Abstract

The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.

© 2009 OSA

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References

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  1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
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    [CrossRef] [PubMed]
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  5. 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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    [CrossRef] [PubMed]
  7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990).
    [CrossRef]
  8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).
  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

1990

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990).
[CrossRef]

1983

1974

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Elssner, K. E.

Estrada, J. C.

Freischlad, K.

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990).
[CrossRef]

Gallagher, J. E.

Grzanna, J.

Herriott, D. R.

Koliopoulos, C. L.

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990).
[CrossRef]

Merkel, K.

Mosiño, J. F.

Quiroga, J. A.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Spolaczyk, R.

White, A. D.

Appl. Opt.

J. Opt. Soc. Am

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990).
[CrossRef]

Opt. Express

Other

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).

J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).

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

Detuned components c and ε.

Tables (1)

Tables Icon

Table 1. Detuning Phase Shift error for several TPS algorithms

Equations (27)

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I(x,y,t,ω0)=a(x,y)+b(x,y)cos[ϕ(x,y)+ω0t].
I(ω)=aδ(ω)+πbexp(iϕ)δ(ωω0)+πbexp(iϕ)δ(ω+ω0).
g(t)=h(t)I(x,y,t).
G(ω)=aH(ω)δ(ω)+πbH(ω)exp(iϕ)δ(ωω0)+πbH(ω)exp(iϕ)δ(ω+ω0).
G(ω)=aH(ω)δ(ω)+πb|H(ω)|exp(iϕ+iθ)δ(ωω0)+πb|H(ω)|exp(iϕ+iθ)δ(ω+ω0).
G(ω0)=πb|H(ω0)|exp[iϕ+θ(ω0)].
G(ω0)=πb|H(ω0)|exp[iϕ+θ(ω0)].
G(ω0Δ)=exp[iθ(ω0+Δ)]{cexp(iϕ)±εexp(iϕ)}.
c=πb|H(ω0+Δ)|;ε=πb|H(ω0Δ)|.
G(ω0Δ)=exp[iθ(ω0+Δ)]{cos(ϕ)(c+ε)+isin(ϕ)(cε)}.
tan(ϕ)=cεc+εtan(ϕ)=σtan(ϕ).
r=εc=1σ1+σ=|H(ω0Δ)||H(+ω0+Δ)|.
Δϕ=tan1[σtan(ϕ)]ϕ.
tan(Δϕ)=tan(ϕ)σtan(ϕ)1+σtan2(ϕ).
tan(Δϕ)=   (1σ1+σ)2tan(ϕ)1+tan2(ϕ)1+(1σ1+σ)(1tan2(ϕ)1+tan2(ϕ)).
tan(Δϕ)=rsin(2ϕ)1+rcos(2ϕ).
Δϕ=rsin(2ϕ)=|H(ω0Δ)||H(ω0+Δ)|sin(2ϕ)=(σ1σ+1)sin(2ϕ).
|Δϕmax|=sin1|H(ω0Δ)/H(ω0+Δ)|.
|Δϕmax|=sin1|H(ω0+Δ)/H(ω0Δ)|.
tan[ϕ(x,y,α=π/2)]=I(α)I(0)I(0)I(α).
h(t,α=π/2)=[δ(t)δ(tα)]+i[δ(t+α)δ(t)].
H(ω,α)=iexp[ωαi]exp[ωαi]+1i;α=π/2.
Δϕmax=sin1|H(ω=1,α=π/2+Δ)H(ω=1,α=π/2Δ)|.
Δϕmax=sin1|2i(i+1)sin(Δ/2)[sin(Δ/2)+cos(Δ/2)]2i(i1)cos(Δ/2)[sin(Δ/2)+cos(Δ/2)]|=sin1|tan(Δ2)||Δ|/2.
ϕ(x,y,α=π/2)=tan1(I(α)I(α)I(0)I(2α)).
H(ω,α)=12sin(ωα)exp(2ωαi).
Δϕmax=sin1|iexp(Δi)tan(Δ/2)|=sin1|tan(Δ/2)||Δ|/2.

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