Iterative deformation calibration of a transmission flat via the ring-point support on a 300-mm-aperture vertical Fizeau interferometer

Yuntao Wang; Lei Chen; Chenhui Hu; Jia Chen; Zhengyu Zhang; Donghui Zheng; Tuya Wulan

doi:10.1364/OE.411083

1. Introduction

Large-aperture laser interferometers [1,2] are precise, non-contact testing equipment for high-end optics [3–5], such as high-power lasers, space optics, synchrotron radiation, and lithography objectives. When large-diameter transmission flats are mounted on interferometers, they are unavoidably deformed under the forces of clamping [6] and gravity [7], especially for apertures larger than 300 mm. The deformation can be calibrated and compensated accurately through simulations [8] and measurements [9]. At the National Metrology Institute of Japan (NMIJ), the deformation of a 325-mm-aperture transmission flat has been evaluated using the finite element method (FEM) and a scanning profiler. The measurement uncertainty reached λ/77 [8,10]. At Zygo, a 450-mm-aperture flat has been globally corrected. The expanded uncertainty of the interferometer was less than 3 nm [11]. In addition, at Physikalisch-Technische Bundesanstalt (PTB), the X-ray free-electron laser mirrors were characterized with nanometer accuracy on a 300-mm-aperture Fizeau interferometer [12]. Elsewhere, at the Commonwealth Scientific and Industrial Research Organisation (CSIRO), the gravity-induced deformation of the flat has been measured with polymer foam. The error was determined to be less than 4 nm [13]. Then, the three-flat method can be extended through a combination of proper sequencing, rotational averaging, and their symmetry properties. The uncertainty was less than ± 1.5 nm [14]. Moreover, at the National Institute of Standards and Technology (NIST), the three-flat method containing deformation was derived based on rotational and mirror symmetries [6].

Multiple support solutions exist for large-diameter transmission flats, such as adhesive support, tape suspension support, point support, and groove support. Among them, adhesive support [15,16] and point support [17] are the most suitable for transmission flats. The adhesive support can ensure the stability and non-contact working surfaces. However, the materials of the aluminum frame, adhesive, and flat need to be matched to reduce changes caused by stress. Furthermore, surface calibration cannot be performed with adhesive support. Regarding point support, Walter B. Emerson reported as early as 1952 that the bending deformation of a flat could be determined based on differential bending. The bending deformation varies directly depending on the fourth power of the radius [18]. Dew et al. remarked: “Given the frequent use that is made of three-point edge supports for interferometer flat, it is surprising that so few attempts have been made to measure the sag experimentally” [19]. Theoretical elastic deformations of plates have been a significant subject for nearly sixty years. For thin flats, Wan et al. proposed a theory based on multiple axial supports [20]. Alternatively, Ventsel et al. offered a comprehensive systematic representation of thin plates and shells [21]. For thick flats, closed-form expressions of deformations of a circular mirror on a ring support were introduced by Selke [22]. Then, Arnold reported the deformation expressions of a mirror with a central hole. The low-order aberrations were fitted with combined actuator influence functions [23]. Furthermore, Pokorný et al. reported detailed deformation expressions of various point supports [24].Then, they introduced approximate solutions for a spherical lens [25]. Nelson et al. described the deformations of uniform-thickness plates supported by discrete points or continuous rings [26]. However, the aforementioned theoretical expressions lack verification through simulations and experiments.

In the past decade, FEM analyses and experimental measurements have been reported. For example, Gu et al. analyzed the properties of gravitational deformation of a reflection flat based on a three-point support with FEM simulations [27]. Quan et al. built an FEM model and conducted real experiments with three steel balls. The sag and misalignment parameters were retrieved by solving a multivariable unconstrained optimization problem [17]. In addition, Zhou et al. developed an active optics system based on a whiffletree-supported mirror. The FEM analysis was used to compensate for the low-order aberrations caused by thermal changes and gravity relief [28]. Today, simulations and experimental measurements are used predominantly to calibrate reflection flats. However, there is no complete calibration scheme for transmission flats or a detailed calibration analysis of the support process.

In this paper, we proposed a ring-point support scheme for the deformation calibration of a large-diameter transmission flat. First of all, the calibration theory of the ring-point support with elastic deformation was derived. Then, the changes in the surface and stress field of the transmission flat were analyzed quantitatively using FEM modeling, leading to the optimization of the support structure. At last, to validate the proposed calibration approach, we performed an absolute test of the transmission flat using a liquid Ref. [29,30] according to the requirements of ISO 17025 [31–35]. The test result was compared to a measurement of the Zygo interferometer demonstrating the effectiveness of the proposed ring-point support design.

2. Principles

The deformation of transmission flats is closely related to the support mode, material, and manufacturing process. There are two support solutions for transmission flats considered here: adhesive support in the flank, and point support on the bottom, as shown in Fig. 1. The adhesive support can simplify the structure, reduce weight, and realize non-contact testing of the working surface. However, it is difficult to disassemble the flat after adhesives have been applied. In addition, the surface cannot be adjusted with adhesive support. In contrast, a ring-point support can improve the surface accuracy of the flat by enabling adjustment of the microarea.

Fig. 1. Support scheme for a transmission flat. (a) Adhesive support; (b) Ring-point support.

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For large-diameter transmission flats, this paper proposes a uniform distributed ring-point support scheme. When a flat is mounted horizontally, the configuration is illustrated in Fig. 1(b). The origin coordinates are at the center of the flat. According to the theory of elasticity, the deformation equation W = W(r, φ) with polar coordinates (r, φ) is given by

(1)$${\nabla ^\textrm{2}}({{\nabla^\textrm{2}}W({r,\varphi } )} )= \frac{q}{D}, $$

where q denotes the transverse distributed load. The Laplace operator ${\nabla ^\textrm{2}}$and flexural rigidity D can be expressed as

(2)$$\left\{ \begin{array}{l} {\nabla^\textrm{2}}\textrm{ = }\frac{{{\partial^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial^2}}}{{\partial {\varphi^2}}}\\ D = \frac{{E{h^3}}}{{12({1 - {\nu^2}} )}} \end{array} \right., $$

where E denotes the Young’s modulus, h represents the thickness of the flat, and v is Poisson’s ratio.

When the flat is supported by ring points with k-fold symmetry, it is in static equilibrium with the gravity G of the flat. Each discrete support j is characterized by its forces f_j and its polar coordinates (b_j, ϕ_j). Here, it is assumed that the diameter of the points is negligible relative to the diameter of the flat. Therefore, the sums of the applied forces and moments are zero:

(3)$$\left\{ \begin{array}{l} P = G\\ \sum\limits_{j = 1}^k {{f_j} - P} = 0\\ \sum\limits_{j = 1}^k {{f_j}{b_j}\exp ({i{\varphi_j}} )} = 0 \end{array} \right., $$

For boundary freedom, the boundary conditions of a flat are

(4)$$ \left\{\begin{array}{l} M_{r \mid r=a}=-D\left[\frac{\partial^{2} W}{\partial r^{2}}+v\left(\frac{\partial W}{r \partial r}+\frac{\partial^{2} W}{r^{2} \partial \varphi^{2}}\right)\right]_{r=a}=0 \\ V_{r \mid r=a}=-D\left[\frac{\partial}{\partial r} \nabla^{2} W+\frac{1-v}{r} \frac{\partial^{2}}{\partial r \partial \varphi}\left(\frac{1}{r} \frac{\partial W}{\partial \varphi}\right)\right]_{r=a}=0 \\ \left.W\right|_{r=b_{j}, \varphi=\varphi_{j}=0}(j=1,2, \ldots, k) \end{array}\right. $$

where M_r represents the radial bending couple, and V_r is the transverse shear-stress resultant.

To obtain the deformation W(r, φ), it is necessary to obtain the laterally distributed load q:

(5)$$q = \frac{{ - G}}{{\pi {a^2}}} + \frac{{G\delta ({r - b} )}}{{2\pi b}} + \frac{{G\delta ({r - b} )}}{{\pi b}}\sum\limits_{m = 1}^\infty {\cos [{km({\varphi - {\varphi_0}} )} ]} , $$

where G is the gravity of the flat, a is the radius of the flat, and b is the average distance from the center of the flat for all points, that is, b = Σb_j/k. The flat is supported with radius b = βa. In addition, φ₀ is the azimuth of the nearest point relative to the origin axis.

Under static balance (Eq. (3) conditions and boundary constraints (Eq. (4), the uniformly distributed lateral load (Eq. 5) can be used to solve the differential deformation equation (Eq. (1). Therefore, the surface distribution of the flat is obtained as follows:

(6)$$W({r,\varphi } )= {W_0}(r )+ \sum\limits_{m = 1}^\infty {{W_m}(r )\cos ({km\varphi } )} - K, $$

where the constant K ensures zero deformation at all points. Furthermore, W₀(r) represents the fundamental frequency (m = 0), while W_m(r) denotes the frequency components when m ≥ 1.

The surface W(r, φ) is considered in two parts, that is, the flat has different surface distributions for r > b and r ≤ b.

First, for r > b:

(7)$$\begin{array}{c} W({r,\varphi } )= \frac{{q{a^4}}}{{64D}}\left\{ {\left[ {1 - {{\left( {\frac{r}{a}} \right)}^2}} \right]\left[ {\frac{{7 + 3\nu }}{{1 + \nu }} - 4\frac{{1 - \nu }}{{1 + \nu }}{\beta^2} + {{\left( {\frac{r}{a}} \right)}^2}} \right]} \right.\left. { + 8\left[ {{\beta^2} + {{\left( {\frac{r}{a}} \right)}^2}} \right]\ln \left( {\frac{r}{a}} \right)} \right\}\\ + \sum\limits_{m = 1}^\infty {[{{A_m}{r^n} + {B_m}{r^{n + 2}} + {C_m}{r^{ - n}} + {D_m}{r^{ - n + 2}}} ]\cos ({km\varphi } )} - K \end{array}, $$

where the coefficients are expressed as follows:

(8)$$\left\{ \begin{array}{l} {A_m} = \frac{q}{{8D}}\frac{{({1 - \nu } )}}{{({3 + \nu } )}}\frac{{{\beta^n}}}{{n({n - 1} )}}\left[ {\frac{{8({1 + \nu } )}}{{n{{({1 - \nu } )}^2}}} + n - ({n - 1} ){\beta^2}} \right]{a^{4 - n}}\\ {B_\textrm{m}}\textrm{ = } - \frac{q}{{\textrm{8}D}}\frac{{({1 - \nu } )}}{{({3 + \nu } )}}{\beta^n}\left( {\frac{1}{n} - \frac{{{\beta^2}}}{{n + 1}}} \right){a^{2 - n}}\\ {C_m} ={-} \frac{q}{{8D}}\frac{{{\beta^{n + 2}}}}{{n({n + 1} )}}{a^{4 + n}}\\ {D_m} = \frac{q}{{8D}}\frac{{{\beta^n}}}{{n({n - 1} )}}{a^{2 + n}} \end{array} \right., $$

Second, for r ≤ b:

(9)$$\begin{array}{c} W({r,\varphi } )= \frac{{q{a^4}}}{{16D}}\left\{ {({1 - {\beta^2}} )\left[ {\frac{{3 + v }}{{1 + v }} - \frac{{1 - v }}{{1 + v }}{{\left( {\frac{r}{a}} \right)}^2}} \right] - \frac{1}{4}\left[ {1 - {{\left( {\frac{r}{a}} \right)}^2}} \right]\left[ {\frac{{5 + v }}{{1 + v }} - {{\left( {\frac{r}{a}} \right)}^2}} \right]} \right.\\ \left. { + 2\left[ {{\beta^2} + {{\left( {\frac{r}{a}} \right)}^2}} \right]\ln \beta } \right\} + \sum\limits_{m = 1}^\infty {[{{E_m}{r^n} + {F_m}{r^{n + 2}}} ]\cos ({km\varphi } )} - K \end{array}, $$

where the coefficients can be expressed as follows:

(10)$$\left\{ \begin{array}{l} {E_m} = {A_m} + \frac{q}{{8D}}\frac{{{\beta^{ - n + 2}}}}{{n({n - 1} )}}{a^{4 - n}}\\ {F_m} = {B_m} - \frac{q}{{8D}}\frac{{{\beta^{ - n}}}}{{n({n + 1} )}}{a^{2 - n}} \end{array} \right., $$

where n = km is the m-order expression of the k points.

Finally, the deformation W(r, φ) must be minimized. By subtracting low-order rotational symmetric aberrations with Zernike polynomials, the calibrated surface W_calibration(r, φ) is generated. Furthermore, the root mean square σ_rms of the calibrated surface can be used to determine whether the calibration process is complete:

(11)$${W_{calibration}}({r,\varphi } )\textrm{ = }W({r,\varphi } )+ {p_2}{({r/a} )^2} - \sum\limits_{n.m} {({{c_n}^m{Z_n}^m + {p_{2nm}}{{({r/a} )}^2}} )}, $$

where r/a is the normalized radius and n and m represent the set of low-order rotational symmetric aberrations. The small parabolic deformation p₂(r/a)² is tolerated when the flat moves in the vertical direction. At this stage, p₂ and p_2m are zero if there is no movement.

In summary, under static balance conditions and boundary constraints, the uniformly distributed load can be used to solve the differential deformation equation, which enables the surface distribution of the flat to be determined. The surface distribution is considered for two regimes, namely r > b and r ≤ b. The number k, distance b, and force f_j of the points are the key factors affecting the deformation, which require detailed analysis. Furthermore, by subtracting the low-order rotational symmetric aberrations, the calibration surface W_calibration is generated. The calibrated root mean square σ_rms can be used to determine whether the calibration process is complete.

3. Simulation

To obtain the optimized support structure and force operation, the simulations are designed to compare the theory and FEM, the adhesive support and ring-point support, various numbers k of support points, various distances b from the center of the flat to the support points, and various forces f_j. The deformation simulations were performed using COMSOL Multiphysics 5.4 software. The flat comprises fused silica with a 350-mm diameter and a 90-mm thickness. The frame is made of aluminum, the adhesive is made of adhesive silicone, and a polytetrafluoroethylene (PTFE) plastic coating is used to protect the flat from scratching. The number of triangular elements was 5,700,000. Table 1 lists the properties of the materials used in the system.

Table 1. Material properties of the large-diameter flat.

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Before deformation simulations, a comparison between deformation theory and FEM was performed, as shown in Fig. 2. The deformation theory can be applied to both thin and thick flats. Hence, a flat having thickness-to-diameter ratios equal to one-tenth (h = 35 mm, 2a = 350 mm) is chosen for versatility. In addition, the support structure is a three-point support with b = 0.68a. Figure 2(a) shows the theoretical calculation, and Fig. 2(b) shows the FEM analysis. In Fig. 2(a), it is worth noting that high-frequency components are neglected for polynomial decomposition. Hence, they appear as invalid data, which can be compensated for by interpolation methods [36,37]. Comparing the two results, the topographies are consistent, and the difference in amplitude is small. This proves that the theory is consistent with FEM.

Fig. 2. Comparison between theory and FEM. (a) Theory: PV = 87.60 nm, RMS = 18.55 nm; (b) FEM: PV = 89.33 nm, RMS = 18.60 nm.

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First, the ultimate deformation of the flat was compared for the adhesive support and ring-point support. In Fig. 3(a) and 3(b), 26 adhesive spots can achieve the ultimate deformation of the adhesive support. In addition, an overall deformation occurs for global surface. The edge shifts down by 10.88 nm, indicating that the adhesive support cannot keep the flat completely fixed. In contrast, 100 points can achieve the ultimate deformation of the ring-point support, as shown in Fig. 3(c) and 3(d). In addition, the flat is completely fixed when the minimum deformation is zero. Furthermore, with the 300-mm aperture, the deformation amount supported by adhesive support (20.31 nm, PV) is larger than the ring-point support (16.77 nm, PV). This proves that the ring-point support provides more effective support. Therefore, the ring-point support was selected for further investigation of the large-diameter flat.

Fig. 3. Two different supports in a single ring configuration. (a) 26 adhesive spots in flank with 350-mm aperture; (b) Convergent tendency of deformation via adhesive support with 300-mm aperture; (c) 100 points on the bottom with 350-mm aperture; (d) Convergent tendency of deformation via ring-point support with 300-mm aperture.

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Then, several key parameters of the ring-point support were analyzed. In practice, the number of points is finite and discrete. There must be a certain separation distance to satisfy feasible operations. To ensure a reasonable point distribution and small deformation, simulations were performed for different numbers. In Fig. 4, in order to maintain balance in the flat, an 8-point support is considered first. The flat with 8-point support exhibits the largest deformation as well as the most extensive sag area (97.86%). As the number of support points increases, the size of the sag area decreases, and the rotational asymmetric aberration gradually disappears, indicating that gravity relief is achieved. For a 32-point support, the deformation at the regions surrounding the support points gradually synchronizes. As the number of points continues to increase, the topography no longer demonstrates significant change, although the deformation amplitude continues to decrease. The curve in Fig. 4(e) is generated by the deformation simulations in Figs. 4(a)-(d). The deformation gradually reduces and converges to 16.77 nm for the 300-mm aperture. For a 340-mm point separation distance (i.e., b = 170 mm), the 100-point support is sufficient to achieve minimal deformation. However, acceptable deformation is achieved when the number of support points exceeds 60.

Fig. 4. Influence of the number of points. (a) 8 points, PV = 133.8 nm; (b) 32 points, PV = 50.45 nm; (c) 64 points, PV = 35.05 nm; (d)100 points, PV = 30.07 nm; (e) Deformation curves for 300-mm aperture (red dashed) and 350-mm aperture (blue solid).

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Furthermore, the variation of the distance b also affects the deformation of the flat, as shown in Fig. 5. Although the topography only changes by a small amount, differences are observed for the sag range and deformation amplitude. For the 300-mm aperture, the minimum in the deformation curve corresponds to a 314-mm point separation (i.e., b = 157 mm). As the point separation distance increases further, the deformation amplitude gradually increases. However, to achieve a larger observation range, point separations exceeding 310 mm also have an excellent support effect.

Fig. 5. Influence of the distance b. (a) 300 mm (b = 150 mm), PV = 24.83 nm; (b) 310 mm (b = 155 mm), PV = 26.73 nm; (c) 320 mm (b = 160 mm), PV = 28.88 nm; (d) 340 mm (b = 170 mm), PV = 33.86 nm; (e) Deformation curves for 300-mm aperture (red dashed) and 350-mm aperture (blue solid).

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Finally, during calibration, the stress change within the flat was studied with various applied forces f_j. It is necessary to limit the force applied to each support point to reduce the stress change inside the flat, which can both protect the flat and improve the measurement accuracy. In the case of an axisymmetric stress field, four types of stress components (shear, axial, radial, and hoop stresses) are schematically represented in cylindrical coordinates in Fig. 6(a). Based on the stress values of the four types of stress components, the maximum principal stress σ_p and principal directions θ_p [38,39] are calculated:

(12)$${\sigma _p} = ({{\sigma_r} + {\sigma_z}} )/2 + {\{{{{[{({{\sigma_r} - {\sigma_z}} )/2} ]}^2} + {\tau_{zr}}^2} \}^{{1 / 2}}}, $$

(13)$$\tan {\theta _p} ={-} ({{\sigma_r} - {\sigma_z}} )/2{\tau _{zr}} + {\{{{{[{({{\sigma_r} - {\sigma_z}} )/2{\tau_{zr}}} ]}^2} + 1} \}^{{1 / 2}}}. $$

Fig. 6. Stress field for the ring-point support design. (a) Stress components active at the support points; (b) Principal stress line of the transmission flat; (c) Principal stress corresponding to a 0-N loading for each support; (d) Principal stress corresponding to a 0.2-N loading for each support; (e) Principal stress corresponding to a 2.6 N-loading for each support; (f) Three-dimensional stress field corresponding to a 90-N loading for one support; (g) Three-dimensional stress field corresponding to a 5.8-N loading for each support (32 points); (h) Three-dimensional stress field corresponding to a 2.6-N loading for each support (72 points).

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The maximum principal stress distributions are shown globally and partly in Figs. 6(b)-(e). When no force (0 N) is applied to the flat by the support points, the flat is supported by an aluminum frame. The stress occurs at the edge of the clear aperture of the frame. In addition, the stress direction is perpendicular to the deformation outline of the flat. Hence, it is vital to protect the flat from damage using a buffer film at the clear aperture. Upon application of a slight force (0.2 N) to each of the 72 points simultaneously, the inner edge of the frame is no longer stressed. The force on the support points is increased further, until each support point is subject to a 2.6-N force. At this stage, the edge of the clear aperture no longer exhibits stress, indicating complete support provided by the points. Concurrently, the maximum principal stress is 4.41×10⁵ N/m² in the wedge direction of the flat.

The stress change inside the flat caused by the support points was also studied. The isometric tomography results for the stress components are synthesized into a three-dimensional stress field, as shown in Figs. 6(f)-(h). When a 90-N force is applied to a single support point, the stress area within the flat is very large (4.23%). In particular, in polarization interference, an additional phase delay can disrupt the initial phase, which results in measurement errors. When the 32-point support is used, although the continuous ring effect is shown in Fig. 4(b), the stress region of the internal flat is still discrete in Fig. 6(g). In response to doubling the number of support points, the stress change area inside the flat retains a ring-shaped zone (0.0037%). As the number of support points increases, the force required at each point decreases, resulting in internal stress being reduced. In addition, the main stress area also shrinks to achieve gravity relief. However, the stress caused by gravity remains significant, which needs a concave topography in the processing. Hence, a uniformly distributed 72-point support with a 2.6-N force for each point can reduce the internal stress changes of the flat.

In summary, during the calibration of the transmission flat with a 350-mm diameter and a 90-mm thickness, a 314-mm point separation distance and 100 support points yielded the smallest deformation. However, to achieve a larger observation range, a point separation greater than 310 mm and more than 60 points also provide effective support. In this study, the ring-point support structure using 72 uniformly distributed points and a 340-mm separation distance was selected, with each point subjected to a 2.6-N loading. In addition, for actual operation, it is still necessary to consider the material matching for the flat and aluminum frame, the absolute reference test, and the control of the environmental temperature and humidity.

4. Measurement

4.1 System design of 300-mm-aperture vertical Fizeau interferometer

The calibration of large-diameter flats is a problem that must be considered for a large-aperture interferometer. For a 300-mm-aperture vertical Fizeau interferometer (see Fig. 7(a)), the beam emitted from a wavelength tunable laser (TLB6800LN, λ = 632.8nm, New Focus, USA) passes through the aperture stop. Then, the beam is reflected by the beam splitter and reflectors. Next, the beam is collimated by a collimation lens. Finally, the beam reaches the transmission flat and the test sample, a liquid for example, which generates a coherent beam. After returning to the vertical Fizeau system, the coherent beam propagates through the beam splitter and imaging lens. Finally, interference fringes are recorded using a charge-coupled device (Point Grey Research Grasshopper3 GS3-U3-51S5M, 2448 × 2048, FLIR, USA). As depicted in Fig. 7(b), to protect the optical system from contamination and ensure the stability of the interference cavity, the air float vibration reduction device, stable sample table, isolation hood, and support structure were all carefully designed. The total weight of the interferometer was close to 1 ton. During operation, from the power supply and the control box to the motor and the circuit board, the stability is achieved through the precise control of each component. The absolute reference standard was dimethyl silicone. The liquid tray containing dimethyl silicone was placed between the two marble slabs. The upper marble slab has a hole that corresponds to the transmission flat. The material of the frame is aluminum, which allows for rapid cooling. The support points on the bottom can be used for the deformation calibration. In addition, between the support points and the fused quartz flat, PTFE plastic was added to provide a flat with scratch resistance protection.

Fig. 7. 300-mm-aperture vertical Fizeau interferometer. (a) Light path: (S) fiber output, (AS) aperture stop, (BS) beam splitter, (IL) imaging lens, (CCD) charge-coupled device, (S1) wavelength tunable laser, (TD) turning driver, (PC) computer, (Re) reflectors, (CL) collimator lens, (TF) transmission flat, (SO) dimethyl silicone; (b) Three-dimensional structure.

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4.2 Liquid reference

In theory, a liquid surface has the same radius of the earth. When the diameter of the liquid is 300 mm, the deviation of the liquid surface is only 1.77 nm, which can be regarded as a fixed error. For this reason, a liquid surface is an ideal standard for the absolute test. It is necessary to choose a liquid with moderate viscosity and low surface tension.

The stability and surface accuracy of the liquid reference are key factors for absolute testing. The stabilization time and the infiltration effect were studied with synchronous phase shifting and wavelength phase shifting [40,41], respectively. The tray containing dimethyl silicone was placed on the marble platform for 72 h. A series of images of the liquid surface were recorded rapidly by the CCD camera, as shown in Fig. 8(a). The correlation coefficients of two adjacent surfaces were calculated to obtain the quantified stability index. In Fig. 8(b), the average correlation coefficient of five repeated experiments grows linearly within 40 min. During the 72-h measurement, the correlation coefficient remained at 0.99. Following complete stabilization of the liquid reference, a calibration experiment of the flat was performed, with the ability to restore stabilization after only 3 min when interference is introduced.

Fig. 8. Stability of the liquid reference. (a) Fringe pattern over a 72-h period; (b) Correlation coefficient as a function of time; (c) Temperature and humidity control over a 72-h period.

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In addition, a high-precision temperature and humidity air conditioning system was used to control the ambient temperature and humidity, as shown in Fig. 8(c). The root mean square (RMS) of the temperature was 31 mK, and the RMS of the humidity was 0.75%. To ensure measurement accuracy, the duration of stability was considerably longer than the time required to complete the measurement.

Because of the infiltration effect between the solid and liquid, the size of the liquid tray also affects the surface. The liquid close to the wall of the liquid tray must be outside the usable range. The additional pressure across the curved surface can be described using the Young–Laplace equation. The surface tension and corresponding radius of curvature can be expressed as

(14)$$\frac{\Delta \rho gy}{\delta } = \frac{y^{\prime\prime}}{{(1 + y^{{\prime}2})}^{3/2}} + \frac{{y^{\prime}}}{{x \cdot {(1 + y^{{\prime}2})}^{1/2}}}, $$

where △ρ represents the density difference between dimethyl silicone and air, g is the acceleration due to gravity, and δ represents the surface tension of the liquid. The curvature of a point (x, y) in the horizontal and vertical directions can be expressed as the inverse of the first and second terms on the right-hand side of Eq. (14), respectively.

Figure 9 shows the FEM simulations and measurements of the liquid reference. The 100cs dimethyl silicone with a 3-mm thickness was chosen. The diameter of the tray was 400 mm. In Fig. 9(a) and 9(b), when the dimethyl silicone is in contact with the tray wall, the contact angle between them is acute. The dimethyl silicone wets the tray wall, and the adhesion layer extends along the tray wall. Therefore, at the edge, the liquid surface appears as a peak. Because the liquid surface is affected by the infiltration effect and surface tension simultaneously, the liquid surface drops rapidly from a peak to a valley. Finally, the liquid surface maintains a plane across most of the central area. In Fig. 9(c), the height of the surface deformation is 3.92 nm in the 300-mm range, which is an effective plane datum for the absolute test. Figure 9(d) presents the FEM simulation in a cross-section with an abrupt curve of 11.50 mm and 12.21 mm at the edge. The fringes are impacted by the infiltration effect at the 13.95-mm edge in Fig. 9(e). Concurrently, damping is applied to suppress the infiltration effect and the generation of standing waves. Therefore, there is no distinct fringe change in the 300-mm range (see Fig. 9(f)). In summary, the 100cs dimethyl silicone is reasonable as an absolute reference in the 300-mm range. The dimethyl silicone has a 3-mm thickness and a 400-mm diameter.

Fig. 9. Analyses of infiltration effect and surface tension. (a) Three-dimensional FEM modeling; (b) FEM modeling for a 400-mm-diameter tray; (c) FEM modeling for a 300-mm-diameter range; (d) FEM modeling for a cross-section with a 400-mm diameter; (e) Measurement at the edge of the dimethyl silicone; (f) Measurement under normal operation in a 300-mm range.

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4.3 Iterative deformation calibration

With the liquid reference and a stable environment, iterative deformation calibration was conducted for a transmission flat with a 350-mm diameter and a 90-mm thickness. The flat was supported by a ring-point support using 72 uniformly distributed points and a 340-mm separation distance. Figure 10 shows the process for the calibration experiment. Assuming that the flat will not be deformed, the result is the surface topography of the flat under the condition of no gravity. However, the surface of the flat will bend downward under the condition of gravity, resulting in significant power aberration. To reduce the influence of gravity, according to the deformation theory and simulations, the working surface is modified into a concave surface. Then, iterative calibration is performed using the ring-point support. Of course, it cannot be guaranteed that the obtained concave surface after initial polishing will produce the best calibration effect. The initial calibration result can be used to determine whether iterative polishing and calibration are required.

Fig. 10. Deformation under the gravity. (a) No gravity for a planar surface; (b) No gravity for a concave surface; (c) Influence of gravity on a planar surface; (d) Influence of gravity on a concave surface.

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The calibration can be divided into three steps. The first step is to eliminate astigmatism; the second step is to separate the flat from the aluminum frame to achieve ring-point support; and the third step is the micro calibration of each support point to improve the surface accuracy.

A calibration experiment was performed after initial polishing. First, the initial surface displays obvious astigmatism caused by the uneven contact surface of the aluminum frame in Fig. 11(a), reaching 119.56 nm (λ/5, PV). Moreover, a mutation occurs in a small area, where the protruding support points affect the surface. To calibrate the astigmatism, it is necessary to increase the force on the left and right sides, and to reduce the force on the lower side. In Fig. 11(b), astigmatism was eliminated, which is the dominant factor for deformation calibration. This is the first step. After eliminating the astigmatism, the surface accuracy demonstrated a significant improvement, which decreased from 119.56 to 61.18 nm. Nevertheless, the flat was still supported by the aluminum frame, and the deformation caused by gravity was large. Consequently, it is necessary to continue to increase the force to separate the flat completely from the aluminum frame. In the second step, all directions of the surface were adjusted in Fig. 11(c). Because of the different forces applied at the support points, coma aberration and astigmatism appeared. In Fig. 11(d), the sag was pushed up and the flat was separated from the aluminum frame. However, optimal surface accuracy cannot be guaranteed with this step. In the third step, the lower position was calibrated, leading to the reappearance of the coma and astigmatism (see Fig. 11(e)), albeit these aberrations were smaller than those observed in the second step. When the residual sag was alleviated, the final surface was completed in Fig. 11(f), reaching 36.84 nm (λ/17, PV). Figure 11(g) shows the changes in the overall surface accuracy and the three main aberration components (astigmatism, coma, power) for the calibration process. Following the implementation of the calibration, the astigmatism and coma were reduced significantly. Figure 11(h) shows the proportion of the three main aberration components in the total surface. After the first step, the proportion of power was 0.44. In addition, the deformation caused by gravity was large, and the flat was supported by the aluminum frame. For the second and third steps, the proportion of power dropped to 0.33 and 0.34, respectively. At this stage, the flat was no longer supported by the aluminum frame, but was instead completely supported by points. As shown in Fig. 11(i), σ_rms indicates the elimination effect of the rotational asymmetric aberration. Finally, σ_rms converged to 3.89 nm.

Fig. 11. Deformation calibration for the initial polishing. (a) PV = 119.56 nm, RMS = 14.72 nm; (b) PV = 61.18 nm, RMS = 9.29 nm; (c) PV = 70.62 nm, RMS = 10.76 nm; (d) PV = 40.18 nm, RMS = 5.74 nm; (e) PV = 37.75 nm, RMS = 5.58 nm; (f) PV = 36.84 nm, RMS = 5.60 nm; (g) Change in the aberration amplitude; (h) Proportional change in the different aberrations; (i) Change in the root mean square σ_rms for the calibrated surface W_calibration.

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During calibration, aberrations are calibrated by support points. The mounting force of support points is determined by the real-time fringes and three-dimensional surface. The real-time change in the fringes indicates the magnitude of force f_j and the tilt direction of the flat. The three-dimensional surface indicates the specific effect after calibration. In addition, on the guidance of deformation theory, the root mean square σ_rms of calibrated surface achieved convergence, which indicated that the calibration was completed.

To improve the surface accuracy, the edge of the flat needs to be polished. After the second polishing, the flat was placed in the same frame. The astigmatism appears again in Fig. 12(a), indicating that the contact surfaces of the upper and lower sides of the aluminum frame are higher. At this stage, the surface accuracy of the flat was 127.09 nm (λ/5, PV). After eliminating the astigmatism (see Fig. 12(b)), the flat appeared as a rotational symmetric surface. However, the edge is larger than the initial calibration, indicating that the flat was supported by the frame. After repeating the calibration process, the surface accuracy reached 24.03 nm (λ/25, PV), as shown in Fig. 12(f). As a result, the edge was minimized, which proves the effectiveness of the second polishing. Following the first step of calibration, the proportion of power was 0.46. For the second and third steps of calibration, the proportion of power dropped to 0.25 and 0.26, respectively, implying that the flat was completely supported by the points. As demonstrated in Fig. 12(i), σ_rms converged to 3.12 nm. The calibration was complete.

Fig. 12. Deformation calibration for the second polishing. (a) PV = 127.09 nm, RMS = 22.51 nm; (b) PV = 39.85 nm, RMS = 6.65 nm; (c) PV = 35.98 nm, RMS = 6.06 nm; (d) PV = 29.48 nm, RMS = 4.37 nm; (e) PV = 29.42 nm, RMS = 4.50 nm; (f) PV = 24.03 nm, RMS = 3.86 nm; (g) Changes in the aberration amplitudes; (h) Proportional change in the different aberrations; (i) Change in the root mean square σ_rms for the calibrated surface W_calibration.

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Replication calibration experiments were executed using two different trays respectively. Figure 13(a) shows a calibration result of round tray with a diameter of 400 mm. A total of 100 data were collected within 24 h. The result is 24.03 nm (PV), and the standard deviation is 0.49 nm. Figure 13(b) shows a calibration result of square tray with a diameter of 400 mm. The result is 24.31 nm (PV), and the standard deviation is 1.01 nm. The error is 6.01 nm between a round tray and a square tray, which reached λ/100 in Fig. 13(c). Hence, calibration was effective for different trays, and was stable for a long time.

Fig. 13. Replication experiments via different trays. (a) Round tray, PV = 24.03 nm, RMS = 3.86 nm, standard deviation of PV = 0.49 nm, standard deviation of RMS = 0.09 nm; (b) Square tray, PV = 24.31 nm, RMS = 3.94 nm, standard deviation of PV = 1.01 nm, standard deviation of RMS = 0.11 nm; (c) Error between round tray and square tray: PV = 6.01 nm, RMS = 1.20 nm.

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In summary, a ring-point support using 72 uniformly distributed points and a 340-mm separation distance was selected for iterative deformation calibration. By means of the twice calibration, the accuracy of the flat reached λ/25, and the repeatability reached λ/100.

To verify the effectiveness of the deformation theory, the difference between the calibration and theoretical deformation was obtained, as shown in Figs. 14(a)-(c). The calibrated measurement (see Fig. 14(a)) was performed on the 300-mm-aperture vertical interferometer. The high-frequency components of deformation theory (see Fig. 14(b)) were compensated for complete surface distribution by interpolation. Figures 14(d)-(e) show a measurement result on a 600-mm-aperture horizontal Zygo interferometer.

Fig. 14. Application of deformation theory in experiment. (a) Calibration: PV = 24.03 nm, RMS = 3.86 nm; (b) deformation theory: PV = 17.34 nm, RMS = 2.77 nm; (c) difference: PV = 32.73 nm, RMS = 4.98 nm; (d) Measurement by the Ф600 mm horizontal Zygo interferometer; (e) PV = 36.57 nm, RMS = 5.35 nm.

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Comparing Figs. 14(c) and 14(e), at the edge, Fig. 14(c) exhibits rotational symmetric aberrations while Fig. 14(e) exhibits a clamping deformation. However, the dominating topographies and amplitudes are consistent in Figs. 14(c) and 14(e). This proves that the theoretical deformation and actual deformation are identical. Meanwhile, the ring-point support can be used to calibrate the clamping deformation of horizontal interferometer.

Furthermore, the similarities and differences in the horizontal and vertical interferometers were analyzed. Zernike polynomial decomposition was performed for the measurements of the Zygo interferometer and vertical interferometer.

After initial polishing, in Fig. 15(a), the defocus decreased from 11.88 to 7.63 nm. Both defocus values are positive, indicating that the flat is still concave. Moreover, the spherical aberration increased from 1.94 to 6.78 nm. These changes in defocus and spherical aberration are caused by gravity. For other aberrations, the measurement results of vertical interferometer were significantly smaller than the measurement of the horizontal Zygo interferometer. Figure 15(b) was measured on the horizontal Zygo interferometer. Figure 15(c) was measured on the vertical interferometer. The flat exhibits a clear concave topography, indicating that there is potential for the improvement of accuracy by further polishing. Therefore, the edge of the flat was polished a second time.

Fig. 15. Comparison between horizontal Zygo interferometer and vertical interferometer. (a) Zernike aberration coefficient after initial polishing; (b) Measurement on a horizontal Zygo interferometer: PV = 41.65 nm, RMS = 6.31 nm; (c) Measurement on the vertical interferometer: PV = 36.84 nm, RMS = 5.60 nm; (d) Zernike aberration coefficient after the second polishing (e) Measurement on a horizontal Zygo interferometer: PV = 36.57 nm, RMS = 5.35 nm; (f) Measurement on the vertical interferometer: PV = 24.03 nm, RMS = 3.86 nm.

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After the second polishing, in Fig. 15(d), the defocus decreased from 6.36 to -3.10 nm. Positive and negative results indicate that the concave flat became a convex flat. The spherical aberration increased from 2.07 to 2.96 nm. In addition, compared with the first polishing, the second polishing reduced the defocus and spherical aberration and introduced coma. Although residual coma was not eliminated perfectly, it can be eliminated by further polishing. Consequently, the edge was reduced to a minimal level (see Fig. 15(e)) on the horizontal Zygo interferometer. The remaining clamping deformation can be eliminated by calibration. In Fig. 15(f), the clamping deformation at the edge was completely eliminated on the vertical interferometer, while the middle area began to sag as a result of gravity.

In summary, the gravity-induced deformation is neglected on the horizontal interferometer. The concave surface can be obtained through measurement for the concave flat. However, the asymmetric clamping deformation is inevitable. In contrast, the gravity-induced deformation is significant on the vertical interferometer. After the gravity-induced deformation was compensated, the ring-point support effectively eliminated low-order rotational asymmetric aberrations and avoided the clamping deformation of the horizontal interferometer. Furthermore, the rotational symmetric aberrations detected by the horizontal interferometer can guide the polishing process, which can offset the gravity-induced deformation of the vertical interferometer. For the calibration of a concave flat with a 350-mm diameter and a 90-mm thickness, the defocus should be polished between 6.36 and 11.88 nm, the spherical aberration should be polished between 2.07 and 2.96 nm, and other rotational asymmetric aberrations should be as little as possible with 300-mm aperture.

5. Conclusion

In this paper, a ring-point support scheme is proposed for the deformation calibration of a large-diameter transmission flat. For a flat with a 350-mm diameter and a 90-mm thickness, a ring-point support with a 314-mm point separation distance and 100 points is sufficient to achieve minimal deformation. However, to achieve a larger observation range, the ring-point support structure with a point separation distance of more than 310 mm and more than 60 points provides excellent deformation reduction performance. During the calibration, deformations caused by the frame support and the ring-point support vary, which require further adjustments to separate the flat from the frame. In this study, a ring-point support using 72 uniformly distributed points and a 340-mm separation distance was selected. The surface accuracy of the initial placement was λ/5 (PV). By iterative deformation calibration, the final surface accuracy reached λ/25 (PV) for a 300-mm aperture. In the future, the combination of adhesive support and ring-point support for deformation calibration will be studied. Ultimately, iterative calibration of the transmission flat can satisfy the quality requirements of optical systems.

Funding

National Natural Science Foundation of China (U1731115, 62005122); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX19_0285, KYCX20_0263); China Postdoctoral Science Foundation (2020M671495); Natural Science Foundation of Jiangsu Province (BK20200458).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Material	Young’s modulus [GPa]	Poisson’s ratio	Density [kg/m³]
Fused silica	72.7	0.17	2.201 × 10³
Adhesive silicone	2×10^-3	0.49	1.1 × 10³
Aluminum	70.6	0.33	2.7 × 10³
PTFE plastic	3.2	0.35	1.19 × 10³

Iterative deformation calibration of a transmission flat via the ring-point support on a 300-mm-aperture vertical Fizeau interferometer

Abstract

1. Introduction

2. Principles

3. Simulation

4. Measurement

4.1 System design of 300-mm-aperture vertical Fizeau interferometer

4.2 Liquid reference

4.3 Iterative deformation calibration

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (15)

Tables (1)

Equations (14)

Optics Express