Polarization distortion innately exists in hyper numerical aperture immersion lithography system. Polarization distortion, mainly including polarization aberration (PA) of lithography projection optics and thick mask induced polarization distortion, would seriously impact on lithography imaging quality. Some computational lithography technologies, such as robust optical proximity correction and robust source and mask optimization, have been introduced and developed to reduce the impact of polarization distortion on lithography imaging. In this paper, we innovate a vectorial pupil optimization (VPO) method to further extend degrees of freedom for pupil optimization and compensate polarization distortion for immersion lithography system. An analytical relationship between lithography imaging and active vectorial pupil, and the gradient-based algorithm is adopted to effectively solve VPO. Extensive simulations demonstrate the VPO method simultaneously compensate the PA of projection optics and the thick mask induced polarization distortion sufficiently. Based on PA-aware source mask optimization, the VPO method can further reduce the impact of polarization distortion on lithography imaging. Compared to current pupil wavefront optimization, the proposed VPO effectively reduces the pattern error by 37.2%, which demonstrates the VPO method can improve lithography pattern fidelity.
1. Introduction
Optical lithography is key technology for integrated circuit (IC) manufacture. Nowadays, Argon Fluoride (ArF) lithography are still main lithography technology in industry, although the critical dimension of IC is much shorter than illumination wavelength 193nm [1]. Some computational lithography technologies, such as optical proximity correction (OPC) and source mask optimization (SMO), have been developed and adopted to improve lithography resolution and fidelity [2–5]. Due to hyper-numerical aperture (NA), polarization distortion innately exists in immersion lithography system. The polarization aberration (PA), defined as variations of the amplitude, phase and polarization of an optical wavefront across the exit pupil of an optical system [6,7], is an important polarization distortion for immersion lithography. Generally, PA can be represented as Jones pupil consist of a series of Jones matrix, to describe the modulation of optical system on polarized light [8,9]. Many excellent works systematically and amply analyze the serious impact of PA on hyper-NA lithography imaging [10–12]. Recently, we proposed a PA-aware SMO to reduce the influence of PA on lithography imaging [13].
However, the PA-aware SMO is based on Kirchhoff approximation, which assumes that the mask is very thin and the near field of mask diffraction equals mask pattern. When the linewidth at mask is much shorter than illumination wavelength, the mask topography, or thick mask effects, shouldn’t be ignored in lithography imaging. According to previous research [14,15], thick mask complex diffraction can also induce the polarization distortion. The polarization distortion arising from thick mask are dependent of source pattern and mask pattern with a non-analytical relationship, so it is very difficult to compensate the thick mask induced polarization distortion via SMO. Our earlier work proposed an inverse pupil wavefront optimization (PWO) to compensate thick mask induced aberration [16]. It is noted that the principle of PWO is that optimizing and generating an active wavefront complementary with the distorted wavefront to finally make the actual wavefront as perfect as possible. But the current PWO method can only optimize and adjust the phase distribution, which can compensate the phase distortion of wavefront, but can’t compensate the polarization distortion in theory. As stated above, the polarization distortion, including the PA of projection optics and polarization effects arising from thick mask diffraction, always exists in practical lithography system. Particularly, the thick mask induced polarization distortion would be more serious with the decrease of critical dimension (CD) [14]. Thus, the next node lithography need more advanced wavefront compensation technology
This paper focus on a novel vectorial pupil optimization (VPO) method to compensate polarization distortion on pupil plane of lithography system. We firstly develop a complete vectorial imaging model accounting the PA of projection optics and the complex diffraction of thick mask. Then, we build the analytical relationship between lithography imaging and active vectorial pupil. In this paper, the vectorail pupil is expressed as Jones pupil. A conjugate gradient algorithm is adopted to minimize the pattern error (PAE) via optimizing the distribution of active vectorial pupil. The optimal active vectorial pupil can be realized or generated by the array of sub-wavelength wave plate and polarizer [17,18], however, it is out of the scope of this paper. Compared to the scalar pupil or phase pupil, the vectorial pupil has more adjustable polarization variables and degrees of freedom for optimization. Thus, the VPO framework can effectively reduce the impact of polarization distortion on lithography imaging and improve pattern fidelity. Two test patterns, whose CD are 45nm and 22nm respectively, are used in simulation. The simulation results show the proposed VPO can attain better lithography performance for both two test pattern than current PWO methods. For 45nm CD test pattern, the proposed VPO can reduce the PAE from 596 to 468 compared to PWO method. For 22nm CD test pattern, the proposed VPO can reduce the PAE from 1261 to 792 compared to PWO method. Simulation results demonstrate the superiority of VPO method in lithography performance.
The remainder of this paper is organized as follows. The complete vectorial imaging model is introduced in Section 2. The algorithm and framework of VPO are provided in Section 3. The simulations and discussions are demonstrated in Section 4. Conclusions are presented in Section 5.
2. The complete vectorial imaging model in VPO
In our earlier work [16,19], we have established a lithography vectorial imaging model accounting the polarization characteristics of light and thick mask effects. Here, we develop a complete vectorial imaging model further considering the influence of active vectorial pupil on lithography imaging. Figure 1 shows the schematic of lithography aerial imaging formation based on the complete vector imaging model. Assume unit Jones vector ${{\bf E}_i}$ represents the electric field polarization state emitting from the point light source (xs,ys) and ${\bf G}_{3D}^{{x_s}{y_s}}$ represents thick mask diffraction spectrum corresponding to source point (xs,ys). Considering the PA in hyper-NA immersion projection optics, the electric field vector at entrance pupil can be calculated as ${{\bf E}^{\textrm{ent}}}\textrm{ = }{\bf PA} \odot {\bf G}_{3D}^{{x_s}{y_s}} \odot {{\bf E}_i}$. The notation ${\odot} $ represents entry-by-entry multiplication operation. For VPO method, the electric field vector would be modulated by active vectorial pupil ${{\bf J}_{{\bf VP}}}$. The ${\bf PA}$, ${\bf G}_{3D}^{{x_s}{y_s}}$ and ${{\bf J}_{{\bf VP}}}$ are all vectorial matrices consist of many 2×2 Jones matrices depending on pupil coordinate. At exit pupil, the electric field ${\bf E}$ would be transformed a 3×1 vector via a 3×2 rotation matrix V and correction factor C. The detailed information about rotation matrix V and correction factor C can be found in [16]. Based on Fourier optics, for source point (xs,ys), the electric field at image plane can be calculated as
(1)$${{\bf E}^{{x_s}{y_s}}}\textrm{ = }C{{\mathscr F}^{\textrm{ - }1}}\{{{\bf V} \odot {{\bf J}_{{\bf VP}}} \odot {\bf PA} \odot {\bf G}_{3D}^{{x_s}{y_s}} \odot {{\bf E}_i}} \},$$
where the notation ${{\mathscr F}^{\textrm{ - }1}}$ represents inverse Fourier transform. According to Abbe’s method, the aerial image at wafer plane can be calculated as (2)$${\mathbf I} = \frac{1}{{{J_{sum}}}}\sum\limits_{{x_s}} {\sum\limits_{{y_s}} {\left( {{\mathbf J}({{x_s},{y_s}} )\sum\limits_{p = x,y,z} {{{|{{\bf E}_p^{{x_s}{y_s}}} |}^2}} } \right)} } ,$$
where ${\mathbf J}({{x_s},{y_s}} )$ is the intensity of the source point at (xs,ys), and ${J_{sum}}\textrm{ = }\sum\nolimits_{{x_s},{y_s}} {{\mathbf J}({{x_s},{y_s}} )}$ is a normalization factor.The aerial image described in Eq. (2) would enter into the photoresist and form the printed image through development. In this paper, the sigmoid function [20] is applied to approximate the photoresist effects. The printed image on the wafer can be represented by a sigmoid transformation of the aerial image, that is,
(3)$${\mathbf Z} = sig({\mathbf I} )= \frac{1}{{1 + \exp [{ - a({{\mathbf I} - {t_r}} )} ]}},$$
where a indicates the steepness of the sigmoid function, and tr is the process threshold.3. VPO algorithm and framework
In this paper, we propose a VPO method to further compensate polarization distortion based on current SMO and PWO methods. Thus, the first step of VPO is getting the optimal source pattern and mask pattern by PA-aware SMO, which is described in our previous work [13]. Based on optimal source pattern and mask pattern, we use rigorous electromagnetic field simulator in software PROLITH to calculate the thick mask diffraction spectrum ${\bf G}_{3D}^{{x_s}{y_s}}$. Then we can use complete vectorial imaging model described in Section 2 to calculate the lithography imaging. Generally, the Euclidean distance between printed image ${\mathbf Z}$ and target pattern $\tilde{{\bf Z}}$ is designed as the cost function for inverse lithography. So, the goal of VPO is to find the optimal vectorial pupil ${\hat{{\bf J}}_{{\bf VP}}}$ to make the cost function
(4)$$F = d({{\bf Z,\tilde{Z}}} )= ||{{\bf Z} - \tilde{{\bf Z}}} ||_2^2,$$
minimized, where the notation $|| \cdot ||_{2} $ represents l2-norm. Because ${{\bf J}_{{\bf VP}}}$ is an vectorial matrix consists of many 2×2 Jones matrices, it can be represented by matrix form (5)$${{\bf J}_{{\bf VP}}} = \left( {\begin{array}{cc} {{J_{xx\_real}} + i{J_{xx\_imag}}}&{{J_{xy\_real}} + i{J_{xy\_imag}}}\\ {{J_{yx\_real}} + i{J_{yx\_imag}}}&{{J_{yy\_real}} + i{J_{yy\_imag}}} \end{array}} \right),$$
where ${J_{xx\_real}}$, ${J_{xx\_imag}}$, ${J_{xy\_real}}$, ${J_{xy\_imag}}$, ${J_{yx\_real}}$, ${J_{yx\_imag}}$, ${J_{xx\_real}}$, and ${J_{xx\_real}}$ are all real matrices depending on pupil coordinate. The pupil is circular for lithography projection optics, so all components of ${{\bf J}_{{\bf VP}}}$ can be expanded using Zernike polynomials (6)$${J_{xx\_real}}({\rho ,\theta } ){\bf = }\textrm{1}\textrm{ + }\sum\limits_i {c_i^{xx\_real}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(7)$${J_{xx\_imag}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{xx\_imag}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(8)$${J_{xy\_real}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{xy\_real}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(9)$${J_{xy\_imag}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{xy\_imag}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(10)$${J_{yx\_real}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{yx\_real}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(11)$${J_{yx\_imag}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{yx\_imag}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(12)$${J_{yy\_real}}({\rho ,\theta } ){\bf = }\textrm{1}\textrm{ + }\sum\limits_i {c_i^{yy\_real}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
(13)$${J_{yy\_imag}}({\rho ,\theta } ){\bf = }\sum\limits_i {c_i^{yy\_imag}} {{\bf \Gamma }_i}({\rho ,\theta } ),$$
where ${{\bf \Gamma }_i}({\rho ,\theta } )$ is the ith Zernike polynomial term with $c_i^{xx/xy/yx/yy\_real/imag}$ being the ith Zernike coefficient corresponding to respective components. Typically, 37 terms of Zernike polynomials are accurate enough to fit the components of ${{\bf J}_{{\bf VP}}}$ in Eqs. (6)–(13). In this paper, the unit of $c_i^{xx/xy/yx/yy\_real/imag}$ is always wavelength $\lambda $, which is omitted in the following part for simplicity.Based on Eqs. (5)–(13), the active vectorial pupil ${{\bf J}_{{\bf VP}}}$ can be represented by a series of Zernike coefficients. In this case, the VPO problem can be mathematically modeled as
(14)$$\hat{c}_i^{xx/xy/yx/yy\_real/imag} = \mathop {\arg \; \; \min }\limits_{c_i^{xx/xy/yx/yy\_real/imag}} F.$$
According to our previous work about gradient-based PWO [16,21], the solving process of VPO problem is provided in Table 1 in the form of pseudo-code. The conjugate-gradient algorithm is used to solve VPO problem. Equations (15)–(22) provide the gradient of cost function F to every Zernike coefficients $c_i^{xx/xy/yx/yy\_real/imag}$. The derivation of gradient formula for cost function is shown in the Appendix. (15)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xx\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(16)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xx\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(17)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xy\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(18)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xy\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(19)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yx\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(20)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yx\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(21)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yy\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(22)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yy\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
where (23)$$\Lambda = ({\tilde{{\bf Z}} - {\bf Z}} )\odot {\bf Z} \odot ({\textrm{1} - {\bf Z}} ).$$
Table 1. Pseudo-code of VPO flow.
4. Simulation and discussion
In this section, we would compare the lithography performance for PWO and VPO to demonstrate the validity and superiority of proposed VPO method. In simulation, the parameters of lithography system are as follows: the illumination wavelength is 193nm, the illumination polarization is Y-polarization, the reduction ratio is 1:4, and the NA on the wafer side is 1.35. Figure 2 shows the layout of projection optics used in the simulation. Figure 3 shows the PA of projection optics, which is designed by our group. The PA is extracted in optical design software CODE V [22]. All simulations in this paper are implemented with the existence of this PA.
Figure 4 shows two target patterns used in simulations. The CD of target #1 and target #2 is 45nm and 22nm, respectively. For target #1, the pattern is represented by a 301 × 301 matrix with pixel size of 5nm × 5nm on wafer scale, which means the mask dimension is finite 6020nm × 6020nm in simulation. For target #2, the pattern is represented by a 201 × 201 matrix with pixel size of 2.44nm × 2.44nm on wafer scale, which means the mask dimension is finite 1961.76 nm × 1961.76 nm in simulation.
In order to evaluate the pattern fidelity of lithography imaging, PAE is defined as:
(24)$${\bf PAE}\textrm{ = }||{\tilde{{\bf Z}} - {\bf \Xi }\{{{\bf I} - {t_r}} \}} ||_2^2,$$
where ${\bf \Xi }\{{\cdot} \}= 0$ if the argument is smaller than 0; otherwise, ${\bf \Xi }\{{\cdot} \}= 1$. In simulation, ${t_r}\textrm{ = }0.04$ for target #1 and ${t_r}\textrm{ = }0.02$ for target #2.For target #1, Fig. 5 displays the optimal source pattern and mask pattern by a pixelated PA-aware SMO method [13]. Based on the source pattern and mask pattern, we calculate the thick mask diffraction spectrum in commercial software PROLITH and build VPO optimization problem described in Eq. (14). In simulation, the mask absorbers consist of 55 nm Cr with a refractive index of 1.48 + 1.76i and 18 nm CrO with a refractive index of 1.97 + 1.2i [16]. It should be noted that the source pattern and mask pattern are same and fixed for PWO and VPO.
Then we apply conjugate-gradient algorithm described in Table 1 to solve the VPO problem. For target #1, the maximum of iteration number ${l_{VPO}}$ is 100 in Table 1. Figure 6 shows the convergence curves for PWO and VPO. The comparison between two convergence curves demonstrates that the VPO method can achieve lower PAE.
Figure 7 and Fig. 8 provide the optimization results of PWO and VPO, respectively. We can find the PWO method can only achieve an optimal phase pupil, but the proposed VPO method can get eight optimal pupils. Particularly, the pupils ${J_{xx\_real}}$, ${J_{xx\_real}}$, ${J_{xx\_real}}$, and ${J_{xx\_real}}$ represent the coupling of s and p polarization components for polarized light, which enhances regulation capability and extends degrees of freedom for optimization. Compared to PWO method, the proposed VPO method has more optimization potential for lithography performance improvement, which has been verified in Fig. 6.
Figure 9 demonstrates the printed image for initial case, PWO, and VPO. The printed image of initial case is worst, which is because the thick mask effects are neglected in PA-aware SMO. Thus, it is necessary to adopt pupil optimization to compensate thick mask effects. For PWO and VPO, the VPO achieves higher pattern fidelity, which illustrates the validity and superiority of the proposed VPO. Quantitatively, the PAE of initial case, PWO, and VPO is 886, 596, and 468, respectively. Compared to PWO, the proposed VPO method can reduce the PAE by 21.5%.
To demonstrate the universality of the proposed VPO method, Fig. 10 provides the source pattern and mask pattern used in VPO for target #2. They are also obtained by pixelated PA-aware SMO. We still build and solve VPO optimization problem in similar way to target #1. For target #2, the maximum of iteration number ${l_{VPO}}$ is 40 in Table 1. Figure 11 shows the convergence curves of PWO and VPO for target #2. The comparison between two convergence curves still illustrates that the VPO method can achieve better lithography performance.
Figure 12 and Fig. 13 provide the optimization results of PWO and VPO for target #2, respectively. Figure 14 demonstrates the printed image for initial case, PWO, and VPO. For three printed image, the VPO still achieves the highest pattern fidelity, which illustrates that the proposed VPO are still valid and superior for target #2. Quantitatively, the PAE of initial case, PWO, and VPO is 2786, 1261, and 792, respectively. Compared to PWO, the proposed VPO reduces the PAE by 37.2%. We can find the VPO method is more effective for target #2. This is because the CD of target #2 is 22nm, which is smaller than target #1. The smaller CD would induce larger polarization distortion, so compensation effect of VPO is clearer.
5. Conclusion
This paper proposes a VPO method to compensate polarization distortion in hyper-NA lithography system. Based on complete vectorial imaging model, an analytical relationship between lithography imaging and active vectorial pupil is established. Then, the VPO problem is transformed as a mathematical optimization problem. The conjugate-gradient algorithm is adopted to optimize active vectorial pupil parameters via minimizing cost function. The VPO method extends degrees of freedom for optimization and enhances the compensation capability for polarization distortion. Simulation results demonstrate the proposed VPO method can achieve better lithography performance with the existence of polarization distortion. Particularly, the proposed VPO method is more effective for patterns with small CD. For the test pattern with 22nm CD, compared to current PWO method, the proposed VPO method can reduce the PAE by 37.2% and evidently improve the lithography pattern fidelity.
Appendix
In this section, we would derive the gradients of the cost function Eqs. (15)–(22).
The cost function F described in Eq. (4) can be reformulated as
(25)$$F = {\sum\limits_m {\sum\limits_n {({{z^{mn}} - {{\tilde{z}}^{mn}}} )} } ^2},$$
where
${\tilde{z}^{mn}}$ is the
$({m,n} )$th entry of
$\tilde{{\bf Z}}$, and
${z^{mn}}$ is the
$({m,n} )$th entry of
Z, that is
(26)$${z^{mn}} = \frac{1}{{1 + \exp \left[ { - \frac{a}{{{J_{sum}}}}\sum\limits_{{x_s}} {\sum\limits_{{y_s}} {({{\mathbf J}({{x_s},{y_s}} )({{{|{E_{x,mn}^{{x_s}{y_s}}} |}^2}\textrm{ + }{{|{E_{y,mn}^{{x_s}{y_s}}} |}^2}\textrm{ + }{{|{E_{z,mn}^{{x_s}{y_s}}} |}^2}} )} )+ a{t_r}} } } \right]}}.$$
We rewrite the Eq. (
1) as
(27)$${\bf E}\textrm{ = }C{{\mathscr F}^{\textrm{ - }1}}\{{{\bf V} \odot {{\bf J}_{{\bf VP}}} \odot {{\bf E}_{pupil}}} \},$$
where
${{\bf E}_{pupil}}\textrm{ = }{\bf PA} \odot {\bf G}_{3D}^{{x_s}{y_s}} \odot {{\bf E}_i}$ is a constant in VPO flow. Equation (
28) provides the matrix form of
${\bf V}$,
${{\bf J}_{{\bf VP}}}$, and
${{\bf E}_{pupil}}$ (28)$${\bf V}\textrm{ = }\left({\begin{array}{cc} {V_{11}} & {V_{12}}\\ {V_{21}} & {V_{22}}\\ {V_{31}} & {V_{32}} \end{array}} \right),\; \; \; \; {\bf VP}\textrm{ = }\left( {\begin{array}{cc} {J_{xx}} & {J_{xy}} \\ {J_{yx}} & {J_{yy}} \end{array}}\right),\; \; \; \; \; {{\bf E}_{pupil}}\textrm{ = }\left( {\begin{array}{l} {{E_{pupil\_x}}}\\ {{E_{pupil\_y}}} \end{array}} \right).$$
It should be noted the notations
${\bf V}$ and
${{\bf E}_{pupil}}$ are the function of coordination
$({{x_s},{y_s}} )$, which may be omitted sometimes for simplicity. The
x,
y,
z components of
${\bf E}$ can be represented as
(29)$$E_{x,mn}^{{x_s}{y_s}}\textrm{ = }C{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{J_{xx}}{E_{pupil\_x}}\textrm{ + }{V_{11}}{J_{xy}}{E_{pupil\_y}}\textrm{ + }{V_{12}}{J_{yx}}{E_{pupil\_x}}\textrm{ + }{V_{12}}{J_{yy}}{E_{pupil\_y}}} \},$$
(30)$$E_{y,mn}^{{x_s}{y_s}}\textrm{ = }C{{\mathscr F}^{\textrm{ - }1}}\{{{V_{21}}{J_{xx}}{E_{pupil\_x}}\textrm{ + }{V_{21}}{J_{xy}}{E_{pupil\_y}}\textrm{ + }{V_{22}}{J_{yx}}{E_{pupil\_x}}\textrm{ + }{V_{22}}{J_{yy}}{E_{pupil\_y}}} \},$$
(31)$$E_{z,mn}^{{x_s}{y_s}}\textrm{ = }C{{\mathscr F}^{\textrm{ - }1}}\{{{V_{31}}{J_{xx}}{E_{pupil\_x}}\textrm{ + }{V_{31}}{J_{xy}}{E_{pupil\_y}}\textrm{ + }{V_{32}}{J_{yx}}{E_{pupil\_x}}\textrm{ + }{V_{32}}{J_{yy}}{E_{pupil\_y}}} \},$$
where
(32)$${J_{xx}}\textrm{ = }{J_{xx\_real}} + i{J_{xx\_imag}},$$
(33)$${J_{xy}}\textrm{ = }{J_{xy\_real}} + i{J_{xy\_imag}},$$
(34)$${J_{yx}}\textrm{ = }{J_{yx\_real}} + i{J_{yx\_imag}},$$
(35)$${J_{yy}}\textrm{ = }{J_{yy\_real}} + i{J_{yy\_imag}}.$$
Based on Eqs. (
29)–(
31), we can calculate the gradient of real part or imaginary part for
$E_{x,mn}^{{x_s}{y_s}}$,
$E_{y,mn}^{{x_s}{y_s}}$,
$E_{z,mn}^{{x_s}{y_s}}$ to every Zernike coefficient
$c_i^{xx/xy/yx/yy\_real/imag}$. For simplicity, we only provide the results about
$E_{x,mn}^{{x_s}{y_s}}$ (36)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xx\_real}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(37)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xx\_real}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(38)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xx\_imag}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(39)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xx\_imag}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(40)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xy\_real}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(41)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xy\_real}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(42)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xy\_imag}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(43)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{xy\_imag}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(44)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yx\_real}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(45)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yx\_real}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(46)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yx\_imag}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(47)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yx\_imag}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}}} \}} \},$$
(48)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yy\_real}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(49)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yy\_real}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(50)$$\frac{{\partial {\mathop{\rm Re}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yx\_imag}}} = C\,{\mathop{\rm Re}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \},$$
(51)$$\frac{{\partial {\mathop{\rm Im}\nolimits} \{{E_{x,mn}^{{x_s}{y_s}}} \}}}{{\partial c_i^{yy\_imag}}} = C\,{\mathop{\rm Im}\nolimits} \{{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}}} \}} \}.$$
Based on Eq. (26), we can also calculate the gradient of cost function F to real part or imaginary part of $E_{x,mn}^{{x_s}{y_s}}$, $E_{y,mn}^{{x_s}{y_s}}$, $E_{z,mn}^{{x_s}{y_s}}$. For simplicity, we only provide the results about $E_{x,mn}^{{x_s}{y_s}}$
(52)$$\frac{{\partial F}}{{\partial {\mathop{\rm Re}\nolimits} ({E_{x,mn}^{{x_s}{y_s}}} )}} = \frac{{4a}}{{{J_{sum}}}} \times ({{{\tilde{z}}^{mn}} - {z^{mn}}} )\times {z^{mn}} \times ({1 - {z^{mn}}} )\times {\mathbf J}({{x_s},{y_s}} )\times {\mathop{\rm Re}\nolimits} ({E_{x,mn}^{{x_s}{y_s}}} ),$$
(53)$$\frac{{\partial F}}{{\partial {\mathop{\rm Im}\nolimits} ({E_{x,mn}^{{x_s}{y_s}}} )}} = \frac{{4a}}{{{J_{sum}}}} \times ({{{\tilde{z}}^{mn}} - {z^{mn}}} )\times {z^{mn}} \times ({1 - {z^{mn}}} )\times {\mathbf J}({{x_s},{y_s}} )\times ({ - i} ){\mathop{\rm Im}\nolimits} ({E_{x,mn}^{{x_s}{y_s}}} ).$$
According to chain’s rule, the gradient of cost function
F to every Zernike coefficient
$c_i^{xx/xy/yx/yy\_real/imag}$ can be represented as
(54)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xx\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(55)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xx\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(56)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xy\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(57)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{xy\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{11}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{21}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{31}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(58)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yx\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(59)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yx\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_x}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(60)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yy\_real}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}, \end{array}$$
(61)$$\begin{array}{l} \frac{{\partial F}}{{\partial c_i^{yy\_imag}}} = \sum\limits_{{x_s}{y_s}} {\frac{{4aC}}{{{J_{sum}}}} \times {\mathbf J}({{x_s},{y_s}} )\times \sum\limits_{mn} {\Lambda \odot \{{{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{12}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{x,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} } } \\ \; \; {\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{22}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{y,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]\textrm{ + }{\mathop{\rm Re}\nolimits} [{{{\mathscr F}^{\textrm{ - }1}}\{{i{V_{32}}{{\bf \Gamma }_i}({\rho ,\theta } ){E_{pupil\_y}} \odot {{({E_{z,mn}^{{x_s}{y_s}}} )}^{\ast }}} \}} ]} \}. \end{array}$$
Funding
National Natural Science Foundation of China (61675026, 11627808); National Major Science and Technology Projects of China (2017ZX02101006-001).
Acknowledgments
We gratefully acknowledge KLA-Tencor Corporation for providing academic use of PROLITH.
Disclosures
The authors declare no conflicts of interest.
References
1. M. Neisser, “The 2017 IRDS Lithography Roadmap,” J. Microelectron. Manuf. 1(2), 18010204 (2018).
2. X. Ma, Z. Wang, Y. Li, G. R. Arce, L. Dong, and J. Garcia-Frias, “Fast optical proximity correction method based on nonlinear compressive sensing,” Opt. Express 26(11), 14479–14498 (2018). [CrossRef]
3. 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]
4. Y. Shen, “Lithographic source and mask optimization with a narrow-band level-set method,” Opt. Express 26(8), 10065–10078 (2018). [CrossRef]
5. T. Li and Y. Li, “Lithographic source and mask optimization with low aberration sensitivity,” IEEE Trans. Nanotechnol. 16(6), 1099–1105 (2017). [CrossRef]
6. J. P. McGuire Jr. and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I - Formulation and example,” J. Opt. Soc. Am. A 7(9), 1614–1626 (1990). [CrossRef]
7. J. P. McGuire Jr. and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33(22), 5080–5100 (1994). [CrossRef]
8. 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]
9. X. Xu, W. Huang, and M. Xu, “Orthogonal polynomials describing polarization aberration for M-fold optical systems,” Opt. Express 24(5), 4906–4912 (2016). [CrossRef]
10. J. Kye, G. McIntyre, Y. Norihiro, and H. J. Levinson, “Polarization aberration analysis in optical lithography systems,” Proc. SPIE 6154, 61540E (2006). [CrossRef]
11. B. Geh, J. Ruoff, J. Zimmermann, P. Gräupner, M. Totzeck, M. Mengel, U. Hempelmann, and E. Schmitt-Weaver, “The impact of projection lens polarization properties on lithographic process at hyper-NA,” Proc. SPIE 6520, 65200F (2007). [CrossRef]
12. Y. Tu, X. Wang, S. Li, and Y. Cao, “Analytical approach to the impact of polarization aberration on lithographic imaging,” Opt. Lett. 37(11), 2061–2063 (2012). [CrossRef]
13. T. Li, Y. Sun, E. Li, N. Sheng, Y. Li, P. Wei, and Y. Liu, “Multi-objective lithographic source mask optimization to reduce the uneven impact of polarization aberration at full exposure field,” Opt. Express 27(11), 15604–15616 (2019). [CrossRef]
14. A. Erdmann, “Mask modeling in the low k1 and ultrahigh NA regime: phase and polarization effects,” Proc. SPIE 5835, 69–81 (2005). [CrossRef]
15. A. Erdmann and P. Evanschitzky, “Rigorous electromagnetic field mask modeling and related lithographic effects in the low k1 and ultrahigh numerical aperture regime,” J. Micro/Nanolith. MEMS MOEMS 6(3), 031002 (2007). [CrossRef]
16. C. Han, Y. Li, L. Dong, X. Ma, and X. Guo, “Inverse pupil wavefront optimization for immersion lithography,” Appl. Opt. 53(29), 6861–6871 (2014). [CrossRef]
17. H. Zhao, Y. Yang, Q. Li, and M. Qiu, “Sub-wavelength quarter-wave plate based on plasmonic patch antennas,” Appl. Phys. Lett. 103(26), 261108 (2013). [CrossRef]
18. Y. Takashima, M. Tanabe, M. Haraguchi, and Y. Naoi, “Ultraviolet polarizer with a Ge subwavelength grating,” Appl. Opt. 56(29), 8224–8229 (2017). [CrossRef]
19. 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]
20. A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. on Image Process. 16(3), 774–788 (2007). [CrossRef]
21. T. Li, Y. Liu, Y. Sun, E. Li, P. Wei, and Y. Li, “Multiple-field-point pupil wavefront optimization in computational lithography,” Appl. Opt. 58(30), 8331–8338 (2019). [CrossRef]
22. 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]
Open Access
Add to BibSonomy
