用于液晶光学特性模拟的时域电磁计算方法研究
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摘要
液晶作为一种电控各向异性介质,已经广泛应用于显示领域并取得极大地成功。近年来,液晶在非显示领域的应用,如用于光束控制、波前变换、自适应光学等领域的研究也日益受到关注,并逐步走向实际应用阶段。此外,由于液晶具有很宽的工作光谱范围,使得其在微波和太赫兹波段也有着诱人的应用前景。液晶器件特性数值模拟能够预测器件光学特性,优化器件参数,为器件设计与控制提供依据,是器件研究和开发不可或缺的重要组成部分。
液晶器件数值模拟包含指向矢模拟和光学特性模拟两个方面。光学特性模拟是通过研究光与液晶的相互作用,模拟入射光波在液晶器件中的传播过程。时域有限差分法(FDTD)作为一种严格求解Maxwell方程组的数值方法,是液晶光学特性模拟中应用最广泛的方法。
本课题主要研究用于液晶光学特性模拟的二维时域电磁计算方法,寻求适用于非均匀各向异性介质、具有不同尺度结构细节(相对于入射光的波长)的算法,为液晶光学相控阵等液晶器件的设计和制造奠定理论基础。本课题研究内容如下:
(1)当FDTD方法用于层状各向异性介质的散射分析时,由于任意非对角各向异性介质中电场各分量之间的互相耦合,导致了一维辅助网格上的Maxwell方程组中Dy和Hy沿切向和法向传播的子分量不能独立离散,使得传统的一维辅助FDTD方法无法直接计算层状各向异性介质的入射场场值。为此引入分裂场技术,解决了一维辅助Maxwell方程组的离散问题。
(2)当FDTD方法用于非周期结构液晶器件时,由于左、右边界的不对称性导致了用于平面波源引入的周期边界条件不适用,必须设法解决如何独立计算总场/散射场左、右连接边界上的入射场场值问题。为此根据液晶材料的粘滞特性,将左、右连接边界上入射场场值的计算问题转化为具有不同介电张量分布的层状各向异性介质的入射场场值计算问题,解决了非周期液晶器件FDTD分析时的平面波源引入问题。
(3) FDTD方法为获得精确解,其空间网格采样率需要随着计算区域总体尺寸的增大而增加,导致计算内存消耗大。伪谱时域方法(PSTD)因采用快速傅立叶变换算法求解空间微分,其空间网格采样率与总体尺寸无关,极大地减少了空间采样率。根据液晶盒的薄板结构特点,提出用混合PS-FDTD方法模拟光在液晶中的传播。即在电尺寸小而内部介质变化剧烈的液晶盒厚度方向上采用FDTD方法,而在电尺寸大而内部介质变化又较平滑、且与玻璃基板平行方向上采用PSTD方法,以此减少计算资源消耗。
(4) FDTD方法的稳定条件限定了其最小空间网格尺寸对时间步长的要求,导致了在求解超精细结构细节问题时计算时间过长。针对相位调制型液晶器件中不存在横向磁场和横向电场耦合的情况,提出采用无条件稳定局部一维FDTD方法,以消除稳定条件对时间步长的限制。
通过以上研究,解决了液晶等各向异性介质中FDTD方法的应用问题,为液晶光学特性模拟提供有效手段。
As an electrically controllable anisotropic medium, liquid crystals are widely used for displays and achieved great success. Recently, the application for non-displays such as beam control, wavefront transformation, and adaptive optics has attracted much attention and step into the practical application gradually. Besides, liquid crystals are very promising in the aspect of microwave and Terahertz wave for its wide range of transmittance spectra. Numerical simulation is an essential part in the research and development of liquid crystal devices, which can predict the optical properties of the device, optimize the parameters of the device, and provide the evidence of the device design and control.
Numerical simulation of liquid crystal devices includes director simulation and optical property simulation. Optical property simulation studies about the interaction between light and liquid crystals, and simulates the light propagation through liquid crystal device. Finite-difference time-domain method (FDTD), as a numerical approach for the rigorous solution of Maxwell equations, is the most widely used method for the optical simulation of liquid crystals.
This thesis mainly studied 2D time-domain computational method of electromagnetics for the optical simulation of liquid crystals, and explore the algorithms suitable for inhomogeneous anisotropic media with different spatial details (compared to the wavelength of incident light). The methods studied provide theoretic foundation for design and fabrication of liquid crystal device. The following are the main contents of the study:
(1) When the FDTD method is used for the scattering analysis of layered anisotropic media, the subcomponents of Dy and Hy in the Maxwell equations on 1D auxiliary grid, which propagate in the direction either parallel with or perpendicular to the planar interface, can not be discretized separately. The reason is that the components of electric field are coupled together in the arbitrary off-diagonal anisotropic media. Thus the traditional 1D auxiliary FDTD method can not be applied to calculate the incident fields of layered anisotropic media directly. Therefore, the split-field technique is introduced to solve this problem.
(2) When the FDTD method is used for the liquid crystal devices with non-periodic structure, due to the asymmetry between left and right boundaries, the periodic boundary conditions for the introduction of plane waves is not applicable. Thus, it is necessary to calculate the incident fields on the left boundary and that on the right boundary independently. For this purpose, according to the viscous characteristics of liquid crystals, the problem of the calculation of the incident fields on the left and right connecting boundaries can be simplified to the calculation of the incident fields of layered anisotropic media with different dielectric tensors. Therefore, the problem of the introduction of plane waves for the FDTD analysis of non-periodic liquid crystal devices is solved.
(3) To obtain accurate results, the spatial sampling rate of FDTD method must be increased as the electrical size of the structure being modeled increased, thus leads to large memory consumption. Because a fast Fourier transform algorithm is used to represent the spatial derivatives in pseudospectral time-domain method (PSTD), the spatial sampling rate of PSTD method is independent of the overall electrical size of the structure being modeled, which greatly reduces the spatial sampling rate. In order to reduce the computational cost of FDTD method, hybrid PS-FDTD method is introduced to simulate the propagation of light through liquid crystals based on the characteristics of thin plate structure of liquid crystal cells. The FDTD method is applied to the thickness direction of a cell with a small thickness and fine structures, while the PSTD method is applied to the direction, paralleled with glass substrate, of a cell with a large surface and smooth internal media.
(4) The time step in FDTD method is bounded by the minimum spatial grid for the numerical stability condition, which leads to the large computational time for the solution of problem with fine-scale geometric detail. For the phase-only liquid crystal device without coupling between transverse magnetic waves and transverse electric waves in it, an unconditionally stable locally one-dimensional FDTD method is proposed to remove the restriction of time step limited by the stability condtion.
According to the research above, the application problem of FDTD method in anisotropic medium such as liquid crystals is solved, and it provides the premise and approach to optical simulation of liquid crystals.
液晶器件数值模拟包含指向矢模拟和光学特性模拟两个方面。光学特性模拟是通过研究光与液晶的相互作用,模拟入射光波在液晶器件中的传播过程。时域有限差分法(FDTD)作为一种严格求解Maxwell方程组的数值方法,是液晶光学特性模拟中应用最广泛的方法。
本课题主要研究用于液晶光学特性模拟的二维时域电磁计算方法,寻求适用于非均匀各向异性介质、具有不同尺度结构细节(相对于入射光的波长)的算法,为液晶光学相控阵等液晶器件的设计和制造奠定理论基础。本课题研究内容如下:
(1)当FDTD方法用于层状各向异性介质的散射分析时,由于任意非对角各向异性介质中电场各分量之间的互相耦合,导致了一维辅助网格上的Maxwell方程组中Dy和Hy沿切向和法向传播的子分量不能独立离散,使得传统的一维辅助FDTD方法无法直接计算层状各向异性介质的入射场场值。为此引入分裂场技术,解决了一维辅助Maxwell方程组的离散问题。
(2)当FDTD方法用于非周期结构液晶器件时,由于左、右边界的不对称性导致了用于平面波源引入的周期边界条件不适用,必须设法解决如何独立计算总场/散射场左、右连接边界上的入射场场值问题。为此根据液晶材料的粘滞特性,将左、右连接边界上入射场场值的计算问题转化为具有不同介电张量分布的层状各向异性介质的入射场场值计算问题,解决了非周期液晶器件FDTD分析时的平面波源引入问题。
(3) FDTD方法为获得精确解,其空间网格采样率需要随着计算区域总体尺寸的增大而增加,导致计算内存消耗大。伪谱时域方法(PSTD)因采用快速傅立叶变换算法求解空间微分,其空间网格采样率与总体尺寸无关,极大地减少了空间采样率。根据液晶盒的薄板结构特点,提出用混合PS-FDTD方法模拟光在液晶中的传播。即在电尺寸小而内部介质变化剧烈的液晶盒厚度方向上采用FDTD方法,而在电尺寸大而内部介质变化又较平滑、且与玻璃基板平行方向上采用PSTD方法,以此减少计算资源消耗。
(4) FDTD方法的稳定条件限定了其最小空间网格尺寸对时间步长的要求,导致了在求解超精细结构细节问题时计算时间过长。针对相位调制型液晶器件中不存在横向磁场和横向电场耦合的情况,提出采用无条件稳定局部一维FDTD方法,以消除稳定条件对时间步长的限制。
通过以上研究,解决了液晶等各向异性介质中FDTD方法的应用问题,为液晶光学特性模拟提供有效手段。
As an electrically controllable anisotropic medium, liquid crystals are widely used for displays and achieved great success. Recently, the application for non-displays such as beam control, wavefront transformation, and adaptive optics has attracted much attention and step into the practical application gradually. Besides, liquid crystals are very promising in the aspect of microwave and Terahertz wave for its wide range of transmittance spectra. Numerical simulation is an essential part in the research and development of liquid crystal devices, which can predict the optical properties of the device, optimize the parameters of the device, and provide the evidence of the device design and control.
Numerical simulation of liquid crystal devices includes director simulation and optical property simulation. Optical property simulation studies about the interaction between light and liquid crystals, and simulates the light propagation through liquid crystal device. Finite-difference time-domain method (FDTD), as a numerical approach for the rigorous solution of Maxwell equations, is the most widely used method for the optical simulation of liquid crystals.
This thesis mainly studied 2D time-domain computational method of electromagnetics for the optical simulation of liquid crystals, and explore the algorithms suitable for inhomogeneous anisotropic media with different spatial details (compared to the wavelength of incident light). The methods studied provide theoretic foundation for design and fabrication of liquid crystal device. The following are the main contents of the study:
(1) When the FDTD method is used for the scattering analysis of layered anisotropic media, the subcomponents of Dy and Hy in the Maxwell equations on 1D auxiliary grid, which propagate in the direction either parallel with or perpendicular to the planar interface, can not be discretized separately. The reason is that the components of electric field are coupled together in the arbitrary off-diagonal anisotropic media. Thus the traditional 1D auxiliary FDTD method can not be applied to calculate the incident fields of layered anisotropic media directly. Therefore, the split-field technique is introduced to solve this problem.
(2) When the FDTD method is used for the liquid crystal devices with non-periodic structure, due to the asymmetry between left and right boundaries, the periodic boundary conditions for the introduction of plane waves is not applicable. Thus, it is necessary to calculate the incident fields on the left boundary and that on the right boundary independently. For this purpose, according to the viscous characteristics of liquid crystals, the problem of the calculation of the incident fields on the left and right connecting boundaries can be simplified to the calculation of the incident fields of layered anisotropic media with different dielectric tensors. Therefore, the problem of the introduction of plane waves for the FDTD analysis of non-periodic liquid crystal devices is solved.
(3) To obtain accurate results, the spatial sampling rate of FDTD method must be increased as the electrical size of the structure being modeled increased, thus leads to large memory consumption. Because a fast Fourier transform algorithm is used to represent the spatial derivatives in pseudospectral time-domain method (PSTD), the spatial sampling rate of PSTD method is independent of the overall electrical size of the structure being modeled, which greatly reduces the spatial sampling rate. In order to reduce the computational cost of FDTD method, hybrid PS-FDTD method is introduced to simulate the propagation of light through liquid crystals based on the characteristics of thin plate structure of liquid crystal cells. The FDTD method is applied to the thickness direction of a cell with a small thickness and fine structures, while the PSTD method is applied to the direction, paralleled with glass substrate, of a cell with a large surface and smooth internal media.
(4) The time step in FDTD method is bounded by the minimum spatial grid for the numerical stability condition, which leads to the large computational time for the solution of problem with fine-scale geometric detail. For the phase-only liquid crystal device without coupling between transverse magnetic waves and transverse electric waves in it, an unconditionally stable locally one-dimensional FDTD method is proposed to remove the restriction of time step limited by the stability condtion.
According to the research above, the application problem of FDTD method in anisotropic medium such as liquid crystals is solved, and it provides the premise and approach to optical simulation of liquid crystals.
引文
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