Iterative ripple error suppression algorithm for the dynamic interferometry

Ronggang Zhu; Jianjie Zhou; Jianjie Zhou; Jianjie Zhou; Bo Li; Bo Li; Ya Huang; Ya Huang

doi:10.1364/OE.439721

1. Introduction

The phase-shifting interferometer are widely used in various areas of optical measurement [1–6], whose structures can be divided into temporal phase shift and spatial one [7–10]. In the temporal phase-shifting interferometry, the measurement results are usually affected by air turbulence and environmental vibration, leading to inaccurate measurement results [11–13]. In addition, the calibration process of the phase shifter is complex in the temporal phase-shifting interferometry [14,15]. However, several interferograms can be acquired at the same time in the spatial phase-shifting interferometry, overcoming the above problems.

There are two types of the spatial phase-shifting interferometer. One is based on the principle of polarization interference and realizes simultaneous phase shift with the use of the beam splitting structure and polarized phase-shifting devices [16,17]. The other uses the micro array component while not the beam splitting structure, whose periodically arranged phase shift units are correspondent with pixels of the camera [18,19]. With the introduction of micro array technology, the structure of the dynamic interferometer becomes more compact and its application becomes more extensive. Yoneyama et al. applied a charge-coupled with a microretarder array to the measurement of birefringence parameters [20]. Zhigang Zhang et al. [21] and Yuntian Zhang et al. [22] applied pixelated micropolarizer array to the real-time phase measurement of vortex beam. Katherine Creath applied an pixelated phase measurement camera based on array of wire grid micropolarizers to instantaneous video measurements of dynamic motions within and among live cells [23,24].

There are some problems in the application of the micropolarizer array-based simultaneous phase-shifting interferometer. Novak et al. [25] and Yang Jun et al. [26] discussed the influence of different optical components on the measurement results. Of these, the influence of the fast axis azimuth error and the retardation error of QWP on the measurement results was obviously higher than that of other components, which is usually manifested as periodic ripple error in the measurement results. The fast axis azimuth error can be suppressed by precise alignment. In contrast, the retardation error is the inherent error of wave plate which cannot be eliminated by adjusting itself. Due to the limitation of the current manufacturing process, the retardation error of the wave plate is unavoidable.

An error correction algorithm may be a practical method to eliminate the error caused by QWP retardation error since there is neither extra hardware nor operation. In the past few years, many phase extraction algorithms applied to the pixelated micropolarizer array-based dynamic interferometer have been proposed. Kimbrough presents the extraction accuracy of different phase-shifting algorithms via computer simulations. However, the difference between these algorithms is not obvious in case of the QWP retardation error [27]. Servin et al. proposed the Spatial-Temporal Fringes (STF) method to extract the test phase distribution, which can eliminate high-frequency noise and high-order harmonic error [28]. And its modified version proposed by Bo Li et al. [29,30] can be used to suppress the phase shift error. However, these methods cannot eliminate QWP retardation error either because they assume the background and modulation of phase shift fringes are equal. The iterative algorithms proposed by Okada et al. [31] and Wang et al. [32] can extract phase from randomly phase-shifted interferograms, which is another kind of algorithm immune to the phase shift error, but it is also affected by QWP error since the non-uniform background and modulation assumptions are only valid in their intermediate step.

In this paper, we will deduce the expression of the intensity distribution of the four interferograms introducing the retardation error of QWP in dynamic interferometer. This error results in the inequality of interferograms background and modulation between frames. In addition, due to the property of Gaussian beam, the background and modulation in one frame are non-uniform. Based on this analysis, an iterative algorithm is proposed where the inequal background and modulation between frames and within one frame are both considered in global iterative steps, which is different from former iterative algorithms, therefore it can solve the QWP retardation error problem in dynamic interferometer.

2. Theory

2.1 Influence of retardation error of QWP on phase extraction

A Twyman-Green interferometer based on micro-polarizer array is illustrated in Fig. 1. The system includes three parts. The first part is light source, including laser 1, polarization-maintaining fiber 2 and fiber transmitting terminal 3. The second part is polarization interference, including collimating lens 4, polarizer 5, half-wave plate 6 (HWP), polarizing beam splitter 7 (PBS) and quarter-wave plate 8 (QWP1), 10 (QWP2), 13 (QWP3), reference mirror 9 (RM), lens 11 and test mirror 12 (TM). The third part is image acquisition, including imaging system 14 and pixelated-micropolarizer-array camera 15.

Fig. 1. Twyman-Green dynamic interferometer based on micropolarizer array

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In Fig. 1, the micropolarizer array camera is the key component, which enables each pixel of the detector has a unique phase-shift. The optical axis azimuths of four adjacent micropolarizers are respectively 0°, 45°, 90° and 135°. It is assumed that the fast-axis azimuths of three QWPs are 45° and the RF is absolutely flat. Using the Jones Matrix, the reference beam E_r and the test beam E_t at the output of the micropolarizer array can be respectively represented by

(1)$$\left\{ \begin{array}{l} {E_r} = a(x,y){e^{ - i\theta }}\\ {E_t} = b(x,y){e^{i[{\theta + \varphi (x,y)} ]}} \end{array} \right.$$

where a(x, y), b(x, y) are respectively the amplitude distributions of the reference beam and the test beam at each pixel (x, y), θ is the azimuth of the polarizer, and φ(x, y) is the test phase distribution.

Based on the principle of the polarized phase-shift technology [33], the intensity distributions of the interferograms can be expressed by

(2)$$\begin{aligned} {I_i} &= ({{E_r}\textrm{ + }{E_t}} )\cdot {({{E_r}\textrm{ + }{E_t}} )^{\ast }}\\ &= {a^2}(x,y)\textrm{ + }{b^2}(x,y) + 2a(x,y)b(x,y)\sin [{\varphi (x,y) + 2{\theta_i}} ]\end{aligned}$$

where θ₁∼θ₄ are respectively 0°, 45°, 90° and 135°.

According to Eq. (2), the measuring phase distribution can be demodulated by the four-bucket algorithm, which is expressed by

(3)$$\varphi (x,y) = \arctan \left( {\frac{{{I_1} - {I_3}}}{{{I_2} - {I_4}}}} \right)$$

Finally, the surface height of the test mirror can be expressed by

(4)$$h(x,y) = \frac{\lambda }{{4\pi }}\varphi (x,y)$$

The phase result obtained by Eq. (3) are derived under ideal conditions. Several authors considered three potential error sources of the measurement results [26,34]: firstly, the azimuth error of the micropolarizer array which only changes the contrast of the fringes without introducing errors into the test result; secondly, the azimuth error of the QWP fast axis, which can be eliminated by precise alignment; finally, the QWP retardation error which will result in the ripple phase error but its effect was not given enough attention in references. So in this paper we focus on this error.

The reference beam reflected back by RF passes through QWP1 twice and the PBS directly and then turns to be p-polarized light. The test beam reflected back by TM passes through QWP2 twice and reflects by PBS, and then turns to be s-polarized light. Therefore, the retardation errors of QWP1 and QWP2 only change the contrast of the fringes without introducing measurement errors into the system. However, the retardation error of QWP3 makes the test beam and the reference beam become orthogonally elliptically polarized light rather than circularly polarized light. It is assumed that the retardation deviation of QWP3 is Δδ, that is, the actual retardation can be represented as δ=π/2+Δδ. When the fast-axis azimuths of QWP3 is 45°, the Jones matrix of QWP3 can be represented by

(5)$${G_{QWP3}} = \left[ {\begin{array}{cc} {\cos \left( {\frac{\pi }{4} + \frac{{\Delta \delta }}{2}} \right)}&{ - i\sin \left( {\frac{\pi }{4} + \frac{{\Delta \delta }}{2}} \right)}\\ { - i\sin \left( {\frac{\pi }{4} + \frac{{\Delta \delta }}{2}} \right)}&{\cos \left( {\frac{\pi }{4} + \frac{{\Delta \delta }}{2}} \right)} \end{array}} \right]$$

The Jones vectors of the reference beam and the test beam which have passed through QWP3 and micropolarizer array can be respectively presented by

(6)$$\left\{ \begin{array}{l} {E_r}^{\prime} = {G_{Polarizer}} \cdot {\textrm{G}_{\textrm{QWP3}}} \cdot \left[ {\begin{array}{c} {a(x,y)}\\ 0 \end{array}} \right]\\ {E_t}^{\prime} = {G_{Polarizer}} \cdot {\textrm{G}_{\textrm{QWP3}}} \cdot \left[ {\begin{array}{c} 0\\ {b(x,y)} \end{array}} \right]{e^{i\varphi ({x,y} )}} \end{array} \right.$$

where the Jones matrix of the polarizer unit in the micropolarizer array can be represented as ${G_{Polarizer}} = \left( {\begin{array}{cc} {{{\cos }^2}\theta }&{\frac{1}{2}\sin 2\theta }\\ {\frac{1}{2}\sin 2\theta }&{{{\sin }^2}\theta } \end{array}} \right).$

Using the Jones calculus, the intensity distributions of the four interferograms can be represented by

(7)$$\left\{ \begin{array}{l} {I_1}^{\prime} = \frac{{{a^2}(x,y)\textrm{ + }{b^2}(x,y)}}{2} + \frac{{\sin \Delta \delta [{b^2}(x,y) - {a^2}(x,y)] }}{2} + a (x,y)b(x,y) \cos \Delta \delta \sin \varphi (x,y)\\ {I_2}^{\prime} = \frac{{{a^2}(x,y)\textrm{ + }{b^2}(x,y)}}{2} - a (x,y)b(x,y)\cos \varphi (x,y) \\ {I_3}^{\prime} = \frac{{{a^2}(x,y)\textrm{ + }{b^2}(x,y)}}{2} - \frac{{\sin \Delta \delta [{b^2}(x,y) - {a^2}(x,y)] }}{2} - a (x,y)b(x,y)\cos \Delta \delta \sin \varphi (x,y)\\ {I_4}^{\prime} = \frac{{{a^2}(x,y)\textrm{ + }{b^2}(x,y)}}{2} + a (x,y)b(x,y)\cos \varphi (x,y) \end{array} \right.$$

According to Eq. (7), the retardation error of QWP3 causes inconsistency of the background intensity and the modulation between the four interferograms. Both subsequent simulation and experimental results suggested that this error appears as the ripple error with the same period as the fringes.

2.2 Iterative algorithm to suppress ripple error

It is assumed that the background intensity and the modulation of each interferogram are different, and Eq. (7) can be presented by

(8)$${I_{ij}} = {a_i}{A_j} + {b_i}{B_j}\cos [{\varphi _j} + (i - 1)\frac{\pi }{2}]$$

where the subscript i denotes the ith phase-shifted interferogram (i=1,2,3,4), j denotes the jth pixel in each interferogram (j=1,2,…,M), I_ij is the intensity of an interferogram, A_j is the background intensity, a_i is the constant coefficient of A_j, B_j is the modulation, b_i is the constant coefficient of B_j, φ_j is the test phase, and M is the total number of pixels in each interferogram.

According to Eq. (8), the phase of each pixel can be calculated by

(9)$${\varphi _j} = \arctan [\frac{{({a_2}{I_{4j}} - {a_4}{I_{2j}})({a_3}{b_1} + {a_1}{b_3})}}{{({a_3}{I_{1j}} - {a_1}{I_{3j}})({a_2}{b_4} + {a_4}{b_2})}}]$$

Therefore, the precise solution of a₁∼a₄, b₁∼b₄ should be obtained before the final test phase is extracted. That is, the inequal background and modulation of interferograms between frames should be solved. In this paper, an iterative algorithm is proposed to obtain more accurate values of a₁∼a₄, b₁∼b₄ step by step through the iterative process. The specific iteration steps of the proposed algorithm are described below.

Step 1. Solving φ_j

In the initial iteration, assuming a₁=…=a₄, b₁=…=b₄, Eq. (9) can be expressed as ${\varphi _j} = \arctan \left( {\frac{{{I_{4j}} - {I_{2j}}}}{{{I_{1j}} - {I_{3j}}}}} \right)$, which is the ideal phase extraction without the retardation error of QWP3. We here assume a₁=…=a₄=b₁=…=b₄=1 and then the phase value of each pixel can be solved by substituting I_1j, I_2j, I_3j, I_4j into Eq. (9).

Step 2. Solving A_j and B_j

Substituting φ_j solved in Step 1, the equations can be obtained as

(10)$$\left[ {\begin{array}{cc} {{a_1}}&{{b_1}\cos {\varphi_j}}\\ {{a_2}}&{ - {b_2}\sin {\varphi_j}}\\ {{a_3}}&{ - {b_3}\cos {\varphi_j}}\\ {{a_4}}&{{b_4}\sin {\varphi_j}} \end{array}} \right]\left[ {\begin{array}{c} {{A_j}}\\ {{B_j}} \end{array}} \right] = \left[ {\begin{array}{c} {{I_{1j}}}\\ {{I_{2j}}}\\ {{I_{3j}}}\\ {{I_{4j}}} \end{array}} \right]$$

For each pixel, the equation group Eq. (10) is an overdetermined equation group composed of four equations and two unknowns. Therefore, the unknowns A_j and B_j can be solved by use of the overdetermined least-squares method:

(11)$$\left[ {\begin{array}{c} {{A_j}}\\ {{B_j}} \end{array}} \right] = {({C^T}C)^{ - 1}}{C^T}\left[ {\begin{array}{c} {{I_{1j}}}\\ {{I_{2j}}}\\ {{I_{3j}}}\\ {{I_{4j}}} \end{array}} \right]$$

where $C = \left[ {\begin{array}{cc} {{a_1}}&{{b_1}\cos {\varphi_j}}\\ {{a_2}}&{ - {b_2}\sin {\varphi_j}}\\ {{a_3}}&{ - {b_3}\cos {\varphi_j}}\\ {{a_4}}&{{b_4}\sin {\varphi_j}} \end{array}} \right].$

By solving A_j and B_j of each pixel, the initial solutions of background intensity distribution and the modulation distribution of the whole interferogram can be obtained.

Step 3. Refining A_j′ and B_j′

In general, since the beam emitted by a laser light source obeys the Gaussian distribution after passing through collimating lens, the background intensity A_j and the modulation B_j can be approximately expressed by

(12)$$\left\{ \begin{array}{l} {A_j} \approx {k_0} - {k_1}{e^{ - \frac{{{{({x_j} - {x_0})}^2} + {{({y_j} - {y_0})}^2}}}{{2{\sigma_a}^2}}}}\\ {B_j} \approx {k_0}^{\prime} - {k_1}^{\prime}{e^{ - \frac{{{{({x_j} - {x_0})}^2} + {{({y_j} - {y_0})}^2}}}{{2{\sigma_b}^2}}}} \end{array} \right.$$

where x_j and y_j are respectively lateral ordinates of the corresponding pixel, and x₀, y₀, k₀, k₀′, k₁, k₁′ are the parameters of the Gaussian function.

The coarse solutions of A_j and B_j obtained in Step 2 are solved under the condition that a₁∼a₄, b₁∼b₄ and φ_j are not accurate, so the ripple error similar to the error in the measurement results would appear in the traditional phase shift algorithm. We can do a polynomial expansion of Eq. (12) and find that the ripple error term is mostly reflected in the higher-order terms and does not affect the low-order terms. As we tested, the 2^nd order polynomial is a good balance between the sufficient fit precision for A_j and B_j and suppression the ripple error.

Solving lower-order terms in A_j and B_j is to solve coefficients of lower-order terms. By making use of the Taylor expansion of the exponential function and only choosing low-order (e.g., linear, quadratic) polynomials for polynomial fitting, A_j and B_j can be approximately expressed by

(13)$$\left\{ \begin{array}{l} {A_j} \approx {D_{00}} + {D_{10}}{x_j} + {D_{01}}{y_j} + {D_{20}}{x_j}^2 + {D_{02}}{y_j}^2\\ {B_j} \approx {E_{00}} + {E_{10}}{x_j} + {E_{01}}{y_j} + {G_{20}}{x_j}^2 + {G_{02}}{y_j}^2 \end{array} \right.$$

where D₀₀∼D₀₂ and E₀₀∼E₀₂ are coefficients of low-order terms. Equation (13) can be written as the following two 5×M equation groups:

(14)$$\left[ {\begin{array}{@{}ccccc@{}} 1&{{x_1}}&{{y_1}}&{{x_1}^2}&{{y_1}^2}\\ 1&{{x_2}}&{{y_2}}&{{x_2}^2}&{{y_2}^2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{{x_M}}&{{y_M}}&{{x_M}^2}&{{y_M}^2} \end{array}} \right]\left[ {\begin{array}{@{}c@{}} {{D_{00}}}\\ {{D_{10}}}\\ {{D_{01}}}\\ {{D_{20}}}\\ {{D_{02}}} \end{array}} \right] = \left[ {\begin{array}{@{}c@{}} {{A_1}}\\ {{A_2}}\\ \vdots \\ {{A_M}} \end{array}} \right], \left[ {\begin{array}{@{}ccccc@{}} 1&{{x_1}}&{{y_1}}&{{x_1}^2}&{{y_1}^2}\\ 1&{{x_2}}&{{y_2}}&{{x_2}^2}&{{y_2}^2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{{x_M}}&{{y_M}}&{{x_M}^2}&{{y_M}^2} \end{array}} \right]\left[ {\begin{array}{@{}c@{}} {{E_{00}}}\\ {{E_{10}}}\\ {{E_{01}}}\\ {{E_{20}}}\\ {{E_{02}}} \end{array}} \right] = \left[ {\begin{array}{@{}c@{}} {{B_1}}\\ {{B_2}}\\ \vdots \\ {{B_M}} \end{array}} \right]$$

Then coefficients D₀₀∼D₀₂ and E₀₀∼E₀₂ can be solved as:

(15)$$\left[ {\begin{array}{c} {{D_{00}}}\\ {{D_{10}}}\\ {{D_{01}}}\\ {{D_{20}}}\\ {{D_{02}}} \end{array}} \right] = {({F^T}F)^{ - 1}}{F^T}\left[ {\begin{array}{c} {{A_1}}\\ {{A_2}}\\ \vdots \\ {{A_M}} \end{array}} \right], \left[ {\begin{array}{c} {{E_{00}}}\\ {{E_{10}}}\\ {{E_{01}}}\\ {{E_{20}}}\\ {{E_{02}}} \end{array}} \right] = {({G^T}G)^{ - 1}}{G^T}\left[ {\begin{array}{c} {{B_1}}\\ {{B_2}}\\ \vdots \\ {{B_M}} \end{array}} \right]$$

where $F = \left[ {\begin{array}{ccccc} 1&{{x_1}}&{{y_1}}&{{x_1}^2}&{{y_1}^2}\\ 1&{{x_2}}&{{y_2}}&{{x_2}^2}&{{y_2}^2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{{x_M}}&{{y_M}}&{{x_M}^2}&{{y_M}^2} \end{array}} \right],G = \left[ {\begin{array}{ccccc} 1&{{x_1}}&{{y_1}}&{{x_1}^2}&{{y_1}^2}\\ 1&{{x_2}}&{{y_2}}&{{x_2}^2}&{{y_2}^2}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{{x_M}}&{{y_M}}&{{x_M}^2}&{{y_M}^2} \end{array}} \right].$

Using coefficients of low-order terms solved in Eq. (15), A_j′ and B_j′ can be obtained by:

(16)$$\left[ {\begin{array}{c} {{A_1}^{\prime}}\\ {{A_2}^{\prime}}\\ \vdots \\ {{A_M}^{\prime}} \end{array}} \right] = F\left[ {\begin{array}{c} {{D_{00}}}\\ {{D_{10}}}\\ {{D_{01}}}\\ {{D_{20}}}\\ {{D_{02}}} \end{array}} \right], \left[ {\begin{array}{c} {{B_1}^{\prime}}\\ {{B_2}^{\prime}}\\ \vdots \\ {{B_M}^{\prime}} \end{array}} \right] = G\left[ {\begin{array}{c} {{E_{00}}}\\ {{E_{10}}}\\ {{E_{01}}}\\ {{E_{20}}}\\ {{E_{02}}} \end{array}} \right]$$

Step 4. Solving coefficients a₁∼a₄ and b₁∼b₄

Substituting the refined A_j′ and B_j′ in Step 3 and φ_j solved in Step 1 into Eq. (8), the overdetermined 2×M equations group for the i-th interferogram can be obtained:

(17)$$\left[ {\begin{array}{cc} {{A_1}^{\prime}}&{{B_1}^{\prime}\cos {\varphi_1}}\\ {{A_2}^{\prime}}&{{B_2}^{\prime}\cos {\varphi_2}}\\ \vdots & \vdots \\ {{A_M}^{\prime}}&{{B_M}^{\prime}\cos {\varphi_M}} \end{array}} \right]\left[ {\begin{array}{c} {{a_i}}\\ {{b_i}} \end{array}} \right] = \left[ {\begin{array}{c} {{I_{i1}}}\\ {{I_{i2}}}\\ \vdots \\ {{I_{iM}}} \end{array}} \right]$$

And then a_i and b_i of the i-th interferogram can be solved by:

(18)$$\left[ {\begin{array}{c} {{a_i}}\\ {{b_i}} \end{array}} \right] = {({H^T}H)^{ - 1}}{H^T}\left[ {\begin{array}{c} {{I_{i1}}}\\ {{I_{i2}}}\\ \vdots \\ {{I_{iM}}} \end{array}} \right]$$

where $H = \left[ {\begin{array}{cc} {{A_1}}&{{B_1}\cos {\varphi_1}}\\ {{A_2}}&{{B_2}\cos {\varphi_2}}\\ \vdots & \vdots \\ {{A_M}}&{{B_M}\cos {\varphi_M}} \end{array}} \right].$

Although φ_j substituted into Eq. (17) is not accurate enough, the results of A_j′ and B_j′ are relatively more accurate than the initial solutions. Therefore, the values of a₁∼a₄ and b₁∼b₄ solved in Eq. (18) are also more accurate.

Then substitute the results obtained in Step 4 into Step 1 for iterative calculation and the final measurement results with higher precision can be obtained, where the flow is shown in Fig. 2. When the root-mean-square (RMS) value of the deviation of the reconstructed phases between two iterative steps is less than a set threshold, the iteration terminates.

Fig. 2. Flow chart of the iterative algorithm

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3. Simulation and discussion

To validate the algorithm, a simulated surface with the peaks function is shown in Fig. 3.

Fig. 3. The simulated surface

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It is assumed that the intensity of both reference beam a(x, y) and test beam b(x, y) obeys the Gaussian distribution and the amplitude ratio between them is 5:1. According to Eq. (7), the phase shift fringes of the simulated surface after adding a λ/300 retardation error into QWP3 are shown in Fig. 4.

Fig. 4. The phase shift fringes of the simulated surface

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The simulated results are as shown in Fig. 5, where (a) and (b) are respectively the retrieved surface before and after iterations, and (c) and (d) are respectively deviations between the original simulated surface and the retrieved surface before and after iterations. The Error Root Mean Square (ERMS) values of Fig. 5(c) and (d) are respectively 0.002952λ and 0.000013λ, where the relative error compared with the simulated surface RMS are respectively 6.02% and 0.03%. It can be seen that there is obvious ripple error in the retrieved surface before iterations, while this error can be eliminated to a large extent in the retrieved surface after iterations.

Fig. 5. (a) The retrieved surface before iterations and (c) its error (ERMS value: 0.002952λ); (b) The retrieved surface after iterations and (d) its error (ERMS value: 0.000013λ)

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The performance of the algorithm may be affected by the fringe number since its period is the same as the ripple error. To give the analysis, the convergence curves of ERMS values of retrieved surfaces with different numbers of fringes are shown in Fig. 6. Firstly, it can be seen that no matter how many fringes are included in the interferogram, the iterative algorithm in this paper can converge even in the case of 0 fringe, but the iterative efficiency is different. Secondly, when the number of fringes is equal to or more than 3, the iterative curves of different fringes coincide basically. Only about 10 iterations are needed and the retrieved surface tends to be stable. Thirdly, when there is only one fringe in the interferogram, the ripple error in the measurement result is similar as the A_j’ and B_j’ with Gaussian shape, and this interference make the iterative efficiency decrease evidently.

Fig. 6. Convergence curves of ERMS values of the retrieved surfaces with different numbers of fringes (curves of 3 fringes and 30 fringes are overlapped)

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4. Experiments

The experiment setup is built according to Fig. 1. A standard spherical mirror is used as the specimen in the test arm and the interferograms can be acquired as shown in Fig. 7, where (a)∼(d) are fringe patterns obtained by rearranging the image collected by pixelated-micropolarizer-array camera and their phase shift are respectively 0, π/2, π and 3π/2.

Fig. 7. (a)∼(d) Interferograms obtained by rearrangement of image collected by pixelated-micropolarizer-array camera

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The measurement results are shown in Fig. 8. The retrieved surface before iterations is shown in Fig. 8(a) where the ripple error is significant in the result. The stable retrieved surface after 10 iterations is shown in Fig. 8(b), where the ripple error is suppressed significantly compared with the former. The deviation between Fig. 8(a) and (b) is shown in Fig. 8(c), which is basically consistent with the simulation result in Fig. 5(c), presented as the ripple error with the same period as the fringes.

Fig. 8. (a) Surface shape result reconstructed from Fig. 7; (b) Surface shape result after iterations and (c) Deviations of surface before and after iterations

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5. Conclusion

In this paper, an iteration algorithm to eliminate the influence of the QWP retardation error in the dynamic interferometry is proposed. This error causes inconsistency of the background intensity and the modulation of the four phase shift interferograms, further resulting in inaccuracy of measurement results, which is usually shown as periodic ripple error in the measurement results. To solve this problem, the background intensity, modulation of each interferogram and the test phase should be considered as the unknowns, which can be all solved by the proposed iterative algorithm. Both the simulation and experiment validate the accuracy and convergence speed of the algorithm. The algorithm performance in the cases of different fringes number are analyzed, the results show that only the “one fringe” interferogram will evidently reduce the convergence efficiency, which can be avoided in actual measurement easily. So this work is a practical method to improve the measurement accuracy of dynamic interferometry without any additional hardware and operation.

Funding

National Natural Science Foundation of China (11873070); Natural Science Foundation of Jiangsu Province (BK20191095).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Iterative ripple error suppression algorithm for the dynamic interferometry

Abstract

1. Introduction

2. Theory

2.1 Influence of retardation error of QWP on phase extraction

2.2 Iterative algorithm to suppress ripple error

3. Simulation and discussion

4. Experiments

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (18)

Optics Express