Modelling and correction for polarization errors of a 600&#x2005;mm aperture dynamic Fizeau interferometer

Xinyu Miao; Jun Ma; Yifan Yu; Jianxin Li; Cong Wei; Rihong Zhu; Lei Chen; Caojin Yuan; Qing Wang; Donghui Zheng

doi:10.1364/OE.409983

1. Introduction

The measurement of large aperture optical components becomes much more critical as large aperture components are increasingly being used in high power systems and astronomical systems, such as NIF laser systems containing 7,648 (meter-level) large aperture components [1,2]. At present, the mainstream phase shifting interferometers for testing large aperture flats are based on the traditional PZT technique [3,4] and wavelength tuning technique [5,6]. Compared with the use of PZT to push a standard flat with large aperture and heavy weight, it is easier to be implemented using the wavelength tuning technique, but with higher cost. However, the measurement accuracy of both is easily affected by the environment, especially vibration and airflow disturbance, because it always takes a particular time to acquire the interferograms needed. On the other hand, both of them are affected by coherent noise in the system and errors are shown in the measurement results. In order to avoid the influence of the environment, dynamic interferometer systems are proposed, which can collect multiple phase-shifted interferograms simultaneously to measure the instantaneous wavefront results.

One traditional dynamic interferometer produces a high frequency carrier through the large angle between the test and reference beams, and the phase distribution can be reconstructed with Fourier transform [7,8]. Nevertheless, the high density carrier restricts the spatial resolution of the image and produces systematic errors for the test and reference beams travelling different paths. Most of the currently used dynamic interferometers are based on polarization phase shifting systems [9–11]. James Millerd [12] proposed a dynamic polarization phase shifting interferometer using a pixelated phase-mask method with four micropolarizers at different polarization directions in a small cell. Hui-Kang Tenga [13] proposed a polarization-shifting Michelson interferometer based on an interference microscopy device, which used a combination of a quarter-wave plate(QWP) and a polarizer to realize the phase shift. Sanjib Chatterjee [14] proposed a new technique using a circular path optical configuration. The orthogonal linear polarization components reflected from the reference and test surfaces were used to introduce a polarization phase shift.

However, when the synchronous polarization phase shifting technique is applied to the measurement for large aperture optical elements, the polarization errors of the system introduced by the polarized device are often more apparent [15,16]. Besides, the birefringence phenomenon of large aperture components caused by stress inevitably affects the measurement results, and it also destroys the characteristics of the common optical path of the Fizeau interferometer because the effects of birefringence on P light and S light are different [17]. The analysis of various phase shifting errors in the interferometer using Lissajous curves has been discussed [18–21]. Chunyu Zhao models and analyzes the residual birefringence of components, and finds out that the error is much smaller if circular polarization is used rather than linear polarization [17]. But his work is limited to simulation, and the analysis for large aperture interferometer is insufficient. Wenhua Zhu [22] proposed a 600mm aperture dynamic interferometer based on a point source system, using four independent imaging systems to acquire four separate phase-shifted interferograms on the detector to achieve dynamic measurement, thereby effectively avoiding environmental factors such as airflow and vibration.

This paper focuses on the correction of polarization errors based on a 600mm dynamic interferometer consisting of source module and interference module. By analyzing the characteristics of the optical devices in the two modules, the polarization errors can be divided into two different types: one is single and double frequency print through errors related to polarized elements, and the other is distribution defects related to birefringence. The light source module is mainly used to realize the path matching with the delay line stage. Lissajous ellipse fitting method is used to correct the single and double frequency print through errors caused by imperfect polarized elements. In the interference module, the main source of polarization error is the birefringence phenomenon of large aperture optical components. A wave plate model is present to analyze and correct the influence of the birefringence. On the basis of theoretical analysis, experiments are carried out on a 600 mm diameter Fizeau interferometer and then compared with the results of the wavelength tuning phase shifting method, which is free of polarization errors. Experiments show that the polarization errors correction algorithm proposed in this paper can effectively eliminate multiple polarization errors in the system.

2. Theoretical analysis and simulation on polarization errors

In order to reasonably describe the sources of polarization errors in the overall large aperture interference system and then correct them, this article divides the interferometer into a front-end source module and a back-end interference module, as shown in Fig. 1.

Fig. 1. Large-aperture interferometer system, including source module and interference module

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In the source module, the collimated light emitted from the low coherence laser passes through the polarizer to become linearly polarized light and is then divided into two light paths of P light and S light by PBS with two QWPs behind. Assuming that the X_g axis of the global coordinate system (X_g, Y_g) is consistent with the polarization direction of the P light, the polarization direction of the other S light is 90 °. The two beams of lights are reflected by the retro reflectors placed on the delay line stage and PZT, respectively, and then pass through the QWPs again. They are coupled into the fiber coupler and then enter the interference module together. Besides, one can easily change the source of the interference module into a wavelength tuning laser by switching the fiber coupled into the fiber coupler.

The P light and S light entering the interference module are reflected by TF and RF, respectively, and after passing through the QWP in front of the CCD, they become orthogonally right-handed and left-handed circularly polarized lights to interfere. The micro-polarizers in the pixel mask in front of the CCD is placed at different angles and can act as phase shifters. Four phase-shifting interferograms can be acquired on the CCD at the same time.

The two modules include polarizers, half-wave plates(HWP), PBS, QWP, and other polarizing elements. When the actual device and its ideal parameters are inconsistent, it will affect the measurement results, which is often related to phase distribution, named single and double frequency print through errors. For example, when the phase delay of a HWP is not halfwave, it will lead to the inconsistency of the two light beams that interfere, so that the background light intensity distribution of the four interferograms collected by the same camera is inconsistent. Single frequency error often appears in the measurement results. Several error sources have been discussed [16] before including polarizer angle and extinction error, retardance error of wave plates, and instruments retrace error. So we mainly focus on the correction for these errors in this paper. In the interference module, the main source of polarization error is the birefringence caused by the gravity of the large aperture optical element, and the effect on the result is independent of phase distribution, only related to the nature of the large-aperture element.

2.1 Correction for polarization errors caused by polarized devices

No matter which phase shifting interferometry is used, the final phase result can be obtained by calculating the tangent value, as shown in Eq. (1)

(1)$$\tan ({\varphi ^{\prime}}) = \frac{{N(x,y)}}{{D(x,y)}},$$

where

(2)$$\begin{array}{l} N(x,y,\varphi ) = {y_0} + {a_y}\sin (\varphi + {\delta _N}),\\ D(x,y,\varphi ) = {x_0} + {a_x}\cos (\varphi + {\delta _D}). \end{array}$$

Here, N and D can be regarded as a parametric equation of an ellipse, where (x, y) corresponds to different points in an interferogram. When φ changes, the curve drawn by the coordinates of (D, N) is an ellipse whose center is (x0, y0), and its long and short axes are ax and ay. Ideally, the background light intensity and modulation distribution of multiple interferograms collected by the same camera are consistent, and the following conditions can be obtained at this time:

(3)$${x_0} = 0,\begin{array}{cc} {}&{{y_0} = 0} \end{array},\begin{array}{cc} {}&{{a_x} = {a_y},} \end{array}\begin{array}{cc} {}&{{\delta _N} = {\delta _D}.} \end{array}\begin{array}{ccc} {}&{}&{} \end{array}$$

In this case, the ellipse determined by N and D is a circle. As shown in Fig. 2, the result calculated by Eq. (1) is φ ‘= φ, and there is no error in the measurement result. However, there always are deviations between the parameters of the various polarized elements and their ideal parameters and also the alignment angle error of the QWP and HWP in the interference system. All these imperfect factors will lead to an inconsistency in the background light intensity and the modulation among four interferograms. Under this condition, the center of the ellipse determined by N and D deviates from the origin, and the long and short axes are also inconsistent, and the single and double frequency errors will be observed in the result of the solution.

Fig. 2. Lissajous figure with no errors, inconsistency in light intensity and inconsistency in modulation distribution

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In order to correct the print through errors caused by the inconsistent light intensity and modulation, this paper uses the conception of Lissajous curve and the method of ellipse fitting to firstly fit the (D, N) coordinates to get the parameters of an ellipse, on which we perform a specific transformation to acquire (D’, N’), and then combined with Eq. (1) to obtain the corrected phase distribution without print through errors. The equation of ellipse can be transformed from Eq. (2) to Eq. (4):

(4)$${\left( {\frac{{D - {x_0}}}{{{a_x}}}} \right)^2} + {\left( {\frac{{N - {y_0}}}{{{a_y}}}} \right)^2} - 2\frac{{({D - {x_0}} )({N - {y_0}} )}}{{{a_x}{a_y}}}\sin ({{\delta_N} - {\delta_D}} )= \cos {({{\delta_N} - {\delta_D}} )^2}.$$

Rewrite Eq. (4) as a general quadratic equation as follows:

(5)$$F = {A_0}{x^2} + {A_1}xy + {A_2}{y^2} + {A_3}x + {A_4}y + {A_5}.$$

There are six unknown coefficients in Eq. (5), and can be obtained with six different sets of (D, N) coordinates by changing the value of δ_N in Eq. (5). On this basis, the six unknown numbers A₀ ∼ A₅ can be solved, which are used to calculate the actual values of x₀, y₀, a_x and a_y, recorded as x₀’, y₀’, a_x’ and a_y’.

(6)$$\left\{ {\begin{array}{{c}} {{x_0}^{\prime} = \frac{{2{A_\textrm{2}}{A_3} - {A_1}{A_4}}}{{{A_1}^2 - 4{A_0}{A_2}}},}\\ {{y_0}^{\prime} = \frac{{2{A_\textrm{0}}{A_4} - {A_\textrm{1}}{A_3}}}{{{A_1}^2 - 4{A_0}{A_2}}},}\\ {{a_x}^{\prime} = \sqrt {\frac{{4{A_2}}}{{{A_1}^2 - 4{A_0}{A_2}}}\left( {{A_\textrm{5}}\textrm{ + }\frac{{{A_2}{A_\textrm{3}}^2\textrm{ + }{A_\textrm{0}}{A_\textrm{4}}^2\textrm{ - }{A_\textrm{1}}{A_\textrm{3}}{A_\textrm{4}}}}{{{A_1}^2 - 4{A_0}{A_2}}}} \right)} ,}\\ {{a_y}^{\prime} = \sqrt {\frac{{4{A_\textrm{0}}}}{{{A_1}^2 - 4{A_0}{A_2}}}\left( {{A_\textrm{5}}\textrm{ + }\frac{{{A_2}{A_\textrm{3}}^2\textrm{ + }{A_\textrm{0}}{A_\textrm{4}}^2\textrm{ - }{A_\textrm{1}}{A_\textrm{3}}{A_\textrm{4}}}}{{{A_1}^2 - 4{A_0}{A_2}}}} \right)} .} \end{array}} \right.$$

Combined with Eq. (6), the Eq. (1) can be rewritten as:

(7)$$\varphi ^{\prime} = \arctan (\frac{{{y_0} + {a_y}\sin (\varphi + {\delta _N})\textrm{ - }{y_0}^{\prime}}}{{{x_0} + {a_x}\cos (\varphi + {\delta _D}) - {x_0}^{\prime}}} \times \frac{{{a_x}^{\prime}}}{{{a_y}^{\prime}}}).$$

In order to obtain multiple (D, N) distributions, we use PZT in the light source module for phase shift, each time shifted by π/6, and the simulation interferograms are shown in Fig. 3.

Fig. 3. Six different (D, N) pairs with phase-shifting of π/6 obtained by PZT in the source module by simulation

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Using the multiple (D, N) values obtained in Fig. 3, combined with Eq. (6) and Eq. (7), the light intensity and the modulation distribution in the interferograms are corrected, so that the phase distribution φ’ obtained in Eq. (1) eliminates the print through errors. The simulation result of using Lissajous ellipse fitting to correct one of the pixels is shown in Fig. 4.

Fig. 4. The Lissajous curves before correction (a) and after correction (b)

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The corrected phase distribution of the entire profile is shown in Fig. 5(c), and shows no influence of print through errors.

Fig. 5. (a) One of the simulation interferograms, (b) the results with double frequency print through errors and (c) results after correction

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Once the related devices in the system are adjusted to the working state, their positions and angles stay fixed. Although the background light intensity and modulation degree may be different at every point in space, the single and double frequency errors are corrected pixel-by-pixel to achieve global polarization errors correction in this paper.

2.2 Correction for polarization errors caused by birefringence and its wave plate model

The source of polarization error that we are most interested in is the effect of stress-induced birefringence of large aperture collimating objectives on the measurement results. In a natural state without internal stress, glass is optically isotropic, and the light propagating in it is uniquely determined by the law of refraction, and its polarization state does not change when passing through the glass. However, the large aperture collimating objective is usually under stress, and the glass density changes from an isotropic body to an anisotropic body and birefringence occurs when polarized light passes through it. At this time, the polarization state of P light and S light required by the polarization phase shifting method changes when propagating through the collimator, which destroys the common path characteristic of the Fizeau interferometer, and finally affects the measurement result. This kind of polarization error is inevitable, and it remains a key obstacle for apply polarization-based dynamic phase shifting technique to a large aperture interferometer. It is not easy to test the birefringence state of the large flat. To find a simple solution for the correction is of great importance. In order to theoretically analyze the effect of birefringence on the measurement results, this article simplifies the effect of the collimator into a wave plate model, where the direction of the fast axis and the amount of phase retardance varies pixel by pixel.

The Jones matrix of P light, S light, the QWP at angles of 45 °and the polarizer at different angles ω are:

(8)$$\begin{array}{l} P = \left[ {\begin{array}{{c}} 1\\ 0 \end{array}} \right],\begin{array}{{cc}} {}&{S = \left[ {\begin{array}{{c}} 0\\ 1 \end{array}} \right]} \end{array}\begin{array}{{cc}} {,\begin{array}{{cc}} {}&{QWP\left( {\frac{\pi }{4}} \right) = \frac{1}{{\sqrt 2 }}\left[ \begin{array}{l} 1\begin{array}{{c}} {\begin{array}{{cc}} {}&{ - i} \end{array}} \end{array}\\ - i\begin{array}{{cc}} {}&1 \end{array} \end{array} \right],} \end{array}}&{\begin{array}{{cc}} {}&{} \end{array}} \end{array}\\ P(\omega ) = \left( {\begin{array}{{cc}} {{{\cos }^2}\omega }&{\cos \omega \sin \omega }\\ {\cos \omega \sin \omega }&{{{\sin }^2}\omega } \end{array}} \right). \end{array}$$

Ideally, the Jones matrix of interference light that reaches the QWP in front of the CCD camera is:

(9)$$E = \left[ {\begin{array}{{c}} a\\ {b\exp (i2\gamma )} \end{array}} \right],$$

where γ is the single pass phase difference between TF and RF caused by cavity length and the surface map of the RF. The Jones matrix after the interference light passes through the QWP and polarizer array in front of CCD is:

(10)$$\begin{aligned} E(\alpha )&= P(\omega )\times QWP\left( {\frac{\pi }{4}} \right) \times E\\ \begin{array}{{cc}} {}&{} \end{array}&\begin{array}{{c}} = \end{array}\{{a\exp [ - i\omega ] - ib\exp [i(\omega + 2\gamma )]} \}\left[ \begin{array}{l} \cos (\omega )\\ \sin (\omega ) \end{array} \right]. \end{aligned}$$

On this basis, the light intensity distribution on the CCD is:

(11)$$\begin{array}{l} I = {|{a\exp [ - i\omega ] - ib\exp [i(\omega + 2\gamma )]} |^2}\\ \begin{array}{{cc}} {}&{ = {a^2} + {b^2} + 2ab\sin (2\gamma + 2\omega )} \end{array}. \end{array}$$

When the angle ω of the micro-polarizers changes, it is equivalent to shift the phase of the interferogram to solve the desired surface shape distribution γ.

Nevertheless, polarized light will be affected by birefringence. Assume that the angle between the fast axis of the system's wave plate model and the polarization direction of P light is α, and the phase retardance is Φ. The fast axis of the wave plate is defined as the X_w of the local coordinate system; the slow axis of the wave plate is Y_w. As shown in Fig. 6.

Fig. 6. The global coordinate system (X_g, Y_g) and the local coordinate system of the wave plate model (X_w, Y_w)

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In the local coordinate system (X_w, Y_w), the Jones matrix of the model and the P light and S light can be expressed as:

(12)$$W = \left[ {\begin{array}{{cc}} 1&0\\ 0&{\exp (i\phi )} \end{array}} \right],\begin{array}{{cc}} {}&{{P_w} = \left[ {\begin{array}{{c}} {\cos \alpha }\\ { - \sin \alpha } \end{array}} \right]} \end{array},\begin{array}{{cc}} {}&{{S_w} = \left[ {\begin{array}{{c}} {\sin \alpha }\\ {\cos \alpha } \end{array}} \right]} \end{array}.$$

Assuming that Pw light is the reference light reflected by TF and Sw is the test light reflected by RF, the Jones matrix of the two after passing through the wave plate model twice can be expressed as:

(13)$$\left\{ {\begin{array}{{c}} {{P_{TF}} = W{P_w} = \left( {\begin{array}{{c}} {\cos \alpha }\\ { - \sin \alpha \exp (i2\Phi )} \end{array}} \right)\begin{array}{{cccc}} {}&{}&{}&{\begin{array}{{ccc}} {\begin{array}{{cc}} {}&{} \end{array}}&{}&{(a),} \end{array}} \end{array}}\\ {{P_{RF}} = W{S_w}\exp (i2\gamma ) = \left( {\begin{array}{{c}} {\sin \alpha }\\ {\cos \alpha \exp (i2\Phi )} \end{array}} \right)\exp (i2\gamma )\begin{array}{{cc}} {}&{(b).} \end{array}} \end{array}} \right.$$

The Jones matrices of the interference light reaching the QWP in front of the CCD camera are:

(14)$$\left\{ \begin{array}{ll} E = \left( {\begin{array}{c} {{E_x}}\\ {{E_y}\exp (i2\beta )} \end{array}} \right) & (a), \\ E = \left( {\begin{array}{c} {\cos \alpha + \sin \alpha \exp (i2\gamma )}\\ { - \sin \alpha \exp (i2\Phi ) + \cos \alpha \exp (i2\Phi )\exp (i2\gamma )} \end{array}} \right) & (b). \end{array} \right.$$

Equation (14) is similar to Eq. (9). If the Jones matrix in Eq. (14b) containing α and γ can be transformed into Eq. (14a), β can be calculated, which is the distribution of the surface shape affected by the birefringence. The polarization error caused by birefringence can be expressed as:

(15)$$\delta = \beta - \gamma .$$

Now we will mainly focus on transforming (14-b) to (14-a). Based on Eq. (14), we can get:

(16)$$\left\{ {\begin{array}{{l}} \begin{array}{l} {E_x} = \cos \alpha + \sin \alpha \exp (i2\gamma ) = \cos \alpha + \sin \alpha \cos 2\gamma + i\sin \alpha \sin 2\gamma \\ = {E_{xa}} + i{E_{xb}} = \sqrt {E_{xa}^2 + E_{xb}^2} \ast \exp ({i{\theta_x}} ), \end{array}\\ \begin{array}{l} {E_y} ={-} \sin \alpha \exp (i2\Phi ) + \cos \alpha \exp (i2\Phi )\exp (i2\gamma )\\ = ({ - \sin \alpha \cos 2\Phi + \cos \alpha \cos (2\Phi + 2\gamma )} )+ i({ - \sin \alpha \sin 2\Phi + \cos \alpha \sin (2\Phi + 2\gamma )} )\\ = {E_{ya}} + i{E_{yb}} = \sqrt {E_{ya}^2 + E_{yb}^2} \ast \exp ({i{\theta_y}} ),\\ \beta = {\theta_y} - {\theta_x}. \end{array} \end{array}} \right.$$

As shown in Eq. (16), both elements in the Jones matrix E can be written in a form containing amplitude and phase, and the desired phase β can be regarded as the difference between two elements after animating the common phase factor exp(iθ_x). From Eq. (16), we can know that θ_x and θ_y can be expressed as:

(17)$$\left\{ {\begin{array}{{l}} {\tan {\theta_x} = \frac{{{E_{xb}}}}{{{E_{xa}}}} = \frac{{\sin \alpha \sin 2\gamma }}{{\cos \alpha + \sin \alpha \cos 2\gamma }},}\\ {\tan {\theta_y} = \frac{{{E_{yb}}}}{{{E_{ya}}}} = \frac{{ - \sin \alpha \sin 2\Phi + \cos \alpha \sin (2\Phi + 2\gamma )}}{{ - \sin \alpha \cos 2\Phi + \cos \alpha \cos (2\Phi + 2\gamma )}}.} \end{array}} \right.$$

Now we get that:

(18)$$\tan ({{\theta_y} - {\theta_x}} )= \frac{{\tan {\theta _y} - \tan {\theta _x}}}{{1 + \tan {\theta _y}\tan {\theta _x}}} = \frac{{\frac{{{E_{yb}}}}{{{E_{ya}}}} - \frac{{{E_{xb}}}}{{{E_{xa}}}}}}{{1 + \frac{{{E_{yb}}}}{{{E_{ya}}}}\frac{{{E_{xb}}}}{{{E_{xa}}}}}} = \frac{{{E_{yb}}{E_{xa}} - {E_{xb}}{E_{ya}}}}{{{E_{ya}}{E_{xa}} + {E_{yb}}{E_{xb}}}}.$$

The tangent value of β can be derived from Eq. (17) and Eq. (18):

(19)$$\left\{ {\begin{array}{{c}} {{E_{yb}}{E_{xa}} - {E_{xb}}{E_{ya}} = \cos {{(\alpha )}^2}\sin (2\gamma + 2\Phi ) - \sin {{(\alpha )}^2}\sin (2\gamma \textrm{ - }2\Phi ),}\\ {{E_{ya}}{E_{xa}} + {E_{yb}}{E_{xb}} = \cos {{(\alpha )}^2}\cos (2\gamma + 2\Phi ) - \sin {{(\alpha )}^2}\cos (2\gamma \textrm{ - }2\Phi ),}\\ {\tan ({{\theta_y} - {\theta_x}} )= \frac{{{E_{yb}}{E_{xa}} - {E_{xb}}{E_{ya}}}}{{{E_{ya}}{E_{xa}} + {E_{yb}}{E_{xb}}}} = \frac{{\cos {{(\alpha )}^2}\sin (2\gamma + 2\Phi ) - \sin {{(\alpha )}^2}\sin (2\gamma \textrm{ - }2\Phi )}}{{\cos {{(\alpha )}^2}\cos (2\gamma + 2\Phi ) - \sin {{(\alpha )}^2}\cos (2\gamma \textrm{ - }2\Phi )}}.} \end{array}} \right.$$

So far, we have obtained the relationship between the actual phase β=θ_y-θ_x and the ideal phase distribution γ. Consider several special cases: if α = mπ, then β = 2γ + 2Φ; if α = mπ +π / 2, then β = 2γ-2Φ, which means that when the fast axis of the wave plate model and the X_g or Y_g axis of the global coordinate system coincides, the effect of the wave plate model on the result will be simply the addition and subtraction of the amount of phase retardance. If Φ = 2mπ, then tan(θ_y-θ_x) = tan (2γ), at this time, because the wave plate model does not produce a phase retardance, it does not affect the measurement results. From another point of view, when the plane shape of the reflective flat crystal is perfect, γ = 0 and α is small, then the result of Eq. (19) is entirely a polarization error caused by the model. The error at this time can be expressed as:

(20)$$\tan ({{\theta_y} - {\theta_x}} )= \frac{{\tan (2\Phi )}}{{\cos (\textrm{2}\alpha )}}.$$

As of now, as long as the angle α and phase retardance Φ of the wave plate model can be obtained, we can eliminate the polarization error from the actual measurement results and obtain the correct surface distribution.

2.3 Calculation of parameters of the wave plate model

In order to obtain the two critical parameters of the wave plate model established above, we continue to use the Jones matrix for analysis and derivation. Taking general elliptically polarized light as an example, its Jones matrix T is:

(21)$$T = \left( {\begin{array}{{c}} {{a_\textrm{1}}}\\ {{a_\textrm{0}}\exp (i\psi )} \end{array}} \right).$$

In the system shown in Fig. 1, when the QWP in front of the CCD is removed, the polarized light becomes the elliptically polarized light shown in Eq. (21) after passing through the interference system. After passing through the micro-polarizer array, it directly hits on the CCD. The matrix is:

(22)$$\begin{array}{l} {T_p} = P(\omega )\times T = \left( {\begin{array}{{cc}} {{{\cos }^2}\omega }&{\cos \omega \sin \omega }\\ {\cos \omega \sin \omega }&{{{\sin }^2}\omega } \end{array}} \right)\left( {\begin{array}{{c}} {{a_\textrm{1}}}\\ {{a_\textrm{0}}\exp (i\psi )} \end{array}} \right)\\ \begin{array}{{l}} {\begin{array}{{l}} {} \end{array}\begin{array}{{l}} {} \end{array} = } \end{array}({{a_\textrm{1}}\cos \omega + {a_\textrm{0}}\sin \omega \exp (i\psi )} )\left( {\begin{array}{{c}} {\cos \omega }\\ {\sin \omega } \end{array}} \right). \end{array}$$

Therefore, we can know that the light intensity distribution collected by the CCD camera is:

(23)$$I(\omega ,\psi ) = {a_\textrm{1}}^2{\cos ^2}\omega + {a_\textrm{0}}^2{\sin ^2}\omega + 2{a_0}{a_\textrm{1}}\sin \omega \cos \omega \cos \psi .$$

When the angle ω of the micro polarizer is 0, π/4, π/2, π3/4, the corresponding light intensity distribution is:

(24)$$\left\{ {\begin{array}{{l}} {{I_1} = {a_\textrm{1}}^2,}\\ {{I_2} = \frac{1}{2}{a_0}^2 + \frac{1}{2}{a_1}^2 + {a_0}{a_1}\cos \psi ,}\\ {{I_3} = {a_\textrm{0}}^2,}\\ {{I_\textrm{4}} = \frac{1}{2}{a_0}^2 + \frac{1}{2}{a_1}^2\textrm{ - }{a_0}{a_1}\cos \psi .} \end{array}} \right.$$

From Eq. (24), we can calculate the parameters of the elliptically polarized light we want:

(25)$$\left\{ {\begin{array}{{l}} {\xi \textrm{ = }\arctan \left( {\sqrt {{I_\textrm{1}}/{I_\textrm{3}}} } \right),}\\ {\psi = \arccos \left( {\frac{{{I_2} - {I_4}}}{{2\sqrt {{I_1}{I_3}} }}} \right).} \end{array}} \right.$$

Combining Eq. (13a) and Eq. (21), we can obtain the angle α and phase retardance Φ of the wave plate model. It can be known that -45°<ξ <45°, while 0°<Ψ<180°.

(26)$$\left\{ {\begin{array}{{l}} {\alpha ={-} \xi ,}\\ {\phi = \psi /2.} \end{array}} \right.$$

On this basis, the Eq. (19) can be used to correct the polarization error caused by birefringence. We know that -45°<α<45°, and 0°<Φ<90°.

3. Experimental verification of the polarization error correction procedure

In this paper, a 600mm Fizeau interferometer is used to carry out synchronous interferometer experiments based on polarization phase shifting method and combined with the content discussed above, the polarization errors in the source module and the interference module are corrected and then compared with the results acquired with a wavelength tuning laser.

3.1 Parameter calculation of the wave plate model

According to the discussion in Section 2.3, the parameters of the wave plate model can be obtained by calculating the light intensity distribution collected by the CCD without QWP for a single polarized light. CCD collected pictures, as shown in Fig. 7.

Fig. 7. (a)The intensities of the P light passing through the micro polarizers with different angles (b) the intensity of the 90° polarizer.

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It can be seen from Fig. 7 and Eq. (24) that if the large aperture element in the interference module does not have birefringence, the incident P light will still be linearly polarized after passing through a series of elements. The intensity distribution should be 0 in the 90° interferogram. However, in fact, although the light intensity at the 90 ° position in the collected image is close to 0, the light intensity at the position near the middle and the edge is significantly larger, and the cause of this phenomenon is the birefringence phenomenon.

Combining Eq. (23) and Eq. (24), the two parameter distributions of the polarization model can be obtained as follows:

It can be seen from Fig. 8(a) that the value of cos (α) is mostly close to 1, which is close to the theoretical value, and the phase retardance Φ is very different between pixels, close to λ/10.

Fig. 8. (a) The cos(α) values and (b) the Φ values of the wave plate model

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3.2 Polarization errors and correction in experiments

In actual experiments, the single frequency print through error is very obvious, while double frequency is almost invisible. After eliminating the print through errors, the phase distribution has some depressions and protrusions compared with the results of the wavelength tuning method. The interferograms of 600mm aperture interferometer collected in experiments and its single-step processing results are as follows:

It can be clearly seen from Fig. 9(a) that the single frequency print through error is consistent with the distribution of the interferogram fringes, which is mainly due to the inconsistent background light intensity of the four interferograms. In Fig. 9(b), after being processed with the correction method described in Section 2.1, single frequency error is well eliminated. The two depressions and one protrusion surrounded by the red dotted circle Fig. 9(b) are measurement errors caused by birefringence in the interference module, and the remaining work now is mainly to combine the wave plate model parameters of Fig. 7 and Eq. (20) to eliminate the effect of birefringence. To simplify the problem, we directly subtract the polarization error represented by Eq. (20) from the calculated wave surface. The surface shape result is as follows:

Fig. 9. (a) the surface map with print through errors and (b) the surface map after correction with Lissajous ellipse fitting method

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It can be seen from Fig. 10(a) that the place severely affected by the birefringence is significantly eliminated compared with the test result of Fig. 9(c). The PV value is also reduced from 0.121λ to 0.109λ and RMS value from 0.019λ to 0.014λ. In order to verify the repeatability of this method, ten different experiments were completed, the statistical results are as follows:

Fig. 10. The surface maps acquired from (a) dynamic interferometer eliminating errors from birefringence effects and (b) wavelength tuning method

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Figure 11 shows the statistics of ten corrections results. The average value of the 10 PVs is 0.107λ, the RMS is 0.014λ, and the repeatability reaches 0.0006λ. In order to verify the correctness of the correction method proposed in the paper, the same large aperture flat are also measured by the wavelength tuning phase shifting method, by only changing the light connected to the interference module from the source module to the wavelength tuning laser. The single measurement result of wavelength shifting is shown in Fig. 10(b). Because the reference light and the test light have the same polarization state, the result will not be affected by the birefringence of the system. The results of the ten repeated test results of the wavelength tuning method are as follows:

Fig. 11. Statistics of repeatability data of polarization error correction results

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As shown in Fig. 12, the average PV of the ten results of the wavelength method is 0.105λ, and the RMS is 0.013λ, and its repeatability accuracy can reach 0.0009λ. Comparing the correction results of the polarization method and the results of wavelength method, the PV deviation of the two is 0.002λ, and of RMS is 0.001λ. The repeatability accuracy of both is below λ/1000, which verifies the repeatability and reproducibility of the measurements.

Fig. 12. Statistics of repeatability data via wavelength method

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3.3 Discussion and analysis of residuals errors after corrections

It can be seen from Fig. 10 that although the correction method proposed in this paper can eliminate most of the errors, including the effect of frequency print through errors and birefringence, there are still some residual errors in the center area. The main reasons may be: (a) the process of eliminating the frequency errors may also have a small effect on the actual distribution of the surface shape, resulting in residual errors in the results after birefringence correction; (b) the QWP before the CCD is not needed when calculating the parameters of the wave plate model, resulting in a little difference in parameters between the model and the actual situation if the QWP is imperfect; (c) in order to simplify the model in this paper, directly subtracting the theoretical error in Eq. (20) from the phase calculated by the interferogram may result in the failure to complete the correction; (d) the retardance of the plate model is restricted between 0 to 180°, and it does not work if the retardance Φ>180°. In fact, when the angle of the fast axis in the wave plate model is not 0, the error caused by the birefringence is related to the theoretical phase γ distribution and the phase retardance amount Φ. We conducted relevant numerical simulations on this.

As shown in Fig. 13(b), after correcting the polarization error according to the method of this paper, there is still some inevitable residual error compared with the theoretical profile, which is related to the phase distribution. In order to reduce this effect, the interference cavity should be adjusted to zero fringe position as much as possible during measurement.

Fig. 13. The phase distribution affected by birefringence before correction(a) and its residual error after correction(b) with the global fast axis 0.1rad and phase retardance 1.2rad

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On the other hand, the linearly polarized P light or S light incident into the interference module is not completely linearly polarized, because the extinction ratio of PBS is approximately 1:50 to 1:10. Also, when the linearly polarized light is coupled into the optical fiber and then inserted into the interference module, the depolarization phenomenon may also occur due to the bending of the polarization-maintaining fiber. In this case, when the light intensity distribution collected on the CCD is inconsistent with the ideal value, it is difficult to distinguish whether it is the result of its own light properties or the influence of the birefringence in the interference module, which will affect the calculation of the wave plate model, mainly on angle α. It is interesting but a great challenge to decouple the fiber depolarization and the birefringence, because the wave plate model is based on a ideally linear start. Two thoughts are considering to be the potential solutions for this problem. One is to take the ellipsoidal light as the initial input, and see how it goes in the result. This will make the relationship between ideal phase distribution and the calculated one much more complicated. Another way is try to calibrate the influence of the depolarized light from fiber by directly hitting on the polarization camera. This could be easier but needs some changes in structure. Future work can be done to analyze the wave plate model further to eliminate the polarization errors completely.

4. Conclusion

It is a huge challenge to imply the dynamic phase shifting technique based on polarization to a Fizeau interferometer with a diameter of 600mm. This paper analyzes the sources of polarization errors of large aperture dynamic interferometers, including the effects of polarized elements and birefringence of large aperture optical elements, and establishes a wave plate model for corrections. We are using the Lissajous ellipse fitting method to correct the influence of the polarized devices, effectively eliminating the single and double frequency print through errors in the result. The core of this paper is to propose a wave plate model with fast axis angle α and phase retardation Φ for analyzing the birefringence effect of large aperture optical components. On this basis, we establish the relationship between the actually calculated phase and the ideal phase distribution, which shows excellent effects for birefringence correction. Compared with the results of the wavelength tuning phase shifting method, the deviation of PV values is 0.002λ, and of RMS values is 0.001λ, and the repeatability accuracy of the two is below λ/1000, which thoroughly verifies the correctness and repeatability of the method proposed in this paper. The correction method in this paper is easy to realize all by the system itself with no extra auxiliary equipment or large-scale structural adjustment. As far as we know, this article presents the first 600mm dynamic interferometer based on polarization phase shifting method.

Funding

National Natural Science Foundation of China (61775097, 61975081).

Disclosures

The authors declare no conflicts of interest.

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Modelling and correction for polarization errors of a 600 mm aperture dynamic Fizeau interferometer

Abstract

1. Introduction

2. Theoretical analysis and simulation on polarization errors

2.1 Correction for polarization errors caused by polarized devices

2.2 Correction for polarization errors caused by birefringence and its wave plate model

2.3 Calculation of parameters of the wave plate model

3. Experimental verification of the polarization error correction procedure

3.1 Parameter calculation of the wave plate model

3.2 Polarization errors and correction in experiments

3.3 Discussion and analysis of residuals errors after corrections

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (13)

Equations (26)

Optics Express