Generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surfaces with global optimized wavefront reconstruction algorithm

Zhongming Yang; Jinyu Du; Chao Tian; Jiantai Dou; Qun Yuan; Zhishan Gao

doi:10.1364/OE.25.026133

1. Introduction

The spherical surfaces with high numerical aperture are commonly used in lithography optics. With the demand for the nanometer accuracy, various interferometric testing techniques, such as point diffraction interferometer [1–5] and absolute measurement method [6–10] have been developed. Due to the stringent working condition requirements and the expensive manufacturing cost, the point diffraction interferometer is rarely used in the majority of the conventional optical shops.

Several absolute measurement methods for spherical surface have been proposed in calibrating the reference surface deviation. The measurement accuracy in the two sphere method is sensitive to the astigmatism aberration, which introduced by the center of cat’s eye position determine [6]. The random ball averaging method is used in calibration of the interferometric system, however, it is very laborious [7]. A lot of work has been done in the field of the shift-rotation absolute measurement technique, such as N-position method [8], Zernike polynomial fitting method [9], pixel-level spatial frequency data-reduction method [10]. In all of the absolute measurement techniques, the testing surface shape obtained from the measurement results in different positions and we need to adjust the test spherical surface many times. The lateral and longitudinal displacement between the test spherical surface and the reference surface, which caused by the adjusting error of mechanical devices, will introduce high order misalignment aberrations in high numerical aperture spherical surface measurement [11,12]. As the nanometer quality is a requirement for high numerical aperture spherical surface, we need to obtain the testing surface shape from the measurement results by excluding the reference surface deviation and the high order misalignment aberrations [13].

In this paper, we propose a generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surface with global optimized wavefront reconstruction algorithm. Section 2 presents the theoretical analysis of our proposed absolute measurement method and the wavefront reconstruction algorithm which is based on the global optimization. Section 3 shows the simulation of the process generalized shift-rotation absolute measurement. Section 4 experimentally validates the accuracy and feasibility of our generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surface. Section 5 presents the error analysis and discussions. Some concluding remarks are shown in section 6.

2. Theoretical analysis

Optical interferometry is the most commonly employed method to measure the high numerical aperture spherical surface with high accuracy. The precision of the high numerical aperture spherical surface measurement is limited by the accuracy of the reference surface and the high order misalignment aberrations. The result of the high numerical aperture spherical surface measurement can be represented as

T (x, y) = W_{t e s t} (x, y) + W_{r e f e r e n c e} (x, y) + W_{a d j u s t m e n t} (x, y) + τ_{n o i s e} (x, y)

where

W_{t e s t} (x, y)

is the test high numerical aperture spherical surface shape,

W_{r e f e r e n c e} (x, y)

is the reference surface deviation,

W_{a d j u s t m e n t} (x, y)

is the high order misalignment aberrations, and

τ_{n o i s e} (x, y)

is the measurement noises. Assuming the measurement time is short, the measurement noise is usually small and ignored. So the result of the high numerical aperture spherical surface measurement can be written as

T (x, y) = W_{t e s t} (x, y) + W_{r e f e r e n c e} (x, y) + W_{a d j u s t m e n t} (x, y)

2.1 High order misalignment aberration in high numerical aperture spherical surface measurement

In high numerical aperture spherical surface measurement, the lateral, tilt and longitudinal displacement between test spherical surface and the reference surface will introduce high order misalignment aberrations. High order misalignment aberrations introduced by the lateral and tilt misalignment are equivalent in spherical surface measurement, as shown in Fig. 1(a). High order misalignment aberrations introduced by the longitudinal misalignment in high numerical aperture spherical surface measurement is shown in Fig. 1(b). In the high numerical aperture spherical surface measurement, the high order misalignment aberration can be written as [12,13]

\begin{array}{l} W_{a d j u s t m e n t} (x, y) = a_{1} \cdot Z_{1} - \frac{1}{4 R} \cdot (^{a_{1} \cdot Z_{1}) 2} + \frac{1}{8 R^{2}} \cdot (^{a_{1} \cdot Z_{1}) 3} \\ + a_{2} \cdot Z_{2} - \frac{1}{4 R} \cdot (^{a_{2} \cdot Z_{2}) 2} + \frac{1}{8 R^{2}} \cdot (^{a_{2} \cdot Z_{2}) 3} \\ + a_{3} \cdot Z_{3} + a_{8}' \cdot Z_{8} + a_{15}' \cdot Z_{15} + a_{24}' \cdot Z_{24} + a_{35}' \cdot Z_{35} + a_{48}' \cdot Z_{48} \end{array}

in which,

R

is the radius of the curvature of the test spherical surface,

Z_{1}

is the

x

direction tilt term in Zernike polynomials,

Z_{2}

is the

y

direction tilt term in Zernike polynomials,

Z_{3}

is the Zernike defocus term,

Z_{8}

,

Z_{15}

,

Z_{24}

,

Z_{35}

, and

Z_{48}

are the Zernike primary, secondary, tertiary, fourth, and fifth spherical terms.

a_{1}

,

a_{2}

,

a_{3}

,

a_{8}'

,

a_{15}'

,

a_{24}'

,

a_{35}'

, and

a_{48}'

are the corresponding coefficients.

Fig. 1 Misalignment aberration introduced by the test spherical surface misalignment.

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2.2 Generalized shift-rotation absolute measurement method

In order to calibrate the reference surface deviation and misalignment aberration in high numerical aperture spherical surface testing, a generalized shift-rotation absolute measurement method is proposed, as shown in Fig. 2. The position of the reference surface in our absolute measurement process is unchanged, the test position of the surface is shifted three times during the measurement–two original positions, a rotation position, a translation positions. First, there are two measurements in original positions, which have a longitudinal shift between those two measurements. And the results are

Fig. 2 The measurement process: (a) Basic measurement, (b) Rotational measurement, (c) Translational measurement.

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{\begin{cases} T_{0} (x, y) = W_{t e s t} (x, y) + W_{r e f e r e n c e} (x, y) + W_{a d j u s t m e n t 0} (x, y) \\ T_{0}' (x, y) = W_{t e s t} (x, y) + W_{r e f e r e n c e} (x, y) + W_{a d j u s t m e n t 0'} (x, y) \end{cases}

The difference between the $T_{0} (x, y)$ and $T_{0}' (x, y)$ can be written as

\begin{array}{l} T_{0} (x, y) - T_{0}' (x, y) = W_{a d j u s t m e n t 0} (x, y) - W_{a d j u s t m e n t 0'} (x, y) \\ = Δ a_{1} Z_{1} - \frac{1}{4 R} (^{Δ a_{1} Z_{1}) 2} + \frac{1}{8 R^{2}} (^{Δ a_{1} Z_{1}) 3} \\ + Δ a_{1} Z_{1} - \frac{1}{4 R} (^{Δ a_{1} Z_{1}) 2} + \frac{1}{8 R^{2}} (^{Δ a_{1} Z_{1}) 3} \\ + Δ a_{3} Z_{3} + r_{1} \cdot Δ a_{3} Z_{8} + r_{2} \cdot Δ a_{3} Z_{15} + r_{3} \cdot Δ a_{3} Z_{24} \\ + r_{4} \cdot Δ a_{3} Z_{35} + r_{5} \cdot Δ a_{3} Z_{48} \end{array}

In Eq. (5), the amount of $r_{1}$ , $r_{2}$ , $r_{3}$ , $r_{4}$ and $r_{5}$ , which depends on $N A$ (numerical aperture) of the test spherical surface, can be calculated by the Zernike fitting. With known $r_{1}$ , $r_{2}$ , $r_{3}$ , $r_{4}$ and $r_{5}$ , spherical aberrations introduced by longitudinal misalignments can be calculated based on the coefficient of Zernike defocus $a_{3}$ in four measurement positions in our measurement process.

Then, the position of the reference surface remaining unchanged, the test position of the surface is rotated with an angle $Δ θ$ . The measurement results can be written as

T_{1} (r, θ) = W_{t e s t} (r, θ + Δ θ) + W_{r e f e r e n c e} (r, θ) + W_{a d j u s t m e n t 1} (r, θ)

where

T_{1} (r, θ)

is the polar coordination transformation of

T_{1} (x, y)

. At last, the position of the reference surface is unchanged, the test position of the surface is shifted with a certain translation

δ_{y}

along the

y

axis and a certain translation

δ_{x}

along the

x

axis, which is based on the rotation position of the test surface. The measurement results can be written as

T_{2} (x, y) = W_{t e s t} (θ + Δ θ; x + δ_{x}, y + δ_{y}) + W_{r e f e r e n c e} (x, y) + W_{a d j u s t m e n t 2} (x, y)

With known $r_{1}$ , $r_{2}$ , $r_{3}$ , $r_{4}$ , $r_{5}$ and the radius of the curvature of the test spherical surface, $R$ the high order misalignment aberrations in Eq. (3) can be removed from the measurement results. The measurement results in our three positions can be written as

{\begin{cases} T_{0} (x, y) = W_{t e s t} (x, y) + W_{r e f e r e n c e} (x, y) \\ T_{1} (r, θ) = W_{t e s t} (r, θ + Δ θ) + W_{r e f e r e n c e} (r, θ) \\ T_{2} (x, y) = W_{t e s t} (θ + Δ θ; x + δ_{x}, y + δ_{y}) + W_{r e f e r e n c e} (x, y) \end{cases}

The wavefront difference data of the high numerical spherical surface can be obtained by

{\begin{cases} Δ W_{1} (r, θ) = W_{t e s t} (r, θ + Δ θ) - W_{t e s t} (r, θ) = T_{1} (x, y) - T_{0} (x, y) \\ Δ W_{2} (x, y) = W_{t e s t} (θ + Δ θ; x + δ_{x}, y + δ_{y}) - W_{t e s t} (x, y) = T_{2} (x, y) - T_{0} (x, y) \end{cases}

Based on Eq. (9), the shape of the test high numerical aperture spherical surface $W_{t e s t} (x, y)$ can be obtained by our global optimized wavefront reconstruction method.

2.3 Generalized shift-rotation absolute measurement method

The wavefront reconstruction method plays an important role in shift-rotation absolute measurement of the spherical surface. Wavefront reconstruction methods based on Fourier transform [14], Zernike polynomial fitting [15] and Hudgin model [16] are widely used in interferometric testing. However, those wavefront reconstruction methods are sensitive to the translation error and rotation error in the measurement by adjusting device, especially in the measurement by an imperfect translation fixture. In order to get the absolute test surface shape, the wavefront reconstruction method based on differential evolution algorithm with strategic adaptation is introduced in our work, which expects to eliminate testing errors caused by translation error and rotation error.

It is well known that the absolute test surface shape $W_{t e s t} (x, y)$ can be described by the sum of Zernike polynomials $Z_{i} (x, y)$ or $Z_{i} (r, θ)$ , which is the Zernike polynomials in the polar coordination system, with the their corresponding coefficients $a_{i}$ :

{\begin{cases} W_{t e s t} (x, y) = \sum_{i = 1}^{N} a_{i} Z_{i} (x, y) \\ W_{t e s t} (x, y) = \sum_{i = 1}^{N} a_{i} Z_{i} (r, θ) \end{cases}

Based on Eq. (10), the wavefront difference of the high numerical spherical surface can be described by:

{\begin{cases} Δ W_{1} (r, θ) = \sum_{i = 1}^{N} a_{i} \cdot [Z (r, θ + Δ θ) - Z (r, θ)] \\ Δ W_{2} (x, y) = \sum_{i = 1}^{N} a_{i} \cdot [Z (θ + Δ θ, x + δ_{x}, y + δ_{y}) - Z (x, y)] \end{cases}

where the wavefront difference of the Zernike polynomials

Z (r, θ + Δ θ) - Z (r, θ)

and

Z (θ + Δ θ, x + δ_{x}, y + δ_{y}) - Z (x, y)

can be obtained from the point clouds of Zernike polynomials

Z_{i} (x, y)

. Based on Eq. (9) and Eq. (11), we can construct the cost function to estimate the Zernike polynomials coefficient

a_{i}

of the absolute test surface shape

W_{t e s t} (x, y)

, the rotation angle

Δ θ

, the translation

δ_{x}

along the

x

axis and the translation

δ_{y}

along the

y

axis:

\begin{array}{l} f (a_{1}, a_{2}, \dots, a_{N}, Δ θ, δ_{x}, δ_{y}) \\ = \sum_{(x, y) \in S} {{[T_{1} (x, y) - T_{0} (x, y)] - \sum_{i = 1}^{N} a_{i} \cdot [Z (r, θ + Δ θ) - Z (r, θ)]} + \dots \\ \dots + {[T_{2} (x, y) - T_{0} (x, y)] - \sum_{i = 1}^{N} a_{i} \cdot [Z (θ + Δ θ, x + δ_{x}, y + δ_{y}) - Z (x, y)]}} \end{array}

where S is the mask that defines the region of valid data. Next, we will discuss the optimization algorithm used to seek the

X = (a_{1}, a_{2}, \dots, a_{N}, δ_{x}, δ_{y})

to minimize the cost function Eq. (12):

X (a_{1}, a_{2}, \dots, a_{N}, Δ θ, δ_{x}, δ_{y}) = \arg \min f (X)

The differential evolution algorithm is a powerful population based stochastic search technique, inspired by the nature of the evolution of species, which has been successfully applied in global optimization [17]. Global optimization has been successfully used in optical testing, such as fringe patterns demodulation [18]. To achieve the most satisfactory optimization performance to our problem, we use the self-adaptive differential evolution algorithm, in which introduce both trial vector generation strategy and control parameter adaptation schemes into the conventional differential evolution framework [19]. The self-adaptive differential evolution algorithm aims at evolving a solution vector, so-called individuals, which have a population size $N P \times D$ , towards the global optimum. The solution vector can be written as:

\begin{array}{l} X_{i, G} = {x_{i, G}^{1}, x_{i, G}^{2}, \dots, x_{i, G}^{D}} \\ = {a_{1 i, G}^{1}, a_{2 i, G}^{2}, \dots, a_{N i, G}^{N}, Δ θ_{i, G}^{N + 1} δ_{x i, G}^{N + 2}, δ_{y i, G}^{N + 3}} \end{array}

where

D = N + 3

is the dimensionality of the solution vector,

i = 1, 2, \dots, N P

,

N P

is the population size. In the self-adaptive differential evolution algorithm,

N P

is a user-specified parameter, which is relies on the complexity of a given problem. The initial population should better cover the entire search space and the minimum and maximum parameter bounds within the search space can be denoted as:

{\begin{cases} X_{\max} = {x_{\max}^{1}, x_{\max}^{2}, \dots, x_{\max}^{D}} \\ X_{\min} = {x_{\max}^{1}, x_{\min}^{2}, \dots, x_{\min}^{D}} \end{cases}

The initial value of the $j$ th parameter in the $i$ th individual at the 0th generation can be initialized as follows:

x_{i, 0}^{j} = x_{\min}^{j} + r a n d (0, 1) \times (x_{\max}^{j} - x_{\min}^{j})

where

j = 1, 2, \dots, D

, and rand(0,1) is a uniformly random variable distributed in range [0,1]. A. Trial Vector Generation

After initialization, the strategy adaptation is employed to generate a trial vector $U_{i, G} = (u_{i, G}^{1}, u_{i, G}^{2}, \dots, u_{i, G}^{D})$ with respect to $X_{i, G}$ in the current population. Using the effective trial vector generation strategy, such as “DE/rand/1/bin”, the value of the $j$ th parameter in the $i$ th individual in the trial vector $U_{i, G}$ , which at the generation G, can be written as DE/rand/1/bin:

u_{i, j} = {\begin{matrix} x_{r_{1}, j} + F \cdot (x_{r_{2}, j} - x_{r_{3}, j}), i f r a n d [0, 1] < C R o r j = j_{r a n d} \\ x_{i, j}, o t h e r w i s e \end{matrix}

where

r_{1}

,

r_{2}

,

r_{3}

are mutually exclusive integers randomly generated by

r a n d [0, 1]

. Scaling factor

F

is approximated by a normal distribution with mean value 0.5 and standard deviation 0.3, which denoted by

N (0.5, 0.3)

. In self-adaptive differential evolution algorithm, the proper choice of the value of crossover rate

C R

can lead to successful optimization, which generally falls into a small range for a given problem.

C R

obeys a normal distribution with mean value

C R m

and standard deviation

S t d

, which denoted by

N (C R m, S t d)

. The

C R m

can be initialized as 0.5 and

S t d

can be set as 0.1.

j_{r a n d}

is a randomly chosen integer by

r a n d [0, D]

. B. Selection Operation

After the trial vector generation, the value of the cost function about trial vector $f (U_{i, G})$ is compared to the value of its corresponding cost function about target vector $f (X_{i, G})$ in current population, which used to determine the new individual of the population at next generation. The selection operation can be written as follows:

X_{i, G + 1} = {\begin{matrix} U_{i, G}, i f f (U_{i, G}) \leq f (X_{i, G}) \\ X_{i, j}, o t h e r w i s e \end{matrix}

The steps of trial vector generation and selection operation are repeated generation after generation until the preset convergence criterions of cost function are satisfied, as shown in Fig. 3. If we seek the $(a_{1}, a_{2}, \dots, a_{N}, Δ θ, δ_{x}, δ_{y})$ to minimize the cost function Eq. (12), the absolute test surface shape $W_{t e s t} (x, y)$ can be got, which described by the sum of Zernike polynomials $Z_{i} (x, y)$ with the corresponding coefficient $a_{i}$ .

Fig. 3 The process of Wavefront reconstruction process based on self-adaptive differential evolution algorithm.

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3. Numerical simulation and analysis

Numerical simulation and analysis of the high order misalignment aberration in high numerical aperture spherical surface measurement and the comparison with previous shift-rotation absolute measurement methods have been proposed by us [12,13]. Now, let’s focus on our generalized shift-rotation method based on self-adaptive differential evolution algorithm. In order to validate the accuracy and feasibility of the proposed generalized shift-rotation method, the simulated test surface and reference surface are generated by the sum of Zernike polynomials $Z_{i} (x, y)$ , the corresponding coefficient of Zernike polynomials are shown in Fig. 4(a), the test surface shape and reference surface shape are shown in Fig. 4(b) and Fig. 4(c), in which the high order misalignment aberrations have been removed.

Fig. 4 The slope of the ideal surface (units: nm). (a) The coefficients of Zernike polynomials of the test surface and reference surface, (b) The test surface slope, (c) The reference surface slope.

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According to the process of our generalized shift-rotation absolute measurement method, as shown in Fig. 2, measurement results are got in turn. Firstly, on the original position, the test results $T_{02} (x, y)$ on the CCD is shown in Fig. 5(a). Secondly, the position of the reference surface is unchanged. The test position of the surface is rotated with an angle of ${11.25}^{\circ}$ . The measurement results $T_{1} (r, θ)$ on the CCD is shown in Fig. 5(b). Lastly, the position of the reference surface is unchanged. The test position of the surface is shifted with a 17 pixels translation along the $y$ axis and a 9 pixels translation along the $x$ axis, which is based on the rotation position of the test surface. The measurement results $T_{2} (x, y)$ on the CCD is being shown in Fig. 5(c).

Fig. 5 The measurement results on the three positions (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

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Wavefront difference data between rotational and original position is shown in Fig. 6(a), and wavefront difference data between translational and original position is shown in Fig. 6(b). According to the process of wavefront reconstruction method based on self-adaptive differential evolution algorithm, as shown in Fig. 3, the Zernike polynomials coefficient $a_{i}$ of the test surface shape $W_{t e s t} (x, y)$ , the rotation angle $Δ θ$ , the translation $δ_{x}$ along the $x$ axis and the translation $δ_{y}$ along the $y$ axis are got. The Zernike polynomials coefficients of the test surface shape are consistent with the Zernike polynomials coefficients of simulated test surface, which are shown in Fig. 7(a). The rotation angle $Δ θ = {11.25}^{\circ}$ , the translation along the positive half of $x$ axis $δ_{x} = 9$ and the translation along the negative half of $x$ axis $δ_{y} = - 17$ . The absolute test surface shape $W_{t e s t} (x, y)$ , which described by the sum of Zernike polynomials $Z_{i} (x, y)$ with the corresponding coefficient $a_{i}$ is shown in Fig. 7(b). The comparison of the surface deviation of the simulated surface and test surface shape obtained by our simple shift rotation measurement method is shown in Fig. 7(c), with the PV $6.33 \times 10^{- 12}$ nm and RMS $7.54 \times 10^{- 13}$ nm.

Fig. 6 Wavefront difference data (units: nm). (a) Between rotational and original position, (b) Between translational and original position.

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Fig. 7 The simulation results (units: nm). (a) The Zernike polynomials coefficients of the test surface shape compared with the Zernike polynomials coefficients of simulated test surface, (b) Absolute test surface shape, (c) The comparison of the surface deviation of the simulated surface and test surface shape.

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4. Experiments and results

To verify the generalized shift-rotation absolute measurement method described above, experiments were carried out in a 6-inch Fizeau interferometer with F/0.75 sphere transmission and two test spherical surface. Commercial Fizeau interferometer is working at 632.8 nm. In the experiment, the test spherical surface was tested at three positions (basic, rotational, translational position). The radius curvature of the first test spherical surface is 63.84 mm and the clear aperture is 87.00 mm. On the original position, the test results $T_{02} (x, y)$ is shown in Fig. 8(a). And then, the position of the reference surface is unchanged, the test surface is rotated with an angle. The measurement results $T_{1} (r, θ)$ is shown in Fig. 8(b). Lastly, the position of the reference surface is unchanged. The test position of the surface is shifted translate to a translational position. The measurement results $T_{2} (x, y)$ is shown in Fig. 8(c). Wavefront difference data between rotational and original position is shown in Fig. 9(a), the wavefront difference data between translational and original position is shown in Fig. 9(b). According to the process of wavefront reconstruction method based on self-adaptive differential evolution algorithm, the Zernike polynomials coefficient $a_{i}$ of the first test surface shape $W_{t e s t} (x, y)$ , the rotation angle $Δ θ$ , the translation $δ_{x}$ along the $x$ axis and the translation $δ_{y}$ along the $y$ axis are got. The rotation angle $Δ θ = - {3.21}^{\circ}$ , the translation along the $x$ axis $δ_{x} = 7$ and the translation along the axial $y$ $δ_{y} = - 9$ . The absolute test surface shape $W_{t e s t} (x, y)$ and the reference surface shape $W_{r e f e r e n c e} (x, y)$ , which are described by the sum of Zernike polynomials $Z_{i} (x, y)$ , are shown in Fig. 10(a) and Fig. 10(b), respectively.

Fig. 8 The measurement results of the first test spherical surface on the three positions in experiment (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

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Fig. 9 The wavefront difference data of the first test spherical surface in experiment (units: nm). (a) Between rotational and original position, (b) Between translational and original position.

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Fig. 10 The measurement results of the first test spherical surface in experiment (units: nm). (a) The absolute test surface shape, (b) The reference surface shape.

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The radius curvature of the second test spherical surface is 50.04 mm and the clear aperture is 70.00 mm. On the original position, the test results $T_{02} (x, y)$ is shown in Fig. 11(a). And then, the position of the reference surface is unchanged, the test surface is rotated with an angle. The measurement results $T_{1} (r, θ)$ is shown in Fig. 11(b). Lastly, the position of the reference surface is unchanged. The test position of the surface is shifted translate to a translational position. The measurement results $T_{2} (x, y)$ is shown in Fig. 11(c). Wavefront difference data between rotational and original position is shown in Fig. 12(a), the wavefront difference data between translational and original position is shown in Fig. 12(b). According to the process of wavefront reconstruction method described above, the Zernike polynomials coefficient $a_{i}$ of the first test surface shape $W_{t e s t} (x, y)$ , the rotation angle $Δ θ$ , the translation $δ_{x}$ along the $x$ axis and the translation $δ_{y}$ along the $y$ axis are got. The rotation angle $Δ θ = {5.12}^{\circ}$ , the translation along the $x$ axis $δ_{x} = 11$ and the translation along the axial $y$ $δ_{y} = - 8$ . The absolute test surface shape $W_{t e s t} (x, y)$ and the reference surface shape $W_{r e f e r e n c e} (x, y)$ , which are described by the sum of Zernike polynomials $Z_{i} (x, y)$ , are shown in Fig. 13(a) and Fig. 13(b), respectively. Deference of reference surface shapes between two testing is shown in Fig. 14, with the PV $0.90$ nm and RMS $0.12$ nm, which mainly caused by the random error between two testing.

Fig. 11 The measurement results of the second test spherical surface on the three positions in experiment (units: nm). (a) The original position, (b) The rotational position, (c) The translational position.

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Fig. 12 The wavefront difference data of the second test spherical surface in experiment (units: nm). (a) Wavefront difference data between rotational and original position, (b) Wavefront difference data between translational and original position.

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Fig. 13 The measurement results of the second test spherical surface in experiment (units: nm). (a) The absolute test surface shape, (b) The reference surface shape.

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Fig. 14 Deference of reference surface shapes between two testing (units: nm). (a) The Zernike polynomials coefficients of the reference surface shape, (b) The wavefront difference data between two testing.

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5. Error analysis and discussions

According to the process of wavefront reconstruction method based on self-adaptive differential evolution algorithm, the Zernike polynomials coefficient $a_{i}$ of the test surface shape $W_{t e s t} (x, y)$ , the rotation angle $Δ θ$ , the translation $δ_{x}$ along the $x$ axis and the translation $δ_{y}$ along the $y$ axis are got. So the translation error and rotation error in other absolute measurement method are avoided in our generalized shift-rotation absolute measurement method.

The CCD noise is one of the major error sources, which is a sort of random error during the absolute test for optical surfaces, such as cylindrical surface measurement using the conjugate differential absolute method [20]. In order to investigate the influence of the CCD noise to the measurement results. The random noise amplitude is set to between −1nm and 1nm, which add to the measurement results on the three positions in our simulations. After the process of wavefront reconstruction method based on self-adaptive differential evolution algorithm, the Zernike polynomials coefficients of the test surface shape are got, which are shown in Fig. 15(a), with the Zernike polynomials coefficients of simulated test surface. The rotation angle $Δ θ = {11.25}^{\circ}$ , the translation along the $x$ axis $δ_{x} = 9$ and the translation along the $y$ axis $δ_{y} = - 17$ . The absolute test surface shape $W_{t e s t} (x, y)$ is shown in Fig. 15(b). The comparison of surface deviation of the simulated surface and test surface shape obtained by our simple shift rotation measurement method is shown in Fig. 15(c), with the PV $0.15$ nm and RMS $0.04$ nm. Based on the simulation results with CCD noise, the accuracy of our generalized shift-rotation absolute measurement method is still under the sub-nanometer level with the normal noise conditions, which is the same order of magnitude of the CCD noise. Besides, our generalized shift-rotation absolute measurement method is not designed to be performed in real time, the overall execution time of our algorithm is depend on the image size, the parameters of the self-adaptive differential evolution algorithm, such as search ranges and generations.

Fig. 15 The simulate results with noise (units: nm). (a) The Zernike polynomials coefficients of the test surface shape compared with the Zernike polynomials coefficients of simulated test surface, (b) Absolute test surface shape, (c) The comparison of the surface deviation of the simulated surface and test surface shape.

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6. Conclusions

We have presented a generalized shift-rotation absolute measurement method to measure the absolute surface shape of high-numerical-aperture spherical surface. High order misalignment aberrations can be removed from the measurements by the wavefront difference method. Our generalized shift-rotation absolute measurement process only needs one rotational measurement position and one translational measurement position, which is convenient compared with previous shift-rotation absolute measurement methods. The translation error and rotation error in other absolute measurement method are avoided in our shift-rotation absolute measurement method. A wavefront reconstruction method based on self-adaptive differential evolution algorithm is proposed to calculate the Zernike polynomials coefficient $a_{i}$ of the absolute surface shape $W_{t e s t} (x, y)$ , the rotation angle $Δ θ$ , the translation $δ_{x}$ along the $x$ axis and the translation $δ_{y}$ along the $y$ axis. Experimental absolute results of the test surface and reference surface are given, which indicate the validity of the proposed absolute measurement method.

Funding

National Natural Science Foundation of China (NSFC) (61377015, 61505080); Natural Science Foundation of Jiangsu Province (BK20150788); The Fundamental Research Funds of Shandong University (11170077614092).

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Generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surfaces with global optimized wavefront reconstruction algorithm

Abstract

1. Introduction

2. Theoretical analysis

2.1 High order misalignment aberration in high numerical aperture spherical surface measurement

2.2 Generalized shift-rotation absolute measurement method

2.3 Generalized shift-rotation absolute measurement method

3. Numerical simulation and analysis

4. Experiments and results

5. Error analysis and discussions

6. Conclusions

Funding

References and links

Cited By

Figures (15)

Equations (18)

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