Abstract

The objective of this paper is to design phase shifting algorithms error-resistant to the nonlinearity of phase-shift error and photoelectric detector simultaneously. An effective construction approach is proposed based on self-convolution of the rectangle window to design algorithms with perfect zero point distribution, according to the fact that the error-resistant capability is entirely determined by the number and order of zero points of Fourier transform of the related window function. Theoretical analysis and numerical simulations compared to the commercial 13-frame algorithm demonstrate the validity of the approach to design algorithms with enhanced error-resistant capability not only to CCD-caused harmonics but also to PZT ramping nonlinearity.

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References

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  1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing Third Edition, D. Malacara, (New York, 2007), pp. 547–666.
  2. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [Crossref]
  3. H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
    [Crossref]
  4. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
    [Crossref] [PubMed]
  5. K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express 12(23), 5579–5594 (2004).
    [Crossref] [PubMed]
  6. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
    [Crossref] [PubMed]
  7. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
    [Crossref]
  8. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for non-sinusoidal waveforms with phase shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995).
    [Crossref]
  9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref] [PubMed]
  10. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997).
    [Crossref] [PubMed]
  11. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [Crossref] [PubMed]
  12. J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009).
    [Crossref] [PubMed]
  13. Zygo is a registered trademark of Zygo Corporation.

2009 (2)

2004 (1)

2000 (1)

1999 (1)

H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
[Crossref]

1997 (1)

1996 (1)

1995 (2)

1990 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Burke, J.

Burton, D. R.

H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
[Crossref]

de Groot, P.

Estrada, J. C.

Farrant, D. I.

Freischlad, K.

Groot, P.

Hanayama, R.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Hibino, K.

Koliopoulos, C. L.

Lalor, M. J.

H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
[Crossref]

Larkin, K. G.

Oreb, B. F.

Phillion, D. W.

Quiroga, J. A.

Servin, M.

Surrel, Y.

Zhang, H.

H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Other (2)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing Third Edition, D. Malacara, (New York, 2007), pp. 547–666.

Zygo is a registered trademark of Zygo Corporation.

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Figures (5)

Fig. 1
Fig. 1

The signal spectrum

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Fig. 2
Fig. 2

The amplitude spectrum of windows

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Fig. 3
Fig. 3

The correlative curve of the designed phase shifting algorithm in Eq. (17). (a) The window function; (b) its amplitude spectrum.

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Fig. 4
Fig. 4

The calibration curve of hardware devices. (a) PZT calibration curve; (b) CCD calibration curve. Inside each curve, the red marks mean measured points and the blue line mean fitted polynomial curve.

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Fig. 5
Fig. 5

Comparison of Zygo 13-frame and self-designed algorithm. (a) The amplitude spectrum of two algorithms; (b) the log-amplitude spectrum of two algorithms; (c) the PV phase error versus the miscalibration coefficient; (d) the PV phase error versus the second-order nonlinearity coefficient; (e) the PV phase error versus the third-order nonlinearity coefficient; (f) the PV phase error versus the coefficient of second order signal harmonics.

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Tables (1)

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Table 1 Performance of several algorithms for higher harmonic suppression, compensation for phase-shift error and sensitivity to random noise.

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Equations (23)

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θ ′ = tan − 1 ( ∑ n s n I n / ∑ n c n I n ) = arg ( ∑ n ( c n + i s n ) I n ) = arg ( ∑ n w n I n ) .
θ ′ = arg ( ∫ w ( t ) I ( t ) d t ) ,
θ ′ = arg ( ∫ W ˜ ( v ) I ˜ * ( v ) d v ) ,
I ( t ) = ∑ k = 0 q γ k ⋅ cos ( θ k + k ( v 0 t + ∑ j = 1 p ε j t j ) ) ,
I ( t ) ≈ ∑ k = − q q γ k 2 exp ( i θ k ) ( exp ( i k v 0 t ) ( 1 + i k ∑ j = 1 p ε j t j ) ) ,
I ˜ ( v ) = ∑ k = − q q γ k 2 exp ( i θ k ) ( δ ( v − k v 0 ) + k ∑ j = 1 p i j + 1 ε j δ ( j ) ( v − k v 0 ) ) .
θ ′ = arg ( ∑ k = − q q γ k 2 exp ( i θ − k ) ( W ˜ ( k v 0 ) + k ∑ j = 1 p ( − i ) j + 1 ε j W ˜ ( j ) ( k v 0 ) ) ) .
W ˜ ( j ) ( k v 0 ) = 0 , j = 0 , 1 , ⋯ , p , k = − q , ⋯ , − 2 , 0 , 1 , ⋯ , q .
x ( t ) = { 1 , | t | ≤ T 0 / 2 0 , | t | > T 0 / 2 , T 0 = 2 π v 0 ,
X ˜ ( v ) = 2 π v 0 sin c ( v v 0 π ) ,
y ( t ) = x ( t ) ∗ ⋯ ∗ p x ( t ) ,
Y ˜ ( v ) = X ˜ p + 1 ( v ) ,
w ( t ) = exp ( − i v 0 t ) y ( t ) ,
W ˜ c ( v ) = Y ˜ ( v + v 0 ) ,
W ˜ d ( v ) = 1 T s ∑ m = − ∞ + ∞ W ˜ c ( v − m v s ) , v s = 2 π T s .
r = v s v 0 ≥ q + 2     and   r   i s   integer .
s = imag ( w ) = [ 0 4 0 − 20 0 40 0 − 40 0 20 0 − 4 0 ] c = real ( w ) = [ 1 0 − 10 0 31 0 − 44 0 31 0 − 10 0 1 ] Δ φ = π 2
W ˜ ( j ) ( k v 0 ) = 0 , ( j = 0 , 1 , 2 , 3 , k = − 2 , 0 , 1 , 2 )   and   W ˜ ( v )   is   of   period 4 .
P ( x ) = ( p 0 + x ) + p 1 x + p 2 x 2 + p 3 x 3 = ( − 15.79 + x ) − 0.25 x + 0.00057 x 2 − 0.00000031 x 3
C ( x ) = a 0 + a 1 x + a 2 x 2 = 8.93 + 180.30 x + 42.85 x 2 ,
I r = C ( I o ) = a 0 + a 1 I o + a 2 I o 2 = ( a 0 + a 1 A 0 + a 2 A 0 2 + a 2 A 1 2 2 ) + ( a 1 A 1 + 2 a 2 A 0 A 1 ) cos ( θ ) + a 2 A 1 2 2 cos ( 2 θ ) = B 0 + B 1 cos ( θ ) + B 2 cos ( 2 θ )
= B 0 ⋅ I n 0 + B 1 2 exp ( i θ ) ⋅ I n 1 ( 1 + i p 1 v 0 ⋅ n + i p 2 v 0 2 λ 4 π ⋅ n 2 + i p 3 v 0 3 ( λ 4 π ) 2 ⋅ n 3 ) + B 1 2 exp ( − i θ ) ⋅ I n − 1 ( 1 − i p 1 v 0 ⋅ n − i p 2 v 0 2 λ 4 π ⋅ n 2 − i p 3 v 0 3 ( λ 4 π ) 2 ⋅ n 3 ) + B 2 2 exp ( 2 i θ ) ⋅ I n 2 ( 1 + 2 i p 1 v 0 ⋅ n + 2 i p 2 v 0 2 λ 4 π ⋅ n 2 + 2 i p 3 v 0 3 ( λ 4 π ) 2 ⋅ n 3 ) + B 2 2 exp ( − 2 i θ ) ⋅ I n − 2 ( 1 − 2 i p 1 v 0 ⋅ n − 2 i p 2 v 0 2 λ 4 π ⋅ n 2 − 2 i p 3 v 0 3 ( λ 4 π ) 2 ⋅ n 3 )
I v ′ = B 0 I ˜ v 0 + B 1 2 exp ( i θ ) ( I ˜ v 1 + i 2 p 1 v 0 ⋅ d I ˜ v 1 d v + i 3 p 2 v 0 2 λ 4 π ⋅ d 2 I ˜ v 1 d v 2 + i 4 p 3 v 0 3 ( λ 4 π ) 2 ⋅ d 3 I ˜ v 1 d v 3 ) + B 1 2 exp ( − i θ ) ( I ˜ v − 1 − i 2 p 1 v 0 ⋅ d I ˜ v − 1 d v − i 3 p 2 v 0 2 λ 4 π ⋅ d 2 I ˜ v − 1 d v 2 − i 4 p 3 v 0 3 ( λ 4 π ) 2 ⋅ d 3 I ˜ v − 1 d v 3 ) + B 2 2 exp ( 2 i θ ) ( I ˜ v 2 + 2 i 2 p 1 v 0 ⋅ d I ˜ v 2 d v + 2 i 3 p 2 v 0 2 λ 4 π ⋅ d 2 I ˜ v 2 d v 2 + 2 i 4 p 3 v 0 3 ( λ 4 π ) 2 ⋅ d 3 I ˜ v 2 d v 3 ) + B 2 2 exp ( − 2 i θ ) ( I ˜ v 2 − 2 i 2 p 1 v 0 ⋅ d I ˜ v 2 d v − 2 i 3 p 2 v 0 2 λ 4 π ⋅ d 2 I ˜ v 2 d v 2 − 2 i 4 p 3 v 0 3 ( λ 4 π ) 2 ⋅ d 3 I ˜ v 2 d v 3 )

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