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Research Article Evaluation of the Trapped Surface Wave of a Vertical Electric Dipole Based on Undetermined Coefficient Method in the Presence of N-Layered Region: A Graphical Approach HongLeiXu , 1 Ting Ting Gu , 2 Yong Zhu, 1 Xiao Wei, 1 Liang Sheng Li, 1 and Hong Cheng Yin 1 1 Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854, China 2 Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China Correspondence should be addressed to Hong Cheng Yin; [email protected] Received 7 June 2019; Revised 21 August 2019; Accepted 20 September 2019; Published 20 October 2019 Academic Editor: Giuseppina Monti Copyright © 2019 Hong Lei Xu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. e 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. Ac- cordingly, 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. e general integral representation of the electromag- netic 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 in- vestigators to derive analytical expressions for the Som- merfeld 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 1/ρ 2 . Un- fortunately, 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 de- termined by residue sums of the poles, can be excited Hindawi International Journal of Antennas and Propagation Volume 2019, Article ID 1657587, 12 pages https://doi.org/10.1155/2019/1657587

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  • Research ArticleEvaluation of the Trapped Surface Wave of a Vertical ElectricDipole Based on Undetermined Coefficient Method in thePresence of N-Layered Region: A Graphical Approach

    Hong Lei Xu ,1 Ting Ting Gu ,2 Yong Zhu,1 Xiao Wei,1 Liang Sheng Li,1

    and Hong Cheng Yin 1

    1Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854, China2Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China

    Correspondence should be addressed to Hong Cheng Yin; [email protected]

    Received 7 June 2019; Revised 21 August 2019; Accepted 20 September 2019; Published 20 October 2019

    Academic Editor: Giuseppina Monti

    Copyright © 2019 Hong Lei Xu et al. +is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    In previous studies, the trapped surface wave, which is defined by the residue sums, has been addressed in the evaluation of theSommerfeld 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 whenN> 4.+e scope of this paper is to overcome the limitations in root finding algorithm implied by the previousapproach and provide solution of poles in stratified media. A set of pole equations following with explicit expressions are derivedbased on the undetermined coefficient method, which enable a graphical approach to obtain initial values of real roots. Ac-cordingly, the generated trapped surface wave components are computed when both the observation point and the electric dipolesource are on or near the surface of a dielectric-coated conductor. Validity, efficiency, and accuracy of the proposed method areillustrated by numerical examples.

    1. Introduction

    It is known that the electromagnetic fields radiated by avertical or horizontal electric dipole in stratified media areinteresting and practically important in many cases [1–3]. Inthe past decades, this problem has led to many publishedpapers and available achievements on analytical solutions[4–21], especially the three-layered or four-layered cases.

    +e general integral representation of the electromag-netic field due to a dipole source has been addressed by Kinget al. [6] in the presence of the N-layered region, which wasdeveloped from the original expressions firstly formulatedby Sommerfeld in 1909 [7] in the presence of half-spaces.Efforts have been made in a series of works [6–8] by in-vestigators to derive analytical expressions for the Som-merfeld integrals, which leads to a better understanding ofdescribing electromagnetic radiation from the dipole sourcethan numerical solution, as well as time savings with respect

    to conventional techniques used to evaluate the integrals. InChapter 15 of the monograph [6], the propagation of theelectromagnetic pulses radiated by a horizontal electricdipole with delta-function excitation in the presence of thethree-layered region was processed analytically, whichdemonstrated that the total field on or near the air-dielectricboundary is determined primarily by lateral wave, where theamplitude of the field along the boundary is 1/ρ2. Un-fortunately, the integrals cannot be evaluated by means ofthe mentioned analytical procedure for electromagnetic fieldin 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 includingCollin [11, 12] and Zhang and Pan [13], the three-layeredstructure was reconsidered by the use of asymptoticmethods, contour integration, and branch cuts, where it ispointed out that the trapped surface wave, which is de-termined by residue sums of the poles, can be excited

    HindawiInternational Journal of Antennas and PropagationVolume 2019, Article ID 1657587, 12 pageshttps://doi.org/10.1155/2019/1657587

    mailto:[email protected]://orcid.org/0000-0002-4949-182Xhttps://orcid.org/0000-0002-2877-7191https://orcid.org/0000-0003-1685-7394https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/1657587

  • e�ciently by a dipole source with the amplitude of the �eld1/ρ1/2 and should not be neglected over a dielectric-coatedconductor [13]. To extend the study, the Sommerfeld in-tegrals have been evaluated in the four-layered case by asimilar method [14, 15], correspondingly.  e details aresummarized in a recent book by Li [16].

     ese new developments on the analytical results for theelectromagnetic �eld in three-layered and four-layeredstructures [9–16], where it was proved that the trappedsurface wave de�ned by the residue sums as the dominantwave propagates along the surface of air-dielectric boundaryat long propagation distance [13], aroused interest in thestudy on properties of the trapped surface wave in the N-layered structure. However, few literatures relate to acomputational scheme to solve the poles, for which thesurface impedance at the air-dielectric boundary is in ex-pression of a recursive form due to multireections [6, 17].Specially, the discrete pole root λ∗j is hard to solve by an-alytical and numerical solutions due tomultivalue propertiesof the recursive equation over four-layered media.

    In a recent study by Cross, the solution of poles relying onnumerical root �nding algorithms is addressed in the four-layered structure to approximate the scenario of a leakingwater-pipe, buried in the shallow subsurface [18], where it issuggested that the analytical solution of poles requires e�-ciency and enhancement of accuracies in strati�ed media.

    Following the research line, we are attempting to derivethe pole equation to release the di�culties in root �nding inthe N-layered structure. In the analysis, a set of poleequations with explicit expressions is developed for acomputational scheme, so as to make it possible analyticalevaluation of the trapped surface wave based on the un-determined coe�cient method.  e solution of obtainedequations in three-layered to six-layered structures is carriedout, respectively, as illustrative examples in lossless case by agraphical approach. e obtained equations oer advantagesin terms of time savings with respect to standard poleequation in root �nding procedures. In addition, compu-tation and discussion are carried out by investigating thefull-wave analytical expressions, as well as their interferingbehaviour, which guarantees correctness by evaluating theresidue sums’ contributions to the �elds, but also allows us togain useful insight into the physics of the problem.

    2. Formulation of the Problem

     e 3D geometry under consideration and its 2D cylindricalcoordinate system are shown in Figures 1 and 2, respectively,where the vertical electric dipole in the ϕ � 0 direction islocated at (0, 0, d). Region z> 0 is the upper half-spaceoccupied by air, and the lower half-space is composed of asuccess of n + 1 horizontal layers, each with arbitrary thicknesslj(j � 1, 2, . . . , n) and arbitrary wave number kj, in which

    k0 � ω����μ0ε0

    √,

    kj � ω

    �������������

    μ0 ε0εrj +iσjω

    ( )

    , j � 1, 2, . . . , n + 1.(1)

    In the monograph by King et al. [6], the general integralformulas are addressed for the electromagnetic �eld due to avertical electric dipole in the presence of N-layered media. e integrated formulas in Region 0 (Air) are written informs of

    B0ϕ �iμ04π∫∞

    0[eic0|z− d| + eic0(z+d)

    − Qn + 1( ) eic0(z+d)]c− 10 J1(λρ)λ

    2dλ,

    (2)

    E0ρ �iωμ04πk20

    ∫∞

    0[±eic0|z− d| + eic0(z+d)

    − Qn + 1( )eic0(z+d)]J1(λρ)λ

    2dλ,z≥d,0< z

  • E0z � −ωμ04πk20

    0e

    ic0|z− d| + eic0(z+d)

    − Qn + 1( eic0(z+d)c

    − 10 J0(λρ)λ

    3dλ.(4)

    In aforementioned formulas, n represents for thenumber of intermediate dielectric layers and the reflectioncoefficient Qn is expressed by

    Qn � −c0 − k

    20/ ωμ0( Zs0(0)

    c0 + k20/ ωμ0( Zs0(0)

    , (5)

    in which Zs0(0) represents the surface impedance. If theprocedure is continued to the boundary at z′ � z � 0 be-tween Region 0 and Region 1, the desired surface impedanceis obtained. It is

    Zs0′ (0) �ωμ0c0

    k20tanhtanh− 1

    c1k20

    c0k21

    tanh− ic1l1

    + tanh− 1c2k

    21

    c1k22

    tanh− ic2l2

    + tanh− 1c3k

    22

    c2k23

    tanh− ic3l3 + · · ·

    + tanh− 1cnk

    2n− 1

    cn− 1k2n

    tanh− icnln

    + tanh− 1cn+1k

    2n

    cnk2n+1

    . . . ,

    (6)

    ci �

    ������

    k2i − λ2

    ; i � 0, 1, 2, . . . , n + 1. (7)

    +e first and second terms in (2)–(4) stand for the directwave and the ideal reflected wave, respectively, and havebeen previously solved in [6] by King et al. Considering thatc0 and c1 are functions of λ, which is made of the relationsbetween Bessel and Hankel function:

    Jn(λρ) �12

    H(1)n (λρ) + H

    (2)n (λρ) ,

    H(1)n (− λρ) � H

    (2)n (λρ)(− 1)

    n+1.

    (8)

    +us, the third terms in (2)–(4) are rewritten as follows:

    B(3)0ϕ � −

    ik204πω

    − ∞

    Zs0(0)H(1)1 (λρ)eic0(z+d)λ

    2

    c0 + k20/ ωμ0( Zs0′ (0) c0

    dλ, (9)

    E(3)0ρ � −

    i

    4π∞

    − ∞

    Zs0(0)H(1)1 (λρ)eic0(z+d)λ

    2

    c0 + k20/ ωμ0( Zs0′ (0)

    dλ, (10)

    E(3)0z �

    14π

    − ∞

    Zs0(0)H(1)0 (λρ)eic0(z+d)λ

    3

    c0 + k20/ ωμ0( Zs0′ (0) c0

    dλ. (11)

    By similar treatment of asymptotic methods, contourintegration and branch cuts applied in three-layered andfour-layered structures, which is addressed in detail by Li[16], the terms from (9)–(11) can be considered as thecombination of the branch cut integrals and contribution ofthe 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 atλ1 � k0, λ2 � k1, . . ., λj � kn+1, respectively. Since the ap-proximation of each branch cut integral refers to a lot ofmathematical derivations, which is beyond the main scopeof this paper, the lateral-wave-term is not addressed in il-lustrative examples for simplicity. In the next, attention ismade on evaluation of the trapped-surface-wave-term byinviting analytical techniques to evaluate the discrete poleresidues.

    3. Evaluation of the Trapped Surface Wave inthe 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], thetrapped-surface-wave-term is defined by the sum of poleresidues. In the N-layered structure, the terms of trappedsurface wave due to a vertical electric dipole are written as

    BSur0ϕ �

    k202ω

    j

    Zs0′ λ∗j H

    (1)1 λ∗j ρ e

    ic0 λ∗j (z+d) λ∗j

    2

    qN′ λ∗j c0 λ

    ∗j

    ,

    ESur0ρ �

    12

    j

    Zs0′ λ∗j H

    (1)1 λ∗j ρ e

    ic0 λ∗j (z+d) λ∗j

    2

    qN′ λ∗j

    ,

    ESur0z �

    i

    2

    j

    Zs0′ λ∗j H

    (1)0 λ∗j ρ e

    ic0 λ∗j (z+d) λ∗j

    3

    qN′ λ∗j c0 λ

    ∗j

    ,

    (12)

    in which the function qN′ is expressed by

    qN′ �z

    zλc0 +

    k20ωμ0

    Zs0′ (0)n�N− 2 , (13)

    and λ∗j (j � 1, 2, . . .) are the discrete roots of the poleequation of the electromagnetic field in the presence of theN-layered region, which is in a variant of the surface im-pedance function Zs0′ (0) at the air-to-dielectric boundarywhere n � N − 2 represents the number of intermediatedielectric layers of the N-layered structure, written as

    qN � c0 +k20ωμ0

    Zs0′ (0)n�N− 2 � 0. (14)

    Considering the pole equation by (14) with substitutionof (6) in the form of a recursive expression, it is necessary toderive the explicit pole equation first. Suppose (14) can berewritten with respect to the function G(n)N , so that

    qN � c0 +c1k

    20

    k21G

    (n)N (λ) � 0, (15)

    International Journal of Antennas and Propagation 3

  • in which G(n)N is defined by

    G(n)N (λ) �

    k21ωμ0c1

    Zs0′ (0). (16)

    +e superscript n of function G(n)N represents the re-cursive order of the function that is in a variant of λ. In themeanwhile, (15) can be expanded if the order n≥ 1 bytaking into account of (6) with the function G(n− 1)N . Spe-cifically, it is expressed by a set of sequential equations, asfollows:

    q|N�2 � c0 +c1k

    20

    k21G

    (0)2 (λ),

    q|N�3 � c0 +c1k

    20

    k21G

    (1)3 (λ)

    � c0 +c1k

    20

    k21tanh − ic1l1 + tanh

    − 1 c2k21

    c1k22

    G(0)3 (λ) ,

    q|N�4 � c0 +c1k

    20

    k21G

    (2)4 (λ)

    � c0 +c1k

    20

    k21tanh − ic1l1 + tanh

    − 1 c2k21

    c1k22

    G(1)4 (λ)

    � c0 +c1k

    20

    k21tanh − ic1l1 + tanh

    − 1

    c2k21

    c1k22

    · tanh − ic2l2 + tanh− 1 c3k

    22

    c2k23

    G(0)4 (λ),

    . . .

    q|N � c0 +c1k

    20

    k21G

    (n)N (λ)

    � c0 +c1k

    20

    k21tanh − ic1l1 + tanh

    − 1 c2k21

    c1k22

    G(n− 1)N (λ) .

    (17)

    It is seen from equation (17) that the recursive order n offunction G(n)N is reduced if the following identity is satisfied:

    G(j+1)N (λ) � tanh − icmlm + tanh

    − 1 cm+1k2m

    cmk2m+1

    G(j)

    N (λ) ,

    m � N − j − 1.(18)

    Combining equations from (17), it is inferred thatG

    (j)N (λ)|j�0 ≡ 1 is applied in each equation when n � N − 2.

    Consequently, the equations are adapted by exploiting both(18) and G(j)N (λ)|j�0 ≡ 1, as follows:

    N � 3 : tanh ic1l1( + G(0)3 (λ)

    �c2k

    21

    c1k22

    + G(0)3 (λ)

    c2k21

    c1k22tanh ic1l1( , whenG

    (0)3 (λ) � 1,

    (19)

    N � 4 : tanh ic2l2( + G(1)4 (λ)

    �c2k

    21

    c1k22

    + G(1)4 (λ)

    c2k21

    c1k22tanh ic2l2( , whenG

    (0)4 (λ) � 1,

    (20)

    N � 5 : tanh ic3l3( + G(2)5 (λ)

    �c2k

    21

    c1k22

    + G(2)5 (λ)

    c2k21

    c1k22tanh ic3l3( , whenG

    (0)5 (λ) � 1,

    . . .

    (21)

    In (19), the explicit expression of pole equation in thethree-layered structure is easy to obtain by G(0)3 � 1. Sim-ilarly, for general case with n> 3, such as in the four-layeredand five-layered structures by (20) and (21), respectively,where the function G(n− 1)N with respect to the variable λ hasbeen applied, the explicit equation is defined by suppressingG

    (0)N � 1. +rough mathematical derivations, the variant

    coefficients are derived accordingly in expression of re-cursive form, as follows:

    N≥ 4 : G(0)3 (λ) � −c0k

    21

    c1k20,

    N≥ 5 : G(1)4 (λ) �tanh ic1l1( + G

    (0)3 (λ)

    c2k21/c1k

    22 + G

    (0)3 (λ) c2k

    21/c1k

    22 tanh ic1l1(

    ,

    N≥ 6 : G(2)5 (λ) �tanh ic2l2( + G

    (1)4 (λ)

    c3k22/c2k

    23 + G

    (1)4 (λ) c3k

    22/c2k

    23 tanh ic2l2(

    .

    . . .

    (22)

    +erefore, it is concluded that the general expression ofpole equation for electromagnetic field in the presence of N-layered region is derived from (14), in expression of

    tanh icnln( + G(n− 1)N (λ) �

    c2k21

    c1k22

    + G(n− 1)N (λ)

    c2k21

    c1k22tanh icnln( , whenG

    (0)N � 1,

    (23)

    4 International Journal of Antennas and Propagation

  • where

    G(j)

    N (λ) �tanh icjlj + G

    (j− 1)N (λ)

    cj+1k2j/cjk

    2j+1 + G

    (j− 1)N (λ) cj+1k

    2j/cjk

    2j+1 tanh icjlj

    .

    (24)

    3.2. Solution of Poles: A Graphical Approach. In order toevaluate the pole residues analytically, the set of equationsfrom (19)–(21) are developed into explicit expressions withsubstitution of equations from (22). For convenience, thepole equation in the presence of three-layered region ob-tained from (19) is rewritten as

    N � 3 : tanh ic1l1( �c2k

    21

    c1k22

    +c0k

    21

    c1k20

    −c0k

    21

    c1k20

    c2k21

    c1k22tanh ic1l1( .

    (25)

    Analogously, the pole equation is derived for electro-magnetic field of a vertical electric dipole in the presence ofN-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:

    N � 4 : tanh ic2l2( �c3k

    22

    c2k23

    +c1k

    22

    c2k21

    ·c1k

    20

    c0k21

    ×tanh ic1l1( − A1(λ)tanh ic1l1( − B1(λ)

    −c3k

    22

    c2k23

    c1k22

    c2k21

    ·c1k

    20

    c0k21

    ×tanh ic1l1( − A1(λ)tanh ic1l1( − B1(λ)

    · tanh ic2l2( ,

    (26)

    N � 5 : tanh ic3l3( �c4k

    23

    c3k24

    +c2k

    23

    c3k22

    ·c0k

    21

    c1k20

    ×tanh ic2l2( − A2(λ)tanh ic2l2( − B2(λ)

    −c4k

    23

    c3k24

    c2k23

    c3k22

    ·c0k

    21

    c1k20

    ×tanh ic2l2( − A2(λ)tanh ic2l2( − B2(λ)

    · tanh ic3l3( ,

    (27)

    N � 6 : tanh ic4l4( �c5k

    24

    c4k25

    +c3k

    24

    c4k23

    ·c0k

    21

    c1k20

    ×tanh ic3l3( − A3(λ)tanh ic3l3( − B3(λ)

    −c5k

    24

    c4k25

    c3k24

    c4k23

    ·c0k

    21

    c1k20

    ×tanh ic3l3( − A3(λ)tanh ic3l3( − B3(λ)

    · tanh ic4l4( .

    . . .

    (28)

    By a few mathematical efforts, the functions Aj(λ) andBj(λ) from (26)–(28) are derived through iterative sub-stitutions, which are listed in Appendix, correspondingly. Itis noted that the pole equation can be expressed in analogousform for the electromagnetic field of a vertical dipole in thepresence of N-layered media. Accordingly, substituting theequations from (26)–(28) with functions Aj(λ), the explicitexpression of equations are obtained. By transpositions andrearrangements, the expression of pole equation in thepresence of N-layered region can be described as

    1 +cn+1k

    2n

    cnk2n+1

    ·cn− 1k

    2n

    cnk2n− 1

    ·

    n− 1m�1

    mj�1C∗n (j, m)tanh icjlj

    n− 1k�1

    mj�1C∗n (m − j, n − m)tanh icjlj

    ⎡⎢⎢⎣ ⎤⎥⎥⎦

    · tanh icnln(

    �cn+1k

    2n

    cnk2n+1

    +cn− 1k

    2n

    cnk2n− 1

    ·1 − n− 1m�1

    mj�1C∗n (j, m)tanh icjlj

    n− 1m�1

    mj�1C∗n (m − j, n − m)tanh icjlj

    ,

    (29)

    in which the coefficients C∗n (j, m), j� 1, . . ., m; m� 1, . . ., nare a set of coefficients defined by positive or negative ratio ofcjkm and cmkj, which are determined from (26)–(28). If thebottom half-space is considered as perfectly conductinglayer as depicted in Figure 3, the expression of (29) aresimplified in condition of kn+1⟶∞, which reduces to

    tanh icnln( �cn− 1k

    2n

    cnk2n− 1

    ·1 − n− 1m�1

    mj�1C∗n (j, m)tanh icjlj

    1 − n− 1m�1mj�1C∗n (m − j, n − m)tanh icjlj

    . (30)

    It is seen from (29) and (30) that the obtained equations forthe fields are expressed in form of addition, subtraction, andmultiplication of functions tanh(icjlj) with coefficients C∗n ,which offer advantages in terms of time savings with respect tostandard numerical root finding procedures by (14). Specifi-cally, the obtained pole equations for the electromagnetic fieldover a conductor coated by two-layered to four-layered di-electrics are in terms of the following equations:

    N � 4 : tanh ic2l2( �c1k

    22

    c2k21

    ·c1k

    20

    c0k21

    tanh ic1l1( − A1(λ)tanh ic1l1( − B1(λ)

    ,

    (31)

    N � 5 : tanh ic3l3( �c2k

    23

    c3k22

    ·c0k

    21

    c1k20

    tanh ic2l2( − A2(λ)tanh ic2l2( − B2(λ)

    ,

    (32)

    International Journal of Antennas and Propagation 5

  • N � 6 : tanh ic4l4( ) �c3k

    24

    c4k23·c0k

    21

    c1k20

    ( )tanh ic3l3( ) − A3(λ)tanh ic3l3( ) − B3(λ)

    .

    . . .

    (33)

    It is convenient to validate (31) in derivation followed,which is exactly the same with (4.24) presented in [16] forthe electromagnetic �eld of a vertical electric dipole on thesurface of a conductor coated by two dielectric layers.

    qPerN�4 � c0 − ic1k

    20

    k21tan c1l1 + tanh

    − 1 c2k21

    c1k22

    ( )tan c2l2( )[ ][ ]

    � c0 − ic1k

    20

    k21tan c1l1( ) +

    c2k21

    c1k22

    ( )tan c2l2( )[ ]

    × 1 −c2k

    21

    c1k22tan c1l1( )tan c2l2( )[ ]

    − 1

    � 0.

    (34)

     rough the aforementioned analysis, the computationalscheme to evaluate the residue sums for the trapped surfacewave is proposed by exploiting the derived equations from(25)–(28). A graphical approach is applied in the present studyto obtain initial values of real roots. For instance of a �ve-layeredstructure interpreted in Figure 4, the real roots are obtainedfrom intersections of functions f(λ) and g(λ), where theintermediate dielectric layers are assumed to be lossless.

    fN�5(λ) � tan c3l3( );

    gN�5(λ) �right side of equation(27)

    i.

    (35)

    4. Computation and Discussion

    4.1. Illustrative Examples by a Graphical Approach.Primary objective of this section is to show the graphicalapproach 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 obtainedin lossless case, where the bottom half-space is consideredas perfectly conducting and the electric lengths of in-termediate dielectrics are identical.  e computation co-e�cients of relative permittivity in the intermediatedielectrics are chosen by εr1 � 2.65, εr2 � 4.0, εr3 � 6.0, andεr4 � 8.0, respectively, where each electric length of theintermediate dielectrics are chosen by kili � 0.45π andkili � 1.45π, respectively.

    As illustrated in Figure 5, the roots of equations from(25) to (27) are obtained from intersections of functionstan(cnln) and the right side terms gN(λ) on the condition ofkn+1⟶∞. It is seen from Figures 5(c)–5(h), the terms ofright side of (27)/i representing �ve-layered structure haveimaginary part in the region of k1 ≤ c∗j ≤ k2, since thefunction c1 � (k

    21 − c∗2j )

    1/2 is imaginary in the region ofcj > k1. In Figure 6, the roots of (28) are plotted on con-dition of k5⟶∞ for electromagnetic �eld of a verticaldipole in the six-layered structure. In the computations, theintermediate dielectrics are chosen as lossless and thebottom half-space is perfectly conducting, whereas in thegeneral cases where the roots may not be always real, theresulted roots are considered as initial values for numericaliterations.

    In Figure 7, by applying the roots of poles, the electro-magnetic �eld of a vertical dipole is computed, correspond-ingly. e total �eld, the trapped surface wave, and DRL waves(direct wave, reected waves, and lateral wave) are com-puted with the same parameters in Figures 5(b) and 5(f ),respectively, at the operating frequency f � 100MHz. It isseen from Figure 7 that the trapped surface wave propa-gates along the air-to-dielectric boundary as the dominatewave on the condition of k1 < k2 < k3, and k3⟶∞, in thepresence of four-layered region. When k0 < λ< k1, cj(λ) �i(λ2 − k2j)

    1/2 is always a positive imaginary number, and theterms including the factor will attenuate exponentially asin the z direction. In Figure 8, the curves of the trappedsurface wave versus propagation distance are plotted inthree-layered to �ve-layered structures, respectively, whenboth the observation point and dipole source are placed onthe surface of air-dielectric boundary.  e electric lengthsof each layer of intermediate dielectrics are identical with

    ε0, μ0, σ0; k0

    ε1, μ1, σ1; k1

    ε2, μ2, σ2; k2

    ε3, μ3, σ3; k3

    εi, μi, σi; ki

    εn, μn, σn; kn

    Air

    γ1l1

    γ2l2

    γ3l3

    γili

    γnln

    zZ↑ + Z↓ = 0

    Z↓ = ωμ0γ0/k02

    Z↑ = Zs0N=n

    z′1 = l1

    z′ Perfect conducting base

    Figure 3:  e N-layered model with the bottom half-space asconducting.

    ε0, μ0, σ0; k0

    ε1, μ1, σ1; k1

    ε2, μ2, σ2; k2

    ε3, μ3, σ3; k3

    Air

    γ1l1

    γ2l2

    γ3l3

    Z↑ + Z↓ = 0

    Z↓ = ωμ0γ0/k02

    Z↑ = Zs0N=4

    z′1 = l1

    z′ Perfect conducting base

    z

    Figure 4: A �ve-layered model with the bottom half-space asperfectly conducting.

    6 International Journal of Antennas and Propagation

  • 0

    10

    20

    30

    40

    2.5 3.53λ

    tan(γ1l1)

    Right-hand side ofequation (25)/i

    (a)

    –5

    0

    5

    10

    15

    20

    2.5 43.53λ

    tan(γ2l2)

    Right-hand side ofequation (26)/i

    (b)

    –40

    –20

    0

    20

    40

    60

    80

    100

    2.5 4 4.5 53.53λ

    Real of tan(γ3l3)

    Real of right-hand side of equation (27)/i

    k0 k1 k2 k3

    (c)

    –40

    –20

    0

    20

    40

    60

    80

    100

    2.5 4 4.5 53.53λ

    Real of tan(γ3l3)

    Real of right-hand side of equation (27)/i

    k0 k1 k2 k3

    (d)

    –20

    –10

    0

    10

    20

    2.5 3.53λ

    tan(γ1l1)

    Right-hand side ofequation (25)/i

    (e)

    –10

    –5

    0

    5

    10

    15

    20

    2.5 43.53λ

    tan(γ2l2)

    Right-hand side ofequation (26)/i

    (f )

    –40

    –20

    0

    20

    40

    60

    80

    100

    2.5 4 4.5 53.53λ

    Imaginary of tan(γ3l3)

    Imaginary of right-hand side of equation (27)/i

    k0 k1 k2 k3

    (g)

    –40

    –20

    0

    20

    40

    60

    80

    100

    2.5 4 4.5 53.53λ

    Imaginary of tan(γ3l3)

    Imaginary of right-hand side of equation (27)/i

    k0 k1 k2 k3

    (h)

    Figure 5: Graphical approach for root finding in evaluations of poles by equations from (25)–(27) at the operating frequency f� 100MHzfor the electromagnetic field due to a vertical dipole over a perfect conductor coated by multilayered lossless dielectrics, with (a) N� 3,kili � 0.45π; (b) N� 4, kili � 1.45π; (c) N� 5, kili � 0.45π; (d) N� 5, kili � 1.45π; (e) N� 3, kili � 1.45π; (f ) N� 4, kili � 1.45π; (g) N� 5,kili � 0.45π; and (h) N� 5, kili � 1.45π.

    –400

    –200

    0

    200

    400

    600

    800

    1000

    2 5 643

    Real of tan(γ4l4)Real of right-hand side of equation (28)/i

    k0 k1 k2 k4k3

    λ

    (a)

    0

    –400

    –200

    200

    400

    600

    800

    1000

    2 5 643

    Imaginary of tan(γ4l4)Imaginary of right-hand side ofequation (28)/i

    k0 k1 k2 k4k3

    λ

    (b)

    Figure 6: Continued.

    International Journal of Antennas and Propagation 7

  • kili � 0.45π. It is seen that field strength for trapped surfacewave disperses as increased by layers of coating dielectrics.

    One would ask whether the derived equation in ex-pression of (23) is advantageous in terms of computationover previous numerical procedures. +is aspect is illus-trated 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 ofthe trapped surface wave generated by a vertical electricdipole lying on a horizontal layered medium at z� 0.

    4.2. Interfering Behaviour of the Electromagnetic Field abovethe Surface of a Dielectric-Coated Conductor. To investigatethe propagation properties of the electromagnetic field of avertical electric dipole over a dielectric-coated conductor, thespatial distributions in ρ − z plane are plotted in Figures 9 to11 at the operating frequency f� 100MHz, when both theobservation point and radiating source are located at thesurface of air-dielectric boundary. In Figure 9, the DR fields(including the direct wave and ideal reflected wave) arecomputed by the first two terms in equations from (2)–(4). It isnoted that the DR terms in the multilayered structures are thesamewith those for the uniformhalf-space. For a three-layeredstructure, the total fields (including the DRwaves, lateral wave,and trapped surface wave) are plotted in Figure 10 with therelative permittivity and electric length of the intermediatedielectric chosen as εr1 � 2.65, and k1l1 � 0.45π, respectively.It is seen that the DR wave, lateral wave, and the trappedsurface wave are combined in the total field to produce aninterference pattern.

    In order to investigate the properties of the electromagneticfield, the total field (including the DR waves, lateral wave, andtrapped surface wave) is computed in Figures 11(a) and 11(b),respectively. +e lateral wave and trapped surface wave arecomputed in Figures 11(c) and 11(d), respectively. +e com-putation coefficients of relative permittivity in the intermediatedielectrics are chosen as the same in Figures 5(c) and 5(d),while each electric length of the intermediate dielectrics are

    2 5 643–200

    –100

    0

    100

    200

    300

    400

    Real of tan(γ4l4)Real of right-hand side ofequation (28)/i

    k0 k1 k2 k4k3

    λ

    (c)

    2 5 643–200

    –100

    0

    100

    200

    300

    Imaginary of tan(γ4l4)Imaginary of right-hand side ofequation (28)/i

    k0 k1 k2 k4k3

    λ

    (d)

    Figure 6: Graphical approach for root finding in evaluations of poles by (33) for the electromagnetic field due to a vertical dipole over aperfect conductor coated by four-layered lossless dielectrics, with (a) kili � 0.45π; (b) kili � 0.45π; (c) kili � 1.45π; and (d) kili � 1.45π.

    400

    350

    300

    250

    200

    150

    100

    50

    0100 200 300 400 500

    Propagation distance ρ (m)

    Elec

    tric

    fiel

    d E z

    (V/m

    )

    Total fieldTrapped surface waveDRL waves

    l1 = 1.6932m, εr1 = 2.65l2 = 1.2557m, εr2 = 4.0

    Figure 7: +e electric component |E0z(ρ, z)| of electromagneticfield due to a vertical dipole on the conductor coated by two di-electric layers.

    8 International Journal of Antennas and Propagation

  • Table 1: Comparison of computational feasibility and complexity for root finding algorithms. (n: represents the order of computations).

    Method Analytical solution by graphicalapproachComputation complexity of root finding

    algorithmNested

    recursion+ree-layered case In lossless case Easy 1Four-layered case In lossless case Complicated 2N≥ 5 Unavailable Complicated N − 2N≤ 6 by proposed equations(25)–(28) In lossless case Easy 0

    N> 6 by proposed equations (23) and(24) In lossless case Complicated 0

    50

    45

    40

    35

    30

    25

    20

    15

    10

    5

    0–100 –50 0 50 100

    –110

    –120

    –130

    –140

    –150

    –160

    –170

    Hei

    ght z

    (m)

    Propagation distance ρ (m)

    Figure 9: Spatial distributions in ρ − z plane of the DR terms of the magnetic field component |B0ϕ| in dB due to a vertical electric dipole inthe presence of N-layered regions.

    103

    102

    101

    1000 200 400 600 800 1000

    Propagation distance ρ (m)

    Trap

    ped

    surfa

    ce w

    ave E

    zs (V

    /m)

    l1 = 0.5255m, εr1 = 2.65l2 = 0.3897m, εr2 = 4.0l3 = 0.3019m, εr3 = 6.0

    N = 3, kili = 0.45πN = 4, kili = 0.45πN = 5, kili = 0.45π

    Figure 8: +e electric component of trapped surface wave |ES0z(ρ, z)| generated by a vertical dipole in the presence of three-layered to five-layered regions.

    International Journal of Antennas and Propagation 9

  • chosen by kili � 0.45π and kili � 1.45π, respectively. It is seenfrom Figures 11(c) and 11(d) that in the z direction and thetrapped surface, wave attenuates rapidly away from the surfaceof air and dielectrics.

    5. Conclusion

    In summary, a computational scheme is presented to evaluatethe residue sums for the fields of a vertical electric dipole orientedperpendicular to a stratified medium. +e pole equation is

    reconstructed with explicit expression in the presence of the N-layered region.+erefore, a set of explicit equations is derived toenable analytical solutions on roots finding algorithms. Com-putations and analysis of electromagnetic field and its spatialdistributions are carried out on and above a planar multidi-electric-coated perfect conductor. +e lateral wave and thetrapped surface wave are combined in the total field to producean interference pattern. It has been shown how the proposedequations are accurate and have significantly less computationcomplexity than previous numerical procedures.

    10

    8

    6

    4

    2

    0

    6050403020100

    –100 –50 0 50 100

    Hei

    ght z

    (m)

    Propagation distance ρ (m)

    (a)

    10

    8

    6

    4

    2

    0

    605040302010010

    –100 –50 0 50 100

    Hei

    ght z

    (m)

    Propagation distance ρ (m)

    (b)

    10

    8

    6

    4

    2

    0

    Hei

    ght z

    (m)

    –100 –50 0 50 100Propagation distance ρ (m)

    50

    0

    –50

    –100

    –150

    –200

    (c)

    10

    8

    6

    4

    2

    0

    Hei

    ght z

    (m)

    –100 0–50 50 100Propagation distance ρ (m)

    35302520151050–5

    (d)

    Figure 11: Spatial distributions in ρ − z plane of the electric field |Ez(ρ, z)| in dB due to a vertical electric dipole above the conductor coatedby two dielectric layers at f � 100MHz; εr1 � 2.65, εr2 � 4, and z � d � 0mwith (a) k1l1 � k2l2 � 0.45π; (b) k1l1 � k2l2 � 1.45π; (c) trappedsurface wave; and (d) lateral wave.

    1086

    20

    4

    Hei

    ght z

    (m)

    Propagation distance ρ (m)–100 –50 50 1000

    60

    40

    20

    0

    (a)10

    86

    24

    0

    Hei

    ght z

    (m)

    Propagation distance ρ (m)–100 –50 50 1000

    –120

    –140

    –160

    –180

    (b)

    Figure 10: Spatial distributions in ρ − z plane of the electromagnetic field in dB due to a vertical electric dipole above a dielectric-coatedconductor. (a) |E0z(ρ, z)| and (b) |B0Φ(ρ, z)|.

    10 International Journal of Antennas and Propagation

  • Appendix

    +e coefficients in A1(λ) to A3(λ) and B1(λ) to B3(λ) aredefined as follows:

    A1(λ) �c0k

    21

    c1k20,

    B1(λ) �c1k

    20

    c0k21,

    A2(λ) �c0k

    22

    c2k20

    −c1k

    22

    c2k21

    · tan ic1l(

    +c0k

    21

    c1k20

    · tan ic1l( · tan ic2l( ,

    B2(λ) �c2k

    20

    c0k22

    −c2k

    21

    c1k22

    · tan ic1l(

    +c1k

    20

    c0k21

    · tan ic1l( · tan ic2l( ,

    A3(λ) �c0k

    23

    c3k20

    −c1k

    23

    c3k21

    · tan ic1l(

    −c2k

    23

    c3k22

    · tan ic2l( +c0k

    22

    c2k20

    · tan ic2l( · tan ic3l(

    +c2k

    23

    c3k22

    ·c0k

    21

    c1k20

    · tan ic2l( · tan ic1l(

    +c0k

    21

    c1k20

    · tan ic1l( · tan ic3l(

    −c1k

    22

    c2k21

    · tan ic1l( · tan ic2l( · tan ic3l( ,

    B3(λ) �c3k

    20

    c0k23

    −c3k

    21

    c1k23

    · tan ic1l(

    −c3k

    22

    c2k23

    · tan ic2l( +c2k

    20

    c0k22

    · tan ic2l( · tan ic3l(

    +c3k

    20

    c0k23

    ·c1k

    22

    c2k21

    · tan ic1l( · tan ic2l(

    +c1k

    20

    c0k21

    · tan ic1l( · tan ic3l(

    −c2k

    21

    c1k22

    · tan ic1l( · tan ic2l( · tan ic3l( .

    (A.1)

    Data Availability

    +e data used to support the findings of this study areavailable from the corresponding author upon request.

    Conflicts of Interest

    +e authors declare that they have no conflicts of interest.

    Acknowledgments

    +is work was supported by the National Natural ScienceFoundation of China under Grant number 61490695.

    References

    [1] V. Koju and W. M. Robertson, “Leaky Bloch-like surfacewaves in the radiation-continuum for sensitivity enhancedbiosensors via azimuthal interrogation,” Scientific Reports,vol. 7, no. 1, p. 3233, 2017.

    [2] A. Delfan, M. Liscidini, and J. E. Sipe, “Surface enhancedRaman scattering in the presence of multilayer dielectricstructures,” Journal of the Optical Society of America B, vol. 29,no. 8, pp. 1863–1874, 2012.

    [3] W. Y. Pan and K. Li, “ELF wave propagation along sea-rockboundary and mCSEM method,” in Propagation of SLF/ELFElectromagneticWaves, pp. 201–207, Zhejiang University Press,Springer-Verlag, Heidelberg, Germany, 1st edition, 2014.

    [4] K. Li, S. O. Park, and H. Q. Zhang, “Electromagnetic field overthe spherical Earth coated with N-layered dielectric,” RadioScience, vol. 39, no. 2, pp. 1–10, 2004.

    [5] J. R. Wait, Electromagnetic Waves in Stratified Media, Per-gamon Press, New York, NY, USA, 2nd edition, 1970.

    [6] R. W. P. King, M. Owens, and T. T. Wu, Lateral Electro-magnetic Waves:Ceory and Applications to Communications,Geophysical Exploration, and Remote Sensing, Springer, NewYork, NY, USA, 1992.

    [7] A. Sommerfeld, “Propagation of waves in wireless telegra-phy,” Annalen der Physik, vol. 333, no. 4, pp. 665–736, 1909.

    [8] D. Margetis and T. T. Wu, “Exactly calculable field compo-nents of electric dipoles in planar boundary,” Journal ofMathematical Physics, vol. 42, no. 2, pp. 713–745, 2001.

    [9] J. R. Wait, “Comment on “+e electromagnetic field of avertical electric dipole in the presence of a three-layered re-gion” by Ronold W. P. King and Sheldon S. Sandler,” RadioScience, vol. 33, no. 2, pp. 251–253, 1998.

    [10] S. F. Mahoud, R. W. P. King, and S. S. Sandler, “Remarks on“+e electromagnetic field of a vertical electric dipole over theearth or sea” [and reply],” IEEE Transactions on Antennas andPropagation, vol. 47, no. 11, pp. 1745–1747, 1999.

    [11] R. E. Collin, “Hertzian dipole radating over a lossy earth or sea:some early and late 20th-century controversies,” IEEE Antennasand Propagation Magazine, vol. 46, no. 2, pp. 64–79, 2004.

    [12] R. E. Collin, “Some observations about the near zone electricfield of a Hertzian dipole above a lossy earth,” IEEE Trans-actions on Antennas and Propagation, vol. 52, no. 11,pp. 3133–3137, 2004.

    [13] H. Q. Zhang and W. Y. Pan, “Electromagnetic field of avertical electric dipole on a perfect conductor coated with adielectric layer,” Radio Science, vol. 37, no. 4, pp. 1–7, 2001.

    [14] Y. H. Xu,W. Ren, L. Liu, and K. Li, “Trapped surface wave andlateral wave in the presence of a four-layered region,” Progressin Electromagnetics Research, vol. 82, pp. 271–285, 2008.

    [15] Y. L. Lu, Y.-L. Wang, Y. H. Xu, and K. Li, “Electromagneticfield of a horizontal electric dipole buried in a four-layeredregion,” Progress in Electromagnetics Research B, vol. 16,pp. 247–275, 2009.

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  • [16] K. Li, Electromagnetic Fields in Stratified Media, ZhejiangUniversity Press, Springer-Verlag, Heidelberg, Germany,2009.

    [17] J. R. Wait, “Excitation of surface waves on conducting,stratified, dielectric-clad, and corrugated surfaces,” Journal ofResearch of the National Bureau of Standards, vol. 59, no. 6,pp. 365-366, 1957.

    [18] J. D. Cross and P. R. Atkins, “Electromagnetic propagation infour-layered media due to a vertical electric dipole: a clari-fication,” IEEE Transactions on Antennas and Propagation,vol. 63, no. 2, pp. 866–870, 2015.

    [19] Z. He and R. S. Chen, “A fast marching-on-in-degree solutionfor analysis of conductors coated with thin dispersive di-electric,” IEEE Transactions on Antennas and Propagation,vol. 65, no. 9, pp. 4751–4758, 2017.

    [20] S. He, Z.-P. Nie, and J. Hu, “Numerical solution of scatteringfrom thin dielectric-coated conductors based on TDS ap-proximation and EM boundary conditions,” Progress inElectromagnetics Research, vol. 93, pp. 339–354, 2009.

    [21] Z. He, D. Z. Ding, and R. S. Chen, “An efficient marching-on-in-degree solver of surface integral equation for multilayerthin medium-coated conductors,” IEEE Antennas andWireless Propagation Letters, vol. 15, pp. 1458–1461, 2016.

    12 International Journal of Antennas and Propagation

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