Interferometric measurement of high-order aspheric surface parameter errors based on a virtual-real combination iterative algorithm

Qun Hao; Xin Tao; Yao Hu; Tengfei Li; Tengfei Li; Weiqian Zhao

doi:10.1364/OE.435252

1. Introduction

Aspheric surfaces have more degrees of freedom than spheric surfaces, which make optical instruments with aspheric surfaces have less aberrations, better image quality, and less weight and volume than instruments without aspheric surfaces [1–3]. Aspheric surfaces are widely used in advanced optical systems, such as astronomical telescopes, outer space telescopes, lithography, satellite remote sensing cameras, and ignition facilities of manageable nuclear fusion because of these benefits [4–9]. For example, large reflective telescopes are usually multi-mirror aspherical systems. The primary mirrors are generally meter-sized large concave aspheric surfaces, whereas the secondary mirrors are typically convex aspheric surfaces. Another example is that high-order aspheric surfaces are utilized in lithography to obtain better optical performance.

Aspheric surface parameters are widely utilized to characterize aspheric surfaces, especially in optical design [10–12]. These parameters constrain the shape freedom of aspheric surfaces, which makes utilizing aspheric surfaces in optical design and modeling easier. For example, high-order rotational aspheric surfaces can be expressed by the vertex radius of curvature, conic constant, and high-order aspheric coefficients [10]. In optical fabrications, the aspheric surface parameter errors (SPEs) of optical components are measured to estimate whether the components satisfy the demand of design. With the widely utilization of aspheric surfaces, the SPE measurement faces new challenges when aspherics with large apertures and high-order SPEs are measured.

Many methods have been proposed to measure SPEs of aspheric surfaces. Among them, four significant categories of methods are geometric optical tracing methods, feature-based methods, scanning methods and interferometry methods.

Geometric optical tracing methods [13–18] are firstly proposed. N-Bench methods [13,14] are proposed to measure vertex radius of curvature and conic constant. A ray tracing method [15] is proposed to measure vertex radius of curvature. Null-screen methods [16–18] are proposed to measure vertex radius of curvature and conic constant. Feature-based methods [19–21] utilize the feature of conic surface to measure the SPEs of conic surfaces. A method utilizes three confocal positions to measure vertex radius of curvature and conic constant [19]. Laser differential confocal methods [20,21] utilize the conjugate foci of conic surfaces to measure vertex radius of curvature. The above-mentioned two categories of methods can measure vertex radius of curvature of conic surfaces with a high accuracy but still cannot measure high-order aspheric surfaces.

Scanning methods [22–27] including coordinate measuring machine methods [22–24] and swing arm profilometer method [25–27] are most common methods in aspheric surface measurement. Scanning methods have several benefits such as high precision and high flexibility and can be applied to concave and convex aspheric surfaces as well as conic and high-order aspheric surfaces. However, the measurement time is long, typically several hours, due to point wise scanning, and the tactile measurement may scratch the surfaces under test. Thus, high accurate, noncontact, and fast methods are still desired.

Interferometry methods for surface figure error measurements [28–50] are noncontact and allow precise measurements in a short time, typically several seconds. These benefits result in the great potential of interferometry methods in aspheric SPE measurement compared with scanning methods. Full-field interferometry methods [28–44] generally utilize compensators to generate wavefronts matching the aspheric surface under test because the wavefronts at the exit pupil of standard interferometers are usually planar or spherical, which cannot be utilized to measure aspheric surfaces directly. The generated wavefronts can completely match the aspheric surface under test in null interferometry methods [28–36] and cannot completely match the aspheric surface under test in non-null interferometry methods [37–44] where a residual wavefront remains and a retrace error exists. Meanwhile, the wavefronts generated by a null compensator still cannot completely match the aspheric surface under test if SPEs are significant. In this case, the existence of retrace error makes null methods to be non-null. Thus, non-null interferometry methods are more representative than null methods in SPE measurement.

In full-field interferometry methods, the distance between the compensator and the surface under test, which is defined as the compensation distance, distinctly impacts the measured wavefront error. As the compensation distance is adjusted, when the peak-to-valley value of the wavefront error is minimum, the wavefront error is defined as the best compensation wavefront. Meanwhile, the compensation distance corresponding to the best compensation wavefront is defined as the best compensation distance.

Since both the compensation distance and the corresponding wavefront error vary with the SPEs of the surface under test, full-filed interferometry methods utilize the compensation distance and the wavefronts measured by the interferometer to calculate the SPEs of aspheric surfaces [45–50]. A null compensation method [45] utilizes a null compensator to measure the surface figure error and a laser tracker to measure the compensation distance. Then, vertex radius of curvature and conic constant can be calculated from the surface figure error and the compensation distance. High-order surface parameters are supposed to be zero in the model because the surface under test is conic. A non-null method [46] utilizes multiple measurements of the aspheric surface under test shifting along the optical axis with precise displacements to obtain the compensation distances. Vertex radius of curvature and surface figure error can be optimized simultaneously, while similarly other SPEs are not considered in the mathematic model. When conic constant error and high-order SPEs exist, the impact of these SPEs cannot be corrected. Thus, only surfaces without conic constant error and high-order SPEs can be measured in this method. Several non-null methods [47–50] measure the displacement of the surface under test from the cat’s-eye position, which is generated by the partial compensator and a removable combined aplanat, to the measurement position to obtain the compensation distance. Vertex radius of curvature and conic constant errors of a surface under test with high-order surface parameters can be calculated through an iterative process because the mathematic model considers nominal high-order surface parameters. However, when high-order SPEs exist, the mathematic model becomes inaccurate because the errors will couple with each other.

Consequently, in these interferometry methods, several issues have not yet been solved. First, high-order SPEs cannot be calculated using the previous mathematic model which only considers conic aspheric surfaces and high-order aspheric surfaces without high-order SPEs. The calculated results of vertex radius of curvature and conic constant will also be affected if high-order SPEs exist. Second, the accuracy of compensation distance measurement has a great influence on the solution accuracy of SPEs. If compensation distances are measured by laser trackers, the mechanical mountings of the compensators and the surface under test should be customized and is complex to align. If compensation distances are measured from the axial displacement of the surface under test, moving a surface under test over a large distance precisely is difficult, especially when the aperture of the surface under test is large. Moreover, the accuracy of the compensation distance measurement is limited by the accuracy of the precision linearity rail. Third, in previous interferometry methods, only concave aspheric surface parameters can be measured. One reason is that the design of compensators and alignment of compensation systems for concave aspheric surfaces is easier than those for convex aspheric surfaces. Meanwhile the mathematic models in previous methods aim at concave aspheric surfaces without considering the suitability for convex aspheric surfaces.

In this paper, an interferometric measurement method for high-order aspheric SPEs based on virtual-real combination iterative algorithm (VRCIA) is proposed to solve the issues above. VRCIA is proposed to calculate SPEs through the best compensation wavefront and the best compensation distance. This algorithm is adaptive to concave aspheric surfaces and convex aspheric surfaces with high order SPEs. In the measurement system, a partial compensation interferometer system is introduced to measure the best compensation wavefront. Meanwhile, an axial space measurement based on laser differential confocal technique [51–53] is introduced to measure the best compensation distance precisely.

2. Measurement system and virtual-real combination iterative algorithm

The following section describes the interferometric measurement method for high-order aspheric SPEs based on VRCIA in three sections: 1) SPE measurement system configuration, 2) the theory of VRCIA, and 3) flowchart of aspheric SPE measurement.

2.1 Surface parameter error measurement system configuration

In order to measure the SPEs, the best compensation wavefront and the best compensation distance are significant. The schematic sketch of SPE measurement system as shown in Fig. 1 is proposed to measure the best compensation wavefront and the best compensation distance, respectively. The system can be divided into two subsystems: the real partial compensation interferometer system and the laser differential confocal system. The part enclosed by a solid line is the real partial compensation interferometer system measuring the best compensation wavefront, whereas the part enclosed by dashed line is the laser differential confocal system measuring the best compensation distance. The two subsystems with different-wavelength lasers can work independently or together through a dichroic mirror. The components enclosed by dot-dash line are the surface under test and the compensator that are the same in the two subsystems.

Fig. 1. Schematic of surface parameter error measurement system.

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In the partial compensation interferometer system, a Fizeau interferometer is adopted to measure the best compensation wavefront. The planar wavefront of the collimated light from the Fizeau interferometer, after passing through the compensator for the first time, is transformed to the testing wavefront of the surface under test. And the testing wavefront, after reflected by the surface under test and passing through the compensator for the second time, is transformed to a residual wavefront. The compensation distance is adjusted until that the peak-to-valley of the residual wavefront reaches the minimum. Then, the wavefront that is the real best compensation wavefront can be obtained by phase-shifting method, and the distance between the compensator and the aspheric surface under test is the best compensation distance to be measured. The compensator and the surface under test are mounted together and the best compensation distance is fixed. Then the compensator and the surface under test are moved to the laser differential confocal system integrally to measure the best compensation distance.

In the laser differential confocal system, the laser passes through an object lens (OL3) and a pinhole to generate a point source. The light from the point source passes through Lens 1 to generate a collimated light. The parallel light passes through a polarizer (P), a polarized beam splitter (PBS), a quarter-wave plate (1/4 WP) and an objective (Lens 2) to form a converging beam. When Lens 2 moves axially, the convergence point of the beam is moved. The displacement of Lens 2 can be measured by an additional displacement measurement system. When the convergence point of the beam is moved to the last surface of the compensator or the aspheric surface under test, the measurement beam is reflected and then reflected by the PBS to the collecting lens (Lens 3) and beam splitter (BS). The split two measurement beams by BS are received by virtual pinholes (VPHs). Two VPHs with opposite offset M receive the intensity signals, respectively. The difference of the signals by two VPHs is zero when the convergence point focuses on a surface. When the convergence point of the beam is moved from the last surface of the compensator to the aspheric surface under test, the displacement of Lens 2 is measured. Then, the best compensation distance can be calculated by axial space ray tracing measurement’s theory [51].

In summary, the best compensation wavefront can be obtained directly from an interferometer, and the best compensation distance can be measured by laser differential confocal system and axial space ray tracing measurement’s theory.

2.2 Theory of the virtual-real combination iterative algorithm

This subsection illustrates the theory of virtual-real combination iterative algorithm which utilizes the best compensation wavefront and the best compensation distance to calculate SPEs of the aspheric surface under test.

In partial compensation interferometer system, the best compensation wavefront and the best compensation distance are determined by SPEs. To realize the SPEs measurement by interferometric method, the relationship among the best compensation wavefront, the best compensation distance and SPEs need to be deduced.

The relationship between the best compensation wavefront and SPEs are deduced first. Aspheric surfaces, especially rotationally symmetric aspheric surfaces, can be expressed as Eq. (1),

(1)$$z = \frac{{{r^2}}}{{R + \sqrt {{R^2} - (1 + K){r^2}} }} + \sum\limits_{i = 2}^n {{A_{2i}}{r^{2i}}} ,\qquad i = 2,3,4, \cdots ,N$$

where R is the vertex radius of curvature, K is the conic constant, ${A_{2i}}$ are high-order aspheric coefficients, ${r^2} = {x^2} + {y^2}$, and z is the sag of the aspheric surfaces. Considering McLaurin Series Expansion of Eq. (1), z can be expressed as Eq. (2).

(2)$$z = \sum\limits_{i = 1}^n {{D_{2i}}{r^{2i}}} \textrm{ = }{D_2} \cdot {r^2} + {D_4} \cdot {r^4} + {D_6} \cdot {r^6} + {D_8} \cdot {r^8} + \sum\limits_{i = 5}^n {{D_{2i}}{r^{2i}}} ,\quad \;\;\;i = 1,2,3, \cdots ,N, $$

where the relationship between aspheric surface parameters and the first four coefficients ${D_{2i}}$ is listed in Eq. (3).

(3)$$\left\{ \begin{array}{l} {D_2}\textrm{ = }1/2R\\ {D_4} = {A_4} + (K + 1)/8{R^3}\\ {D_6} = {A_6} + {(K + 1)^2}/16{R^5}\\ {D_8} = {A_8} + 5{(K + 1)^3}/128{R^7} \end{array} \right.$$

Considering the influence of SPEs such as $\Delta R$, $\Delta K$ and $\Delta {A_{2i}}$, the deviation of the surface sag can be calculated as Eq. (4).

(4)$$\begin{array}{ll} \Delta z &= z^{\prime} - z = \sum\limits_{i = 1}^n {{{D^{\prime}}_{2i}}{r^{2i}}} - \sum\limits_{i = 1}^n {{D_{2i}}{r^{2i}} = } \sum\limits_{i = 1}^n {\Delta {D_{2i}}{r^{2i}}} \\ &\textrm{ = }\Delta {D_2} \cdot {r^2} + \Delta {D_4} \cdot {r^4} + \Delta {D_6} \cdot {r^6} + \Delta {D_8} \cdot {r^8} + \sum\limits_{i = 5}^n {\Delta {D_{2i}}{r^{2i}}} \end{array}$$

where the relationship between aspheric SPEs and the first four coefficient deviations $\Delta {D_{2i}}$ of the surface sag can be derived from Eqs. (2) to (4) as Eq. set (5).

(5)$$\left\{ \begin{array}{l} \Delta {D_2} = \frac{1}{{2({R + \Delta R} )}} - \frac{1}{{2R}} ={-} \frac{{\Delta R}}{{2R({R + \Delta R} )}}\\ \Delta {D_4} = \frac{{(K + \Delta K + 1)}}{{8{{({R + \Delta R} )}^3}}} - \frac{{(K + 1)}}{{8{R^3}}} + \Delta {A_4}\\ \Delta {D_6} = \frac{{{{(K + \Delta K + 1)}^2}}}{{16{{({R + \Delta R} )}^5}}} - \frac{{{{({K + 1} )}^2}}}{{16{R^5}}} + \Delta {A_6}\\ \Delta {D_8} = \frac{{5{{(K + \Delta K + 1)}^3}}}{{128{{({R + \Delta R} )}^7}}} - \frac{{5{{({K + 1} )}^3}}}{{128{R^7}}} + \Delta {A_8} \end{array} \right.$$

On the other hand, the residual wavefront’s deviation reflects the deviation of the surface sag. Considering the best compensation wavefront is the residual wavefront at the best compensation distance, we define the best compensation wavefront’s deviation $\Delta \varphi$ as the deviation between the real best compensation wavefront $\varphi ^{\prime}$ measured in the real partial compensation interferometer system and the ideal best compensation wavefront $\varphi$ simulated in the virtual partial compensation interferometer system. Then the best compensation wavefront’s deviation can be considered as twice the surface sag’s deviation plus the retrace error. If the retrace error is omitted, the relationship between the best compensation wavefront’s deviation and the surface sag’s deviation can be approximately expressed as

(6)$$\begin{aligned} \Delta \varphi &= \varphi ^{\prime} - \varphi \approx 2 \cdot \frac{{2\pi }}{\lambda } \cdot \sum\limits_{i = 1}^n {\Delta {D_{2i}}{{({\beta r} )}^{2i}}} \\ & \textrm{ = }\frac{{4\pi }}{\lambda } \cdot \left( {\Delta {D_2} \cdot {{({\beta r} )}^2} + \Delta {D_4} \cdot {{({\beta r} )}^4} + \Delta {D_6} \cdot {{({\beta r} )}^6} + \Delta {D_8} \cdot {{({\beta r} )}^8} + \sum\limits_{i = 5}^n {\Delta {D_{2i}}{{({\beta r} )}^{2i}}} } \right), \end{aligned}$$

where $\lambda $ is the wavelength of the interferometer and $\beta $ is the zoom index equal to the ratio of the aperture of the residual wavefront and that of the aspheric surface under test. Then the coefficient deviations can be solved by polynomial fitting of the best compensation wavefront’s deviation. This approximation will introduce retrace error in the solution. Retrace error is the deviation between the residual wavefront and the surface figure error of the aspheric surface under test in a non-null measurement, which will be corrected by a retrace error correction method in the subsequent procedure.

In consequence, the relationship between the best compensation wavefront’s deviation and SPEs is set up through Eqs. (5) and (6). However, in the Eq. set (5), $\Delta {D_{2i}}(i = 1,2,3,4)$ are known from Eq. (6), and $\Delta R$, $\Delta K$, and $\Delta {A_{2i}}(i = 2,3,4)$ are unknown. The number of the equations is one less than that of the unknowns. Therefore, another equation describing the change in the best compensation distance is introduced to solve SPEs.

Figure 2(a) illustrates the definition of the best compensation distance in the compensation system of the nominal aspheric surface under test. The black solid curve line is the profile of a convex aspheric surface in the cross section of the optical axis, whereas the blue dash arc line is the best fit sphere. ${d_0}$ is the compensation distance defined as the distance between the last surface of the compensator and the aspheric surface under test. ${P_0}$ is the vertex point of the nominal aspheric surface under test and is set to be the origin of coordinate. Based on the theory of slope asphericity, the best-fit sphere can be calculated by the aperture and parameters of the aspheric surface under test [47]. The key points of the best-fit sphere are the center of the best-fit sphere O and the tangent point $A({{A_ \bot },{y_A}} )$ of the best-fit sphere and the aspheric surface under test. When the focus of the partial compensator is located at the center of the best-fit sphere O, compensation distance ${d_0}$ is the best compensation distance that conforms to the following Eq. (7),

(7)$${d_0} = {L_P} \mp O{P_0} = {L_P} \mp (O{A_ \bot } + {P_0}{A_ \bot }), $$

where the sign is minus when the aspheric surface under test is convex and plus when the aspheric surface under test is concave; ${P_0}{A_ \bot }$ is the z value of tangent point A which can be calculated by Eq. (8),

(8)$$\begin{array}{cc} {{P_0}{A_ \bot } = {z_A} = \frac{{y_A^2}}{{R + \sqrt {{R^2} - (1 + K)y_A^2} }} + \sum\limits_{i = 2}^n {{A_{2i}}y_A^{2i}} ,}&{i = 2,3,4 \cdots } \end{array},$$

and $O{A_ \bot }$ can be calculated by geometrical relationship as Eq. (9)

(9)$$\begin{aligned} O{A_ \bot } &= \frac{{A{A_ \bot }}}{{\tan ({\angle AO{A_ \bot }} )}} = \frac{{{y_A}}}{{{{\left. {\frac{{dz}}{{dr}}} \right|}_{r = {y_A}}}}}\\ & = {\left( {\frac{2}{{R + \sqrt {{R^2} - ({1 + K} )y_A^2} }} + \frac{{({1 + K} )y_A^2}}{{\sqrt {{R^2} - ({1 + K} )y_A^2} }} + 2i \cdot \sum\limits_{i = 2}^n {{A_{2i}}y_A^{2i - 2}} } \right)^{ - 1}}. \end{aligned}$$

When SPEs exist in the real aspheric surface under test, the best compensation distance is different from that of the nominal aspheric surface under test, as shown in Fig. 2(b).The new tangent point is $A^{\prime}({{{A^{\prime}}_ \bot },{{y^{\prime}}_A}} )$, ${P_R}{A^{\prime}_ \bot }$ can be calculated by Eq. (10),

(10)$${P_R}{A^{\prime}_ \bot } = \frac{{y^{\prime 2}_A}}{{({R + \Delta R} )+ \sqrt {{{({R + \Delta R} )}^2} - (1 + K + \Delta K)y^{\prime 2}_A} }} + \sum\limits_{i = 2}^n {({{A_{2i}} + \Delta {A_{2i}}} )y^{\prime {2i}}_A} ,$$

and $O{A^{\prime}_ \bot }$ can be calculated as Eq. (11)

(11)$$\begin{array}{ll} O{{A^{\prime}}_ \bot } &= \left( {\frac{2}{{({R + \Delta R} )+ \sqrt {{{({R + \Delta R} )}^2} - ({1 + K + \Delta K} )y^{\prime 2}_A} }}} \right.\\ &{\left. { + \frac{{({1 + K + \Delta K} )y^{\prime 2}_A}}{{\sqrt {{{({R + \Delta R} )}^2} - ({1 + K + \Delta K} )y^{\prime 2}_A} }} + 2i \cdot \sum\limits_{i = 2}^n {({{A_{2i}} + \Delta {A_{2i}}} )y^{\prime ^{2i - 2}}_A} } \right)^{ - 1}}. \end{array}$$

Figure 2(c) shows that the best compensation distance of the real aspheric surface deviates from $V{P_0}$ to $V{P_R}$, where ${P_0}$ is the vertex of nominal aspheric surface and ${P_R}$ is the vertex of real aspheric surface with SPEs. The deviation of the best compensation distance can be calculated as Eq. (12),

(12)$$\Delta d = {d_R} - {d_0} ={\mp} [{O{{A^{\prime}}_ \bot } + {P_\textrm{R}}{{A^{\prime}}_ \bot } - O{A_ \bot } - {P_0}{A_ \bot }} ]$$

where the choice of the minus or plus sign is in accordance with Eq. (7).

Fig. 2. (a) Schematic of the best compensation distance d₀ when the surface under test is ideal. (b) Schematic of the best compensation distance d_R when the surface under test is the real surface with SPEs. (c) Schematic of the deviation of the best compensation distance when the surface under test is with SPEs.

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If ${y_A}$ is considered as an estimation of ${y^{\prime}_A}$ approximately, i.e. ${y_A}$ is substitute for ${y^{\prime}_A}$ in Eqs. (10) and (11), Eq. (12) will contribute a new equation relating the known compensation distance change to the unknown SPEs. Combining Eq. (12) with Eq. set (5) makes the number of equations equal to the number of unknowns and SPEs are solvable.

It is important to note that two approximations exist in above mathematical deduction. One approximation is in Eq. (12), where ${y_A}$ is considered as an estimation of ${y^{\prime}_A}$ approximately. The other approximation is in Eq. (6), where the best compensation wavefront’s deviation is approximately considered as twice the surface sag’s deviation to calculate $\Delta {D_{2i}}$ without considering the retrace error. A VRCIA is proposed to eliminate the impact of the two approximations by two iterative optimization processes, respectively.

The first iterative optimization process of VRCIA eliminates the impact of approximation of ${y^{\prime}_A}$ as ${y_A}$. The SPEs in virtual partial compensation interferometer system are set to be zero as the initial state. After each calculation of Eqs. (12) and (5), the newly calculated SPEs will replace the SPEs in virtual partial compensation interferometer system as an update. Levenberg–Marquardt algorithm can be adopted to solve SPEs, where the objective function is expressed as,

(13)$$\begin{array}{l} \min {O_1}(\Delta {R_j},\Delta {K_j},\Delta {A_{2i}}_{,j})\\ = \min \left[ {{{\left( {\frac{{\Delta {R_j} - \Delta {R_{j - 1}}}}{R}} \right)}^2} + {{\left( {\frac{{\Delta {K_j} - \Delta {K_{j - 1}}}}{K}} \right)}^2} + \mathop \sum \limits_{i = 2}^N {{\left( {\frac{{\Delta {A_{2i}}_{,j} - \Delta {A_{2i}}_{,j - 1}}}{{{A_{2i}}}}} \right)}^2}} \right] \end{array}$$

where $\Delta {R_j}$, $\Delta {K_j}$ and $\Delta {A_{2i,j}}$ are the jth calculation results of the SPEs. The SPEs are iteratively optimized according to the objective function until the deviation of the last two calculated SPEs is less than the threshold $\varepsilon$.

The second iterative optimization process of VRCIA eliminates the impact of retrace error in Eq. (6) with an modified iterative reverse optimization algorithm with reference to Ref. [46]. In the traditional iterative reverse optimization algorithm, the variables are the surface figure errors that are usually described by Zernike fringe polynomial coefficients, whereas in SPEs measurement, the variables are the SPEs. To construct the optimization objective function, we decompose the real residual wavefront $\varphi$ in the real partial compensation interferometer system and the virtual residual wavefront $\varphi ^{\prime}$ in the virtual partial compensation interferometer system with the finite terms of Zernike fringe polynomials as

(14)$$\left\{ \begin{array}{l} \varphi = \mathop \sum \limits_{j = 1}^N {B_j}{Z_j}\\ \varphi^{\prime} = \mathop \sum \limits_{j = 1}^N {{B^{\prime}}_j}{Z_j}, \end{array} \right.$$

where ${B_j}$ and ${B^{\prime}_j}$ are the Zernike coefficients of $\varphi$ and $\varphi ^{\prime}$, respectively. In a certain partial compensation interferometer system, the residual wavefront is determined by the aspheric surface under test when the compensation distance is fixed. SPEs of a rotationally symmetric aspheric surface under test will only affect the rotationally symmetric Zernike fringe coefficients, which are consequently set as the optimization target, and the SPEs are set as the optimization variables. When the SPEs in virtual partial compensation interferometer system is equal to the SPEs in real partial compensation interferometer system, ${B_j}$ will be equal to ${B^{\prime}_j}$. Thus, the objective function can be expressed as,

(15)$$\min {O_2}(\Delta R,\Delta K,\Delta {A_{2i}}) = \min \left[ {\sum {{({{B_j} - {{B^{\prime}}_j}} )}^2}} \right]\quad \;\;\;j = 1,4,9,16,25,36,37.$$

Among 37 items of Zernike fringe polynomials, only seven items are rotationally symmetric when j = 1, 4, 9, 16, 25, 36, 37. Thus only seven items are considered in the objective function.

The final SPEs can be obtained by the two iterative optimization processes in VRCIA after the errors from the approximation in Eq. (12) and the retrace error in Eq. (6) are eliminated.

2.3 Aspheric surface parameter errors measurement

Based on the proposed VRCIA and measurement system, the flowchart of aspheric SPEs measurement is described as follows and shown in Fig. 3.

A. The virtual partial compensation interferometer system consisting of a virtual aspheric surface under test and its virtual compensator is designed. The parameters of virtual surface under test are initialized by the nominal parameters. The partial compensator is designed [37].
B. The real partial compensation interferometer system including an interferometer, a real aspheric surface under test, and a real partial compensator is constructed. The real partial compensator is manufactured or selected according to the parameters of the virtual partial compensator.
C. The real best compensation wavefront of the real partial compensation interferometer system is measured. In the measurement, the compensation distance is adjusted until the peak-to-valley value of the real residual wavefront is the minimum. Then, the compensation distance is the real best compensation distance, and the real residual wavefront at the best compensation distance is the best compensation wavefront.
D. The real best compensation distance is measured by laser differential confocal system.
E. The virtual residual wavefront and the virtual best compensation distance are simulated by the ray tracing algorithm in the virtual partial compensation interferometer system with an optical simulation software, such as ZEMAX and CODEV, etc.
F. $\Delta {D_{2i}}$ and $\Delta d$ are calculated. $\Delta {D_{2i}}$ can be fitted by Eq. (6), and $\Delta d$ is the deviation of the real best compensation distance to the virtual best compensation distance.
G. Intermediate SPEs are calculated by the first iterative optimization process of VRCIA. Intermediate SPEs can be calculated by solving Eq. set (5) and Eq. (12). Then, the parameters of the aspheric surface under test in virtual partial compensation interferometer system are updated. Steps E to step F are repeated until ${O_1}$ in Eq. (12) is optimized to be small enough.
H. Final SPEs are solved by the second iterative optimization process of VRCIA to optimize ${O_2}$ as in Eq. (14).

Fig. 3. Flowchart of SPE measurement

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3. Experiments and results

3.1 Measurement system

Figure 4 shows the hardware of the measuring system, where the components on the left of the orange dash line is the partial compensation interferometer system and the components on the right of the orange dash line is the laser differential confocal system.

Fig. 4. Surface parameter error measurement system.

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In partial compensation interferometer system, a Fizeau interferometer, Zygo Dynafiz 4” with a 632.8 nm laser, is adopted to measure the residual wavefront. The interferometer in the experiment is a precisely calibrated 4” Dynafiz Zygo interferometer with a peak pixel deviation less than 0.5 nm (λ/1200) in Quick Phase Shift Interferometer mode. The reference flat mirror offered by Zygo has a surface quality better than 1/20 λ.

Table 1 shows the nominal parameters of the aspheric surface under test. The aspheric surface parameter fitted from the point cloud measured by UA3P-5 is considered as the standard result. UA3P-5 of Panasonic has a resolution of 0.3 nm, an accuracy of 0.05 μm and a repeatability of 0.05 µm.

Table 1. Nominal parameters of aspheric surface

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The compensator is designed in virtual partial compensation interferometer system based on the nominal parameters of the aspheric surface under test. The compensator is a doublet whose parameters are shown in Table 2. The best compensation distance in the initial virtual partial compensation interferometer system is 16.135 mm. In real partial compensation interferometer system, the real compensator is manufactured according to the same parameters. After manufacture, the virtual partial compensation interferometer system is updated with the calibration results of the real compensator to keep the accordance with the real partial compensation interferometer system. The best compensation distance in real partial compensation interferometer system is measured by laser differential confocal system.

Table 2. Parameters of the compensator

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In laser differential confocal system, the wavelength of the laser is 532 nm. Object Lens 3 is 40 X/0.65. The size of the pinhole is 10 microns. Lens 2 and Lens 3 are doublets with a focal length of 200 mm. Object Lens 1 and object Lens 2 are 40 X/0.60. CCD 1 and CCD 2 are Point Grey Blackfly S BFS-U3-51S5M-C. The sizes of the virtual pinholes are both 50 pixels. A Michelson heterodyne interferometer is utilized to measure the displacement of the Lens 2 with a high accuracy to measure the best compensation distance accurately. In Michelson heterodyne interferometer, the dual-frequency laser is PT-1105C, and the digital phase card is PT-1313B.

Currently, the best compensation wavefront and the best compensation distance are measured respectively. In real partial compensation interferometer system, the best compensation distance can be fixed by locking the adjusting device of the surface under test, and the best compensation wavefront at the best compensation distance can be measured using the interferometer. Then, the compensator and surface under test are moved together towards the laser differential confocal system to measure the best compensation distance.

3.2 Measurement results

3.2.1 Residual wavefront and best compensation distance

The real best compensation wavefront was measured by partial compensation interferometer system, as shown in Fig. 5(a). Figure 5(b) shows the initial virtual best compensation wavefront simulated in the virtual partial compensation interferometer system, where the aspheric parameters are nominal. The wavefront deviation in Fig. 5(c) is the point-to-point deviation between the real best compensation wavefront and the initial virtual best compensation wavefront. The wavefront deviation contains rotationally symmetric and non-rotationally symmetric components. In aspheric SPE measurement, only the rotational wavefront deviation is considered as discussed in Section 2.1. Figure 5(d) shows the rotationally symmetric wavefront deviation and Table 3 lists the corresponding rotationally symmetric Zernike fringe coefficients of the real wavefront.

Fig. 5. The best compensation wavefront and initial deviations, (a) the real best compensation wavefront, (b) the initial virtual best compensation wavefront, (c) the deviation between the real best compensation wavefront and the initial virtual best compensation wavefront, and (d) the rotationally symmetric wavefront deviation fitted with Zernike fringe polynomial from the wavefront deviation.

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Table 3. Fitting results of the real residual wavefront

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The best compensation distance was measured by laser differential confocal system. Ten measurements were made to measure the displacement of the objective Lens 2, and the best compensation distances were calculated afterwards. Table 4 shows the measurement results of the best compensation distance. The average value of best compensation distance is 16.1891, and the uncertainty of the average value of the best compensation distance is 0.0010 mm with a confidence coefficient k = 3. Thus, the best compensation distance measured by laser differential confocal system is 16.1891 ± 0.0010 mm.

Table 4. The measured best compensation distance

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3.2.2 Parameters calculation

SPEs can be calculated with the best compensation wavefront measured by partial compensation interferometer system and the best compensation distance measured by laser differential confocal system. The solution follows the flowchart shown in Fig. 3.

Figure 6 shows the results of the first iterative optimization process of VRCIA. Figure 6(a) shows the best compensation distance calculated in each iteration cycle. Figures 6(b), (c) and (d) show the vertex radius of curvature, the conic constant and the fourth-order aspheric coefficient in each iteration cycle. The best compensation distance and SPEs in virtual partial compensation interferometer system are all convergent in several steps.

Fig. 6. Key parameters in iteration process. (a) The best compensation distance in the iterations, (b) vertex radius of curvature in the iterations, (c) conic constant in the iterations, and (d) fourth-order aspheric coefficients in the iterations.

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Table 5 compares the convergent intermediate SPEs after the first iterative optimization process of VRCIA and the final SPEs after the second iterative optimization process of VRCIA.

Table 5. Intermediate and final surface parameter errors

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The intermediate error of vertex radius of curvature is very small with a relative deviation of approximately only 2.517×10⁻²%. The intermediate error of conic constant is also very small with a relative deviation of approximately 0.175%. The relative deviation of intermediate fourth-order aspheric coefficient is approximately 3.375%.

The final error of vertex radius of curvature remains the same as the intermediate error. The final error of the conic constant reduces to a relative deviation of approximately 9.537×10⁻²%. The relative deviation of final fourth-order aspheric coefficient reduces to approximately 3.02%.

Figure 7 shows the final wavefront and final surface figure error. Figure 7(a) is the real best compensation wavefront measured by partial compensation interferometer system, which is the same as Fig. 5 (a). Figure 7(b) shows the final virtual best compensation wavefront. The wavefront deviation in Fig. 7(c) is the point-to-point deviation between the real and the virtual wavefront. The wavefront mainly contains non-rotationally symmetric components. Figure 7(d) shows the rotationally symmetric wavefront deviation. Compared with Fig. 5(d), the rotationally symmetric wavefront error decreases considerably.

Fig. 7. The best compensation wavefront and final deviations after calculation, (a) the real best compensation wavefront, (b) the final virtual best compensation wavefront, (c) the deviation between the real best compensation wavefront and the final virtual best compensation wavefront, and (d) the rotationally symmetric wavefront deviation between the real and the final virtual best compensation wavefront.

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However, some errors still exist in the calculated results. The ring of corrugation is caused by the errors as well as higher order aspheric surface parameters. In this experiment, we only considered fourth-order aspheric surface parameter. If sixth- or higher order aspheric surface parameters are considered, the PV of the residual deviation will be less.

3.3 Accuracy analysis

The measurement error mainly includes two categories of errors. One category is the system error including the non-rotational surface sag error and adjustment error during the measurement of the best compensation wavefront, and the Abbe error and cosine error during the measurement of the best compensation distance. The other error category is random error which exists in the measurement of the best compensation wavefront and the best compensation distance. And the error in the best compensation distance measurement is more obvious than that in the best compensation wavefront measurement.

Considering the random error in the best compensation distance measurement, the accuracy of the SPEs is numerically analyzed. According to the result from Section 3.2.1, the best compensation distance is 16.1891 ± 0.0010 mm. Thus, the SPEs are calculated at each best compensation distance which varies from 16.1881 mm to 16.1901 mm with a step of 0.0001 mm. Figure 8 shows the results.

Fig. 8. Influence of the accuracy of best compensation distance on the measured SPEs. (a) the variation of vertex radius of curvature with the best compensation distance, (b) the variation of vertex radius of conic constant and fourth-order aspheric coefficient with the best compensation distance.

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Figure 8 (a) shows the calculation results of vertex radius of curvature under the variation of the best compensation distance error. Vertex radius of curvature varies from 35.7469 mm to 35.7489 mm and is linear to the best compensation distance. Figure 8(b) shows the simulation results of the conic constant and fourth-order coefficient under the variation of the best compensation distance error. The conic constant varies from −0.6286 to −0.6284, and the fourth-order coefficient varies from −4.8890×10⁻⁷ to −4.8810×10⁻⁷. The conic constant and fourth-order aspheric coefficient are approximately negatively correlated because of coupling.

Thus, the measurement results show that vertex radius of curvature is 35.7479 ± 0.0010 mm, conic constant is −0.6285 ± 0.0001, and fourth-order aspheric coefficient is −(4.8850 ± 0.0040) ×10⁻⁷. The results are basically in correspondence with the results measured by UA3P-5. The deviation between the measurement results and the real parameters measured by UA3P-5 is mainly caused by the system error which needs to be compensated in more precise measurement.

4. Discussions

As for the McLaurin series utilized in Eq. (2) to describe aspheric surfaces, the convergency should be noticed. Under normal conditions, Eq. (2) is convergent when the measured aperture diameter of the surface under test satisfies Eq. (16)

(16)$$r < \frac{{|R |}}{{\sqrt {|{1 + K} |} }}.$$

As for the scope of application, VRCIA proposed in Sec. 2.1 can apply to concave aspheric surfaces and convex aspheric surfaces as well as high-order SPEs. In the experiment, the SPEs of a convex aspheric surface with fourth-order coefficient were measured for demonstration because the measurement of convex surfaces is generally more difficult than that of concave surfaces. When a concave aspheric surface is measured, the center of the best-fit sphere is in front of the surface under test, and only a minus or plus sign will simply be changed in Eq. (12).

In large-aperture aspheric surface parameter measurement, the compensator and the surface under test are large, which makes it difficult to measure the compensation distance by moving the surface under test or the compensator. The proposed measurement system utilizes a laser differential confocal system to measure the compensation distance. In laser differential confocal system, the converging lens is moved instead of the surface under test or the compensator. Meanwhile, the aperture of the converging lens could be much smaller than the surface under test or the compensator. Thanks to the development of laser differential confocal technology, the compensation distance can be measured precisely even when the displacement of the converging lens is long. Furthermore, combining the two measurement systems through a dichroic mirror to measure the residual wavefront and compensation distance simultaneously has application prospects. However, the optical axis of the laser differential confocal system and the optical axis of the compensation interferometer system are required to be coincident with each other.

5. Conclusions

An aspheric SPE measurement method is proposed to solve the bottleneck of SPE measurement of high-order aspheric surfaces and convex aspheric surfaces. A VRCIA is proposed to calculate the SPEs through two iterative optimization processes. Based on VRCIA, the SPEs of concave and convex aspheric surfaces with high-order SPEs can be calculated by the best compensation wavefront and the best compensation distance. An aspheric SPE measurement system with two subsystems is also proposed. The SPE measurement system utilizes a partial compensation interferometer system to measure the best compensation wavefront and a laser differential confocal system to measure the best compensation distance. An experiment was carried out to measure a convex aspheric surface with a fourth-order coefficient. The SPEs of the surface under test, including vertex radius of curvature, conic constant, and fourth-order aspheric parameter, were measured with a high accuracy. Accuracy analysis shows that the error of vertex radius of curvature is linear to the error of the best compensation wavefront, whereas the conic constant and fourth order aspheric coefficient are approximately negatively correlated.

Funding

National Natural Science Foundation of China (51735002); Strategic Priority Program of Chinese Academy of Science (XDA25020317).

Acknowledgments

We want to thank Mr. Jiahang Lv and Mr. Lei Cao for their helps in calibrating the refraction index of the compensator. Mr. Xiang Ding is appreciated for the aspheric surface measurement by UA3P-5. Mr. Shuai Yang is appreciated for his suggestions in the setup of laser differential confocal system.

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Parameters	R (mm)	K	A₄
Nominal value	35.8258	−0.6291	1.4398×10⁻⁷
Real value measured by UA3P-5	35.7569	−0.6291	−4.741×10⁻⁷

Surface	Radius (mm)	Thickness (mm)	Aperture (mm)	Glass
1	58.86	15.3	46.0	ZF4
2	−41.54	20.0	50.8	ZF7
3	−1074.58	—-		—-

Parameters	R (mm)	K (mm)	A₄ (mm)
Real value by UA3P-5	35.7569	−0.6291	−4.741×10⁻⁷
Intermediate surface parameters	35.7479	−0.6280	−4.901×10⁻⁷
Intermediate error	−0.0090	0.0011	−1.60×10⁻⁸
Intermediate relative error	2.517×10⁻²%	0.175%	3.375%
Final surface parameters	35.7479	−0.6285	−4.884×10⁻⁷
Final error	−0.0090	0.0006	−1.43×10⁻⁸
Final relative error	2.517×10⁻²%	9.537×10⁻²%	3.02%

Parameters	R (mm)	K	A₄
Nominal value	35.8258	−0.6291	1.4398×10⁻⁷
Real value measured by UA3P-5	35.7569	−0.6291	−4.741×10⁻⁷

Surface	Radius (mm)	Thickness (mm)	Aperture (mm)	Glass
1	58.86	15.3	46.0	ZF4
2	−41.54	20.0	50.8	ZF7
3	−1074.58	—-		—-

Interferometric measurement of high-order aspheric surface parameter errors based on a virtual-real combination iterative algorithm

Abstract

1. Introduction

2. Measurement system and virtual-real combination iterative algorithm

2.1 Surface parameter error measurement system configuration

2.2 Theory of the virtual-real combination iterative algorithm

2.3 Aspheric surface parameter errors measurement

3. Experiments and results

3.1 Measurement system

3.2 Measurement results

3.2.1 Residual wavefront and best compensation distance

3.2.2 Parameters calculation

3.3 Accuracy analysis

4. Discussions

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (5)

Equations (16)

Optics Express