Abstract
In previous studies, the trapped surface wave, which is defined by the residue sums, has been addressed in the evaluation of the Sommerfeld integrals describing electromagnetic field of a vertical dipole in the presence of three-layered or four-layered region. But unfortunately, the existing computational scheme cannot provide analytical solution of the field in the presence of the N-layered region when N > 4. The scope of this paper is to overcome the limitations in root finding algorithm implied by the previous approach and provide solution of poles in stratified media. A set of pole equations following with explicit expressions are derived based on the undetermined coefficient method, which enable a graphical approach to obtain initial values of real roots. Accordingly, the generated trapped surface wave components are computed when both the observation point and the electric dipole source are on or near the surface of a dielectric-coated conductor. Validity, efficiency, and accuracy of the proposed method are illustrated by numerical examples.
1. Introduction
It is known that the electromagnetic fields radiated by a vertical or horizontal electric dipole in stratified media are interesting and practically important in many cases [1–3]. In the past decades, this problem has led to many published papers and available achievements on analytical solutions [4–21], especially the three-layered or four-layered cases.
The general integral representation of the electromagnetic field due to a dipole source has been addressed by King et al. [6] in the presence of the N-layered region, which was developed from the original expressions firstly formulated by Sommerfeld in 1909 [7] in the presence of half-spaces. Efforts have been made in a series of works [6–8] by investigators to derive analytical expressions for the Sommerfeld integrals, which leads to a better understanding of describing electromagnetic radiation from the dipole source than numerical solution, as well as time savings with respect to conventional techniques used to evaluate the integrals. In Chapter 15 of the monograph [6], the propagation of the electromagnetic pulses radiated by a horizontal electric dipole with delta-function excitation in the presence of the three-layered region was processed analytically, which demonstrated that the total field on or near the air-dielectric boundary is determined primarily by lateral wave, where the amplitude of the field along the boundary is . Unfortunately, the integrals cannot be evaluated by means of the mentioned analytical procedure for electromagnetic field in the presence of the N-layered region.
In the comments by Wait and Mahoud et al. in 1998 [9, 10], and studies by other pioneers, particularly including Collin [11, 12] and Zhang and Pan [13], the three-layered structure was reconsidered by the use of asymptotic methods, contour integration, and branch cuts, where it is pointed out that the trapped surface wave, which is determined by residue sums of the poles, can be excited efficiently by a dipole source with the amplitude of the field and should not be neglected over a dielectric-coated conductor [13]. To extend the study, the Sommerfeld integrals have been evaluated in the four-layered case by a similar method [14, 15], correspondingly. The details are summarized in a recent book by Li [16].
These new developments on the analytical results for the electromagnetic field in three-layered and four-layered structures [9–16], where it was proved that the trapped surface wave defined by the residue sums as the dominant wave propagates along the surface of air-dielectric boundary at long propagation distance [13], aroused interest in the study on properties of the trapped surface wave in the N-layered structure. However, few literatures relate to a computational scheme to solve the poles, for which the surface impedance at the air-dielectric boundary is in expression of a recursive form due to multireflections [6, 17]. Specially, the discrete pole root is hard to solve by analytical and numerical solutions due to multivalue properties of the recursive equation over four-layered media.
In a recent study by Cross, the solution of poles relying on numerical root finding algorithms is addressed in the four-layered structure to approximate the scenario of a leaking water-pipe, buried in the shallow subsurface [18], where it is suggested that the analytical solution of poles requires efficiency and enhancement of accuracies in stratified media.
Following the research line, we are attempting to derive the pole equation to release the difficulties in root finding in the N-layered structure. In the analysis, a set of pole equations with explicit expressions is developed for a computational scheme, so as to make it possible analytical evaluation of the trapped surface wave based on the undetermined coefficient method. The solution of obtained equations in three-layered to six-layered structures is carried out, respectively, as illustrative examples in lossless case by a graphical approach. The obtained equations offer advantages in terms of time savings with respect to standard pole equation in root finding procedures. In addition, computation and discussion are carried out by investigating the full-wave analytical expressions, as well as their interfering behaviour, which guarantees correctness by evaluating the residue sums' contributions to the fields, but also allows us to gain useful insight into the physics of the problem.
2. Formulation of the Problem
The 3D geometry under consideration and its 2D cylindrical coordinate system are shown in Figures 1 and 2, respectively, where the vertical electric dipole in the direction is located at . Region is the upper half-space occupied by air, and the lower half-space is composed of a success of horizontal layers, each with arbitrary thickness and arbitrary wave number , in which
In the monograph by King et al. [6], the general integral formulas are addressed for the electromagnetic field due to a vertical electric dipole in the presence of N-layered media. The integrated formulas in Region 0 (Air) are written in forms of
In aforementioned formulas, represents for the number of intermediate dielectric layers and the reflection coefficient is expressed byin which represents the surface impedance. If the procedure is continued to the boundary at between Region 0 and Region 1, the desired surface impedance is obtained. It is
The first and second terms in (2)–(4) stand for the direct wave and the ideal reflected wave, respectively, and have been previously solved in [6] by King et al. Considering that and are functions of , which is made of the relations between Bessel and Hankel function:
Thus, the third terms in (2)–(4) are rewritten as follows:
By similar treatment of asymptotic methods, contour integration and branch cuts applied in three-layered and four-layered structures, which is addressed in detail by Li [16], the terms from (9)–(11) can be considered as the combination of the branch cut integrals and contribution of the residues of poles, which are defined by the trapped-surface-wave-term and the lateral-wave-term, respectively. It is necessary to shift the contours around all the branches at , , …, , respectively. Since the approximation of each branch cut integral refers to a lot of mathematical derivations, which is beyond the main scope of this paper, the lateral-wave-term is not addressed in illustrative examples for simplicity. In the next, attention is made on evaluation of the trapped-surface-wave-term by inviting analytical techniques to evaluate the discrete pole residues.
3. Evaluation of the Trapped Surface Wave in the Presence of the N-Layered Region
3.1. Residue of the Poles for the Trapped-Surface-Wave-Term
Following the integration procedure addressed in [16], the trapped-surface-wave-term is defined by the sum of pole residues. In the N-layered structure, the terms of trapped surface wave due to a vertical electric dipole are written asin which the function is expressed byand are the discrete roots of the pole equation of the electromagnetic field in the presence of the N-layered region, which is in a variant of the surface impedance function at the air-to-dielectric boundary where represents the number of intermediate dielectric layers of the N-layered structure, written as
Considering the pole equation by (14) with substitution of (6) in the form of a recursive expression, it is necessary to derive the explicit pole equation first. Suppose (14) can be rewritten with respect to the function , so thatin which is defined by
The superscript n of function represents the recursive order of the function that is in a variant of . In the meanwhile, (15) can be expanded if the order by taking into account of (6) with the function . Specifically, it is expressed by a set of sequential equations, as follows:
It is seen from equation (17) that the recursive order n of function is reduced if the following identity is satisfied:
Combining equations from (17), it is inferred that is applied in each equation when . Consequently, the equations are adapted by exploiting both (18) and , as follows:
In (19), the explicit expression of pole equation in the three-layered structure is easy to obtain by . Similarly, for general case with n > 3, such as in the four-layered and five-layered structures by (20) and (21), respectively, where the function with respect to the variable has been applied, the explicit equation is defined by suppressing . Through mathematical derivations, the variant coefficients are derived accordingly in expression of recursive form, as follows:
Therefore, it is concluded that the general expression of pole equation for electromagnetic field in the presence of N-layered region is derived from (14), in expression ofwhere
3.2. Solution of Poles: A Graphical Approach
In order to evaluate the pole residues analytically, the set of equations from (19)–(21) are developed into explicit expressions with substitution of equations from (22). For convenience, the pole equation in the presence of three-layered region obtained from (19) is rewritten as
Analogously, the pole equation is derived for electromagnetic field of a vertical electric dipole in the presence of N-layered region by substituting (23) with (24), iteratively. Specifically, the pole equations for the fields in the four-layered to six-layered structures can be written as follows:
By a few mathematical efforts, the functions and from (26)–(28) are derived through iterative substitutions, which are listed in Appendix, correspondingly. It is noted that the pole equation can be expressed in analogous form for the electromagnetic field of a vertical dipole in the presence of N-layered media. Accordingly, substituting the equations from (26)–(28) with functions , the explicit expression of equations are obtained. By transpositions and rearrangements, the expression of pole equation in the presence of N-layered region can be described asin which the coefficients , j = 1, …, m; m = 1, …, n are a set of coefficients defined by positive or negative ratio of and , which are determined from (26)–(28). If the bottom half-space is considered as perfectly conducting layer as depicted in Figure 3, the expression of (29) are simplified in condition of , which reduces to
It is seen from (29) and (30) that the obtained equations for the fields are expressed in form of addition, subtraction, and multiplication of functions with coefficients , which offer advantages in terms of time savings with respect to standard numerical root finding procedures by (14). Specifically, the obtained pole equations for the electromagnetic field over a conductor coated by two-layered to four-layered dielectrics are in terms of the following equations:
It is convenient to validate (31) in derivation followed, which is exactly the same with (4.24) presented in [16] for the electromagnetic field of a vertical electric dipole on the surface of a conductor coated by two dielectric layers.
Through the aforementioned analysis, the computational scheme to evaluate the residue sums for the trapped surface wave is proposed by exploiting the derived equations from (25)–(28). A graphical approach is applied in the present study to obtain initial values of real roots. For instance of a five-layered structure interpreted in Figure 4, the real roots are obtained from intersections of functions and , where the intermediate dielectric layers are assumed to be lossless.
4. Computation and Discussion
4.1. Illustrative Examples by a Graphical Approach
Primary objective of this section is to show the graphical approach to evaluate the poles by equations from (25)–(28) representing the three-layered to six-layered structures, respectively. In Figure 5 and 6, the real roots are obtained in lossless case, where the bottom half-space is considered as perfectly conducting and the electric lengths of intermediate dielectrics are identical. The computation coefficients of relative permittivity in the intermediate dielectrics are chosen by , , , and , respectively, where each electric length of the intermediate dielectrics are chosen by and , respectively.
(a)
(b)
(c)
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As illustrated in Figure 5, the roots of equations from (25) to (27) are obtained from intersections of functions and the right side terms on the condition of . It is seen from Figures 5(c)–5(h), the terms of right side of (27)/i representing five-layered structure have imaginary part in the region of , since the function is imaginary in the region of . In Figure 6, the roots of (28) are plotted on condition of for electromagnetic field of a vertical dipole in the six-layered structure. In the computations, the intermediate dielectrics are chosen as lossless and the bottom half-space is perfectly conducting, whereas in the general cases where the roots may not be always real, the resulted roots are considered as initial values for numerical iterations.
In Figure 7, by applying the roots of poles, the electromagnetic field of a vertical dipole is computed, correspondingly. The total field, the trapped surface wave, and DRL waves (direct wave, reflected waves, and lateral wave) are computed with the same parameters in Figures 5(b) and 5(f), respectively, at the operating frequency f = 100 MHz. It is seen from Figure 7 that the trapped surface wave propagates along the air-to-dielectric boundary as the dominate wave on the condition of , and , in the presence of four-layered region. When , is always a positive imaginary number, and the terms including the factor will attenuate exponentially as in the z direction. In Figure 8, the curves of the trapped surface wave versus propagation distance are plotted in three-layered to five-layered structures, respectively, when both the observation point and dipole source are placed on the surface of air-dielectric boundary. The electric lengths of each layer of intermediate dielectrics are identical with . It is seen that field strength for trapped surface wave disperses as increased by layers of coating dielectrics.
One would ask whether the derived equation in expression of (23) is advantageous in terms of computation over previous numerical procedures. This aspect is illustrated in Table 1, which shows the computation complexity and nested recursion times taken by the proposed method, and the previous numerical solution, to calculate the poles of the trapped surface wave generated by a vertical electric dipole lying on a horizontal layered medium at z = 0.
4.2. Interfering Behaviour of the Electromagnetic Field above the Surface of a Dielectric-Coated Conductor
To investigate the propagation properties of the electromagnetic field of a vertical electric dipole over a dielectric-coated conductor, the spatial distributions in plane are plotted in Figures 9 to 11 at the operating frequency f = 100 MHz, when both the observation point and radiating source are located at the surface of air-dielectric boundary. In Figure 9, the DR fields (including the direct wave and ideal reflected wave) are computed by the first two terms in equations from (2)–(4). It is noted that the DR terms in the multilayered structures are the same with those for the uniform half-space. For a three-layered structure, the total fields (including the DR waves, lateral wave, and trapped surface wave) are plotted in Figure 10 with the relative permittivity and electric length of the intermediate dielectric chosen as , and , respectively. It is seen that the DR wave, lateral wave, and the trapped surface wave are combined in the total field to produce an interference pattern.
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(b)
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In order to investigate the properties of the electromagnetic field, the total field (including the DR waves, lateral wave, and trapped surface wave) is computed in Figures 11(a) and 11(b), respectively. The lateral wave and trapped surface wave are computed in Figures 11(c) and 11(d), respectively. The computation coefficients of relative permittivity in the intermediate dielectrics are chosen as the same in Figures 5(c) and 5(d), while each electric length of the intermediate dielectrics are chosen by and , respectively. It is seen from Figures 11(c) and 11(d) that in the direction and the trapped surface, wave attenuates rapidly away from the surface of air and dielectrics.
5. Conclusion
In summary, a computational scheme is presented to evaluate the residue sums for the fields of a vertical electric dipole oriented perpendicular to a stratified medium. The pole equation is reconstructed with explicit expression in the presence of the N-layered region. Therefore, a set of explicit equations is derived to enable analytical solutions on roots finding algorithms. Computations and analysis of electromagnetic field and its spatial distributions are carried out on and above a planar multidielectric-coated perfect conductor. The lateral wave and the trapped surface wave are combined in the total field to produce an interference pattern. It has been shown how the proposed equations are accurate and have significantly less computation complexity than previous numerical procedures.
Appendix
The coefficients in to and to are defined as follows:
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant number 61490695.