A nonlinear measurement method of polarization aberration in immersion projection optics by spectrum analysis of aerial image

Enze Li; Yanqiu Li; Naiyuan Sheng; Tie Li; Yiyu Sun; Pengzhi Wei

doi:10.1364/OE.26.032743

Optics Express
Vol. 26,
Issue 25,
pp. 32743-32756
(2018)
•https://doi.org/10.1364/OE.26.032743

A nonlinear measurement method of polarization aberration in immersion projection optics by spectrum analysis of aerial image

Enze Li, Yanqiu Li, Naiyuan Sheng, Tie Li, Yiyu Sun, and Pengzhi Wei

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Enze Li, Yanqiu Li, Naiyuan Sheng, Tie Li, Yiyu Sun, and Pengzhi Wei, "A nonlinear measurement method of polarization aberration in immersion projection optics by spectrum analysis of aerial image," Opt. Express 26, 32743-32756 (2018)

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About this Article
History
- Original Manuscript: October 9, 2018
- Revised Manuscript: November 13, 2018
- Manuscript Accepted: November 14, 2018
- Published: November 28, 2018

Abstract

Polarization aberrations (PA) can be presented by Jones pupil and can also impact the imaging performance of immersion projection optics significantly. Precise PA measurement is most important for resolution enhancement technology and holistic lithography at 7nm node and below, in order to improve the pattern fidelity and processing stability. However, the current imaging-based measurement method of PA by linear approximation has not taken the coupling effect of the PA coefficients into account. This paper proposes a nonlinear measurement method of PA based on a rigorous nonlinear model to improve the measurement accuracy significantly. In this invention, the new spectrum modulation theory is developed to establish a rigorous quadratic form of PA and aerial image spectrum. A hybrid genetic algorithm is developed to solve the quadratic form inversely to obtain the PA accurately. An overall simulation validates that this method provides a superior quality of PA measurement with very high precision of $10^{-4} λ$ .

1. Introduction

Polarization aberration (PA) characterizes the changes in the phase, intensity, and polarization of light after passing through the projection optics (PO) [1,2]. It can be represented by the Jones pupil, and can also be expanded into multiple forms, such as orientation Zernike polynomials [3], field-orientation Zernike polynomials [4,5], and pseudo-Zernike polynomials [6]. As the critical dimensions of integrated circuits (IC) continuously shrink to 7 nm and beyond, PA cannot be neglected due to its impact to imaging performance in immersion projection optics (IPO) [7,8]. And PA needs to be measured and conducted to various resolution enhancement technologies such as source and mask optimization (SMO) [9,10], hybrid SMO (HSMO) [11], and source polarization mask optimization (SPMO) [12,13] to improve the pattern fidelity and processing stability. Therefore, there is a need for IPO to develop techniques and systems to accurately measure the PA.

The imaging-based measurements are widely used in aberration measurement for IPO, which have been proposed to measure the wave aberration [14–16] and PA [17]. However, these methods established the relationship between wave aberrations and aerial image errors (e.g focus shift, lateral shift, and image placement error) based on a linear model that neglects the coupling effect of PA coefficients on imaging, thus resulting in theoretical errors. Therefore, SY Liu in [18] proposed an aberration measurement based on a quadratic aberration model to avoid the theoretical errors and provide a more accurate PA estimation. Similarly, it is necessary to establish a PA measurement based on the nonlinear model for a superior quality of PA estimation.

In our previous work [19], we established a small-scale nonlinear model and developed a method to measure the PA Zernike coefficients up to 10th, which provided higher accuracy than linear models. However, there is an incomplete analysis of the spectrum and a low utilization of spectrum information in this model. In addition, the method is used in the measurement of the PA coefficients up to 10th, so when extended it up to 37th, the solution will be more likely to fall into a local minimal value, resulting in the inability to solve the real PA.

In this paper, a nonlinear measurement of PA in IPO by spectrum analysis of aerial image is proposed. Through analysis, the spectrum modulation theory, which is the mechanism of PA impacting the imaging, is obtained to derive a quadratic form about PA coefficients and the aerial image spectrum. Based on this quadratic form, the overdetermined equations can be built by measuring multiple groups of aerial images. the PA coefficients up to the 37th order can be estimated accurately by developing a hybrid genetic algorithm and using it to solve the overdetermined equations inversely. An overall simulation is used to validate the validity and accuracy of the proposed method.

2. Spectrum modulation of PA

A critical part of the imaging-based PA measurement method is to establish the relationship between the PA coefficients and some image information. Therefore, we need to choose a kind of image information with the following characteristic: There is a simple and explicit analytical relationship between the image information and the PA, and this image information is easy to extract from the aerial image with high accuracy. For this purpose, the physical mechanism of the impact of the PA on the imaging needs to be analyzed to find the image information that satisfies these characteristics. In this Section, the vector imaging model is given first in 2.1. In 2.2, the spectrum modulation, which is the mechanism of PA's impact on imaging, is derived, and the aerial image spectrum is selected as the image information.

2.1 Vector imaging model with PA

Under the Abbe imaging principle, the rigorous vector imaging model [20,21] can be expressed in the following form:

\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \iint T (f, g) \cdot K_{f; k} (f, g) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f x + g y + γ z}} d f d g |}^{2} d f_{s} d g_{s}, \end{array}

where

(x_{a}, y_{a})

(f, g)

, and

(f_{s}, g_{s})

are the coordinates of the image plane, the pupil plane, and the source plane, respectively. Then

S (f_{s}, g_{s})

T (f, g)

, and

{\vec{E}}_{i} (f_{s}, g_{s})

are the effective source intensity, the Fourier spectrum of the mask transmission function, and the polarization state of incident light denoted by Jones vector, respectively. And K is the PO part of the lithography system, which can be expressed as:

{\vec{K}}_{f} (f, g) = A (f, g) \cdot M_{o} (f, g) \cdot J (f, g),

where

A (f, g)

M_{o} (f, g)

, and

J (f, g)

are the correction factor, the transfer matrix in the exit pupil of PO, and the PA expressed in form of Jones pupil, respectively. Their specific form is given in Appendix A.

2.2 Spectrum modulation

If a mask with good periodicity is used in imaging process, the Fourier spectrum $T (f, g)$ will be a discrete form. Then the Eq. (1) becomes the sum of each spectrum point of $T (f, g)$ , which can be expressed as:

\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \sum_{f, g = \pm q_{i}} q (f, g) \cdot K_{f; k} (f, g) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f x + g y + γ z}} |}^{2} d f_{s} d g_{s}, \end{array}

where

q_{i}

is the coordinate of the spectrum point of

T (f, g)

and

q (f, g)

is the distribution coefficient of the spectrum. And in Fig. 1, we show the impact mechanism of the PA on imaging, where

\pm t

order is the cutoff diffraction order. It shows that

K_{f; k} (f, g)

, which contains the PA of PO, impacts the aerial image by directly impacting the spectrum points. Therefore, the spectrum of the aerial image contains the PA information of the PO.

Fig. 1 the impact mechanism of the PA on imaging.

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If the analytical relationship between the spectrum of the aerial image and the PA of the PO is established, the PA can be measured by spectrum analysis of aerial image. Since the PA directly impacts the intensity of the imaging spectrum, the relationship between PA and the imaging spectrum can be more succinct and rigorous than the relationship between PA and some imaging errors (e.g focus shift, lateral shift, and image placement error). In addition, the spectrum analysis method extracts the period information of the image, so many random systematic errors can be eliminated by this extraction method, and their impact on accuracy is reduced to a low level.

3. The nonlinear measurement of PA

It shows that, in Eq. (3), the Fourier spectrum of the mask transmission function $T (f, g)$ directly determines the analytical form of the lithography system, so selecting and constructing a reasonable mask can reduce the computational complexity and improve the measurement accuracy. In this paper, we select the one-dimensional dense line mask for the derivation of the theoretical model for the test mask. In the follow-up work, we will also select or construct other masks that have better characteristics in Fourier spectrum for this theory.

For the one-dimensional dense line masks (binary mask, alternating phase shift mask, and attenuation phase shift mask), their Fourier spectrums can be uniformly written in the following form:

\begin{matrix} T (f, g) = q_{0} δ (0, 0) + q_{1} δ (f_{+ 1}, g_{+ 1}) + q_{1} δ (- f_{+ 1}, - g_{+ 1}) + \dots \\ + q_{t} δ (t f_{+ 1}, t g_{+ 1}) + q_{t} δ (- t f_{+ 1}, - t g_{+ 1}), \end{matrix}

where the cutoff diffraction order t can be expressed as

{\begin{matrix} t^{2} f_{_{+ 1}}^{2} + t^{2} g_{_{+ 1}}^{2} \leq {(N A / λ_{0})}^{2} \\ {(t + 1)}^{2} f_{_{+ 1}}^{2} + {(t + 1)}^{2} g_{_{+ 1}}^{2} > {(N A / λ_{0})}^{2} \end{matrix}, t \in N_{+} .

That is, diffraction orders higher than t will not be received by the pupil and have no contribute to imaging. Then, substitute Eq. (4) into Eq. (3):

\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \begin{array}{l} q_{0} \cdot K_{f; k} (0, 0) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \\ + q_{1} \cdot K_{f; k} (f_{+ 1}, g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f_{+ 1} \cdot x + g_{+ 1} \cdot y + γ z}} \\ + q_{1} \cdot K_{f; k} (- f_{+ 1}, - g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- f_{+ 1} \cdot x - g_{+ 1} \cdot y + γ z}} \\ + \dots \\ + q_{t} \cdot K_{f; k} (t f_{+ 1}, t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ + q_{t} \cdot K_{f; k} (- t f_{+ 1}, - t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{array} |}^{2} d f_{s} d g_{s} . \end{array}

It can be seen that this method does not limit the diffraction mode to three-beam interference. When t = 2 or more, the diffraction mode can be five or more beam interference. This can effectively avoid the introduction of the three-dimensional photomask topography effects and reduce the measurement error.

Traditional coherent illumination with X and Y polarization states is choose in this theory, so the effective light sources $S$ can be formulated using the delta function:

S (f_{s}, g_{s}) = δ (f_{s}, g_{s}),

and

{\vec{E}}_{i} (f_{s}, g_{s})

can be formulated as

{\vec{E}}_{i} (f_{s}, g_{s}) = [\begin{matrix} E_{i, x} (f_{s}, g_{s}) \\ E_{i, y} (f_{s}, g_{s}) \end{matrix}] = {\begin{matrix} {[1, 0]}^{T}, X polarization \\ {[0, 1]}^{T}, Y polarization \end{matrix} .

Substituting Eq. (7) into Eq. (6), the discrete form of the aerial image is obtained:

\begin{array}{l} I (x_{a}, y_{a}) = \\ \sum_{k = x, y, z} {| \begin{array}{l} q_{0} \cdot K_{f; k} (0, 0) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \\ + q_{1} \cdot K_{f; k} (f_{+ 1}, g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f_{+ 1} \cdot x + g_{+ 1} \cdot y + γ z}} \\ + q_{1} \cdot K_{f; k} (- f_{+ 1}, - g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- f_{+ 1} \cdot x - g_{+ 1} \cdot y + γ z}} \\ + \dots \\ + q_{t} \cdot K_{f; k} (t f_{+ 1}, t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ + q_{t} \cdot K_{f; k} (- t f_{+ 1}, - t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{array} |}^{2} . \end{array}

From Eq. (8) and Eq. (9), it can be seen that only $J_{x x} (f, g)$ and $J_{y x} (f, g)$ contribute to the imaging in X-polarized illumination, so we can get the relationship between $a_{n}^{m}, b_{n}^{m}$ and $I (x_{a}, y_{a})$ formulated according to Eq. (9):

\begin{matrix} I (x_{a}, y_{a}) = I_{1} + I_{2} + I_{3} \\ = {| C_{11} \cdot a + C_{12} \cdot b |}^{2} + {| C_{21} \cdot a + C_{22} \cdot b |}^{2} + {| C_{31} \cdot a + C_{32} \cdot b |}^{2} \\ = [a^{H}, b^{H}] \cdot \sum_{k = 1, 2, 3} [\begin{matrix} C_{k 1}^{H} \cdot C_{k 1} & C_{k 1}^{H} \cdot C_{k 2} \\ C_{k 2}^{H} \cdot C_{k 1} & C_{k 2}^{H} \cdot C_{k 2} \end{matrix}] \cdot [\begin{matrix} a \\ b \end{matrix}] \\ = [a^{H}, b^{H}] \cdot S_{x - p o l} \cdot [\begin{matrix} a \\ b \end{matrix}], \end{matrix}

where

{\begin{matrix} a = {[a_{1}, a_{2}, \dots, a_{j \max}]}^{T} \\ b = {[b_{1}, b_{2}, \dots, b_{j \max}]}^{T} \end{matrix}, j \max is the expasion order .

Similarly in Y-polarized illumination, only

J_{x y} (f, g)

and

J_{y y} (f, g)

contribute to imaging, and the relationship between

a'_{n}^{m}, b'_{n}^{m}

and

I (x_{a}, y_{a})

has a similar form to X-polarized illumination:

\begin{matrix} I (x_{a}, y_{a}) = I'_{1} + I'_{2} + I'_{3} \\ = {| C'_{11} \cdot a' + C'_{12} \cdot b' |}^{2} + {| C'_{21} \cdot a' + C'_{22} \cdot b' |}^{2} + {| C'_{31} \cdot a' + C'_{32} \cdot b' |}^{2} \\ = [a'^{H}, b'^{H}] \cdot \sum_{k = 1, 2, 3} [\begin{matrix} C'_{k 1}^{H} \cdot C'_{k 1} & C'_{k 1}^{H} \cdot C'_{k 2} \\ C'_{k 2}^{H} \cdot C'_{k 1} & C'_{k 2}^{H} \cdot C'_{k 2} \end{matrix}] \cdot [\begin{matrix} a' \\ b' \end{matrix}] \\ = [a'^{H}, b'^{H}] \cdot S_{y - p o l} \cdot [\begin{matrix} a' \\ b' \end{matrix}] . \end{matrix}

where

{\begin{matrix} a' = {[a'_{1}, a'_{2}, \dots, a'_{j \max}]}^{T} \\ b' = {[b'_{1}, b'_{2}, \dots, b'_{j \max}]}^{T} \end{matrix}, j \max is the expasion order .

The form of

C_{11}

C_{12}

C_{21}

C_{22}

C_{31}

C_{32}

and

C'_{11}

C'_{12}

C'_{21}

C'_{22}

C'_{31}

C'_{32}

are given in Appendix B. From Eq. (10) and Eq. (12), the following conclusion is obtained: The relation between PA coefficients and aerial images is quadratic and can be written as follows

I (x_{a}, y_{a}) = p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i}, w h e r e, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}] .

In order to obtain the relation between the aerial image spectrum and the PA coefficients, a Fourier transformation is carried out to Eq. (14):

\begin{matrix} \tilde{I} (f, g) = F {I (x_{a}, y_{a})} \\ = F {p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i}}, w h e r e, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}] . \\ = p_{i}^{H} \cdot F {S_{i - p o l}} \cdot p_{i} \\ = p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i} \end{matrix}

This is the quadratic form of the spectrum of the aerial image and the PA coefficients, and $S_{i - p o l}$ is its sensitivity matrix. The form of $\tilde{I} (f, g)$ is discrete due to the periodicity of the aerial image $I (x_{a}, y_{a})$ , and the Eq. (15) shows that each matrix element of $S_{i - p o l}$ has a same form as $\tilde{I} (f, g)$ . Therefore, each of its elements can be expressed as:

\begin{matrix} s_{n, m} = s_{0; n, m} \cdot δ (0, 0) + s_{+ 1; n, m} \cdot δ (f_{+ 1}, g_{+ 1}) + s_{- 1; n, m} \cdot δ (- f_{+ 1}, - g_{+ 1}) \\ + s_{+ 2; n, m} \cdot δ (2 f_{+ 1}, 2 g_{+ 1}) + s_{- 2; n, m} \cdot δ (-2 f_{+ 1}, - 2 g_{+ 1}) + \dots . \end{matrix}

Each order of the spectrum can be selected as the feature information to establish its relationship with the PA coefficients, and according to the Eq. (16), the sensitivity matrix of

l

order of spectrum

S_{l}

can be defined as a matrix composed of

s_{l; n, m}

. Therefore, the relationship between PA and any order spectrum of aerial image can be expressed as

\tilde{I} (f_{l}, g_{l}) = p_{i}^{H} \cdot S_{l; i - p o l} \cdot p_{i}; where, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}], l = 0, + 1, - 1, + 2, - 2 \dots .

Based on this quadratic form, the overdetermined equations can be built by measuring multiple groups of aerial images, and the PA can be obtained by solving the equations reversely.

4. Hybrid genetic algorithm

This set of overdetermined equations is a nonlinear problem, which is easy to fall into a local minimum points when solved by some classical algorithms, such as gradient descent, nonlinear least-squares algorithm, etc. Therefore, a hybrid genetic algorithm is developed to improve the convergence speed and the calculation accuracy. This algorithm is a hybrid of genetic algorithm and classical algorithm, and draws on the advantages of both algorithms. The specific process of this algorithm is shown in Fig. 2. The initial population is set according to the initial value of PA and the error range, and the largescale genetic algorithm is used to guarantee that the final solution is the global minimum solution. The classical algorithm is used to accelerate the evolution speed of excellent individuals to improve the convergence speed and accuracy of the algorithm. The small scale genetic algorithm and optimization of optimal term are designed to reduce the number of populations to speed up the convergence of the algorithm when some individuals in the populations are very close to the target solution. Therefore, this algorithm can solve the overdetermined equations to estimate the PA quickly and accurately.

Fig. 2 Process of the hybrid genetic algorithm.

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5. Simulations

This section presents an overall simulation to verify this method in measuring the PA coefficients up to the 37th order of an arbitrary field of view in PO, and compare it to the methods based on liner approximation models. The simulation is divided into two parts, namely the imaging measurement simulation and the PA solution simulation. Figure 3 shows the process of the simulation.

Fig. 3 process of the simulation experiment.

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In the first part, the design value of PA is obtained by the ray tracing of CODE V to an arbitrary field of view in PO designed by the laboratory, as shown in Fig. 4. Then, adding a random error to this design value represents the deviation of the true value of the PA from the design value during the production, assembly, and use of the PO. We use this as the true value of the PA of the PO in this simulation. Take it into the vector lithography imaging model and solve its aerial image. And Fourier transform it to obtain the observations (spectrum of aerial image) required by this theory. Change the mask pitch, angle and other parameters, repeat the above process to obtain a set of observations of the lens at this field of view.

Fig. 4 PA of our designed immersion optics extracted from ray tracing by code V.

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In the second part, the spectrum of aerial image is input to the proposed PA measurement method to obtain the PA expansion coefficients up to the 37th order. Table 1 shows some of the simulation parameters. Figure 5 gives the comparison of the PA obtained by this method with the true values, and Fig. 6 shows the errors of measurements relative to true values.

Table 1. simulation parameters.

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Fig. 5 the comparison of the measurements with the true values. (a) is obtained under X-polarized illumination, and (b) is obtained under Y-polarized illumination.

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Fig. 6 the errors between the measurements and the true values.

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As can be seen from the Figs. 5(a) and 5(b) and Fig. 6, the measurements obtained by this method has a high accuracy. The errors of all PA coefficients are $10^{-4} λ$ orders of magnitude, and most of them are $10^{-5} λ$ orders of magnitude. In contrast, the linear approximation method proposed in [17] established the linear relationship between Zernike coefficients of PA and the phase shift and intensity distribution of aerial image with first order approximation. By measuring the aerial images of test masks in different orientations and pitches under different illumination settings, the PA coefficients can be obtained with errors of $10^{-2} λ$ to $10^{-3} λ$ orders of magnitude. Therefore, compared to this linear approximation method, the accuracy of the proposed method is improved by two orders of magnitude. Convert these PA expansion coefficients to the Jones pupil, and the root mean square error (RMSE) of the measured pupil and true pupil are shown in Table 2. It can be seen that the RMSE of Jones pupils obtained by proposed method are $10^{-4} λ$ orders of magnitude, which is also one orders of magnitude smaller than the linear approximation method. Therefore, this method provides a superior quality estimation of the PA expansion coefficients up to the 37th order.

Table 2. The RMSE of measured pupil and true pupil.

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6. Conclusion

A nonlinear measurement of PA in IPO by spectrum analysis of aerial image has been proposed to improve the quality of PA estimation. The spectrum modulation that is the mechanism of PA impacting imaging is unearthed, and the quadratic form that is the sensitivity matrix of PA coefficients and the aerial image spectrum is derived. The overdetermined equations can be built by measuring multiple groups of aerial images, and by using the hybrid genetic algorithm we developed, the PA coefficients up to the 37th order are determined by solving the overdetermined equations in reverse. An overall PA measurement simulation proves that the errors of PA coefficients are $10^{-4} λ$ to $10^{-5} λ$ orders of magnitude and the RMSE of Jones pupil are $10^{-4} λ$ orders of magnitude. It shows that the method has a very high accuracy, which is one or two orders of magnitude higher than the linear approximation method.

Appendix A Specific analysis form of each item in ${\vec{K}}_{f} (f, g)$

As presented in Section 2, the PO part of the lithography system can be described as

{\vec{K}}_{f} (f, g) = A (f, g) \cdot M_{o} (f, g) \cdot J (f, g),

where

A (f, g)

is the corrections of the radiometric correction factor which can be described as

A (f, g) = M \cdot \sqrt{\frac{n_{1} \sqrt{1 - {(\frac{M λ_{0} f}{n_{1}})}^{2} - {(\frac{M λ_{0} g}{n_{1}})}^{2}}}{n_{2} \sqrt{1 - {(\frac{λ_{0} f}{n_{2}})}^{2} - {(\frac{λ_{0} g}{n_{2}})}^{2}}}},

M_{o} (f, g)

is the direction of the entrance and exit electric vector of the large NA PO which can be described as

M_{o} (f, g) = [\begin{matrix} 1 - \frac{α^{2}}{1 + κ} & - \frac{α \cdot β}{1 + κ} \\ - \frac{α \cdot β}{1 + κ} & 1 - \frac{β^{2}}{1 + κ} \\ - α & - β \end{matrix}],

where

{\begin{cases} α = \frac{λ_{0}}{n_{2}} f \\ β = \frac{λ_{0}}{n_{2}} g \\ κ = \sqrt{1 - α^{2} - β^{2}} \end{cases},

and

J (f, g)

is the PA expressed in form of Jones's pupil which can be described as

J (f, g) = {\begin{cases} [\begin{matrix} J_{x x} (f, g) & J_{x y} (f, g) \\ J_{y x} (f, g) & J_{y y} (f, g) \end{matrix}], f^{2} + g^{2} \leq {(\frac{N A}{λ_{0}})}^{2} \\ [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \end{cases} .

Otherwise, we choose pseudo-Zernike basis [6] to expand it. pseudo-Zernike basis can be described as

P Z_{n}^{m} (ρ, θ) = R_{n}^{| m |} (ρ) \cdot \exp {i m θ} .

And then, the Jones pupil

J (f, g)

can be represented as

\begin{matrix} J (f, g) = [\begin{matrix} J_{x x} (f, g) & J_{x y} (f, g) \\ J_{y x} (f, g) & J_{y y} (f, g) \end{matrix}] \\ = [\begin{matrix} A_{x x} (ρ, θ) \cdot \exp {i Θ_{x x} (ρ, θ)} & A_{x y} (ρ, θ) \cdot \exp {i Θ_{x y} (ρ, θ)} \\ A_{y x} (ρ, θ) \cdot \exp {i Θ_{y x} (ρ, θ)} & A_{y y} (ρ, θ) \cdot \exp {i Θ_{y y} (ρ, θ)} \end{matrix}] \\ = [\begin{matrix} \sum_{m, n} a_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) & \sum_{m, n} b'_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) \\ \sum_{m, n} b_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) & \sum_{m, n} a'_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) \end{matrix}], \end{matrix}

where

a_{n}^{m}, b_{n}^{m}, a'_{n}^{m}, b'_{n}^{m}

are unknown PA coefficients to be measured. In fringe labeling, these PA coefficients also can be re-sort and labeling in one subscript

{a_{i}, b_{i}, a'_{i}, b'_{i} | i \in {1, 2, 3, \dots}} .

Appendix B Specific analysis form of $C_{11}$ , $C_{12}$ , $C_{21}$ , $C_{22}$ , $C_{31}$ , $C_{32}$ and $C'_{11}$ , $C'_{12}$ , $C'_{21}$ , $C'_{22}$ , $C'_{31}$ , $C'_{32}$

In Eq. (10) and Eq. (12) in Section 3, we give the relationship between PA coefficients and areal image. The specific forms of $C_{11}$ , $C_{12}$ , $C_{21}$ , $C_{22}$ , $C_{31}$ , and $C_{32}$ are:

{\begin{matrix} C_{11} (i) = A (0, 0) \cdot δ_{m 0} \cdot q_{0} \cdot {(-1)}^{i - 1} \\ + A (f_{+ 1}, g_{+ 1}) \cdot C_{1} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{1} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{21} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{2} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{2} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{31} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{3} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {- Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{3} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} {- Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \end{matrix},

and

{\begin{matrix} C_{12} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{2} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{2} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{22} (i) = A (0, 0) \cdot δ_{m 0} \cdot q_{0} \cdot {(-1)}^{i - 1} \\ + A (f_{+ 1}, g_{+ 1}) \cdot C_{4} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{4} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{32} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{5} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {- Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{5} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} {- Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \end{matrix},

where

i \in {1, 2, \dots, j \max},

{\begin{matrix} E X P (t) = e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ E X P (- t) = e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{matrix}, t \in N_{+},

{\begin{matrix} Z_{i} (t) = P Z_{n}^{m} (t \frac{λ_{0} \cdot f_{+ 1}}{N A}, t \frac{λ_{0} \cdot g_{+ 1}}{N A}) \\ Z_{i} (- t) = P Z_{n}^{m} (- t \frac{λ_{0} \cdot f_{+ 1}}{N A}, - t \frac{λ_{0} \cdot g_{+ 1}}{N A}) \end{matrix}, t \in N_{+},

{\begin{cases} C_{1} (f, g) = 1 - \frac{α^{2}}{1 + κ} \\ C_{2} (f, g) = - \frac{α \cdot β}{1 + κ} \\ C_{4} (f, g) = 1 - \frac{β^{2}}{1 + κ} \end{cases}; {\begin{cases} C_{3} (f, g) = α \\ C_{5} (f, g) = β \end{cases},

C'_{11}

C'_{12}

C'_{21}

C'_{22}

C'_{31}

,and

C'_{32}

are the same vectors as

C_{11}

C_{12}

C_{21}

C_{22}

C_{31}

, and

C_{32}

in form and can be obtained by simply swapping α and β in the latter formula.

Funding

General Program of National Natural Science Foundation of China (No. 61675026); Major Scientific Instrument Development Project of National Natural Science Foundation of China (No. 11627808); National Science and Technology Major Project (No. 2017ZX02101006-001).

References

1. J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33(22), 5080–5100 (1994). [CrossRef] [PubMed]

2. J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 2. Tilted and decentered optical systems,” Appl. Opt. 33(22), 5101–5107 (1994). [CrossRef] [PubMed]

3. J. Ruoff and M. Totzeck, “Orientation Zernike polynomials: a useful way to describe the polarizationeffects of optical imaging systems,” J. Microlith., Microfabr. Microsyst. 8, 031404 (2009).

4. X. Xu, W. Huang, and M. Xu, “Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems,” Opt. Express 23(21), 27911–27919 (2015). [CrossRef] [PubMed]

5. X. Xu, W. Huang, and M. Xu, “Orthonormal polynomials describing polarization aberration for M-fold optical systems,” Opt. Express 24(5), 4906–4912 (2016). [CrossRef] [PubMed]

6. J. Haddadnia, M. Ahmadi, and K. Faez, “An efficient feature extraction method with pseudo-Zernike moment in RBF neural network-based human face recognition system,” EURASIP J. Adv. Signal Process. 2003(9), 267692 (2003). [CrossRef]

7. N. Yamamoto, J. Kye, and H. J. Levinson, “Polarization aberration analysis using Pauli-Zernike representation,” Proc. SPIE 6520, 65200Y (2007). [CrossRef]

8. Y. Li, X. Guo, X. Liu, and L. Liu, “A technique for extracting and analyzing the polarization aberration of hyper-numerical aperture image optics,” Proc. SPIE 9042, 904204 (2013). [CrossRef]

9. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013). [CrossRef] [PubMed]

10. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014). [CrossRef] [PubMed]

11. X. Ma, C. Han, Y. Li, B. Wu, Z. Song, L. Dong, and G. R. Arce, “Hybrid source mask optimization for robust immersion lithography,” Appl. Opt. 52(18), 4200–4211 (2013). [CrossRef] [PubMed]

12. X. Ma, J. Gao, C. Han, Y. Li, L. Dong, and L. Liu, “Efficient source polarization optimization for robust optical lithography,” Proc. SPIE 9052, 90520T (2014). [CrossRef]

13. X. Ma, L. Dong, C. Han, J. Gao, Y. Li, and R. G. Arce, “Gradient-based joint source polarization mask optimization for optical lithography,” J. Micro/Nanolith. MEMS MOEMS 14(2), 023504 (2015). [CrossRef]

14. B. Zhu, X. Wang, S. Li, G. Yan, L. Shen, and L. Duan, “Wavefront aberration measurement method for a hyper-NA lithographic projection lens based on principal component analysis of an aerial image,” Appl. Opt. 55(12), 3192–3198 (2016). [CrossRef] [PubMed]

15. H. van der Laan, M. Dierichs, H. van Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001). [CrossRef]

16. Z. Qiu, X. Wang, Q. Yuan, and F. Wang, “Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width,” Appl. Opt. 48(2), 261–269 (2009). [CrossRef] [PubMed]

17. L. Dong, Y. Li, X. Dai, H. Liu, and K. Liu, “Measuring the polarization aberration of hyper-NA lens from the vector aerial image,” Proc. SPIE 9283, 928313 (2014). [CrossRef]

18. S. Liu, S. Xu, X. Wu, and W. Liu, “Iterative method for in situ measurement of lens aberrations in lithographic tools using CTC-based quadratic aberration model,” Opt. Express 20(13), 14272–14283 (2012). [CrossRef] [PubMed]

19. Z. Xiang and Y. Li, “Retrieve polarization aberration from image degradation: a new measurement method in DUV lithography,” Proc. SPIE 10460, 84 (2017). [CrossRef]

20. X. Ma, Y. Li, and L. Dong, “Mask optimization approaches in optical lithography based on a vector imaging model,” J. Opt. Soc. Am. A 29(7), 1300–1312 (2012). [CrossRef] [PubMed]

21. D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE 7640, 76402Y (2010). [CrossRef]

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Fig. 1 the impact mechanism of the PA on imaging.

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Fig. 2 Process of the hybrid genetic algorithm.

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Fig. 3 process of the simulation experiment.

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Fig. 4 PA of our designed immersion optics extracted from ray tracing by code V.

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Fig. 5 the comparison of the measurements with the true values. (a) is obtained under X-polarized illumination, and (b) is obtained under Y-polarized illumination.

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Fig. 6 the errors between the measurements and the true values.

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Tables (2)

Table 1 simulation parameters.

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Table 2 The RMSE of measured pupil and true pupil.

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Equations (31)

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\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \iint T (f, g) \cdot K_{f; k} (f, g) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f x + g y + γ z}} d f d g |}^{2} d f_{s} d g_{s}, \end{array}

{\vec{K}}_{f} (f, g) = A (f, g) \cdot M_{o} (f, g) \cdot J (f, g),

\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \sum_{f, g = \pm q_{i}} q (f, g) \cdot K_{f; k} (f, g) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f x + g y + γ z}} |}^{2} d f_{s} d g_{s}, \end{array}

\begin{matrix} T (f, g) = q_{0} δ (0, 0) + q_{1} δ (f_{+ 1}, g_{+ 1}) + q_{1} δ (- f_{+ 1}, - g_{+ 1}) + \dots \\ + q_{t} δ (t f_{+ 1}, t g_{+ 1}) + q_{t} δ (- t f_{+ 1}, - t g_{+ 1}), \end{matrix}

{\begin{matrix} t^{2} f_{_{+ 1}}^{2} + t^{2} g_{_{+ 1}}^{2} \leq {(N A / λ_{0})}^{2} \\ {(t + 1)}^{2} f_{_{+ 1}}^{2} + {(t + 1)}^{2} g_{_{+ 1}}^{2} > {(N A / λ_{0})}^{2} \end{matrix}, t \in N_{+} .

\begin{array}{l} I (x_{a}, y_{a}) \\ = \iint S (f_{s}, g_{s}) \cdot \sum_{k = x, y, z} {| \begin{array}{l} q_{0} \cdot K_{f; k} (0, 0) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \\ + q_{1} \cdot K_{f; k} (f_{+ 1}, g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f_{+ 1} \cdot x + g_{+ 1} \cdot y + γ z}} \\ + q_{1} \cdot K_{f; k} (- f_{+ 1}, - g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- f_{+ 1} \cdot x - g_{+ 1} \cdot y + γ z}} \\ + \dots \\ + q_{t} \cdot K_{f; k} (t f_{+ 1}, t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ + q_{t} \cdot K_{f; k} (- t f_{+ 1}, - t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{array} |}^{2} d f_{s} d g_{s} . \end{array}

S (f_{s}, g_{s}) = δ (f_{s}, g_{s}),

{\vec{E}}_{i} (f_{s}, g_{s}) = [\begin{matrix} E_{i, x} (f_{s}, g_{s}) \\ E_{i, y} (f_{s}, g_{s}) \end{matrix}] = {\begin{matrix} {[1, 0]}^{T}, X polarization \\ {[0, 1]}^{T}, Y polarization \end{matrix} .

\begin{array}{l} I (x_{a}, y_{a}) = \\ \sum_{k = x, y, z} {| \begin{array}{l} q_{0} \cdot K_{f; k} (0, 0) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \\ + q_{1} \cdot K_{f; k} (f_{+ 1}, g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {f_{+ 1} \cdot x + g_{+ 1} \cdot y + γ z}} \\ + q_{1} \cdot K_{f; k} (- f_{+ 1}, - g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- f_{+ 1} \cdot x - g_{+ 1} \cdot y + γ z}} \\ + \dots \\ + q_{t} \cdot K_{f; k} (t f_{+ 1}, t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ + q_{t} \cdot K_{f; k} (- t f_{+ 1}, - t g_{+ 1}) \cdot {\vec{E}}_{i} (f_{s}, g_{s}) \cdot e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{array} |}^{2} . \end{array}

\begin{matrix} I (x_{a}, y_{a}) = I_{1} + I_{2} + I_{3} \\ = {| C_{11} \cdot a + C_{12} \cdot b |}^{2} + {| C_{21} \cdot a + C_{22} \cdot b |}^{2} + {| C_{31} \cdot a + C_{32} \cdot b |}^{2} \\ = [a^{H}, b^{H}] \cdot \sum_{k = 1, 2, 3} [\begin{matrix} C_{k 1}^{H} \cdot C_{k 1} & C_{k 1}^{H} \cdot C_{k 2} \\ C_{k 2}^{H} \cdot C_{k 1} & C_{k 2}^{H} \cdot C_{k 2} \end{matrix}] \cdot [\begin{matrix} a \\ b \end{matrix}] \\ = [a^{H}, b^{H}] \cdot S_{x - p o l} \cdot [\begin{matrix} a \\ b \end{matrix}], \end{matrix}

{\begin{matrix} a = {[a_{1}, a_{2}, \dots, a_{j \max}]}^{T} \\ b = {[b_{1}, b_{2}, \dots, b_{j \max}]}^{T} \end{matrix}, j \max is the expasion order .

\begin{matrix} I (x_{a}, y_{a}) = I'_{1} + I'_{2} + I'_{3} \\ = {| C'_{11} \cdot a' + C'_{12} \cdot b' |}^{2} + {| C'_{21} \cdot a' + C'_{22} \cdot b' |}^{2} + {| C'_{31} \cdot a' + C'_{32} \cdot b' |}^{2} \\ = [a'^{H}, b'^{H}] \cdot \sum_{k = 1, 2, 3} [\begin{matrix} C'_{k 1}^{H} \cdot C'_{k 1} & C'_{k 1}^{H} \cdot C'_{k 2} \\ C'_{k 2}^{H} \cdot C'_{k 1} & C'_{k 2}^{H} \cdot C'_{k 2} \end{matrix}] \cdot [\begin{matrix} a' \\ b' \end{matrix}] \\ = [a'^{H}, b'^{H}] \cdot S_{y - p o l} \cdot [\begin{matrix} a' \\ b' \end{matrix}] . \end{matrix}

{\begin{matrix} a' = {[a'_{1}, a'_{2}, \dots, a'_{j \max}]}^{T} \\ b' = {[b'_{1}, b'_{2}, \dots, b'_{j \max}]}^{T} \end{matrix}, j \max is the expasion order .

I (x_{a}, y_{a}) = p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i}, w h e r e, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}] .

\begin{matrix} \tilde{I} (f, g) = F {I (x_{a}, y_{a})} \\ = F {p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i}}, w h e r e, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}] . \\ = p_{i}^{H} \cdot F {S_{i - p o l}} \cdot p_{i} \\ = p_{i}^{H} \cdot S_{i - p o l} \cdot p_{i} \end{matrix}

\begin{matrix} s_{n, m} = s_{0; n, m} \cdot δ (0, 0) + s_{+ 1; n, m} \cdot δ (f_{+ 1}, g_{+ 1}) + s_{- 1; n, m} \cdot δ (- f_{+ 1}, - g_{+ 1}) \\ + s_{+ 2; n, m} \cdot δ (2 f_{+ 1}, 2 g_{+ 1}) + s_{- 2; n, m} \cdot δ (-2 f_{+ 1}, - 2 g_{+ 1}) + \dots . \end{matrix}

\tilde{I} (f_{l}, g_{l}) = p_{i}^{H} \cdot S_{l; i - p o l} \cdot p_{i}; where, i = x o r y, p_{x} = [\begin{matrix} a \\ b \end{matrix}], p_{y} = [\begin{matrix} a' \\ b' \end{matrix}], l = 0, + 1, - 1, + 2, - 2 \dots .

{\vec{K}}_{f} (f, g) = A (f, g) \cdot M_{o} (f, g) \cdot J (f, g),

A (f, g) = M \cdot \sqrt{\frac{n_{1} \sqrt{1 - {(\frac{M λ_{0} f}{n_{1}})}^{2} - {(\frac{M λ_{0} g}{n_{1}})}^{2}}}{n_{2} \sqrt{1 - {(\frac{λ_{0} f}{n_{2}})}^{2} - {(\frac{λ_{0} g}{n_{2}})}^{2}}}},

M_{o} (f, g) = [\begin{matrix} 1 - \frac{α^{2}}{1 + κ} & - \frac{α \cdot β}{1 + κ} \\ - \frac{α \cdot β}{1 + κ} & 1 - \frac{β^{2}}{1 + κ} \\ - α & - β \end{matrix}],

{\begin{cases} α = \frac{λ_{0}}{n_{2}} f \\ β = \frac{λ_{0}}{n_{2}} g \\ κ = \sqrt{1 - α^{2} - β^{2}} \end{cases},

J (f, g) = {\begin{cases} [\begin{matrix} J_{x x} (f, g) & J_{x y} (f, g) \\ J_{y x} (f, g) & J_{y y} (f, g) \end{matrix}], f^{2} + g^{2} \leq {(\frac{N A}{λ_{0}})}^{2} \\ [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \end{cases} .

P Z_{n}^{m} (ρ, θ) = R_{n}^{| m |} (ρ) \cdot \exp {i m θ} .

\begin{matrix} J (f, g) = [\begin{matrix} J_{x x} (f, g) & J_{x y} (f, g) \\ J_{y x} (f, g) & J_{y y} (f, g) \end{matrix}] \\ = [\begin{matrix} A_{x x} (ρ, θ) \cdot \exp {i Θ_{x x} (ρ, θ)} & A_{x y} (ρ, θ) \cdot \exp {i Θ_{x y} (ρ, θ)} \\ A_{y x} (ρ, θ) \cdot \exp {i Θ_{y x} (ρ, θ)} & A_{y y} (ρ, θ) \cdot \exp {i Θ_{y y} (ρ, θ)} \end{matrix}] \\ = [\begin{matrix} \sum_{m, n} a_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) & \sum_{m, n} b'_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) \\ \sum_{m, n} b_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) & \sum_{m, n} a'_{n}^{m} \cdot P Z_{n}^{m} (ρ, θ) \end{matrix}], \end{matrix}

{a_{i}, b_{i}, a'_{i}, b'_{i} | i \in {1, 2, 3, \dots}} .

{\begin{matrix} C_{11} (i) = A (0, 0) \cdot δ_{m 0} \cdot q_{0} \cdot {(-1)}^{i - 1} \\ + A (f_{+ 1}, g_{+ 1}) \cdot C_{1} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{1} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{21} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{2} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{2} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{31} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{3} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {- Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{3} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} {- Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \end{matrix},

{\begin{matrix} C_{12} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{2} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{2} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{22} (i) = A (0, 0) \cdot δ_{m 0} \cdot q_{0} \cdot {(-1)}^{i - 1} \\ + A (f_{+ 1}, g_{+ 1}) \cdot C_{4} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{4} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} \cdot {Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \\ C_{32} (i) = A (f_{+ 1}, g_{+ 1}) \cdot C_{5} (f_{+ 1}, g_{+ 1}) \cdot q_{1} \cdot {- Z_{i} (1) \cdot E X P (1) + Z_{i} (- 1) E X P (- 1)} + \dots \\ + A (t f_{+ 1}, t g_{+ 1}) \cdot C_{5} (t f_{+ 1}, t g_{+ 1}) \cdot q_{t} {- Z_{i} (t) \cdot E X P (t) + Z_{i} (- t) E X P (- t)} \end{matrix},

i \in {1, 2, \dots, j \max},

{\begin{matrix} E X P (t) = e^{2 π i {t f_{+ 1} \cdot x + t g_{+ 1} \cdot y + γ z}} \\ E X P (- t) = e^{2 π i {- t f_{+ 1} \cdot x - t g_{+ 1} \cdot y + γ z}} \end{matrix}, t \in N_{+},

{\begin{matrix} Z_{i} (t) = P Z_{n}^{m} (t \frac{λ_{0} \cdot f_{+ 1}}{N A}, t \frac{λ_{0} \cdot g_{+ 1}}{N A}) \\ Z_{i} (- t) = P Z_{n}^{m} (- t \frac{λ_{0} \cdot f_{+ 1}}{N A}, - t \frac{λ_{0} \cdot g_{+ 1}}{N A}) \end{matrix}, t \in N_{+},

{\begin{cases} C_{1} (f, g) = 1 - \frac{α^{2}}{1 + κ} \\ C_{2} (f, g) = - \frac{α \cdot β}{1 + κ} \\ C_{4} (f, g) = 1 - \frac{β^{2}}{1 + κ} \end{cases}; {\begin{cases} C_{3} (f, g) = α \\ C_{5} (f, g) = β \end{cases},

Parameters	Configuration	Parameters	Configuration
NA	1.35	Test mask pitch	600 ~1100nm
λ	193.368nm	Test mask orientation	$0 °, 22.5 °, 45 °, 67.5 °,$ $90 °, 112.5 °, 135 °, 157.5 °$
Defocus, z	0nm, 50nm	Illumination type	Traditional coherent
Test mask type	Binary	source polarization	X and Y

Jones pupil	linear approximation method (λ)	Proposed method (λ)
Re(Jxx)	6.35 $\times 10^{- 3}$	4.83 $\times 10^{- 4}$
Im(Jxx)	3.55 $\times 10^{- 3}$	4.77 $\times 10^{- 4}$
Re(Jxy)	5.00 $\times 10^{- 3}$	4.51 $\times 10^{- 4}$
Im(Jxy)	3.73 $\times 10^{- 3}$	4.43 $\times 10^{- 4}$
Re(Jyx)	5.00 $\times 10^{- 3}$	2.86 $\times 10^{- 4}$
Im(Jyx)	3.73 $\times 10^{- 3}$	3.23 $\times 10^{- 4}$
Re(Jyy)	7.12 $\times 10^{- 3}$	2.71 $\times 10^{- 4}$
Im(Jyy)	3.51 $\times 10^{- 3}$	3.21 $\times 10^{- 4}$
average	4.92 $\times 10^{- 3}$	3.82 $\times 10^{- 4}$

Parameters	Configuration	Parameters	Configuration
NA	1.35	Test mask pitch	600 ~1100nm
λ	193.368nm	Test mask orientation	$0 °, 22.5 °, 45 °, 67.5 °,$ $90 °, 112.5 °, 135 °, 157.5 °$
Defocus, z	0nm, 50nm	Illumination type	Traditional coherent
Test mask type	Binary	source polarization	X and Y

Jones pupil	linear approximation method (λ)	Proposed method (λ)
Re(Jxx)	6.35 $\times 10^{- 3}$	4.83 $\times 10^{- 4}$
Im(Jxx)	3.55 $\times 10^{- 3}$	4.77 $\times 10^{- 4}$
Re(Jxy)	5.00 $\times 10^{- 3}$	4.51 $\times 10^{- 4}$
Im(Jxy)	3.73 $\times 10^{- 3}$	4.43 $\times 10^{- 4}$
Re(Jyx)	5.00 $\times 10^{- 3}$	2.86 $\times 10^{- 4}$
Im(Jyx)	3.73 $\times 10^{- 3}$	3.23 $\times 10^{- 4}$
Re(Jyy)	7.12 $\times 10^{- 3}$	2.71 $\times 10^{- 4}$
Im(Jyy)	3.51 $\times 10^{- 3}$	3.21 $\times 10^{- 4}$
average	4.92 $\times 10^{- 3}$	3.82 $\times 10^{- 4}$

Abstract

1. Introduction

2. Spectrum modulation of PA

2.1 Vector imaging model with PA

2.2 Spectrum modulation

3. The nonlinear measurement of PA

4. Hybrid genetic algorithm

5. Simulations

6. Conclusion

Appendix A Specific analysis form of each item in K→f(f,g)

Appendix B Specific analysis form of C11,C12,C21,C22,C31, C32 andC'11,C'12,C'21,C'22,C'31,C'32

Funding

References

Cited By

Figures (6)

Tables (2)

Equations (31)

Optics Express

Appendix A Specific analysis form of each item in ${\vec{K}}_{f} (f, g)$

Appendix B Specific analysis form of $C_{11}$ , $C_{12}$ , $C_{21}$ , $C_{22}$ , $C_{31}$ , $C_{32}$ and $C'_{11}$ , $C'_{12}$ , $C'_{21}$ , $C'_{22}$ , $C'_{31}$ , $C'_{32}$