Abstract

Systematic spectral analysis of Phase Shifting Interferometry (PSI) algorithms was first proposed in 1990 by Freischlad and Koliopoulos (F&K). This analysis was proposed with the intention that “in a glance” the main properties of the PSI algorithms would be highlighted. However a major drawback of the F&K spectral analysis is that it changes when the PSI algorithm is rotated or its reference signal is time-shifted. In other words, the F&K spectral plot is different when the PSI algorithm is rotated or its reference is time-shifted. However, it is well known that these simple operations do not alter the basic phase demodulation properties of PSI algorithms, except for an unimportant piston. Here we propose a new way to analyze the spectra of PSI algorithms which is invariant to rotation and/or reference time-shift among other advantages over the nowadays standard PSI spectral analysis by F&K.

© 2009 OSA

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References

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  1. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).
  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. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007).
    [PubMed]
  7. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express 17(11), 8789–8794 (2009).
    [CrossRef] [PubMed]

2009

1997

1996

1995

1990

Creath, K.

Cywiak, M.

Estrada, J. C.

Freischlad, K.

Koliopoulos, C. L.

Mosiño, J. F.

Phillion, D. W.

Quiroga, J. A.

Schmit, J.

Servin, M.

Surrel, Y.

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Figures (3)

Fig. 1
Fig. 1

Spectral magnitude |H(ω)| of two PSI algorithms. We show the input interferogram as I(t), and the output complex signal as Ic(t). We can clearly see that these two filters pass only the left side analytical signal at frequency -ω 0, while rejecting the background a(x,y) at ω = 0 and the right side analytical signal at + ω 0.

Fig. 2
Fig. 2

Spectral magnitude of the two 4-step PSI algorithms in Eq. (17). We see that they are actually proportional and as a consequence, have identical phase demodulation properties. These PSI filters let pass the left side analytical signal at ω = −1.0 while rejecting the background at ω = 0 and the complex signal at ω = 1.0.

Fig. 3
Fig. 3

Spectral magnitude |H(ω)| of the 5-step Schwider-Hariharan algorithm and its rotated twin. Both are of course identical. Only the left side analytical signal at ω = −1.0 passes through, while rejecting the background and the complex signal at ω = + 1.0.

Equations (28)

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I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ω0t].
I(x,y,t)=a+b2exp[i(φ+ω0t)]+b2exp[i(φ+ω0t)].
Ic(t)=[b(x,y)/2]exp{i[φ(x,y)+ω0t]}=h(t)*I(x,y,t).
h(t)=hr(t)+ihi(t).
H(ω0)0,    and      H(ω0)=H(0)=0.
{F[hr(t)]iF[hi(t)]   }|ω=ω0=0,      and      F[hr(t)]|ω=0=F[hi(t)]|ω=0=0.
tan[φ(x,y)]=hi(t)*I(x,y,t)hr(t)*I(x,y,t)|t=0.
h(t)exp[iΔ0],    and    h(tt0).
h(tt0)=exp[iω0(tt0)]{   nanδ(tnT)}     =h(t)exp[iω0t0].
h(t)exp[iΔ0]=hr(t)cos(Δ0)hi(t)sin(Δ0)+i[hr(t)sin(Δ0)+hi(t)cos(Δ0).
tan[φ(x,y)+Δ0]=[hr(t)sin(Δ0)+hi(t)cos(Δ0)]*I(t)[hr(t)cos(Δ0)hi(t)sin(Δ0)]*I(t)|t=0.
Hi(ω)=F[hi(t)]Hr(ω)=F[hr(t)].
Hi(ω)=F[hr(t)sin(Δ0)+hi(t)cos(Δ0)]Hr(ω)=F[hr(t)cos(Δ0)hi(t)sin(Δ0)].
H(ω)=|H(ω)|exp[iΔ(ω)]=Ic(ω)I(ω).
|H(ω)|=|F[h(t)]|=Re(ω)2+Im(ω)2.
|H(ω)|=|F[h(t)]|=|F{h(t)exp[iΔ0]}|=|F[h(tt0)]|.
tan[φ1(x,y)]=I1+I2I3I4I1I2I3+I4,  and  tan[φ2(x,y)]=I2I4I1I3.
h1(t)=δ(t+3π/4)δ(t+π/4)δ(tπ/4)+δ(t3π/4)+                   i[δ(t+3π/4)+δ(t+π/4)δ(tπ/4)δ(t3π/4)]h2(t)=δ(tπ/2)δ(t+π/2)+i   [δ(t)δ(t+π)].
Hr1(ω)=2cos(πω/4)2cos(3πω/4),Hi1(ω)=2i[sin(πω/4)+sin(3πω/4)]Hr2(ω)=2isin(πω/2),             Hi2(ω)=1exp(iπω).
|H1(ω)|=2|H2(ω)|.
tan[φ(x,y)]=2(I2I4)I12I3+I5.
h(t)=δ(tπ)2δ(t)+δ(t+π)+i[2δ(tπ/2)2δ(t+π/2)].
Hr(ω)=F[δ(tπ)2δ(t)+δ(t+π)]=2+2cos(πω)Hi(ω)=F[2δ(tπ/2)2δ(t+π/2)]=4isin(πω/2).
h(t)exp[iΔ0]={δ(tπ)2δ(t)+δ(t+π)+i[2δ(tπ/2)2δ(t+π/2)]}exp[iΔ0].
2   h(t)exp[iπ/4]=δ(tπ)+2δ(tπ/2)2δ(t)2δ(t+π/2)+δ(t+π)+                     i[δ(tπ)2δ(tπ/2)2δ(t)+2δ(t+π/2)+δ(t+π)].
tan[φ(x,y)+π/4]=I1+2I22I32I4+I5I12I22I3+2I4+I5.
2Hr(ω)=2+2cos(πω)4isin(πω/2),2Hi(ω)=22cos(πω)4isin(πω/2).
|H(ω)|=|2+2cos(πω)+4sin(πω/2)|.

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