Blind self-calibrating algorithm for phaseshifting interferometry by use of crossbispectrum Hongwei Guo* Lab of Applied Optics and Metrology, Department of Precision Mechanical Engineering, Shanghai University, Shanghai, 200072, China *
[email protected]
Abstract: A blind self-calibrating algorithm for phase-shifting interferometry is presented, with which the nonlinear interaction introduced by phase shift errors, between the reconstructed phases and the reconstructed amplitudes of the reference wave, is measured with crossbispectrum. Minimizing an objective function based on this crossbispectrum allows accurately estimating the true phase shifts from only three interferograms in the absence of any supplementary assumptions and knowledge about these interferograms. ©2011 Optical Society of America OCIS codes: (100.2650) Fringe analysis; (120.3180) Interferometry; (120.5050) Phase measurement.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
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20. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007). 21. K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
1. Introduction In phase-shifting interferometry, miscalibration of phase shifter is a major factor decreasing measurement accuracy, and a number of algorithms have been developed for solving this problem. For example, the least-squares methods allow minimizing the measurement error induced by random phase shift errors [1,2], and averaging techniques and some optimized techniques can also compress the influence of phase shift errors [3–5]. A kind of more effective methods is to retrieve phases and phase shifts simultaneously, so that the errors caused by miscalibrations of phase shifters can be eliminated [6–11]. With these algorithms, however, capturing at least five interferogarms has been demonstrated to be the sufficient and necessary condition under which the phases and phase shifts are uniquely determined [11]. In recent years, some attempts have been made in order to break through this limitation, and several algorithms using three or four interferogarms have been proposed. These algorithms are useful in the situation that only fewer than five fringe patterns are available, for example in spatial phase-shifting interferometry [12] or when measuring a moveable profile so that the image capturing time is very tight. Among these algorithms, the algorithm of Cai et al. [13] exploits the statistics of fringes and is effective when the amplitudes of beams have been known prior to measurements. Also proposed by Cai et al. [14], the technique assumes the amplitude of the reference wave to be a constant over fringe patterns. The methods proposed by Xu et al. [15] and by Gao et al. [16] are also based on the same assumption that the amplitudes of the reference waves are uniform. The least-squares iterative algorithm of Wang and Han [17] can give satisfactory results when background intensities and modulations are evenly distributed. All these algorithms allow estimating phase shifts from only three interferograms, but the supplementary constraints in them, such as those concerning uniformities of the beam amplitudes or background intensities, imply that the number of unknowns decreases. These algorithms share a drawback that the supplementary restraints are usually very strong and not always satisfied in measurement practice, typically when the beam is not perfectly shaped or when directly a Gaussian beam is used. Without any supplementary constraints, it is not easy to determine the unknowns such as phases and phase shifts when only three fringe patterns are captured, because the aforementioned sufficient and necessary condition is valid. In other words, the number of independent equations in this case is smaller than that of unknowns, thus making the equation system under-determined. A practically effective solution for this problem was proposed by Goldberg and Bokor [18]. It uses Fourier transforms to determine the phase shift between two interferograms with a carrier frequency. In the technique of Soloviev and Vdovin [19], properly tilting the reference mirror and using Hough transforms of the image differences can estimate the phase shifts between consecutive frames. The algorithm in Ref [20]. permits blindly estimating phase shifts by minimizing the cross power spectrum between the background and fringes. However, it is not stable enough when the problem is poorly conditioned. The reason is that the crosstalk introduced by phase shift errors between the calculated background and fringes is not linear but complicatedly nonlinear; and it is impossible to fully describe such nonlinear interactions by simply using a second order correlation (e.g. cross power spectrum). This reason makes the reliability of this algorithm significantly decreased. From the aforementioned facts, we know that deriving a self-calibrating algorithm depending on only three interferograms in the absence of any preconditions or knowledge about interferograms still remains substantially challenging. To solve this problem, this paper presents, to the best of our knowledge, a novel blind self-calibrating algorithm, that minimizes the high order correlations between phases and amplitudes of the reference wave.
#142138 - $15.00 USD
(C) 2011 OSA
Received 7 Feb 2011; revised 30 Mar 2011; accepted 30 Mar 2011; published 7 Apr 2011
11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7808
2. Principle 2.1 Reconstructions of phase and beam amplitudes In phase-shifting interferometry, the intensities of the kth interferogram is often described with a function of the form I k r o 2 r o cos( k ), 2
2
k 0,1,
, K 1
(1)
where K is the number of phase shifts, |r| and |o| denote the amplitudes of the reference and object beams, respectively, φ is the phase to be measured, and δk are phase shifts with δ0 = 0. All these parameters except δk are functions of pixel coordinates, and δk for each fringe pattern is commonly a constant (assuming no tilt errors exist). When K = 3, by defining c0 = |r|2 + |o|2, c1 = 2|r||o|cosφ, and c2 = 2|r||o|sinφ, Eq. (1) is restated as
I k c0 c1 cos k c2 sin k ,
k 0, 1, 2 .
(2)
We presume for the moment that the phase shifts, δk, are known. Then solving the system based on Eq. (2) results in
I 0 sin( 2 1 ) I1 sin 2 I 2 sin 1 , sin( 2 1 ) sin 2 sin 1
(3)
I 0 ( sin 1 sin 2 ) I1 sin 2 I 2 sin 1 , sin( 2 1 ) sin 2 sin 1
(4)
I 0 (cos 2 cos 1 ) I1 (1 cos 2 ) I 2 (1 cos 1 ) . sin( 2 1 ) sin 2 sin 1
(5)
c0 c1 and
c2 Further we have
arctan(
r
c2 ), c1
(6)
c0 c0 2 c12 c2 2 , 2
(7)
and c0 c0 c1 c2 2
o
2
2
2
.
(8)
Through Eqs. (3-8), the errors in phase shifts will introduce distortions in the reconstructed φ, |r|, and |o|. See Fig. 1 for example, (a) simulates a phase map and (b) is one of three interferograms generated with phase shifts being δk = {0, 1.7, 2.9} and the amplitude of the reference wave being Gaussian-shaped (with a 70% decrease in magnitude at the corners). The size of the interferogram is 256 × 256 pixels. Because the true phase shifts are unknown in measurement practice, we use their nominal values δk = {0, π/2, π} to recover φ and |r|. The resulting errors are displayed in (c) and (d), respectively, where the maximum phase error is 0.1676 radians and the RMS (root-mean-square) phase error is 0.1098 radians. From (c) and (d) we can observe that the shapes of artifacts in φ and |r| are similar to each other and to that of the fringes in (b), but they may have different spatial frequencies. This phenomenon typically reveals the effect of a nonlinear interaction between φ and |r| in the presence of
#142138 - $15.00 USD
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Received 7 Feb 2011; revised 30 Mar 2011; accepted 30 Mar 2011; published 7 Apr 2011
11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7809
phase shift errors. In other words, the phase shift errors introduce special correlations between φ and |r|. Because |r| is mainly determined by the light source and the optics in the reference arm, whereas φ depends on the optical path difference, naturally there should be no correlations between their distributions. For this reason, the actual phase shifts can be estimated by minimizing these correlations, but a problem arisen is how these correlations can be measured.
Fig. 1. Numerical simulation results. (a) Phase map. (b) One of three interferograms. (c) The errors in the reconstructed amplitudes of the reference wave and (d) the errors in the reconstructed phase map by using nominal phase shifts.
2.2 Cross-bispectrum as a measure of correlations As mentioned in Section 1, the algorithm from Ref [20] employs a cross power spectrum to estimate the correlations between the background intensities and fringes. This algorithm is not stable enough, because the cross power spectrum can only measure the second-order correlations between two signals and is blind to higher order ones introduced by nonlinear interactions [For example, there are two signals, i.e. f1(t) = cos(ωt) and its square f2(t) = cos2(ωt). The frequency of the first one is ω. The second signal can be restated as f2(t) = [1 + cos(2ωt)]/2, so its frequencies are 0 and 2ω. The spectra of f1(t) and f2(t) are not overlapped, thus their cross power spectrum equals 0. This fact means that the cross power spectrum cannot describe the nonlinear correlations between these two signals]. We overcome the above problem by using high-order correlations. Considering the cis function of φ
exp(i ) cos i sin,
(9)
the third-order correlations between ψ and |r| are estimated by a cross-bispectrum [21]
B(1 , 2 ) E{Ψ (1 )Ψ (2 ) R (1 2 )},
(10)
where E{·} is the expectation operator and * denotes complex conjugate; ω denotes the spatial frequency in rad/sample; and Ψ () FT ( ) and R( ) FT ( r r ) are onedimensional Fourier transforms of ψ and |r|, respectively, with ψ and |r| being calculated using
#142138 - $15.00 USD
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Received 7 Feb 2011; revised 30 Mar 2011; accepted 30 Mar 2011; published 7 Apr 2011
11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7810
Eqs. (3-9). The purpose of subtracting the averages and r from ψ and |r| is to make them zero-mean normalized. For a pair of phase shifts (δ1, δ2), we select N cross-sections in the fringe patterns, calculate the corresponding ψ and |r| first and then Ψ(ω) and R(ω). Finally we take the following average for estimating the cross-bispectrum: N
Bˆ (1 , 2 ) (1 , 2 ) Ψ n (1 )Ψ n(2 ) Rn (1 2 ),
(11)
n 1
which is a two-dimensional complex-valued function. Because the angular frequency for a digital signal has a range from –π to π, Eq. (11) is defined in the hexagonal field |ω1|
