填充多层介质的柱形波导传播特性研究
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摘要
在电磁场理论研究中，波导的传播特性是重要的课题之一。尽管对填充介质的规则柱形波导和同轴线的传播特性已作了大量的研究，但都主要涉及填充一层或两层介质的波导或同轴线，对于填充多层介质的柱形波导研究不多。本文主要研究几种类型的填充多层介质的金属柱形波导和同轴线的传播特性。
我们知道，材料对电磁波的响应是通过本构关系中材料参数来反映的，本构关系描写了电磁场量之间的函数依从关系。手征介质本构关系在文献中有多种形式，导致在应用时容易发生错误。
论文首先总结描写手征介质的本构关系，给出描写互易和非互易手征介质本构关系的四种基本形式，这四种基本形式又由于在应用中所取时谐因子的不同，其具体的形式有略有不同。另外，论文还介绍一些其它手征类介质的本构关系。
其次，我们详细而全面地介绍了在椭圆柱坐标系中求解齐次标量亥姆霍兹方程的方法以及所涉及到的角向马修函数和径向马修函数。马修函数是角向和径向马修方程的解，是由法国数学家E．L．Mathieu于1868年在分析椭圆形膜的振动时提出来的，在许多物理和天文学问题中都要用到此函数，在椭圆柱形物体的电磁散射、电磁波在椭圆波导及同轴线传播等问题中也要用到此函数。马修方程实际上是在椭圆柱坐标系中，用分离变量法，由齐次标量亥姆霍兹方程得到的两个横向方程，它分为角向马修方程和径向马修方程，其解称为角向马修函数和径向马修函数。如果按阶次是整数或非整数来划分，马修函数又可分为整数阶马修函数和非整数阶马修函数；如果按奇偶性来分，每种角向马修函数和径向马修函数又有奇、偶两种形式。如果按是否是周期函数来分，马修函数又可分为周期和非周期马修函数。角向马修函数有第一类和第二类角向马修函数(非周期)，整数阶径向马修函数按其和贝塞尔函数的对应关系可分为第一类径向马修函数、第二类径向马修函数和马修-汉克尔函数。由于马修函数本身的复杂性，其应用比较难，函数符号的使用上也没有统一规定，不同学者采用不同的符号，常常产生混乱。我们对过去文献中所使用的杂乱的马修函数符号进行重新规范，提出的一套表示整数阶角向和径向马修函数的符号，这一工作有助于认识和理解马修函数，规范马修函数符号的使用。为计算有关椭圆波导问题，我们还介绍了马修函数的数值计算
The investigations on the propagation characteristics of waveguides are an important major in electromagnetic theory. Though many investigations have taken to the regular cylinderical metal waveguides or coaxial lines, these investigations mainly focused on the waveguides or coaxial lines filled with single layer medium or two layers of media. No investigations are related to the regular cylinderical metal waveguides or coaxil lines filled with multilayered media. These propagation characteristics will be given in this thesis.
We know that their constitutive relations of the materials represent the respondences of the materials to the electromagnetic wave. The constitutive relations of the media describe the relationships between electric and magnetic fields. Since there are many types of constitutive relations of chiral media in the scientific literatures, some mistakes are easily made when we use them.
Firstly, in this thesis, we will summarize the types of the chiral constitutive relations and introduce four elementary types of constitutive relations for the reciprocal and non-reciprocal chiral medium. The forms will be changed as different time conventions are used in the electromagnetic theory. In addition, some other constitutive relations for chiral medium are given.
Secondly, the methods to solve the homogeneous scalar Helmholtz equation in the elliptical-cylindrical coordinate system and the angular and the radial Mathieu functions are introduced. Mathieu functions are the solutions of the angular and radial Mathieu equations. These functions are used in many physical and astronomical problems, and were inroduced by French mathematician Emile Leonard Mathieu in 1868, as the result of investigating the vibrating modes in an elliptic membrane. These functions are also used to solve the problems of the electromagnetic waves scattered by an elliptical cylinder and of electromagnetic wave propagation in the elliptical waveguide or elliptical coaxial line. Actually, Mathieu equations are two transversal equations derived from the homogeneous scalar Helmholtz equation by using the method of separation of variables in the elliptical-cylindrical coordinate system. These two equations are called angular Mathieu and radial Mathieu equations, respectively, and their solutions are called angular Mathieu and radial Mathieu functions. They can be divided into integral
我们知道，材料对电磁波的响应是通过本构关系中材料参数来反映的，本构关系描写了电磁场量之间的函数依从关系。手征介质本构关系在文献中有多种形式，导致在应用时容易发生错误。
论文首先总结描写手征介质的本构关系，给出描写互易和非互易手征介质本构关系的四种基本形式，这四种基本形式又由于在应用中所取时谐因子的不同，其具体的形式有略有不同。另外，论文还介绍一些其它手征类介质的本构关系。
其次，我们详细而全面地介绍了在椭圆柱坐标系中求解齐次标量亥姆霍兹方程的方法以及所涉及到的角向马修函数和径向马修函数。马修函数是角向和径向马修方程的解，是由法国数学家E．L．Mathieu于1868年在分析椭圆形膜的振动时提出来的，在许多物理和天文学问题中都要用到此函数，在椭圆柱形物体的电磁散射、电磁波在椭圆波导及同轴线传播等问题中也要用到此函数。马修方程实际上是在椭圆柱坐标系中，用分离变量法，由齐次标量亥姆霍兹方程得到的两个横向方程，它分为角向马修方程和径向马修方程，其解称为角向马修函数和径向马修函数。如果按阶次是整数或非整数来划分，马修函数又可分为整数阶马修函数和非整数阶马修函数；如果按奇偶性来分，每种角向马修函数和径向马修函数又有奇、偶两种形式。如果按是否是周期函数来分，马修函数又可分为周期和非周期马修函数。角向马修函数有第一类和第二类角向马修函数(非周期)，整数阶径向马修函数按其和贝塞尔函数的对应关系可分为第一类径向马修函数、第二类径向马修函数和马修-汉克尔函数。由于马修函数本身的复杂性，其应用比较难，函数符号的使用上也没有统一规定，不同学者采用不同的符号，常常产生混乱。我们对过去文献中所使用的杂乱的马修函数符号进行重新规范，提出的一套表示整数阶角向和径向马修函数的符号，这一工作有助于认识和理解马修函数，规范马修函数符号的使用。为计算有关椭圆波导问题，我们还介绍了马修函数的数值计算
The investigations on the propagation characteristics of waveguides are an important major in electromagnetic theory. Though many investigations have taken to the regular cylinderical metal waveguides or coaxial lines, these investigations mainly focused on the waveguides or coaxial lines filled with single layer medium or two layers of media. No investigations are related to the regular cylinderical metal waveguides or coaxil lines filled with multilayered media. These propagation characteristics will be given in this thesis.
We know that their constitutive relations of the materials represent the respondences of the materials to the electromagnetic wave. The constitutive relations of the media describe the relationships between electric and magnetic fields. Since there are many types of constitutive relations of chiral media in the scientific literatures, some mistakes are easily made when we use them.
Firstly, in this thesis, we will summarize the types of the chiral constitutive relations and introduce four elementary types of constitutive relations for the reciprocal and non-reciprocal chiral medium. The forms will be changed as different time conventions are used in the electromagnetic theory. In addition, some other constitutive relations for chiral medium are given.
Secondly, the methods to solve the homogeneous scalar Helmholtz equation in the elliptical-cylindrical coordinate system and the angular and the radial Mathieu functions are introduced. Mathieu functions are the solutions of the angular and radial Mathieu equations. These functions are used in many physical and astronomical problems, and were inroduced by French mathematician Emile Leonard Mathieu in 1868, as the result of investigating the vibrating modes in an elliptic membrane. These functions are also used to solve the problems of the electromagnetic waves scattered by an elliptical cylinder and of electromagnetic wave propagation in the elliptical waveguide or elliptical coaxial line. Actually, Mathieu equations are two transversal equations derived from the homogeneous scalar Helmholtz equation by using the method of separation of variables in the elliptical-cylindrical coordinate system. These two equations are called angular Mathieu and radial Mathieu equations, respectively, and their solutions are called angular Mathieu and radial Mathieu functions. They can be divided into integral
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