Vertex radius of curvature error measurement of aspheric surface based on slope asphericity in partial compensation interferometry

Qun Hao; Tengfei Li; Yao Hu; Shaopu Wang; Yan Ning; Yifeng Tan; Xinmu Zhang

doi:10.1364/OE.25.018107

1. Introduction

Aspheric surfaces, which deviate from the shape of a sphere, are used in optical instruments to change light path, correct aberrations, improve image quality, and decrease the weight and volume of optical systems [1–3]. Aspheric surfaces are important in optical surfaces and have been widely used in many kinds of products [4]. In particular high-precision aspheric surfaces play key roles in astronomical telescopes, outer space telescopes, and satellite remote sensing cameras [5].

The machining error of aspheric surfaces can be categorized into surface figure error (SFE) and surface parameter error. SFE is the irregular difference between the measured and nominal surface and can be measured by interferometry [3] or profilometry [6]. Surface parameter error affects the basic shape of the aspheric surface, and it mainly includes vertex radius of curvature (VROC) error and conic constant (denoted as K) error. VROC error affects basic properties, such as focal length of an optical curved surface, aberration, and imaging quality of an optical system [7]. Thus, precisely measuring VROC error is critical for manufacturing and aligning optical surfaces and remains challenging.

VROC error of aspheric surfaces can be measured using contact and noncontact methods.

Contact measurement methods are similar to those used for measuring VROC error of spheric surfaces. The absolute shape of the surface is first determined with profilometers or coordinate measuring machines. Coefficients of the shape formula can be directly calculated from surface fitting. VROC error is then obtained by comparing the measured and nominal VROC. But the test process is time-consuming and may tend to scratch the test surface [8].

Noncontact measurement methods can be further divided into two categories, namely, with or without using interferometry. Without using interferometry, Diaz–Uribe [9, 10] calculated K and VROC with experimental data for longitudinal aberration and corresponding angles of the normals to the surface. This method exhibits relative accuracy of 1% when the nominal VROC and K values are 44.95 mm and −1, respectively. Methods with high accuracy require high-precision instruments to precisely measure longitudinal aberration and corresponding angles. Wang [3] proposed ray tracing for measuring VROC error of aspheric surfaces. The relative accuracy of the method is higher than 0.5% when the nominal VROC is 1300 mm. However, this method is unsuitable for measuring nonparaxial aspheric surfaces or aspheric surfaces, whose high-order longitudinal normal aberration cannot be neglected [3].

Generally, most interferometry-based methods adopt null compensators or null computer-generated holograms (CGHs) to obtain VROC error [11]. Small errors in surface figure measurements will result in large errors in VROC calculation [8]. Each unique aspheric surface requires one single null compensator or null CGH, which increases the development costs of the method [12]. To overcome the deficiency of methods using null interferometer, Yang [13] simultaneously calculated the actual VROC error and SFE of the test surface in a non-null interferometry system by using a simultaneous optimization process in a multiconfiguration model. The relative accuracy of this method is higher than 0.025% when the nominal VROC is 816 mm. This method would accumulate the wavefront phase measurement error by using the multiconfiguration of an interferometer model [13].

VROC error can be measured with relative accuracy higher than 0.01% by using some special properties of aspheric surfaces; these properties include characteristic interferograms generated by off-axis segment of a conic surface [14–16]; local vertex aperture of aspheric surfaces, which can be regarded as a spherical surface [8, 17]; and aberration-free point of paraboloid where collimating beam converges to the focus without aberration [7].

Most noncontact methods for measuring VROC error are restricted to some specific aspheric surfaces; in this regard, few theoretical studies have characterized VROC error. These limitations are the focus of this study.

In our previous work, we proposed partial compensation interferometry (PCI) to measure SFE of aspheric surfaces by using digital Moiré method [4, 5, 18, 19]. Slope asphericity was used to describe the gradient of aspheric surfaces and characterize the difficulty of aspheric surface measurements [20]. Further study shows that slope asphericity can be used to locate the best compensation distance (BCD) for aspheric surfaces, based on which the VROC error of aspheric surfaces can be measured.

In this paper, we propose a method for VROC error measurement in rotational symmetric conicoid surfaces in PCI by using slope asphericity theory. Prior to measurement, the theory of slope asphericity and BCD is established, and the relationship between the BCD and VROC error is deduced. The proposed method is used to measure the VROC error of aspheric surfaces. This method utilizes the obtained relationship to decrease the complexity of calculation and increase the universality of the VROC error measurement of aspheric surfaces; one partial compensator (PC) can measure aspheric surfaces in some ranges and with many parameters [18]. The method can also be used to simultaneously measure the VROC error and the decoupled SFE in high precision and establish a theoretical equation for characterizing the VROC error. Furthermore, this method can be extended to measure high-order aspheric surfaces.

2. Theory of slope asphericity and BCD

The following section describes the theory of slope asphericity and BCD in detail and is divided into two parts: 1) slope asphericity and BCD; and 2) relationship between BCD and VROC error.

2.1 Slope asphericity and BCD

In conventional null interferometry for rotational symmetric aspheric surface under test (SUT) [21,22], the longitudinal spherical aberration of null compensator completely compensates the longitudinal aberration of SUT. In PCI, PC is designed based on the nominal SUT to partially compensate the longitudinal aberration of the SUT [18]; some residual wave aberration is allowed in the wavefront after partial compensation, and this wavefront is called residual wavefront. The slope of the residual wavefront reflects the gradient change in the residual wave aberration and determines the density of the interference fringes; among which, the detectable parts are limited by the spatial resolution of the CCD detector. The density of the interference fringes increases with the increasing slope of the residual wavefront and gradient of the residual wave aberration.

Difficulty in measuring SUT must be evaluated because the residual wavefront is nonzero and its slope is limited in PCI. As such, in [20], Xie et al. proposed slope asphericity, which is defined as follows: on one point of the SUT, slope asphericity is the absolute angle between the normal to the SUT and the normal to the fit sphere. Apparently, the maximum slope asphericity across the SUT varies among different fit spheres. Moreover, the fit sphere that converts the maximum slope asphericity to the minimum is defined as the best-fit sphere. For a conicoid SUT, the best-fit sphere is tangent with the SUT on the points of 0.86 aperture, where the corresponding slope asphericity is zero (the slope asphericity on the vertex is also zero) [20]. Figure 1 shows the geometry of the slope-asphericity-based best-fit sphere for a specific conicoid SUT. The solid curve denotes the SUT. The dotted curve denotes the best-fit sphere, which is tangent with the SUT on the points of 0.86 aperture, i.e., points A and B. O is the center of the best-fit sphere, O’ is the vertex curvature center of the SUT, and P is the vertex of the SUT. The solid lines OA and OB are the normals to both the best-fit sphere and the SUT, respectively. For a certain SUT, the best-fit sphere is uniquely determined; thus, the geometrical relationship between O and SUT can also be determined. The SUT and O can be regarded as a whole.

Fig. 1 Geometry of slope-asphericity-based best-fit sphere for a specific conicoid SUT.

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For a certain PC and SUT pair as shown in Fig. 2, the partial compensation result differs with changes in the distance between PC and SUT. Thus, BCD should be determined. The maximum slope of the residual wavefront changes gradually with the SUT moving along the optical axis. When the maximum slope reaches the minimum, the density of the interference fringes is at the minimum; the distance between the second surface of PC and the vertex of SUT is defined as BCD (denoted as $\bar{H P}$ in Fig. 2), where H is the vertex of the second surface of PC, A’ is the projection of A on the x-axis, and the red dashed lines denote the collimating rays propagating through the PC and then incident on the SUT.

Fig. 2 Geometry of BCD for a specific PC–SUT pair. $\bar{H P}$ is the BCD.

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By employing slope asphericity and the best-fit sphere, the BCD can be divided into two parts, i.e., $\bar{H O}$ and $\bar{O P}$ can be calculated for a certain SUT with the best-fit sphere [20], and it changes when the VROC error is introduced. By utilizing optical design software, we can obtain $\bar{H P}$ , then $\bar{H O}$ through $\bar{H O} = \bar{H P} - \bar{O P}$ . To a concave paraboloid SUT (nominal VROC is 889 mm; aperture diameter is 68.1 mm) and the PC shown in Table 1, the changes in $\bar{H O}$ induced by VROC error are listed in Table 2, which shows that $\bar{H O}$ can be thought of as a constant because it changes less than 5 μm when the VROC error of up to 2.5 mm is introduced. Other simulations for different PC–SUT pairs are made with the VROC error of 2.5 mm introduced, and the results are: 1) for conicoid SUT (nominal VROC is 1179.447 mm, K is –0.499365; aperture diameter is 580 mm) and its corresponding PC-1 in Table 3, $\bar{H O}$ changes less than 8 μm; 2) for conicoid SUT (nominal VROC is 1725.2 mm, K is –1; aperture diameter is 108 mm) and its corresponding PC-2 in Table 3, $\bar{H O}$ changes less than 5 μm. The results are close.

Table 1. Paraeters of the PC

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Table 2. Change in $\bar{H O}$ induced by VROC error for a certain PC

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Table 3. Parameters of the PC-1 and PC-2

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Hence, the relationship between the change in $\bar{H P}$ and VROC error can be deduced based on slope asphericity and the best-fit sphere. In this paper, we utilize the above theory to measure the VROC error of SUT.

2.2 The relationship between the BCD and VROC error

Based on the theory of slope asphericity and BCD proposed in Section 2.1, the relationship between the BCD and VROC error is deduced in this section, which forms the theoretical basis for VROC error measurement of SUT in PCI. It contains two parts: 1) based on the definition and the theory of best-fit sphere, the geometrical relationship between $\bar{O P}$ and VROC is deduced; 2) when the VROC error is introduced to the SUT, the change in BCD is deduced.

For a conicoid SUT, the relationship between the BCD and VROC error is easier to obtain compared with the high-order SUT, resulting in simpler deduction process. The present study mainly focuses on measuring VROC error of conicoid SUT; in our future study, we aim to measure VROC error of high-order SUT. In this paper, we adopt concave conicoid SUT to validate the measurement method.

The conicoid SUT is rotational symmetric, so we only need to analyze one plane crossing the rotational axis.

First, $\bar{O P}$ denoted as d is deduced as follows. In the Cartesian coordinate system as shown in Fig. 2, point P is located at the origin and the x-axis is the optical axis. Then, the basic shape of the conicoid SUT can be described as

x = - \frac{y^{2}}{R_{0} + \sqrt{R_{0}^{2} - (K + 1) y^{2}}},

where R₀ is the nominal VROC and K is the conic constant.

To calculate the angle between the normal to the conicoid SUT and the normal to the best-fit sphere, the slope equations of the normals are necessary. The slope equations have simpler expressions in the polar coordinate system. Hence, we establish polar coordinate system and locate the point O as the origin. The slope equation of the normal to the conicoid SUT can be denoted as

k_{1} = \frac{ρ \sin θ}{(1 + K) (ρ \cos θ - d) + R_{0}} .

The slope equation of the normal to the best-fit sphere can be denoted as

k_{2} = \tan θ .

The conicoid SUT and the best-fit sphere are tangent with each other, so the normals of them are coincident, i.e., $k_{1} = k_{2}$ . Based on Eqs. (2) and (3), we obtain

\frac{ρ_{A} \sin θ_{A}}{(1 + K) (ρ_{A} \cos θ_{A} - d) + R_{0}} = \tan θ_{A} .

On point A, the radius of the best-fit sphere, denoted as $R_{r}$ , can be calculated [20]. We obtain $ρ_{A} = R_{r}$ and $θ_{A} = \sin^{- 1} \frac{y_{A}}{R_{r}}$ , where $y_{A}$ is the y-axis of point A.

Based on the geometrical relationship in Fig. 2, we obtain

ρ_{A} \cos ρ_{A} = \bar{Ο Α^{'}} = d - \bar{Α^{'} Ρ},

where

\bar{Α^{'} Ρ}

is the sagittal of A, which is equal to

| x_{A} |

. Simplifying Eq. (4) by substituting Eq. (5) leads to

d = R_{0} - K \cdot | x_{A} | .

Second, when the VROC error is introduced to the SUT, which is denoted as $Δ R$ , the conicoid SUT equation can be expressed as

x^{'} = - \frac{y^{2}}{R_{0} + Δ R + \sqrt{{(R_{0} + Δ R)}^{2} - (K + 1) y^{2}}} .

If $Δ R$ is much less than R₀, the longitudinal shape change caused by $Δ R$ can be deduced as

Δ x = | x^{'} | - | x | = - \frac{y^{2} Δ R}{\sqrt{R_{0}^{2} - (K + 1) y^{2}} (R_{0} + \sqrt{R_{0}^{2} - (K + 1) y^{2}})} = g (y) \cdot Δ R,

where

g (y) = - \frac{y^{2}}{\sqrt{R_{0}^{2} - (K + 1) y^{2}} (R_{0} + \sqrt{R_{0}^{2} - (K + 1) y^{2}})} .

The longitudinal shape change on A caused by the VROC error is denoted as

Δ x_{A} = | x_{A}^{'} | - | x_{A} | = g (y_{A}) \cdot Δ R .

d after VROC error is introduced is denoted as d', which is

d^{'} = R_{0} + Δ R - K \cdot | x_{A}' | .

The BCD for the conicoid SUT with VROC error introduced is denoted as ${\bar{H P}}^{'}$ , and the change in the BCD is obtained by ${\bar{H P}}^{'} - \bar{HP}$ . Hence, when $Δ R$ is much less than R₀, the change in BCD caused by $Δ R$ can be calculated as

{\bar{HP}}^{'} - \bar{HP} \approx d^{'} - d = Δ R - K \cdot (| x_{A}^{'} | - | x_{A} |) = (1 - K \cdot g (y_{A})) \cdot Δ R,

where

g (y_{A})

is a constant determined by the nominal conicoid SUT.

Equation (12) shows the relationship between the BCD and VROC error: for a certain PC and conicoid SUT pair, the change in the BCD caused by VROC error is proportional to the VROC error. Based on the relationship, VROC error of the conicoid SUT can be calculated by measuring the change in BCD between the conicoid SUT and the nominal conicoid SUT.

Equation (6) is the deformation of relational expression of longitudinal aberration [9, 10]. Hence, for a general SUT, Eq. (6) can be extended to a general approximate equation [22]

d = R_{0} - (\frac{K}{2 R_{0}} + 4 R_{0}^{2} A_{4}) \cdot y_{A^{'}}^{2},

where

y_{A^{'}}

is the y value of

A^{'}

on which the best-fit sphere is tangent with the general SUT,

A_{4}

is the fourth order aspheric coefficient (or aspheric deformation constant). Based on Eq. (13), measuring the VROC error of high-order SUTs becomes possible, which increases the universality of the method.

3. Simultaneous measurement of VROC error and decoupled SFE

In the following sections, based on theoretical analysis of slope asphericity and BCD proposed above, the VROC error of rotational symmetric conicoid SUT is calculated by measuring the change in the BCD in PCI system, and the decoupled SFE is measured simultaneously.

The measurement system consists of two parts: real PCI system and virtual PCI system. The residual wavefront interferes with the reference light to form interference fringes. The difference between the real interference fringes formed by the conicoid SUT in the real interferometer (real PCI system) and the virtual interference fringes formed by the nominal conicoid SUT in the virtual interferometer (virtual PCI system) reflects the machining error of conicoid SUT [4, 5, 18, 19, 23], the fabrication and alignment error of PC, and other system errors. The fabrication and alignment error of PC and other system errors can be calibrated [18, 24–26], and it is not discussed in the present paper.

3.1 The method to eliminate the influence of the measured SFE

The BCD for the SUT is obtained by moving the SUT along the optical axis, thus some residual wave aberration is introduced [27], which is a small curvature to the residual wavefront, that is, defocus aberration [22]. Hence, in PCI, we move the SUT to introduce some defocus aberration to partially compensate the VROC error. And the method proposed in Section 2.2 is only appropriate for the situation where the machining error of conicoid SUT contains only VROC error.

SFE is the irregular difference between the measured and nominal surface. In most interferometry-based methods, the measured SFE is coupled with some residual VROC error [13]. It contains some extra defocus aberration, so the position of BCD will be changed, which may introduce some deviation in the measurement of the change in BCD. To practical VROC error measurement, the defocus aberration in the measured SFE needs to be eliminated by removing the second order term with quadric surface fitting. Thus, the measured SFE with defocus aberration eliminated can be obtained, which is defined as the decoupled SFE in our study. Then the decoupled SFE is added to the nominal SUT to generate the reference SUT to eliminate the influence of SFE. And the change in BCD between the SUT and the reference SUT can be used to measure the VROC error of SUT.

3.2 The PCI method for measuring VROC error of conicoid SUT

Before measuring the VROC error of conicoid SUT, based on the principle proposed in Section 3.1, the decoupled SFE is obtained by PCI method [4,5,18,19,23]. Then the decoupled SFE is added to the nominal conicoid SUT to form a reference conicoid SUT.

According to the analyses in Section 2.2 and 3.1, VROC error of conicoid SUT can be calculated from the change in the BCD between the conicoid SUT and the reference conicoid SUT according to Eq. (12). Hence, this section focuses on how to experimentally measure the change in the BCD, the key of which is to measure the BCD for the conicoid SUT.

In real PCI system, PC is used to partially compensate the large wave aberration produced by conicoid SUT, and some residual longitudinal spherical aberration exists. The collimating beam propagating through the PC cannot be converged to the same point on the optical axis, so it is difficult to exactly determine the distance between the second surface of PC and the vertex of the conicoid SUT. The solution to this problem is adding two removable lenses after PC to make up a removable combined aplanat [28], as shown in Fig. 3. By observing the corresponding characteristic interferogram with straight fringe [28], we can locate the cat’s eye position, which also is the back focus point of the combined aplanat, where all the collimating rays propagate though the aplanat and converge on the vertex of conicoid SUT. The distance between the second surface of PC and the cat’s eye position can be easily obtained based on the design of the aplanat.

Fig. 3 Sketch map of removable combined aplanat to locate the cat’s eye position of the SUT. The red rectangle marks denote the removable combined aplanat.

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Afterwards, the two lenses in the rear are removed, and the conicoid SUT is moved from the cat’s eye position to locate the position of BCD for the conicoid SUT. In reality, directly calculating BCD for the conicoid SUT is difficult. We can only locate the position of BCD by moving the conicoid SUT along the optical axis while observing the interferogram fringes to search for the position at which the density of the interference fringes is as sparse as possible. However, the exact position cannot be located precisely for the reason that the interferogram fringes are almost the same near the position of BCD, so the calibration needs to be taken.

When the position of conicoid SUT changes, the residual wavefront will also change with the new residual wave aberration introduced [27]. Suppose the change in the position is $Δ l$ and the corresponding change in residual wave aberration is $Δ W$ . Then $Δ W$ can be calculated as [29]

Δ W = \frac{1}{2} n^{'} \cdot Δ l \int_{0}^{u^{'}} d {u^{'}}^{2},

where

n^{'}

denotes the refractive index of image space and

u^{'}

denotes the aperture angle of image space, and these two parameters remain the same in both real PCI system and virtual PCI system. Based on Eq. (14), when the position of conicoid SUT changes, the corresponding change in residual wave aberration is only proportional to the position change.

Hence, the change in the BCD between the conicoid SUT and the reference conicoid SUT can be calibrated as follows: (1) move the reference conicoid SUT in virtual PCI system along the optical axis, (2) calculate the difference of residual wavefront between the real PCI and the virtual PCI by digital Moiré method [19], (3) locate the position of the reference conicoid SUT to minimize the difference of residual wavefront, and (4) measure the change in the distance between the conicoid SUT and the reference conicoid SUT, i.e., the change in the BCD after calibration. Then the VROC error can be calculated.

Finally, the principle of PCI method for measuring VROC error is summarized as shown in Fig. 4. In real PCI system in Fig. 4(a), Nominal position P₀ denotes the position of the BCD for nominal conicoid SUT, to which the conicoid SUT is moved from the cat’s eye position determined by the above removable combined aplanat. Measurement position P₁ denotes the position of the BCD for the conicoid SUT. In virtual PCI system in Fig. 4(b), Nominal position P₂ also denotes the position of the BCD for nominal conicoid SUT. Calibrated position P₃ denotes the position which is used to calibrate the position of BCD in real PCI system. In addition, P₃ is also used to eliminate the deviation of the BCD between the reference conicoid SUT and nominal conicoid SUT.

Fig. 4 The principle of PCI method for measuring VROC error. (a) Real PCI system. (b) Virtual PCI system

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In real PCI system as shown in Fig. 4(a), the conicoid SUT is located at P₀ at first. Then it is moved to locate P₁. The displacement between the two positions is denoted as $Δ L_{1}$ , which is equal to the change in the BCD between the conicoid SUT and nominal conicoid SUT. At P₁, to meet the requirement of image detection and digital Moiré method, some tilt should be added to the real interferogram by adjusting the reference flat mirror in real PCI [5], and the real interferogram after adding tilt denoted as I_r is captured.

In virtual PCI system as shown in Fig. 4(b), the reference conicoid SUT is located at P₂ at first. Then move the reference conicoid SUT along the optical axis, calculate the difference of residual wavefront with I_r and virtual interferograms generated in virtual PCI system by digital Moiré method [4, 5, 18, 19], and search for the position at which the peak-to-valley (PV) value of the difference of residual wavefront is minimum. Thus, P₃ is located. The displacement between the two positions is denoted as $Δ L_{2}$ , which is used to calibrate $Δ L_{1}$ .

Hence, the difference of the two displacements, $Δ L_{1} - Δ L_{2}$ , denotes the calibrated change in the BCD between the conicoid SUT and the reference conicoid SUT. Then based on Eq. (12), the VROC error can be calculated as

Δ R = \frac{Δ L_{1} - Δ L_{2}}{1 - K \cdot g (y_{A})} .

Furthermore, iteration method can be adopted in the measurement of decoupled SFE and VROC error of a conicoid SUT to improve the accuracy of results.

4. Experiment and results

An experiment was carried out to validate the VROC error measurement of conicoid SUT based on slope asphericity in PCI. A concave paraboloid with a nominal VROC of 889 mm and an aperture diameter of 68.1 mm was tested in PCI system.

4.1 Measurement system parameters and setup

The virtual PCI system was set up in optical design software, Zemax in our case, according to the structure shown in Fig. 4(b). The parameters of PC are listed in Table 4, and the nominal paraboloid was used to measure the SFE by PCI method [4, 5, 18, 19, 23].

Table 4. Parameters of the removable combined aplanat

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The real PCI system layout is presented in Fig. 5, which consists of a precisely calibrated 4” DynaFiz Zygo interferometer, a removable combined aplanat, a precision linearity rail, magnetic grid displacement measurement modules, and the test paraboloid with its adjustment mechanism. The laser wavelength in the experiment was 632.8 nm. The reference flat mirror offered by Zygo in the interferometer had a surface quality better than $1 / 20 λ$ . The removable combined aplanat consisted of the PC and two removable lenses and could limit the maximum longitudinal aberration below 1 μm. The parameters of the removable combined aplanat designed in Zemax are also listed in Table 4. The distance from the second surface of PC to the cat’s eye position was 1570.92 mm. The precision linearity rail had a 0.01 mm positioning accuracy with a straightness better than $\pm 1 mm / 6 m$ , and the magnetic grid displacement measurement modules with an accuracy of $\pm 0.03 mm$ would show the displacement of the test paraboloid.

Fig. 5 Real PCI system. (a) Diagram of the Real PCI system. (b) Setup of the Real PCI system.

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4.2 The location of nominal position P₀ and measurement position P₁

First, the test paraboloid was moved to locate P₀, which was 1836.04 mm from the second surface of PC. In the real PCI system with the removable combined aplanat, when the characteristic interferogram with straight fringe as shown in Fig. 6(b) was observed with naked eyes [28], the collimating beam incident on PC was converged on the vertex of test paraboloid, and the cat’s eye position was located. The removable combined aplanat could locate the cat’s eye position within 1 μm, and the interferograms at the positions 0.01 mm before and after the cat’s eye position are shown in Fig. 6(a) and (c) for comparison. Then the two removable lenses were removed, and the test paraboloid was moved from the cat’s eye position to P₀ for a distance of $1836.04 - 1570.92 = 265 .12 mm$ .

Fig. 6 Interferograms at different positions when using removable combined aplanat. The circle marks denote the fringes deviating from the straight ones. (a) The interferogram at the position 0.01 mm before the cat’s eye position. (b) The characteristic interferogram at the cat’s eye position. (c) The interferogram at the position 0.01 mm after the cat’s eye position

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Second, the test paraboloid was moved to locate P₁ by observing the interferogram fringes to search for the position at which the density of the interference fringes was as sparse as possible, which is shown in Fig. 7. Then the displacement from P₀ to P₁ was measured, $Δ L_{1} = - 1.90 mm$ , and I_r was captured after tilt was added.

Fig. 7 The interferogram before adding tilt at the measurement position P₁

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4.3 The measurement of the decoupled SFE and VROC error of test paraboloid

Based on the method to eliminate the influence of the measured SFE proposed in Section 3.1, the decoupled SFE can be obtained with quadric surface fitting. The SFE measurement result is shown in Fig. 8. As shown in Fig. 8(b), the PV value of the decoupled SFE was 111.98 nm or 0.1770 λ, and the RMS value of the result was 42.96 nm or 0.0679 λ. Then the SFE was added to the nominal paraboloid to generate the reference paraboloid.

Fig. 8 SFE measurement result of the test paraboloid. (a) The SFE without been decoupled. (b) The decoupled SFE. The circle mark denotes the deviation induced by the fixator.

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According to the measurement method proposed in Section 3.2, the reference paraboloid was located at P₂ at first. Then the reference paraboloid was moved along the optical axis, and the PV value of the difference of residual wavefront was calculated to locate P₃. The relationship between the PV value and $Δ L_{2}$ is shown in Fig. 9. The minimum PV value was 0.4582 λ, and the corresponding displacement from P₂ to P₃ was obtained, $Δ L_{2} = 0.09 mm$ .

Fig. 9 The relationship between the PV value of the difference of residual wavefront and $Δ L_{2}$

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Based on Eq. (15), the calculated VROC error of test paraboloid was −1.99 mm.

As a comparison result, the surface was measured with a coordinate measuring machine, the UA3P from Panasonic. The UA3P used a probe for contact and scan measurements. The Z-axial resolution of the UA3P is 10 nm. The X-axial and Y-axial resolutions are 0.01 mm and 2 mm, respectively. The result is presented in Fig. 10. As shown in Fig. 10, the PV value of the decoupled SFE was 167.93 nm or 0.2654 λ, and the RMS value of the result was 61.54 nm or 0.0973 λ. With surface fitting, the VROC of test paraboloid was 887.00 mm, so the VROC error was −2.00 mm.

Fig. 10 The decoupled SFE of the test paraboloid measured by the UA3P.

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The relative measurement accuracy ${Δ R_{r} / R}_{0}$ is $| \frac{Δ R_{r}}{R_{0}} | = | \frac{- 1.99 - (- 2.00)}{889} | < 0.01 % .$

Comparing Fig. 8(b) and Fig. 10, the decoupled SFE measured by the PCI method and the UA3P was analyzed. The PV value and the RMS value measured by the two methods are similar, with the deviation less than λ/10 and λ/30, respectively. However, their distribution has some deviation.

The difference might be caused by three factors. First, because the fixator of the test paraboloid shown in Fig. 5 may cause surface deformation, some deviation is introduced to the result, which is marked in Fig. 8(b). Second, the retrace error caused by alignment errors in the real PCI affects the result. Third, the SFE measured by the UA3P has some abnormal points which affect the surface fitting on the VROC error and the decoupled SFE.

To check repeatability, two other experiments were carried out to measure the VROC error of the same paraboloid. The results were −2.07 mm and −2.03 mm, and the average VROC error was −2.03 mm.

In conclusion, the PCI method is feasible to simultaneously measure the decoupled SFE and the VROC error precisely.

5. Discussion

The measurement method proposed in the present paper is focused on measuring VROC error of conicoid SUT. By modifying the expression of Eq. (6), the relationship between the BCD and VROC error for conicoid SUT can be extended to high-order SUT after solving these limitations: 1) the influence of conic constant error; 2) the influence of the error of $A_{4}$ ; 3) the influence of the change in $\bar{HO}$ induced by VROC error with $Δ R / R_{0}$ larger than 1%.

Error sources in the measurement of SFE and VROC error affect the measurement accuracy.

Error sources of the SFE measurement include the modeling error in virtual PCI system with Zemax, fabrication and alignment errors of the interferometer body and the PC, Moiré processing errors, and residual alignment errors due to phase compensation of the figure error and alignment errors. All the sources above were analyzed in our previous work [23], and we follow the conclusion of it.

Error sources in the measurement of VROC error can be classified as mechanical errors, optical errors, and system errors. The accuracy of VROC error measurement depends on the measurement of $Δ L_{1}$ and $Δ L_{2}$ , which are related to the location of P₀, P₁, P₂, and P₃.

5.1 Mechanical errors of VROC error measurement

Mechanical errors mainly refer to the positioning offset of precision linearity rail, which affects the location of P₀ and P₁. The positioning accuracy is 0.01 mm, so the accuracy of VROC error measurement in this experiment is limited to 0.01 mm, which can be improved by using higher-precision linearity rail and adopting laser differential confocal positioning technology to locate the corresponding positions [7].

5.2 Optical errors of VROC error measurement

Optical errors include: (1) alignment errors of the interferometer, the PC, and the test paraboloid, (2) alignment error of the optical axis and the precision linearity rail, and (3) positioning error of removable combined aplanat to locate P₀.

Alignment errors of the interferometer, the PC, and the test paraboloid may generate off-axis aberration such as coma and astigmatism. And the astigmatism will affect the measurement of VROC error. In addition, alignment error of the optical axis and the precision linearity rail directly affects the measurement accuracy of VROC error by

Δ L_{1} = Δ l^{'} \cdot \cos α,

where

Δ L_{1}

is the real displacement between P₀ and P₁,

Δ l^{'}

is the displacement measured by the magnetic grid displacement measurement modules which would be substituted into Eq. (15) to calculate VROC error, and α is the angle between the optical axis and the precision linearity rail. Hence, the alignment of the whole system is essential before the experiment, and it can be limited below 1°, and the corresponding error is well below 1 μm.

Positioning error of removable combined aplanat directly affects the positioning accuracy of P₀. P₁ is located by observing the density of the interference fringes, which means that the location of P₁ is not affected, and the deviation between P₁ and the BCD can be calibrated in virtual PCI system, so the displacement between P₀ and P₁, $Δ L_{1}$ , cannot be measured precisely. The removable combined aplanat is designed to limit the maximum longitudinal aberration below 1 μm, which is less than the positioning accuracy of the precision linearity rail, so it can meet the measurement requirement.

5.3 System errors of VROC error measurement

System errors include fabrication errors of the optical components and modeling error in virtual PCI system with Zemax.

Fabrication errors of the optical components are limited below 1/10λ, which affects the measurement of SFE but is negligible in the measurement of VROC error.

Modeling error in virtual PCI system with Zemax affects the position of P₂. P₂ is located through optimization in Zemax, and the position offset is limited below 5 μm.

The PCI method for measuring VROC error relies on the precise location of P₀ and P₁. Hence, by improving the positioning accuracy of the two positions and the alignment accuracy of the whole system, the method can achieve high measurement accuracy.

6. Conclusion

In the present paper, slope asphericity is used to define and locate BCD for the SUT, and the relationship between the BCD and VROC error is deduced. Then the VROC error measurement of SUT based on the slope asphericity theory in PCI is proposed to measure the decoupled SFE and VROC error simultaneously. The main advantages of this method include simple structure and calculation process, wide measurement range, simple instruments, and simultaneous measurement of decoupled SFE and VROC error. Experimental results indicate that with this method, the measurement accuracy ΔR/R₀ is 0.01%, and the decoupled SFE is λ/10 of the peak-to-valley value. The method can achieve high measurement accuracy by improving the positioning accuracy. Furthermore, this method establishes the theoretical equation to characterize the VROC error, which can also be extended to the measurement of high-order SUTs. In addition, conic constant K can be measured by improving the theory of slope asphericity and BCD, which is also the direction for further study.

Funding

National Natural Science Foundation of China (NSFC) (51327005); National Key Basic Research Program of China (2015CB059900); National Major Scientific Instruments and Equipment Development Project of China (2014YQ350461).

Acknowledgments

The authors thank Xiang Ding for his help with measuring the surfaces with the UA3P.

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VROC error (mm)	${\bar{HO}}^{'} - \bar{HO}$ (mm)	VROC error (mm)	${\bar{HO}}^{'} - \bar{HO}$ (mm)
−5.00	−0.0060	1.00	0.0012
−4.00	−0.0048	2.00	0.0024
−3.00	−0.0036	3.00	0.0036
−2.00	−0.0024	4.00	0.0048
−1.00	−0.0012	5.00	0.0060

Component	Surface	Radius (mm)	Thickness (mm)	Glass	Aperture (mm)	Figure error (λ)
PC-1	1 2	220.75 −151.63	19.03	K3	40.00	$1 / 10 λ$
PC-2	1 2	1037.50 −3981.00	12.00	K9	41.10	$1 / 10 λ$

Component	Surface	Radius (mm)	Thickness (mm)	Glass	Aperture (mm)	Figure error (λ)
PC	1 2	578.40 −3350.00	22.00 14.96	K9	100.00	$1 / 10 λ$
Removable Lens 1	1 2	250.00 250.00	11.50 8.01	ZF6	50.80	$1 / 10 λ$
Removable Lens 2	1 2	350.00 272.30	11.00	ZF6	50.80	$1 / 10 λ$

VROC error (mm)	${\bar{HO}}^{'} - \bar{HO}$ (mm)	VROC error (mm)	${\bar{HO}}^{'} - \bar{HO}$ (mm)
−5.00	−0.0060	1.00	0.0012
−4.00	−0.0048	2.00	0.0024
−3.00	−0.0036	3.00	0.0036
−2.00	−0.0024	4.00	0.0048
−1.00	−0.0012	5.00	0.0060

Component	Surface	Radius (mm)	Thickness (mm)	Glass	Aperture (mm)	Figure error (λ)
PC-1	1 2	220.75 −151.63	19.03	K3	40.00	$1 / 10 λ$
PC-2	1 2	1037.50 −3981.00	12.00	K9	41.10	$1 / 10 λ$

Vertex radius of curvature error measurement of aspheric surface based on slope asphericity in partial compensation interferometry

Abstract

1. Introduction

2. Theory of slope asphericity and BCD

2.1 Slope asphericity and BCD

2.2 The relationship between the BCD and VROC error

3. Simultaneous measurement of VROC error and decoupled SFE

3.1 The method to eliminate the influence of the measured SFE

3.2 The PCI method for measuring VROC error of conicoid SUT

4. Experiment and results

4.1 Measurement system parameters and setup

4.2 The location of nominal position P₀ and measurement position P₁

4.3 The measurement of the decoupled SFE and VROC error of test paraboloid

5. Discussion

5.1 Mechanical errors of VROC error measurement

5.2 Optical errors of VROC error measurement

5.3 System errors of VROC error measurement

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (4)

Equations (16)

Optics Express

Abstract

1. Introduction

2. Theory of slope asphericity and BCD

2.1 Slope asphericity and BCD

2.2 The relationship between the BCD and VROC error

3. Simultaneous measurement of VROC error and decoupled SFE

3.1 The method to eliminate the influence of the measured SFE

3.2 The PCI method for measuring VROC error of conicoid SUT

4. Experiment and results

4.1 Measurement system parameters and setup

4.2 The location of nominal position P0 and measurement position P1

4.3 The measurement of the decoupled SFE and VROC error of test paraboloid

5. Discussion

5.1 Mechanical errors of VROC error measurement

5.2 Optical errors of VROC error measurement

5.3 System errors of VROC error measurement

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (4)

Equations (16)

Optics Express

4.2 The location of nominal position P₀ and measurement position P₁