Problems on polarization aberrations in large aperture dynamic interferometry based on the polarization phase shifting technique

Xinyu Miao; Jun Ma; Cong Wei; Nianfeng Wang; Hui Ding; Rihong Zhu; Caojin Yuan

doi:10.1364/OE.471634

1. Introduction

Polarization based phase-shifting technique is free of the influence of vibration and airflow disturbance, and has been widely applied in various interferometer systems to realize simultaneous measurements [1–4]. However, the polarization aberrations caused by imperfect polarized elements and the birefringence effect become inevitable for the measurements of large optics in a Fizeau interferometer for their heavy weight and the mechanical mounting structure. The stress-induced birefringence destroys the common path configuration when two orthogonally polarized lights propagate through the beam expander system including the large aperture collimator and transmission flat. The birefringence of large optics is inevitable and annealing times of several months are necessary to reach 4 nm/cm birefringence [5]. The birefringence effect usually stays stable if the mounting structure is fixed and the temperature unchanged [6]. It is of great importance to determine the birefringence distribution and then find a proper approach to control and reduce its impact on the measurement result, especially in large aperture dynamic interferometers.

The methods used to evaluate the stress through optical path difference measurement can be classified into two kinds: matrix analysis method and interference method, both of which are based on the analysis of effects that birefringence brings into the intensity maps or interference fringes maps captured by the detector. The matrix analysis method inserts the birefringence sample between polarization analyzers and a series of intensity maps are collected when the analyzer is placed at different orientations [2,7–14]. Rudolf Oldenbourg proposed intensity processing algorithms using different recorded frames to measure the distribution of linear birefringence with variable retarders [7]. Connor Lane presented a new procedure applying the Mueller matrix calculus to calculate the birefringence information after deriving the first three Stokes parameters from the intensities [12]. Aijun Zeng used a beam splitter and Wollaston prisms to capture four intensities combined with four detectors, which made it possible to resolve the birefringence magnitude and direction in real time [15]. Zeng also proposed and verified a method with refractive index matching liquid to rectify the optical path when measuring the distribution in a lens [14]. Most of the matrix analysis methods focus on detecting the changes in the polarization state modulated by the polarization optical elements. However, the aperture of the tested sample is limited by the analyzer and the rotation of the analyzer introduces mechanical errors into the results.

Laser interferometry has been widely applied in various fields for its advantage of noncontact and high precision, where the birefringence information can be obtained from the interference fringes. Compared with the matrix analysis method that only needs one polarization state in the system, some interference methods require two orthogonally polarized lights, with only one light passing through the sample [16–22]. Nandini Ghosh presented a full-field technique for simultaneous measurement of the magnitude and orientation of birefringence using a monolithic birefringence sensitive interferometer consisting of a suitably polarization-masked cube beam splitter to generate a pair of orthogonally polarized and collinearly propagating light beams [22]. Taking the advantage of the dependence of measurements on the amplitude and phase, Yukitoshi Otani described a phase shifting technique for two dimensional birefringence distribution measurement without moving the sample mechanically [16]. Jennifer L. Rouke examined the birefringence in the GRIN rod, which was fabricated by ion exchange and may contain stress from two different processes [23]. A phase-stepping Twyman-Green interferometer was used to measure the optical path difference between rays of two orthogonal polarizations. Andreas Berger presented a general measurement algorithm for locally linear polarization distribution to characterize the properties of the elements manipulating both polarization and phase of wave fields [19,24]. All the interferometric methods mentioned above require two separate beams, employing a Twyman or Mach-Zehnder-like configuration, which makes the system suffer from vibration and system errors. Although these methods performed well in the test for birefringent samples on small scale, they are not suitable for the measurement of the large aperture collimator in the Fizeau type interferometer.

It is not realistic to move the collimator with a large diameter after being placed in the dynamic Fizeau interferometer. The birefringence distribution of the large optics could change after being moved during measurements. Measuring the birefringence of large optics without movements is the best way to keep the birefringence stable. Inspired by the method proposed by Eugene R. Cochran where a Fizeau interferometer was used to measure an optical window [25], we present an interferometric method to obtain the full field birefringence distribution based on the polarization rotation and collimated wavefront calculation. For a system with a beam expander, the large aperture dynamic system keeps unmoved and an interference cavity is established. The large optics are placed in the cavity and treated as the birefringent material. The theoretical analysis of this method is given and the measurement of the birefringence is transformed into two independent measurements of the optical path of the transmitted wavefront when lights with orthogonal polarization states incident. This method is especially suitable for the birefringence measurement of the large aperture dynamic Fizeau interferometer with linearly polarized lights. The method proposed here allows the birefringence of large optics to be determined with high efficiency and without movements, after which the errors that birefringence brings into the interferometric results can be corrected. We believe that this determination and correction method could help the polarization based large aperture dynamic interferometer be more widely used with high precision.

2. Configuration and principle

2.1 Measurement configuration transformed from dynamic Fizeau interferometer

The layout of the measurement system is depicted in Fig. 1. The main elements are the same as the dynamic interferometer described before [2], except the light source and a TF (transmission flat) with a diameter of 100 mm. Two linearly polarized beams with orthogonally polarization states are generated in the source module, only one of which enters the interference module at one time with another light blocked out. For an interferometer with an expander system, either Galileo or Kepler type, the large aperture collimator and the TF can be viewed as birefringent samples placed in an interference cavity consisting of a RF (reflection flat) with a diameter of 600 mm in the large end and a TF with a diameter of 100 mm in the small end. The polarization state of the incident lights entering the interference module can be switched between P polarized light (along with the horizontal axis) and S polarized light (along with the vertical axis) by simply blocking one beam in the arm above or right to the PBS in the source module. If no source module is available, this switching step can also be done by rotating a HWP if we have a polarizer (oriented at horizontal direction) one HWP placed between the pinhole and the BS in the interference module. The extinction of the output polarization state at the 100-mm collimator is ensured by the polarized components in the source module. The extinction ration can reach 1000:1 for a commercial PBS.

Fig. 1. The measurement configuration transformed from a dynamic interferometer. The small end consists of the 100-mm collimator and a 100-mm TF. The large end consists of large collimator, TF and RF with a diameter up to 600 mm or 800 mm.

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No polarized elements are used in the interference module to avoid the potential error introduced by imperfect components. A tunable laser is implemented to acquire phase shifting interferograms for wavefront calculation. The 100-mm TF inserted into the interference module serves as the reference surface and the 600-mm RF as the test wavefront. Two independent measurements of the transmitted wavefront will be taken, and the birefringent distribution is deduced through the wavefront changes between P and S light. The theoretical basis of this correcting method will be introduced later.

2.2 Analysis on the birefringence effects on different incident lights

The interferometer systems can be divided into linearly polarized system and circularly polarized system according the polarization state of the incident lights. Two beams with orthogonally polarized states are required in a Fizeau dynamic interferometer, but the birefringence destroys the common path characteristic for the difference between the optical paths that the birefringence introduces into incident lights. The beam expander system consisting of mainly the large aperture collimator and TF is simplified as a phase retarder with a retardance Φ with its fast axis at the orientation of angle θ against the X axis. Some folded mirrors that are ignored in the expander may cause depolarization effects too, but all the influences can be described in the Jones matrix as a whole. Both the influences on linearly polarized lights and circularly polarized lights are explained in the following contents. We will start with the basic theory of polarization phase shifting technique. The Jones matrixes of incident lights, QWP, polarizer and the phase retarder W_S are expressed as

(1)$$\begin{array}{l} \tilde{P} = \left[ {\begin{array}{c} 1\\ 0 \end{array}} \right],\textrm{ }\tilde{S} = \left[ {\begin{array}{c} 0\\ 1 \end{array}} \right],\textrm{ }\tilde{L} = \left[ {\begin{array}{c} 1\\ j \end{array}} \right],\textrm{ }\tilde{R} = \left[ {\begin{array}{c} 1\\ { - j} \end{array}} \right],\\ {E_{QWP}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{cc} 1&{ - j}\\ { - j}&1 \end{array}} \right]\textrm{ }{E_{polarizer}} = \left[ {\begin{array}{cc} {{{\cos }^2}\alpha }&{\sin \alpha \cos \alpha }\\ {\sin \alpha \cos \alpha }&{{{\sin }^2}\alpha } \end{array}} \right]\\ {W_s} = \left[ {\begin{array}{cc} {{c^2} + {s^2}{e^{ - i\Phi }}}&{cs - cs{e^{ - i\Phi }}}\\ {cs - cs{e^{ - i\Phi }}}&{{s^2} + {c^2}{e^{ - i\Phi }}} \end{array}} \right] = \left[ {\begin{array}{cc} A&B\\ C&D \end{array}} \right], \end{array}$$

where c = cosθ and s = sinθ. $\tilde{P}$ and $\tilde{S}$ represents the Jones matrixes of incident lights that are linearly polarized at horizontal and vertical directions. $\tilde{L}$ and $\tilde{R}$ represents the Jones matrixes of incident lights that are left-handed and right-handed circularly polarized. α is the angle between the transmitting direction of polarizer and X axis. It is worth noting that the distribution of W_S is uneven, which means that the phase retardance Φ and the orientation angle θ varies from pixel to pixel.

For a dynamic interferometer based on polarization, two orthogonally polarized lights, served as test light and reference light, are modulated by the birefringence of the large optics. The difference is that the linearly polarized lights will pass through a QWP and a micro-polarizer array (served as the phase shifter) before reaching the camera, while the circularly polarized lights pass through the array only, as shown in Fig. 2.

Fig. 2. The elements before the lights reaching the camera. 1: imaging lens; 2: QWP; 3: polarization camera with a micro-polarizer array.

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Firstly, we consider the condition that no birefringence exists in the system. For a linearly polarized system, the Jones matrix arriving at the CCD and its intensity can be expressed as:

(2)$$\begin{array}{l} {{\tilde{E}}_{summary}} = \tilde{P} + \tilde{S}{e ^{ - i\gamma }} = \left[ {\begin{array}{c} 1\\ {{e^{ - i\gamma }}} \end{array}} \right],\\ \tilde{E}(\alpha )= {E_{polarizer}} \times {E_{QWP}} \times {{\tilde{E}}_{summary}} = \{{{e^{ - i\alpha }} - i{e^{ - i({\alpha - \gamma } )}}} \}\left[ \begin{array}{l} \cos (\alpha )\\ \sin (\alpha ) \end{array} \right],\\ I = {|{{e^{ - i\alpha }} - i{e^{ - i({\alpha - \gamma } )}}} |^2} = 2 + 2\sin (2\alpha - \gamma ), \end{array}$$

where γ represents the surface map of the test flat and four phase shifting interferograms can be acquired when α equals to 0, 45°,90° and 135°. We can see from Eq. (2) that the combination of QWP and polarizer only serves as the phase shifter and will not change the surface information γ. This means that we don’t have to calculate the intensity distribution every time. The phase calculated by the intensity maps is equal to the phase retardance difference between the two components of ${\tilde{E}_{summary}}$.

Now the condition changes and birefringence appears in the system. According to Eq. (1), the Jones matrixes of the lights pass through the imaging lens are expressed as:

(3)$$\begin{array}{l} {{\tilde{E}}_P} = {W_s}\tilde{P} = \left[ {\begin{array}{c} A\\ C \end{array}} \right],\textrm{ }{{\tilde{E}}_S} = {W_s}\tilde{S} = \left[ {\begin{array}{c} B\\ D \end{array}} \right],\textrm{ }{{\tilde{E}}_{linear}} = {{\tilde{E}}_P} + {{\tilde{E}}_S} = \left[ {\begin{array}{c} {A + B}\\ {C + D} \end{array}} \right]\\ {{\tilde{E}}_L} = {W_s}\tilde{L} = \left[ {\begin{array}{c} {A + jB}\\ {C + jD} \end{array}} \right],\textrm{ }{{\tilde{E}}_R} = {W_s}\tilde{R} = \left[ {\begin{array}{c} {A - jB}\\ {C - jD} \end{array}} \right],\textrm{ }{{\tilde{E}}_{circular}} = {{\tilde{E}}_L} + {{\tilde{E}}_R} = \left[ {\begin{array}{c} {2A}\\ {2C} \end{array}} \right]. \end{array}$$

The summary the two lights E_P and E_S pass through the QWP and the polarizer to interfere, while the ${\tilde{E}_L}$ and ${\tilde{E}_R}$ passing through the polarizer only. For the linearly polarized system, according to the analysis above, the phase retardation difference between the two components of the summary matrix ${\tilde{E}_{linear}}$ can be treated as the error that the birefringence brings into the measurement results. Because both components of ${\tilde{E}_{linear}}$ should be 1 if no birefringence exists. For the circularly polarized system, the summary matrix ${\tilde{E}_{circular}}$ needs subsequent processing, which will be introduced later. Now the problem is to find out how much impact the birefringence has on the interferometric results. Combining Eq. (1) and Eq. (3), we have:

(4)$$\begin{array}{l} A + B = {c^2} + {s^2}\cos \Phi - j{s^2}\sin \Phi + cs - cs\cos \Phi + jcs\sin \Phi \\ \textrm{ } = ({{c^2} + {s^2}\cos \Phi + cs - cs\cos \Phi } )+ j({cs\sin \Phi - {s^2}\sin \Phi } ),\\ \frac{{{\mathop{\rm Im}\nolimits} ({A + B} )}}{{Re ({A + B} )}} = \frac{{cs\sin \Phi - {s^2}\sin \Phi }}{{{c^2} + {s^2}\cos \Phi + cs - cs\cos \Phi }} = \tan ({{\Phi _{L1}}} ), \end{array}$$

and

(5)$$\begin{array}{l} C + D = {s^2} + {c^2}\cos \Phi - j{c^2}\sin \Phi + cs - cs\cos \Phi + jcs\sin \Phi \\ \textrm{ } = ({{s^2} + {c^2}\cos \Phi + cs - cs\cos \Phi } )+ j({cs\sin \Phi - {c^2}\sin \Phi } ),\\ \frac{{{\mathop{\rm Im}\nolimits} ({C + D} )}}{{Re ({C + D} )}} = \frac{{cs\sin \Phi - {c^2}\sin \Phi }}{{{s^2} + {c^2}\cos \Phi + cs - cs\cos \Phi }} = \tan ({{\Phi _{L2}}} ), \end{array}$$

where Φ_L1 and Φ_L2 represents the phase retardation of the two components of matrix ${\tilde{E}_{linear}}$. We will know the error of the birefringence once we determine amount of ΔΦ_L:

(6)$$\tan ({\Delta {\Phi _L}} )= \tan ({{\Phi _{L2}} - {\Phi _{L1}}} )= \frac{{\tan ({{\Phi _{L2}}} )- \tan ({{\Phi _{L1}}} )}}{{1 + \tan ({{\Phi _{L2}}} )\tan ({{\Phi _{L1}}} )}} = \frac{{{N_L}}}{{{D_L}}},$$

and

(7)$$\begin{array}{l} {N_L} ={-} \sin \Phi ({{c^2} - {s^2}} )={-} \sin \Phi \cos 2\theta ,\\ {D_L} = \cos \Phi + 4{c^2}{s^2}({1 - \cos \Phi } )= \cos \Phi {\cos ^2}2\theta + {\sin ^2}2\theta . \end{array}$$

Combining Eq. (4)∼(7), we can get the expression for ΔΦ_L:

(8)$$\tan ({\Delta {\Phi _L}} )= \frac{{ - ({{c^2} - {s^2}} )\sin \Phi }}{{\cos \Phi + 4{c^2}{s^2}({1 - \cos \Phi } )}} = \frac{{ - \sin \Phi \cos 2\theta }}{{\cos \Phi {{\cos }^2}2\theta + {{\sin }^2}2\theta }}.$$

This is the tangent value of the polarization aberration that the birefringence brings into the measurements results. Its amount is modulated by the orientation θ, which makes it not equal to the phase retardance Φ. But for some special values of θ, such as 0 or the integer multiple of π/2, the error is equal to the phase retardance or with a minus sign.

For circularly polarized system, the retardation difference of ${\tilde{E}_{circular}}$’s components is not the polarization error. Two different approaches are proposed in this paper that can be applied to processing the summary matrix ${\tilde{E}_{circular}}$: one is orthogonal decomposition method (ODM), another is inverse transformation method (ITM). Both of the two methods aim at obtaining the error induced into the interference results. The formula derivation process of both methods are shown in the following contents and they can mutually verify the correctness of them two.

The ODM decomposes matrix ${\tilde{E}_{circular}}$ into the sum of two orthogonal matrices as below:

(9)$$\begin{array}{l} \left[ {\begin{array}{c} {2A}\\ {2C} \end{array}} \right] = ({{c^2} + {s^2}\cos \Phi + cs\sin \Phi + j({ - {s^2}\sin \Phi - ({cs - cs\cos \Phi } )} )} )\left[ {\begin{array}{c} 1\\ j \end{array}} \right]\\ \textrm{ } + ({{c^2} + {s^2}\cos \Phi - cs\sin \Phi + j({ - {s^2}\sin \Phi + cs - cs\cos \Phi } )} )\left[ {\begin{array}{c} 1\\ { - j} \end{array}} \right]\\ \textrm{ } = {L_0}{e^{j{\Phi _L}}}\left[ {\begin{array}{c} 1\\ j \end{array}} \right] + {R_0}{e^{j{\Phi _R}}}\left[ {\begin{array}{c} 1\\ { - j} \end{array}} \right], \end{array}$$

where L₀ and R₀ are the modulus of the terms before the two decompositions, Φ_L and Φ_R are the phase retardation terms.

(10)$$\begin{array}{l} \tan ({{\Phi _L}} )= \frac{{ - {s^2}\sin \Phi - ({cs - cs\cos \Phi } )}}{{{c^2} + {s^2}\cos \Phi + cs\sin \Phi }},\\ \tan ({{\Phi _R}} )= \frac{{ - {s^2}\sin \Phi + cs - cs\cos \Phi }}{{{c^2} + {s^2}\cos \Phi - cs\sin \Phi }}. \end{array}$$

The phase difference of the two orthogonal components is the error information we want to know.

(11)$$\begin{array}{l} \tan ({\Delta {\Phi _C}} )= \tan ({{\Phi _R} - {\Phi _L}} )= \frac{{\tan ({{\Phi _R}} )- \tan ({{\Phi _L}} )}}{{1 + \tan ({{\Phi _R}} )\tan ({{\Phi _L}} )}} = \frac{{{N_C}}}{{{D_C}}},\\ {N_C} = 2cs({{s^2} - {c^2}} )({1 - \cos \Phi } )={-} \sin 2\theta \cos 2\theta ({1 - \cos \Phi } ),\\ {D_C} = \cos \Phi {\sin ^2}2\theta + {\cos ^2}2\theta . \end{array}$$

The phase difference can be derived from Eq. (10) and Eq. (11).

(12)$$\tan ({\Delta {\Phi _C}} )={-} \frac{{\sin 2\theta \cos 2\theta ({1 - \cos \Phi } )}}{{{{\cos }^2}2\theta + {{\sin }^2}2\theta \cos \Phi }}.$$

This is the value stands for the polarization aberration in the circularly polarized system. The derivation process of ITM will also be shown to verify the correctness of the result ΔΦ_C. As we know that for a linearly polarized system, the QWP is used to transform the linearly polarized lights into circularly polarized lights to interfere. The concept of ITM is to reverse this transformation. The inverse matrix of QWP is introduced to manipulate the summary matrix ${\tilde{E}_{circular}}$. The inverse matrix of QWP is E_QI, and the resulting matrix is E_IC.

(13)$$\begin{array}{l} {E_{QI}} = \left[ {\begin{array}{cc} 1&j\\ j&1 \end{array}} \right]\textrm{, }{{\tilde{E}}_{circular}} = \left[ {\begin{array}{c} {2[{{c^2} + {s^2}\cos \Phi + j({ - {s^2}\sin \Phi } )} ]}\\ {2[{cs - cs\cos \Phi + j({cs\sin \Phi } )} ]} \end{array}} \right],\\ {{\tilde{E}}_{IC}} = \left[ {\begin{array}{c} {{c^2} + {s^2}\cos \Phi - cs\sin \Phi + j[{cs - cs\cos \Phi - {s^2}\sin \Phi } ]}\\ {cs - cs\cos \Phi + {s^2}\sin \Phi + j[{{c^2} + {s^2}\cos \Phi + cs\sin \Phi } ]} \end{array}} \right] = \left[ {\begin{array}{c} {{P_C}{e^{i{\Phi _{PC}}}}}\\ {{S_C}{e^{i{\Phi _{SC}}}}} \end{array}} \right], \end{array}$$

where P_C and R_C are the modulus of the terms before the two decompositions, and Φ_PC and Φ_SC are the phase retardation terms. The matrix E_IC now has the same function with the summary matrix ${\tilde{E}_{linear}}$ in Eq. (3). The phase difference between the two components of ${\tilde{E}_{IC}}$ is the error induced into the measurement result. The phase retardation Φ_PC and Φ_SC can be derived as:

(14)$$\begin{array}{l} \tan {\Phi _{{P_C}}} = \frac{{cs - cs\cos \Phi - {s^2}\sin \Phi }}{{{c^2} + {s^2}\cos \Phi - cs\sin \Phi }},\\ \tan {\Phi _{{S_C}}} = \frac{{{c^2} + {s^2}\cos \Phi + cs\sin \Phi }}{{cs - cs\cos \Phi + {s^2}\sin \Phi }}. \end{array}$$

Compared with Eq. (10), we can find that:

(15)$$\begin{array}{l} \tan ({{\Phi _L}} )= \tan \left( {{\Phi _{{S_C}}} - \frac{\pi }{2}} \right),\\ \tan ({{\Phi _R}} )\textrm{ = }\tan {\Phi _{{P_C}}}. \end{array}$$

These two methods we proposed here are equivalent, with only a constant phase difference, which can be neglected. The consistency of the results of the two methods also verifies the correctness of the processing approaches.

Until now, we know how the birefringence of large optics affects the interference measurement results, and the polarization aberrations are named as ΔΦ_L and ΔΦ_C respectively for linearly and circularly polarized systems. And for small birefringence, we can get that:

(16)$$\begin{array}{l} \Delta {\Phi _L} \approx{-} \Phi \cos 2\theta ,\\ \Delta {\Phi _C} \approx \frac{{\sin 4\theta {\Phi ^2}}}{4}. \end{array}$$

2.3 Analysis on the correcting method based on rotating the polarization states

The main principle that the birefringence destroys the common path characteristic of Fizeau interferometers is the difference between the effects that the birefringence has on the orthogonally polarized test and reference lights. In order to correct the polarization errors in dynamic interferometers, in this paper, we propose a method measuring the effects on test and reference lights independently, and establishing the relationship between the independent measurement results and the errors obtained in Eq. (8) and Eq. (12). In the system introduced in Fig. 1, two consecutive measurements are conducted with orthogonally polarized incident lights. Based on Eq. (3), the transmitted light can be expressed as

(17)$$\begin{array}{l} {{\tilde{P}}_{ind}} = \left[ {\begin{array}{c} {{c^2} + {s^2}{e^{ - i\Phi }}}\\ {cs - cs{e^{ - i\Phi }}} \end{array}} \right] = ({{c^2} + {s^2}{e^{ - i\Phi }}} )\left[ {\begin{array}{c} 1\\ 0 \end{array}} \right] + ({cs - cs{e^{ - i\Phi }}} )\left[ {\begin{array}{c} 0\\ 1 \end{array}} \right],\\ {{\tilde{S}}_{ind}} = \left[ {\begin{array}{c} {cs - cs{e^{ - i\Phi }}}\\ {{s^2} + {c^2}{e^{ - i\Phi }}} \end{array}} \right] = ({cs - cs{e^{ - i\Phi }}} )\left[ {\begin{array}{c} 1\\ 0 \end{array}} \right] + ({{s^2} + {c^2}{e^{ - i\Phi }}} )\left[ {\begin{array}{c} 0\\ 1 \end{array}} \right]. \end{array}$$

Due to the existence of the birefringence, the incident P linearly polarized light is modulated into ${\tilde{P}_{ind}}$ by large optics and then reflected by 600-mm RF, which can be decomposed into two orthogonal parts shown as the right hand in Eq. (17). And S linearly polarized light becomes ${\tilde{s}_{ind}}$. The reference light, which is reflected by the 100-mm TF, is still P light (or S light) that only interferes with the P component of ${\tilde{P}_{ind}}$ (or S component of ${\tilde{S}_{ind}}$). The phase change introduced by the expander system for the transmitted light is determined by the ratio of the complex factor, which is given as follows:

(18)$$\begin{array}{l} {\Phi ^B}_p = {\tan ^{ - 1}}(\frac{{ - {s^2}\sin \Phi }}{{{c^2} + {s^2}\cos \Phi }}),\\ {\Phi ^B}_s = {\tan ^{ - 1}}(\frac{{ - {c^2}\sin \Phi }}{{{s^2} + {c^2}\cos \Phi }}), \end{array}$$

where the Φ^B_P and Φ^B_S represent the phase retardance introduced when the incident light is P and S light. The large optics become birefringent due to the stress and mechanical force, and the incident linearly polarized light becomes elliptically polarized if its orientation is not parallel or perpendicular to the directions of the fast axis of the birefringent elements. We can know from Eq. (18) that the effect of the birefringence on P and S light is different, which is not a problem in traditional Fizeau interferometer for its common path characteristic. But for the dynamic interferometer based on the polarization phase shifting method, the difference between the effects on P and S light introduces polarization errors into the measurement results. The phase difference ΔΦ^B_L between the Φ^B_P and Φ^B_S can be written as

(19)$$\begin{array}{l} \tan (\Delta {\Phi ^B}_L) = \tan ({\Phi ^B}_S - {\Phi ^B}_P) ={-} \frac{{({{c^2} - {s^2}} )\sin \Phi }}{{\cos \Phi + 2{c^2}{s^2}({1 - \cos \Phi } )}},\\ \tan ({\Phi ^B}_p + {\Phi ^B}_s) ={-} \tan \Phi . \end{array}$$

It is shown in Eq. (19) that the difference ΔΦ^B_L is quite close to the error ΔΦ_L for linearly polarized system in Eq. (8). A simulation is conducted to find out the numerical difference between ΔΦ^B_L and ΔΦ_L. According to the experiments in the previous paper [2], the phase retardation in the 600-mm interferometer is about λ/10 level.

As we can see from Fig. 3 that for retardation Φ up to π/4, the maximum phase difference between the ΔΦ^B_L and ΔΦ_L is about 0.0016π (0.008λ), which can be neglected compared with the large flat under test with 0.2π rad (λ/10 PV). This is the principle that we use to correct the polarization errors brought by stress-induced birefringence through the measured wavefronts switching the polarization state of the incident light. Another simulation is conducted to find out how the phase difference changes with varying Φ and θ. As shown in Fig. 4 that the amount of the difference becomes larger when Φ increases, but the maximum value is different according to different values of θ. In Fig. 4(b) we can see that, at given value of Φ, regular changes can be observed when θ changes at a period of about π/4. Another interesting discovery in Eq. (19) is that the summary of the two effect results, ΔΦ^B_P and ΔΦ^B_S, only depends on the phase retardance Φ, and is equal to the phase retardance regardless of the orientation varying among the large optics.

Fig. 3. The simulation results of the phase difference between ΔΦ_BL and ΔΦ_L with a diameter of 600 mm. (a) the phase retardation distribution of Φ; (b) shows the 2D distribution of the difference between ΔΦ^B_L, and ΔΦ_L with Φ varies from –π/4 to π/4 in (a) and random θ from –π/2 to π/2; (c) shows the 1D distribution of the difference.

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Fig. 4. The simulation results of the phase difference between ΔΦ^B_L and ΔΦ_L. (a) shows the phase difference varying with Φ at different constant θ. (b) shows the phase difference varying with θ at different constant Φ.

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The condition for a circularly polarized system is more complicated. The same processing steps are applied, and we get:

(20)$$\begin{array}{l} {{\tilde{L}}_{ind}} = {W_s}\tilde{L} = \left[ {\begin{array}{c} {A + jB}\\ {C + jD} \end{array}} \right] = ({L{A_{11}} + jL{B_{11}}} )\left[ {\begin{array}{c} 1\\ j \end{array}} \right] + ({L{A_{22}} + jL{B_{22}}} )\left[ {\begin{array}{c} 1\\ { - j} \end{array}} \right],\\ {{\tilde{R}}_{ind}} = {W_s}\tilde{R} = \left[ {\begin{array}{c} {A - jB}\\ {C - jD} \end{array}} \right] = ({R{A_{11}} + jR{B_{11}}} )\left[ {\begin{array}{c} 1\\ j \end{array}} \right] + ({R{A_{22}} + jR{B_{22}}} )\left[ {\begin{array}{c} 1\\ { - j} \end{array}} \right], \end{array}$$

where ${\tilde{L}_{ind}}$ and ${\tilde{R}_{ind}}$ represents the matrix of left-handed and right-handed circularly polarized lights after being manipulated by W_S. Both ${\tilde{L}_{ind}}$ and ${\tilde{R}_{ind}}$ can be decomposed into the combination of two parts with orthogonal polarization states. And the coefficients for ${\tilde{L}_{ind}}$ in Eq. (20) can be calculated as:

(21)$$\left\{ {\begin{array}{l} {L{A_{11}} = \frac{{1 + \cos \Phi }}{2},}\\ {L{B_{11}} ={-} \frac{{\sin \Phi }}{2},}\\ {L{A_{22}} = \frac{{({{c^2} - {s^2}} )({1 - \cos \Phi } )- 2cs\sin \Phi }}{2},}\\ {L{B_{22}} = \frac{{({{c^2} - {s^2}} )\sin \Phi + 2cs({1 - \cos \Phi } )}}{2}.} \end{array}} \right.\textrm{ }$$

And for ${\tilde{R}_{ind}}$ can be calculated as:

(22)$$\left\{ {\begin{array}{l} {R{A_{11}} = \frac{{({{c^2} - {s^2}} )({1 - \cos \Phi } )+ 2cs\sin \Phi }}{2},}\\ {R{B_{11}} = \frac{{({{c^2} - {s^2}} )\sin \Phi - 2cs({1 - \cos \Phi } )}}{2},}\\ {R{A_{22}} = \frac{{1 + \cos \Phi }}{2},}\\ {R{B_{22}} ={-} \frac{{\sin \Phi }}{2}.} \end{array}} \right.$$

Same as the analysis for linearly polarized system, the reference light only interferes with the component that has the same polarization state. And the effects that birefringence has on circularly polarized lights can be expressed as:

(23)$$\begin{array}{l} {\Phi ^B}_{Left} = {\tan ^{ - 1}}(\frac{{L{B_{11}}}}{{L{A_{11}}}})\textrm{ = }{\tan ^{ - 1}}(\frac{{ - \frac{{\sin \Phi }}{2}}}{{\frac{{1 + \cos \Phi }}{2}}})\textrm{ = } - \frac{\Phi }{2},\\ {\Phi ^B}_{Right} = {\tan ^{ - 1}}(\frac{{R{B_{22}}}}{{R{A_{22}}}})\textrm{ = }{\tan ^{ - 1}}(\frac{{ - \frac{{\sin \Phi }}{2}}}{{\frac{{1 + \cos \Phi }}{2}}})\textrm{ = } - \frac{\Phi }{2}. \end{array}$$

The two effects Φ^B_Left and Φ^B_Right has the same value means that we cannot correct the errors induced by birefringence in a circularly polarized dynamic interferometer by the proposed method in this paper. A similar approach described in the previous work is more suitable for the correction in the circularly polarized system [2].

The following correcting procedure is shown only for linearly polarized systems. For the independent measurement conducted in the system shown in Fig. 1, the interferometric measurement results contain more information including the 600-mm RF surface distribution, the defocus of the collimator, and the 100-mm TF surface distribution. The total phase distribution is calculated through phase shifting interferograms with a wavelength tuning laser.

(24)$$\begin{array}{l} {M_P} = {\Phi _{Co}} + {\Phi _T} + {\Phi _R} + {\Phi ^B}_P,\\ {M_S} = {\Phi _{Co}} + {\Phi _T} + {\Phi _R} + {\Phi ^B}_S, \end{array}$$

where M_P and M_S represent the measurement wavefront acquired by the four-step phase shifting method when the incident light is P and S linearly polarized respectively, Φ_Co, Φ_T, and Φ_R represents the wavefront contribution of the collimator, TF, and RF. One of the advantages of this measurement configuration is that the expander system under test in the interference module needs no movement during the two measurement procedures because the polarization state is switched in the source module in Fig. 1. And as a result, the distribution of Φ_Co, Φ_T, and Φ_R is the same in M_P and M_S. By subtracting the results of the two measurements, the relationship between the effects that the birefringence has on P light and S light can be obtained, and this value is also the key data for correcting the measurement results of the dynamic interferometer. The subtracting result is shown as follows:

(25)$${M_{Diff}} = {M_S} - {M_P} = {\Phi ^B}_S - {\Phi ^B}_P = \Delta {\Phi _L}.$$

Here, we transform the polarization errors that the birefringence brings into the interferometric result into the phase distribution difference between two individual tests with two polarized states. The movement of large optics and the consumption of the symmetry of the birefringence are not needed.

3. Experiment

The measurement schematic diagram is shown in Fig. 1. The main difference between the interferometer module and the ordinary Fizeau interferometer is that the beam expander system is treated as a sample, and is placed in the interference cavity composed of 100-mm TF and 600-mm RF to be tested. The effect of stress-induced birefringence can be obtained by subtracting the two independent measurement wavefronts of the sample, thereby correcting the polarization aberration of the large aperture dynamic interferometer.

The interferograms of two independent measurements and the calculated phase maps are shown in Fig. 5. The ring fringes come from the defocus of the expander system. It should be noted that the measurement configuration is based on the initial dynamic interferometer system and elements change are made as little as possible. The coherent noise in the interferograms is obvious since a diffuser is not used in the system which destroys the polarized state of the test lights in a polarization based dynamic interferometer. However, when a wavelength tuning laser with good coherence is used, the coherent noise is not negligible. A Zernike fitting algorithm with 37 Zernike polynomials is applied to eliminate the influence brought by coherent noise, considering that the noise is distributed in the high-frequency region while the birefringence is low frequency. The residual wavefront subtracting the two wavefronts is shown in Fig. 6(b).

Fig. 5. Two sets of measurement results: (a) the set up with P incident light; (b) the interferogram with P incident light; (c) the phase map with P incident light; (d) the set up with S incident light; (e) the interferogram with S incident light; (f) the phase map with S incident light;

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Fig. 6. (a) the measurement result with polarization aberrations obtained via the dynamic interferometer based on the polarization phase shifting method; (b) the difference phase map of P light and S light measurement results after Zernike fitting; (c) the phase map after correction by combining the two results in (a) and (b).

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Figure 6(a) is the result of the dynamic interferometer based on the polarization phase shifting method with aberrations [2], Fig. 6(b) is the residual wavefront after Zernike fitting, which is also the phase retardance ΔΦ^B_L in Eq. (19), Fig. 6(c) is the measurement result after correction. In order to make it more obvious, only the central area with a diameter of 400 mm is shown in Fig. 6(a) and Fig. 6(b). The two dimensional result of the phase retardance can be easily obtained and then used for the polarization errors calibration. It can be found that the area affected by the stress-induced birefringence in the uncorrected result is significantly compensated.

The calibrated dynamic result of the 600-mm RF is compared with the directly interferometric result via the wavelength tuning method as shown in Fig. 7, in which case a reliable and mature phase-shifting technique is applied. The RMS difference is 0.011λ. The Zernike comparison between the calibration result and wavelength tuning method is also shown in Fig. 8. The comparison of the results before and after the correction procedure shows good removal of the polarization aberrations in Figs. 6(a) and (c). However, there are still some residual aberrations can be observed in the difference result in Fig. 7(c) and the Zernike coefficients in Fig. 8. The main reason for the residual aberrations is that it takes two measurements of the transmitted wavefront of the expander system and the system instability brings environmental vibration and airflow disturbance into the wavefront results. As mentioned in Eq. (24) and (25) that this method assumes that the system remains stable. A better result can be obtained if better control of the environment is realized. The elimination of the polarization aberrations in Fig. 6(c) and the comparison between the two methods proves the correctness and the accuracy of this calibration principle.

Fig. 7. (a) the measurement result after calibration in a dynamic interferometer; (b) the measurement result via wavelength tuning method, (c) the difference between the results in (a) and (b).

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Fig. 8. The Zernike polynomials comparison between wavelength tuning method and interferometry method proposed in this work.

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4. Analysis

The birefringence of large optics is inevitable and annealing times of several months are necessary to reach 4 nm/cm birefringence. Lower target values would increase this time considerably. The problem is what amount of birefringence is acceptable in interferometers and whether it can be corrected. The analysis and simulation results provides groundwork and theoretical guide for the application of interferometers.

4.1 Errors arising from birefringence in polarization based dynamic interferometers

A large number of formula derivations have been completed above. Simulation results are presented below to illustrate the error arising from the birefringence in polarization based dynamic interferometers more intuitively.

From Eq. (16), we can find that the error ΔΦ_L is proportional to the phase retardation Φ while ΔΦ_C is proportional to the square of Φ, which means the error induced by birefringence is always smaller for a circularly polarized system when Φ is small. The difference of ΔΦ_L and ΔΦ_C is shown in Fig. 9.

Fig. 9. Blue line: the errors in the linearly polarized system; red line: the errors in circularly polarized system.

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As illustrated in Fig. 9, for the same retardation Φ=0.5, the error in the linear system ΔΦ_L is larger than ΔΦ_C as θ varies from –π/2 to π/2. As a result, a circularly polarized system is recommended for dynamic interferometers with small birefringence and no correction procedure is needed.

Another simulation is conducted to estimate how ΔΦ_L and ΔΦ_C changes along with the phase retardance Φ at certain angles of θ. We want to theoretically know how much stress birefringence affects the measurement results of a dynamic interferometer, and to see how much it will affect the results.

As we can see from Fig. 10, for small θ, the errors ΔΦ_L and ΔΦ_C shows cyclical changes when Φ changes. Both the maximum values of ΔΦ_L and ΔΦ_C can be larger than 0.2π (λ/10), which definitely cannot be ignored in interferometric measurements. For a large optics with a diameter up to 600 mm or even 800 mm, a total retardance less than λ/30 is required if we want to ignore the errors, which means a 2 nm/cm birefringence is needed for a thickness of 10cm. This places extremely high demands on the annealing process. And this is why a correction procedure is needed to eliminate the birefringence influence.

Fig. 10. The changes of ΔΦ_L and ΔΦ_C along with the phase retardance of the birefringence

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4.2 Errors in a Twyman-Green interferometer

The polarization based Twyman-Green dynamic interferometer is usually used in a spherical mirror measurement. But in real inspection environment, a folded mirror, usually coted, is often inserted before the test surface to make the light path more compact. Such a mirror will cause some depolarization effect into the test light. At oblique incidence, the reflectivity, transmittance and phase shift of P light and S light are generally different, resulting in that the polarization state of incident light cannot be maintained in transmitted light and reflected light [26]. In this case, we characterize the phenomenon with a waveplate. The system configuration is different from a Fizeau type interferometer and a different result is obtained.

As shown in Fig. 11(a) that for a Twyman-Green dynamic interferometer, the input light is divided into P and S polarized lights and reaches reference mirror and test mirror separately after pass through the QWP. The problem occurs if the folded mirror changes the polarization state of the incident light. Only test light is depolarized and the reference light stays circularly polarized. This procedure can also be expressed using Jones matrix as follows:

(26)$$\begin{array}{l} {{\tilde{E}}_R} = \left[ {\begin{array}{cc} 0&0\\ 0&1 \end{array}} \right]{E_Q}{E_Q}{{\tilde{E}}_P} = \left[ {\begin{array}{c} 0\\ 1 \end{array}} \right],\\ {{\tilde{E}}_T} = \left[ {\begin{array}{cc} 1&0\\ 0&0 \end{array}} \right]{E_Q}{W_s}{E_Q}{{\tilde{E}}_S} = \left[ {\begin{array}{c} { - j({1 + {e^{\textrm{ - }j\Phi }}} )}\\ 0 \end{array}} \right], \end{array}$$

where the reference light become S linearly polarized after being reflected by reference mirror and pass through the QWP and PBS again, while the test light becomes P linearly polarized but with an extra phase term. Both reference and test lights will pass through a QWP and polarizer arrays to interfere, which is the same case in Section 3.2. We can obtain the errors induced by the depolarization effect of the large mirror with the same calculation approach. And the result is expressed as:

(27)$$\Delta {\Phi _{TG}} = 0 - \arctan \left( {\frac{{ - \sin \Phi }}{{1 + \cos \Phi }}} \right) = \frac{\Phi }{2}.$$

Fig. 11. (a) Configuration of the Twyman-Green interferometer; (b) the interference cavity when a Fizeau type dynamic interferometer is used to measure the transmitted wavefront of a large plate or optical system with birefringence.

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In this case, the polarization states of the test and reference light are different. The test light becomes left-handed circularly polarized light and is manipulated by the mirror while the reference light stays unaffected. A similar case is that when the Fizeau type dynamic interferometer is used to measure the transmitted wavefront of a large flat, as shown in Fig. 11 (b). Here, the flat is placed between the interference cavity consists of the RF and TF. The reference light is directly reflected by the RF and the test light pass through the flat under test with birefringence and then is reflected by TF. The birefringence only affects the test light, which is S polarized light. The same procedure is repeated to calculate the measurement errors. The Jones matrix of the reference and test lights are:

(28)$$\begin{array}{l} {{\tilde{E}}_R} = {{\tilde{E}}_P} = \left[ {\begin{array}{c} 1\\ 0 \end{array}} \right],\\ {{\tilde{E}}_T} = {W_s}{{\tilde{E}}_S} = \left[ {\begin{array}{c} {cs({1 - {e^{\textrm{ - }j\Phi }}} )}\\ {{s^2} + {c^2}{e^{\textrm{ - }j\Phi }}} \end{array}} \right]. \end{array}$$

And the error in this case is:

(29)$$\Delta {\Phi _{TW}} = \frac{{ - c({c + s} )\sin \Phi }}{{1 + {c^2}\cos \Phi + cs ({{c^2} - {s^2}} )\cos \Phi }}.$$

5. Conclusion

In conclusion, we present a method that is convenient for practical application for the correction of birefringence distribution of large optics with diameters up to 800 mm by detecting and calculating the wavefront difference when switching the polarization state of the incident light. The theoretical analysis on the effects brought by birefringence on the interferometric results is given, and the principle of the correcting method proposed in this paper is derived. The measurement configuration is transformed from the dynamic interferometer, where a 100-mm TF is inserted to form an interference cavity with a 600-mm RF, and the expander system is tested as a whole without movements. This method is proved to be a convenient and reliable technique to acquire the polarization aberrations that the birefringence brings to the interferometry results in the linearly polarized system, and experiments are conducted to correct the polarization aberration in the large aperture dynamic interferometer. The corrected result is compared with the result obtained through the wavelength tuning method and the accuracy of the correction is verified.

Funding

National Natural Science Foundation of China (61975081, 62175112).

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.

References

1. C. Wei, J. Ma, L. Chen, J. Li, F. Chen, R. Zhu, R. Guo, C. Yuan, J. Meng, Z. Wang, and D. Gao, “Null interferometric microscope for ICF-capsule surface-defect detection,” Opt. Lett. 43(21), 5174–5177 (2018). [CrossRef]

2. X. Miao, J. Ma, Y. Yu, J. Li, C. Wei, R. Zhu, L. Chen, C. Yuan, Q. Wang, and D. Zheng, “Modelling and correction for polarization errors of a 600 mm aperture dynamic Fizeau interferometer,” Opt. Express 28(22), 33355–33370 (2020). [CrossRef]

3. D. Wang, C. Wang, X. Tian, H. Wu, J. Liang, and R. Liang, “Snapshot phase-shifting lateral shearing interferometer,” Opt. Lasers Eng. 128, 106032 (2020). [CrossRef]

4. X. Tian and R. Liang, “Snapshot phase-shifting diffraction phase microscope,” Opt. Lett. 45(12), 3208–3211 (2020). [CrossRef]

5. T. Doehring, P. Hartmann, H. F. Morian, and R. Jedamzik, “Optical glasses and glass ceramics for large optical systems,” Proc. SPIE 4842, 56 (2003). [CrossRef]

6. L. Sun and S. Edlou, “Low-birefringence lens design for polarization sensitive optical systems,” Proc. SPIE 6289, 62890H (2006). [CrossRef]

7. M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42(16), 3009–3017 (2003). [CrossRef]

8. J. Yu and C. Cui, “Residual stress measurement of glass based on Muller matrix ellipsometer,” Proc. SPIE 1134, 113430O (2019). [CrossRef]

9. X. Zhang, J. Miao, D. Liu, J. Zhu, J. Hu, O. Test, and M. Technology, “Phase error analysis for birefringent homogeneity measurements of largescale optical materials with spherical mirror,” Proc. SPIE 8417, 84170B (2012). [CrossRef]

10. J. Lin, “Simultaneous measurement of optical rotation angle and retardance,” Opt. Commun. 281(5), 940–947 (2008). [CrossRef]

11. C. Bei, Z. Xuejie, L. Cheng, and J. Zhu, “Full-field stress measurement based on polarization ptychography,” J. Opt. 21(6), 065602 (2019). [CrossRef]

12. C. Lane, D. Rode, and T. Rösgen, “Optical characterization method for birefringent fluids using a polarization camera,” Opt. Lasers Eng. 146, 106724 (2021). [CrossRef]

13. R. Fang, A. Zeng, L. Zhu, L. Liu, and H. Huang, “Simultaneous measurement of retardation and fast axis angle of eighth-wave plate in real time,” Opt. Commun. 285(24), 4884–4886 (2012). [CrossRef]

14. L. Zhang, K. Xia, L. Zhu, A. Zeng, and H. Huang, “Direct Stress Birefringence Distribution Measurement in Lens Using Refractive Index Matching Liquid,” IEEE Photonics Technol. Lett. 33(13), 637–640 (2021). [CrossRef]

15. L. Liu, A. Zeng, B. Chen, F. Li, L. Zhu, and H. Huang, “Simultaneous measurement of small birefringence magnitude and direction in real time,” Opt. Lasers Eng. 53, 19–24 (2014). [CrossRef]

16. Y. Otani, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33(5), 1604 (1994). [CrossRef]

17. S. Sircar and K. Bhattacharya, “Measurement of birefringence using polarization phase-shifting Mach–Zehnder interferometer,” Opt. Eng. 54(11), 113112 (2015). [CrossRef]

18. H. K. Teng, C. Chou, and H. F. Chang, “Polarization-shifting interferometry on two-dimensional linear birefringent parameters measurement,” Opt. Commun. 224(4-6), 197–204 (2003). [CrossRef]

19. A. Berger, V. Nercissian, K. Mantel, and I. Harder, “Evaluation algorithms for multistep measurement of spatially varying linear polarization and phase,” Opt. Lett. 37(19), 4140 (2012). [CrossRef]

20. S. Rothau, C. Kellermann, V. Nercissian, A. Berger, K. Mantel, and N. Lindlein, “Simultaneous measurement of phase and local orientation of linearly polarized light: implementation and measurement results,” Appl. Opt. 53(14), 3125–3130 (2014). [CrossRef]

21. K. H. Chen, C. H. Lin, and P. C. Liu, “An interferometric method for simultaneously determination the phase retardation and fast-axis azimuth angle of a wave plate,” J. Mod. Opt. 67(11), 992–997 (2020). [CrossRef]

22. N. Ghosh and K. Bhattacharya, “Polarization phase-shifting interferometric technique for complete evaluation of birefringence,” Appl. Opt. 50(15), 2179–2184 (2011). [CrossRef]

23. J. L. Rouke and D. T. Moore, “Birefringence in gradient-index rod lenses: a direct measurement method and interferometric polarization effects,” Appl. Opt. 40(28), 4971–4980 (2001). [CrossRef]

24. S. Rothau, X. Rao, and N. Lindlein, “Simultaneous measurement of phase transmission and linear or circular dichroism of an object under test,” Appl. Opt. 58(7), 1739–4856 (2019). [CrossRef]

25. E. R. Cochran and C. Ai, “Interferometric stress birefringence measurement,” Appl. Opt. 31(31), 6702 (1992). [CrossRef]

26. K. Rabinovitch, “The Application Of Thin Films Technique To Compensate Polarization Effects On Total Internal Reflection,” Proc. SPIE 1038, 337 (1988). [CrossRef]

Problems on polarization aberrations in large aperture dynamic interferometry based on the polarization phase shifting technique

Abstract

1. Introduction

2. Configuration and principle

2.1 Measurement configuration transformed from dynamic Fizeau interferometer

2.2 Analysis on the birefringence effects on different incident lights

2.3 Analysis on the correcting method based on rotating the polarization states

3. Experiment

4. Analysis

4.1 Errors arising from birefringence in polarization based dynamic interferometers

4.2 Errors in a Twyman-Green interferometer

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (29)

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