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A novel phase-shifts nπ/2 calibration method for phase-stepping interferometry, in which the sum intensities and the squared sum of a series of phase-shifted interferograms are firstly calculated, and then by the minimization of the variances of these sums perform the nπ/2 phase-shifts calibration, is proposed in this paper. The proposed method can overcome effectively the effect of the variations of background and modulation intensities in interferograms with any phase structure, and it is also insensitive to the nonlinearity of phase shifter. Numerical simulation and experiments are implemented to verify the effectiveness of this method.
©2009 Optical Society of America
1. Introduction
Phase-stepping interferometry (PSI) [1,2], as a wavefront measurement method with high accuracy, is widely used in optical testing. In PSI, phase shifts are introduced between two interfering beams and the object phase can be retrieved from series of interferograms using an appropriate algorithm. It is normal practice to restrict the number of frames to 3, 4 or even 2 [3,4], which practically save time and space significantly and reduces requirement on system stability. However, the accuracy of phase determination from fewer frames extremely relies on the accurate values of phase-shifts. Therefore, a sufficiently exact phase-shift calibration is a necessary and important job.
Many calibration techniques [5–12] have been developed to measure an unknown phase-shift. The techniques using the Carré algorithm [5] or its variations [6] calculate the phase step from four interferograms with constant phase step, but they are only applicable to the linear phase shifter. In Ref. 7, phase-shift are determined from only two interferograms by analyzing the intensity correlation, and the calibration may be carried out with a strict requirement that an integral number of fringes should be contained. Additionally, the methods in Refs. 7 and 8 assume that there are no changes presenting in background and modulation intensities. Some authors apply the Fourier-transform method (FTM) to determine the phase-shift between two interferograms with high carrier-frequency [9,10], but spatial spectrum analysis causes a burdensome computational load, and the phase evaluation with the FTM in Ref. 9 produces noticeable inaccuracies at the border of the measured pattern. Goldberg and Bokor [10] developed a simple Fourier-transform method [11,12] to measure the global phase-shift value between two interferograms. In this method the phase-shift was extracted by comparing the phase of the first-order maximum of one interferogram in the Fourier-domain with another. The uncertainty in phase-shift determination by use of the spatial frequency domain depends primarily on the disturbance of the zero order and the sidelobes relative to each other. Choosing a higher spatial carrier-frequency could reduce the overlap of the orders, but the carrier-frequency is often limited by the bandwidth of the detectors and the measured waves. Ensuring that the calculated point occurs at the peak value of the first-order sidelobes is a simple approach to lowering the error [10], but it is difficult to overcome the influence of any random noise on this single point. Recently, two-run-times-two-frame phase-shift method is proposed, but a good mirror is required and a known phase-step needs to be introduced [13].
Generally, in PSI the popular phase-step is π/2 even for some single-phase-step methods [3,4], and hence the only calibration phase-shifts nπ/2 for the phase shifter is sufficient. In this paper we present a phase-shifts calibration technique that removes some needs for such a priori knowledge of the phase shifter, and it also does not require the establishment of a special phase structure in interferogram (e.g., linearity, integral number of fringes and high carrier-frequency). With it the sums and the squared sums of the first and the subsequent interferograms at the same spatial location are calculated, and then nπ/2 phase-shifts can be determined by minimization of the variances of these sums. The performance of the algorithm is firstly verified by computer simulations, in which we take into account the presence of nonuniform background and modulation intensities in the phase-shifted interferograms. Additionally, two cases with different fringes, square and circular apertures are also investigated. Then we apply the proposed method as well as correlation method and FTM to real interferograms and find that the experimental results are in agreement with numerical simulations predictions. The proposed method allows phase-shifts nπ/2 calibration to be made to high accuracy without consideration of optics quality, tilting modulation amplitude and interferogram characteristics, and it covers no complicated mathematical operation.
2. Theory
Let Ik(x,y) be the intensities of the kth interference patterns of a two-beam interferometer:
where ϕ(x,y) is the initial phase distribution; δk denotes phase-shift amount of the kth interferogram with δ 1 = 0; A(x,y) and B(x,y) are the background and the modulation intensities; (x, y) is an arbitrary point in the interference pattern and will be omitted for brevity.Let us consider now the sum of interference intensities I 1 and Ik:
We average the above equation over full frame, and assume independence between B and ϕ to obtain the expectation E[S]where the signmeans spatial averaging. In general, the background intensity A and the modulation amplitude B are constants over the region of interest. If a tilting modulation is introduced or the inherent aberration ϕ in the optical system can be considered as nearly random, one has the approximationsand. And hence Eq. (3) is reduced toand the variance Var[S] of the sum intensity S is given by Equation (5) shows that when the phase-shift tends to π the variance of the sum intensity falls to the minimum. Based on this idea, phase-shift π can be calibrated by finding the minimum variance of all the sum intensities.To calibrate other two phase-shifts π/2 and 3π/2, the background intensity A need be obtained, whereas it can be extracted directly from , where denotes the interferogram with phase-shift π. Then we subtract A from each phase-shifted interferograms, yielding the background-suppressed phase-shifted intensities . The sum of the squared intensities removed the background is expressed as
Similarly, one has the approximationsand . Consequently, we can obtain the expectation and the variance of Eq. (6) as Equation (8) shows that when δ equals to π/2 or 3π/2 Var[Ss] reaches to the minimum, so phase-shifts π/2 and 3π/2 can be calibrated by finding the minimum variance of Ss.The calibration process of phase-shifts nπ/2 is summarized as follows.
- 1) A series of phase-shifted interferograms are recorded as the applied voltage increases;
- 2) Calculate the sums of the first and the subsequent interferograms at the same spatial location within the region of interest;
- 3) Calculate the variances of these sum intensities, and fit the calculated data to a best polynomial function;
- 4) By finding the minimum of fitted function phase-shift π is found;
- 5) Extract the background intensity from, and calculate the background-suppressed phase-shifted intensities by subtracting A from each phase-shifted pattern;
- 6) Calculate the sums of the squared intensities and the variances of these sums;
- 7) Individually fit the variance data within two segments before and after phase-shift π, two best polynomial functions can be obtained;
- 8) By finding the minimum of the two fitted functions phase-shifts π/2 and 3π/2 are obtained.
In addition, if A and B are not uniform over the whole frame, the variances of the sums from Eq. (2) and the squared sums from Eq. (6) can be expressed respectively as
From two above equations we can find that the variance of A is a direct current term appearing in Eq. (9), and it is not contained in Eq. (10), so the variation of the background intensity does not affect this calibration. Additionally, the term in Eq. (9) scarcely varies from frame to frame, and variation of Var[S] is mainly decided by the term . Thus the variation of B will not affect the calibration of phase-shift π. At the same time, as long as a requirement, i.e., , is met, the variation of B would also not affect the calibrations of other two phase-shifts π/2 and 3π/2, which can be gotten from Eq. (10) if δ equals kπ and kπ + π/2 (k is an integer) Var[Ss] equals respectively and . Meanwhile, this requirement is easily met in the experimental system with the introduction of a tilting modulation because generally approaches zero or equals a small value.3. Simulation
Some numerical simulations were performed to test the ability of standing against the variation of the background and the modulation intensities. A phase map with random noise ranging from –π/4 to π/4 is used. Both the background and the modulation intensities change according to Gaussian function: exp[-r(x 2 + y 2)], where x and y are normalized to −1 to 1, and r is the varying parameter and is set to be 0, 0.3 and 0.6, respectively. By varying the assumed phase-shifts from 0 to 2π three sets of phase-shifted interferograms with three types of background and modulation intensities are generated and quantized to 8-bit gray level. Three representative interferograms are shown in Fig. 1 .
Fig. 1 Interferograms with varying background intensity and modulation amplitude both described by Gaussian function, the variable r equals 0 (a), 0.3 (b) and 0.6 (c).
The variances of S and Ss for the first and subsequent interferograms can be calculated and then are plotted in Fig. 2 . General conclusion that follows from Fig. 2 is that Var[S] in phase-shift π [Fig. 2(a)] as well as Var[Ss] in phase-shifts π/2 and 3π/2 [Fig. 2(b)] reaches to the minimum values. Note that three groups of curves in each figure present the same trend except the difference in the variance’ magnitudes, thus this technique is not sensitive to the variations of background and modulation intensities.
Fig. 2 The variances of the sum intensities (a) and squared sums (b) as a function of assumed phase shifts.
To verify the robust applicability of the proposed method to the phase-shifted interferograms with any phase structure, we performed two following simulations and compared it with correlation method [7]. In the two simulations a monotonically increasing phase map with random noise ranging from –π/4 to π/4 is used, and a nonlinear relationship between phase-shift and voltage is defined as δ = 5 × 10−4 V 2 + 0.101V, where δ and V represent phase-shift and applied voltage, respectively. Each phase-shifted interferogram, with constant background intensity, and modulation distribution described by Gaussian function (minimal amplitude is about 37 percent of its maximum) has individually added an independent Gaussian white noise with a mean of zero and variance σ2 = 1 (the maximum noise gray level is about 7). The first case, simulated interferogram contains four full fringes, and a square aperture is considered, and one of them is shown in Fig. 3 (a) ; the second case, in which nonintegral number of fringes and circular aperture are considered, one interferogram is shown in Fig. 3 (b).
Fig. 3 Simulated interferograms: (a) Case 1, with four full fringes and square aperture, (b) Case 2, with nonintegral fringes and circular aperture.
Figure 4 shows the calibration results with the proposed method, and the results with the correlation method are shown in Fig. 5 . The resulting relationship between phase-shift and applied voltage is plotted in Fig. 6 . In Fig. 6 the gray dotted curve is 2-order polynomial fitted to the unwrapped phase for case 1 in Fig. 5. Table 1 summarizes the calibration results of two methods for two cases for which the average error of calibration voltages is defined as , where Vm is the measured value, and Vt is the theoretical value.
Fig. 4 Calibration results for π (a) and for π/2 and 3π/2 (b) with the proposed method for two cases. Sign ‘*’,’ × ’: Computed from data; Curves: 5-order polynomial fits.
Fig. 6 Relationship between phase-shift and voltage. Black real curve, ideal phase-shift, gray dotted curve, fitted result of unwrapped phase for case 1 in Fig. 5; squares, results of proposed method for two cases.
Table 1. Comparison of calibration results of the proposed method and correlation method.
For the first case, we can see from Fig. 6 and Table 1 that the calibration results of both methods show good agreement between them. The average calibration errors of two methods are almost equal. For the second case, however, the correlation method cannot carry out an accurate calibration (for example, phase-shifts π and 3π/2 cannot be found) (see Fig. 5) mainly due to the appearance of incomplete fringes. On the contrary, the proposed method can still be used, and the calibration results are nearly same as that from the first case.
4. Experiments and discussion
To perform a realistic demonstration and to verify the simulation results, a Mach-Zehnder interferometric optical configuration (see Fig. 7 ) is used to calibrate a piezoelectric transducer (PZT) phase shifter with an increment of one volt. The phase shifts between the object and the reference beams are introduced by the phase shifter, which is placed in the object beam for convenient adjustment. There is no strong requirement on the flatness or quality of mirrors needed to perform this phase calibration. Two experiments corresponding to the simulations in Section 3 have been performed.
In the first experiment, a series of interferograms with about eight full fringes are acquired as the applied voltage increases, and one of them is shown in Fig. 8(a) . In the case, correlation method [7] and FTM [10] can also be used. In the second experiment, much less fringes are contained in interferograms by tilting slightly the mirror. In addition, the circular aperture is also considered. One of the recorded interferograms is shown in Fig. 8(b). Other experimental condition is same as that in the first experiment.
The calibration results with correlation method and FTM are shown in Figs. 9 and 10 . For the first experiment, by fitting each set of unwrapped phase-shift data two best linear functions corresponding to correlation method and FTM can be found to be δ = 0.0435V-0.0192, and δ = 0.0439V-0.086, respectively. Therefore, the driving voltages for the phase-shifts {0, π/2, π and 3π/2} equal {0, 36, 72, 108} and {0, 36, 72, 107}, respectively. For the second experiment, it is shown from Fig. 9 that correlation method results in a wrong calibration. At the same time, FTM cannot also be used due to low carrier-frequency being involved.
With the proposed method described in Section 2, the calibration results for two experiments are shown in Fig. 11 . These calculated data are respectively fitted to a 5-order polynomial function according to the least-squares method. By finding the minimum of the fitted functions the driving voltages for nπ/2 phase-shifts can be determined. To minimize the influence of the PZT nonlinearity the calibrations for phase-shifts π/2 and 3π/2 are individually implemented with respect to two segments before and after phase-shift π. As a result, the driving voltages for phase-shifts {0, π/2, π and 3π/2} can be determined and equal {0, 36, 73, 107} for Exp.1 and {0, 35, 73, 106} for Exp.2.
Fig. 11 Calibration results for π (a) and for π/2 and 3π/2 (b) with the proposed method for two experiments. Sign ‘ + ’,’ × ’: Computed from data; curves: 5-order polynomial fits.
The results from the first experiment show good agreement between three methods. At the same time, the results given by the proposed method for the second experiment also accords with that for the first experiment. The difference is mainly due to the slightly different environmental condition in which each experiment was carried out. In addition, it is also shown that the experimental results are in agreement with numerical simulations predictions.
5. Conclusion
We have presented a novel phase-shifts nπ/2 calibration method, and the performance of this method is tested numerically and experimentally. Without considerations of quality of the optical components, phase shifter nonlinearity and tilting modulation amplitude allow this calibration method to be implemented conveniently and accurately. At the same time, the proposed method can also overcome effectively the effect of the variations of background and modulation intensities in interferograms with any phase structure and optical aperture. Based on these advantages we reasonably believe that the proposed method is an accurate and feasible method for phase-shifts nπ/2 calibration in PSI, and it can be completed faster than Fourier-transform method due to only sum and variance operations being included.
Acknowledgments
The authors thank Prof. Wenhan Jiang and the reviewers for their helpful suggestions; also thank Drs. Dongmei Cai and Linhai Huang for their heuristic discussion and valuable assistance in practical experiments.
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