Abstract

A phase shifting device (PZT) which is commonly applied in interferometry for phase measurement, unfortunately, has a nonlinear response to the applied voltage. In certain configurations such as, holographic moiré, where incorporation of two PZTs yields multiple phase information regarding the two orthogonal displacement components, the nonlinear response of the two PZTs yields highly erroneous result. In this context, we present for the first time a method for compensating multiple nonlinearities in the PZTs. Experimental results show the feasibility of the proposed method. The statistical performance of this method is also verified by comparing with the Cramér-Rao lower bound.

© 2006 Optical Society of America

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References

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  1. P. K. Rastogi, "Phase shifting applied to four-wave holographic interferometers," Appl. Opt. 31,1680-1681 (1992).
    [CrossRef] [PubMed]
  2. P. K. Rastogi, "Phase-shifting holographic moir´e: Phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moir´e," Appl. Opt. 32,3669-3675 (1993).
    [CrossRef] [PubMed]
  3. S. M. Kay, "Modern spectral estimation: theory and application," Prentice Hall, Englewood cliffs, New Jersey.
  4. B. Raphael, and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146,729-758 (2003).
    [CrossRef]
  5. D. C. Rife, and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Transactions on Information Theory,  IT-20,591-598 (1974).
    [CrossRef]

2003

B. Raphael, and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146,729-758 (2003).
[CrossRef]

1993

1992

1974

D. C. Rife, and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Transactions on Information Theory,  IT-20,591-598 (1974).
[CrossRef]

Boorstyn, R. R.

D. C. Rife, and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Transactions on Information Theory,  IT-20,591-598 (1974).
[CrossRef]

Raphael, B.

B. Raphael, and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146,729-758 (2003).
[CrossRef]

Rastogi, P. K.

Rife, D. C.

D. C. Rife, and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Transactions on Information Theory,  IT-20,591-598 (1974).
[CrossRef]

Smith, I. F. C.

B. Raphael, and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146,729-758 (2003).
[CrossRef]

Appl. Math. Comput.

B. Raphael, and I. F. C. Smith, "A direct stochastic algorithm for global search," Appl. Math. Comput. 146,729-758 (2003).
[CrossRef]

Appl. Opt.

IEEE Transactions on Information Theory

D. C. Rife, and R. R. Boorstyn, "Single-tone parameter estimation from discrete-time observations," IEEE Transactions on Information Theory,  IT-20,591-598 (1974).
[CrossRef]

Other

S. M. Kay, "Modern spectral estimation: theory and application," Prentice Hall, Englewood cliffs, New Jersey.

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Figures (4)

Fig. 1.
Fig. 1.

Fringe patterns obtained experimentally from optical set up in Reference 1: a) Moiré, and b) fringes corresponding to the phase difference φ 1.

Fig. 2.
Fig. 2.

Histogram representations for phase steps a) α = 25°, and b) β = 50°; and for nonlinear coefficients c) ε 12 = 0.15, and ε 22 = 0.20.

Fig. 3.
Fig. 3.

Wrapped phases a) φ 1, and b) φ 2.

Fig. 4.
Fig. 4.

a) Comparison of the MSE in the estimation of phase φ 1 of the our proposed method with CRB for various levels of nonlinearity. Same nonlinearity is assumed in both the PZTs a) ε 12 = ε 22 = 0.01, b) ε 12 = ε 22 = 0.1, c) ε 12 = ε 22 = 0.2, and d) ε 12 = ε 22 = 0.4.

Equations (11)

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I ̄ n = I d c 1 [ 1 + γ 1 cos ( φ 1 + α n ) ] + I d c 2 [ 1 + γ 2 cos ( φ 2 + β n ) ] I n + η n
α n = n α + ε 12 ( n α ) 2 / π
β n = n β + ε 22 ( n β ) 2 / π
I ̄ = I + η = S ( ξ ) C + η
p ( I ̄ ; ξ ) = 1 π N σ N exp { 1 σ 2 [ I ̄ S ( ξ ) C ] H [ I ̄ S ( ξ ) C ] }
C = { [ S ( ξ ) ] H S ( ξ ) } 1 S ( ξ ) H I ̄
D ( ξ ) = I ̄ H I ̄ I ̄ H S ( ξ ) { [ S ( ξ ) ] H S ( ξ ) } 1 S ( ξ ) H I ̄
ξ = max I ̄ H S ( ξ ) { [ S ( ξ ) ] H S ( ξ ) } 1 S ( ξ ) H I ̄
J = E { [ ψ log p ( I ̄ ) ] [ ψ log p ( I ̄ ) ] T }
J r , s = 1 σ 2 n = 0 N 1 I n ψ r I n ψ s
E { ψ ̂ r 2 } J r , r 1

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