)b1sin(3ω/2)b2sin(ω/2).
H(0)=H(α)=H(α)=0.
H(0)=1+a1=0.
H(α)=cos(α/2)+a1cos(3α/2)+b1sin(3α/2)+b2sin(α/2)=0.
2H(α)=3sin(α/2)3a1sin(3α/2)3b1cos(3α/2)b2cos(α/2)=0.
a1=1;   b1=[1+2cos(α)]/sin(α);   b2=[2+4cos(α)+3cos(2α)]/sin(α).
tan(φ)=[1+2cos(α)](I1I4)+[2+4cos(α)+3cos(2α)](I2I3)sin(α)(I1I2I3+I4).
Δφ=tan(Δ/2)Δ/2.
2H(α)=3sin(α/2)3a1sin(3α/2)+3b1cos(3α/2)+b2cos(α/2)=0.
a1=1;   b1=cos(α)/sin(α);   b2=[2+cos(α)]/sin(α).
tan(φ)=cos(α)(I1I4)[2+cos(α)](I2I3)sin(α)(I1I2I3+I4).
2H(0)=3b1b2=0.
a1=1;   b1=1/tan(α/2);   b2=3/tan(α/2).
tan(φ)=1tan(α/2)(I13I2+3I3I4I1I2I3+I4).

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