Self-Calibration of Precision XYθ<sup>z</sup> Metrology Stages

Chuxiong Hu, Yu Zhu and Luzheng Liu

### Abstract

This chapter studies the on-axis calibration for precision XYθ<sup>z</sup> metrology stages and presents a holistic XYθ<sup>z</sup> self-calibration approach. The proposed approach uses an artifact plate, specially designed with XY grid mark lines and angular mark lines, as a tool to be measured by the XYθ<sup>z</sup> metrology stages. In detail, the artifact plate is placed on the uncalibrated XYθ<sup>z</sup> metrology stages in four measurement postures or views. Then, the measurement error can be modeled as the construction of XYθ<sup>z</sup> systematic measurement error (i.e. stage error), artifact error, misalignment error, and random measurement noise. With a new property proposed, redundance of the XYθ<sup>z</sup> stage error is obtained, while the misalignment errors of all measurement views are determined by rigid mathematical processing. Resultantly, a least squarebased XYθ<sup>z</sup> self-calibration law is synthesized for final determination of the stage error. Computer simulation is conducted, and the calculation results validate that the proposed scheme can accurately realize the stage error even under the existence of various random measurement noise. Finally, the designed artifact plate is developed and illustrated for explanation of a standard XYθ<sup>z</sup> self-calibration procedure to meet practical industrial requirements.

Keywords: XYθ<sup>z</sup> stage, self-calibration, measurement system, least square, stage error

### 1. Introduction

Precision XYθ<sup>z</sup> motion stages are ubiquitously utilized in industrial mechanical systems to meet the requirement of high-performance manufacture [1]. As automatical servo systems, these stages have both precision linear encoders and angle encoders for measurement and motion feedback control [2–7]. In practice, the measurement accuracy inevitably suffers from surface non-flatness and un-roundness, axis nonorthogonality, scale graduation nonuniformity, encoder installation eccentricity, read-head misalignment, and so on, which resultantly generate systematic measurement error, i.e. stage error. The stage error can in principle be eliminated through calibration technology [8–10]. Due to the difficulty on finding a more accurate standard tool in traditional calibration technologies, self-calibration technology has been developed with utilization of an artifact with mark positions not precisely known. As an alternative of intelligent calibration processes, self-calibration is an effective and economical approach especially for micro-/nano-level mechanical systems [11–14].

Existing self-calibration technologies were developed for X, XY, XYZ, and angular metrology stages, respectively. For example, Takac studied one-dimensional

self-calibration and developed a scheme that made a set of tool graduation marks appear to have identical spacing with relative scale [15]. In [16], self-calibration method for single-axis dual-drive nanometer positioning stage was presented. In [17], an XY self-calibration strategy was presented for two-dimensional metrology stages, which used an artifact plate as assistance measured by three views to construct equations of stage error, misalignment error, and artifact error. Fourier transformation was employed in the scheme to meet the challenge of random measurement noise. This method is popularly followed by many engineers and researchers [18–21]. In [18], a self-calibration algorithm was developed to test the out-of-plane error of two-dimensional profiling stages. The algorithm suppresses artifact-related errors in consideration of the geometrical congruence of three profile measurement views. Computer simulation and experimental results both showed that the calibration accuracy was free from artifact imperfection and only minimally affected by random measurement errors. In [19], a self-calibration method was proposed for mapping the errors in XY plane and the squareness error between Z-axis and XY plane of the scanning probe microscopes. In [22, 23], selfcalibration approach for three-dimensional metrology stages was completely provided with experimental validation.

On the other hand, lots of self-calibration technologies have been developed for angular metrology systems [14, 24] in US National Institute of Standards and Technology (NIST), National Metrology Institute of Japan (NMIJ), Germany's National Metrology Institute the Physikalisch-Technische Bundesanstalt (PTB), Korea Research Institute of Standards and Science, etc. Specifically, circle closure principle was frequently used to cross-calibrate index tables in NIST [25, 26]. A high-precision rotary encoder self-calibration system was built based on equaldivision-averaged method and had been adopted as the angular national standard system in NMIJ [27, 28]. The equal-division-averaged method was also expanded for self-calibration of the scale error in an angle comparator [29]. In addition, a known prime factor algorithm-based method was presented for self-calibration of divided circles in PTB [30, 31].

designed artifact plate is manufactured, and a standard on-axis XYθ<sup>z</sup> self-calibration

presented in Section V. Finally, the conclusion is provided in Section VI.

For a XYθ<sup>z</sup> metrology system, linear encoders are employed for measuring movement along X and Y axes, and a rotary encoder for measuring rotation along θ<sup>z</sup> axis. Thus, the systematic errors along X-axis, Y-axis, and θ<sup>z</sup> axis are independent. And once the metrology system is set, the geometric relationship among X-axis, Yaxis, and θ<sup>z</sup> axis is also determined, which will be described in detail later. In the Cartesian grid, define Glð Þ x; y as the linear stage error at ð Þ x; y where ð Þ x; y is the true location. And Grð Þ θ<sup>z</sup> is the rotary stage error at θ<sup>z</sup> where θ<sup>z</sup> is the true angle value. Herein, the uncalibrated XYθ<sup>z</sup> field consists of XY with L � L and θ<sup>z</sup> with 360°, while the XYθ<sup>z</sup> origin point is set as the same at the center of the L � L field. In

Glð Þ� x; y Gxð Þ x; y e<sup>x</sup> þ Gyð Þ x; y e<sup>y</sup>

(1)

Grð Þ� θ<sup>z</sup> G<sup>θ</sup><sup>z</sup> ð Þ θ<sup>z</sup> e<sup>θ</sup><sup>z</sup>

The proposed scheme mainly features the following two benefits: (1) Departing from previous self-calibration technologies, the proposed scheme first solves the onaxis self-calibration problem of XYθ<sup>z</sup> metrology stages and (2) complicated mathematical manipulations, especially the calculations of misalignment errors in previous XY self-calibration schemes, are significantly avoided in the proposed strategy. The remainder of this chapter is organized as follows. In Section II, the stage error of XYθ<sup>z</sup> metrology stage is explained, and a newly designed artifact plate and related artifact error are also described. The principle of the developed XYθ<sup>z</sup> self-calibration scheme with four measurement views is presented in Section III. In Section IV, computer simulation is conducted to show the calibration performance of the proposed method. And the procedure for performing a standard XYθ<sup>z</sup> self-calibration is

procedure following the proposed scheme is introduced.

An artifact plate with mark lines on an XYθ<sup>z</sup> metrology stage.

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

2. Self-calibration problem formulation

2.1 Stage error

Figure 1.

the following, we define

25

In summary of previous self-calibration strategies, a systematic self-calibration strategy for calibration of XYθ<sup>z</sup> metrology stage is seldom published up to present. To address this problem, we have proposed a preliminary framework to selfcalibrate the XYθ<sup>z</sup> stage error in [32], assuming that the angular coordinate and the XY coordinate are uncorrelated while the XY stage error and θ<sup>z</sup> stage error are solved separately. This assumption leads to the final XYθ<sup>z</sup> calibration being not in a uniform coordinate, which means that it is not a complete and accurate XYθ<sup>z</sup> self-calibration strategy. In this chapter, we further study the self-calibration of precision XYθ<sup>z</sup> metrology stages and present a complete and accurate on-axis selfcalibration approach. Specifically, a new artifact plate is designed as the assistant tool, and four measurement views of the designed artifact plate on the uncalibrated XYθ<sup>z</sup> metrology stage are constructed to provide measurement information. The detailed specification of the artifact plate on the XYθ<sup>z</sup> stage is shown in Figure 1. Combining with symmetry, transitivity, and circle closure principle, certain redundance of the XYθ<sup>z</sup> stage error is established, while the misalignment errors of all measurement views are determined by rigid mathematical manipulation. Resultantly, a least square-based XYθ<sup>z</sup> self-calibration law is proposed for the final determination of the stage error. Computer simulation is conducted, and the calculation results validate that scheme proposed in this paper can figure out the stage error rather accurately in the absence of random measurement noise. The self-calibration accuracy of the proposed scheme is also tested to meet the challenge of various random measurement noises, and the calibration results validate that the scheme can effectively alleviate the effects of random measurement noise. Finally, the

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

self-calibration and developed a scheme that made a set of tool graduation marks appear to have identical spacing with relative scale [15]. In [16], self-calibration method for single-axis dual-drive nanometer positioning stage was presented. In [17], an XY self-calibration strategy was presented for two-dimensional metrology stages, which used an artifact plate as assistance measured by three views to construct equations of stage error, misalignment error, and artifact error. Fourier transformation was employed in the scheme to meet the challenge of random measurement noise. This method is popularly followed by many engineers and researchers [18–21]. In [18], a self-calibration algorithm was developed to test the out-of-plane error of two-dimensional profiling stages. The algorithm suppresses artifact-related errors in consideration of the geometrical congruence of three profile measurement views. Computer simulation and experimental results both showed that the calibration accuracy was free from artifact imperfection and only minimally affected by random measurement errors. In [19], a self-calibration method was proposed for mapping the errors in XY plane and the squareness error between Z-axis and XY plane of the scanning probe microscopes. In [22, 23], selfcalibration approach for three-dimensional metrology stages was completely pro-

On the other hand, lots of self-calibration technologies have been developed for

In summary of previous self-calibration strategies, a systematic self-calibration strategy for calibration of XYθ<sup>z</sup> metrology stage is seldom published up to present. To address this problem, we have proposed a preliminary framework to selfcalibrate the XYθ<sup>z</sup> stage error in [32], assuming that the angular coordinate and the XY coordinate are uncorrelated while the XY stage error and θ<sup>z</sup> stage error are solved separately. This assumption leads to the final XYθ<sup>z</sup> calibration being not in a uniform coordinate, which means that it is not a complete and accurate XYθ<sup>z</sup> self-calibration strategy. In this chapter, we further study the self-calibration of precision XYθ<sup>z</sup> metrology stages and present a complete and accurate on-axis selfcalibration approach. Specifically, a new artifact plate is designed as the assistant tool, and four measurement views of the designed artifact plate on the uncalibrated XYθ<sup>z</sup> metrology stage are constructed to provide measurement information. The detailed specification of the artifact plate on the XYθ<sup>z</sup> stage is shown in Figure 1. Combining with symmetry, transitivity, and circle closure principle, certain redundance of the XYθ<sup>z</sup> stage error is established, while the misalignment errors of all measurement views are determined by rigid mathematical manipulation. Resultantly, a least square-based XYθ<sup>z</sup> self-calibration law is proposed for the final determination of the stage error. Computer simulation is conducted, and the calculation results validate that scheme proposed in this paper can figure out the stage error rather accurately in the absence of random measurement noise. The self-calibration accuracy of the proposed scheme is also tested to meet the challenge of various random measurement noises, and the calibration results validate that the scheme can effectively alleviate the effects of random measurement noise. Finally, the

angular metrology systems [14, 24] in US National Institute of Standards and Technology (NIST), National Metrology Institute of Japan (NMIJ), Germany's National Metrology Institute the Physikalisch-Technische Bundesanstalt (PTB), Korea Research Institute of Standards and Science, etc. Specifically, circle closure principle was frequently used to cross-calibrate index tables in NIST [25, 26]. A high-precision rotary encoder self-calibration system was built based on equaldivision-averaged method and had been adopted as the angular national standard system in NMIJ [27, 28]. The equal-division-averaged method was also expanded for self-calibration of the scale error in an angle comparator [29]. In addition, a known prime factor algorithm-based method was presented for self-calibration of

vided with experimental validation.

Standards, Methods and Solutions of Metrology

divided circles in PTB [30, 31].

24

Figure 1. An artifact plate with mark lines on an XYθ<sup>z</sup> metrology stage.

designed artifact plate is manufactured, and a standard on-axis XYθ<sup>z</sup> self-calibration procedure following the proposed scheme is introduced.

The proposed scheme mainly features the following two benefits: (1) Departing from previous self-calibration technologies, the proposed scheme first solves the onaxis self-calibration problem of XYθ<sup>z</sup> metrology stages and (2) complicated mathematical manipulations, especially the calculations of misalignment errors in previous XY self-calibration schemes, are significantly avoided in the proposed strategy. The remainder of this chapter is organized as follows. In Section II, the stage error of XYθ<sup>z</sup> metrology stage is explained, and a newly designed artifact plate and related artifact error are also described. The principle of the developed XYθ<sup>z</sup> self-calibration scheme with four measurement views is presented in Section III. In Section IV, computer simulation is conducted to show the calibration performance of the proposed method. And the procedure for performing a standard XYθ<sup>z</sup> self-calibration is presented in Section V. Finally, the conclusion is provided in Section VI.

### 2. Self-calibration problem formulation

#### 2.1 Stage error

For a XYθ<sup>z</sup> metrology system, linear encoders are employed for measuring movement along X and Y axes, and a rotary encoder for measuring rotation along θ<sup>z</sup> axis. Thus, the systematic errors along X-axis, Y-axis, and θ<sup>z</sup> axis are independent. And once the metrology system is set, the geometric relationship among X-axis, Yaxis, and θ<sup>z</sup> axis is also determined, which will be described in detail later. In the Cartesian grid, define Glð Þ x; y as the linear stage error at ð Þ x; y where ð Þ x; y is the true location. And Grð Þ θ<sup>z</sup> is the rotary stage error at θ<sup>z</sup> where θ<sup>z</sup> is the true angle value. Herein, the uncalibrated XYθ<sup>z</sup> field consists of XY with L � L and θ<sup>z</sup> with 360°, while the XYθ<sup>z</sup> origin point is set as the same at the center of the L � L field. In the following, we define

$$\begin{aligned} \mathbf{G}\_l(\mathbf{x}, \boldsymbol{\mathcal{y}}) & \equiv \mathbf{G}\_\mathbf{x}(\mathbf{x}, \boldsymbol{\mathcal{y}}) \mathbf{e}\_\mathbf{x} + \mathbf{G}\_\boldsymbol{\mathcal{y}}(\mathbf{x}, \boldsymbol{\mathcal{y}}) \mathbf{e}\_\mathbf{y} \\ \mathbf{G}\_r(\boldsymbol{\theta}\_\mathbf{z}) & \equiv \mathbf{G}\_{\boldsymbol{\theta}\_\mathbf{z}}(\boldsymbol{\theta}\_\mathbf{z}) \mathbf{e}\_{\boldsymbol{\theta}\_\mathbf{z}} \end{aligned} \tag{1}$$

where ex, ey, and e<sup>θ</sup><sup>z</sup> are the unit vectors of the stage axes. For notation, we combine linear and rotary stage errors and define Gð Þ x; y; θ<sup>z</sup> is the stage error at ð Þ x; y; θ<sup>z</sup> where ð Þ x; y is the true location and θ<sup>z</sup> is the true angle in the Cartesian grid:

$$\begin{split} \mathbf{G}(\mathbf{x}, \boldsymbol{y}, \boldsymbol{\theta}\_{\boldsymbol{x}}) & \equiv \mathbf{G}\_{l}(\mathbf{x}, \boldsymbol{y}) + \mathbf{G}\_{r}(\boldsymbol{\theta}\_{\boldsymbol{x}}) \\ &= \mathbf{G}\_{\mathbf{x}}(\mathbf{x}, \boldsymbol{y}) \mathbf{e}\_{\mathbf{x}} + \mathbf{G}\_{\boldsymbol{y}}(\mathbf{x}, \boldsymbol{y}) \mathbf{e}\_{\boldsymbol{y}} + \mathbf{G}\_{\boldsymbol{\theta}\_{\boldsymbol{x}}}(\boldsymbol{\theta}\_{\boldsymbol{x}}) \mathbf{e}\_{\boldsymbol{\theta}\_{\boldsymbol{x}}} \end{split} \tag{2}$$

Suppose the X-Y sample sites are in an N � N square array (N is odd) covering the L � L field and the θ<sup>z</sup> sample lines are in a K array (K is a multiple of 4) covering the 360° field. Then in the Cartesian grid, the positions of the sample sites are

$$\propto\_m = m\Delta, \mathcal{y}\_n = n\Delta, \theta\_k = k\ominus \tag{3}$$

radius vector where the points lie in. Angle deviation of points here means the angle between position vector of actual point and that of ideal point. Take Gθz,<sup>0</sup> as an

, xm; <sup>y</sup><sup>0</sup>

, xð Þ <sup>m</sup>; <sup>0</sup> <sup>&</sup>gt;

Gθz,<sup>0</sup> ¼ Eð Þ ϕ<sup>m</sup>

Gy,m,<sup>0</sup> xm þ Gx,m,<sup>0</sup> 

, xð Þ <sup>m</sup>; <sup>0</sup> <sup>&</sup>gt;

Gy,m,<sup>0</sup> xm

Similarly, along X-axis and Y-axis, we can obtain the following four equations:

<sup>m</sup> ¼ �1; �2; <sup>⋯</sup>; � <sup>N</sup> � <sup>1</sup>

<sup>n</sup> ¼ �1; �2; <sup>⋯</sup>; � <sup>N</sup> � <sup>1</sup>

The goal of the proposed self-calibration method is to determine Gm,n,k through different measurement postures, through which the measurement accuracy can be

In this chapter, an artifact plate which possesses mark lines different from previous researches in [17, 20] is designed specifically for XYθ<sup>z</sup> self-calibration. Figure 2 shows the details of artifact plate on the stage. In detail, an N � N grid mark array is on the artifact plate with the same size as the stage sample site array. Furthermore, it has K mark lines with equal angle interval. The plate XYθ<sup>z</sup> coordinate axis' origin is located on the center of the mark array. During the plate

<sup>m</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>;

<sup>n</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>;

2

>

(8)

N � 1 2

N � 1 2

2

2

(9)

, define ϕ<sup>m</sup> as the angle

example. For the point ð Þ <sup>m</sup>; <sup>0</sup> on +X-axis, <sup>m</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>; <sup>N</sup>�<sup>1</sup>

ϕ<sup>m</sup> ¼ < xm þ Gx,m,0; y<sup>0</sup> þ Gy,m,<sup>0</sup>

¼ < xm þ Gx,m,0; Gy,m,<sup>0</sup>

ϕ<sup>m</sup> ¼ < xm þ Gx,m,0; Gy,m,<sup>0</sup>

Gy,m,<sup>0</sup> xm 

Gθz,<sup>0</sup> ¼ E

where < a, b> is the angle between vectors a and b and, then,

Noting that Gy,m,<sup>0</sup> ≪ xm and Gx,m,<sup>0</sup> ≪ xm, one can obtain

¼ arctan

≐ arctan

Gy,m,<sup>0</sup> xm

Gy,m,<sup>0</sup> xm

yn

yn

Gy,m,<sup>0</sup> xm

<sup>4</sup> <sup>¼</sup> <sup>E</sup> � Gx,0,n

<sup>4</sup> <sup>¼</sup> <sup>E</sup> � Gx,0,n

≐

which subsequently results in

Gθz,<sup>0</sup> ¼ E

Gθz,K

Gθz,K <sup>2</sup> ¼ E

Gθz, <sup>3</sup><sup>K</sup>

compensated directly.

2.2 Artifact error

27

deviation of the point ð Þ m; 0 , i.e.

where Eð Þ ϕ<sup>m</sup> is the expectation of ϕm.

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , and Δ ¼ L=N which is called sample site interval; <sup>k</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, …, K � 1 and <sup>⊖</sup> <sup>¼</sup> <sup>360</sup>=K<sup>∘</sup> . For notation simplicity, through Eq. (2), we can denote

$$\mathbf{G}\_{m,n,k} \equiv G\_{\mathbf{x},m,n} \mathbf{e}\_{\mathbf{x}} + G\_{\mathbf{y},m,n} \mathbf{e}\_{\mathbf{y}} + G\_{\theta\_{\mathbf{z}},k} \mathbf{e}\_{\theta\_{\mathbf{z}}} \tag{4}$$

where Gθz,k � Gθ<sup>z</sup> ð Þ θ<sup>k</sup> , Gx,m,n � Gx xm; yn , and Gy,m,n � Gy xm; yn .

Similar to the detailed explanation in [17], to define the coordinates' origin, orientation, and grid scale of the XY axes stage, there are no translation property, no rotation property, and no magnification property for Gx,m,n and Gy,m,n, which can be expressed mathematically as

$$\begin{aligned} \sum\_{m,n} \mathcal{G}\_{\mathbf{x},m,n} &= \sum\_{m,n} \mathcal{G}\_{\mathbf{y},m,n} = \mathbf{0} \\ \sum\_{m,n} \left( \mathcal{G}\_{\mathbf{y},m,n} \mathfrak{x}\_m - \mathcal{G}\_{\mathbf{x},m,n} \mathfrak{y}\_n \right) &= \mathbf{0} \\ \sum\_{m,n} \left( \mathcal{G}\_{\mathbf{x},m,n} \mathfrak{x}\_m + \mathcal{G}\_{\mathbf{y},m,n} \mathfrak{y}\_n \right) &= \mathbf{0} \end{aligned} \tag{5}$$

For X-Y axes, two dimensionless parameters O and R are defined as the XY nonorthogonality and the XY scale difference of Gx,m,n and Gy,m,n, respectively. As a result, Gx,m,n and Gy,m,n are

$$\begin{aligned} G\_{\mathbf{x},m,n} &= O\mathbf{y}\_n + R\mathbf{x}\_m + F\_{\mathbf{x},m,n} \\ G\_{\mathbf{y},m,n} &= O\mathbf{x}\_m - R\mathbf{y}\_n + F\_{\mathbf{y},m,n} \end{aligned} \tag{6}$$

Therefore, one can obtain Gx,m,n and Gy,m,n by the first calculation of the firstorder components O and R, and the later determination of the residual error Fx,m,n and Fy,m,n. Noting that the origin of XY axes is the center of the sample array, we also can get the following properties of Fx,m,n and Fy,m,n which is also detailed in [17, 20]:

$$\begin{aligned} \sum\_{m,n} F\_{\mathbf{x},m,n} &= \sum\_{m,n} F\_{\mathbf{x},m,n} \mathbf{x}\_m = \sum\_{m,n} F\_{\mathbf{x},m,n} \mathbf{y}\_n = \mathbf{0} \\ \sum\_{m,n} F\_{\mathbf{y},m,n} &= \sum\_{m,n} F\_{\mathbf{y},m,n} \mathbf{x}\_m = \sum\_{m,n} F\_{\mathbf{y},m,n} \mathbf{y}\_n = \mathbf{0} \end{aligned} \tag{7}$$

Besides, for rotary self-calibration, an important property, i.e. the circle closure principle, could directly bridge the gap between Gkþ<sup>K</sup> and Gk, i.e. Gkþ<sup>K</sup> ¼ Gk for k ¼ 0; 1; 2, …, K � 1, which significantly facilitates the self-calibration process. To calculate the stage error components at θz, i.e. Gθz,k, a new property must be pointed out as follows:

Gθz,k is definitely related to the errors of XY orientations. In other words, the expected value of angle deviation of points is exactly the θ<sup>z</sup> orientation error of the

where ex, ey, and e<sup>θ</sup><sup>z</sup> are the unit vectors of the stage axes. For notation, we combine linear and rotary stage errors and define Gð Þ x; y; θ<sup>z</sup> is the stage error at ð Þ x; y; θ<sup>z</sup> where ð Þ x; y is the true location and θ<sup>z</sup> is the true angle in the Cartesian grid:

¼ Gxð Þ x; y e<sup>x</sup> þ Gyð Þ x; y e<sup>y</sup> þ G<sup>θ</sup><sup>z</sup> ð Þ θ<sup>z</sup> e<sup>θ</sup><sup>z</sup>

xm ¼ mΔ, yn ¼ nΔ, θ<sup>k</sup> ¼ k⊖ (3)

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

<sup>2</sup> , and Δ ¼ L=N

.

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

Gm,n,k � Gx,m,ne<sup>x</sup> þ Gy,m,ne<sup>y</sup> þ Gθz,keθ<sup>z</sup> (4)

, and Gy,m,n � Gy xm; yn

Suppose the X-Y sample sites are in an N � N square array (N is odd) covering the L � L field and the θ<sup>z</sup> sample lines are in a K array (K is a multiple of 4) covering the 360° field. Then in the Cartesian grid, the positions of the sample sites are

<sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup>

Similar to the detailed explanation in [17], to define the coordinates' origin, orientation, and grid scale of the XY axes stage, there are no translation property, no rotation property, and no magnification property for Gx,m,n and Gy,m,n, which can

> ∑m,nGx,m,n ¼ ∑m,nGy,m,n ¼ 0 ∑m,n Gy,m,nxm � Gx,m,nyn

For X-Y axes, two dimensionless parameters O and R are defined as the XY nonorthogonality and the XY scale difference of Gx,m,n and Gy,m,n, respectively. As

> Gx,m,n ¼ Oyn þ Rxm þ Fx,m,n Gy,m,n ¼ Oxm � Ryn þ Fy,m,n

Therefore, one can obtain Gx,m,n and Gy,m,n by the first calculation of the firstorder components O and R, and the later determination of the residual error Fx,m,n and Fy,m,n. Noting that the origin of XY axes is the center of the sample array, we also can get the following properties of Fx,m,n and Fy,m,n which is also detailed in [17, 20]:

∑m,nFx,m,n ¼ ∑m,nFx,m,nxm ¼ ∑m,nFx,m,nyn ¼ 0

Besides, for rotary self-calibration, an important property, i.e. the circle closure principle, could directly bridge the gap between Gkþ<sup>K</sup> and Gk, i.e. Gkþ<sup>K</sup> ¼ Gk for k ¼ 0; 1; 2, …, K � 1, which significantly facilitates the self-calibration process. To calculate the stage error components at θz, i.e. Gθz,k, a new property must be

Gθz,k is definitely related to the errors of XY orientations. In other words, the expected value of angle deviation of points is exactly the θ<sup>z</sup> orientation error of the

<sup>∑</sup>m,nFy,m,n <sup>¼</sup> <sup>∑</sup>m,nFy,m,nxm <sup>¼</sup> <sup>∑</sup>m,nFy,m,nyn <sup>¼</sup> <sup>0</sup> (7)

∑m,n Gx,m,nxm þ Gy,m,nyn

<sup>¼</sup> <sup>0</sup>

<sup>¼</sup> <sup>0</sup>

which is called sample site interval; <sup>k</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, …, K � 1 and <sup>⊖</sup> <sup>¼</sup> <sup>360</sup>=K<sup>∘</sup>

(2)

. For nota-

(5)

(6)

Gð Þ� x; y; θ<sup>z</sup> Glð Þþ x; y Grð Þ θ<sup>z</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup>

be expressed mathematically as

a result, Gx,m,n and Gy,m,n are

pointed out as follows:

26

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

Standards, Methods and Solutions of Metrology

tion simplicity, through Eq. (2), we can denote

where Gθz,k � Gθ<sup>z</sup> ð Þ θ<sup>k</sup> , Gx,m,n � Gx xm; yn

radius vector where the points lie in. Angle deviation of points here means the angle between position vector of actual point and that of ideal point. Take Gθz,<sup>0</sup> as an example. For the point ð Þ <sup>m</sup>; <sup>0</sup> on +X-axis, <sup>m</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>⋯</sup>; <sup>N</sup>�<sup>1</sup> 2 , define ϕ<sup>m</sup> as the angle deviation of the point ð Þ m; 0 , i.e.

$$\begin{aligned} \phi\_m &= < (\mathfrak{x}\_m + G\_{\mathfrak{x},m,0}, \mathfrak{y}\_0 + G\_{\mathfrak{y},m,0}), (\mathfrak{x}\_m, \mathfrak{y}\_0) > 0 \\ &= < (\mathfrak{x}\_m + G\_{\mathfrak{x},m,0}, G\_{\mathfrak{y},m,0}), (\mathfrak{x}\_m, \mathfrak{O}) > \end{aligned}$$

where < a, b> is the angle between vectors a and b and, then,

$$G\_{\theta\_{\mathfrak{z}},0} = E(\phi\_m)$$

where Eð Þ ϕ<sup>m</sup> is the expectation of ϕm. Noting that Gy,m,<sup>0</sup> ≪ xm and Gx,m,<sup>0</sup> ≪ xm, one can obtain

$$\begin{aligned} \phi\_m &= < (\mathfrak{x}\_m + \mathrm{G}\_{\mathrm{x},m,0}, \mathrm{G}\_{\mathfrak{y},m,0}), (\mathfrak{x}\_m, \mathbf{0}) > \\ &= \arctan\left(\frac{\mathrm{G}\_{\mathfrak{y},m,0}}{\mathfrak{x}\_m + \mathrm{G}\_{\mathrm{x},m,0}}\right) \\ &\doteq \arctan\left(\frac{\mathrm{G}\_{\mathfrak{y},m,0}}{\mathfrak{x}\_m}\right) \\ &\doteq \frac{\mathrm{G}\_{\mathfrak{y},m,0}}{\mathfrak{x}\_m} \end{aligned}$$

which subsequently results in

$$G\_{\theta\_{\mathbf{t}},0} = E\left(\frac{G\_{\mathbf{y},m,0}}{\varkappa\_m}\right) \tag{8}$$

Similarly, along X-axis and Y-axis, we can obtain the following four equations:

$$\begin{aligned} G\_{\theta\_z, 0} &= E\left(\frac{G\_{\mathbf{y}, m, 0}}{\mathbf{x}\_m}\right) \qquad \left(m = 1, 2, \dots, \frac{N - 1}{2}\right) \\ G\_{\theta\_z, \frac{N}{4}} &= E\left(-\frac{G\_{\mathbf{x}, 0, n}}{\mathbf{y}\_n}\right) \quad \left(n = 1, 2, \dots, \frac{N - 1}{2}\right) \\ G\_{\theta\_z, \frac{N}{4}} &= E\left(\frac{G\_{\mathbf{y}, m, 0}}{\mathbf{x}\_m}\right) \quad \left(m = -1, -2, \dots, -\frac{N - 1}{2}\right) \\ G\_{\theta\_z, \frac{N}{4}} &= E\left(-\frac{G\_{\mathbf{x}, 0, n}}{\mathbf{y}\_n}\right) \quad \left(n = -1, -2, \dots, -\frac{N - 1}{2}\right) \end{aligned} \tag{9}$$

The goal of the proposed self-calibration method is to determine Gm,n,k through different measurement postures, through which the measurement accuracy can be compensated directly.

#### 2.2 Artifact error

In this chapter, an artifact plate which possesses mark lines different from previous researches in [17, 20] is designed specifically for XYθ<sup>z</sup> self-calibration. Figure 2 shows the details of artifact plate on the stage. In detail, an N � N grid mark array is on the artifact plate with the same size as the stage sample site array. Furthermore, it has K mark lines with equal angle interval. The plate XYθ<sup>z</sup> coordinate axis' origin is located on the center of the mark array. During the plate

movement, the plate axis will move with the plate. The locations of the nominal mark in the plate coordinate system are totally the same with that of the sample site in the stage coordinate system. Due to the unavoidable imperfection of the artifact plate, all the actual marks at ð Þ m; n; k deviate from their nominal location by Am,n,k which is defined as artifact error expressed by

$$\mathbf{A}\_{m,n,k} \equiv A\_{\mathbf{x},m,n} \mathbf{e}\_{\mathbf{p}\mathbf{x}} + A\_{\mathbf{y},m,n} \mathbf{e}\_{\mathbf{p}\mathbf{y}} + A\_{\theta\_{\mathbf{z}},k} \mathbf{e}\_{\mathbf{p}\mathbf{z}}$$

$$A\_{\mathbf{x},m,n} \equiv A\_{\mathbf{x}}(\mathbf{x}\_m, \mathbf{y}\_n), A\_{\mathbf{y},m,n} \equiv A\_{\mathbf{y}}(\mathbf{x}\_m, \mathbf{y}\_n), \tag{10}$$

$$A\_{\theta\_{\mathbf{z}},k} \equiv A\_{\theta\_{\mathbf{z}}}(\theta\_k)$$

which will be considered as a misalignment error, which consists of a small rotation between their orientations and a small offset between their origins. Besides, random measurement noise also exists in the measurement process, but the effects of noise

a. In View 0, which is the initial view, the XYθ<sup>z</sup> axes of the plate are aligned as closely in line with those of the stage as possible; in this view, both grid and

b. In View 1, the artifact plate is rotated 90°, around the origin from View 0 on

c. In View 2, the artifact plate is rotated 360/K°, i.e., ⊖, around the origin from View 0 on the stage; in this view, only angular marks are measured.

For each measurement view, the artifact plate is firmly fixed on the stage, and a

In the following, we would present the measurement and mathematical manipulations of each measurement view and then the reconstruction of the stage error map, in which V stands for the measured deviation for a mark from its nominal

d.In View 3, the artifact plate is translated by one sample site, i.e., Δ, along þX-axis from View 0 on the stage; in this view, both grid and angular marks

mark alignment system is needed to help the XYθ<sup>z</sup> metrology stage to precisely

measure the mark lines. The detailed instruments are presented later.

the stage; in this view, both grid and angular marks are measured.

can be assumed to be completely attenuated by repeated measurements:

angular marks are measured.

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

are measured.

position in the stage coordinate.

Figure 3.

29

Independent measurement views for XYθ<sup>z</sup> self-calibration.

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , and k ¼ 0, 1, …, K � 1; epx, epy, and epz are the unit vectors of the plate axes.

It should be noted that every mark on the artifact plate has an identification number (m, n, k). During the motions of the plate on the stage, the identification number of the mark will not change. This characteristic is also utilized to identify each physical mark of the plate in the following comparison of different measurement views. Ax,m,n and Ay,m,n also have no translation property and no rotation property [17, 20], which essentially have defined the axis origin and axis orientation, i.e.

$$\begin{aligned} \sum\_{m,n} A\_{\mathbf{x},m,n} &= \sum\_{m,n} A\_{\mathbf{y},m,n} = \mathbf{0} \\ \sum\_{m,n} (A\_{\mathbf{y},m,n} \mathbf{x}\_m - A\_{\mathbf{x},m,n} \mathbf{y}\_n) &= \mathbf{0} \end{aligned} \tag{11}$$

### 3. XYθ<sup>z</sup> self-calibration principle

#### 3.1 The measurement views

The self-calibration method is based on four different postures or views of the designed artifact plate on the uncalibrated XYθ<sup>z</sup> metrology stage, which is shown in Figure 3. As shown in Figure 3, the XYθ<sup>z</sup> stage is the gray part, while the artifact plate is the white part. The 3-D specification can also be found in Figures 1 and 2.

Without the loss of generality, for each view, there inevitably exists a misalignment error; for that these coordinate axes cannot be aligned completely,

Figure 2. A designed artifact plate with mark lines for XYθ<sup>z</sup> self-calibration.

### Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

movement, the plate axis will move with the plate. The locations of the nominal mark in the plate coordinate system are totally the same with that of the sample site in the stage coordinate system. Due to the unavoidable imperfection of the artifact plate, all the actual marks at ð Þ m; n; k deviate from their nominal location by Am,n,k

Am,n,k � Ax,m,nepx þ Ay,m,nepy þ Aθz,kepz

<sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup>

It should be noted that every mark on the artifact plate has an identification number (m, n, k). During the motions of the plate on the stage, the identification number of the mark will not change. This characteristic is also utilized to identify each physical mark of the plate in the following comparison of different measurement views. Ax,m,n and Ay,m,n also have no translation property and no rotation property [17, 20], which essentially have defined the axis origin and axis orienta-

> ∑m,nAx,m,n ¼ ∑m,nAy,m,n ¼ 0 ∑m,n Ay,m,nxm � Ax,m,nyn

The self-calibration method is based on four different postures or views of the designed artifact plate on the uncalibrated XYθ<sup>z</sup> metrology stage, which is shown in Figure 3. As shown in Figure 3, the XYθ<sup>z</sup> stage is the gray part, while the artifact plate is the white part. The 3-D specification can also be found in Figures 1 and 2. Without the loss of generality, for each view, there inevitably exists a misalignment error; for that these coordinate axes cannot be aligned completely,

, Ay,m,n � Ay xm; yn

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

,

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

<sup>¼</sup> <sup>0</sup> (11)

<sup>2</sup> , and

(10)

which is defined as artifact error expressed by

Standards, Methods and Solutions of Metrology

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

3. XYθ<sup>z</sup> self-calibration principle

A designed artifact plate with mark lines for XYθ<sup>z</sup> self-calibration.

3.1 The measurement views

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup>

tion, i.e.

Figure 2.

28

Ax,m,n � Ax xm; yn

Aθz,k � A<sup>θ</sup><sup>z</sup> ð Þ θ<sup>k</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

k ¼ 0, 1, …, K � 1; epx, epy, and epz are the unit vectors of the plate axes.

which will be considered as a misalignment error, which consists of a small rotation between their orientations and a small offset between their origins. Besides, random measurement noise also exists in the measurement process, but the effects of noise can be assumed to be completely attenuated by repeated measurements:


For each measurement view, the artifact plate is firmly fixed on the stage, and a mark alignment system is needed to help the XYθ<sup>z</sup> metrology stage to precisely measure the mark lines. The detailed instruments are presented later.

In the following, we would present the measurement and mathematical manipulations of each measurement view and then the reconstruction of the stage error map, in which V stands for the measured deviation for a mark from its nominal position in the stage coordinate.

Figure 3. Independent measurement views for XYθ<sup>z</sup> self-calibration.

For View 0,

$$\begin{cases} V\_{0,\mathbf{x},m,n} = G\_{\mathbf{x},m,n} + A\_{\mathbf{x},m,n} - \rho\_0 \mathbf{y}\_n + t\_{0\mathbf{x}} \\\\ V\_{0,\mathbf{y},m,n} = G\_{\mathbf{y},m,n} + A\_{\mathbf{y},m,n} + \rho\_0 \mathbf{x}\_m + t\_{0\mathbf{y}} \\\\ V\_{0,\theta\_x,k} = G\_{\theta\_x,k} + A\_{\theta\_x,k} + \rho\_0 \end{cases} \tag{12}$$

<sup>φ</sup><sup>2</sup> <sup>¼</sup> <sup>∑</sup>kV2,θz,k � <sup>∑</sup>kV0, <sup>θ</sup>z,k

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

<sup>φ</sup><sup>3</sup> <sup>¼</sup> <sup>∑</sup>kV3, <sup>θ</sup>z,k � <sup>∑</sup>nV0,θz,k

<sup>K</sup> <sup>þ</sup> <sup>∑</sup>m,n <sup>V</sup>0, y,m,nxm � <sup>V</sup>0,x,m,nyn

<sup>K</sup> <sup>þ</sup> <sup>∑</sup>m,n <sup>V</sup>0, y,m,nxm � <sup>V</sup>0,x,m,nyn

After elimination of the misalignment error in View 0, View 1, and View 2,

U0,x,m,n ¼ V0,x,m,n � t0<sup>x</sup> þ φ0yn ¼ Fx,m,n þ Ax,m,n þ Oyn þ Rxm

U0, y,m,n ¼ V0, y,m,n � t0<sup>y</sup> � φ0xm ¼ Fy,m,n þ Ay,m,n þ Oxm � Ryn

U1,x,m,n ¼ V1,x,m,n þ φ1xm � t1<sup>x</sup> ¼ Fx,�n,m � Ay,m,n þ Oxm � Ryn

U1, y,m,n ¼ V1, y,m,n þ φ1yn � t1<sup>y</sup> ¼ Fy,�n,m þ Ax,m,n � Oyn � Rxm

And to keep the notation consistent with the previous views, we define

U3,x,m,n ¼ V3,x,m,n þ φ3yn ¼ Fx,mþ1,n þ Ax,m,n þ Oyn þ Rxm þ ξ<sup>x</sup>

U3, y,m,n ¼ V3, y,m,n � φ3xm ¼ Fy,mþ1,n þ Ay,m,n þ Oxm � Ryn þ ξ<sup>y</sup>

in [17], the stage error components O and R can be calculated out as:

Comparing Eq. (18) of View 0 with Eq. (19) of View 1, with the same procedure

� � " #

� � " #

A least square-based self-calibration algorithm is synthesized to determinate Gm,n,k. In this algorithm, the computation of the misalignment error components ξ<sup>x</sup> and ξ<sup>y</sup> is unnecessary, while φ<sup>3</sup> is determined by Eq. (17). Because O and R are known by Eq. (22), the algorithm is constructed to just calculate out Fx,m,n, Fy,m,n,

> Fx,m,n � Fy,�n,m ¼ U0,x,m,n � U1, y,m,n � 2Oyn � 2Rxm Fy,m,n þ Fx,�n,m ¼ U0, y,m,n þ U1,x,m,n � 2Oxm þ 2Ryn

� � <sup>þ</sup> <sup>∑</sup>m,n <sup>U</sup>1,x,m,nxm � <sup>U</sup>1, y,m,nyn

� � <sup>þ</sup> <sup>∑</sup>m,n �U1,x,m,nyn � <sup>U</sup>1, y,m,nxm

� �

<sup>m</sup> þ y<sup>2</sup> n

� �

<sup>m</sup> þ y<sup>2</sup> n

∑m,n x<sup>2</sup>

∑m,n x<sup>2</sup>

<sup>4</sup> þ Aθz,k

combining Eq. (6), the measurement data are to be rearranged as:

U0, <sup>θ</sup>z,k ¼ V0, <sup>θ</sup>z,k � φ<sup>0</sup> ¼ Gθz,k þ Aθz,k

<sup>U</sup>1, <sup>θ</sup>z,k <sup>¼</sup> <sup>V</sup>1, <sup>θ</sup>z,k � <sup>φ</sup><sup>1</sup> <sup>¼</sup> <sup>G</sup>θz,kþ<sup>K</sup>

U3, <sup>θ</sup>z,k ¼ V3, y,m,n � φ<sup>3</sup>

<sup>O</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>R</sup> <sup>¼</sup> <sup>1</sup> 2

and Gθz,k.

31

where ξ<sup>x</sup> ¼ t3<sup>x</sup> þ RΔ and ξ<sup>y</sup> ¼ t3<sup>y</sup> þ OΔ.

∑m,n U0,x,m,nyn þ U0, y,m,nxm � �

∑m,n U0,x,m,nxm � U0, y,m,nyn � �

<sup>m</sup> þ y<sup>2</sup> n

<sup>m</sup> þ y<sup>2</sup> n

∑m,n x<sup>2</sup>

∑m,n x<sup>2</sup>

Comparing Eq. (18) with (19), one obtains

3.2 XYθ<sup>z</sup> self-calibration algorithm

∑m,n x<sup>2</sup>

∑m,n x<sup>2</sup>

U2, <sup>θ</sup>z,k ¼ V2, <sup>θ</sup>z,k � φ<sup>2</sup> ¼ Gθz,kþ<sup>1</sup> þ Aθz,k (20)

� �

<sup>m</sup> þ y<sup>2</sup> n � �

(17)

(18)

(19)

(21)

(22)

(23)

� �

<sup>m</sup> þ y<sup>2</sup> n � �

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , and k ¼ 1, 2, …, K. For View 1,

$$\begin{cases} V\_{1,\mathbf{x},m,n} = G\_{\mathbf{x},-n,m} - A\_{\mathbf{y},m,n} - \rho\_1 \mathbf{x}\_m + t\_{1\mathbf{x}} \\\\ V\_{1,\mathbf{y},m,n} = G\_{\mathbf{y},-n,m} + A\_{\mathbf{x},m,n} - \rho\_1 \mathbf{y}\_n + t\_{1\mathbf{y}} \\\\ V\_{1,\theta\_z,k} = G\_{\theta\_{z,k}, \frac{K}{4}} + A\_{\theta\_z, k} + \rho\_1 \end{cases} \tag{13}$$

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , and k ¼ 1, 2, …, K. For View 2,

$$V\_{\mathcal{Q},\theta\_k,k} = G\_{\theta\_k,k+1} + A\_{\theta\_k,k} + \rho\_2 \tag{14}$$

where k ¼ 1, 2, …, K. For View 3,

$$\begin{cases} V\_{3,x,m,n} = G\_{\mathbf{x},m+1,n} + A\_{\mathbf{x},m,n} - \rho\_3 y\_n + t\_{3\mathbf{x}} \\\\ V\_{3,y,m,n} = G\_{\mathbf{y},m+1,n} + A\_{\mathbf{y},m,n} + \rho\_3 \mathbf{x}\_m + t\_{3\mathbf{y}} \\\\ V\_{3,\theta\_z,k} = G\_{\theta\_z,k} + A\_{\theta\_{\mathbf{z}},k} + \rho\_3 \end{cases} \tag{15}$$

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>3</sup> <sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup> <sup>2</sup> , � <sup>N</sup>�<sup>3</sup> <sup>2</sup> , …, <sup>N</sup>�<sup>1</sup> <sup>2</sup> , and k ¼ 1, 2, …, K. It shall be pointed out that φ<sup>0</sup> and t<sup>0</sup> ¼ t0<sup>x</sup>; t0<sup>y</sup> are the rotation and offset of the misalignment error of View 0 and the notations of other views are similar. And φ<sup>0</sup> is a small angle, for which the 'small angle' approximation can be adopted. For the rotation misalignment error of other views, this approximation is still tenable.

Similar to the presentation of [17], combining (5) and (11), summing over all the sites of (12) and (13), we can obtain the misalignment error, i.e. the offset components t0x, t0y, t1x, t1<sup>y</sup>, and the rotation components φ0, φ1, i.e.

$$\begin{aligned} t\_{0x} &= \frac{\sum\_{m,n} V\_{0,\ge,m,n}}{N^2}, t\_{0y} = \frac{\sum\_{m,n} V\_{0,\ge,m,n}}{N^2} \\ t\_{1x} &= \frac{\sum\_{m,n} V\_{1,\ge,m,n}}{N^2}, t\_{1y} = \frac{\sum\_{m,n} V\_{1,\ge,m,n}}{N^2} \\ \varphi\_0 &= \frac{\sum\_{m,n} \left( V\_{0,\ge,m,n} \mathbf{x}\_m - V\_{0,\ge,m,n} \mathbf{y}\_n \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} \\ \end{aligned} \tag{16}$$
 
$$\begin{aligned} \varphi\_1 &= \frac{\sum\_{m,n} \left( -V\_{1,\ge,m,n} \mathbf{y}\_n - V\_{1,\ge,m,n} \mathbf{x}\_m \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} \end{aligned}$$

Noting Eqs. (5), (11), and (12), summing over all sites of (14) and (15), φ<sup>2</sup> and φ<sup>3</sup> can be determined, and the detailed result is

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

For View 0,

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup>

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup>

where k ¼ 1, 2, …, K.

where <sup>m</sup> ¼ � <sup>N</sup>�<sup>1</sup>

For View 2,

For View 3,

For View 1,

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

Standards, Methods and Solutions of Metrology

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

φ<sup>3</sup> can be determined, and the detailed result is

30

It shall be pointed out that φ<sup>0</sup> and t<sup>0</sup> ¼ t0<sup>x</sup>; t0<sup>y</sup>

V0,x,m,n ¼ Gx,m,n þ Ax,m,n � φ0yn þ t0<sup>x</sup> V0, y,m,n ¼ Gy,m,n þ Ay,m,n þ φ0xm þ t0<sup>y</sup>

<sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup>

V1,x,m,n ¼ Gx,�n,m � Ay,m,n � φ1xm þ t1<sup>x</sup>

V1, y,m,n ¼ Gy,�n,m þ Ax,m,n � φ1yn þ t1<sup>y</sup>

<sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup>

V3,x,m,n ¼ Gx,mþ1,n þ Ax,m,n � φ3yn þ t3<sup>x</sup>

V3, y,m,n ¼ Gy,mþ1,n þ Ay,m,n þ φ3xm þ t3<sup>y</sup>

<sup>2</sup> , <sup>n</sup> ¼ � <sup>N</sup>�<sup>1</sup>

misalignment error of View 0 and the notations of other views are similar. And φ<sup>0</sup> is a small angle, for which the 'small angle' approximation can be adopted. For the rotation misalignment error of other views, this approximation is still tenable.

Similar to the presentation of [17], combining (5) and (11), summing over all the sites of (12) and (13), we can obtain the misalignment error, i.e. the offset compo-

<sup>N</sup><sup>2</sup> , t0<sup>y</sup> <sup>¼</sup> <sup>∑</sup>m,nV0, y,m,n

<sup>N</sup><sup>2</sup> , t1<sup>y</sup> <sup>¼</sup> <sup>∑</sup>m,nV1, y,m,n

<sup>m</sup> þ y<sup>2</sup> n 

<sup>m</sup> þ y<sup>2</sup> n 

<sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>∑</sup>m,n <sup>V</sup>0, y,m,nxm � <sup>V</sup>0,x,m,nyn

∑m,n x<sup>2</sup>

<sup>φ</sup><sup>1</sup> <sup>¼</sup> <sup>∑</sup>m,n �V1, y,m,nyn � <sup>V</sup>1,x,m,nxm

∑m,n x<sup>2</sup>

Noting Eqs. (5), (11), and (12), summing over all sites of (14) and (15), φ<sup>2</sup> and

N2

N2

V3, <sup>θ</sup>z,k ¼ Gθz,k þ Aθz,k þ φ<sup>3</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>3</sup>

nents t0x, t0y, t1x, t1<sup>y</sup>, and the rotation components φ0, φ1, i.e.

<sup>t</sup>0<sup>x</sup> <sup>¼</sup> <sup>∑</sup>m,nV0,x,m,n

<sup>t</sup>1<sup>x</sup> <sup>¼</sup> <sup>∑</sup>m,nV1,x,m,n

<sup>4</sup> þ Aθz,k þ φ<sup>1</sup>

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

<sup>2</sup> , � <sup>N</sup>�<sup>3</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

are the rotation and offset of the

V2, <sup>θ</sup>z,k ¼ Gθz,kþ<sup>1</sup> þ Aθz,k þ φ<sup>2</sup> (14)

(12)

(13)

(15)

(16)

<sup>2</sup> , and k ¼ 1, 2, …, K.

<sup>2</sup> , and k ¼ 1, 2, …, K.

<sup>2</sup> , and k ¼ 1, 2, …, K.

V0, <sup>θ</sup>z,k ¼ Gθz,k þ Aθz,k þ φ<sup>0</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

V1, <sup>θ</sup>z,k ¼ Gθz,kþ<sup>K</sup>

<sup>2</sup> , …, <sup>N</sup>�<sup>1</sup>

$$\begin{aligned} \rho\_2 &= \frac{\sum\_k V\_{2,\theta\_2,k} - \sum\_k V\_{0,\theta\_2,k}}{K} + \frac{\sum\_{m,n} \left(V\_{0,\mathbf{y},m,n} \mathbf{x}\_m - V\_{0,\mathbf{x},m,n} \mathbf{y}\_n\right)}{\sum\_{m,n} \left(\mathbf{x}\_m^2 + \mathbf{y}\_n^2\right)} \\\\ \rho\_3 &= \frac{\sum\_k V\_{3,\theta\_2,k} - \sum\_n V\_{0,\theta\_2,k}}{K} + \frac{\sum\_{m,n} \left(V\_{0,\mathbf{y},m,n} \mathbf{x}\_m - V\_{0,\mathbf{x},m,n} \mathbf{y}\_n\right)}{\sum\_{m,n} \left(\mathbf{x}\_m^2 + \mathbf{y}\_n^2\right)} \end{aligned} \tag{17}$$

After elimination of the misalignment error in View 0, View 1, and View 2, combining Eq. (6), the measurement data are to be rearranged as:

$$U\_{0,\mathbf{x},m,n} = V\_{0,\mathbf{x},m,n} - t\_{0\mathbf{x}} + \rho\_0 \mathbf{y}\_n = F\_{\mathbf{x},m,n} + A\_{\mathbf{x},m,n} + O\mathbf{y}\_n + R\mathbf{x}\_m$$

$$U\_{0,\mathbf{y},m,n} = V\_{0,\mathbf{y},m,n} - t\_{0\mathbf{y}} - \rho\_0 \mathbf{x}\_m = F\_{\mathbf{y},m,n} + A\_{\mathbf{y},m,n} + O\mathbf{x}\_m - R\mathbf{y}\_n \tag{18}$$

$$U\_{0,\theta\_k,k} = V\_{0,\theta\_k,k} - \rho\_0 = G\_{\theta\_k,k} + A\_{\theta\_k,k}$$

$$U\_{1,\mathbf{x},m,n} = V\_{1,\mathbf{x},m,n} + \rho\_1 \mathbf{x}\_m - t\_{1\mathbf{x}} = F\_{\mathbf{x},-n,m} - A\_{\mathbf{y},m,n} + O\mathbf{x}\_m - R\mathbf{y}\_n$$

$$U\_{1,\mathbf{y},m,n} = V\_{1,\theta\_k,m} + \rho\_0 \mathbf{y}\_n - t\_{1\mathbf{y}} = F\_{\mathbf{y},-n,m} + A\_{\mathbf{x},m,n} - O\mathbf{y}\_n - R\mathbf{x}\_m \tag{19}$$

$$U\_{1,\theta\_k,k} = V\_{1,\theta\_k,k} - \rho\_1 = G\_{\theta\_k,k} + \frac{\kappa}{4} + A\_{\theta\_k,k}$$

$$U\_{2,\theta\_k,k} = V\_{2,\theta\_k,k} - \rho\_2 = G\_{\theta\_k,k+1} + A\_{\theta\_k,k} \tag{20}$$

And to keep the notation consistent with the previous views, we define

$$\begin{aligned} U\_{3,x,m,n} &= V\_{3,x,m,n} + \rho\_3 \mathbf{y}\_n = F\_{x,m+1,n} + A\_{x,m,n} + O \mathbf{y}\_n + R \mathbf{x}\_m + \xi\_x \\\\ U\_{3,y,m,n} &= V\_{3,y,m,n} - \rho\_3 \mathbf{x}\_m = F\_{y,m+1,n} + A\_{y,m,n} + O \mathbf{x}\_m - R \mathbf{y}\_n + \xi\_y \\\\ U\_{3,\theta,k} &= V\_{3,y,m,n} - \rho\_3 \end{aligned} \tag{21}$$

where ξ<sup>x</sup> ¼ t3<sup>x</sup> þ RΔ and ξ<sup>y</sup> ¼ t3<sup>y</sup> þ OΔ.

Comparing Eq. (18) of View 0 with Eq. (19) of View 1, with the same procedure in [17], the stage error components O and R can be calculated out as:

$$O = \frac{1}{2} \left[ \frac{\sum\_{m,n} \left( U\_{0,x,m,n} \mathbf{y}\_n + U\_{0,y,m,n} \mathbf{x}\_m \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} + \frac{\sum\_{m,n} \left( U\_{1,x,m,n} \mathbf{x}\_m - U\_{1,y,m,n} \mathbf{y}\_n \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} \right]$$

$$R = \frac{1}{2} \left[ \frac{\sum\_{m,n} \left( U\_{0,x,m,n} \mathbf{x}\_m - U\_{0,y,m,n} \mathbf{y}\_n \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} + \frac{\sum\_{m,n} \left( -U\_{1,x,m,n} \mathbf{y}\_n - U\_{1,y,m,n} \mathbf{x}\_m \right)}{\sum\_{m,n} \left( \mathbf{x}\_m^2 + \mathbf{y}\_n^2 \right)} \right] \tag{22}$$

### 3.2 XYθ<sup>z</sup> self-calibration algorithm

A least square-based self-calibration algorithm is synthesized to determinate Gm,n,k. In this algorithm, the computation of the misalignment error components ξ<sup>x</sup> and ξ<sup>y</sup> is unnecessary, while φ<sup>3</sup> is determined by Eq. (17). Because O and R are known by Eq. (22), the algorithm is constructed to just calculate out Fx,m,n, Fy,m,n, and Gθz,k.

Comparing Eq. (18) with (19), one obtains

$$\begin{aligned} F\_{\mathbf{x},m,n} - F\_{\mathbf{y},-n,m} &= U\_{0,\mathbf{x},m,n} - U\_{1,\mathbf{y},m,n} - 2O\mathbf{y}\_n - 2R\mathbf{x}\_m \\ F\_{\mathbf{y},m,n} + F\_{\mathbf{x},-n,m} &= U\_{0,\mathbf{y},m,n} + U\_{1,\mathbf{x},m,n} - 2O\mathbf{x}\_m + 2R\mathbf{y}\_n \end{aligned} \tag{23}$$

Then combining Eqs. (18) and (21), define

$$\begin{aligned} L\_{\mathbf{x},m,n} &= U\_{\mathbf{3},\mathbf{x},m,n} - U\_{\mathbf{0},\mathbf{x},m,n} = F\_{\mathbf{x},m+1,n} - F\_{\mathbf{x},m,n} + \xi\_{\mathbf{x}} \\ L\_{\mathbf{y},m,n} &= U\_{\mathbf{3},\mathbf{y},m,n} - U\_{\mathbf{0},\mathbf{y},m,n} = F\_{\mathbf{y},m+1,n} - F\_{\mathbf{y},m,n} + \xi\_{\mathbf{y}} \end{aligned} \tag{24}$$

where <sup>M</sup> <sup>¼</sup> <sup>N</sup>�<sup>1</sup>

Gθz, <sup>3</sup><sup>K</sup>

4. Computer simulation

standard deviation of 0:01<sup>∘</sup>

measurement noises.

33

revolution, while the sample site internal is <sup>Δ</sup> <sup>¼</sup> <sup>15</sup><sup>∘</sup>

generated by mean value of 0 and standard deviation of 0:01<sup>∘</sup>

Gθz,kþ<sup>K</sup>

where k ¼ 1, 2, …, K and Gθz,kþ<sup>K</sup> ¼ Gθz,k.

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

<sup>2</sup> . Noting Eqs. (18), (19), and (20), we consequently obtain

(28)

4 , Gθz, <sup>K</sup> 2 , and

. And the nominal Gθz,k is

, which are utilized as the nominal Aθz,k. The nominal

. And minor modifi-

<sup>4</sup> � Gθz,k ¼ U1, <sup>θ</sup>z,k � U0, <sup>θ</sup>z,k

Gθz,kþ<sup>1</sup> � Gθz,k ¼ U2, <sup>θ</sup>z,k � U0, <sup>θ</sup>z,k

The proposed method features certain benefits remarked as follows:

that complicated algebraic manipulations are significantly avoided.

and θ<sup>z</sup> orientation. By a least square method, values of Gθz, 1, Gθz,K

• We propose a new property to construct connection between XY orientation

In this section, we use MATLAB software to simulate the self-calibration process. First, arbitrary stage linear error maps on a 11 � 11 sample site array with the sample site interval Δ ¼ 10 mm are generated using the command 'normrnd' with a mean of 0 and standard deviation of 0.2 μm, which are utilized as the nominal Gx,m,n and Gy,m,n. Besides, any stage rotary error is mapped within a

cation has been made to the data to satisfy the relevant requirements like Eqs. (5), (9), and (11). Then, arbitrary artifact linear error maps are generated with mean of 0 and standard deviation of 0.3 μm, which are employed as the nominal Am,n, and arbitrary artifact rotary error maps are generated with a mean of 0 and

stage error component Gm,n is shown in Figure 4 where the red lines are Gm,n � 10000. The nominal stage error component Gθz,k is shown in Figure 5 where Gθz,k between the actual measurement system and perfect measurement system has been zoomed in for 360/π times. In addition, for each view, we add a random misalignment which is made up of a rotation and an offset. And the standard deviation value in the misalignment is 0.3° for rotation and 30 μm for offset. Since Eq. (26) has some redundance, it is clear that Gm,n,k can be figured out rather accurately if there is no random measurement noise. In addition, as there are no reported complete on-axis XYθ<sup>z</sup> self-calibration strategies in published papers, we just test their own effectiveness of the proposed strategy. Herein, we focus on testing the calibration accuracy of the proposed strategy in various random

<sup>4</sup> are determined by Eq. (27), which is quite important for the calculation algorithm of Gθz,k. With full utilization of the measurement of View 0, View 1, and View 2, we can construct a simple algorithm with strong robustness.

• In previous self-calibration schemes for XY stages and XYZ stages [17, 18, 20–23], the properties of no translation and no rotation for the stage error cannot be used in View 3. Resultantly, it needs complicated algebraic manipulations to determine the misalignment error component φ3. In this chapter, Eq. (17) directly determines the value of φ3, and it is so convenient

Combining (27) and (28), the stage error Gθz,k can be determined through a least square solution as the above equation group has K unknowns and 2K + 4 equations.

Consequently, we can obtain

$$\begin{aligned} F\_{\mathbf{x},m+2,n} - 2F\_{\mathbf{x},m+1,n} + F\_{\mathbf{x},m,n} &= L\_{\mathbf{x},m+1,n} - L\_{\mathbf{x},m,n} \\ F\_{\mathbf{x},m+1,n+1} - F\_{\mathbf{x},m+1,n} - F\_{\mathbf{x},m,n+1} + F\_{\mathbf{x},m,n} &= L\_{\mathbf{x},m,n+1} - L\_{\mathbf{x},m,n} \\ F\_{\mathbf{y},m+2,n} - 2F\_{\mathbf{y},m+1,n} + F\_{\mathbf{y},m,n} &= L\_{\mathbf{y},m+1,n} - L\_{\mathbf{y},m,n} \\ F\_{\mathbf{y},m+1,n+1} - F\_{\mathbf{y},m+1,n} - F\_{\mathbf{y},m,n+1} + F\_{\mathbf{y},m,n} &= L\_{\mathbf{y},m,n+1} - L\_{\mathbf{y},m,n} \end{aligned} \tag{25}$$

From previous subsections, Eqs. (7), (23), and (25) can yield

$$\begin{aligned} \sum\_{m,n} F\_{x,m,n} &= \sum\_{m,n} F\_{x,m,n} x\_m = \sum\_{m,n} F\_{x,m,n} y\_n = 0 \\ \sum\_{m,n} F\_{y,m,n} &= \sum\_{m,n} F\_{y,m,n} x\_m = \sum\_{m,n} F\_{y,m,n} y\_n = 0 \\ F\_{x,m,n} - F\_{y,-n,m} &= U\_{0,x,m,n} - U\_{1,y,m,n} - 2Oy\_n - 2Rx\_m \\ F\_{y,m,n} + F\_{x,-n,m} &= U\_{0,y,m,n} + U\_{1,x,m,n} - 2Ox\_m + 2Ry\_n \\ F\_{x,m+2,n} - 2F\_{x,m+1,n} + F\_{x,m,n} &= L\_{x,m+1,n} - L\_{x,m,n} \\ F\_{x,m+1,n+1} - F\_{x,m+1,n} - F\_{x,m,n+1} + F\_{x,m,n} &= L\_{x,m,n+1} - L\_{x,m,n} \\ F\_{y,m+2,n} - 2F\_{y,m+1,n} + F\_{y,m,n} &= L\_{y,m+1,n} - L\_{y,m,n} \\ F\_{y,m+1,n+1} - F\_{y,m+1,n} - F\_{y,m,n+1} + F\_{y,m,n} &= L\_{y,m,n+1} - L\_{y,m,n} \end{aligned} \tag{26}$$

which actually can determinate Fx,m,n and Fy,m,n with certain redundancy. Then a least square estimation law for Fx,m,n and Fy,m,n can be synthesized through a least square solution of the set of Fx,m,n and Fy,m,n equations to meet the challenge of random measurement noise [17, 20, 21]. Thus, combining the solved parameters O and R, we can obtain Gx,m,n and Gy,m,n.

Afterward, according to (9), we can obtain the determination of Gθz,0, Gθz, <sup>K</sup> 4 , Gθz,K 2 , and Gθz, <sup>3</sup><sup>K</sup> <sup>4</sup> with certain redundancy. Here we use the method of least squares as follows:

Gθz, <sup>1</sup> ¼ ∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>Gy,m,0xm � <sup>1</sup> <sup>M</sup> <sup>∑</sup><sup>M</sup> <sup>m</sup>¼<sup>1</sup>Gy,m,0∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>xm ∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>x<sup>2</sup> <sup>m</sup> � <sup>1</sup> <sup>M</sup> <sup>∑</sup><sup>M</sup> <sup>m</sup>¼<sup>1</sup>xm <sup>2</sup> Gθz, <sup>1</sup>þ<sup>K</sup> <sup>4</sup> ¼ � ∑<sup>M</sup> <sup>n</sup>¼<sup>1</sup>Gx,0,nyn � <sup>1</sup> <sup>M</sup> <sup>∑</sup><sup>M</sup> <sup>n</sup>¼<sup>1</sup>Gx,0,n∑<sup>M</sup> <sup>n</sup>¼<sup>1</sup>yn ∑<sup>M</sup> <sup>n</sup>¼<sup>1</sup>y<sup>2</sup> <sup>n</sup> � <sup>1</sup> <sup>M</sup> <sup>∑</sup><sup>M</sup> <sup>n</sup>¼<sup>1</sup>yn <sup>2</sup> Gθz, <sup>1</sup>þ<sup>K</sup> <sup>2</sup> ¼ ∑�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>Gy,m,0xm � <sup>1</sup> <sup>M</sup> <sup>∑</sup>�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>Gy,m,0∑�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>xm ∑�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>x<sup>2</sup> <sup>m</sup> � <sup>1</sup> <sup>M</sup> <sup>∑</sup>�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>xm <sup>2</sup> Gθz, <sup>1</sup>þ3<sup>K</sup> <sup>4</sup> ¼ � ∑�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>Gx,0,nyn � <sup>1</sup> <sup>M</sup> <sup>∑</sup>�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>Gx,0,n∑�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>yn ∑�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>y<sup>2</sup> <sup>n</sup> � <sup>1</sup> <sup>M</sup> <sup>∑</sup>�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>yn <sup>2</sup> (27)

Then combining Eqs. (18) and (21), define

Standards, Methods and Solutions of Metrology

Fx,mþ2,n � 2Fx,mþ1,n þ Fx,m,n ¼ Lx,mþ1,n � Lx,m,n

Fy,mþ2,n � 2Fy,mþ1,n þ Fy,m,n ¼ Ly,mþ1,n � Ly,m,n

From previous subsections, Eqs. (7), (23), and (25) can yield

∑m,nFx,m,n ¼ ∑m,nFx,m,nxm ¼ ∑m,nFx,m,nyn ¼ 0 ∑m,nFy,m,n ¼ ∑m,nFy,m,nxm ¼ ∑m,nFy,m,nyn ¼ 0 Fx,m,n � Fy,�n,m ¼ U0,x,m,n � U1, y,m,n � 2Oyn � 2Rxm Fy,m,n þ Fx,�n,m ¼ U0, y,m,n þ U1,x,m,n � 2Oxm þ 2Ryn Fx,mþ2,n � 2Fx,mþ1,n þ Fx,m,n ¼ Lx,mþ1,n � Lx,m,n

Fy,mþ2,n � 2Fy,mþ1,n þ Fy,m,n ¼ Ly,mþ1,n � Ly,m,n

Fx,mþ1,nþ<sup>1</sup> � Fx,mþ1,n � Fx,m,nþ<sup>1</sup> þ Fx,m,n ¼ Lx,m,nþ<sup>1</sup> � Lx,m,n

Fy,mþ1,nþ<sup>1</sup> � Fy,mþ1,n � Fy,m,nþ<sup>1</sup> þ Fy,m,n ¼ Ly,m,nþ<sup>1</sup> � Ly,m,n

Fx,mþ1,nþ<sup>1</sup> � Fx,mþ1,n � Fx,m,nþ<sup>1</sup> þ Fx,m,n ¼ Lx,m,nþ<sup>1</sup> � Lx,m,n

Fy,mþ1,nþ<sup>1</sup> � Fy,mþ1,n � Fy,m,nþ<sup>1</sup> þ Fy,m,n ¼ Ly,m,nþ<sup>1</sup> � Ly,m,n

which actually can determinate Fx,m,n and Fy,m,n with certain redundancy. Then a least square estimation law for Fx,m,n and Fy,m,n can be synthesized through a least square solution of the set of Fx,m,n and Fy,m,n equations to meet the challenge of random measurement noise [17, 20, 21]. Thus, combining the solved parameters O

Afterward, according to (9), we can obtain the determination of Gθz,0, Gθz, <sup>K</sup>

<sup>m</sup> � <sup>1</sup>

<sup>M</sup> <sup>∑</sup><sup>M</sup>

<sup>M</sup> <sup>∑</sup><sup>M</sup>

<sup>M</sup> <sup>∑</sup><sup>M</sup>

<sup>M</sup> <sup>∑</sup><sup>M</sup>

<sup>M</sup> <sup>∑</sup>�<sup>1</sup>

<sup>M</sup> <sup>∑</sup>�<sup>1</sup>

<sup>M</sup> <sup>∑</sup>�<sup>1</sup>

<sup>M</sup> <sup>∑</sup>�<sup>1</sup>

<sup>m</sup> � <sup>1</sup>

<sup>n</sup> � <sup>1</sup>

<sup>4</sup> with certain redundancy. Here we use the method of least squares

<sup>m</sup>¼<sup>1</sup>Gy,m,0∑<sup>M</sup>

<sup>n</sup>¼<sup>1</sup>Gx,0,n∑<sup>M</sup>

<sup>m</sup>¼�<sup>M</sup>Gy,m,0∑�<sup>1</sup>

<sup>m</sup>¼�<sup>M</sup>xm <sup>2</sup>

<sup>n</sup>¼�<sup>M</sup>Gx,0,n∑�<sup>1</sup>

<sup>n</sup>¼�<sup>M</sup>yn <sup>2</sup>

<sup>m</sup>¼<sup>1</sup>xm <sup>2</sup>

<sup>n</sup>¼<sup>1</sup>yn <sup>2</sup> <sup>m</sup>¼<sup>1</sup>xm

<sup>n</sup>¼<sup>1</sup>yn

<sup>m</sup>¼�<sup>M</sup>xm

<sup>n</sup>¼�<sup>M</sup>yn

Consequently, we can obtain

and R, we can obtain Gx,m,n and Gy,m,n.

Gθz, <sup>1</sup> ¼

Gθz, <sup>1</sup>þ<sup>K</sup>

Gθz, <sup>1</sup>þ<sup>K</sup> <sup>2</sup> ¼

Gθz, <sup>1</sup>þ3<sup>K</sup>

<sup>4</sup> ¼ �

<sup>4</sup> ¼ �

∑<sup>M</sup>

∑<sup>M</sup>

∑�<sup>1</sup>

∑�<sup>1</sup>

<sup>m</sup>¼<sup>1</sup>Gy,m,0xm � <sup>1</sup>

<sup>n</sup>¼<sup>1</sup>Gx,0,nyn � <sup>1</sup>

∑<sup>M</sup> <sup>n</sup>¼<sup>1</sup>y<sup>2</sup> <sup>n</sup> � <sup>1</sup>

<sup>m</sup>¼�<sup>M</sup>Gy,m,0xm � <sup>1</sup>

<sup>n</sup>¼�<sup>M</sup>Gx,0,nyn � <sup>1</sup>

∑�<sup>1</sup> <sup>n</sup>¼�<sup>M</sup>y<sup>2</sup>

∑�<sup>1</sup> <sup>m</sup>¼�<sup>M</sup>x<sup>2</sup>

∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>x<sup>2</sup>

Gθz,K 2

32

as follows:

, and Gθz, <sup>3</sup><sup>K</sup>

Lx,m,n ¼ U3,x,m,n � U0,x,m,n ¼ Fx,mþ1,n � Fx,m,n þ ξ<sup>x</sup> Ly,m,n ¼ U3, y,m,n � U0, y,m,n ¼ Fy,mþ1,n � Fy,m,n þ ξ<sup>y</sup>

(24)

(25)

(26)

4 ,

(27)

where <sup>M</sup> <sup>¼</sup> <sup>N</sup>�<sup>1</sup> <sup>2</sup> . Noting Eqs. (18), (19), and (20), we consequently obtain

$$\begin{aligned} G\_{\theta\_z, k + \frac{k}{4}} - G\_{\theta\_z, k} &= U\_{1, \theta\_z, k} - U\_{0, \theta\_z, k} \\ G\_{\theta\_z, k + 1} - G\_{\theta\_z, k} &= U\_{2, \theta\_z, k} - U\_{0, \theta\_z, k} \end{aligned} \tag{28}$$

where k ¼ 1, 2, …, K and Gθz,kþ<sup>K</sup> ¼ Gθz,k.

Combining (27) and (28), the stage error Gθz,k can be determined through a least square solution as the above equation group has K unknowns and 2K + 4 equations.

The proposed method features certain benefits remarked as follows:


### 4. Computer simulation

In this section, we use MATLAB software to simulate the self-calibration process. First, arbitrary stage linear error maps on a 11 � 11 sample site array with the sample site interval Δ ¼ 10 mm are generated using the command 'normrnd' with a mean of 0 and standard deviation of 0.2 μm, which are utilized as the nominal Gx,m,n and Gy,m,n. Besides, any stage rotary error is mapped within a revolution, while the sample site internal is <sup>Δ</sup> <sup>¼</sup> <sup>15</sup><sup>∘</sup> . And the nominal Gθz,k is generated by mean value of 0 and standard deviation of 0:01<sup>∘</sup> . And minor modification has been made to the data to satisfy the relevant requirements like Eqs. (5), (9), and (11). Then, arbitrary artifact linear error maps are generated with mean of 0 and standard deviation of 0.3 μm, which are employed as the nominal Am,n, and arbitrary artifact rotary error maps are generated with a mean of 0 and standard deviation of 0:01<sup>∘</sup> , which are utilized as the nominal Aθz,k. The nominal stage error component Gm,n is shown in Figure 4 where the red lines are Gm,n � 10000. The nominal stage error component Gθz,k is shown in Figure 5 where Gθz,k between the actual measurement system and perfect measurement system has been zoomed in for 360/π times. In addition, for each view, we add a random misalignment which is made up of a rotation and an offset. And the standard deviation value in the misalignment is 0.3° for rotation and 30 μm for offset. Since Eq. (26) has some redundance, it is clear that Gm,n,k can be figured out rather accurately if there is no random measurement noise. In addition, as there are no reported complete on-axis XYθ<sup>z</sup> self-calibration strategies in published papers, we just test their own effectiveness of the proposed strategy. Herein, we focus on testing the calibration accuracy of the proposed strategy in various random measurement noises.

4.1 Simulation with random measurement noise of standard deviation 0.02 μm

For the simulation in this subsection, we add an independent random Gaussian measurement noise to every grid mark's each measurement and independent angular measurement noise to every angular mark's each measurement. Gx,m,n and Gy,m,n's random measurement noise is generated by a mean value 0 and standard deviation 0.02 μm, and Gθz,k's random measurement noise is generated by a mean value 0 and standard deviation 0.001°. The reconstructed stage error Gm,n,k can thus be determined according to the proposed four measurement views and self-calibration algorithm. So as to test the robustness of the algorithm, the calibration errors EGx, EGy, and EG<sup>θ</sup> are calculated, which are defined as the deviation between the actual and the recalculated stage error, i.e. EGx <sup>¼</sup> Gx,m,n � <sup>G</sup>^ x,m,n, EGy <sup>¼</sup> Gy,m,n � <sup>G</sup>^ y,m,n, and EG<sup>θ</sup> <sup>¼</sup> <sup>G</sup>θ,k � <sup>G</sup>^ <sup>θ</sup>,k. In Table 1, the maximum, the minimum, and the standard deviation for EGx, EGy, and EG<sup>θ</sup> are all listed. It is obvious that the stage error can be accurately recalculated by the proposed method even in the case of random measurement noise—when there is measurement noise by standard deviation 0.02 μm and 0.001° and the standard deviations of the calibra-

for Gx,m,<sup>n</sup> and Gy,m,<sup>n</sup> and standard deviation 0.001° for G<sup>θ</sup><sup>z</sup> ,<sup>k</sup>

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

tion errors are smaller than 0.02 μm and 0.001°, respectively.

the challenge of random measurement noise effectively.

In addition, the algorithm's accuracy is also tested to meet the challenge of various random measurement noises. We generate the random measurement noises for 20 times. As a result, the calibration errors'standard deviations are shown in Figures 6 and 7. All the results illustrate that all the 20 standard deviations keep in the same level with the measurement noises themselves. The simulation results demonstrate that the algorithm is robust and accurate. In addition, it can deal with

4.2 Simulation with random measurement noise of standard deviation

0 and standard deviation of 0.0001°. Through the proposed scheme, the reconstructed stage error Gm,n,k can be calculated out, and the maximum value max(�), the minimum value min(�), and the standard deviation std(�) of EGx, EGy, and EG<sup>θ</sup> are detailed in Table 2. Furthermore, the algorithm's accuracy and robustness are tested for arbitrary 20 times; the results are shown in Figures 8 and 9. It can be observed that the calibration error is also about the same size as the random measurement noises themselves. All these results further verify that the proposed

EGx (μm) 0.0541 �0.0490 0.0195 EGy (μm) 0.0511 �0.0506 0.0196 EG<sup>θ</sup> (°) 1.5877e�003 �1.2065e�003 7.1184e�004

Calculation performance indexes (with random measurement noise std = 0.02 μM).

Table 1.

35

0.002 μm for Gx,m,<sup>n</sup> and Gy,m,<sup>n</sup> and standard deviation 0.0001° for G<sup>θ</sup><sup>z</sup> ,<sup>k</sup>

max(�) min(�) std(�)

The simulation in this subsection is set up exactly the same way as in the previous subsection, except for adding a different random Gaussian measurement noise to every site's measurement, which is to test the consistency of the proposed scheme's robustness to random measurement noise. The random measurement noise for Gx,m,n and Gy,m,n is generated with a mean of 0 and standard deviation of 0.002 μm, and the random measurement noise for Gθz,k is generated with a mean of

Figure 4. Gm,n 10000 with max(Gm,n) = 0.5494 μm, min(Gm,n) = 0.5806 μm.

Figure 5. <sup>G</sup><sup>θ</sup><sup>z</sup> , <sup>k</sup> <sup>360</sup>=π<sup>∘</sup> with max(G<sup>θ</sup><sup>z</sup> ,k) = <sup>0</sup>:0243<sup>∘</sup> , min(G<sup>θ</sup><sup>z</sup> ,k) = <sup>0</sup>:01282<sup>∘</sup> .

### 4.1 Simulation with random measurement noise of standard deviation 0.02 μm for Gx,m,<sup>n</sup> and Gy,m,<sup>n</sup> and standard deviation 0.001° for G<sup>θ</sup><sup>z</sup> ,<sup>k</sup>

For the simulation in this subsection, we add an independent random Gaussian measurement noise to every grid mark's each measurement and independent angular measurement noise to every angular mark's each measurement. Gx,m,n and Gy,m,n's random measurement noise is generated by a mean value 0 and standard deviation 0.02 μm, and Gθz,k's random measurement noise is generated by a mean value 0 and standard deviation 0.001°. The reconstructed stage error Gm,n,k can thus be determined according to the proposed four measurement views and self-calibration algorithm. So as to test the robustness of the algorithm, the calibration errors EGx, EGy, and EG<sup>θ</sup> are calculated, which are defined as the deviation between the actual and the recalculated stage error, i.e. EGx <sup>¼</sup> Gx,m,n � <sup>G</sup>^ x,m,n, EGy <sup>¼</sup> Gy,m,n � <sup>G</sup>^ y,m,n, and EG<sup>θ</sup> <sup>¼</sup> <sup>G</sup>θ,k � <sup>G</sup>^ <sup>θ</sup>,k. In Table 1, the maximum, the minimum, and the standard deviation for EGx, EGy, and EG<sup>θ</sup> are all listed. It is obvious that the stage error can be accurately recalculated by the proposed method even in the case of random measurement noise—when there is measurement noise by standard deviation 0.02 μm and 0.001° and the standard deviations of the calibration errors are smaller than 0.02 μm and 0.001°, respectively.

In addition, the algorithm's accuracy is also tested to meet the challenge of various random measurement noises. We generate the random measurement noises for 20 times. As a result, the calibration errors'standard deviations are shown in Figures 6 and 7. All the results illustrate that all the 20 standard deviations keep in the same level with the measurement noises themselves. The simulation results demonstrate that the algorithm is robust and accurate. In addition, it can deal with the challenge of random measurement noise effectively.

### 4.2 Simulation with random measurement noise of standard deviation 0.002 μm for Gx,m,<sup>n</sup> and Gy,m,<sup>n</sup> and standard deviation 0.0001° for G<sup>θ</sup><sup>z</sup> ,<sup>k</sup>

The simulation in this subsection is set up exactly the same way as in the previous subsection, except for adding a different random Gaussian measurement noise to every site's measurement, which is to test the consistency of the proposed scheme's robustness to random measurement noise. The random measurement noise for Gx,m,n and Gy,m,n is generated with a mean of 0 and standard deviation of 0.002 μm, and the random measurement noise for Gθz,k is generated with a mean of 0 and standard deviation of 0.0001°. Through the proposed scheme, the reconstructed stage error Gm,n,k can be calculated out, and the maximum value max(�), the minimum value min(�), and the standard deviation std(�) of EGx, EGy, and EG<sup>θ</sup> are detailed in Table 2. Furthermore, the algorithm's accuracy and robustness are tested for arbitrary 20 times; the results are shown in Figures 8 and 9. It can be observed that the calibration error is also about the same size as the random measurement noises themselves. All these results further verify that the proposed


#### Table 1. Calculation performance indexes (with random measurement noise std = 0.02 μM).

Figure 4.

Figure 5.

34

<sup>G</sup><sup>θ</sup><sup>z</sup> , <sup>k</sup> <sup>360</sup>=π<sup>∘</sup> with max(G<sup>θ</sup><sup>z</sup> ,k) = <sup>0</sup>:0243<sup>∘</sup>

, min(G<sup>θ</sup><sup>z</sup> ,k) = <sup>0</sup>:01282<sup>∘</sup>

.

Gm,n 10000 with max(Gm,n) = 0.5494 μm, min(Gm,n) = 0.5806 μm.

Standards, Methods and Solutions of Metrology

Figure 6.

Standard deviation of calibration error EGx & EGy for arbitrary 20 times (with random measurement noise std = 0.02 μm).

#### Figure 7.

Standard deviation of calibration error EG<sup>θ</sup> for arbitrary 20 times (with random measurement noise std = 0.001°).


strategy can accurately determine the stage error even under the existence of

Standard deviation of calibration error EG<sup>θ</sup> for arbitrary 20 times (with random measurement noise

Standard deviation of calibration error EGx & EGy for arbitrary 10 times (with random measurement noise

4.3 The artifact plate and the procedure for performing a standard XYθ<sup>z</sup>

Based on the proposed theory principle, we design and manufacture an artifact plate for XYθ<sup>z</sup> self-calibration. The specification of the practical artifact plate is shown in Figure 10. Specifically, the artifact plate is made in square shape with

random measurement noise.

self-calibration

Figure 9.

37

std = 0.0001°).

Figure 8.

std = 0.002 μm).

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

#### Table 2.

Calculation performance indexes (with random measurement noise std = 0.02 μm).

#### Figure 8.

Standard deviation of calibration error EGx & EGy for arbitrary 10 times (with random measurement noise std = 0.002 μm).

#### Figure 9.

Figure 7.

Table 2.

36

Figure 6.

std = 0.02 μm).

Standards, Methods and Solutions of Metrology

std = 0.001°).

Standard deviation of calibration error EG<sup>θ</sup> for arbitrary 20 times (with random measurement noise

EGx (μm) 0.0051 0.0051 0.0020 EGy (μm) 0.0051 0.0053 0.0020 EG<sup>θ</sup> (°) 2.9963e004 6.9733e004 7.6657e005

Calculation performance indexes (with random measurement noise std = 0.02 μm).

max() min() std()

Standard deviation of calibration error EGx & EGy for arbitrary 20 times (with random measurement noise

Standard deviation of calibration error EG<sup>θ</sup> for arbitrary 20 times (with random measurement noise std = 0.0001°).

strategy can accurately determine the stage error even under the existence of random measurement noise.

### 4.3 The artifact plate and the procedure for performing a standard XYθ<sup>z</sup> self-calibration

Based on the proposed theory principle, we design and manufacture an artifact plate for XYθ<sup>z</sup> self-calibration. The specification of the practical artifact plate is shown in Figure 10. Specifically, the artifact plate is made in square shape with

reference location, which is named as View 0 of Figure 3. The artifact plate's mark locations are measured in the XYθ<sup>z</sup> metrology stage with the help of some mark alignment system such as those in [21, 23]. The mark alignment system should be designed to obtain the measurement data of each mark position accurately. Get V0,x,m,n, V0, y,m,n, and V0,θz,k for P (e.g. P ¼ 5) times, and average the results, and

Step 1: After rotating 90° around the origin shown as View 1 in Figure 3, the artifact plate is replaced to the rotated reference location on the stage. The XYθ<sup>z</sup> metrology stage is utilized to measure the artifact plate mark locations. Obtain V1,x,m,n, V1, y,m,n, and V1,θz,k for P (e.g. P ¼ 5) times and average the results. Get U1,x,m,n, U1, y,m,n, and U1, <sup>θ</sup>z,k by Eq. (19). And calculate out O and R by Eq. (22). Step 2: After rotating 360=K<sup>∘</sup> around the origin shown as View 2 in Figure 3, the

artifact plate is replaced to the rotated reference location in the stage. The XYθ<sup>z</sup> metrology stage is utilized to measure the angles of the artifact plate mark lines, and get V2, <sup>θ</sup>z,k for P (e.g. P ¼ 5) times, and average the results, and then obtain U2, <sup>θ</sup>z,k

and average the results. Obtain U3,x,m,n, U3, y,m,n, and U3, <sup>θ</sup>z,k by Eq. (21). Step 4: After above steps, the equation groups (26), (27), and (28) can be obtained. Therefore, a least square solution can be used for the determination of Fx,m,n, Fy,m,n, and Gθz,k. Then Gx,m,n and Gy,m,n can be determined by Eq. (6) as O and R are previously computed in Step 1, which completes the final determination

Step 3: After translating one grid interval Δ along +X-axis of the stage shown as View 3 in Figure 3, the artifact plate is replaced to the translated reference location in the stage. The XYθ<sup>z</sup> metrology stage is used to measure the location of the marks of the artifact plate, and get V3,x,m,n, V3, y,m,n, and V3, <sup>θ</sup>z,k for P (e.g. P ¼ 5) times,

In this chapter, an on-axis self-calibration approach has been first developed for precision XYθ<sup>z</sup> metrology stages to solve the calibration problem. The proposed scheme uses a new artifact plate designed with special mark lines as the assistant tool for calibration. In detail, the artifact plate is placed on the uncalibrated XYθ<sup>z</sup> metrology stages in four measurement postures or views. Then, the measurement error can be modeled as the construction of XYθ<sup>z</sup> systematic measurement error (i.e. stage error) and other errors or noise. Based on the redundance of the XYθ<sup>z</sup> stage error, a least square-based XYθ<sup>z</sup> self-calibration law is resultantly synthesized

for final determination of the stage error. Computer simulations have been

metrology stages in practical applications.

conducted to verify that the proposed method can figure out the stage error rather accurately even under existence of various random measurement noises. Finally, a standard on-axis XYθ<sup>z</sup> self-calibration procedure with the designed artifact plate is introduced. As an integration of XY self-calibration and θ self-calibration, the proposed scheme solves the XYθ<sup>z</sup> self-calibration problem without complicated mathematical processing for misalignment errors. The developed approach has essentially provided a fundamental principle for on-axis self-calibration of precision XYθ<sup>z</sup>

The proposed scheme has pros and cons. It provides a significant theory fundamental for self-calibration of XYθ<sup>z</sup> metrology or motion stages. However, the calibration accuracy is seriously affected by the mark alignment systems which may be a little complex. Therefore, the proposed self-calibration scheme is very suitable for standard calibration in national standard institutes but a little limited for wide

then obtain U0,x,m,n, U0, y,m,n, and U0, <sup>θ</sup>z,k by Eq. (18).

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

by Eq. (20).

of Gm,n,k.

39

5. Conclusions

Figure 10. A designed artifact plate for XYθ<sup>z</sup> self-calibration.

material of optical glass. The thickness is 3 mm. The origin of the plate is located at the center. The radius or the circle in the plate is 80 mm. The circle of 360° is divided into 72 sectors by 72 lines with a width of 5 μm and accuracy of 5". The 21 21 sample site array is with the sample site interval Δ = 5 mm. The accuracy of the circle lines and the straight lines is 1 μm and the width is 5 μm.

For presentation, convenience, and clarity, an example of an artifact plate on a XYθ<sup>z</sup> stage is also provided and shown in Figure 11. In the following, we list the procedure of performing a standard XYθ<sup>z</sup> self-calibration following the proposed scheme, which may be useful for engineers in practical implementations:

Step 0: As shown in Figures 3 and 11, put the artifact plate shown in Figure 10 in the XYθ<sup>z</sup> stage. The artifact plate's array marks are consequently at the stage's origin

Figure 11. An artifact plate on a XYθ<sup>z</sup> stage for self-calibration.

reference location, which is named as View 0 of Figure 3. The artifact plate's mark locations are measured in the XYθ<sup>z</sup> metrology stage with the help of some mark alignment system such as those in [21, 23]. The mark alignment system should be designed to obtain the measurement data of each mark position accurately. Get V0,x,m,n, V0, y,m,n, and V0,θz,k for P (e.g. P ¼ 5) times, and average the results, and then obtain U0,x,m,n, U0, y,m,n, and U0, <sup>θ</sup>z,k by Eq. (18).

Step 1: After rotating 90° around the origin shown as View 1 in Figure 3, the artifact plate is replaced to the rotated reference location on the stage. The XYθ<sup>z</sup> metrology stage is utilized to measure the artifact plate mark locations. Obtain V1,x,m,n, V1, y,m,n, and V1,θz,k for P (e.g. P ¼ 5) times and average the results. Get U1,x,m,n, U1, y,m,n, and U1, <sup>θ</sup>z,k by Eq. (19). And calculate out O and R by Eq. (22).

Step 2: After rotating 360=K<sup>∘</sup> around the origin shown as View 2 in Figure 3, the artifact plate is replaced to the rotated reference location in the stage. The XYθ<sup>z</sup> metrology stage is utilized to measure the angles of the artifact plate mark lines, and get V2, <sup>θ</sup>z,k for P (e.g. P ¼ 5) times, and average the results, and then obtain U2, <sup>θ</sup>z,k by Eq. (20).

Step 3: After translating one grid interval Δ along +X-axis of the stage shown as View 3 in Figure 3, the artifact plate is replaced to the translated reference location in the stage. The XYθ<sup>z</sup> metrology stage is used to measure the location of the marks of the artifact plate, and get V3,x,m,n, V3, y,m,n, and V3, <sup>θ</sup>z,k for P (e.g. P ¼ 5) times, and average the results. Obtain U3,x,m,n, U3, y,m,n, and U3, <sup>θ</sup>z,k by Eq. (21).

Step 4: After above steps, the equation groups (26), (27), and (28) can be obtained. Therefore, a least square solution can be used for the determination of Fx,m,n, Fy,m,n, and Gθz,k. Then Gx,m,n and Gy,m,n can be determined by Eq. (6) as O and R are previously computed in Step 1, which completes the final determination of Gm,n,k.

### 5. Conclusions

material of optical glass. The thickness is 3 mm. The origin of the plate is located at the center. The radius or the circle in the plate is 80 mm. The circle of 360° is divided into 72 sectors by 72 lines with a width of 5 μm and accuracy of 5". The 21 21 sample site array is with the sample site interval Δ = 5 mm. The accuracy of

For presentation, convenience, and clarity, an example of an artifact plate on a XYθ<sup>z</sup> stage is also provided and shown in Figure 11. In the following, we list the procedure of performing a standard XYθ<sup>z</sup> self-calibration following the proposed

Step 0: As shown in Figures 3 and 11, put the artifact plate shown in Figure 10 in the XYθ<sup>z</sup> stage. The artifact plate's array marks are consequently at the stage's origin

the circle lines and the straight lines is 1 μm and the width is 5 μm.

Figure 10.

Figure 11.

38

An artifact plate on a XYθ<sup>z</sup> stage for self-calibration.

A designed artifact plate for XYθ<sup>z</sup> self-calibration.

Standards, Methods and Solutions of Metrology

scheme, which may be useful for engineers in practical implementations:

In this chapter, an on-axis self-calibration approach has been first developed for precision XYθ<sup>z</sup> metrology stages to solve the calibration problem. The proposed scheme uses a new artifact plate designed with special mark lines as the assistant tool for calibration. In detail, the artifact plate is placed on the uncalibrated XYθ<sup>z</sup> metrology stages in four measurement postures or views. Then, the measurement error can be modeled as the construction of XYθ<sup>z</sup> systematic measurement error (i.e. stage error) and other errors or noise. Based on the redundance of the XYθ<sup>z</sup> stage error, a least square-based XYθ<sup>z</sup> self-calibration law is resultantly synthesized for final determination of the stage error. Computer simulations have been conducted to verify that the proposed method can figure out the stage error rather accurately even under existence of various random measurement noises. Finally, a standard on-axis XYθ<sup>z</sup> self-calibration procedure with the designed artifact plate is introduced. As an integration of XY self-calibration and θ self-calibration, the proposed scheme solves the XYθ<sup>z</sup> self-calibration problem without complicated mathematical processing for misalignment errors. The developed approach has essentially provided a fundamental principle for on-axis self-calibration of precision XYθ<sup>z</sup> metrology stages in practical applications.

The proposed scheme has pros and cons. It provides a significant theory fundamental for self-calibration of XYθ<sup>z</sup> metrology or motion stages. However, the calibration accuracy is seriously affected by the mark alignment systems which may be a little complex. Therefore, the proposed self-calibration scheme is very suitable for standard calibration in national standard institutes but a little limited for wide

industrial applications. In the next step, the development of this calibration system is an important topic, which will do help for wide applications of the proposed scheme.

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Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

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## Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 51775305 and 51475262, Autonomous Scientific Research Project of Tsinghua University under Grant 20151080363, and Autonomous Research Project of State Key Lab of Tribology at Tsinghua University under Grant SKLT2018C02.

### Author details

Chuxiong Hu1,2\*, Yu Zhu1,2 and Luzheng Liu1,2

1 State Key Lab of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, China

2 Beijing Key Lab of Precision/Ultra Precision Manufacture Equipments and Control, Tsinghua University, Beijing, China

\*Address all correspondence to: cxhu@tsinghua.edu.cn

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Self‐Calibration of Precision XYθ<sup>z</sup> Metrology Stages DOI: http://dx.doi.org/10.5772/intechopen.85539

### References

industrial applications. In the next step, the development of this calibration system is an important topic, which will do help for wide applications of the proposed

This work was supported in part by the National Natural Science Foundation of China under Grant 51775305 and 51475262, Autonomous Scientific Research Project of Tsinghua University under Grant 20151080363, and Autonomous Research Project of State Key Lab of Tribology at Tsinghua University under Grant

scheme.

Acknowledgements

Standards, Methods and Solutions of Metrology

SKLT2018C02.

Author details

40

University, Beijing, China

Chuxiong Hu1,2\*, Yu Zhu1,2 and Luzheng Liu1,2

Control, Tsinghua University, Beijing, China

provided the original work is properly cited.

\*Address all correspondence to: cxhu@tsinghua.edu.cn

1 State Key Lab of Tribology, Department of Mechanical Engineering, Tsinghua

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Beijing Key Lab of Precision/Ultra Precision Manufacture Equipments and

[1] Zhong G, Shao Z, Deng H, Ren J. Precise position synchronous control for multi-axis servo systems. IEEE Transactions on Industrial Electronics. 2017;64(5):3707-3717

[2] Hu C, Wang Z, Zhu Y, et al. Accurate three-dimensional contouring error estimation and compensation scheme with zero-phase filter. International Journal of Machine Tools and Manufacture. 2018;128:33-40

[3] Hu C, Hu Z, Zhu Y, Wang Z. Advanced GTCF based LARC contouring motion control on an industrial X-Y linear-motor-driven stage with experimental investigation. IEEE Transactions on Industrial Electronics. 2017;64(4):3308-3318

[4] Chen Z, Yao B, Wang Q. μ-Synthesis based adaptive robust control of linear motor driven stages with highfrequency dynamics: A case study with comparative experiments. IEEE/ASME Transactions on Mechatronics. 2015; 20(3):1482-1490

[5] Chen Z, Yao B, Wang Q. Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME Transactions on Mechatronics. 2013;18(3):1122-1129

[6] Hu C, Wang Z, Zhu Y, Zhang M, Liu H. Performance oriented precision LARC tracking motion control of a magnetically levitated planar motor with comparative experiments. IEEE Transactions on Industrial Electronics. 2016;63(9):5763-5773

[7] Liu G, Dass R, Nguang S-K, Partridge A. Principles, design, and calibration for a genre of irradiation angle sensors. IEEE Transactions on Industrial Electronics. 2013;60(1): 210-216

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[14] Lu X-D, Trumper DL. Selfcalibration of on-axis rotary encoders. CIRP Annals: Manufacturing Technology. 2007;56(1):499-504

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[16] Jeong YH, Dong J, Ferreira PM. Selfcalibration of dual-actuated single-axis nanopositioners. Measurement Science and Technology. 2008;19(4):1-13

[17] Ye J, Takac MT, Berglund CN, et al. An exact algorithm for self-calibration of precision metrology stages. Precision Engineering. 1997;20(1):16-32

[18] Yoo S, Kim SW. Self-calibration algorithm for testing out-of-plane errors of two-dimensional profiling stages. International Journal of Machine Tools & Manufacture. 2004;44:767-774

[19] Xu M, Dziomba T, Dai G, Koenders L. Self-calibration of scanning probe microscope: Mapping the errors of the instrument. Measurement Science and Technology. 2008;19(2):1-6

[20] Hu C, Zhu Y, Hu J, Zhang M, Xu D. A holistic self-calibration algorithm for XY precision metrology systems. IEEE Transactions on Instrumentation and Measurement. 2012;61(9):2492-2500

[21] Zhu Y, Hu C, Hu J, Yang K. Accuracy and simplicity oriented selfcalibration approach for twodimensional precision stages. IEEE Transactions on Industrial Electronics. 2012;60(6):2264-2272

[22] Dang QC, Yoo S, Kim S-W. Complete 3-D self-calibration of coordinate measuring machines. Annals of the CIRP. 2006;55(1):1-4

[23] Hu C, Zhu Y, Hu J, Xu D, Zhang M. A holistic self-calibration approach for determination of three-dimensional stage error. IEEE Transactions on Instrumentation and Measurement. 2013;62(2):483-494

[24] Lu X-D, Graetz R, Amin-Shahidi D, Smeds K. On-axis self-calibration of angle encoders. CIRP Annals: Manufacturing Technology. 2010;59(1): 529-534

[25] Estler T, Queen H. An advanced angle metrology system. Annals of the CIRP. 1993;42(1):573-576

Chapter 3

Abstract

low threshold.

1. Introduction

2. Physics of quantum dots

43

Khalil Ebrahim Jasim

Third-Order Nonlinear Optical

Quantum dots (QDs) are semiconducting nanocrystalline particles. QDs are attractive photonic media. In this chapter, we introduce third-order nonlinear optical properties and a brief idea about the physics of QDs. Z-scan technique and theoretical analysis adopted to obtain nonlinear parameters will be discussed. Analysis of third-order nonlinear optical parameters for PbS QDs suspended in toluene with radii 2.4 and 5.0 nm under different excitation beam power level and three different wavelengths (488, 514, and 633 nm) will be detailed. Third-order optical susceptibility χ(3) and optical-limiting behavior of PbS QD suspended in toluene are presented. Irrespective of their size, QDs are a good example of optical limiters with

Keywords: quantum dot, semiconductor, nonlinear optics, Z-scan technique,

Generally speaking, quantum dots (QDs) are nanocrystalline structures that can be grown using physical or chemical methods. QDs are an essential medium for extensive range of applications in photonics. Some attention-grabbing applications of these materials include fluorescence imaging, electroluminescence, frequency up conversion lasing, stabilization of laser power fluctuations, photon microscopy, solar cells, optical signal reshaping, and nonlinear optics. QDs demonstrate a strong confinement for both electrons and holes due to size effect (quantum confinement). Therefore, nonlinear optical properties are expected to be greatly enhanced in QD media, due to flexibility of their electronic and optical properties that are controlled via choosing the precise size and concentration in the system. This chapter will discuss the physics and capacity of QDs as a nonlinear medium. Introductory idea about nonlinear optics will be outlined. Then, theoretical background of the Z-scan technique used to investigate QDs' third-order nonlinear optical properties will be detailed. As an example, characterization of lead sulfide (PbS)'s nonlinear optical properties will be presented.

In bulk structure of semiconductors, charge carriers (electrons and holes) have continuous distribution of energy states and their motion is not confined. However,

nonlinear index of refraction, nonlinear absorption coefficient, third-order optical susceptibility, optical limiting, optical switching

Properties of Quantum Dots

[26] Estler WT. Uncertainty analysis for angle calibrations using circle closure. Journal of Research of the National Institute of Standards and Technology. 1998;103(2):141-151

[27] Masuda T, Kajitani M. An automatic calibration system for angular encoders. Precision Engineering. 1989;11(2): 95-100

[28] Watanabe T, Fujimoto H, Masuda T. Self-calibratable rotary encoder. In: Proceedings of the 7th Int., Sym, Meas., Techol. Intellig. Instrum. 2005. pp. 240-245

[29] Kim J-A, Kim JW, Kang C-S, Jin J, Eom TB. Precision angle comparator using self-calibration of scale errors based on the equal-division-averaged method. In: MacroScale 2011 – Proceedings. Recent Developments in Traceable Dimensional Measurements. 2011. pp. 1-4

[30] Probst R. Self-calibratable of divided circles on the basis of a prime factor algorithm. Measurement Science and Technology. 2008;19(1):1-11

[31] Just A, Krause M, Probst R, Bosse H, Haunerdinger H, Spaeth C, et al. Comparison of angle standards with the aid of a high-resolution angle encoder. Precision Engineering. 2009;33:530-533

[32] Zhu Y, Hu C, Hu J, Zhang M, Xu D. On-axis self-calibration of precision XYθ<sup>z</sup> metrology systems: An approach framework. In: Proceedings of 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM). 2013. pp. 1078-1083

### Chapter 3

[16] Jeong YH, Dong J, Ferreira PM. Selfcalibration of dual-actuated single-axis nanopositioners. Measurement Science and Technology. 2008;19(4):1-13

Standards, Methods and Solutions of Metrology

[25] Estler T, Queen H. An advanced angle metrology system. Annals of the

[26] Estler WT. Uncertainty analysis for angle calibrations using circle closure. Journal of Research of the National Institute of Standards and Technology.

[27] Masuda T, Kajitani M. An automatic calibration system for angular encoders. Precision Engineering. 1989;11(2):

[28] Watanabe T, Fujimoto H, Masuda T. Self-calibratable rotary encoder. In: Proceedings of the 7th Int., Sym, Meas.,

[29] Kim J-A, Kim JW, Kang C-S, Jin J, Eom TB. Precision angle comparator using self-calibration of scale errors based on the equal-division-averaged method. In: MacroScale 2011 – Proceedings. Recent Developments in Traceable Dimensional Measurements.

Techol. Intellig. Instrum. 2005.

[30] Probst R. Self-calibratable of divided circles on the basis of a prime factor algorithm. Measurement Science and Technology. 2008;19(1):1-11

Haunerdinger H, Spaeth C, et al. Comparison of angle standards with the aid of a high-resolution angle encoder. Precision Engineering. 2009;33:530-533

(AIM). 2013. pp. 1078-1083

[31] Just A, Krause M, Probst R, Bosse H,

[32] Zhu Y, Hu C, Hu J, Zhang M, Xu D. On-axis self-calibration of precision XYθ<sup>z</sup> metrology systems: An approach framework. In: Proceedings of 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics

CIRP. 1993;42(1):573-576

1998;103(2):141-151

95-100

pp. 240-245

2011. pp. 1-4

[17] Ye J, Takac MT, Berglund CN, et al. An exact algorithm for self-calibration of precision metrology stages. Precision

Engineering. 1997;20(1):16-32

Technology. 2008;19(2):1-6

[21] Zhu Y, Hu C, Hu J, Yang K. Accuracy and simplicity oriented self-

calibration approach for twodimensional precision stages. IEEE Transactions on Industrial Electronics.

[22] Dang QC, Yoo S, Kim S-W. Complete 3-D self-calibration of

of the CIRP. 2006;55(1):1-4

coordinate measuring machines. Annals

[23] Hu C, Zhu Y, Hu J, Xu D, Zhang M. A holistic self-calibration approach for determination of three-dimensional stage error. IEEE Transactions on Instrumentation and Measurement.

[24] Lu X-D, Graetz R, Amin-Shahidi D, Smeds K. On-axis self-calibration of angle encoders. CIRP Annals:

Manufacturing Technology. 2010;59(1):

2012;60(6):2264-2272

2013;62(2):483-494

529-534

42

[18] Yoo S, Kim SW. Self-calibration algorithm for testing out-of-plane errors of two-dimensional profiling stages. International Journal of Machine Tools & Manufacture. 2004;44:767-774

[19] Xu M, Dziomba T, Dai G, Koenders L. Self-calibration of scanning probe microscope: Mapping the errors of the instrument. Measurement Science and

[20] Hu C, Zhu Y, Hu J, Zhang M, Xu D. A holistic self-calibration algorithm for XY precision metrology systems. IEEE Transactions on Instrumentation and Measurement. 2012;61(9):2492-2500
