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Virtual-real combination Ritchey-Common interferometry

Yiming Liu, Yao Hu, Shaohui Zhang, Shen Huang, Jinpeng Li, Limin Yan, Jiahang Lv, Zhen Wang, Xu Chang, and Qun Hao

Open Access Open Access

Abstract

Large optical flats play a remarkable role in advanced large-aperture optical systems and the testing of the surface shape error is indispensable for the fabrication. The widely adopted Ritchey-Common test for large optical flats will fail without the rigorous test configurations including a large F/# prerequisition and a flat-to-interferometer distance invariance. A virtual-real combination Ritchey-Common interferometry is proposed to avoid the large F/# prerequisition by accurately modelling the optical path in a virtual interferometer. Furthermore, a virtual-real combination iterative algorithm is proposed in this method to break the flat-to-interferometer distance invariance. Measurement experiments for 100 mm and 422 mm aperture flats were performed to demonstrate the feasibility of this method. Compared with a direct testing in a standard Zygo interferometer, the peak to valley (PV) and root mean square (RMS) errors were less than 0.1 λ and 0.01 λ (λ=632.8 nm), respectively, in different Ritchey angles and flat-to-interferometer distances. Further numerical simulations demonstrate that RMS errors for various Zernike aberrations in arbitrary F/# are less than 0.01 λ. This method can break the distance invariance restriction and achieve high accuracy with an arbitrary F/#, thus providing substantial freedom in the design of test configurations to accommodate various test scenarios.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, interest in advanced optical systems, especially in astronomical optics, high-power laser, and military field, is increasing [1–4]. The aperture of advanced optical systems has been increased to improve the performance of the system. Large optical flats are indispensable in large optical systems. For example, the M3 mirror in the Thirty Meter Telescope is a 3.5 m × 2.5 m elliptical flat mirror [5]. Meanwhile, the M4 mirror in the European Extremely Large Telescope is a Ф2.4 m flat mirror [6,7]. The National Ignition Facility contains 7648 large-aperture optics, and the aperture of the laser mirror ranges from 0.5 m to 1.0 m [8]. The Navy of U.S. developed a Mid-InfraRed Advanced Chemical Laser and a beam control subsystem with a 1800 mm aperture output mirror on tactical servo mount [4]. The surface shape of large optical flats is the guarantee of performance of the optical system. Therefore, the development of a test method for large optical flats is of great significance.

Interferometry is an efficient optical method, which can accurately test the mirror surface, and commercial interferometers play an important role in the testing and fabrication of optical flats [9]. The aperture of commercial interferometer should be larger than the test flats to cover full aperture surface shape. Nevertheless, the development of large aperture interferometers is very difficult and expensive. At present, the ZYGO 36-inch interferometer is the largest commercial interferometer [10]. And it can hardly meet the testing requirements of large optical flats in meter size.

The testing methods for large optical flats mainly include sub-aperture stitching interferometry, pentaprism scanning test, oblique incidence interferometry, and Ritchey–Common test.

Sub-aperture stitching interferometry [11,12] divides the entire surface of the test flat into several sub-apertures. The surface shape of each sub-aperture is tested by a small-aperture interferometer mounted on a high-precision scanning mechanism, and then full aperture surface shape of the test flat is reconstructed by stitching the divided sub-apertures. Sub-aperture stitching interferometry does not need a large reference flat and can reduce the test cost. Nevertheless, the number of sub-apertures will be greatly increased when the aperture of test flat is much larger than the reference flat in interferometer. In the process of sub-aperture stitching, errors will be transmitted and accumulated. And during the time-consuming scanning, the test results will be strongly affected by the precision of the scanning mechanism and environmental factors, such as temperature instability, vibration, and air disturbance. Pentaprism scanning test [13] combines a pentaprism, an auto-collimator and also a high-precision scanning mechanism to test the angle change in the normal direction of each point on the surface under test. This method can achieve high accuracy in the test of low-order aberrations but still suffers from the same problems, i.e., insufficient mechanical scanning accuracy and long test time, as the sub-aperture stitching interferometry. The oblique incidence interferometry [14] utilized a well-polished flat mirror as the return flat. The collimated beam emitted by the interferometer is incident on the test flat obliquely and then reflected back to the interferometer by the return flat. Different from the above two scanning methods, the oblique incidence interferometry can realize transient full-field measurement. However, this method can only extend the testing area in one direction, which means this method cannot break the aperture restriction in interferometry completely.

Ritchey–Common test [15] is a special oblique incidence interferometry. In Ritchey–Common test, the return flat is replaced by a well-polished concave spherical mirror to completely break the aperture restriction. The spherical mirror is relatively easy to fabricate and to test with an accuracy of a few hundredths of a wavelength without the use of large auxiliary optics [15]. Thus, this method has been widely used to measure large flat mirrors in meter size [16–21]. The configuration of Ritchey-Common test consists of an interferometer, a test flat and a spherical mirror. The principle of the Ritchey-Common test includes the following requirements. The divergent beam emitted by the interferometer is obliquely incident on the test flat, and the incident angle of the ray along the main optical axis is called the Ritchey angle and denoted as θ. The beam is then reflected back to the interferometer by the spherical mirror, and the curvature center of which should be coincident with the focal point of the interferometer. The above requirements pose the following challenges to the implementation of the Ritchey-Common test.

The first challenge lies in the compression of the test data originally in a circular area into an approximately elliptical aperture because of the oblique incidence of the divergent beam of light on the flat. Such a drawback complicated the mapping relationship between the flat mirror and system pupil. Shu [15] derived the influence function matrix (IFM) between the Zernike polynomial representations of the flat mirror and the system wavefront at the pupil that obtained in the Ritchey–Common test configuration to retrieve the surface shape under test (SSUT). The IFM method has been developed and widely used in the Ritchey–Common test thus far. In 2001, Han [16] updated the mathematical expressions derived by Shu and measured the large astronomical flats with the IFM method. In 2007, Yuan [17] applied the IFM method in the Large Sky Area Multi-Object Fiber Spectroscopic Telescope to test hexagonal submirrors of reflecting Schmidt plate. In 2020, Kim [19] tested a Ф1.2 m flat mirror with a smaller reference sphere by combining the IFM method and a dual sub-aperture stitching method to reduce the cost. In the IFM method, the derivation of the IFM is remarkably complicated and is simplified by a large F/# prerequisition, i.e., the ratio between the distance between focal point and test flat and the aperture of the surface under test, as an approximate precondition $({\textrm{tan}\theta /({2\ast \textrm{F}/\# } )\ll 1} )$. Thus, a long optical path up to tens of meters is generally required given the aperture of the test is up to meter-size. However, the large F/# prerequisition cannot be easily realized in real tests due to the limited measurement space, thus decreasing the accuracy. Even if a large F/# is satisfied, then the long optical path will have stringent requirements on the experimental environment, and uncontrollable factors, such as airflow and vibration, will cause errors in the test results.

The second challenge is the defocus of spherical mirror. The misalignment of spherical mirror is inevitable and defocus aberration will be introduced into the system wavefront. Nevertheless, power of the test flat will also appear as defocus aberration in the system wavefront. To distinguish the effects of the power and misalignment of spherical mirror, Doerband [20] proposed a mathematical algorithm to eliminate the influence of the defocus by changing the Ritchey angle and applied it in the test of the tertiary mirrors of Very Large Telescope. Zhu [21] further developed this method and analyzed the effect of Ritchey angle accuracy. This method can eliminate the influence of defocus, nevertheless, this method requires the distance between the interferometer and the flat keeps invariant during the test with two Ritchey angles, which places a high demand on the mounting system and sufficiently increases the test cost. The data processing of this method is also intricate, and the interpolation error will be introduced in the test results, thus limiting its application. The above two challenges place rigorous demands on the test configuration.

In this study, a virtual-real combination Ritchey-Common interferometry (VRCRI) is proposed to provide remarkable freedom in the design of test configurations to accommodate various test scenarios. A virtual interferometer is built to characterize the mapping relationship between the flat mirror and system pupil, as well as the influence of defocus by accurately modelling the Ritchey-Common optical path. Virtual interferometer is a tool that proposed in our previous work to accurately describe the optical systems and eliminate system errors [22,23], and it is adopted in this study by combining the principle of Ritchey-Common test. The large F/# prerequisition in the conventional method is avoided by the accurate virtual interferometer modeling. A virtual-real combination iterative algorithm is proposed to calculate the defocus of spherical mirror and remove the restriction of distance invariance. A set of verification experiments and numerical simulations are conducted to validate the feasibility of this method.

2. Theory of VRCRI

The following section describes the theory of VRCRI in two parts: 1) modeling of virtual interferometer, and 2) virtual-real combination iterative algorithm.

2.1 Modeling of virtual interferometer

The first part is to model a virtual interferometer. The basic principle of the VRCRI is shown in Fig. 1. The red part in the left is a real interferometer, and the blue part in the right is the virtual interferometer. The red part shows the principle of Ritchey-Common test. The divergent beam emitted by the interferometer is incident on the test flat obliquely. The beam is then reflected back to the interferometer by a spherical mirror. The focal point of the interferometer coincides with the curvature center of the spherical mirror. The test flat forms an angle θ with the main optical axis, which is called the Ritchey angle. r is half of the clear aperture of the test flat; d1 is the distance between the focal point of the interferometer and the test flat; d2 is the distance between the test flat and the spherical mirror; Δs is the defocus of spherical mirror; R is the curvature radius of the spherical mirror; SW is the system wavefront that collected by the interferometer. The optical path parameters of real interferometer, i.e., d1, r, θ, R and SW, will be accurately measured and then adopted in the virtual interferometer. The virtual interferometer is identical with the real interferometer, and then the SSUT and Δs can be calculated in the virtual interferometer by our method. In our implementation, the internal error and structure of the interferometer is ignored to simplify the model. The focal point of the interferometer is set as the shared initial position and preset end position of each ray in the virtual interferometer. Then the virtual interferometer is established by ray-tracing method based on an accurate full-field optical path model derived from the principle of Ritchey-Common test as follows.

 figure: Fig. 1.

Fig. 1. Principle of VRCRI.

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The system wavefront SW that collected by the interferometer comprises the following three parts

$$SW = {W_{\textrm{sph}}} + {W_{\textrm{def}}}\textrm{ + }{W_{\textrm{SSUT}}}. $$

The first part Wsph is introduced by the surface shape of the spherical mirror;

The second part Wdef is introduced by the defocus of spherical mirror;

The third part WSSUT is introduced by the SSUT.

  • a. Wsph
The error introduced by the surface shape of spherical mirror should be calibrated as a system error in the test [20]. After the point-to-point subtraction, the spherical mirror can be considered to be a perfect sphere surface and the first part Wsph will be neglected in the following mathematical derivation.
  • b. Wdef
To study the influence of the defocus, a rectangular coordinate system is established by neglecting the test flat. As shown in Fig. 2 (a) and (b), Fig. 2 (a) is the principle of full-field optical path in Ritchey-Common test; Fig. 2 (b) is the principle of defocus by neglecting the SSUT; the focal point in Fig. 1 is set as the origin; x1 and y1 represent the system pupil coordinate; x2 and y2 represent the test flat coordinate; z-axis is set as the main optical axis; Δs is the defocus of spherical mirror; point a (x1, y1) is set as an arbitrary point on the spherical mirror; L1 is the optical path of incident light, and L2 is the optical path of reflected light. The incident light is supposed to backtrack in ideal condition; however, it will deviate from the ideal path due to the defocus of spherical mirror. The red dotted line indicates the ideal position with no defocus, and the blue line indicates the real position.

 figure: Fig. 2.

Fig. 2. Schematic sketch of the Ritchey-Common test (a) for the full-field optical path model and (b) for the analysis of defocus.

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The surface shape formula of spherical mirror can be expressed as

$$ x_{1}^{2}+y_{1}^{2}+\left(z_{1}-\Delta s\right)^{2}=R^{2}, $$
and
$$ z_{1}=\sqrt{R^{2}-x_{1}^{2}-y_{1}^{2}}+\Delta s. $$

The optical path of the incident light is

$$ L_{1}=\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}=\sqrt{x_{1}^{2}+y_{1}^{2}+\left(\sqrt{R^{2}-x_{1}^{2}-y_{1}^{2}}+\Delta s\right)^{2}}, $$
and the optical path of reflected light can be calculated by its length along x, y and z axis as:
$${L_2} = \sqrt {{{\left( {{z_1} \ast {{\left( {\frac{{\partial x}}{{\partial z}}} \right)}_{{L_2}}}} \right)}^2} + {{\left( {{z_1} \ast {{\left( {\frac{{\partial y}}{{\partial z}}} \right)}_{{L_2}}}} \right)}^2}\textrm{ + }{z_1}^2} = {z_1} \ast \sqrt {{{\left( {\frac{{\partial x}}{{\partial z}}} \right)}_{{L_2}}}^2 + {{\left( {\frac{{\partial y}}{{\partial z}}} \right)}_{{L_2}}}^2\textrm{ + 1}}, $$
where ${\left( {\frac{{\partial x}}{{\partial z}}} \right)_{{L_2}}}$ and ${\left( {\frac{{\partial y}}{{\partial z}}} \right)_{{L_2}}}$ are the partial derivative of x and y along z direction in L2. According to the geometrical relationship of optical set up and Fresnel formula, we can get
$$\left\{ \begin{array}{l} {\left( {\frac{{\partial x}}{{\partial z}}} \right)_{{L_2}}}\textrm{ = }2\arctan (\frac{{{x_1}}}{{{z_1} - \Delta s}}) - \arctan (\frac{{{x_1}}}{{{z_1}}})\\ {\left( {\frac{{\partial \textrm{y}}}{{\partial z}}} \right)_{{L_2}}}\textrm{ = }2\arctan (\frac{{{y_1}}}{{{z_1} - \Delta s}}) - \arctan (\frac{{{y_1}}}{{{z_1}}}) \end{array} \right., $$
and Eq. (5) can be written as:
$${L_2} = {z_1} \ast \sqrt {{{(2\arctan (\frac{{{x_1}}}{{{z_1} - \Delta s}}) - \arctan (\frac{{{x_1}}}{{{z_1}}}))}^2} + {{(2\arctan (\frac{{{y_1}}}{{{z_1} - \Delta s}}) - \arctan (\frac{{{y_1}}}{{{z_1}}}))}^2} + 1}. $$

The system wavefront introduced by the defocus can be calculated by Eq. (8)

$${W_{\textrm{def}}}({{x_1},{y_1}} )= {L_1} + {L_2} - L, $$
where L is the optical path of main optical light
$$L = 2R + 2\Delta s = 2({{d_1} + {d_2}} ), $$
and then the Wdef can be calculated by the Eqs. (2)–(9).
  • c. WSSUT
The WSSUT can be calculated with Eq. (10) [14]
$${W_{\textrm{SSUT}}}({{x_1},{y_1}} )= 4\cos I \ast S({{x_2},{y_2}} ), $$
where S (x2, y2) is SSUT; the transform relationship between (x1, y1) and (x2, y2) is [21]
$$\left\{ \begin{array}{l} {x_2} = \frac{{{d_1} \cdot {x_1}}}{{{d_1} \cdot \cos \theta - {x_1} \cdot \sin \theta }} \\ {y_2} = \frac{{{y_1} \cdot ({{d_1} + {x_2} \cdot \sin \theta } )}}{{{d_1}}} \\ {x_2}^2 + {y_2}^2 \le {r^2} \end{array} \right.;$$
I is the incident angle of an arbitrary ray incident to the flat
$$ \cos I=\frac{d_{1} \cdot \cos \theta-x_{1} \cdot \sin \theta}{\sqrt{x_{1}^{2}+y_{1}^{2}+d_{1}^{2}}}, $$
then the WSSUT can be calculated by combining Eqs. (10)–(12).

As a summary, the system wavefront SW has been defined as

$$SW = {W_{\textrm{def}}}\textrm{ + }{W_{\textrm{SSUT}}}, $$
and can be expanded as
$$ \left\{\begin{array}{l} S W=L_{1}+L_{2}-2 R-2 \Delta s+4 \cos I * S\left(x_{2}, y_{2}\right) \\ L_{1}=\sqrt{x_{1}^{2}+y_{1}^{2}+\left(\sqrt{R^{2}-x_{1}^{2}-y_{1}^{2}}+\Delta s\right)^{2}} \\ L_{2}=z_{1} * \sqrt{\left(2 \arctan \left(\frac{x_{1}}{z_{1}-\Delta s}\right)-\arctan \left(\frac{x_{1}}{z_{1}}\right)\right)^{2}+\left(2 \arctan \left(\frac{y_{1}}{z_{1}-\Delta s}\right)-\arctan \left(\frac{y_{1}}{z_{1}}\right)\right)^{2}+1} \\ z_{1}=\sqrt{R^{2}-x_{1}^{2}-y_{1}^{2}}+\Delta s \\ x_{2}=\frac{d_{1} \cdot x_{1}}{d_{1} \cdot \cos \theta-x_{1} \cdot \sin \theta} \\ y_{2}=\frac{y_{1} \cdot\left(d_{1}+x_{2} \cdot \sin \theta\right)}{d_{1}} \\ x_{2}^{2}+y_{2}^{2} \leq r^{2} \\ \cos I=\frac{d_{1} \cdot \cos \theta-x_{1} \cdot \sin \theta}{\sqrt{x_{1}^{2}+y_{1}^{2}+d_{1}^{2}}} \end{array}\right. $$

The equation set (14) illustrates the essence of the full-field Ritchey-Common optical path model. The virtual interferometer can be built by combining this model and the optical path parameters of the real interferometer. Obviously, the F/# is not adopted in the modeling process, and thus the large F/# prerequisition in the conventional method is avoided. Based on this model, the VRCRI is potential to freely adjust the optical path to accommodate various test scenarios.

Theoretically the SSUT can be calculated by solving the equation set (14) directly. However, two challenges limit the accuracy of the current Ritchey-Common test as proposed in the Introduction section. Herein we readdress the challenges in the context of the proposed virtual interferometer model. First, the transformation from (x1, y1) to (x2, y2) in equation set (14) with an unchanged spatial resolution means to stretch the approximate elliptical aperture into a fully circular area. The lack of pixels is inevitable and will lead to the loss of surface information. In Ref. [21], an interpolation algorithm is utilized to fill the pixel gap, which decrease the confidence of test result. Second, d1, r, θ, R and SW in equation set (14) can be accurately measured in the real tests, but it is difficult to measure Δs with the current technique. The influence of Δs is eliminated by fitting the test result in two Ritchey angles [20]. If the test result is calculated by solving the equation set (14) directly, the relationship between the two sets of coordinates should stay in good agreement. It means d1 is required to be invariant when rotating the test flat, which is very difficult. The ‘invariant d1 restriction’ places a high demand on the mounting system [21]. Thus, a novel algorithm to calculate the SSUT is required to face the above challenges.

2.2 Virtual-real combination iterative algorithm

The second part is to calculate the SSUT in two Ritchey angles by a virtual-real combination iterative algorithm. The virtual-real combination iterative algorithm includes two steps: iterative reverse optimization (IRO) algorithm and parameter optimization in virtual interferometer.

  • (1) Iterative reverse optimization algorithm

The IRO algorithm is based on the frame work built in our previous publications [22,23], and can be used for different optimization target of different systems. Hereby, the IRO algorithm is aimed at eliminating the interpolation error and breaking the ‘invariant d1 restriction’.

The theory of this algorithm for Ritchey-Common test is presented in Fig. 3. The real SSUT S and virtual SSUT S′ are given by the finite terms of Zernike polynomials as follows:

$$\left\{ \begin{array}{l} S = \mathop \sum \limits_{i = 1}^N {a_i}{Z_i}\\ S\prime = \mathop \sum \limits_{i = 1}^N {a_i}\prime {Z_i} \end{array} \right., $$
where i is the number of Zernike coefficient, Zi is Zernike Standard polynomials, ai and ai′ are the Zernike coefficients of S and S′, respectively. Similarly, the real system wavefront SW and virtual system wavefront SW′ can be expressed as
$$\left\{ \begin{array}{l} SW = {W_{\textrm{SSUT}}} = \mathop \sum \limits_{j = 1}^N {b_j}{Z_j}\\ SW\prime = {W_{\textrm{SSUT}}}\prime = \mathop \sum \limits_{j = 1}^N {b_j}\prime {Z_j} \end{array} \right., $$
where bj and bj′ are the Zernike coefficients of SW and SW′, respectively.

 figure: Fig. 3.

Fig. 3. Theory of IRO algorithm.

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The real system wavefronts are measured from the real interferometer and characterized by the Zernike Standard polynomials as Eq. (16) shows. And then the Zernike coefficients bj of WSSUT are set as the optimization targets, and the ai are set as the optimization variables. In virtual interferometer, each ray launches from the focal point, impinges on the test flat and is reflected back by the spherical mirror. Based on the equation set (14) and Eq. (16), bj′ are determined by ai′. The SW is uniquely determined by the SSUT when the test configurations are fixed. Herein, the defocus is neglected and is set to 0 temporarily. The details of eliminating the defocus will be present in next step. The nominal parameters for the real and virtual interferometers are identical. Thus, when the virtual SW is equal to the real SW, i.e.,

$${b_j}\mathrm{\prime } = {b_j}, $$
we can get:
$${a_i}\mathrm{\prime } = {a_i}. $$

The SSUT is iteratively optimized in accordance with the optimization targets, and the SSUT can be obtained until the difference between the real and virtual system wavefronts is minimum. The optimization targets M is expressed as

$$\min M\begin{array}{{cc}} { = \min \left[ {\sum {{({{b_j} - {b_j}\prime } )}^2}} \right]}&{j = 1,2,3,4 \ldots } \end{array}. $$

Then the SSUT can be obtained by Zernike fitting with ai after the optimization. The relationship between the two sets of coordinates can be easily adjusted to keep identical in the Zernike fitting process of IRO algorithm, and thus ‘invariant d1 restriction’ is broken. This allows the variation of d1 and requires a remeasurement of d1 after the changing of the Ritchey angle, which is technically easy and accurate.

  • (2) Parameter optimization of virtual interferometer

The key point in the virtual-real combination iterative algorithm is to ensure the accordance between the virtual and real interferometer. Because the defocus of spherical mirror is unknown and is neglected in the model adopted in the first step, measurement error will be introduced into the optimization result. Thus, the second step of virtual-real combination iterative algorithm is to modify the virtual interferometer by calculating the defocus of spherical mirror to retrieve a more accurate result.

In our method, the defocus amount can be derived from the solved SSUT and the incident angle I on the flat. Two Ritchey angles are necessary and the detailed derivation is as follows. According to equation set (14), the surface results without consideration of the unknown defocus errors corresponding to two Ritchey angles after the IRO algorithm in the first step can be expressed as:

$$\left\{ \begin{array}{l} {S_1}({x_1},{y_1}) = \mathop \sum \limits_{i1 = 1}^N {a_{i1}}{Z_{i1}} = \frac{{S{W_1} - ({{L_{11}} + {L_{21}} - 2R} )}}{{4\cos {I_1}}}\\ {S_2}({x_1},{y_1}) = \mathop \sum \limits_{i2 = 1}^N {a_{i2}}{Z_{i2}} = \frac{{S{W_2} - ({{L_{12}} + {L_{22}} - 2R} )}}{{4\cos {I_2}}} \end{array} \right., $$
where the subscript of S, SW, I, and the second subscript of L1 and L2 correspond to the two Ritchey angles, and the spatial dependence (x1, y1) is omitted for conciseness. It is worthy noticing that Eq. (20) is the theoretical solution and is presented here only for the derivation of the defocus error. Because the defocuses Δs1 and Δs2 corresponding to Ritchey angles θ1 and θ2, respectively, are not included in this equation, the solved S1 and S2 are generally not similar with each other, meaning that the surface results compromise the following two parts:
  • a. Real SSUT.
  • b. Error introduced by the defocus.

For the first part, the real SSUT S0 should be the same in the two Ritchey angles

$$\left\{ \begin{array}{l} {S_0}({x_1},{y_1}) = \frac{{S{W_1} - ({{L_{11}} + {L_{21}} - 2R - 2\Delta {s_1}} )}}{{4\cos {I_1}}}\\ {S_0}({x_1},{y_1}) = \frac{{S{W_2} - ({{L_{12}} + {L_{22}} - 2R - 2\Delta {s_2}} )}}{{4\cos {I_2}}} \end{array} \right.. $$

For the second part, we can get the error introduced by Δs1 and Δs2, denoted as D1(x1, y1) and D2(x1, y1), from the combinatorial formulas (20) and (21) as

$$\left\{ \begin{array}{l} {D_1}({{x_1},{y_1}} )= {S_1}({x_1},{y_1}) - {S_0}({x_1},{y_1}) = \frac{{\Delta {s_1}}}{{2\cos {I_1}}}\\ {D_2}({{x_1},{y_1}} )= {S_2}({x_1},{y_1}) - {S_0}({x_1},{y_1}) = \frac{{\Delta {s_2}}}{{2\cos {I_2}}} \end{array} \right.. $$

Then Eq. (23) can be derived from Eq. (22)

$${D_1}({{x_1},{y_1}} )- {D_2}({{x_1},{y_1}} )= {S_1}({x_1},{y_1}) - {S_2}({x_1},{y_1}) = \left[ {\begin{array}{{cc}} {\Delta {s_1}}&{ - \Delta {s_2}} \end{array}} \right] \bullet \left[ {\begin{array}{{c}} {\frac{1}{{2\cos {I_1}}}}\\ {\frac{1}{{2\cos {I_2}}}} \end{array}} \right] = {\boldsymbol s} \bullet {\boldsymbol { I}}({{x_1},{y_1}} ), $$
where I is a vector including the incident angles in Ritchey angle θ1 and θ2, which can be calculated according to the last formula in equation set (14); ${\bullet}$ is the symbol of vector dot product. Using the least-square method we can yield the defocus vector s = [Δs1 −Δs2] as
$${\boldsymbol s} = {{[{{S_1}({x_1},{y_1}) - {S_2}({x_1},{y_1})} ]} / {\boldsymbol I}}({{x_1},{y_1}} ). $$

Finally, the virtual interferometers are remodeled with Δs1 and Δs2 respectively, and the real SSUT can be obtained by performing the IRO algorithm proposed in first step again.

As a summary, based on the proposed method, the flowchart of the VRCRI is described as follows and shown in Fig. 4.

  • a. Test the surface shape of the spherical mirror.
  • b. Build the test setup in Ritchey angle 1. Measure d1, r, θ1, R and SW1.
  • c. Build the test setup in Ritchey angle 2. Measure d1, r, θ2, R and SW2.
  • d. Deduct Wsph from the system wavefronts SW1 and SW2.
  • e. Model the virtual interferometer without Δs1 and Δs2.
  • f. Calculate S1(x1, y1) and S2(x1, y1) by the IRO algorithm.
  • g. Calculate the defocus of the spherical mirror in two Ritchey angles, Δs1 and Δs2, by equations (24).
  • h. Model the virtual interferometer again with Δs1 and Δs2, and the real SSUT can be obtained by IRO algorithm.

 figure: Fig. 4.

Fig. 4. Flowchart of VRCRI.

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3. Experiments

In section 3.1, a verification experiment testing a Φ100 mm flat mirror was conducted and the result was compared with the direct test result of Zygo interferometer to validate the performance of the VRCRI comprehensively.

In section 3.2, a verification experiment with a Φ422 mm flat mirror was performed to assess the difficulty and feasibility of VRCRI in the test of large flat mirror.

3.1 Φ100 mm experiment

The experimental setup is shown in Fig. 5. This setup comprised a precisely calibrated 4” Zygo DynaFiz interferometer, a precision linearity rail, magnetic grid displacement measurement modules, a spherical mirror, and a test flat with its adjustment mechanism. The precision linearity rail had a 0.01 mm positioning accuracy with straightness better than ±1 mm/6 m, and the magnetic grid displacement measurement modules with an accuracy of ±0.03 mm would show d1. F/# would not affect the test result in this experiment and was set to 7.8 to accommodate the transmission sphere of interferometer and the spherical mirror in the lab. In this experiment, r was 50 mm, R was 1052 mm, d1 was 779.98 mm when θ1 was 20°, d1 was 780.02 mm when θ2 was 30°, and laser wavelength λ was 632.8 nm.

 figure: Fig. 5.

Fig. 5. Experimental setup of Φ100 mm flat mirror.

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The surface shape of the spherical mirror was tested by the Zygo interferometer and was calibrated as a system error. The test result is shown in Fig. 6, wherein PV is 0.1462 λ, and RMS is 0.0068 λ.

 figure: Fig. 6.

Fig. 6. Surface shape of the spherical mirror.

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The measured system wavefronts SW1 and SW2 from the interferometer when the Ritchey angles are 20° and 30° are shown in Fig. 7. The spatial resolutions of the two wavefronts are 956×1003 and 882×1003, respectively. The apertures are distorted to different elliptical ones.

 figure: Fig. 7.

Fig. 7. System wavefront (a) in 20° and (b) in 30°.

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After deducting Wsph from the system wavefronts SW1 and SW2, the virtual interferometer was preliminarily modeled by neglecting Δs, and S1(x1, y1) and S2(x1, y1) were calculated by the IRO algorithm as shown in Figs. 8 (a) and (b). Obviously, the apertures are restored to circular ones and the spatial resolutions of the two surface results are the same, which avoids the unreasonable interpolation operation and reserves the coordinate correspondence between different Ritchey angles. Meanwhile, the surface results are neither similar in shape nor in quantity, which indicates the existence of defocus errors.

 figure: Fig. 8.

Fig. 8. Surface results (a) is S1(x1, y1), (b) is S2(x1, y1), (c) in 20° with 78 terms, (d) in 30° with 78 terms, (e) in 20° with 231 terms, (f) in the direct test with ZYGO interferometer and (g) the surface shape deviation between (e) and (f).

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The defocus of the spherical mirror in two Ritchey angles, Δs1 and Δs2, were calculated by Eq. (24). In this test, Δs1 was 0.0203 µm and Δs2 was -0.1729 µm. Virtual interferometer was then accurately modeled with Δs1 and Δs2, and the real SSUT was obtained by IRO algorithm. The surface results of VRCRI and direct test in Zygo interferometer are shown in Figs. 8 (c), (d) and (f), and their PV and RMS are presented in Table 1. Compared with Figs. 8 (a) and (b), the results of 20° and 30° in Figs. 8 (c) and (d) are in close agreement, which shows that the calculation and removal of the defocus error are very successful. For quantitative comparison, in Table 1, the error is defined as the difference between the measured value of VRCRI and direct test in Zygo interferometer. It shows that the accuracy of the VRCRI reaches the upper limit of the interferometer in engineering (error of PV < 0.1λ).

Tables Icon

Table 1. Test results of VRCRI and direct test in Zygo interferometer (Unit: λ)

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Although the PV and RMS value of the measured results are close, Fig. 8(f) measured with Zygo interferometer reveals a dip (marked with the red arrow) in the center of the surface result while Figs. 8 (c) and (d) measured with 78 terms of Zernike polynomials (eleventh order) lose the high-frequency information, i.e., the dip. A total of 231 terms of Zernike polynomials (twentieth order) in the VRCRI were used to recover the SSUT, and the result is shown in Fig. 8(e). The PV of the result is 0.3822 λ, and its RMS is 0.0402 λ. The surface shape deviation between the direct measured and the calculated surface shape by 231 terms of Zernike polynomials is shown in Fig. 8(g). The PV of the deviation is 0.0930 λ, and its RMS is 0.0140 λ. The dip can be effectively characterized with 231 terms. Thus, the performance of the VRCRI for high-order Zernike aberrations can be effectively improved by freely adding the polynomial terms in the test. Meanwhile, only the terms up to the fourth-order, corresponding to 15 Zernike terms, have been included in the IFM method [15], thus limiting its performance in the test for high-order Zernike aberrations. What’s more, there are many periodic residual grooves caused by the grinding wheel in the processing of the test flat in Fig. 8(g). These grooves are of relatively high spatial frequency, and can hardly be characterized even with 231 terms of Zernike polynomials. This is the major source of the surface shape deviation. Generally, the term number required by the Zernike polynomial fitting process in VRCRI is not fixed and is determined by the SSUT. For a SSUT with high-frequency information, more polynomials can effectively improve the accuracy of VRCRI, nevertheless, it will also significantly increase the data processing time and lead to a decrease in test efficiency. The fitting error of SSUT determines the upper limit of the accuracy of VRCRI, and thus, a better fitting method rather than Zernike polynomial method may contribute to improve the performance of VRCRI in accuracy and test efficiency.

3.2 Φ422 mm experiment

It has been mentioned in the Introduction section that Ritchey-Common test suffers from the large F/# prerequisition and distance invariance. The feasibility of VRCRI has been verified in section 3.1 with the small flat to address the above challenges, but the test environment of a large flat mirror is quite different from that of the small flat mirror. In this section, we try to assess the problems as follows and further demonstrate the feasibility of the VRCRI in the test of a large flat mirror.

First, the optical path in the test of a large flat mirror is much longer than that in the test of a small flat mirror, which means the test result of traditional temporal phase shifting interferometry will be strongly affected by the airflow and vibration [24,25]. Second, the measurement of optical path parameter is more difficult [26]. A long precision linearity rail will significantly increase the testing costs and it is also very difficult and dangerous to move a large flat mirror. Third, the flat mirror should keep perpendicular during the test to avoid the test data compression in vertical direction, which is difficult in the adjustment of a large flat mirror [20].

As a good approaching to large flat mirror, a Φ422 mm flat mirror was measured and the diameter was beyond the measurement range of the interferometer in our lab. Thus, the test result of VRCRI was compared with the result of IFM method to verify the correctness. The experimental setup was changed to adapt the test requirement of a Φ422 mm test flat and is shown in Fig. 9. For the first problem, a dynamic interferometer (4D PhaseCam 6000) was utilized to realize the instantaneous measurement. System wavefront was averaged through 200 tests to eliminate the influence of airflow. For the second problem, the optical path parameters required by the virtual interferometer was accurately measured by the laser tracker and sphere-mounted retroreflector (SMR) [18]. For the third problem, a laser level was utilized to keep the optical elements staying perpendicular and the main optical path propagating in the horizontal direction. Considering the test requirement of IFM method and the influence of airflow, the F/# was set to 8.6 in the test. It was hard to keep d1 invariant in different Ritchey angles, and d1 was measured separately in two tests. When θ1 was 24.01°, d1 was measured to be 3671.25 mm. When θ2 was 39.59°, d1 was measured to be 3659.18 mm. r was 211 mm, R was 4500 mm, and laser wavelength λ was 632.8 nm.

 figure: Fig. 9.

Fig. 9. Experimental setup of Φ422 mm flat mirror.

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Since d1 in two tests was measured separately, the defocus error cannot be estimated by conventional method due to the ‘invariant d1 restriction’. Thus, the defocus error in the surface result of IFM method and VRCRI was both calculated by the second step proposed in section 2.2. In this test, Δs1 was -0.0178 µm and Δs2 was -0.0080 µm. The surface results of IFM method and VRCRI are shown in Fig. 10 and the shapes of the two results show close agreement. The PV of the test flat is 0.1182 λ, and its RMS is 0.0224 λ in IFM method. And the PV is 0.1217 λ, RMS is 0.0230 λ in VRCRI. The PV difference is 0.0035 λ, and the RMS difference is 0.0006 λ. Since the F/# had been set to 8.6 to realize the large F/# prerequisition, IFM method can also achieve relatively high accuracy in this test. The test results of both methods are in close agreement, it further proves the feasibility and performance of VRCRI in real test.

 figure: Fig. 10.

Fig. 10. Surface results of Ф422 mm flat mirror (a) by IFM method and (b) by VRCRI.

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4. Discussions

4.1 Influence of F/#

In this section, the potential of VRCRI in different F/# are discussed with numerical simulations, and the influence of F/# in the IFM method and VRCRI is analyzed.

Each SSUT was simulated by a single Zernike polynomial respectively. The 4–15 Zernike Standard polynomials, including defocus, primary astigmatism at 0°, primary astigmatism at 45°, primary x coma, primary y coma, primary x trefoil, primary y trefoil, primary spherical aberration, secondary astigmatism at 0°, secondary astigmatism at 45°, primary x tetrafoil, and primary y tetrafoil, were then utilized in simulation. r was 50 mm; d2 was 272 mm; the Ritchey angle θ was 20°, the influence of defocus was neglected in this simulation, thus only the condition in one Ritchey angle is analyzed here; d1 = 2r × F/#; R = d1 + d2 (Δs = 0); F/# is 4, 6, 8, 10, and 20; the Zernike coefficient was 1 λ.

The system wavefront was constructed on the basis of Eq. (14). The SSUT was calculated by IFM method and VRCRI respectively. The influence of F/# in the IFM method and VRCRI can be obtained by comparing the calculation results with the preset SSUT.

The RMS errors of two methods in different F/# are shown in Fig. 11. The errors for all the Zernike aberrations in arbitrary F/# are less than 0.01 λ in the VRCRI, while only when the F/# is greater than 20 will the RMS error of most Zernike aberrations be less than 0.01 λ in IFM method. Considering the actual test requirements, when the aperture of SSUT is 1 m, d1 is less than 4 m in VRCRI when F/# is less than 4. Meanwhile, to achieve the same accuracy in the test of same flat mirror, F/# in IFM method should be larger than 20 and d1 would be larger than 20 m. Thus, the VRCRI has great potential to reduce the impact of environmental factors.

 figure: Fig. 11.

Fig. 11. RMS errors in different F/# (a) in IFM method and (b) in VRCRI.

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The accuracy of IFM method gradually increases as F/# increases, which is in accordance with the theoretical prediction, while the accuracy of VRCRI increases progressively as the F/# decreases. However, since theoretically the virtual interferometer has been accurately modeled and the F/# is not adopted in the modeling process, so the accuracy of VRCRI is supposed to be independent from the F/#. The reason for this phenomenon is worth discussing and we try to analyze it as follows. The tracing rays in virtual interferometer can be characterized by its initial position and an optical vector [27]. In our implementation, the focal point of the interferometer is set as the shared initial position of each ray, and thus the ray is characterized only by its optical vector. The optical vector is further denoted by the directional cosines in x, y and z directions. The adjacent rays are sampled with the same angular increment. When the angle increment in three directions is determined, the number of the tracing rays is determined by the maximum angle that the marginal ray deviates from the main optical axis. For the same test flat, the maximum angle and the number of tracing rays will decrease as the F/# increases, and it will decrease the accuracy of VRCRI. This question can be solved by decreasing the angle increment to deal with the case where F/# is greater than 20, which is usually not necessary. In actual test scenarios, F/# is usually set to less than 20 to reduce the influence of air disturbance. As analyzed above, a larger F/# is not conducive to obtain accurate measurement results with VRCRI just because of the insufficient sampling. The theoretical error of VRCRI will gradually decrease with the increase of the sampling rate and tend to be 0.

As a result, the VRCRI can significantly reduce the optical path and eliminate the large F/# dependence in conventional Ritchey–Common test.

4.2 Error analysis of optical path parameters

The optical path parameters, i.e., d1, θ, r, R and SW, is utilized in the virtual interferometer to accurately describe the optical systems. All the parameters should be accurately measured and the measurement errors will influence the accuracy of VRCRI. In this section, the errors caused by the optical path parameters are analyzed based on Eq. (14).

SSUT is randomly generated with 78 terms of Zernike polynomials, and its surface shape is shown in Fig. 12. The PV of SSUT is 1.8926 λ, and its RMS is 0.1948 λ.

 figure: Fig. 12.

Fig. 12. Simulated SSUT for error analysis.

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The optical path parameters in section 3.2 are utilized in this simulation. It is recommended to set the Ritchey angle between 20° and 50° in the Ritchey-Common test [21]. Thus, the Ritchey angle θ was set at 20°, 30°, 40°and 50°; d1 was 3660 mm; r was 211 mm; R was 4500 mm; λ was 632.8 nm, and Δs was 0 µm. The influence of the defocus can be removed by the virtual-real combination iterative algorithm; thus, the virtual interferometer with no defocus was modeled, and the system wavefronts in different Ritchey angles were constructed on the basis of Eq. (14).

Then the measured SSUT results are calculated by adding errors in optical path parameters respectively. The RMS deviation between the measured SSUT and the preset SSUT is defined as the RMS error, which is plotted in Figs. 13 and 14 against different parameter errors.

(1) Surface errors caused by distance σd and Ritchey angle σθ

 figure: Fig. 13.

Fig. 13. Surface error caused by (a) d1 and (b) θ.

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 figure: Fig. 14.

Fig. 14. Surface error caused by r.

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Due to the difficulty in precise positioning of the mirror center and focal point, the measurement of distance d1 and Ritchey angle θ are both difficult and require multiple measurements. The distance measurement accuracy of a laser tracker can reach micron scale. Considering the positioning error, the distance error is less than 10 mm when d1 is 3660 mm. It can be seen from Fig. 13 (a) that the surface error caused by d1 satisfies

$${\sigma _d} \le 1.5 \times {10^{ - 4}}\lambda . $$

Direct measurement of the Ritchey angle is difficult, but it can be calculated by the width-height-ratio of interferogram [21], the Ritchey angle error can be less than 0.2°. It can be deduced from Fig. 13 (b) that the surface error caused by θ satisfies

$${\sigma _\theta } \le 4.1 \times {10^{ - 3}}\lambda . $$

(2) Surface error caused by half of the clear aperture of the test flat σr and curvature radius

Half of the clear aperture of the test flat r defines the limit of x2 and y2 and it can be easily measured with an accuracy better than 1 mm. As shown in Fig. 14, the surface error caused by r satisfies

$${\sigma _r} \le 2.4 \times {10^{ - 4}}\lambda . $$

The curvature radius R is not adopted in the Eq. (14), which means R will hardly introduce surface errors in real tests. But an exact value of R would make the adjustment of test set up easier and faster.

(3) Surface error caused by SW σSW

Since the defocus is neglected in this simulation, SW can be expressed as

$$SW = 4\cos I \ast {S_0}, $$
the real SSUT is
$${S_0} = \frac{{SW}}{{4\cos I}}. $$

The error in SW is defined as ErrSW and it satisfies

$$SW + Er{r_{SW}} = 4\cos I \ast {S_1}, $$
so SSUT with error is
$${S_1} = \frac{{SW + Er{r_{sw}}}}{{4\cos I}}, $$
and the surface error caused by the SW is
$${\sigma _{SW}} = {S_1} - {S_0} = \frac{{SW + Er{r_{\textrm{sw}}}}}{{4\cos I}} - \frac{{SW}}{{4\cos I}} = \frac{{Er{r_{\textrm{sw}}}}}{{4\cos I}}. $$

When Ritchey angle θ is less than 50°, the incident angle I of any ray will be generally less than 60° and the cosI will be larger than 0.5. Considering the typical RMS error of SW is 0.01 λ, we can get

$${\sigma _{SW}} \le 5.0 \times {10^{ - 3}}\lambda $$

(4) Synthesis error

Considering the effect of the aforementioned several errors on the measurement results, the upper limit of the total measurement error caused by optical path parameters is

$${\sigma _{\max }} = \sqrt {{\sigma _d}^2 + {\sigma _\theta }^2 + {\sigma _r}^2 + {\sigma _{SW}}^2} \approx 0.0065\lambda . $$

It is worth noting that all errors caused by optical path parameters increase with the increase of the Ritchey angle θ. A smaller Ritchey angle can reduce the surface errors caused by the measurement errors of optical path parameters.

5. Conclusions

In this study, a virtual-real combination Ritchey-Common interferometry is proposed for large optical flat testing in high accuracy. In this method, a virtual interferometer is accurately modeled by ray-tracing method to avoid the large F/# prerequisition in Ritchey-Common test. A virtual-real combination iterative algorithm is proposed to calculate the defocus of spherical mirror and remove the restriction of distance invariance in conventional method. The feasibility of this method has been proved by a set of verification experiments without the distance invariance requirement and numerical simulations in different F/#. In comparison with the direct test result of a standard Zygo interferometer, the PV error of the VRCRI is less than 0.1 λ, the RMS error is less than 0.01 λ. This method is engineeringly potential to break the distance invariance requirement, shorten the optical path and provide remarkable freedom in the design of test configurations to accommodate various test scenarios for large optical flat testing.

Funding

National Natural Science Foundation of China (51735002, 12003067, 12141304); Nanjing International Science and Technology Cooperation Project (2020SX00100193).

Acknowledgement

We would like to thank Dr. Shuo Zhu for her helpful suggestion.

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 maybe obtained from the authors upon reasonable request.

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Data availability

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

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Figures (14)
Fig. 1.
Fig. 1. Principle of VRCRI.

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Fig. 2.
Fig. 2. Schematic sketch of the Ritchey-Common test (a) for the full-field optical path model and (b) for the analysis of defocus.

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Fig. 3.
Fig. 3. Theory of IRO algorithm.

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Fig. 4.
Fig. 4. Flowchart of VRCRI.

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Fig. 5.
Fig. 5. Experimental setup of Φ100 mm flat mirror.

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Fig. 6.
Fig. 6. Surface shape of the spherical mirror.

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Fig. 7.
Fig. 7. System wavefront (a) in 20° and (b) in 30°.

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Fig. 8.
Fig. 8. Surface results (a) is S1(x1, y1), (b) is S2(x1, y1), (c) in 20° with 78 terms, (d) in 30° with 78 terms, (e) in 20° with 231 terms, (f) in the direct test with ZYGO interferometer and (g) the surface shape deviation between (e) and (f).

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Fig. 9.
Fig. 9. Experimental setup of Φ422 mm flat mirror.

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Fig. 10.
Fig. 10. Surface results of Ф422 mm flat mirror (a) by IFM method and (b) by VRCRI.

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Fig. 11.
Fig. 11. RMS errors in different F/# (a) in IFM method and (b) in VRCRI.

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Fig. 12.
Fig. 12. Simulated SSUT for error analysis.

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Fig. 13.
Fig. 13. Surface error caused by (a) d1 and (b) θ.

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Fig. 14.
Fig. 14. Surface error caused by r.

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Tables (1)

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Table 1. Test results of VRCRI and direct test in Zygo interferometer (Unit: λ)

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Equations (34)

Equations on this page are rendered with MathJax. Learn more.

S W = W sph + W def  +  W SSUT .
x 1 2 + y 1 2 + ( z 1 − Δ s ) 2 = R 2 ,
z 1 = R 2 − x 1 2 − y 1 2 + Δ s .
L 1 = x 1 2 + y 1 2 + z 1 2 = x 1 2 + y 1 2 + ( R 2 − x 1 2 − y 1 2 + Δ s ) 2 ,
L 2 = ( z 1 ∗ ( ∂ x ∂ z ) L 2 ) 2 + ( z 1 ∗ ( ∂ y ∂ z ) L 2 ) 2  +  z 1 2 = z 1 ∗ ( ∂ x ∂ z ) L 2 2 + ( ∂ y ∂ z ) L 2 2  + 1 ,
{ ( ∂ x ∂ z ) L 2  =  2 arctan ⁡ ( x 1 z 1 − Δ s ) − arctan ⁡ ( x 1 z 1 ) ( ∂ y ∂ z ) L 2  =  2 arctan ⁡ ( y 1 z 1 − Δ s ) − arctan ⁡ ( y 1 z 1 ) ,
L 2 = z 1 ∗ ( 2 arctan ⁡ ( x 1 z 1 − Δ s ) − arctan ⁡ ( x 1 z 1 ) ) 2 + ( 2 arctan ⁡ ( y 1 z 1 − Δ s ) − arctan ⁡ ( y 1 z 1 ) ) 2 + 1 .
W def ( x 1 , y 1 ) = L 1 + L 2 − L ,
L = 2 R + 2 Δ s = 2 ( d 1 + d 2 ) ,
W SSUT ( x 1 , y 1 ) = 4 cos ⁡ I ∗ S ( x 2 , y 2 ) ,
{ x 2 = d 1 ⋅ x 1 d 1 ⋅ cos ⁡ θ − x 1 ⋅ sin ⁡ θ y 2 = y 1 ⋅ ( d 1 + x 2 ⋅ sin ⁡ θ ) d 1 x 2 2 + y 2 2 ≤ r 2 ;
cos ⁡ I = d 1 ⋅ cos ⁡ θ − x 1 ⋅ sin ⁡ θ x 1 2 + y 1 2 + d 1 2 ,
S W = W def  +  W SSUT ,
{ S W = L 1 + L 2 − 2 R − 2 Δ s + 4 cos ⁡ I ∗ S ( x 2 , y 2 ) L 1 = x 1 2 + y 1 2 + ( R 2 − x 1 2 − y 1 2 + Δ s ) 2 L 2 = z 1 ∗ ( 2 arctan ⁡ ( x 1 z 1 − Δ s ) − arctan ⁡ ( x 1 z 1 ) ) 2 + ( 2 arctan ⁡ ( y 1 z 1 − Δ s ) − arctan ⁡ ( y 1 z 1 ) ) 2 + 1 z 1 = R 2 − x 1 2 − y 1 2 + Δ s x 2 = d 1 ⋅ x 1 d 1 ⋅ cos ⁡ θ − x 1 ⋅ sin ⁡ θ y 2 = y 1 ⋅ ( d 1 + x 2 ⋅ sin ⁡ θ ) d 1 x 2 2 + y 2 2 ≤ r 2 cos ⁡ I = d 1 ⋅ cos ⁡ θ − x 1 ⋅ sin ⁡ θ x 1 2 + y 1 2 + d 1 2
{ S = ∑ i = 1 N ⁡ a i Z i S ′ = ∑ i = 1 N ⁡ a i ′ Z i ,
{ S W = W SSUT = ∑ j = 1 N ⁡ b j Z j S W ′ = W SSUT ′ = ∑ j = 1 N ⁡ b j ′ Z j ,
b j ′ = b j ,
a i ′ = a i .
min M = min [ ∑ ( b j − b j ′ ) 2 ] j = 1 , 2 , 3 , 4 … .
{ S 1 ( x 1 , y 1 ) = ∑ i 1 = 1 N ⁡ a i 1 Z i 1 = S W 1 − ( L 11 + L 21 − 2 R ) 4 cos ⁡ I 1 S 2 ( x 1 , y 1 ) = ∑ i 2 = 1 N ⁡ a i 2 Z i 2 = S W 2 − ( L 12 + L 22 − 2 R ) 4 cos ⁡ I 2 ,
{ S 0 ( x 1 , y 1 ) = S W 1 − ( L 11 + L 21 − 2 R − 2 Δ s 1 ) 4 cos ⁡ I 1 S 0 ( x 1 , y 1 ) = S W 2 − ( L 12 + L 22 − 2 R − 2 Δ s 2 ) 4 cos ⁡ I 2 .
{ D 1 ( x 1 , y 1 ) = S 1 ( x 1 , y 1 ) − S 0 ( x 1 , y 1 ) = Δ s 1 2 cos ⁡ I 1 D 2 ( x 1 , y 1 ) = S 2 ( x 1 , y 1 ) − S 0 ( x 1 , y 1 ) = Δ s 2 2 cos ⁡ I 2 .
D 1 ( x 1 , y 1 ) − D 2 ( x 1 , y 1 ) = S 1 ( x 1 , y 1 ) − S 2 ( x 1 , y 1 ) = [ Δ s 1 − Δ s 2 ] ∙ [ 1 2 cos ⁡ I 1 1 2 cos ⁡ I 2 ] = s ∙ I ( x 1 , y 1 ) ,
s = [ S 1 ( x 1 , y 1 ) − S 2 ( x 1 , y 1 ) ] / I ( x 1 , y 1 ) .
σ d ≤ 1.5 × 10 − 4 λ .
σ θ ≤ 4.1 × 10 − 3 λ .
σ r ≤ 2.4 × 10 − 4 λ .
S W = 4 cos ⁡ I ∗ S 0 ,
S 0 = S W 4 cos ⁡ I .
S W + E r r S W = 4 cos ⁡ I ∗ S 1 ,
S 1 = S W + E r r s w 4 cos ⁡ I ,
σ S W = S 1 − S 0 = S W + E r r sw 4 cos ⁡ I − S W 4 cos ⁡ I = E r r sw 4 cos ⁡ I .
σ S W ≤ 5.0 × 10 − 3 λ
σ max = σ d 2 + σ θ 2 + σ r 2 + σ S W 2 ≈ 0.0065 λ .
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