## Abstract

An overview of the problems involved in the study of electromagnetic power transmission between lossy media is presented. Starting from the well-known problem of the transmission at a dielectric–conductor interface, the different representations of the complex propagation vector of the plane waves are introduced. Analytical expressions to convert from one formulation to the other are obtained. Moreover, the transmission of a plane wave at the interface between two lossy media is taken into account. An explanation of the strange behavior of the transmitted wave is developed by means of power considerations. Finally, the interesting effect of the parallel-attenuated transmitted wave is presented, and its properties as a function of the incident phase vector amplitude are deduced.

© 2012 Optical Society of America

## 1. INTRODUCTION

The transmission of electromagnetic plane wave on the plane interface between lossy media is a widely studied problem in optics and electromagnetic theory. We consider the propagation of an inhomogeneous plane wave in a lossy medium, which impinges on the interface with another lossy medium. We assume the media are linear, homogeneous, and isotropic; with these exceptions, we define the problem in the most general manner.

The complex plane-wave propagation vectors are represented with two different formulations: the phase and attenuation vectors, i.e., the Adler–Chu–Fano formulation [1], and the complex angle formulation [2]. In Fig. 1, the phase and attenuation vectors of the incident, ${\mathit{\beta}}_{1}$, ${\mathit{\alpha}}_{1}$, and of the transmitted, ${\mathit{\beta}}_{2}$, ${\mathit{\alpha}}_{2}$, waves are shown. The angles that these vectors form with the normal direction to the interface are, respectively ${\xi}_{1}$, ${\zeta}_{1}$, and ${\xi}_{2}$, ${\zeta}_{2}$. Moreover, we define the angles between the phase and the attenuation vectors, in the two media, as ${\eta}_{1}={\zeta}_{1}-{\xi}_{1}$ and ${\eta}_{2}={\zeta}_{2}-{\xi}_{2}$, respectively.

To understand the origin of these different formulations, we start from the classic problem of the refraction between a lossless and a lossy medium, which is well established in the literature [3–7]. The difference between the complex angle of the transmitted wave vector and the angle of the transmitted phase vector is well understood. However, some confusion between these angles may occur. To emphasize the possible mistake that can be made between these formulations, we consider the arguments in [8], where the expressions of the Fresnel coefficients found in the literature are questioned. The mistake has been pointed out and corrected in [9], but allows us to clarify the differences between the possible representations of the complex wave vector of an inhomogeneous plane wave in a lossy medium. Moreover, the relations between the real and the imaginary parts of the complex angle with the phase and the attenuation vectors are presented. The connection between these two formulations has been explored before, in [10], where a numerical result, which needs different determinations between the solution of a polynomial equation of the fourth order, and the inversion of a cosine function, is presented.

We consider a complex wave vector lying on the $(x,y)$ coordinate plane, and we find an analytical expression that connects the phase and the attenuation vectors, with its real and imaginary parts. This result is achieved by considering the polarization vectors of the complex vector, connected to its polarization ellipse [11,12].

Afterward, the transmission of a plane wave between two lossy media is considered. This problem has been considered in [10], where it has been found that the transmitted angle of the phase vector, for some incident wave, may exceed $\pi /2$; i.e., the phase vector can be directed toward the half plane of origin of the incident wave. This effect is interpreted as a total reflection. We believe that the angle’s determination has been chosen without any physical, or mathematical, reason. We apply energy conservation to prove the determinations are correct. Furthermore, we analyze the direction of propagation of the power associated to the incident and the transmitted plane waves. When the angle of the transmitted power direction exceeds $\pi /2$, the angle of the incident power direction exceeds it, too. Therefore, we find that there is no total-reflection phenomenon between two lossy media, interpreting the effect through the ill posedness of the problem.

The analytical expressions of the incident critical angles ${\xi}_{1}^{\xi}$ and ${\xi}_{1}^{\zeta}$, for which the angles of the phase and attenuation transmitted vectors match $\pi /2$, respectively, have been presented in [13]. We analyze their expressions and we deduce some physical considerations on the behavior of these particular transmitted waves.

In Section 2, some historical remarks on the problem of refraction by a dielectric–conductor interface are presented. In Section 3, we consider the two possible representations of a complex wave vector in a lossy medium and we use the polarization vectors to calculate the real and the imaginary parts of its complex angle as a function of the phase and attenuation vectors. In Section 4, the results found in [10] are considered. We find a physical reason to justify the determinations of the transmitted angles and, by considering the power flow directions, we show how there is no total-reflection phenomenon between two lossy media. In Section 5, we analyze the case when the incident phase vector has an angle ${\xi}_{1}^{\zeta}$, for which the transmitted attenuation vector is parallel to the interface, discussing some properties of this wave. Finally, in Section 6, conclusions are drawn.

In this paper, we consider a time dependence of the form $\mathrm{exp}(i\omega t)$. Moreover, for any complex vector $\mathit{v}$ we define the dot products $\mathit{v}\xb7\mathit{v}={v}^{2}$ and $\mathit{v}\xb7{\mathit{v}}^{*}={|\mathit{v}|}^{2}$. With the symbols ${\u03f5}_{i}$, ${\mu}_{i}$, and ${\sigma}_{i}$, we refer to the permittivity, permeability, and conductivity of medium *i*, respectively.

## 2. THE DIELECTRIC–CONDUCTOR INTERFACE

The problem of the refraction at the interface between a lossless medium and a lossy one is a classic one in optics and electromagnetic theory, treated in many popular books [3–7]. The problem is faced by considering a homogeneous incident plane wave and an inhomogeneous transmitted plane wave. The Snell law is exactly the same of the case of lossless media, but the wave number and the transmitted angle are complex quantities. The generalized Snell law can be written as follows:

The imaginary part is orthogonal to the interface, because of the continuity of the tangential part of the propagation vector on the interface. The real part of the propagation vector forms an angle ${\xi}_{2}$ with the normal to the interface. The connection between this angle and the real and imaginary parts of the complex angle ${\theta}_{2}$ does not have a simple expression, and it has been found in [3,4]. Particularly, in [3], it is pointed out that the angle ${\xi}_{2}$ is the “true” transmission angle, because ${\theta}_{2}$, being complex, does not have a well-established physical meaning. The confusion between these angles can often happen when the interaction with lossy media is considered. For instance, in [8], the conclusions drawn in [3] are denied, asserting the transmission angle ${\theta}_{2}$ has to be purely real; in fact the incident wave vector has no tangential component of the attenuation vector, so the transmitted wave has none, too. When the problem is considered with more attention, the confusion in [8] appears evident and, in [9], this mistake has been corrected, showing the Fresnel coefficients computed are identical to the expressions found in [3].

When the transmitted propagation vector, and its complex angle, are known, the reflection and transmission Fresnel coefficients can easily be written as an analytic continuation of the coefficients of the lossless case. A summary of the properties of these coefficients for the incidence on the dielectric–conductor interface is presented in [14]. These coefficients can never vanish. Therefore, the total-transmission effect can never happen, or the Brewster angle cannot exist for an interface between a lossless and a lossy medium. Actually, in the literature there have been many attempts to define a pseudo-Brewster angle in this case [15–18].

In the lossless case, the Brewster angle is defined as the incident angle for which the transmitted coefficient in $H$ polarization, ${\mathrm{\Gamma}}_{H}$, vanishes. Here, we call $H$ polarization the case in which the magnetic field of the incident wave is orthogonal to the plane of incidence, and $E$ polarization the dual case. The Brewster angle can also be defined as the angle for which the ratio between the reflection coefficients in the two polarizations, ${\mathrm{\Gamma}}_{H}/{\mathrm{\Gamma}}_{E}$, is zero, which is obviously the same as the previous definition in the lossless case. In this way, the pseudo-Brewster angle can be defined in two different ways, as the angle for which the magnitude of ${\mathrm{\Gamma}}_{H}$ is minimum, or for which the magnitude of ${\mathrm{\Gamma}}_{H}/{\mathrm{\Gamma}}_{E}$ is minimum. These two conditions result in two different angles when a lossy medium is considered. The importance of the second condition is because of its connection with the degree of polarization [16]: when the magnitude of this quantity is minimum, the degree of polarization is maximum; i.e., the reflected wave by the interface tends to be $E$ polarized.

As well as the total-transmission, the total-reflection effect disappears, too, when a lossy medium is considered. The transmitted wave, in fact, is always an inhomogeneous wave, with a phase vector that cannot be parallel to the interface, because of the medium losses.

Another interesting effect at the interface between a lossless and a lossy medium, found in the literature, is the nondeviated wave. There is a particular incident angle for which the transmitted angle of the phase vector is equal to the incident one. Therefore, the phase vector does not deviate because of the transmission. This effect was pointed out first in [19], as noted in [20].

## 3. ON THE COMPLEX WAVE VECTOR

Let us consider again the generalized Snell law. As mentioned earlier, by manipulating Eq. (1), the propagation vector may be written as a superposition of phase and attenuation vectors. Therefore, the transmitted propagation vector has two possible representations: the complex angle formulation, used in [3,4], and the formulation with the phase and attenuation vectors, used in [1]. In the first case, the wave vector is represented by two complex numbers, the wave number ${k}_{2}={k}_{2R}+i{k}_{2I}$ and the transmitted angle ${\theta}_{2}={\theta}_{2R}+i{\theta}_{2I}$. In the second case, the propagation vector is represented through four real quantities, the amplitudes of the phase and attenuation vectors, ${\beta}_{2}$ and ${\alpha}_{2}$, respectively, and their angles with the normal to the interface, ${\xi}_{2}$ and ${\zeta}_{2}$, respectively. Therefore, in general, we have four degrees of freedom to define the propagation vector. In the case of the dielectric–conductor interface, the transmitted propagation vector has only three degrees of freedom, because ${\zeta}_{2}=0$.

When we start from the complex angle formulation to solve the problem, as in Eq. (1), it is anyway needed to write the equations for the phase and attenuation vectors, called Ketteler’s equations in [3]. These are three independent equations, which is obvious because the transmitted wave vector has three degrees of freedom. As we will see afterward, in the case in which two lossy media are considered, four independent equations are needed, because of the four degrees of freedom of the wave vector. Therefore, to impose the continuity of the wave vector at the interface, the phase and attenuation vectors have to be considered anyway. This is the reason why in [1] the problem is faced directly with this formulation, avoiding the definition of a complex angle altogether.

When two lossy media are considered, the same problem arises, and the Adler–Chu–Fano formulation seems best suited to face it. However, in some cases, the plane wave has to be represented with the complex-angle formulation; this occurs, for instance, when the plane wave must be decomposed in cylindrical or spherical harmonics [21–23]. Therefore, it is important to have a tool that allows us to pass from one formulation to the other.

In [10], imposing the equivalence of the complex wave vectors in the two representations, an equation for $\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{2R}$ is achieved. The three-dimensional case, when the phase and attenuation vectors and the normal to the interface are not coplanar, is considered. The numerical solution of the equation gives four different possible values for $\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{R2}$, and its inversion leads to another indetermination, because the inverse cosine is a multivalued function. The right value among all the possible solutions is chosen; however, the reason for this choice is not clear. This indetermination is connected to the difficulty to compare the complex vectors.

When we impose the equality between two real vectors, first of all, we force the direction of the vectors be the same. Complex vectors do not have a direction in the common meaning; the effective direction can be considered through the associated polarization ellipse [11,12]. Two complex vectors are parallel if their polarization ellipses have the same ellipticity.

A generic complex vector can be represented as a superposition of two real vectors or as the product of a complex number $k$ and a complex unit vector $\mathit{n}$, where the term “complex unit vector” means the vector satisfies $\mathit{n}\xb7\mathit{n}=1$. These two representations coincide with the two formulations considered above, when the unit complex vector is represented on the plane $(x,y)$ with a complex angle. Let us consider the generic wave vector $\mathit{k}$ and its representation with both the complex angle and the Adler–Chu–Fano formulations:

To describe the polarization of a complex vector, there is an important tool that can be used: the polarization vectors $\mathit{p}$ and $\mathit{q}$, which are real vectors. The polarization vectors of the vector $\mathit{k}$ may be written as follows [12]:

Therefore, the real and the imaginary parts of the complex angle $\theta $ may be written as a function of the phase and attenuation vectors:

Expressions (8), (9), and (10) allow us to compute the complex angle of a complex wave vector, knowing its phase and attenuation vectors. Expression (10) is of particular interest, because the imaginary part ${\theta}_{I}$, which is the inhomogeneity degree of the wave [2], is connected only to the lossy tangent of the medium, i.e., the ratio between the imaginary and the real parts of the number ${k}^{2}$, and to the tangent of the angle $\eta $, which can be considered as a degree of the wave inhomogeneity in the Adler–Chu–Fano formulation. Moreover, expression (10) is valid for $\eta \in (-\pi /2,\pi /2)$, i.e., for a lossy medium. In a lossless medium, an inhomogeneous wave can be considered, too. In this case, the expression (10) becomes indeterminate. An alternative expression can easily be found by considering the imaginary part of the dispersion equation, i.e., $\mathrm{Im}({k}^{2})=2\beta \alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\eta $, and may be written as follows:

## 4. POWER TRANSMISSION BETWEEN TWO LOSSY MEDIA

The problem of the transmission of a plane wave between two lossy media has been deeply analyzed in [10]. The complex vectors of both the incident and the transmitted waves with their phase and attenuation vectors are represented, as defined in Fig. 1. The continuity of the tangential components of the propagation vector on the interface leads to the following conditions:

From the knowledge of the amplitudes (16) and (17), the angles ${\xi}_{2}$ and ${\zeta}_{2}$ are easily found by the inversion of Eqs. (12) and (13); hence

To explain the problem, let us consider the expression (18) with ${\xi}_{1}\in [0,\pi /2]$. When ${\xi}_{1}\in [0,{\xi}_{1}^{\xi}]$ the angle ${\xi}_{2}$ varies in the interval $[0,\pi /2]$. When ${\xi}_{1}\in [{\xi}_{1}^{\xi},\pi /2]$, the angle ${\xi}_{2}$ can follow two different behaviors: it may decrease, coming back in the interval $[0,\pi /2]$, assuming a nonmonotonic behavior, or it may further increase in the interval $[\pi /2,\pi ]$, assuming a monotonic behavior. If the angle ${\eta}_{1}$ is fixed, an analogous behavior may be shown for ${\zeta}_{2}$, as a function of ${\xi}_{1}$, by analyzing expression (19). In Fig. 2 the behavior of ${\xi}_{2}$ in both the nonmonotonic and monotonic determinations is presented; the properties of the media are equal to those in [10]. Different values of ${\eta}_{1}$ are considered. In Fig. 3 the behavior of ${\zeta}_{2}$ is shown under the same conditions. In Figs. 2 and 3 the indetermination on the values of ${\xi}_{2}$ and ${\zeta}_{2}$ in the points ${\xi}_{1}^{\xi}$ and ${\xi}_{1}^{\zeta}$, respectively, is apparent.

The indetermination on the values taken by ${\xi}_{2}$ and ${\zeta}_{2}$ has been solved, in [10], assuming the monotonic behavior, i.e., allowing the angles to go in the interval $[\pi /2,\pi ]$. Actually, there are no mathematical reasons to choose one solution instead of another. Moreover, the solution taken leads to a strange behavior; in fact, the angle ${\xi}_{2}$, which is the angle of the transmitted phase vector, is allowed to exceed the value $\pi /2$, meaning that the constant phase planes may propagate in medium 1. In [10], this behavior is interpreted as a total reflection of the incident wave, obtainable also when ${\u03f5}_{1}<{\u03f5}_{2}$. As we will see, the determination taken is right; however, the behavior is not a total reflection, but is connected with the power flow of the incident plane wave.

First of all, we show that the determinations chosen for the angles ${\xi}_{2}$ and ${\zeta}_{2}$ are correct, pointing out the reasons of the choice, on the basis of a physical constraint. To find the right values of the angles, we impose the energy conservation, i.e., the Poynting theorem, which, in a lossy medium, may be written as follows:

where ${P}_{r}$ is the flow of the radiated power through a closed surface $S$, ${P}_{s}$ is the electric and magnetic power storage in the volume $V$ contained in the surface $S$, and ${P}_{d}$ is the dissipated power in $V$. We take as surface $S$ an infinite square cylinder with axis along the $z$ direction, having the center in the origin of the $(x,y)$ plane, and side $w$.In Fig. 4, the electromagnetic power amplitude, for unit length along $z$, for the two different determinations of the angles ${\xi}_{2}$ and ${\zeta}_{2}$ is shown, in two cases: when the exceeding of $\pi /2$ occurs, the monotonic case, and when the angles are always lower than $\pi /2$, the nonmonotonic case. The same scenario of Figs. 2 and 3 is considered. In the cases ${\eta}_{1}=\pi /4$ and ${\eta}_{1}=0$, as it is shown in Fig. 3, there is an indetermination on the values of ${\zeta}_{2}$ for ${\xi}_{1}={\xi}_{1}^{\zeta}$; similarly, in the case ${\eta}_{1}=-\pi /4$, as shown in Fig. 2, there is an indetermination on the value of ${\xi}_{2}$ for ${\xi}_{1}={\xi}_{1}^{\xi}$. In Fig. 4, we see that $P\ne 0$ for ${\xi}_{1}>{\xi}_{1}^{\xi ,\zeta}$, when the nonmonotonic determination is chosen. Therefore, the correct determination is the monotonic one, being, in this case, $P=0$ for all values of ${\xi}_{1}$. According with [10], the right determinations of the transmitted angles can be written as follows:

Now, we are going to explain the meaning of the exceeding of $\pi /2$ by the angle of the transmitted phase vector ${\xi}_{2}$. The reason why this effect cannot be a total reflection is simple. In the total reflection phenomenon, between two lossless media, the transmitted wave has an attenuation vector orthogonal to the interface, and a phase vector parallel to it. Therefore, the transmitted wave lies in medium 2, propagating along the interface. In a lossy medium this configuration is not possible, because the angles between the two vectors cannot be equal to $\pi /2$. At most, this effect can be reached as a limit, when a dielectric–conductor interface is considered. The transmitted attenuation vector is purely orthogonal to the interface and, if ${\u03f5}_{1}>{\u03f5}_{2}$, by increasing the incident angle, the transmitted phase vector’s angle can reach asymptotically the value $\pi /2$. In any case, the transmitted wave must lie in medium 2. So, the case ${\xi}_{2}>\pi /2$ cannot be considered a total reflection, because the transmitted wave cannot propagate in medium 1.

To understand this phenomenon, we have to see how the power flow propagates in medium 2. In [1], the Poynting (complex) vector of an inhomogeneous wave in a lossy medium with conductivity $\sigma $ and real permittivity $\u03f5$ is considered, for the two polarizations:

The propagation direction of the real part of the Poynting vector is different from the direction in which “real-power-only” flows [1,6]. From Eq. (23), these two directions, for $E$ polarization, can be found. The real part of the Poynting vector is directed parallel to the phase vector. However, the direction where real-power-only flows is the one orthogonal to $\mathit{\alpha}$, different from the other because $\mathit{\beta}$ and $\mathit{\alpha}$ are not orthogonal. Similarly, in $H$ polarization, the real part of the Poynting vector is parallel to $\mathit{\beta}+\sigma \mathit{\alpha}/(\omega \u03f5)$; in contrast, the real-power-only flows in a direction orthogonal to the vector $\mathit{\alpha}-\sigma \mathit{\beta}/(\omega \u03f5)$, as becomes clear by separating the real and imaginary parts in Eq. (24).

We consider as effective direction of propagation the one where real-power-only propagates, and we call $\rho $ the angle it forms with the normal to the interface. Therefore, considering only the $E$ polarization, we have to check the direction orthogonal to the attenuation vector. In Fig. 5, the direction of the real-power-only is shown in the same scenario of Figs. 2 and 3. In the cases ${\eta}_{1}=0$ and ${\eta}_{1}=\pi /4$, when ${\xi}_{2}$ remains less than $\pi /2$, the angles of real-power-only directions of both the incident and the transmitted waves, ${\rho}_{1}$ and ${\rho}_{2}$, are always less than $\pi /2$, too. However, in the case ${\eta}_{1}=-\pi /4$, when there is a value ${\xi}_{1}={\xi}_{1}^{\xi}$ after which ${\xi}_{2}$ exceeds $\pi /2$, the real-power-only directions of the incident and transmitted waves exceed the value $\pi /2$, too. Moreover, these power directions exceed $\pi /2$ for the same value of ${\xi}_{1}$, which is different from ${\xi}_{1}^{\xi}$.

Now, we can interpret the phenomenon. When an incident inhomogeneous wave from a lossy medium impinges on the interface with another lossy medium, the direction of effective propagation of the wave does not coincide with the phase vector direction, but can be assumed as the real-power-only direction, forming an angle ${\rho}_{1}$ with the normal vector to the interface. When ${\rho}_{1}>\pi /2$, the incident power flows in the direction of the medium 1, i.e., in the negative $x$ direction, as if the power flow does not propagate in the direction of the second medium. In this situation also the angle of the transmitted real-power-only direction, ${\rho}_{2}$, exceeds $\pi /2$, and the reason is evident: if the incident power direction goes in the negative $x$ half plane, the transmitted power also keeps its direction toward the same half plane. Therefore, to consider an incident wave, when lossy media are involved, one needs to check the true direction of the wave, i.e., the direction of its real-power-only flow.

To persuade us of this behavior, we can consider all the cases shown in [10], when ${\xi}_{2}>\pi /2$, and check if the power directions always follow the same behavior. In Fig. 6 all these cases are shown. We may see how ${\rho}_{1}$ and ${\rho}_{2}$ always exceed $\pi /2$ at the same values of ${\xi}_{1}$. It is important to note that the values of ${\xi}_{1}$ for which the real-power-only directions match $\pi /2$ are different from ${\xi}_{1}^{\xi}$. This difference is because ${\xi}_{1}^{\xi}$ tells us when the transmitted phase vector direction matches $\pi /2$, but the real-power-only direction is not linearly connected with the phase vector direction.

## 5. PARALLEL ATTENUATED TRANSMITTED WAVE

In the analysis of the transmission of the electromagnetic radiation between two lossy media, an interesting effect can be shown. When the incident plane-wave phase vector matches the angle ${\xi}_{1}={\xi}_{1}^{\zeta}$, the transmitted wave does not attenuate away from the interface. The analytical expressions of the angles ${\xi}_{1}^{\xi}$ and ${\xi}_{1}^{\zeta}$ are

When the incident wave, at a fixed ${\eta}_{1}$, has a phase vector with ${\xi}_{1}={\xi}_{1}^{\zeta}$, the transmitted wave has the constant-amplitude planes orthogonal to the interface, and the phase vector with an angle ${\xi}_{2}\in [0,\pi /2]$. That means the wave is not attenuated in the direction orthogonal to the interface, and its real-power-only flow, in the case of TE polarization, is orthogonal to it. This result is surprising, because we achieved a transmitted wave in a lossy medium that is not attenuated in the direction where the medium extends. The same effect emerges in the case the interface between a lossless medium and a lossy one is considered, with an inhomogeneous incident wave. In this case, the critical angle has the following expression:

Equation (28) is related to the incidence of an inhomogeneous plane wave, from a lossless medium, impinging on the interface with a lossy one. Therefore, this effect must be considered in the class of the effects of the classic problem of the incidence of a plane wave on a dielectric–conductor interface. The only difference between the present scenario and the classic situation is the inhomogeneity of the incident wave. The possibility to have an inhomogeneous wave in a lossless medium is well established in the literature, e.g., with the leaky waves [25,26]. The generation of this kind of waves has been widely studied, and many leaky-wave antennas have been proposed at microwave frequencies [27], and at optical frequencies, too, in more recent works [28].

Now, we show some interesting properties of the critical angle (28). First of all, it decreases very quickly, increasing the incident phase vector’s amplitude, by starting from a maximum value of $\pi /4$, reached for the minimum value of ${\beta}_{1}$, as shown in Fig. 7. As well as the critical angle, the associated transmitted angle has a sudden decrease, too. Therefore, increasing the amplitude of the incident phase vector, the incidence tends to become orthogonal to the interface. Obviously, because of the separability condition, we get ${\alpha}_{1}^{2}={\beta}_{1}^{2}-\mathrm{Re}({k}_{1}^{2})$. Therefore, the amplitude of the attenuation vector increases with ${\beta}_{1}$, as shown in Fig. 8. In the second medium, where the attenuation is parallel to the interface, also ${\alpha}_{2}$ grows. The behavior of the Fresnel transmission coefficient is of interest, too. As shown in Fig. 8, the amplitude of the transmission coefficient tends to unity increasing ${\beta}_{1}$. Therefore, the amplitude of the electric field in the two media tends to be the same.

Summarizing, increasing the amplitude of the incident phase vector, the attenuation in the first medium increases and the incidence tends to become normal to the interface. The amplitude of the electric field of the transmitted wave tends to be equal to that of the incident wave, while the wave propagates without attenuation in the direction orthogonal to the interface, becoming narrower in the direction parallel to it. The properties of the incident and the transmitted waves can be steered by the amplitude of the incident phase vector. It is of interest to note that, as is known [27], in leaky-wave antennas the radiation properties are strongly connected with this vector’s amplitude.

## 6. CONCLUSIONS

In this article, the transmission of an electromagnetic plane wave between two lossy media is considered. Starting from the classic solutions of the problem, we discussed the effects presented in the literature and the different formulations that can be adopted. The relations between the complex-angle and the Adler–Chu–Fano formulations have been shown through analytical expressions, avoiding indetermination and numerical evaluations.

The results found in [10] have been analyzed, finding physical reasons for the choices made for the determination of the transmitted angles. We tried to understand how the inhomogeneous waves behave at a planar interface between two lossy media, analyzing their power and finding in the real-power-only direction a good candidate to represent an effective direction of propagation of the waves. Moreover, we showed how the exceeding of $\pi /2$ by the transmitted phase vector is connected with the exceeding of $\pi /2$ by the real-power-only direction, and this is an indication that such direction is better suited with respect to the phase-vector direction to understand the direction of propagation of the inhomogeneous wave, when a lossy medium is considered.

Finally, the transmitted wave at the critical angle ${\xi}_{1}^{\zeta}$ is analyzed. This wave is not attenuated in the direction orthogonal to the interface; therefore, its power flow may propagate indefinitely away from the interface. We show some properties of the incident and the transmitted waves as functions of the incident phase vector, which seems to be a good candidate to steer this phenomenon.

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