水平偶极子在涂敷单轴介质的高耗介质表面激励的场
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摘要
对电偶极子在三层介质中激起的电磁场的研究有很多实际意义，比如长波在有覆盖层的海面或地面上的传播，微带天线在导电基底的硅片上激起的电磁场等。但对这个问题的研究出现了不同的看法和分歧，King和Sandler认为：只要中间层的厚度不大，远区沿介质表面传播的波主要是侧面波，它将以ρ~(-2)规律衰减。而Wait在他的评论中却认为，在King和Sandler研究的情况下，远区场应该主要以吸附表面波为主，它将以ρ~(-1／2)规律衰减。后来，Mahmoud也对King的文章给出了评论。
Collin、张红旗、李凯等人对这个问提进行了深入探讨。Collin给出了覆盖一层介质的地球表面上由赫兹偶极子激励的电磁场E_z分量的严格解。张给出了垂直偶极子在涂敷介质的理想和非理想导电基底上产生的场，以及水平偶极子在涂敷介质的理想导电基底上产生的场和埋入基底中的水平偶极子在基底一侧产生的场。在此基础上，李研究了水平偶极子在涂敷单轴各向异性介质层的理想导电基底上激励的场。
在实际情况中，不仅要顾及涂敷层的各向异性特性，同时更一般的情况是有耗基底。针对更实际的情况，本文将研究水平电偶极子在涂敷单轴介质的高耗介质上激励的电磁场。利用张、李论文中所述的类似方法，本文首先给出了六个场分量的由三项组成的完整的积分表达式。前两项分别是直达波和理想反射波，最后一项表示由涂敷层带来的影响。由复变函数理论，把最后这一积分项的求解分解成求被积函数极点的留数和求沿支点形成的割缝的积分两部分。由极点留数得到的那项是吸附表面波，而由沿支点割缝积分得到的是侧面波。
最后求得的这些解包括直达波，理想反射波，吸附表面波和侧面波。而且吸附表面波和侧面波也分别有电型(TM)和磁型(TE)两种。当涂敷层厚度ι满足nπ＜(κ_T)／(κ_L)(κ_L~2-κ_0~2)~(1／2)·ι＜(n+1)π，能激起n+1个模式的电型吸附表面波，其波数在κ_0和κ_L之间，当(n-(1／2))π＜(κ_T~2-κ_0~2)~(1／2)·ι＜(n+(1／2))π，能激起n个模式的磁型吸附表面波，其波数则在κ_0和κ_T之间。在垂直边界的方向上，吸附表面波随指数衰减。在ρ方向上，由于基底的损耗，吸附表面波有较大衰减。侧面波不管是电型还是磁型，其波数都是κ_0，并且受基底导电性的影响较小。
In the past many years, the electromagnetic field generated by a dipole source in the presence of three-layered regions has been visited by many investigators because of its wide useful applications, such as the propagation of the low frequency wave over the earth or sea covered by a layer, and the electromagnetic field generated by a microstrip antenna on the silicon chip with a conductive substrate.
In early 1990's, R. W. P. King, et al. concluded that the dominant field in the far region generated by a electric dipole is the lateral-wave, which attenuates with ρ~(-2). In 1998, J. R. Wait has claimed that under the case addressed by King and Sandier, the main field in the far region is the trapped surface wave, which attenuates with ρ~(-1/2). Lately, Mahmoud also presented comments on King's works. The debates between King et al. and the commenters naturally rekindled the interest in the study on the old problem.
Recently, Collin presented the exact field of a hertzian dipole in the air above a dielectric-coated lossy earth. From 2002 to 2005, Zhang et al. have investigated the electromagnetic field of vertical and horizontal electric dipoles in the three-layered medium. Along the researching line, Li has extended the investigation on the electromagnetic field in a three-layered region consisting of air, a uniaxial lay and a perfect conductor.
Considering the actual situations, in which the substrate layer is always imperfect, it is needed to study the field from a horizontal electric dipole above a high lossy medium coated with a uniaxial layer. In this paper, the explicit formulas were derived for the six components of the electromagnetic field in air excited by a horizontal electric dipole over a planar high lossy dielectric coated with a uniaxial layer. Numerical results were also obtained.
Similar to the perfect case addressed by Li, et al, the complete field in the present case is also composed of the direct wave, the ideal reflected wave, the trapped-surface wave, and the lateral wave. Both trapped-surface wave and
lateral wave have two types - the electric type (TM) and the magnetic type (TE). The wave numbers in the ρ direction of the electric-type trapped surface wave, which are between k_0 and k_L, are different from those of the magnetic-type trapped surface wave, which are between k_0 and k_T. When the thickness l of the uniaxial layer satisfies nπ <[(k_T)/(K_L)]({k_L~2— k)0~2)~(1/2) ? l < (n + 1)π, there are n + 1 modes of the electric-type trapped surface waves. When the thickness l satisfies (n — (1/2))π < (k_T~2 — k_0~2)~(1/2) ? l < (n + (1/2))π, there are n modes of the magnetic-type trapped waves. The wave number of the lateral wave is k_0 no matter how thick the uniaxial layer is.
Collin、张红旗、李凯等人对这个问提进行了深入探讨。Collin给出了覆盖一层介质的地球表面上由赫兹偶极子激励的电磁场E_z分量的严格解。张给出了垂直偶极子在涂敷介质的理想和非理想导电基底上产生的场，以及水平偶极子在涂敷介质的理想导电基底上产生的场和埋入基底中的水平偶极子在基底一侧产生的场。在此基础上，李研究了水平偶极子在涂敷单轴各向异性介质层的理想导电基底上激励的场。
在实际情况中，不仅要顾及涂敷层的各向异性特性，同时更一般的情况是有耗基底。针对更实际的情况，本文将研究水平电偶极子在涂敷单轴介质的高耗介质上激励的电磁场。利用张、李论文中所述的类似方法，本文首先给出了六个场分量的由三项组成的完整的积分表达式。前两项分别是直达波和理想反射波，最后一项表示由涂敷层带来的影响。由复变函数理论，把最后这一积分项的求解分解成求被积函数极点的留数和求沿支点形成的割缝的积分两部分。由极点留数得到的那项是吸附表面波，而由沿支点割缝积分得到的是侧面波。
最后求得的这些解包括直达波，理想反射波，吸附表面波和侧面波。而且吸附表面波和侧面波也分别有电型(TM)和磁型(TE)两种。当涂敷层厚度ι满足nπ＜(κ_T)／(κ_L)(κ_L~2-κ_0~2)~(1／2)·ι＜(n+1)π，能激起n+1个模式的电型吸附表面波，其波数在κ_0和κ_L之间，当(n-(1／2))π＜(κ_T~2-κ_0~2)~(1／2)·ι＜(n+(1／2))π，能激起n个模式的磁型吸附表面波，其波数则在κ_0和κ_T之间。在垂直边界的方向上，吸附表面波随指数衰减。在ρ方向上，由于基底的损耗，吸附表面波有较大衰减。侧面波不管是电型还是磁型，其波数都是κ_0，并且受基底导电性的影响较小。
In the past many years, the electromagnetic field generated by a dipole source in the presence of three-layered regions has been visited by many investigators because of its wide useful applications, such as the propagation of the low frequency wave over the earth or sea covered by a layer, and the electromagnetic field generated by a microstrip antenna on the silicon chip with a conductive substrate.
In early 1990's, R. W. P. King, et al. concluded that the dominant field in the far region generated by a electric dipole is the lateral-wave, which attenuates with ρ~(-2). In 1998, J. R. Wait has claimed that under the case addressed by King and Sandier, the main field in the far region is the trapped surface wave, which attenuates with ρ~(-1/2). Lately, Mahmoud also presented comments on King's works. The debates between King et al. and the commenters naturally rekindled the interest in the study on the old problem.
Recently, Collin presented the exact field of a hertzian dipole in the air above a dielectric-coated lossy earth. From 2002 to 2005, Zhang et al. have investigated the electromagnetic field of vertical and horizontal electric dipoles in the three-layered medium. Along the researching line, Li has extended the investigation on the electromagnetic field in a three-layered region consisting of air, a uniaxial lay and a perfect conductor.
Considering the actual situations, in which the substrate layer is always imperfect, it is needed to study the field from a horizontal electric dipole above a high lossy medium coated with a uniaxial layer. In this paper, the explicit formulas were derived for the six components of the electromagnetic field in air excited by a horizontal electric dipole over a planar high lossy dielectric coated with a uniaxial layer. Numerical results were also obtained.
Similar to the perfect case addressed by Li, et al, the complete field in the present case is also composed of the direct wave, the ideal reflected wave, the trapped-surface wave, and the lateral wave. Both trapped-surface wave and
lateral wave have two types - the electric type (TM) and the magnetic type (TE). The wave numbers in the ρ direction of the electric-type trapped surface wave, which are between k_0 and k_L, are different from those of the magnetic-type trapped surface wave, which are between k_0 and k_T. When the thickness l of the uniaxial layer satisfies nπ <[(k_T)/(K_L)]({k_L~2— k)0~2)~(1/2) ? l < (n + 1)π, there are n + 1 modes of the electric-type trapped surface waves. When the thickness l satisfies (n — (1/2))π < (k_T~2 — k_0~2)~(1/2) ? l < (n + (1/2))π, there are n modes of the magnetic-type trapped waves. The wave number of the lateral wave is k_0 no matter how thick the uniaxial layer is.
引文
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[2] R. W. P. King and S. S. Sandler: The electromagnetic field of a vertical electric dipole in the presence of a three-layered region, Radio Sci., (1994), Vol.29, No.1, pp.97-113.
[3] R. W. P. King: The electromagnetic field of a horizontal electric dipole in the presence of a three-layered region, J. Appl. Phys.(1991), Vol.69, No.12, pp.7987-7995.
[4] R. W. P. King: The electromagnetic field of a horizontal electric dipole in the presence of a three-layered region: Supplement, J. Appl. Phys., (1993), Vol.74, No.8, pp.4845-4548.
[5] J. R. Wait: Comment on "The electromagnetic field of a vertical electric dipole in the presence of a three-layered region" by Ronold, W. P. King and Sheldon S. Sandler, Radio Sci. (1998), Vol.33, No.2, pp.251-253.
[6] R. W. P. King, S. S. Sandler: Reply, Radio Sci., (1998), Vol.33, No.2, pp.255-256.
[7] S. F. Mahmoud: Remarks on "The electromagnetic field of a vertical electric dipole over the earth or sea". IEEE Trans. Antennas Propagat., (1999), Vol.46, No.12, pp. 1745-1946.
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