Calculation of the Transmitted Electromagnetic Field Below a Flat Interface Between Lossless Media in the Far-Field Region Using a Geometrical Optics Approach

Abstract

In this paper, we introduce a novel method for calculating the electromagnetic (EM) field below a flat interface between two lossless media when a radiating vertical Hertzian dipole (VHD) is located far from the interface. The method uses a Geometrical Optics (GOs) approach based on the concept of ‘equal optical lengths’, in the framework of which the location of the ‘virtual image’ of the original source is calculated. Using the well-known formulae for the far EM field of a vertical Hertzian dipole, the EM field at a point below the flat interface is calculated in a closed mathematical form.

1. Introduction

The computation of the electromagnetic (EM) field beneath a flat interface separating two lossless dielectric media, due to a radiating vertical Hertzian dipole (VHD) located above the interface, remains a non-trivial problem in the literature [1,2,3].

The accurate calculation of refracted spherical wave fields has important applications, notably in underwater and underground wireless communications. Among these, the analysis of wave propagation in stratified media poses considerable analytical and computational challenges. In light of this, the present work introduces a simplified and general theoretical framework for the refracted field based on geometrical optics. Central to this approach is the concept of an equivalent imaginary source (IS), which is used to approximate the field configuration below the interface.

The method of images (or image theory) is a well-established analytical tool in electromagnetics, which is particularly effective in solving problems involving boundaries such as ground planes, perfect conductors, or symmetric configurations [4,5].

However, its traditional application is primarily associated with reflection, where the principle of specular reflection (equality of incident and reflected angles) allows for the simple placement of ISs. In contrast, the extension of image theory to problems involving refraction is less straightforward. As noted in prior works, refraction entails a change in the propagation direction governed by Snell’s law, and the concept of a single, fixed IS does not generally apply to all refracted rays. Consequently, the conventional Image Source Method (ISM) lacks a simple, universally valid prescription for the placement of an equivalent source in refractive configurations. In such cases, more general methods—such as ray tracing, beam tracing, or full-wave solutions—are often employed.

It is important to emphasize that the use of IS in this work differs significantly from earlier approaches [3,6,7,8]. In particular, our formulation relies solely on boundary conditions and does not use any additional assumptions for determining the phase and the amplitude of the field. For example, in [3], the amplitude is derived from the law of energy conservation, while the phase is based on the optical path length.

In our method, both quantities emerge naturally from Snell’s law within the far-field approximation. This study is closely linked to the classical Sommerfeld half-space problem [9,10] and geometrical optics, as it leverages Snell’s law in conjunction with asymptotic methods for far-field analysis. Although the Sommerfeld problem has been rigorously solved in integral form, an explicit and practically useful asymptotic evaluation of the Sommerfeld integral—particularly for spherical wave refraction—has remained open. Asymptotic techniques such as the saddle point method provide not only computational efficiency but also physical insight into wave phenomena, offering advantages over purely numerical methods. The primary aim of this work is to present a novel and useful technique for evaluating fields of refracted waves and interpreting wave behavior in the Sommerfeld problem.

Section 2 presents a geometric formulation of the problem, considering a VHD positioned above a planar interface. This section treats the general case, where the dipole may be located at an arbitrary distance from the interface, and derives the expression for the optical path lengths.

In Section 2.1, we analyze two closely spaced incident rays from the dipole and use Snell’s law to determine their refracted paths. The point of intersection of these rays, located above the interface, defines the IS.

Section 2.2 focuses on the far-field regime, where the VHD is located sufficiently far from the interface. Under this condition, the ratio between the optical path lengths from the radiator and their IS is constant, which allows us to derive a closed-form expression for the Cartesian coordinates of the IS as a function of the incident angle.

In Section 3, the far-field EM field at an observation point in the second medium is calculated, using the virtual source position derived earlier. The field amplitude is adjusted using the appropriate transmission coefficient at the boundary between the two media.

Throughout this analysis, we assume harmonic time dependence of the form $e^{-i\omega t}$, and we consider the second medium to be optically transparent, i.e., the wave attenuation is negligible. Additionally, we assume that the refractive index of the second medium is greater than that of the first: ${\epsilon }_2>{\epsilon }_1$ or $n_2>n_1$.

2. Refraction from a Flat Interface

2.1. Coordinates of the Imaginary Source

In this section, we consider the general case of spherical EM waves, emanating from a VHD above a flat interface. To find the field of the refracted wave, it is important to determine the location of the IS $\left(C\right)$ in the first medium. However, according to the boundary condition, the fields from the real and the imaginary sources at the interface of the media should be the same. Therefore, first, we will find the ratio between the optical path length from the imaginary and real sources in the first medium.

Below, we will present some simple equations describing the geometry shown in Figure 1.

In Figure 1, a vertical dipole is positioned at point A on the z-axis, with the coordinates $0,0,z_0$, i.e., at height $z_0$ above the interface. Two adjacent rays, $AD$ and $A{D}^{\prime }$, are selected. The angle between these rays in the first medium is infinitesimal $d\theta$, whereas the angular separation between them in the second medium is $d\xi$. The angles of incidence and refraction of rays $AD$ and $DB$ are denoted by $\theta$ and $\xi$, respectively.

To determine the coordinates of the virtual source C, we calculate the segment $CD=l_{\xi }$ by extending the refracted rays in the reverse direction from points D and $D^{\prime }$ until they intersect at point C.

First, from triangle $AD{P}_{\theta }$, we have

$$D{P}_{\theta }=l_{\theta }d\theta$$

(1)

while from triangle $AD{P}_{\xi }$, we obtain

$$D{P}_{\xi }=l_{\xi }d\xi .$$

(2)

Similarly, from triangles $D{D}^{\prime }{P}_{\theta }$ and $D{D}^{\prime }{P}_{\xi }$ (where $D{D}^{\prime }=dl$), we obtain

$$D{P}_{\theta }=dlcos\theta ,$$

(3)

and

$$D{P}_{\xi }=dlcos\xi .$$

(4)

Equating Equations (1) and (3), we obtain

$$dl=l_{\theta }d\theta /cos\theta .$$

(5)

Similarly, from Equations (2) and (5), we get

$$dl=l_{\xi }d\xi /cos\xi .$$

(6)

Finally, equating Equations (5) and (6), we obtain

$${\displaystyle \frac{d\theta }{d\xi }}={\displaystyle \frac{l_{\xi }}{l_{\theta }}}{\displaystyle \frac{cos\theta }{cos\xi }}.$$

(7)

Next, to find the ratio $d\theta /d\xi$ of the change in the angular size of the ray under refraction, we use Snell’s law:

$${\displaystyle \frac{n_1}{n_2}}={\displaystyle \frac{sin\xi }{sin\theta }}={\displaystyle \frac{k_{01}}{k_{02}}},$$

(8)

showing the relative change of the solid angle $d{\mathrm{\Omega }}_1=sin\theta d\theta d\alpha$ of a thin-ray tube when transmitted into a second medium with a solid angle $d{\mathrm{\Omega }}_2=sin\xi d\xi d\alpha$.

Then, differentiating Equation (8), we find

$${\displaystyle \frac{d\theta }{d\xi }}={\displaystyle \frac{k_{02}}{k_{01}}}{\displaystyle \frac{cos\xi }{cos\theta }}\equiv K.$$

(9)

In order to derive the final equation of this Section, from Equations (7) and (9) we obtain

$${\displaystyle \frac{l_{\xi }}{l_{\theta }}}=K{\displaystyle \frac{cos\xi }{cos\theta }},$$

(10)

and, finally, from Equations (9) and (10), we obtain

$$k_{02}{l}_{\xi }=K^2{k}_{01}{l}_{\theta },$$

(11)

which represents the fundamental equation between the ‘optical lengths’ $k_{01}{l}_{\theta }$ in the upper medium (medium 1) and $k_{02}{l}_{\xi }$ in the lower medium, respectively (also, see at the next Section about our novel ‘image theory’ in the case of plane wave incidence). Moreover, K is a measure of ‘expansion of the cone marked as ‘virtual image’ with vertex C, as compared to the cone marked as the ‘actual source’ A.

Here, two adjacent points D and $D^{\prime }$ that are located very close to each other are considered (see Figure 1). The point C is an intersection of the rays, which are extensions of the rays originating from points D and $D^{\prime }$ in the second medium. The medium above the interface has index of refraction equal to $n_1$, whereas the index of refraction of the lower medium is $n_2$. In Figure 1, the source dipole is located at a distance $z_0$ from the origin in the positive $Oz$ axis and its distance from the refracting point D is $AD=l_{\theta }$ (the angle of incidence is $\theta$). Furthermore, the two refracted rays (the angle of refraction is $\xi$) intersect at point C, and the distance between the virtual image C and the refracting point D is $CD=l_{\xi }$.

The validity of this expression can be checked using the expression (see Appendix A)

$$l_{\xi }=\sqrt{{(\rho -{\rho }_c)}^2+z_c^2},$$

(12)

where the coordinates ${\rho }_c$ and $z_c$ of the IS are found using the intersection of the extensions of the refracted rays in (A6) and (A7)

$$\begin{array}{c}{\rho }_c=-z_0\left(1-k_{01}^2/k_{02}^2\right){tan}^3\theta ,\hfill \\ z_c=z_0{\displaystyle \frac{{\left(k_{02}^2-k_{01}^2{sin}^2\theta \right)}^{\frac{3}{2}}}{k_{01}{k}_{02}^2{cos}^3\theta }}=z_0{\displaystyle \frac{k_{02}}{k_{01}}}{\displaystyle \frac{{cos}^3\xi }{{cos}^3\theta }}.\end{array}$$

(13)

Equation (12) provides an expression for the path of the ray $l_{\xi }$ from the IS, as a function of its coordinates, given by (13). Moreover, (11) connects the path of the ray from the real source to that of its IS. As already mentioned, K determines the relative change in the solid angle of the thin ray tube when refracted from one medium to another. In our case, the ray tube represents the surface of a narrow cone with its vertex located at point $z=z_0$, within which the energy flow is conserved.

2.2. The IS Field in an Infinite Space

The IS (C) field can be conveniently determined in a cylindrical coordinate system using the expressions for a VHD in a homogeneous medium. This approach is justified, as the tangential components of the fields must satisfy the equality

$$E_{\rho }^{LOS}=E_{\rho }^{IM}$$

(14)

at the plane interface ($z=0$) between the two media, according to the boundary condition (see Appendix B)

$$E_{\rho }^{LOS}\simeq -{\displaystyle \frac{p{k}_{01}^2}{4\pi {ϵ}_1}}{\displaystyle \frac{e^{i{k}_{01}{l}_{\theta }}}{l_{\theta }}}sin\theta cos\theta .$$

(15)

The field of the IS at point B, located in a medium with refractive index $n_2$, can be expressed as

$$E_{\rho }^{IM}=A{\displaystyle \frac{e^{i{k}_{02}r}}{r}}sin\xi cos\xi ,$$

(16)

where using $r=l_{\xi }+r_2$ we can easily determine the constant

$$A=-{\displaystyle \frac{p{k}_{02}^2}{4\pi {ϵ}_1}}{\displaystyle \frac{cos\xi }{cos\theta }}{e}^{i(k_{01}{l}_{\theta }-k_{02}{l}_{\xi })}$$

(17)

from (14). Then, taking into account (9), we finally obtain the expression for the electric field in the spherical coordinate system.

$$E_{\theta }^{IM}=-{\displaystyle \frac{p{k}_{01}{k}_{02}}{4\pi {ϵ}_1}}{\displaystyle \frac{e^{i(k_{01}{l}_{\theta }+k_{02}{r}_2)}}{r}}Ksin\xi .$$

(18)

Dividing by the wave impedance, we obtain the expression for the magnetic field radiated from the IS

$$H_{\phi }^{IM}=-{\displaystyle \frac{\omega p{k}_{01}{ϵ}_2}{4\pi {ϵ}_1}}{\displaystyle \frac{e^{i(k_{01}{l}_{\theta }+k_{02}{r}_2)}}{r}}Ksin\xi .$$

(19)

Thus, replacement of the real radiator with its alternative IS enables us to simplify the solution of the boundary value problem, as the fields of the refracted wave and those refracted from the IS are identical due to the boundary condition (14). It should be noted that the fulfillment of other boundary conditions can be easily verified.

3. Actual Calculation of the Transmitted EM Field Below the Flat Interface

Given the position (${\rho }_c,z_c$) as shown above, we can easily calculate the transmitted EM field below the flat interface at the observation point B, with given coordinates (${\rho }_B,z_B$), along the ray $DB$.

As the ray from the source incident on a plane interface splits into two rays, it is first necessary to determine the attenuation of the amplitude of the refracted wave.

To calculate the amplitude of the refracted wave, it is sufficient to multiply the amplitude of the incident wave by the refraction coefficient

$$\begin{array}{c}{E}_{\theta }^T=T_V{E}_{\theta }^{IM},\\ T_V={\displaystyle \frac{2{ϵ}_{1}ncos\theta }{{ϵ}_{2}cos\theta +{ϵ}_{1}ncos\xi }},\end{array}$$

(20)

just as in the case when a plane wave is scattered at a plane interface, where $n=k_{02}/k_{01}$. Indeed, within a thin ray tube, the spherical wave actually becomes locally planar, which, on the one hand, indicates the applicability of Fresnel’s formulas in our problem. On the other hand, the solution to Sommerfeld’s problem describing radiation from a vertical dipole for the reflected wave in the far-field zone is well known [2,11]

$$\begin{array}{c}{E}_{\theta }^R=R_V{E}_{\theta }^{IM},\\ R_V={\displaystyle \frac{{ϵ}_{2}cos\theta -{ϵ}_{1}ncos\xi }{{ϵ}_{2}cos\theta +{ϵ}_{1}ncos\xi }}.\end{array}$$

(21)

As in our case, the transmittance $T_V$ and the reflectivity $R_V$ satisfy the law of energy conservation. A similar representation of the EM field should also be valid for the refracted wave. Below is the well-known radiation formula for VHD in the wave zone (where the radiating dipole is located at the origin; see Appendix B below) applied to IS with coordinates (${\rho }_c,z_c$):

$$\begin{array}{c}\hfill H_{\phi }^T={\displaystyle \frac{Ih{k}_{01}{ϵ}_2}{2\pi i{ϵ}_1}}{\displaystyle \frac{e^{i(k_{01}{l}_{\theta }+k_{02}{r}_2)}}{r}}K{T}_{V}sin\xi ,\end{array}$$

(22)

$$\begin{array}{c}\hfill E_{\theta }^T={\displaystyle \frac{k_{01}{k}_{02}Ih}{2\pi i\omega {ϵ}_1}}{\displaystyle \frac{e^{i(k_{01}{l}_{\theta }+k_{02}{r}_2)}}{r}}K{T}_{V}sin\xi ,\end{array}$$

(23)

$$T_V\left(\xi \right)={\displaystyle \frac{2{ϵ}_{1}n\sqrt{1-{(nsin\xi )}^2}}{{ϵ}_{1}ncos\xi +{ϵ}_2\sqrt{1-{(nsin\xi )}^2}}},$$

(24)

where I is the electric current of the radiating source, $2h$ is the length of the VHD, $T_V$ is the transmission coefficient for the case of vertical polarization, and r is the distance between the virtual source and the observation point B.

4. Discussion

On the other hand, the refracted wave field (23) can be obtained by evaluating the well-known Sommerfeld integral [11] using the saddle point method [2] in the far field (see Appendix C)

$$E_{\theta }^T\left(\xi \right)\sim {\displaystyle \frac{p{k}_{01}{k}_{02}^2}{4\pi {ϵ}_1}}{\displaystyle \frac{e^{i\Psi \left(\xi \right)}}{{\Psi }^{″}\left(\xi \right)}}K\left(\xi \right)T_V\left(\xi \right)sin\xi ,$$

(25)

where $\mathrm{\xi }$ is a root of equation ${\Psi }^{\prime }\left(\xi \right)=0$

$$\begin{array}{c}\Psi \left(\xi \right)=k_{01}{l}_{\theta }+k_{02}{r}_2,\end{array}$$

(26)

$$\begin{array}{c}{\Psi }^{″}\left(\xi \right)=-k_{01}{l}_{\theta }{K}^2-k_{02}{r}_2.\end{array}$$

(27)

To prove the complete equivalence of the formulas, it is sufficient to express the eikonal $\Psi$ in (25), as well as the second derivative ${\Psi }^{″}$ in terms of angular coordinates

$$\begin{array}{c}\Psi =k_{01}{z}_{0}cos\theta +k_{02}({\rho }_{B}sin\xi -zcos\xi )=\hfill \\ k_{01}(z_{0}cos\theta +{\rho }_{D}sin\theta )+k_{02}(({\rho }_B-{\rho }_D)\\ \hfill sin\xi -zcos\xi )=k_{01}{l}_{\theta }+k_{02}{r}_2,\\ {\Psi }^{\prime }=(-z_{0}tan\theta +z{k}_{\rho }/{\varkappa }_2+{\rho }_B){\varkappa }_2=0,\hfill \\ {\Psi }^{″}=-k_{02}\left(l_{\xi }+({\rho }_B-{\rho }_D)sin\xi -zcos\xi \right)=\\ -k_{02}(l_{\xi }+r_2)\equiv -k_{02}r,\end{array}$$

based on Snell’s law (8) and geometric considerations, and taking into account the expression for the optical path length (11) in (27).

It should be noted that expression (25), obtained using the saddle point method, contains the exact values of the eikonal $\mathrm{\Psi }$ along the trajectory $ADB$, as well as the distance r (see ${\mathrm{\Psi }}^{″}$) from IS C to the observation point B, the physical meaning of which is easily explained from the standpoint of geometrical optics, where the underlying idea of ISM is relatively easy to understand.

Therefore, the above evaluation of the Sommerfeld integral in the far field in (25), which represents the exact solution to the Sommerfeld boundary value problem, confirms the validity of the simple approach proposed in geometric optics for constructing the IS with coordinates ${\rho }_c$, $z_c$, as well as for determining the optical path length in (11).

It should be noted that the family of imaginary images of a vertical dipole located on the z-axis ($z_0=5$) forms the focal surface of a rotational hyperboloid around the z-axis (see Figure 2), in contrast to electrostatics, where the image must be located at a single mirror-symmetric point on the z-axis.

The IS coordinates in the first medium ($z_0=5$) can be directly determined from the figure as the intersection point of the hyperbola branch (Figure 2) with a straight line that extends the ray in the second medium. It is evident that the quadrant numbers of the refracted ray and the hyperbola branch must have the same parity (see also Figure 1).

5. Conclusions

A novel approach is proposed within the geometric theory of scattering of a spherical wave at a plane interface between two media, with the primary goal of determining the location of the IS (${\rho }_c,z_c$) of the refracted rays.

Based on Snell’s law (8), relationships for the coordinates of the IS (13) in the first medium, as well as the formula for the optical path length (11) for the ray from the IS, have been derived. The validity of the expressions for the coordinates has been verified by an alternative method (see Appendix A), where a system of two linear equations is proposed. The solution to this system corresponds to the intersection point of two closely spaced rays, which determine the coordinates of the IS (${\rho }_c,z_c$).

It is shown that the focal surface of the ISs represents a rotational hyperboloid (see Figure 2).

At the next stage of solving the problem, the amplitude of the IS is determined, which simplifies the procedure for finding the EM field of the refracted rays at any point in space, since due to the IS, the entire space can now be considered homogeneous.

Thus, from the required condition of field continuity at the interface between the media, considering both the real and imaginary sources, the amplitudes of the magnetic and electric fields (22) and (23) have been determined.

It should be noted that the expressions obtained for the fields in the form of Fresnel formulas indicate that during the transmission (or reflection) of a spherical wave through a plane interface, the entire plane does not play a significant role; only a certain ‘effective zone’ on it has such an impact. This zone encompasses a specific number of Fresnel zones constructed around a narrow ray tube, within which the wavefront is considered planar. In this context, the first Fresnel zone can be taken as the effective zone, while the remaining zones do not play a significant role. Thus, by using the IS, the solution to the problem is reduced to Fresnel’s formulas in (22) and (23). This implies the applicability of the ISM in evaluating the following solutions:

• The problem of spherical wave scattering at a smooth boundary between two media, taking into account the curvature;
• Sommerfeld’s problem of radiation from a horizontal dipole [12].

The validity of the aforementioned electric field formula has been verified by comparing it with the result of the direct computation of the short-wavelength asymptotics of the Sommerfeld integral in the far field using the saddle point method.

As expected, all the formulas obtained within the geometrical optics approximation, as well as the corresponding methodology, agree well with the solution to the Sommerfeld problem for the radiation of a vertical electric dipole above a plane interface between two media.

For the applicability of geometrical optics in our case, it is necessary that the source be sufficiently far from the boundary ($k_{01}r>1$) compared to the wavelength, while the position of the receiver does not play a significant role.

It should be noted that this problem has been solved for transparent, weakly attenuating media, as a more complex problem considering attenuation in the media is to be addressed next. Among the practically important problems, the problem of spherical wave propagation in plane-layered media [13,14] can be mentioned.