Abstract

We present a theoretical analysis to estimate the amount of phase noise due to noisy interferograms in Phase Shifting Interferometry (PSI). We also analyze the fact that linear filtering transforms corrupting multiplicative noise in Electronic Speckle Pattern Interferometry (ESPI) into fringes corrupted by additive gaussian noise. This fact allow us to obtain a formula to estimate the standard deviation of the noisy demodulated phase as a function of the spectral response of the preprocessing spatial filtering combined with the PSI algorithm used. This phase noise power formula is the main result of this contribution.

©2009 Optical Society of America

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References

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  1. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
  2. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
  3. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36, 271–276 (1997).
    [PubMed]
  4. J. Schmit and C. Katherine, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649 (1996).
    [PubMed]
  5. G. Paez and M. Strojnik, “Analysis and minimization of noise effects in phase shifting interferometry,” SPIE Vol. 3744, 295–305 (1999).
  6. K. J. Gasvik, Optical Metrology (John Wiley & Sons Ltd); 2th ed., (1996).
  7. A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, 3th ed., 1991).
  8. 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 Inc., 2001) Chap. 12.
  9. L. W. Couch, Digital & Analog Communication Systems, (Prentice Hall, 2006) 7th ed.
  10. P. Hariharan, B. F. Oreb, and T. Eyui, “Digital phase shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [PubMed]
  11. K. Freischland and C. L. Koliopolous, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–552 (1990).

1999 (1)

G. Paez and M. Strojnik, “Analysis and minimization of noise effects in phase shifting interferometry,” SPIE Vol. 3744, 295–305 (1999).

1997 (1)

1996 (1)

1995 (1)

1990 (2)

1987 (1)

Brophy, C. P.

Couch, L. W.

L. W. Couch, Digital & Analog Communication Systems, (Prentice Hall, 2006) 7th ed.

Eyui, T.

Freischland, K.

Gasvik, K. J.

K. J. Gasvik, Optical Metrology (John Wiley & Sons Ltd); 2th ed., (1996).

Hariharan, P.

Katherine, C.

Koliopolous, C. L.

Kujawinska, M.

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 Inc., 2001) Chap. 12.

Oreb, B. F.

Paez, G.

G. Paez and M. Strojnik, “Analysis and minimization of noise effects in phase shifting interferometry,” SPIE Vol. 3744, 295–305 (1999).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, 3th ed., 1991).

Rathjen, C.

Schmit, J.

Servin, M.

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 Inc., 2001) Chap. 12.

Strojnik, M.

G. Paez and M. Strojnik, “Analysis and minimization of noise effects in phase shifting interferometry,” SPIE Vol. 3744, 295–305 (1999).

Surrel, Y.

Appl. Opt. (3)

J. Opt. Soc. Am. A (3)

SPIE (1)

G. Paez and M. Strojnik, “Analysis and minimization of noise effects in phase shifting interferometry,” SPIE Vol. 3744, 295–305 (1999).

Other (4)

K. J. Gasvik, Optical Metrology (John Wiley & Sons Ltd); 2th ed., (1996).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, 3th ed., 1991).

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 Inc., 2001) Chap. 12.

L. W. Couch, Digital & Analog Communication Systems, (Prentice Hall, 2006) 7th ed.

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Figures (2)

Fig. 1.
Fig. 1. Frequency response of a 3 and 5-step [10] PSI algorithms both having a phase step of π/2 radians. H3(ωt ) represents the 3-step PSI and H5(ωt ) the 5-step one, having the same response be iϕ at ωt , =-1. We also show the analytical signal bδ(ωt +1.0)e iϕ which is the output of these filters to a+bcos[ϕ+(π/2) t]. Finally the value of the square integral of their frequency response is given to obtain a relative noise rejection.

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Fig. 2.
Fig. 2. Phasor representation (spatial and time dependences were omitted) of the estimated analytical signal Ic, and its corrupting noise n. The noiseless amplitude of Ic is represented by the phasor b which has a noiseless angle ϕ. The noise n has an average power σn 2 given by Eq. (15) and its angle is uniformly distributed within [0.2π], so the phasor n may point anywhere within the circle shown. The noisy estimated phase ϕn is obtained by Eq. (10).

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Equations (22)

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I0(x,y)=a(x,y)+b(x,y)cos⁡[ns(x,y)+ϕ(x,y)]+na(x,y).
I1(x,y)=a(x,y)+b(x,y)cos⁡[ns(x,y)]
I2(x,y)=a(x,y)+b(x,y)cos⁡[ns(x,y)+ϕ(x,y)].
I(x,y)=∣I1−I2∣=2b∣cos⁡[ns]−cos⁡[ns+ϕ]∣=2b∣sin⁡(ns+ϕ/2)∣∣sin⁡[ϕ2]∣.
If(x,y)=∑(ξ,η)∊Rh1(x−ξ,y−η)I(ξ,η)=a(x,y)+b(x,y)cos⁡[ϕ(x,y)]+n(x,y).
I(x,y,t)=∑k=−N/2N/2{a+bcos⁡[ϕ+t]+n}δ(t−kα),α=2π/N.
h2(t)=∑k=−N/2N/2hr (k) δ (t−kα)+i ∑k=−N/2N/2hi(k)δ(t−kα),α=2π/N .
hr(k)=hr(−k),hi(k)=−hi(−k)andhi(0)=0.
h3(t)=[2δ(t)−δ(t−π/2)−δ(t+π/2)]+i[δ(t−π/2)−δ(t+π/2)].
H2(ωt)=2∑k=−N/2N/2hr (k) cos⁡(kαωt) −2∑k=−N/2N/2hi(k)sin⁡(kαωt),α=2π/N.
Ic(x,y,t)=If(x,y,t)*hr(t)+iIf(x,y,t)*hi(t)=[I(x,y,t)**h1(x,y)]*h2 (t) .
ϕn(x,y,0)=If(x,y,t)*hi(t)If(x,y,t)*hr(t)∣t=0=[I(x,y,t)**h1(x,y)]*hi(t)[I(x,y,t)**h1(x,y)]*hr(t)∣t=0.
In(x,y,t)=I(x,y,t)+n(x,y,t).
Ic(x,y,t)=In(x,y,t)***h(x,y,t).
Ic(ωx,ωy,t)=In(ωx,ωy,ωt)H(ωx,ωy,ωt).
σn2=E(n2)=Rnn(0,0,0)=η2∫∫∫(−π,π)×(−π,π)×(−π,π)H2(ωx,ωy,ωt)dωxdωydωt.
σn2=E(n2) Rnn (0,0,0) = η2 {∫∫(−π,π)×(−π,π)H12(ωx,ωy)dωxdωy} {∫(−π,π)H22(ωt)dωt}.
{∫(−π,π)HA2(ωt)dωt}<{∫(−π,π)HB2(ωt)dωt} .
Ic(x,y,0)=Re {Ic(x,y,0)}+iIm{Ic(x,y,0)} .
Ic(x,y,0)=b(x,y)cos⁡[ϕ(x,y,0)]+nr(x,y,0)+ib(x,y)sin⁡[ϕ(x,y,0)]+ni(x,y,0).
σϕn=2tan⁡−1(σn2b).
σϕn2≈σn2b2=η2b2{∫∫(0,2π)×(0,2π)H12(ωx,ωy)dωxdωy}{∫(0,2π)H22(ωt)dωt}

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