Asymptotic solution for the electromagnetic scattering of a vertical dipole over plasmonic and non‐plasmonic half‐spaces

Natural Sciences and Engineering Research Council of Canada, Grant/Award Number: RGPIN‐2017‐ 04508; Natural Sciences and Engineering Research Council of Canada, Grant/Award Number: RGPAS‐ 2017‐507962 Abstract A new asymptotic solution for the scattered electromagnetic fields of a vertical Hertzian dipole antenna in the presence of an imperfectly conducting half‐plane for ordinary and plasmonic media is proposed. The scattered electric and magnetic field components are calculated from the intermediate Hertz potential expressed in terms of the Fourier‐Bessel transforms associated with the Sommerfeld‐type integral, which is difficult to evaluate due to the singularities of the integrand near the integration path and its oscillatory and slowly decaying integrand. Using the modified saddle point method, an approximate closed‐form solution of the far‐zone scattered electromagnetic fields including surface waves is presented. The new formulations are applied to calculate radiation patterns of different impedance half‐planes for both ordinary and plasmonic media, that is, seawater, silty clay soil, silty loam soil and lake water as ordinary, and silver and gold as plasmonic media. Furthermore, a numerical evaluation of the proposed solution at various frequencies and comparisons with two alternative state of the art solutions shows that the proposed solution has higher accuracy in terms of the normalised root‐mean‐square error and the normalised maximum absolute error for plasmonic and non‐plasmonic structures.

Sommerfeld first presented a solution including an improper integral with a Bessel function kernel for the EM scattering of a vertical Hertzian dipole over an imperfect ground in [9]. This solution could not be evaluated in a closed form due to the oscillatory, slowly decaying integrand, and several singularities on the integration path. He deformed the integration path and acquired the spectral representation of the Hertz vector using hyperbolic branch cut integrals and the residue of a pole contributing the Zenneck [10] surface wave. He also presented an asymptotic closed-form expression for the term associated with the Zenneck surface wave in the Hertz vector using the error function of a complex argument called numerical distance. Following Sommerfeld's work, Wise [11,12] and Van Der pol [13] reconsidered this classical problem and presented independent solutions for the Hertz vector, which were in accordance with the Sommerfeld's asymptotic solution for high media contrast (i.e. |n 2 | ≫ 1, where n denotes the complex refractive index), and far field regions near the interface of regions (i.e. k 0 ρ ≫ 1 and z/ρ ≪ 1, wherein k 0 represents the free-space wavenumber, z denotes the z-coordinate of the observation point and ρ is the horizontal distance between the source and the observation point). Another ground-breaking study in the development of the scattered EM waves over a half-space was carried out by Norton [14], who developed Van der pole's formulation for an arbitrary height of transmitter and receiver. Nonetheless, Norton's numerical distance parameter in his formulation was not quite accurate for field points far away from the interface caused by the truncation of a binomial expansion of a square root in his derivation. Subsequently, Bannister [15] extended Norton's far field dipole equations to the quasi near field region, while an inaccurate Norton's numerical distance parameter was adopted.
The saddle point method (SPM) of integration is another approach for approximating Sommerfeld integrals (SIs) in far field regions, in which the distance between the antenna and the observation point is considerable [16]. However, for a lossy half-space problem, the SPM method is not capable of approximating SIs since the Sommerfeld pole is close to the saddle point for this problem. Wait [17,18] and Makarov et al. [19] used a multiplicative method for dealing with the pole near the saddle point. Van der Waerden [20] introduced the modified SPM which was applied to the Sommerfeld problem by Collin [21] as well as Bernard and Ishimaru [22]. Although the obtained solutions using SPMs are not similar to the Norton's formulation [14], they are consistent with the Norton surface wave formulation for high contrast media and far field regions near the interface. Further developments on the scattering of EM waves over a lossy half-space have been conducted by numerous researchers, such as King [23][24][25][26], Wait [27,28] and Green [29]. Nevertheless, it was revealed by Mahmoud et al. [30], Wait [31,32] and Yokoyama [33] that the solutions for the scattered EM fields radiated by a VED over a lossy or dielectric coated half-space were not quite accurate due to the trapped surface wave. Consequently, this classical problem was reinvestigated by several researchers in the past few years [34][35][36][37][38]. Sarabandi et al. [39] proposed an analytical method based on the reflection coefficient approximation of integrands employing the Prony method to calculate SIs in near and far field regions. However, the accuracy of the proposed method is restricted when the antenna and observation point are located near the interface due to the number of image points. Eslami Nazari and Huang [40] introduced a new method by decomposing the intermediate Hertz potential into three terms. The first and second term of the Hertz potential associated with the SIs are calculated employing hyperbolic functions, and the third term is approximated using SPM. However, this solution is only applicable for low frequencies, highly conductive surfaces and far field regions. It is worth mentioning that the accuracy and efficiency of all aforementioned methods are limited by the antenna and observation point locations and EM properties of each medium, that is, permittivity and conductivity of the regions.
Several numerical methods have been proposed in order to evaluate Sommerfeld-type integrals. Parhami et al. [41] introduced a method, which is valid for a VED, by deforming the path of integration to the steepest descent path. It should be noted that in this method, the integral is solved asymptotically for large distances between image and observation point, while it is solved numerically for small distances. Michalski [42] developed Parhami's method by proposing a variation over the way of branch cut. Afterwards, Johnson and Dudley [43] proposed a numerical method, in which an analytical technique is utilised in order to reduce the oscillation of the Sommerfeld integrand. However, this method is only valid for small distances between image and observation point. Despite improvements in convergence properties of the SIs using the aforementioned numerical techniques, transformations are required to be applied to the Sommerfeld integrand, which increase the complexity of calculations and computational costs. Moreover, the obtained solutions are not valid for all source and observation point locations and EM properties of layers [39].
Here, the classical Sommerfeld half-space problem is reconsidered and a rigorous closed-form solution for the intermediate Hertz potential and the scattered electric and magnetic field components are presented using the modified SPM method and considering high-order surface wave. The theoretical development is validated by representative numerical results and compared with two alternative state of the art solutions referred to as the King [38] and the Norton-Bannister [15] solutions for ordinary and plasmonic media. The obtained results show that the proposed solution outperforms the conventional solutions at various frequencies and distances from the antenna and even for moderate media contrast.
The article is organised as follows. In Section 2, the scattered electric and magnetic field components using the intermediate Hertz potential are derived in terms of two-dimensional Fourier transforms. A rigorous closed-form solution for the intermediate Hertz potential and scattered electric and magnetic field components in the far field region are proposed in Section 3. Finally, in Section 4, numerical evaluation of the proposed solution at various frequencies and comparisons with the conventional methods, that is, King and Norton-Bannister solutions, for both ordinary and plasmonic media are presented in terms of the normalised root-mean-square error (NRMSE) and the normalised maximum absolute error (NMAE). Finally, the conclusion is presented in Section 5.

| FORMULATION OF THE PROBLEM
The scattered electric ( E ! ) and magnetic (H ! ) field components radiated by a VED located on the z-axis of the cylindrical coordinate system at height h above a lossy half-space, as shown in Figure 1, can be expressed as [28]: in which I denotes the current source, Δl is the antenna length, ω is the angular frequency of the source, ϵ 0 denotes the permittivity of free space and k 0 is the wavenumber in free space. The special Fourier transform of the intermediate Hertz potential Π z can be written as: where in γ m (m = 0, 1) and complex refractive index n 0m can be expressed as: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi y , ϵ 0 is the permittivity of free space and ϵ m denotes the complex permittivity of medium m depending on the frequency. It is worth mentioning that the complex refractive index presented in (4) is valid for both plasmonic and non-plasmonic media at different frequencies. To determine the scattered electric field components (1), the inverse spatial Fourier transform is applied to (3). Therefore, we have and By changing the Cartesian integration variables in (5) and (6) into the polar form, the double integrals are converted to single integrals as follows: wherein In (8), J 0 (kρ) indicates the Bessel function of the first kind of order zero and ρ denotes the horizontal distance between the source and the observation point in the cylindrical coordinate as shown in Figure 1. By using the Sommerfeld identity [44], (7) can be simplified as: According to the geometry of the problem shown in Figure 1, R 0 and R 1 can be calculated as: where z p denotes the z coordinate of the observation point. To present an approximate closed-form solution for the scattered electric and magnetic field components, P, which is a Sommerfeld-type integral, should be evaluated.

| EVALUATION OF THE INTEGRAL
Various analytical solutions have been proposed for seeking an approximate closed-form solution for the scattered electric and magnetic field components. However, proposing a general closed-form solution for different antenna and observation point locations with an arbitrary value of the complex refractive index is the main difficulty in evaluating them.
The zero-order Bessel function in (8) can be written as the sum of two Hankel functions of the first and second kinds with the same argument as By substituting (11) into (8), P can be written as: Moreover, for moderate and high media contrast, (8) can be further simplified as: where β ¼ γ 1 =jk 0 n 2 01 . By changing the integral variable k to ζ = arccos(k/k 0 ), P becomes F I G U R E 1 Dipole source above a lossy half-space in which α 0 = sin −1 (β). By using the first term of the asymptotic expansion of the Hankel function of the first kind in the far field region, (14) can be expressed as: where θ 2 is defined as: Although extra terms for the asymptotic expansion of the Hankel function in (15) may increase the accuracy of the integral evaluation, it is quite accurate in the far-field region since the NRMSE values for the real and imaginary parts of the proposed approximation of the Hankel function are 0.004 and 0.0042, respectively.
By deforming the integration path via the substitution cos (θ 2 − ζ) = 1 −jt 2 and using the modified SPM, P can be expressed as: in which erfc is the Gauss error function, and W e and P e may be expressed as:

| Scattered E-field components
In order to calculate the scattered electric field components over the lossy half-space, the intermediate Hertz potential (9) along with (17) are substituted into (1). The x-component of the scattered electric field may be expressed as: wherein T 1 to T 4 with their sub variables can be acquired from the following equations.
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The y-component of the scattered electric field can be expressed as: The only difference between the prime parameters, T 1 0 , T 3 0 and T 4 0 , and original parameters is the x parameter. In other words, by changing x to y in T 1 , T 3 , T 4 and their sub variables, NAZARI AND HUANG -707 prime parameters can be obtained. By calculating the xand ycomponents of the scattered electric field over the lossy halfspace, cross polarised components are obtained since the polarisation of the antenna is vertical. By using (1), the zcomponent of the scattered electric field can be calculated. Thus, we have in which T 5 to T 8 and their sub variables are obtained from the following equations.

| Scattered H-field components
The scattered magnetic field components can also be calculated using the intermediate Hertz potential (9). In other words, by substituting (9) along with (17) into (2), different components of the scattered magnetic field can be obtained. The x-component of the scattered magnetic field may be expressed as wherein T 9 and C 11 can be expressed as follows: The y-component of the scattered magnetic field over the lossy half-space can also be written as: in which T 9 0 can be obtained by changing C 11 and y to C 3 and x in (25), respectively.

| RESULTS
To evaluate the accuracy and efficiency of the recently developed method for the calculation of the intermediate Hertz potential and the scattered electric and magnetic field components over the lossy half-space, the NRMSE and the NMAE, in where the standard deviation is used for normalisation, are utilised for each frequency. It should be noted that the NRMSE is calculated using the reference and calculated values expressed as 708 -NAZARI AND HUANG

TA B L E 1 Media parameters
Parameter Seawater Silty loam soil Silty clay soil Lake water Silver Gold wherein χ represents the reference values, which are the numerical solutions obtained by the high-order global adaptive quadrature method [45], χ denotes the calculated values, which are the proposed analytical solutions, N is the number of reference or calculated values and σ χ is the standard deviation of the reference values. In order to calculate the NRMSE for each frequency, the amplitude of the scattered electric field (i.e. ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ) is considered as χ while θ (elevation angle in Figure 1) is changed between 0 to π/2 with the resolution of π/200 (N = 100). The numerical solution for the amplitude of the scattered electric field is also considered as χ, while θ is changed between 0 to π/2 with the resolution of π/ 200. The next accuracy metric is the NMAE defined as the maximum deference between the proposed analytical and numerical solutions for the scattered electric field, which is normalised by the standard deviation of the numerical results. Therefore, the NMAE can be expressed as: Here, the proposed solution of the scattered E and H fields are compared with the King and Norton-Bannister solutions, while the rigorous numerical computation of the SIs using the high-order global adaptive quadrature method [45] is considered as reference. Moreover, the elevation pattern of the scattered electric field is compared with the numerical solution. Six various problems have been selected for analysis, with parameters listed in Table 1. Silver and gold illustrate plasmonic media, which have been considered in optical frequency range, and others represent ordinary media.

| Non-plasmonic media
The proposed formulations are applied to six different ordinary media listed in Table 1. In this table, the soil composition is characterised by the percentage of soil constituents, which are sand, clay, silt and water. Silty loam soil consists of roughly equal amounts of silt and sand and a little less clay. On the other hand, silty clay soil has more clay than silt. The complex relative permittivity of seawater as well as lake water are calculated using the Meissner and Wentz model [46], which is based on the double Debye model and is quite accurate at higher frequencies. On the other hand, for silty loam and silty clay soil, the complex relative permittivity is calculated by the developed Dobson model [47,48].
The frequency of the antenna over each medium in Table 1 is related to the application of wave scattering over that region. For seawater, the very high frequency (VHF) band has been selected since the pulsed radars operating at this frequency band are employed for remote sensing of ocean surface in order to extract the speed and direction of ocean surface currents in real time [2,49]. For the silty loam soil scattering problem, the global system for mobile communications (GSM) frequency band has been considered since finding the pattern of the scattered EM fields, coverage area and blind zones over earth surface are of interest [1,50]. Therefore, GSM-1800 (1.8 GHz) has been selected for this scattering problem. On the other hand, the frequency modulation (FM) broadcast band has been selected for the silty clay soil scattering problem due to its application in finding the coverage area of the passive radars and FM broadcast radio systems [51]. For radio oceanography applications, X-band marine radar is commonly used to scan the water surface with high temporal and spatial resolutions [52], and this frequency range has been selected for the lake water scattering problem.
The elevation pattern of the scattered electric field, | E| = (|E x | 2 + |E y | 2 + |E z | 2 ), over the selected ordinary media listed in Table 1 is calculated using the proposed method and compared with the numerical and conventional (Norton-Banister and King solutions) methods. Figure 2 depicts the elevation pattern of the scattered electric field for a VED over selected non-plasmonic media in different frequencies and distances from the antenna. The horizontal and vertical axes, respectively, correspond to θ = 90°and θ = 0°, as shown in Figure 1. As can be seen in Figure 2, the King and Norton-Bannister results are close to each other and become indistinguishable when the media contrast is sufficiently high [53]. In Figure 2(a), which has NAZARI AND HUANG been obtained for seawater at VHF band, the proposed solution has a good agreement with the numerical method not only for low angles, but also near the interface, in which groundwave contribution is high. In order to evaluate the accuracy of the proposed solution when compared with the conventional solutions, the NRMSE and NMAE are calculated shown in Table 2. As can be seen in this table, both NRMSE and NMAE of the proposed solution are lower than the conventional solutions for both scattered E and H fields. The elevation pattern of the scattered electric  field over silty loam soil is shown in Figure 2(b). The proposed solution follows the numerical solution, particularly on the pattern nulls and also on the interface, which corresponds to the groundwave contribution. The NRMSE and NMAE evaluation of the scattered E and H fields reveal that the proposed solution outperforms the conventional solutions even for moderate media contrast. In Figure 2(c), the elevation pattern of the scattered electric field over silty clay soil has been obtained. It can be observed from the figure that the proposed solution agrees well with the numerical solution, particularly in high angles and near the interface, and further, the NRMSE and NMAE values for silty clay soil problem listed in Table 2 are lower than the conventional solutions. Figure 2(d) illustrates the elevation pattern of the scattered electric field for lake water, in which discrepancies between results are not noticeable. However, the NRMSE and NMAE values listed in Table 2 substantiate that the proposed solution is more accurate than the King and Norton-Bannister solutions. To evaluate the accuracy of the proposed solution for the scattered magnetic field, the NRMSE and NMAE values for the H-field over non-plasmonic media listed in Table 1 are calculated, and are shown in Table 2. As evident, all the NRMSE and NMAE values for the H-fields are smaller than the conventional solutions for the ordinary media.
To assess the robustness of the proposed solution in terms of NRMSE at various frequencies, the NRMSE value is calculated for the scattered electric field components for the whole frequency range, which is between 100 MHz and 100 GHz for non-plasmonic media. In other words, the NRMSE value is calculated for each frequency while the observation point angle is changed between θ = 0°and θ = 90°. It should be noted that the real and imaginary parts of the dielectric constant for the ordinary media listed in Table 1 depend on frequency. Therefore, the relative permittivity should be calculated for each frequency for the NRMSE calculation. For seawater, the relative permittivity depends on temperature and also salinity and varies with frequency [54]. In the NRMSE calculation of seawater, the salinity of seawater and lake water has been assumed as 35 and 0, respectively, while the temperature is 17°C. For the silty clay and loam soil, the relative permittivity depends on frequency, temperature and also the texture of the soil [55]. For this scattering problem, the temperature has been assumed as 23°C. Figure 3 illustrates the NRMSE comparison of the proposed and conventional solutions over ordinary media in a wide variety of frequency ranges, while the antenna height (h) and distance of the observation point from the origin of the coordinate system (R) are assumed to be λ/10 and k 0 /10, respectively. As is obvious from this figure, the NRMSE value of the scattered electric field components is better than the conventional solutions (i.e. King and Norton-Bannister solutions) in all frequencies shown in Figure 3(a-d). In order to evaluate the accuracy of each scattered electric field component, that is, E z and E ρ , the NRMSE has been calculated over the presented ordinary media. Figure 4 shows the NRMSE comparison of the scattered electric field for each component, that is, E ρ and E z , over non-plasmonic media at various frequencies. As can be seen in this figure, the E ρ component obtained by the proposed method has better accuracy in comparison with the E z component. In other words, the E z component has more impact on the NRMSE value of the scattered electric field over non-plasmonic media since its NRMSE value is greater than the other component. Also, the NMAE comparison between the proposed and the conventional methods shown in Figure 5 substantiates that the proposed solution outperforms the conventional solution over non-plasmonic media since the value of NMAE is better than the King and the Norton-Bannister solutions in all frequency range.

| Plasmonic media
In order to evaluate the accuracy of the proposed solution for plasmonic media, silver and gold have been considered and the elevation patterns of the scattered electric field have been compared with the numerical solutions at 351.87 and 420.52 THz for silver and gold, respectively. It should be noted that for light-matter interaction in THz frequencies, the real part of the permittivity attains negative value and varies with frequency [56]. Figure 6(a) and (b) depict the elevation patterns of the scattered electric field over the silver and gold, respectively, as plasmonic media. As is obvious by these two figures, the proposed solution agrees well with the numerical method, especially in high angles and near the interface where the surface wave contribution is high. Moreover, the NRMSE and NMAE values mentioned in Table 3 reveal that the proposed solution outperforms the conventional solution for the plasmonic media. In order to evaluate the accuracy of the proposed solution for magnetic field, the NRMSE and NMAE values of the H-field for silver and gold have been shown in Table 3. As evident, all the NRMSE and NMAE values for the H-field are smaller than the conventional solutions.
Similar to non-plasmonic media, the NRMSE evaluation of the scattered electric field for silver and gold are accomplished, while the frequency is changed between 300 and 900 THz. Figure 7 demonstrates the NRMSE value of the scattered electric field at various frequencies for plasmonic media. As can be seen in this figure, the NRMSE value of the scattered electric field is less than the King and Norton-Bannister solutions in the optical frequency range. The NRMSE value of the silver in Figure 7(a) is relatively similar to gold in Figure 7(b) since the real and imaginary parts of the permittivity for both of them are quite similar in the optical frequency range. To evaluate the accuracy of each scattered electric field component over plasmonic media, the NRMSE has been calculated for E ρ and E z components over silver and gold and compared with the conventional solutions, as shown in Figure 8. As observed in this figure, the E ρ and E z components obtained by the proposed method have better accuracy in comparison with the conventional solutions over plasmonic media. In addition, the accuracy of the proposed solution is evaluated in terms of NMAE for plasmonic media. Figure 9 depicts the NMAE of the proposed solution for the scattered electric field intensity over plasmonic media. As is obvious, the proposed solution outperforms the conventional solutions at various frequencies. It is worth mentioning that the complex relative permittivity n 2 for the plasmonic media varies with frequency. Here, the Drude model [57] is employed for the calculation of the complex    relative permittivity at each frequency, which can be expressed as: wherein τ represents the average time between collisions experienced by an electron and can be written as in which m denotes the electron mass, n 0 is the electron density of the metal and e represents the elementary charge. Subsequently, (29) is utilised for each frequency so as to calculate the real and imaginary parts of the complex relative permittivity for the NRMSE and the NMAE calculations at various frequencies.

| CONCLUSION
The authors have developed a new asymptotic solution for the far-zone EM fields of a VED radiating over an imperfectly conducting half-plane. The solution has been evaluated for the both ordinary and plasmonic media. Accuracy comparison indicates that the proposed solution outperforms the conventional solutions like King and Norton-Bannister solutions, in terms of NRMSE and NMAE at various frequencies and distances from the antenna in the far field region. Possible future enhancements to the method NAZARI AND HUANG