1. Introduction

Large optical flats are widely adopted in the auto-collimation optical tests and various advanced optical systems, such as the astronomical telescope, inertial confinement fusion device and laser weapon [1–4]. For example, a 1.25 m × 0.88 m, l.43 m × l.04 m, and 3.50 m × 2.50 m elliptical flat mirror are fabricated for the very large telescope [5], the Keck telescope [6], and the thirty meter telescope [7] as M3 mirror, respectively. Meanwhile, the M4 mirror in the European extremely large telescope is a Ф2.4 m flat mirror [8–10]. There are 7648 large-aperture optics in the National Ignition Facility, and the laser mirror has an aperture of 0.5 m to 1.0 m [11]. The U.S. Navy developed a mid-infrared advanced chemical laser, and a 1.80 m output mirror mounted on a tactical servo is utilized in the beam control subsystem [4]. According to the Rayleigh criterion, the increased aperture can greatly improve system performance. The pursuit of high system performance has led to the continuous increase of the aperture of advanced optical systems, which further increases the aperture of optical flat mirrors. A large optical flat with small surface form deviation is the guarantee of the performance of the optical system. The development of a test method for large optical flats is therefore of great significance, and the increased aperture poses a great challenge in the measurement of the large optical flats.

Interferometry is an efficient optical method for testing flat mirrors [12]. Commercial interferometers are commonly used in the test of optical flats, and the aperture of a commercial interferometer should be larger than the test flats to cover the full aperture in practice. Nevertheless, the development of large aperture interferometers is very difficult and expensive. The aperture of the largest commercial interferometer is 36 inches [13], which can hardly meet the testing requirements of large optical flats in meter size.

Since the stand-alone interferometer can hardly cover the full aperture of a large optical flat in meter size, and only part of the surface can be measured in a single measurement, a widely adopted solution divides the entire surface of the test flat into several sub-apertures. The surface form of each sub-aperture is tested by a small-aperture interferometer mounted on a high-precision scanning mechanism. Then the surface form under test (SFUT) is reconstructed by stitching the divided sub-apertures into the full aperture. This method is named sub-aperture stitching interferometry, which can sufficiently extend the measurable aperture of the interferometer [14,15]. Nevertheless, this method is not sensitive to the low-order aberrations of the test flat and poses high requirements on the precision of the scanning mechanism. And considering the large size of the test flat, the test results will be strongly affected by environmental factors during the multiple measurements to different sub-apertures.

Another widely adopted solution is the Ritchey-Common test. This method adds a well-polished spherical mirror, which is relatively easy to fabricate [16,17], as the return sphere in the test configuration. The configuration of Ritchey-Common test consists of an interferometer, a test flat and a spherical mirror. The divergent beam emitted by the interferometer is obliquely incident on the test flat and then reflected back to the interferometer by the return sphere. The incident angle of the ray along the main optical axis is called the Ritchey angle and is denoted as θ. Unlike the sub-aperture stitching interferometry, this method avoids the time-consuming scanning process and can realize instantaneous full-field measurement with a dynamic interferometer. Due to its practicality and excellent performance, the Ritchey–Common test has been developed and widely used in testing large flat mirrors thus far [5,16–23]. However, Ritchey-Common test still faces many unresolved challenges.

The most troublesome challenge is the multiple tests requirement [5,16,18]. The defocus is inevitable because a return sphere is used in the Ritchey-Common test. Meanwhile, the power of the test flat will also appear as defocus aberration in the system wavefront (SW). In order to distinguish the power of the test flat from the misalignment of the spherical mirror and obtain an accurate result, multiple tests should be undertaken to acquire two more system wavefronts. Multiple tests are implemented by changing the position of the test flat and return sphere. The position change means to change the place or pose of the flat and spherical mirror and are usually done in two ways: changing the Ritchey angle including changing the yaw angle of the flat and the position of the sphere, or rotating the flat along its normal axis.

The multiple tests with position change pose great challenges in the measurement. First, the multiple measurements in different positions greatly increase the test time, which decreases the test efficiency. Second, the multiple tests requirement both deceases the precision and accuracy of the Ritchey-Common test. Because the temperature change and chaotic airflow are inevitable in most test scenarios due to the large measurement space, and a long test time increases the error introduced by environmental factors, which decreases the repeatability and precision of the test [24]. Meanwhile, the traditional Ritchey-Common test performs the least-square-fitting method to the wavefront data obtained in different positions to calculate the power of the flat mirrors [16,19,20,23]. The random errors in the wavefront data obtained in different positions are different, and the least-square-fitting operation between these data will lead to a coupled system error that cannot be accurately estimated in test result, which brings challenges in uncertainty analysis. The random error in collected SW introduces system error in final test result due to the multiple tests requirement, which significantly decreases the measurement accuracy and reliability of the Ritchey-Common test. Third, the multiple test requirement increases the test cost. On the one hand, the poor repeatability means the Ritchey-Common test poses stringent requirements on the environment control which is difficult and expensive. On the other hand, gravitational effect will result in the deformation of the mirror surface and mounting systems when changing the position of the mirrors. The extra deformation of the spherical mirror will cause errors in the test result. The extra deformation of the flat mirror will lead to extra polishing procedures and test failure in multiple tests. To avoid the deformation, the position change of large optical flats requires a fine mounting system, which greatly increases the test cost. Last but not least, the position change of the large aperture mirror is risky and difficult due to its heavy weight. The possible overturning of the mirror will cause great economic loss.

As a result, the multiple tests requirement has troubled the Ritchey-Common test in the optical shop for decades. With the increased aperture of large flats in the advanced optical system, the influence of the multiple tests on the accuracy, cost, efficiency and test environment is increasing. A single-test measurement solution is urgently needed to face the above challenges.

In this study, we propose a single-test Ritchey-Common interferometry (SRI) to achieve high accuracy measurement and avoid the obligatory position change of the test flat. SFUT is reconstructed with the SW obtained in one position, thus improving the repeatability and avoiding the cumbersome steps, coupled system error, extra mirror deformation, and potential overturning problem. The extra SW measurement in different positions is replaced with the measurement of the surface form of a sub-aperture, denoted as sub-SFUT, to eliminate the defocus error. The sub-SFUT is directly measured by a commercial interferometer or any other surface form measurement instrument before the test. In most test scenarios, the sub-aperture size is between 20% to 35% of full aperture and can be further decreased with a fine test environment control, so the measurement of sub-SFUT is easy to implement. A set of verification experiments are conducted to validate the accuracy and feasibility of this method. And the sub-aperture size is discussed with numerical simulations to provide a reference in the design of test configurations.

2. Theory

A flow chart of the proposed SRI for large flat mirrors is depicted in Fig. 1. The measurement step is simple and same as the traditional Ritchey-Common interferometry without position change. Our work focuses on numerical modeling and defocus error elimination. The proposed method can be subdivided into three distinct steps: establishing the relationship between test flat and SW, surface form preliminary reconstruction, and defocus error elimination. In the first step, the relationship between the test flat and SW is established. The established relationship can be applied to any sub-aperture of the test flat. In the second step, a sub-SFUT is preliminary reconstructed without consideration of defocus, denoted as a virtual sub-SFUT, according to the established relationship in the first step. In the third step, the defocus is calculated by combining the virtual and real sub-SFUT. Then the SFUT can be reconstructed after eliminating the defocus error.

 

Fig. 1. Flow chart of SRI. (a) The whole measurement procedure, procedure of (b) surface form preliminary reconstruction and (c) defocus error elimination.

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2.1 Establish the relationship between test flat and system wavefront

This step establishes the relationship between the test flat and SW with a virtual interferometer (VI) in a Ritchey-Common test setup. VI is a tool proposed in our previous work to describe optical systems and eliminate system errors accurately, and it is adopted in this study by combining the principle of the Ritchey-Common test.

The test setup of the Ritchey-Common test is shown in Fig. 2 (a). The test setup comprises an interferometer, a test flat and a spherical mirror. 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 theoretically 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. r₁ is half of the clear aperture of the test flat; r₂ is half of the clear aperture of the sub-aperture; 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; R is the radius of curvature of the spherical mirror; SW is the full aperture system wavefront measured by the interferometer; sub-SW is the corresponding wavefront of sub-SFUT in SW; Δs is the defocus of the spherical mirror.

 

Fig. 2. Schematic diagram of (a) real interferometer, (b) sub-VI, and (c) full-VI.

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The system wavefront SW collected by the interferometer comprises the following four parts

a W_sph

The first part W_sph is introduced by the surface form of the spherical mirror. The surface form of the spherical mirror should be calibrated as a system error in the test. The spherical mirror can be considered as a perfect spherical surface after the point-to-point subtraction, and this part will be neglected in the following mathematical derivation.

b W_SFUT

The second part W_SFUT is introduced by the SFUT. As shown in Fig. 3, a rectangular coordinate system is established by setting the focal point of the interferometer O as the origin point. 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. The red dotted line indicates the ideal position of the spherical mirror with no defocus, and the blue line indicates the real position.

 

Fig. 3. Established coordinate system of the Ritchey-Common test.

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Then the W_SFUT can be expressed with equation set (2) [19]

c W_def

The third part W_def is introduced by the defocus of the spherical mirror and can be expressed as equation set (3) [19]

d W_ran

The last part W_ran is the random errors caused by environmental factors, such as airflow. The random error can be decreased by creating a stable airflow environment or averaging the test results in a rapidly changing environment. The second approach is more commonly used in the test of large flat mirrors. Assuming that the refractive index of the air in the optical path is randomly distributed in a rapidly changing environment, then the mean of the introduced optical path difference is a small quantity $\alpha ({{x_2},{y_2}} )$ compared to the W_def and W_SFUT when the sample number is large enough. The $\alpha ({{x_2},{y_2}} )$ would vary when changing the position of the test setup.

In summary, the system wavefront SW has been defined as

The VI can be built based on Eqs. (2), (3), (4) and the measured optical path parameters of the real interferometer, including d₁, r₁, r₂, θ, Δs and R.

Since the VI has established the relationship between the SFUT and SW, the SFUT can theoretically be directly obtained by solving the above equations. Nevertheless, d₁, r₁, r₂, θ, R and SW in Eqs. (2) and (3) can be accurately measured in real tests, but it is difficult to measure Δs with the current technique. The influence of Δs can be eliminated by fitting the results in multiple tests [5,16,19], which will face the challenges described in the Introduction section.

In our proposed method, the influence of Δs is calculated in a VI modeled with sub-aperture, which is shown in Fig. 2(b) and named sub-VI, meanwhile the VI modeled with full aperture is shown in Fig. 2(c) and named full-VI. The sub-aperture is set as the center part in this paper and shares the defocus Δs and transforming relationship with the full aperture. Theoretically, it can be any part of the test flat by modifying Eqs. (2) and (3). The optical path parameters of the real interferometer, i.e., d₁, r₁, r₂, θ, and R, will be accurately measured and then adopted in the sub-VI and full-VI. The sub-VI is modeled with d₁, r₂, θ, R, sub-SFUT, and sub-SW. It is worth noting that r₂ is not included in Eqs. (2), (3), (4). The sub-VI can be simply established by substituting the r₁ with r₂ in Eq. (2). The sub-SFUT is measured by a small aperture interferometer or other surface form measurement instruments before the test, which is timely and cost-effective.

2.2 Surface form reconstruction

In this step, an iterative reverse optimization (IRO) algorithm [19] is used in the sub-VI to calculate the sub-SFUT based on the established relationship in Section 2.1.

According to the IRO algorithm, the sub-SW picked out from the measured SW can be expressed as

The real system wavefront sub-SW and virtual system wavefront $\textrm{sub - }S{W^\prime }$ can be expressed as

2.3 Defocus error elimination

In this step, the defocus error is eliminated by performing the least-square-fitting operation to the virtual sub-SFUT and measured sub-SFUT.

According to Eqs. (2), (3), and (4), the output virtual SFUT ${S_1}({x_1},{y_1})$ and sub-SFUT $S_1^\mathrm{\prime }(x_1^\mathrm{\prime },y_1^\mathrm{\prime })$ without consideration of Δs after the IRO algorithm can be expressed as

According to Eqs. (2), (3), and (4), the first part real SFUT and sub-SFUT can be expressed as

For the second and third part, the error in sub-SFUT introduced by the defocus and random error can be obtained by combining the second formula in Eq. (9) and Eq. (10)

In Eq. (11), ${W_{\textrm{def}}}^{\prime}$ is approximately linear related to Δs, because the optical path of sub-VI has a large F/# [25]. When the sample number is large enough, $\alpha ({x_1},{y_1})$ can be assumed as a constant α, then Eq. (11) can be written as

In Eq. (12), k is the correlation coefficient between Δs and W_def, Z₄ is the Zernike polynomial of defocus aberration. The last part α in Eq. (12) is similar to the piston aberration, which means this part has little effect in the least-square-fitting operation and can be neglected in the following fitting process [5], then we can get

In Eq. (13), ${{\boldsymbol{I}}^\prime }$ is a vector including the incident angles in Ritchey angle θ, which can be calculated in the VI according to Eq. (13) and the last formula in Eq. (2); ${\bullet}$ is the symbol of the vector dot product. Since the sub-SFUT $S_0^\mathrm{\prime }(x_1^\mathrm{\prime },y_1^\prime )$ has been measured before the test, only $k \cdot \Delta s$ and α are unknown in Eq. (12). Using the least-square-fitting method, we can yield the defocus vector s as

Then, full-VI can be modeled, and the reconstruction process of SFUT is implemented in the full-VI with Eq. (12) by neglecting the α and replacing the sub-aperture data with the full-aperture data.

And the error in SFUT caused by the W_ran can be expressed as follows according to Eq. (12):

Generally, the Ritchey angle θ is less than 50°, the incident angle I of any ray will generally be less than 60° and the cos I will be larger than 0.5. The error introduced by the W_ran in SFUT can be estimated as

Owing to only one SW is needed in the proposed method to eliminate the defocus error, the coupled system error caused by the least-square-fitting operation between the SW in different positions is avoided [16]. According to Eqs. (15) and (16), the error introduced by the W_ran in calculated SFUT is approximately linear related to the W_ran, and can be easily estimated in the error analysis, which sufficiently increases the reliability of the test result.

In summary, based on the proposed method, the procedure of the SRI is described as follows.

However, it is difficult to accurately pick out the sub-SW in SW, especially in low spatial resolution test scenarios, because the SW is distorted into an asymmetric elliptical in the real experiment. An optional alternative is described as follows.

Two approaches differ in the calculation step of virtual sub-SFUT. The first approach obtains the virtual sub-SFUT by establishing the sub-VI directly. The second procedure obtains the virtual SFUT by establishing the full-VI. Because ${S_1}({x_1},{y_1})$ is circular, it is easy to pick out the virtual sub-SFUT in virtual SFUT according to the ratio between r₁ and r₂. The two kinds of procedure are essentially the same since the sub-VI is part of the full-VI. In this paper, the virtual sub-SFUT is obtained by the second approach.

3. Experiments

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

In section 3.2, a verification experiment testing a Φ2050 mm flat mirror was conducted. The result was compared with the test result of the traditional Ritchey-Common interferometry to further validate the feasibility of the SRI in the test of a large optical flat.

3.1 Φ100 mm experiment

The experimental setup is shown in Fig. 4. It comprised a precisely calibrated 4” Zygo DynaFiz interferometer, a linear guide rail, a set of magnetic grid displacement measurement modules, a spherical mirror, and a test flat with its adjustment mechanism. The linear guide 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. In this experiment, r₁ was 50 mm, r₂ was 10 mm, R was 1052 mm, d₁ was 780 mm, θ was 20°, and laser wavelength λ was 632.8 nm.

 

Fig. 4. Experimental setup of Φ100 mm flat mirror.

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The surface form of the spherical mirror was calibrated by the Zygo interferometer before the test. The results are shown in Fig. 5(a). The peak to valley (PV) of the surface form of the spherical mirror is 0.0992 λ, and root mean square (RMS) is 0.0090 λ. The cumbersome position change step is replaced with the measurement of sub-aperture before the test. The sub-SFUT S₀^′ (x₁^′, y₁^′) is also measured by the Zygo interferometer and is shown in Fig. 5(b). The PV of S₀^′ (x₁^′, y₁^′) is 0.0558 λ, and RMS is 0.0092 λ.

 

Fig. 5. (a) is the surface form of the spherical mirror and (b) is S₀^′ (x₁^′, y₁^′).

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Then the SW was measured in the test setup by the interferometer. Because the extra system wavefront measurement in different Ritchey angles is replaced with the measurement of sub-SFUT, only one SW need to be measured in this step. Then the error introduced by the surface form of the spherical mirror was eliminated after deducting W_sph from the SW. The modified SW is shown in Fig. 6(a). The aperture is distorted to an asymmetrical elliptical, and the spatial resolution of the wavefront is 958 × 1021. Then full-VI was preliminarily modeled by neglecting Δs, and S₁(x₁, y₁) was calculated by the IRO algorithm, as shown in Fig. 6(b). The aperture is restored to a circular one, and S₁^′ (x₁^′, y₁^′) is picked out according to the ratio between r₁ and r₂. The S₁^′ (x₁^′, y₁^′) is marked with a red circle in Fig. 6(b) and is shown in Fig. 6(c). Figure 6(b) and Fig. 6(c) share the colorbar to describe the detail furtherly.

 

Fig. 6. Surface results (a) is SW, (b) is S₁(x₁, y₁), (c) is S₁^′ (x₁^′, y₁^′), (d) is SFUT, (e) in the direct test with Zygo interferometer and (f) the surface form deviation between (d) and (e).

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The defocus vector s was calculated by the approach proposed in section 2.2. In this test, s was 0.0277 µm. Then real SFUT can be obtained with s after modeling the full-VI. The surface results of SRI and direct test in the Zygo interferometer are in close agreement and are shown in Figs. 6(d) and (e), and the error is defined as the deviation between them, which is shown in Fig. 6(f). Figures 6(d), (e) and (f) share the colorbar, and the PV and RMS of the surface results in these three figures are presented in Table 1. It shows that the accuracy of the SRI reaches the upper limit of the interferometer in engineering (error of PV < 0.1 λ).

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

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To compared the performance of SRI and traditional Ritchey-Common interferometry, which needs the SWs in two Ritchey angles, another SW is measured by setting θ to 30°. In our previous work, a virtual-real combination Ritchey-Common interferometry (VRCRI) is proposed to realize high accuracy measurement in various test scenarios. Then, the SFUT is calculated by SRI and VRCRI respectively. The test results are shown in Table 2 and are in close agreement because a fine test environment control. Meanwhile, since the coupled system error is avoided in SRI, the measurement error of SRI in 20° and 30° both are smaller than the VRCRI. Our proposed method can increase the accuracy of Ritchey-Common test.

Table 2. Test results of SRI and VRCRI in Φ100 mm experiment (Unit: λ)

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3.2 Φ2050 mm experiment

The accuracy and feasibility of SRI have been verified in section 3.1 with a small flat. Nevertheless, the test environment of a large flat mirror differs from that of a small flat mirror. In this section, we further demonstrate the feasibility of the SRI in the test of a large flat mirror.

In this experiment, r₁ was 1025 mm, R was 25000 mm, d₁ was 18177 mm, θ₁ was 27.19°, θ₂ was 42.91°, the aperture of the spherical mirror was 2700 mm and laser wavelength λ was 632.8 nm. The diameter of SFUT was beyond the measurement range of the commercial interferometer. Thus, the correctness of the test result of SRI was verified by comparing it with the test result of VRCRI in this experiment. As shown in Fig. 7, a dynamic interferometer (4D PhaseCam 6000) was utilized to realize the instantaneous measurement of SW to accommodate the dynamic test scenarios. Creating a stable airflow environment in such a large test scenario is difficult and expensive. Thus, two fans were utilized to create a rapidly changing airflow environment, and the environmental error would be randomly changing. Then, the influence of the airflow in a long optical path was decreased by averaging 200 measurement results of SW.

 

Fig. 7. Experimental setup of Φ2050 mm flat mirror.

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The temperature is strictly controlled to satisfy the requirement of traditional Ritchey-Common interferometry. Temperature stabilization time is more than 24 hours before the test to ensure approximately constant temperature condition during the test. Moreover, the mounting system of the test flat and spherical mirror is specially designed with high costs to avoid mirror deformation when changing the Ritchey angle. Meanwhile, both the weight of the flat mirror and spherical mirror with their mounting system are more than 4 tons. The position change of the mirror needs a series of operations with the help of a crane, which is time-consuming and dangerous. The stringent requirement of multiple tests significantly increases test costs, reduces test efficiency, and threatens manufacturing safety. And all these problems can be avoided in a single-test approach.

In this experiment, the aperture of sub-SFUT is 1400 mm, meaning r₂ was 700 mm. Notably, ‘1400 mm’ is the maximum sub-SFUT size our laboratory can provide. The required sub-SFUT size is much smaller than this size, and the aperture size requirement of sub-SFUT will be discussed in the next section. The aperture of the interferometer in our laboratory is 4” and cannot satisfy the measurement requirement of the sub-SFUT. As a good approach to large aperture measurement, the sub-SFUT was measured by VRCRI with another Ritchey-Common test setup. A Φ1800 mm spherical mirror is utilized as the return sphere in this sub-aperture test setup. The surface form of the Φ1800 mm spherical mirror was calibrated by the 4D interferometer. The results are shown in Fig. 8(a). The PV is 0.1788 λ, and the RMS is 0.0179 λ. Then, the sub-SFUT was measured by the VRCRI and is shown in Fig. 8(b).

 

Fig. 8. (a) is the surface form of the Φ1800 mm spherical mirror, and (b) is S₀^′ (x₁^′, y₁^′).

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The surface form of the Φ2700 mm spherical mirror was calibrated and is shown in Fig. 9(a), wherein PV is 0.2085 λ, and RMS is 0.0209 λ. The SW is shown in Fig. 9(b). Then the S₁(x₁, y₁) was calculated by the IRO algorithm and is shown in Fig. 9(c). Pick out the S₁^′ (x₁^′, y₁^′) in S₁(x₁, y₁), then the s was calculated by modeling the sub-VI. In this test, s was -0.0025 µm. Then real SFUT can be obtained with s after modeling the full-VI. The surface result of SRI is compared with the result of VRCRI, and both results are shown in Figs. 9(d) and (e), and the error is defined as the deviation between them, which is shown in Fig. 9(f). Figures 9(d), (e) and (f) share the colorbar, and the PV and RMS of the surface results in these three figures are presented in Table 3. The test results of SRI and VRCRI are in close agreement. The RMS of the error between them is 0.0057 λ, which means the SRI can sufficiently eliminate the defocus error in the test.

 

Fig. 9. Surface results (a) is the surface form of Φ2700 mm spherical mirror, (b) is SW in 27.19°, (c) is S₁ (x₁, y₁), (d) is SFUT, (e) in the test with VRCRI and (f) the surface form deviation between (d) and (e).

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Table 3. Test results of VRCRI and SRI in Φ2050 mm experiment (Unit: λ)

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4. Discussions

4.1 Sub-aperture size

The sub-aperture size is an important parameter in the SRI. In this section, we assess the influence of sub-aperture size through numerical simulations and experiments.

SFUT is randomly generated with 78 terms of Zernike polynomials, and its surface form is shown in Fig. 10(a). The PV of SFUT is 1.2873 λ, and its RMS is 0.1538 λ. The optical path parameters in section 3.1 are utilized in this simulation. Thus, the Ritchey angle θ was 20°; d₁ was 780 mm; r₁ was 100 mm; R was 1052 mm; λ was 632.8 nm, Δs was 0.1 λ, and sub-aperture size is set as a certain proportion of the full aperture. The SW was constructed based on Eqs. (2), (3), (4) and is shown in Fig. 10(b).

 

Fig. 10. Simulated data for analysis. (a) SFUT, and (b) SW.

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Then, the measured SFUT results are calculated by SRI with different sub-aperture sizes. The RMS of surface form deviation between the measured SFUT and the preset SFUT is defined as the RMS error, which is shown in Fig. 11(a). It can be deduced that the theoretical accuracy of the SRI reaches 4.7 × 10⁻⁴ λ, and the error shows uncorrelated with the sub-aperture size.

 

Fig. 11. RMS errors in different sub-aperture sizes (a) in simulation and (b) in the Φ100 mm experiment.

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To further assess the influence of sub-aperture size in the experiment, the SFUT is calculated with SRI by changing the sub-aperture size in Section 3.1. The influence of sub-aperture size in the experiment is shown in Fig. 11(b). The RMS error will significantly increase if r₂ is less than 20% of r₁. When r₂ is greater than 20% of r₁, the sub-aperture size has little influence on measurement accuracy.

The simulation and experimental results show differences. The reason for this phenomenon is worth discussing, and we try to analyze it as follows. In the ideal condition, the sub-aperture size has no influence on the measurement accuracy and the S₀^′ (x₁^′, y₁^′) is closely related to S₁^′ (x₁^′, y₁^′) as described in Eq. (13). Nevertheless, the measurement of SW will inevitably be affected by some environmental factors such as airflow and vibration. Errors will be introduced in SW and S₁^′ (x₁^′, y₁^′), thus decreasing the accuracy. The s is calculated by the least-square fitting method, and the number of data points in the calculation should be large enough to ensure the algorithm's performance. The SRI can perform best in this experiment when the sub-aperture size exceeds 20% of the full aperture.

The environmental errors can be well controlled in a small flat mirror test. Nevertheless, it is not easy in a large flat mirror test. In order to provide a reference for the testing of the large flat mirror, the SFUT in the Φ2050 mm experiment is calculated with SRI by changing the sub-aperture size in Section 3.2. The influence of sub-aperture size in the Φ2050 mm experiment is shown in Fig. 12. It shows that the SRI can perform best when the sub-aperture size exceeds 35% of the full aperture, which is higher than the requirement in the Φ100 mm experiment. The optical path in the Φ2050 mm experiment is much longer than the Φ100 mm experiment, and the environmental error would significantly increase. It means the required sub-aperture size is highly dependent on environmental error. Generally, the SRI can perform best when the sub-aperture size exceeds 35% of the full aperture in a large flat mirror test. A Φ717.5 mm sub-aperture is required in the test of Φ2050 mm flat mirror, and the sub-SFUT can be measured by a Φ800 mm commercial interferometer. The required sub-aperture size can be decreased with a good test environment control.

 

Fig. 12. RMS errors in different sub-aperture sizes in the Φ2050 experiment.

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4.2 Interpolation algorithm

The procedure of the Ritchey-Common test can be summarized in three steps. First, SW with an approximate elliptical aperture was stretched into a fully circular area with an unchanged spatial resolution. In this step, the lack of pixels is inevitable and will lead to the loss of surface information [19]. Second, the pixel gap is filled with polynomial fitting or an interpolation algorithm. Third, the SUFT is obtained after eliminating the defocus error.

In the third step, the least square fitting method is utilized to calculate the defocus and the data point number of S₀^′ (x₁^′, y₁^′) and S₁^′ (x₁^′, y₁^′) (or the data of two Ritchey angles in traditional Ritchey-Common test) should be same, which means the pixel filling operation in the second step is inevitable.

In the second step, the SRI and VRCRI utilize the Zernike polynomial to fill the pixel gap. Another approach, named the coordinate transform (CT) method, was proposed by Doerband in 1999 [5]. The CT method utilizes some interpolation algorithms to fill the pixel gap, which is commonly adopted in image processing, such as the nearest neighbor method, cubic interpolation method, etc.

As shown in Fig. 13(a), the SW in the Φ2050 mm experiment contains many high spatial frequency information, which can hardly be characterized even with hundreds of terms of Zernike polynomials. The Zernike fitting error with 231 terms of the Zernike standard polynomial is shown in Fig. 13(b). The RMS of the fitting error is 0.0496 λ, which is unacceptable in the test. The insufficiency of the Zernike polynomial means that the test result in Section 3.2, shown in Fig. 13(c), can sufficiently eliminate the defocus error, but it failed to characterize the SFUT accurately.

 

Fig. 13. (a) SW in the Φ2050 mm experiment, (b) Zernike fitting error of the SW, (c) SFUT calculated by SRI, (d) SFUT calculated by combining the CT method and SRI.

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Then, the SFUT in Section 3.2 is calculated by combining the CT method and SRI, shown in Fig. 13(d). The IRO algorithm is replaced with an interpolation algorithm of the CT method. The high spatial frequency information is well characterized in this test result, and the PV is 0.3467 λ, RMS is 0.0226 λ. Meanwhile, the PV of the test result in Section 3.2, shown in Fig. 13(c), is 0.1653 λ, and the RMS is 0.0152 λ. The reason for the difference between the test results of the two methods is analyzed as follows.

In SRI, the Zernike polynomial can accurately characterize the surface form but will lose some high spatial frequency information when the utilized term is limited. In CT method, the interpolated points are calculated with curved surface fitting according to the existing data, while each part of the measured surface exists independently. Obviously, this is not rigorous for a high-precision metrology solution. This method can better restore the high spatial frequency information in the SFUT but will decrease the confidence of test results.

In summary, the IFM method and SRI lose some high spatial frequency information in the test result; meanwhile, the CT method adds some fake information to the test result. This problem is unsolved and needs further study. In order to decrease the influence of the interpolation problem, a small Ritchey angle would be a wise choice in the test. The SRI can be combined with Zernike polynomial or some interpolation algorithms to accommodate various test requirements.

5. Conclusions

In this study, for the first time to our knowledge, we proposed a single-test Ritchey-Common interferometry that can eliminate the defocus error without the position change. The extra system wavefront measurement in different positions is replaced with the direct measurement of sub-SFUT to eliminate the defocus error. In most test scenarios, the sub-aperture size is between 20% to 35% of full aperture and can be further decreased with a fine test environment control, so the measurement of sub-SFUT is easy to implement. The performance and feasibility of this method have been proved by a set of verification experiments. In comparison with the direct test result of a standard Zygo interferometer, the PV error of the SRI is 0.0889 λ, and the RMS error is 0.0126 λ. This method can achieve high accuracy with only one test of SW, thus avoiding the obligatory position change in the test for the large optical flats. We believe that SRI can sufficiently increase the accuracy, reliability, safety and test efficiency, decrease the test cost and influence of environmental factors in the test of large optical flats.