1. Introduction

In traditional phase-shifting interferometry (PSI), a phase shifter is used to achieve N-step phase-shifting while a corresponding N-step phase-shifting algorithm is used to extract the phase [1]. However, an equal phase-shifter step length is required and vibration interference is prohibited. Environment vibration and air turbulence may lead errors in the recovered phase [2,3]. To overcome the effects of environmental vibration, various interference systems [4–18] and phase-extraction algorithms [19–34] have been proposed.

As regards interference systems, changes have been made to improve the interference optical path in order to eliminate the vibration effect [4–14] or to separate and extract the vibration information [16,17]. The vibration can be compensated by directly measuring the cavity motion [4,5], but this method needs complex mechanical structure. Among the various approaches, Sykor et al. realized dynamic measurement by spatial carrier frequency interferometry [6]; however, a high carrier frequency was introduced to the interferogram, which generated a retrace error [8]. To reduce the influence of the retrace error, one way is to calibrate the retrace error for each element to be measured [9]. The calibration process needs more time and effects the measurement accuracy. Moreover, Szwaykowski et al. proposed a simultaneous phase-shifting module for interferometry application using three cameras [10], and Millerd et al. developed a pixelated phase-mask dynamic interferometer that can simultaneously obtain four phase-shifting interferograms [11,12]. Similarly, Kimbrough et al. reported a polarization Fizeau interferometer with a low-coherence source, in which the polarization camera simultaneously obtains four interferograms [14]. Further, a dynamic Fizeau interferometer based on lateral displays of point sources was developed by Zhu et al. [15]. Synchronous PSI such as that employed in the above studies exploits different laser polarization states. For large-aperture measurement, however, optical paths with different polarization states owing to the optical-device nonuniformity and birefringence; this causes error ripple in the results phase [16]. As an alternative approach, Li et al. developed a dynamic low-coherence interferometry system involving a double Fizeau cavity interferometer and a low-coherence laser [17]. In their system, beams with different polarization states are used to measure two Fizeau cavities separately. The advantage is that the common-path Fizeau cavity is measured using the same polarization state. However, the double Fizeau cavity is produced by two sides of the same reference wedge and the carrier frequency cannot be adjusted. In addition, optical path matching is needed when anti-vibration measurement is not required, which increases the conventional measurement complexity.

As regards algorithm application to overcome the effects of environmental vibration, although the random phase-shifting algorithm can solve the phase-shift error problem [19–25], the phase extraction accuracy is affected by the vibration-induced error. Thus, a number of algorithms have been developed [26–34]. For example, Chen et al. proposed an iterative algorithm based on a first-order Taylor expansion to solve the vibration-induced tilt problem [27]. Further, Deck developed a model-based phase-shifting interferometry (MPSI) algorithm based on a multi-parameter physical model, which approximates the physical parameters to a Taylor series expansion [28]. Liu et al. proposed a three-step least-squares iteration algorithm to extract phase and vibration information using least-squares fitting [29]. Liu et al. proposed a novel phase shifting interferometry from two normalized interferograms with random tilt phase-shift [31]. Li et al. reported a phase-tilting interferometry to retrieve the phase from interferograms with random phase tilts [32]. An inter- and intra-frame contrast compensation method is proposed by Liu et al. to retrieve wavefront phase from interferograms subjected to vibration [33]. The above algorithms can solve the vibration problem within a certain range, but their applicability is limited. That is, for an interferogram with a small number of fringes (such as null fringe), uneven background and modulation, and excessive vibration, these algorithms are unreliable. Moreover, most iterative algorithms are affected by initial values, in that an unsuitable initial value yields a local optimal solution and even non-convergence.

To overcome the above problems, in this study, a dual-mode Fizeau interferometer (DFI) and a four-step phase-tilting iteration algorithm are proposed. The DFI achieves dual-mode measurement using a combination of a conventional Fizeau interferometer and a vibration information measurement system. In standard mode, the same-polarization Fizeau interferometer is used to obtain the phase information; this avoids the polarization error caused by optical-device birefringence. In dynamic mode, the added vibration information measurement system extracts the vibration phase. A dichroic mirror combines the two systems and switching between the two measurement modes is unnecessary. In addition, the four-step phase-tilting iteration algorithm proposed in this study is helpful for vibration phase extraction. In this algorithm, the phase, $x$- and $y$-direction coefficients, and phase shift are iterated through least-squares to determine the convergence threshold. High measurement accuracy with null fringe interferograms is also achieved by the DFI, as confirmed through experiment; this solves the null-fringe dynamic measurement problem.

The remainder of this paper is organized as follows. Section 2 introduces the DFI optical system in detail. Section 3 discusses the four-step phase-tilting iteration algorithm and phase extraction algorithm and Section 4 presents a DFI simulation. Section 5 discusses construction of the DFI to verify its robustness and Section 6 discusses the algorithm and experimental results. Finally, Section 7 concludes the study.

2. Optical system of dual-mode Fizeau interferometer

As shown in Fig. 1, the DFI optical system is divided into a conventional Fizeau interferometer and a vibration information measurement system, referred to hereafter as the “standard” and “dynamic” modules, respectively, for convenience. The dynamic module uses different polarizations to separate the reference and test beams and adds an additional carrier frequency to obtain vibration information. This module uses different wavelengths from the standard module, and uses a dichroic mirror to access the measurement part of the standard module. During measurement, the two modules collect information simultaneously to ensure information synchronization.

 

Fig. 1. Schematic of DFI structure.

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As shown in the red-colored section of Fig. 1, the standard module has the same polarization state as a Fizeau interferometer with a narrow-linewidth coherent laser as the light source. After passing through a half-wave plate (HWP1), the polarized beam emitted from the laser is converted into a p-polarized beam. Then, after passing through a beam expander lens, this beam is transmitted through a polarization beam splitter (PBS1). Next, the beam is transmitted through the dichroic mirror and collimated into a large aperture beam, before reaching reference mirror A and test mirror B. The beams returned by A and B are twice transmitted through a quarter-wave plate (QWP1) and transformed into s-polarized beams. The s-polarized beams are reflected by PBS1 and reach a camera (D1) through a collimating lens. The standard module uses two wave-plates and PBS combination to generate the same polarization beam for interference, and to achieve the highest possible energy utilization rate. Further, the standard module is identical to a conventional Fizeau interferometer apart from the inclusion of the dichroic mirror. Note that a phase shifter is employed at reference mirror A.

As shown in the green-colored section of Fig. 1, a short-coherence laser is employed in the dynamic module. Specifically, the Twyman-Green system is used for the light-source component of the dynamic module. A polarized beam is emitted by the short-coherence laser, which is transmitted through a half-wave plate (HWP2), and the polarization direction is adjusted to be 45° to the paper. The beam is then divided into s- and p-polarized beams by a polarization beam splitter (PBS2). The s-polarized beam is reflected by a mirror (M1) and through a quarter-wave plate (QWP2) twice to being p-polarized and transmitted through PBS2. Similarly, the p-polarized beam produced by PBS2 becomes s-polarized by through QWP3 twice after reflection from a different mirror (M2). The separated s- and p-polarized beams are reflected by a beam splitter (BS) and introduced to the measurement part of the standard module by the dichroic mirror. The reflected beams from A and B enter the imaging part of the dynamic module following reflection by the dichroic mirror. The s- and p-polarized beams are separated by another polarization beam splitter (PBS3) and then reflected by mirrors (M3 and M4) to enter the imaging lens group and reach a camera (D2). The four beams incident on D2 are labeled p_A, p_B, s_A, and s_B, respectively.

Note that M1 can be adjusted to satisfy the optical-path matching requirements, causing p_B and s_A (or the other two beams) to interfere. Further, M4 can be adjusted to change the tilt angle, then, the carrier frequency is added between p_B and s_A without affecting the standard module. Although the dynamic module has a large non-common path error, this module is used to extract the relative tilt coefficients only and not the phase. Thus, the non-common path error has no effect on the standard module phase restoration. The dynamic module is attached to the standard module through the dichroic mirror and the modules exhibit no crosstalk.

3. Phase extraction algorithm

Two interferogram groups can be obtained through the optical system described in Section 2. The intensity of the $n\textrm{th}$ interferogram of the two groups is expressed as

In theory, the phases, ${\Phi _S}(x,y)$ and ${\Phi _D}(x,y)$, should be the same. However, the dynamic module has a non-common path error induced by the carrier frequency $E$ and the actual ${\Phi _S}(x,y)$ and ${\Phi _D}(x,y)$ differ; however, the non-common path error has no effect on the relative tilt. Thus, the dynamic-module interferograms are not used for phase extraction but used for relative tilt extraction.

For a group of interferograms with vibration, the first interferogram is usually considered to have no vibration (i.e., ${P_1}(x,y) = 0$), and the tilt planes of subsequent interferograms take the relative quantity of the first interferogram as a reference. As the vibrations of both DFI modules originate from the test-mirror vibration, the relative vibration planes of the dynamic and standard modules are the same. In other words, the relative vibration plane of the dynamic module can be used to extract the phase of the standard module with the interferograms of the standard module.

The complete calculation process is shown in Fig. 2. The relative vibration planes of the interferograms collected by the dynamic module are obtained via the tilt-coefficient extraction algorithm. Then, the phase is extracted through least-squares fitting using the interferograms collected by the standard module. The tilt-coefficient extraction algorithm consists of a Fourier transform and a four-step phase-tilting iteration algorithm. A detailed description of the algorithm is provided below.

 

Fig. 2. Calculation process of proposed method.

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3.1 Tilt coefficients of dynamic-module extraction algorithm

3.1.1 Four-step shift-tilting iteration algorithm

According to Eq. (1), for the $n\textrm{th}$ interferogram with vibration, the intensity ${I_n}$ can be written as

Step 1: Determining $\Phi (x,y)$.

The relative vibration coefficients, ${\alpha _n}$, ${\beta _n}$, and ${\delta _n}$, are known, but the $\Phi (x,y)$ of each pixel $(x,y)$ is unknown. To separate the known quantities from the unknown quantity, Eq. (2) can be rewritten as

The solution of Eq. (5) is given by the linear matrix equation

Finally, $\Phi (x,y)$ can be obtained from the relation

Note that, for matrix $A$ to be nonsingular, $N$ should not be less than three.

Step 2: Determining ${\alpha _n}$.

As shown in Fig. 3, the interferogram size is $X \times Y$ pixels. The ${\alpha _n}x$ of each point in the $x\textrm{th}$ column of the $n\textrm{th}$ vibration plane is the same. Here, $\Phi (x,y)$, ${\beta _n}$, and ${\delta _n}$ in the $x\textrm{th}$ column of the $n\textrm{th}$ interferogram are known. Equation (2) is then rewritten as follows to determine ${\alpha _n}x$:

 

Fig. 3. Schematic diagram of vibration plane sequence.

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According to the least-squares criterion,

Finally, the ${\alpha _n}x$ of the $x\textrm{th}$ column can be obtained from the relation

The ${\alpha _n}x$ of each column in the effective region of the $n\textrm{th}$ interferogram is calculated cyclically. The relative tilt coefficient ${\alpha _n}$ in the $x$-direction can be obtained by calculating the difference of a series of ${\alpha _n}x$ obtained from the $n\textrm{th}$ interferogram.

Step 3: Determining ${\beta _n}$

This step is similar to Step 2, as the ${\beta _n}y$ of each point in the $y\textrm{th}$ row of the $n\textrm{th}$ vibration plane is the same. Here, $\Phi (x,y)$, ${\alpha _n}$, and ${\delta _n}$ in the $y\textrm{th}$ row of the $n\textrm{th}$ interferogram are known. Equation (2) is rewritten as follows to determine ${\beta _n}y$:

Finally, the ${\beta _n}y$ of the $y\textrm{th}$ row can be obtained from the relation

Using the same difference process as described in Step 2, the $y$-direction tilt coefficient ${\beta _n}$ of the $n\textrm{th}$ interferogram is obtained.

Note that, for matrixes $A^{\prime}$ and $A^{\prime\prime}$ to be nonsingular, each interferogram should be at least 3×3 pixels in size and have at least one fringe.

Step 4: Determining ${\delta _n}$

The ${\delta _n}$ of each pixel in an interferogram is the same. For $N$ interferograms, $\Phi (x,y)$, ${\alpha _n}$, and ${\beta _n}$ are known. Equation (2) is then rewritten as follows to determine ${\delta _n}$:

Finally, ${\delta _n}$ can be obtained from

Steps 1–4 are the steps of the iterative cycle of this method. The number of iterations is controlled via a set convergence threshold. The convergence criterion $\varepsilon$ is defined as follows:

The iteration process of the four-step phase-tilting iteration algorithm is shown in Fig. 4. The result of the $k\textrm{th}$ cycle is taken as the initial value of the $(k + 1)\textrm{th}$ cycle. The iterative process is summarized as follows.

 

Fig. 4. Iteration cycle diagram of four-step phase-tilting iteration algorithm.

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The four-step phase-tilting iteration algorithm is identical to other iterative algorithms in that estimated initial values must be supplied before iteration. The accuracy of this initial value affects the convergence rate and the final accuracy of the iteration. Thus, an initial value close to the true value can reduce the number of iterations, improve the convergence speed, and avoid a local optimal solution. For the optical system presented in Section 2, when the carrier frequency of the dynamic-module interferogram is dense, low-accuracy relative tilt coefficients can be extracted using a Fourier transform. Although the coefficients given by this Fourier transform are not sufficiently accurate to constitute the final result, they can be used as initial values for the iteration. The accuracy of the relative tilt coefficients can then be refined through iteration. The method for extracting the relative tilt coefficients using a Fourier transform is described in Section 3.1.2. Further, the accuracies of the four-step phase-tilting iteration algorithm and Fourier transform are compared in Section 6.

3.1.2 Fourier transform for extracting low-accuracy relative tilt coefficients

For Fourier transform, the interferogram is converted to frequency domain for calculation. Equation (2) can be rewritten in complex form as follows:

By performing a 2D Fourier transform of the interferogram of size $X \times Y$, we obtain the following:

Then, ${k_n}$ can be obtained from the relation

After the carrier-frequency coefficients of the interferogram are obtained, the relative tilt coefficients are determined from the following equation:

As noted above, the low-accuracy relative tilt coefficients given by the Fourier transform can be used as initial values of the four-step phase-tilting algorithm for further accurate calculation.

3.2 Standard-module phase extraction algorithm

The two algorithms in Section 3.1 are combined to obtain the relative tilt coefficients ${\alpha _n}$, ${\beta _n}$, and ${\delta _n}$, of the dynamic module. As described at the beginning of Section 3, when the relative tilt coefficient obtained by the dynamic module is applied to the standard module, wavelength conversion is required. The conversion is to multiply each coefficient by the wavelength ratio $e = {\lambda _D}/{\lambda _S}$ (${\lambda _D}$ is the wavelength of dynamic module, ${\lambda _S}$ is the wavelength of standard module). Then, ${P_{S,n}} = {P_{D,n}} \cdot e$. In addition, for the optical system described in Section 2, the relative tilt-plane coefficients of the dynamic module have the same values as those of the standard module. However, owing to the existence of mirrors and lens in the optical system, the interferograms of the standard and dynamic modules may flip in the $x$- or $y$-directions. Therefore, before substituting the coefficients obtained from the dynamic module into the standard module, it is necessary to determine whether opposite values of ${\alpha _n}$ and ${\beta _n}$ are required in accordance with the actual optical system. In this study, the opposite sign of ${\beta _n}$ was used for subsequent calculations. For convenience of expression in this paper, the symbol ${\beta _n}$ is still used to represent the $y$-direction tilt coefficient input into the standard module.

The standard-module phase ${\Phi _S}(x,y)$ is calculated using the least-squares fitting method [35–37]. Equation (1a) can be rewritten as

Here, ${I^{\prime}_{S,n}}$ is the standard-module interferogram obtained from the experiment. Finally, ${\Phi _S}(x,y)$ is determined as

Because the relative tilt coefficients obtained by the dynamic module are sufficiently accurate, the standard module can accurately restore the phase using least-squares fitting. Therefore, even if the standard-module interferogram is a null fringe, the phase can also be recovered. Simulation and experimental verifications are presented in Sections 4 and 5, respectively.

4. Simulations

To verify the DFI feasibility and accuracy, the experimental process was simulated. First, Zernike polynomials were used to fit the test-wavefront phase ${\Phi _S}(x,y)$. The Zernike polynomial coefficients were set as listed in Table 1. To simplify the simulation, ${\Phi _S}(x,y)$ was used to generate interferograms for both modules. In accordance with Eq. (1), a consistent relative vibration ${P_n}(x,y)$ was added to the reference and test wavefront phase. The standard-module interferograms were quasi-zero fringes, as shown in Figs. 5(a1)–(a8). An additional carrier frequency $E(x,y)$ was added to the dynamic module; hence, the dynamic-module interferograms shown in Figs. 5(b1)–(b8) were obtained. In this simulation, each module used twenty interferograms to calculate, and the results of the first eight is used as a presentation.

 

Fig. 5. (a) Standard- and (b) dynamic-module interferograms.

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Table 1. Zernike Polynomial Coefficients of Reference Wavefront.

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Figure 6(a) shows the reference wavefront phase fitted with Zernike polynomials. The peak-to-valley (PV) value was $0.1639\lambda$ (where $\lambda$ is the standard-module laser wavelength) and the root mean square (RMS) value was $0.03116\lambda$. Figure 6 (b) shows the phase recovered by the proposed method, where PV = $0.1637\lambda$ and RMS = $0.03115\lambda$. Figure 6 (c) shows the residual error between Figs. 6(a) and 6(b). The PV and RMS values were $0.0013\lambda$ and $2.1578 \times {10^{ - 4}}\lambda$, respectively. When a few fringes appeared in the standard-module interferogram, high accuracy was achieved for the simulation results.

 

Fig. 6. Simulation results for four fringes in interferogram: (a) reference phase, (b) proposed-method calculated phase, and (c) residual error between (a) and (b).

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According to the algorithm presented in Section 3, the phase in a vibration environment can be measured when the standard-module interferograms are null fringes. Therefore, interferograms with null fringes were simulated. The reference wavefront phase used here is Fig. 6(a). The null fringe interferograms of the standard module with vibration is shown in Figs. 7(a)–(i), and the dynamic-module interferogram is shown in Fig. 7(j) for illustrative purposes. The calculated phase given by the proposed method is shown in Fig. 7(k), with PV =$0.1638\lambda$ and RMS =$0.03116\lambda$. The residual error between Figs. 6(a) and 7(k) is shown in Fig. 7(l), with PV =$0.0010\lambda$ and RMS = $1.7756 \times {10^{ - 4}}\lambda$.

 

Fig. 7. Simulation results for null-fringe interferograms in vibration environment: (a)–(i) standard-module null fringe interferograms, (j) dynamic-module interferogram, (k) calculated phase of proposed method, and (j) residual error between simulated reference phase of Fig. 6(a) and proposed-method calculated phase of (k).

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A comparison of the coefficients given by the tilt-coefficient extraction algorithm and the simulation coefficients is presented in Table 2. The ${\alpha _n}$ and ${\beta _n}$ values are similar to the simulated values, having maximum differences of $1.7035 \times {10^{ - 8}}$ and $2.2623 \times {10^{ - 8}}$ rad/pixel, respectively. The maximum difference in ${\delta _n}$ is 0.0550 rad. The tilt coefficients extracted by the tilt-coefficient extraction algorithm have high accuracy, and the simulation results show that the proposed DFI algorithm is feasible and accurate. In particular, the results indicate that the phase can be recovered accurately when the standard-module interferogram has a null fringe.

Table 2. Tilt-Plane Coefficient Calculation Results and Real Values.

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5. Experiments and analysis

To verify the DFI feasibility and robustness based on the optical system presented in Section 2, a complete DFI system was built to measure a flat plate, which reflectivity is 4%. A He–Ne laser (632.8 nm) and a low-coherence laser (MDL-III-520L-50mW, Changchun New Industries Optoelectronics Tech. Co., Ltd. (CNI), 520 nm) were used as the standard- and dynamic-module light sources, respectively. The coherence length of the low-coherence laser is 0.3mm. The two modules were combined using a dichroic mirror (reflectivity > 98% @470 nm - 590 nm and transmissivity > 90% @620 nm - 700 nm), and the same type of CMOS detectors (GS3-U3-23S6M-C, Point Gray Research Inc.) was used to collect the interferograms. The two cameras collected data simultaneously to ensure that their interferograms reflected the same vibration information.

First, in an environment without vibration interference, a piezoelectric transducer (PZT) was used with the reference mirror A to randomly shift the phase. Only the standard-module interferograms are required in order to recover the phase. Here, the test mirror B was adjusted so that the standard-module interferograms had two fringes. The interferograms are shown in Figs. 8(a)–(h) and the calculated phase given by the advanced iterative algorithm (AIA) is shown in Fig. 8(i). The PV and RMS were $0.1202\lambda$ and $0.0124\lambda$, respectively. This result could be used as the standard reference phase for subsequent vibration-environment measurements.

 

Fig. 8. (a)–(h) Random phase-shifting interferograms and (i) AIA calculated phase.

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The measurement area was not changed and the standard- and dynamic-module interferograms were continuously collected in the vibration environment (vibration frequency is 20 - 200 Hz). The camera exposure time was 0.5 ms and the acquisition frequency was 40 Hz. The standard-module interferograms with vibration are shown in Figs. 9(a)–(g), and one of the dynamic-module interferogram is shown in Fig. 9(h). The recovered phase by proposed method is shown in Fig. 10(a). No ripples due to vibration appeared and the PV and RMS values were $0.1100\lambda$ and $0.0124\lambda$, respectively. This result is consistent with Fig. 8(i). The residual error between the proposed method and AIA is shown in Fig. 10(b), for which the PV and RMS were $0.0421\lambda$ and $0.0039\lambda$, respectively. The ripple in the residual error was due to the difference in background intensity caused by the laser intensity change during the acquisition process. To clearly explain the process of proposed algorithm, Fig. 10(c) is the Fourier transform spectrum diagram of dynamic module. The coordinates of the first-order maximum value are determined by filtering, and the carrier-frequency coefficients are obtained by Eq. (21) and (22).

 

Fig. 9. Interferograms with two fringes (see Visualization 1): (a)–(g) standard-module interferograms with vibration and (h) interferogram of dynamic module synchronized with standard module.

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Fig. 10. Results of comparative experiments: (a) phase given by proposed method and (b) residual error between proposed method and AIA. (c) The Fourier transform spectrum diagram of dynamic module.

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To verify the DFI robustness in the case of a null fringe, test mirror B was adjusted to create standard-module interferograms with null fringes. Note that, when the standard module changes from two to zero fringes, the measurement area can be regarded as unchanged. Standard-module interferograms obtained under the same vibration environment are shown in Figs. 11(a)–(g), and the dynamic-module interferogram is shown in Fig. 11(h), as an example. Figure 12(a) shows the calculated phase given by the proposed method. The PV and RMS were $0.1190\lambda$ and $0.0121\lambda$, respectively. The residual error between the proposed method and AIA is shown in Fig. 12(b), and the PV and RMS of the residual error were $0.0581\lambda$ and $0.0063\lambda$, respectively.

 

Fig. 11. Interferograms without fringes (see Visualization 2): (a)–(g) standard-module interferograms with vibration and (h) interferogram of dynamic module synchronized with standard module.

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Fig. 12. Results of comparative experiments: (a) proposed-method calculated phase and (b) residual error between proposed method and AIA.

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According to the experimental results, the DFI phase in the dynamic mode was consistent with that in the standard mode. Thus, the DFI resolves the problem of phase recovery in a vibration environment. Moreover, regardless of whether the interferograms contain fringes, the DFI can achieve a calculation accuracy of $1/20\lambda$ in terms of the residual error of the reference wave surface.

6. Discussion

The experiments performed in this work show that the DFI produces accurate measurement results in two modes: standard and dynamic. Comparison of the proposed method with the space carrier frequency method [6,7] indicates that the latter introduces a large retrace error between the reference and test wavefronts by tilting the test mirror, and calibration of the retrace error increases the measurement workload. In DFI, the standard module is a Fizeau interferometer, and it don’t need to introduce space carrier frequency on the standard-module interferograms. Although a non-common path error is also introduced to the DFI dynamic module, the dynamic module is only used to extract the relative tilt coefficients, which are unaffected by the non-common path error. The non-common path error on the phase cannot influence the result. As regards comparison with synchronous PSI [10–15], the DFI standard module is a common-path interferometer that utilizes the same polarization beam throughout; hence, the problem of optical path inconsistency between the reference and test wavefronts caused by the birefringence effect is avoided, as well as the problem of contrast inconsistency between interferogram frames.

In the DFI system, the Twyman–Green system is used as the light-source component of the dynamic module for polarization splitting and optical path matching. The Fizeau interferometer splits the beam at the reference mirror so that the reference and test beams come from the same wavefront. In the Twyman–Green system, however, the wavefront information of the p- and s-polarized beams are inconsistent. To solve this problem, a filter is added to the dynamic module to filter out the wavefront error of the p- and s-polarized beams; this filter does not affect the optical path matching. Note also that the magnifications of the two modules were identical in the experiment reported herein. However, a difference in magnification would not affect the transfer of the relative tilt coefficient. The important problem, measuring range and spatial resolution, of interferometer need to be considered. They are same as those in the conventional Fizeau interferometer, the dynamic module doesn’t influence the measuring range and spatial resolution. Additionally, spatial resolution of interferometer also depends on the detector. In the experiment, the pixel size of the detector is 5.86 µm, and resolution is 1920×1200.

A four-step phase-tilting iteration algorithm was proposed for extraction of the relative tilt coefficients from the dynamic module. In this algorithm, the phase distribution, $x$-direction tilt coefficient, $y$-direction tilt coefficient, and phase shift are iterated through least-squares fitting. Similar to other iterative algorithms, estimated initial values are required for the four-step phase-tilting iteration algorithm; these can be set to $\alpha _n^{(0)}\textrm{ = }0$ and $\beta _n^{(0)}\textrm{ = }0$, with $\delta _n^{(0)}$ incremented by $\pi /2$ from 0. The initial values affect the iteration direction, accuracy, and speed. Taking the interferograms simulated in Section 4 as an example, when the initial values of the iteration were set as above, the convergence criterion achieved a convergence threshold of ${10^{ - 5}}$ after 26 iterations. The final result is shown in Fig. 13(a). Although the set convergence threshold was reached, the result was a local optimal solution only. The final phase had a large error because the phase shift under the vibration state did not increase as the initial value was set. The deviation between the initial and real values was excessive and, hence, the convergence direction was incorrect. In Fig. 13(c), the phase shift is taken as an example, the simulation true values and the low-accuracy values extracted by the four-step phase-tilting iteration are compared, as indicated by the blue dotted line and yellow line, respectively.

 

Fig. 13. Algorithm calculation accuracy evaluation: (a) result of four-step phase-tilting iteration algorithm with estimated initial value, (b) Fourier transform result, and (c) comparison of phase shifts obtained using real values, four-step phase-tilting iteration, Fourier transform, and the combination of the two algorithms

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Because the initial values affect the iteration accuracy, in this study, a Fourier transform was used to extract the low-accuracy relative tilt coefficients as initial values. The peak coordinates given by the Fourier transform were integer values, and the relative tilt coefficients determined from the integer coordinates were not sufficiently accurate. Thus, direct application of the relative tilt coefficients extracted by the Fourier transform to the phase recovered from the standard module would increase the overall error through error accumulation. Figure 13(b) shows the recovered phase obtained using the relative tilt coefficients determined from the Fourier transform. Although these values were inaccurate, they were sufficiently accurate to be used as the initial values of the four-step phase-tilting iteration. After iteration, the coefficients were close to the true value and the phase could be effectively recovered (refer to the simulation in Section 4: after 20 iterations, a convergence threshold of ${10^{ - 5}}$ was reached). In Fig. 13(c), the low-accuracy values extracted by the Fourier transform, and the more accurate values obtained by the combination of the two algorithms are compared, as indicated by green line and red line, respectively. Large differences between the true values and those extracted with the Fourier transform are apparent. However, the values obtained by the four-step phase-tilting algorithm are close to the true values. The results achieved through combination of the two algorithms (i.e., the Fourier transform and four-step phase tilting algorithm) were superior than that achieved using the single algorithm (i.e., the Fourier transform or four-step phase tilting algorithm).

As detailed in Section 3.1.1, the four-step phase-tilting iteration algorithm can obtain the phase, $x$-direction tilt coefficient, $y$-direction tilt coefficient, and phase shift. In this study, only the $x$-direction tilt coefficient, $y$-direction tilt coefficient, and phase shift were used. However, the phase based on the corresponding interferogram is accurate. The proposed algorithm can be directly applied to phase extraction when the fringe is sparse (the retrace error is small) in a vibration environment.

There has a problem of non-sinusoidal profile for Fizeau pattern [39], it can cause the phase error, especially in three-bucket two-beam phase-stepping algorithms (TBPSA). The influence of non-sinusoidal profile can be reduced by the least-square fitting algorithm in this paper.

7. Conclusions

In this study, a dual-mode Fizeau interferometer (DFI) was proposed to overcome the difficulty of the Fizeau interferometer in accommodating both anti-vibration and conventional measurements. The proposed DFI consists of a standard module (conventional Fizeau interferometer) and a dynamic module (vibration information measurement system). The standard module can achieve phase-shifting measurement while the dynamic module can measure the vibration phase in a vibration environment. The two modules use different wavelength, and are combined by a dichroic mirror. The DFI preserves the accuracy of Fizeau interferometry and achieves dual-mode measurement. No switching between the two modes is required.

To calculate the dynamic-module vibration phase, a four-step phase-tilting iteration algorithm was proposed. The proposed algorithm can extract the phase and relative tilt coefficients ($x$- and $y$-direction tilt coefficients and phase shift) of the interferogram. Simulation and experimental results show that the DFI has high accuracy and robustness in both modes, and can calculate the null fringe interferogram in a vibration environment. Thus, the proposed DFI constitutes a flexible and high-accuracy solution for optical topography measurement.