Advanced method of global phase shift estimation from two linear carrier interferograms
Phase-shifting interferometry is a kind of important technique used in optical interference metrology. This technique has high precision and good stability, which has been widely used in scientific research and industrial production.
This paper proposes a new method to estimate global phase shift from two interferograms. This method performs algebraic calculation of two interferograms with the assistance of Hilbert transform. An iterative approach is used for the attempted phase to ensure that the minimum of assessment function is obtained.
The simulated result indicate that the maximum calculation error of the global phase-shifting is 1.5%. And then we use experimental data to verify the performance of this method.
The method proposed in this article is simple but precise, and can cope with interferograms with uneven background and modulation, non-periodic apodization, and random noises. It does not require any specific carrier frequency of the measured interferogram or any adjustment of range of integration in accordance with the carrier frequency.
KeywordsGlobal phase-shifting Fringes pattern Hilbert transform Iterative approach
Empirical mode decomposition
Energy-minimum fourier transform algorithm
Phase-shifting interferometry (PSI) is a technique used in optical interference metrology. This technique has high precision and good stability, can be implemented through a variety of hardware, and has been consistently observed by researchers. Many algorithms have been developed to retrieve phase from a group of phase-shifted interferograms. Classical phase-shifting algorithms include fixed steps, variable steps, or random phase-shifting . In recent years, researchers proposed a lot of interesting algorithms, including the two-frame phase shifting algorithms with regularized fringe pattern [2, 3, 4], the unknown or uncalibrated extraction algorithms [5, 6], and the generalized phase shifting method , etc. On some occasions, global phase-shifting value is a known value. The estimated value can be provided through existing information from previous measurements. However, many PSI algorithms need to calibrate the influence of phase-shifted errors from environment vibration, nonlinear response or unbalanced piezo-electric effect. In some other cases, global phase-shifting itself is unknown, which needs to be determined from a series of interferograms.
With respect to solutions of global phase-shifting values among interferograms, Farrell and Player proposed a method based on Lissajous figure fitting . Brug proposed a method based on calculation of the correlation between two images . Goldberg and Bokor, et al., proposed a method based on single-point Fourier transform , which calculates global phase-shifting by comparing the changes in power of the carrier frequency between two interferograms. However, all interference signals have limited length; carrier frequency is not a single frequency; spectral leakage may occur on the + 1 (or − 1) order signal frequency spectrum. Calculated the power change in a single frequency, alone, cannot comprehensively reflect the change in the global phase-shifting and cause the loss of calculation precision. Guo and Rong, et al., proposed an Energy-minimum Fourier Transform algorithm (EMFT) . This method attempts to locate the best range of + 1 (or − 1) order signal frequency spectrum from the power spectrum, and therefore can increase the calculation precision for global phase-shifting under the same conditions. However, the interferograms from the measurements are affected by a variety of factors such as the effect of interferograms apodization, the uneven background, the signal envelope and the random noise. As a result, spectral aliasing may occur between the sideband of + 1 (or − 1) order spectrum and zero order signal frequency spectrum. This issue significantly reduces the precision of the calculation results, especially when the carrier frequency is low. Therefore, some methods for zero order spectrum elimination or suppression were proposed [12, 13]. In recent years, methods with Hilbert Transform (HT) and Hilbert-Huang Transform (HHT), aided by Empirical Mode Decomposition (EMD) have been used to suppress the unevenness of background [14, 15, 16]. On one hand, this issue reduces the robustness of the algorithms; on the other hand, because the generation of interferograms are constraint by the detector and hardware configuration, the choice of carrier frequency is not unlimited. In order to resolve this problem, Vishnyakov and Levin, et al., proposed a method to first do subtraction between two interferograms and then perform the Fourier Transform. This can effectively avoid the spectral aliasing and preserve relatively high calculation precision even when the carrier frequency is low [17, 18]. However, under this method, three interferograms are required for calculation, which limits its applicability.
In this paper, we propose a global phase shifting extraction method which is simple and direct. This method performs algebraic calculation of two interferograms with the assistance of Hilbert transform, and inserts the attempted global phase shift value into assessment function for calculation. The process is repeated until the value of the global phase shift in determined. The proposed method has better precision and robustness in scenarios of spectral aliasing or non-periodic apodization and potential applicability in various aspects of digital holography, interferometry, surface metrology, etc. [19, 20, 21, 22]. This paper introduces the theory behind the method, analyzes the accuracy and adaptability with numerical simulations and experiment.
Capture two linear carrier interferograms, I1 and I2, that include unknown global phase shift;
Set the initial value of δ and incorporate Eq. (7) to calculate the value of the assessment function;
Set the range for δ and the step interval ∆δ, adjust the value of δ, and incorporate F(δ) for additional calculation;
Repeat this step until all F(δ) are solved;
Find the minimum value of F(δ), which is the global phase shift to be determined.
Note that when δ → 2πn(n = 0,1,2,---), cot(δ / 2) → ∞. In this situation, this algorithm is invalid as there seems to be no phase shift between the two interferograms. The difference interferograms should be analyzed if this occurs. If only random noise exists and there is no periodic change in intensity, then it should be concluded that no phase shift occurs between the two interferograms.
Results and discussion
From the perspective of signal shapes, the above described simulated signals include various interfering factors that may impact the calculation of global phase shifts, including, as documented in Literature (, 33–34), unevenness in background intensity and modulation envelopes (α ≠ constant, b ≠ constant), low carrier frequency, signal noises, non-periodic apodization, and etc. For a linear carrier interferogram, if the signal is non-periodic, the sideband of the carrier frequency signal will be broader; if the carrier frequency is not high enough, then it will spectrally aliased with the baseband signal. In practice, however, most interference signals are usually not single frequency. The location of apodization is not readily selected for reasons such as background and modulation unevenness.
This article reports a method to estimation the global phase shift with two interferograms. Compared with existing methods, this method requires no pre-filtering, nor does it have specific requirement for the carrier frequency of the interferograms. During the calculation, this method does not require the selection of a window for integration. It therefore increases the algorithmic adaptability and provides easy automatic processing. The method in this article may resolve the problem of global phase shift calculation when the interference signal faces a variety of factors such as nonperiodic apodization, uneven background and modulation. It also has better precision and robustness than previous methods in scenarios of spectral aliasing between the + 1 order spectrum and the zero order spectrum. The method in this article can calculate the global phase shift with only two interferograms. We believe that this method has broad potential applicability in various aspects of interferogram processing. And it could be extending to process the closed fringe patterns. In practice, this method can be used to screen out one or more sets of interferograms that meet the global phase shift needs from a series of interferograms for further calculation and analysis. This article provides the rationale and process of the calculation under this method, compares it with the existing method that needs Fourier Transform based on simulated interference signals, demonstrates the superiority of the method in this article, and confirms the precision of calculation with this method with data from experimental measurements.
The authors are grateful to Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application and Suzhou Key Laboratory for Precision and Efficient Processing Technology(SZS201712) for their support.
The work reported in this article is supported by the National Natural Science Foundation of China (11503017, 61378056); The “Summit of the Six Top Talents” Program of Jiangsu Province (2015-DZXX-026); Suzhou University of Science and Technology (XKQ201513); Suzhou Key Industry Technology Innovation Plan (SYG201646).
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