Electromagnetic inhomogeneous waves at planar boundaries: tutorial

Fabrizio Frezza; Nicola Tedeschi

doi:10.1364/JOSAA.32.001485

1. INTRODUCTION

The study of inhomogeneous electromagnetic waves at the planar boundary between two media has been an important topic since the early twentieth century. The first efforts to understand the properties of such waves can be ascribed to Sommerfeld and Zenneck [1,2]. The former was involved in the analysis of radiation from a vertical electric dipole located above the planar interface between two homogeneous media, and the latter was about the bounded inhomogeneous plane waves at the interface between dissipative materials. These studies have been carried out for the analysis of guided propagation of electromagnetic waves on stratified structures and the interaction of the electromagnetic sources with Earth’s surface [3,4]. Since the early studies, several kinds of inhomogeneous waves have been recognized; in particular, three waves were individuated: lateral waves, first discovered in acoustic applications and experimentally found also for electromagnetic waves [5,6]; surface waves, bounded waves propagating on the interface [7]; and leaky waves, due to the leakage of energy from guiding structures provided with some kind of aperture. Due to their importance in applications and to their theoretical appeal, these waves have been widely studied and several physical properties have been presented in the literature. In the work by Baños [8], radiation of dipole sources above a planar interface was studied in great depth. Moreover, several reviews have been published on the topic. We refer specifically to the works by Zucker [9], Brown [10], Goubau [11], Schelkunoff [12], Barlow and Brown [13], and Tamir [14–16].

This topic presents great appeal for researchers in recent years, too. In particular, surface waves are of great importance in the analysis of guided propagation in open stratified structures [17] and, recently, in the guided propagation on artificial surfaces, often called metasurfaces [18]. Moreover, they are of great importance in the development of nanophotonics because of their capability of going beyond the diffraction limit [19]. On the other hand, leaky waves are of great importance in the analysis of the so-called leaky-wave antennas. These antennas have great capabilities in terms of gain, directivity, and the possibility of scanning the radiation angle by changing the working frequency [20]. These antennas are historically typical of the microwave frequency regime. However, in recent years, efforts have been made to obtain them at optical frequencies [21,22].

In this paper, we analyze the different fields that can be excited on a planar interface. When a homogeneous plane wave impinges on a planar interface between two dielectrics, a reflected plane wave and a transmitted plane wave, both homogeneous in the lossless case, emerge from the interface in the two media, respectively. This solution of the problem is well known and can be interpreted in simple ray-optical terms involving the concepts of wavefronts, rays, and ray tubes. This description fails in total-reflection conditions, i.e., when the first medium is denser than the second one, and the incident angle is larger than the critical angle [23]. In fact, in this case the refracted wave is along the interface and it is attenuated in the second medium. Here, the concept of surface wave is exploited to indicate a wave propagating along the surface and attenuating away from it. However, this kind of surface wave is not a guided solution of the interface: it exists only if a plane wave, with an infinite wavefront, impinges on it. The homogeneous plane-wave representation proves very useful in understanding the physics of many phenomena, but it cannot account for all the possible phenomena consistent with Maxwell’s equations. From experimental observation, many other kinds of waves strongly connected to a planar interface have been discovered. They can be classified into the following three families:

1. A “lateral wave” is also known as a “head wave” or “refraction arrival wave.” Each of these designations describes a special feature of the wave. The first emphasizes the sideways propagation of the wave parallel to the interface. The second one derives from the fact that, under transient conditions, this wave furnishes the first response in certain regions of the medium containing the source. The third one focuses on the important role played by refraction in establishing the wave [23]. An alternative definition can be the following: the lateral wave is a field variety that is intimately associated with a wave undergoing total reflection when incidence occurs from a denser medium onto the interface with a rarer medium [14]. In other words, the lateral wave is a wave propagating on a planar interface on the side of the bottom medium, which radiates in the top medium.
2. A surface wave is one that propagates along an interface between two different media without radiation. Here, by “radiation” we mean energy converted from the surface-wave field to some other forms [13,15].
3. A leaky wave is an inhomogeneous wave propagating in a lossless material and is generated by a guided structure because of radiation leakage [16,24].

These definitions, being verbal and not in a mathematical form, can easily lead to confusion. For example, one can understand that a surface wave propagates along the interface, in the sense that the real energy flow is parallel to the interface, without radiation in the two media. However, if one considers a surface wave in the case in which one of the two media has losses, then the surface wave will be an inhomogeneous wave in a lossy medium, so its real energy-flow direction is not in the phase direction anymore [25,26], i.e., a portion of the real power must radiate into the lossy medium. On the other hand, a surface wave can be defined in the case of lossy media, too. Just to give another example, if one considers a dielectric two-dimensional slab with a line current parallel to the interface embedded in it, it can be shown, as we will see in the following sections, that this slab may generate a leaky wave in the air half-space above it. However, considering only the previous definitions, this wave can be confused with a lateral one. In fact, it propagates along the interface and radiates in the medium above it. Furthermore, from the above definition of a lateral wave, we see that it is not possible to give a direct definition, but it is defined indirectly by means of its properties.

We want to clarify the definitions of these waves from a mathematical point of view. These definitions are rigorous and unambiguous, but unfortunately they hide the physical characteristics of waves. These characteristics will be investigated in the following sections.

A preliminary step is to define a geometric system. Here, we are interested in studying electromagnetic waves on a planar surface. The simplest case is the planar boundary between two different homogeneous media. However, this surface does not support all the fields that we are interested in, as we are going to see in the following. For this reason, we will consider the more general case of a half-space $z > 0$ , filled by a medium 1, with electric relative permittivity $ε_{1}$ , and we consider the surface $z = 0$ as a general surface that can represent a true boundary with another homogeneous half-space, or it can represent more complicated surfaces. If the surface is independent of both the $x$ and $y$ directions, then it can be represented by an impedance $Z (k_{z})$ . If, for example, we consider the boundary with another homogeneous medium, this impedance will represent the Fresnel reflection coefficient. We suppose that the relative magnetic permeability of all the media involved is equal to unity. Moreover, we suppose linear, homogeneous, isotropic, and stationary media. Actually, many interesting phenomena involving surface waves have been studied in the literature in the presence of either magnetic, anisotropic, nonlinear, or inhomogeneous media [27–30]. However, for the sake of brevity, we are not going to discuss these phenomena in the present paper.

The second step is to define the nature of the excitation. We cannot consider simple plane-wave incidence on the interface because of two reasons. First, this field is not able to excite some of the phenomena that we are interested in. In order to excite these waves, we need a richer spectrum. The second reason lies in the infinite amount of energy of the plane wave that may hide some phenomena. In order to choose an excitation, we must establish what kind of problem we want to face: two-dimensional or three-dimensional? In the first case, the most suitable source is either an electric or a magnetic line current. In the second case, we might consider either an electric or a magnetic elementary dipole. It can be seen that in the two cases, the excited waves on the interface show similar properties and differ only for the different wavefronts [31]. To simplify the mathematical expressions, we will consider the two-dimensional case, with sources independent of the $y$ direction.

Under the previous hypothesis, the electromagnetic field can be expressed by a scalar function, $V (\underset{̲}{r})$ , representing either the electric or magnetic field, in the $y$ direction, depending on the excitation. In particular, we can consider the field in medium 1 as a superposition of two fields: the field $V_{i}$ that the source would emit in free space, and the field $V_{int}$ due to the interaction with the surface. Obviously, the reflected field, $V_{int}$ , contains both the geometrical-optics reflection, valid in the ray-optics approximation, and all the other fields supported by the surface. The field $V_{int}$ can be expanded in plane waves as follows:

V_{int} (x, z) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} s (k_{x}) e^{- j (k_{x} x + k_{z} z)} d k_{x},

where the spectrum

s (k_{x})

depends on the source characteristics and on the impedance of the surface, and

k_{z} = \sqrt{k_{1}^{2} - k_{x}^{2}}

. Moreover, each plane wave of the spectrum must satisfy the radiation condition in medium 1, i.e., the imaginary part of the

z

component,

k_{z}

, of the propagation vector must always be negative. All the fields relevant to the planar surface must be found inside the spectrum in Eq. (1). If we look at the plane-wave spectrum as a complex integral, we know that many different contributions must be taken into account. First of all, the radiation condition constrains the sign of the imaginary part of

k_{z}

. It means that on the complex plane of

k_{x}

there is a branch cut that must never be crossed by the integration path. Moreover, the function

s (k_{x})

presents pole and branch-point singularities. Lateral, surface, and leaky waves are related to these singularities. In particular, the lateral wave is a field with a continuous spectrum and it is not relevant to a pole singularity, but it is connected to the path around a branch cut, i.e., to a branch-point singularity [5]. On the other hand, surface and leaky waves are relevant to pole singularities: the former to proper pole singularities and the latter to improper pole singularities [24]. Here, we talk about proper and improper poles referring to values of

k_{z}

that satisfy or do not satisfy the radiation condition, respectively. In this way, we are connecting the modes of the surface to the singularities of the spectrum representing the reflected field. The modes of the surface are related to the dispersion equation of the structure, i.e., all the modes must be relevant to zeros of the dispersion equation. A simple and fast way to obtain the dispersion equation of a structure is the so-called transverse-resonance method [17,32]. This method yields the dispersion equation of the surface by equating to zero the sum of the surface impedance,

Z (k_{z})

, and of the characteristic impedance of medium 1,

Z_{1} (k_{z})

, obtaining

Z (k_{z}) + Z_{1} (k_{z}) = 0 .

The zeros of this equation give the discrete spectrum of the surface, and they correspond to the poles of the function

s (k_{x})

.

To clarify the above definitions, we now present an example of the trace of the study presented in [33]. We consider the geometry in Fig. 1, where the half-space $z < 0$ is occupied by a dielectric slab, with thickness $d$ and relative permittivity $ε_{d}$ , over a half-space, $z < - d$ filled with a dielectric with relative permittivity $ε_{2}$ . In medium 1, we assume an electromagnetic source with an arbitrary distribution of currents, independent of the direction $y$ . We know the electric field radiated by the source, and we are interested in determining the reflected field by the slab represented by the plane-wave spectrum in Eq. (1).

Fig. 1. Geometry of a slab with thickness $d$ between two half-spaces.

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It can be proved [33] that the spectrum relevant to this structure presents, on the complex plane of $k_{x} = k_{x}^{'} - j k_{x}^{''}$ , two couples of branch points, $\pm k_{1}$ and $\pm k_{2}$ ; two couples of poles on the real axis, $\pm q$ and $\pm r$ ; and two couples of complex poles, $\pm u$ and $\pm v$ ; see Fig. 2. Each branch point is related to a branch cut, as shown in Fig. 2. Then there are two couples of branch cuts: $B_{1}^{\pm}$ , relevant to the branch points $\pm k_{1}$ ; and $B_{2}^{\pm}$ , relevant to the branch points $\pm k_{2}$ . It can be seen that the integral in Eq. (1) is given by the paths around two branch cuts and by the residues due to a couple of poles on the real axis, and due to a couple of complex poles. All these contributions are relevant to the fourth quadrant of the $k_{x}$ complex plane because of the radiation condition. The paths around the branch cuts refer to two different waves: the wave relevant to the branch point $k_{1}$ is the geometrical-optic reflected field by the surface, while the wave relevant to the branch point $k_{2}$ is the lateral-wave contribution. Similarly, the residues due to the poles on the real axis are relevant to surface waves, while the residues due to the complex poles are relevant to leaky waves.

Fig. 2. Complex plane of the variable $k_{x} = k_{x}^{'} - j k_{x}^{''}$ of the integral in Eq. (1), in the case of a line current above a dielectric slab. The black points represent the branch points of the spectrum, and the crosses represent the poles. The four dashed lines represent the branch cuts of the spectrum.

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Fig. 3. (a) Sketch of a plane wave $V_{i}$ incident at the angle of total reflection, with the reflected and transmitted waves. (b) Sketch of a plane wave traveling along the boundary and transmitted in medium 1 at the angle of total reflection. (c) Quasi-optical representation of a lateral wave.

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Fig. 4. Geometrical rays representing the incidence of a cylindrical wave generated by a line current centered on C. Three paths are described: $\bar{CP}$ is the path of the direct ray from the source to the observation point, $\bar{CT} + \bar{TP}$ is the path of the reflected ray, and $\bar{CA} + \bar{AB} + \bar{BP}$ is the path of the lateral ray.

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At this point we have a clear and unambiguous definition of the waves at the interface: lateral waves are fields with a continuous spectrum due to a branch-point singularity, relevant to the medium filling the half-space without sources; surface waves are fields with a discrete spectrum due to real-pole singularities; and leaky waves are fields with a discrete spectrum due to complex-pole singularities.

It is important to note that we considered the structure in Fig. 1 because it has a rich spectrum with several branch points and poles. In general, not all the structures present such singularities. For example, if we consider as a structure the simple boundary between two homogeneous media and as an excitation a single cylindrical wave, then the spectrum will be the product of two elements: the Fresnel reflection coefficient $R (k_{x})$ and the plane-wave spectrum of the cylindrical wave $\hat{CW} (k_{x})$ , well known in both the cases of lossless or dissipative materials [34,35]. It can be seen that the spectrum in this case presents only two paths around two branch cuts and only one real pole singularity in the fourth quadrant [33]. As a consequence, this structure supports only the lateral wave and the surface wave, while the leaky wave cannot be excited.

2. LATERAL WAVE

Lateral-wave propagation is a classical topic in electromagnetic research, with studies spanning nearly the entire twentieth century. Therefore, several reviews specifically addressing the topic have been published. The reader interested in a wider look at lateral-wave properties can refer to a rich bibliography [14,23,33,36]. Early research on lateral waves, which dates back to the first thirty years of the twentieth century, was carried out in the field of elastic waves. In fact, these waves are extremely useful in seismic studies and for the investigation of Earth’s crust. The first interpretations of the phenomena were mostly of a mathematical nature, recognizing lateral waves as solutions due to branch-point singularities of the plane-wave spectrum [4]. The physical interpretation and the experimental verification of lateral waves in acoustics were addressed during the 1930s, while an extension of the results to electromagnetic waves was worked out during the 1940s [33].

Lateral waves occur when a planar-layered medium is considered, so they must be considered in several applications. Interest in lateral waves is also because of the connection with the total internal reflection on the interface between a denser and a rarer medium. As we clarified in Section 1, to excite a lateral wave we must consider a rich spectrum. If we consider a line current above the interface between a denser medium with permittivity $ε_{1}$ and a rarer one with permittivity $ε_{2}$ , then the lateral-wave contribution is due to the integration path around the branch cut relevant to the branch point $k_{2}$ . If we consider a point $P$ in medium 1, we can recognize several contributions to the total electromagnetic field. As explained in Section 1, there are the incident field $V_{i}$ , due to the line current; the geometrical-optics reflected field $V_{r}$ , due to the path around the branch cut relevant to the branch point $k_{1}$ ; the lateral-wave contribution $V_{L}$ , due to the path around the branch cut relevant to the branch point $k_{2}$ ; and the surface-wave contribution $V_{S}$ , due to a pole singularity [33]. Because of the definition of a surface wave, we suppose that it is closely confined to the interface, and here we neglect its contribution. As previously stated, since the lateral wave presents a continuous spectrum, it cannot, in general, be represented as a plane wave. However, in the path around the branch cut due to the branch point $k_{2}$ , the contribution of the branch-point singularity is dominant at a sufficiently large distance from the source. We can see that this contribution corresponds to a geometrical-optics ray impinging on the interface exactly at the critical angle of total reflection. In fact, the branch point corresponds to the following value of the tangential component of the propagation vector:

k_{x} = k_{2} \to \sin θ_{i} = \frac{k_{2}}{k_{1}} \to θ_{i} = θ_{L},

where we suppose

k_{2} < k_{1}

. Hence, the lateral wave is strongly connected to the total internal reflection. This connection allows us to give a quasi-optical interpretation of the lateral wave, as shown in [14]. Let us consider a plane wave

V_{i}

incident at the critical angle

θ_{L}

on the interface between a denser medium 1 and a rarer medium 2, as shown in Fig. 3(a). As is well known, the transmitted wave

V_{L 2}

propagates along the boundary without attenuation and with an infinite wavefront. At this point, we can consider this transmitted wave and imagine it as an incident wave from medium 2 to medium 1. In this case, the reciprocity principle ensures that the transmitted wave in medium 1,

V_{L 1}

, will be a homogeneous plane wave propagating at an angle

θ_{L}

, as shown in Fig. 3(b). If we now connect the two situations in Figs. 3(a) and 3(b), we can see that the wave

V_{i}

, at angle

θ_{L}

, excites the wave parallel to the boundary in medium 2, and such a wave is transmitted again in medium 1 at the same angle, as shown in Fig. 3(c). The waves

V_{L 1}

and

V_{L 2}

together form the lateral wave. In particular, they are the two portions of the lateral wave, respectively, in medium 1 and medium 2. In this way, we explained the behavior of a lateral wave in terms of rays. From this approximate, but intuitive, description, we can understand why the lateral wave cannot be recognized when a plane wave with an infinite wavefront incident at the angle of total reflection is considered. In this case, it would be “hidden” by the reflected wave.

The expression of the transverse field to the plane of incidence of a lateral wave excited by a line current has been obtained in the literature and can be written as follows (here $V_{L}$ represents the electric field in the case of an electric line current and the magnetic field in the case of a magnetic line current, perpendicular to the plane of incidence) [37,38]:

V_{L} = A \frac{- 2 j m ε_{2}}{ε_{1} - ε_{2}} \frac{\exp {- j [k_{1} (L_{1} + L_{2}) + k_{2} L]}}{{(k_{2} L)}^{3 / 2}},

where

L_{1} = \bar{AC}

,

L_{2} = \bar{BP}

, and

L = \bar{AB}

are the paths in Fig. 4, and either

m = 1

if the electric field is transverse to the plane of incidence, i.e., perpendicular polarization, relevant to an electric line current, or

m = ε_{1} / ε_{2}

in the case where the electric field is on the plane of incidence, i.e., parallel polarization, relevant to a magnetic line current. Furthermore, with reference to Fig. 4, we use

r_{1} = \bar{CP}

and

r_{2} = \bar{CT} + \bar{TP}

. We can see that the angle of incidence of the lateral-wave ray is exactly the critical angle

θ_{L}

, while the angle of incidence of the reflected wave is

θ_{2} > θ_{L}

. Equation (4) is valid when

k_{1} r_{2} ≫ 1

and in particular when terms of the order

{(k_{1} r_{2})}^{- 3 / 2}

are negligible. Moreover, terms of the order

{(k_{2} L)}^{- 5 / 2}

are neglected, and both

| {(n_{1} / n_{2})}^{2} - 1 |

and

k_{2} L

are sufficiently large. When

k_{2} L \to 0

, this expression is no longer valid and more accurate expressions must be implemented involving the Weber–Hermite function, i.e., the parabolic cylinder function [5,39].

The amplitude of the lateral wave decays as ${(k_{2} L)}^{- 3 / 2}$ , which is much stronger than the decrease of the geometrical-optics fields which, being cylindrical waves, decay as ${(k r)}^{- 1 / 2}$ . Therefore, the lateral wave weakly contributes to the electromagnetic field and is hard to be observed as a separate component of the total field. We must notice that the behavior ${(k_{2} L)}^{- 3 / 2}$ is strongly connected to the lateral wave and does not depend on the particular excitation that we are considering; in fact, it is possible to see the same dependence with different sources. If we consider, for example, the three-dimensional case with an elemental dipole above the surface instead of the line current, we would find exactly the same expression in Eq. (4) multiplied by a term $x^{- 1 / 2}$ connected to the spherical wavefront [31]. What is exactly the same in both the two- and three-dimensional cases is the expression of the lateral-wave phase:

ϕ_{L} = k_{1} (L_{1} + L_{2}) + k_{2} L .

In this phase, we can notice three different contributions: the path

k_{1} L_{1}

, in medium 1 from the source to the interface; the path

k_{2} L

, in medium 2, below the interface; and the path

k_{1} L_{2}

, from the interface to the observation point

P

. We can prove that this path is shorter than both the path of the incident field,

ϕ_{i} = k_{2} r_{1}

, and the one of the reflected field,

ϕ_{r} = k_{1} r_{2}

. We can write these phases as follows:

ϕ_{i} = \frac{k_{1} x}{\sin θ_{1}},

ϕ_{r} = \frac{k_{1} x}{\sin θ_{2}},

ϕ_{L} = k_{1} (L_{1} + L_{2}) + k_{1} L \sin θ_{L} = k_{1} {\frac{h}{\cos θ_{L}} + \frac{z}{\cos θ_{L}} + [x - (h + z) \tan θ_{L}] \sin θ_{L}} = k_{1} x (\frac{h + z}{x} \cos θ_{L} + \sin θ_{L}) .

Now, we can note that

\frac{h + z}{x} = \cot θ_{2},

\frac{\cos (θ_{L} - θ_{2})}{\sin θ_{2}} = \cos θ_{L} \cot θ_{2} + \sin θ_{L} .

Hence,

ϕ_{L} = \frac{k_{1} x}{\sin θ_{2}} \cos (θ_{L} - θ_{2}) .

At this point, we can compare the different phases. We can note that

θ_{2} < θ_{1}

, then

\sin θ_{2} < \sin θ_{1}

and

ϕ_{i} < ϕ_{r}

, i.e., the direct ray is always shorter than the reflected ray, an obvious consideration. We can also see that

ϕ_{L} < ϕ_{r}

, being

\cos (θ_{L} - θ_{2}) < 1

. Moreover, we can write the lateral-wave phase as follows:

ϕ_{L} = ϕ_{i} \frac{\sin θ_{1}}{\sin θ_{2}} \cos (θ_{L} - θ_{2}) .

Hence, an interval of values of

θ_{1}

that make

ϕ_{L} < ϕ_{i}

surely exists. Therefore, for certain observation points, the lateral wave is the one with the shortest optical path, i.e., is the first wave to reach the observer. This is the reason for some of the different names, such as “refraction arrival wave” or “head wave.” This phenomenon is because the lateral wave propagates for a long distance

L

in the rarer medium 2, so it is faster than the other waves that propagate in medium 1.

This behavior of the lateral wave is extremely useful in acoustics, in the analysis of the seismic phenomena, and in the study of Earth’s structure. We can give an extremely simplified description of Earth’s crust as a layer of thickness $h$ between air and a deeper half-space, called the mantle. In this case, we can consider $z = 0$ on the mantle surface. The elastic-wave velocity in the layer is lower than the one in the mantle. Therefore, when a source generates a wave on Earth’s surface, three waves can be considered: a direct wave, traveling at the air–Earth boundary; a reflected wave on the interface between the crust and the mantle; and a lateral wave, traveling for a distance $L$ in the mantle. If we consider a receiver placed on Earth’s surface, i.e., $θ_{1} = π / 2$ and $z = h$ referring to Fig. 4, at a distance $x$ larger than a critical distance $x_{c}$ , called crossover distance, then the lateral wave reaches the receiver before the direct wave [40]. The crossover distance can be obtained by equating the direct and lateral phases; from Eqs. (6) and (8), we obtain

x_{c} = 2 h \frac{\cos θ_{L}}{1 - \sin θ_{L}} = 2 h {(\frac{v_{2} + v_{1}}{v_{2} - v_{1}})}^{1 / 2} .

Under the condition

x > x_{c}

, the lateral wave precedes the direct wave. This means that a signal, much more attenuated, reaches the receiver before the direct one. This phenomenon led to one of the most important discoveries about Earth’s structure [40]: the determination of the thickness of Earth’s crust, i.e., the distance between Earth’s surface and mantle. This measurement is possible by knowing the position of the source and the starting time of the signal. Measuring the arrival time of the two waves for different distances between the source and the receiver, it is possible to obtain the velocities in the two media. Then, from such velocities, by finding the crossover distance, it is possible to find the thickness of the crust.

This behavior of the lateral wave finds many electromagnetic applications, too, e.g., radio communication over Earth’s surface, submarine communications and detection, or the measurement of the permittivity and conductivity of the oceanic crust. For example, it can be shown that if an electromagnetic source is placed near the interface ( $h \approx 0$ ), then there is a solid angle in the rarer medium, where the lateral wave arrives with a higher power than both the direct and reflected waves. This effect can be used to reduce signal attenuation in communication applications [36]. Communication with submerged submarines is made difficult by the high conductivity of salt water, between 2.8 and 4 S/m, depending on the salt concentration, which causes a strong attenuation of the electromagnetic fields. If communication between two submarines is considered, then the direct path is totally in sea water, and the field is strongly attenuated. On the other hand, if we consider a source in the air, such as an aircraft, the space shuttle, or a satellite, almost all of the incident power would be refracted vertically because of the high permittivity of water; then the communication is possible only if the transmitter and receiver are aligned with the normal to the water interface. However, if the source antenna is placed near the surface of the sea, it excites a lateral wave on the interface, propagating in air and carrying the signal horizontally, reducing the path in the high-conductivity material. A similar application is relevant to radio communications in a forest, where the tree layer is considered as the high-conducting material.

The existence of the lateral wave can be explained by the need of continuity between the wavefronts in the two media. The incident, reflected, and refracted waves, at a sufficient distance from the source, have cylindrical wavefronts. We can imagine what happens when a cylindrical pulse is emitted by the line source in the time domain: the incident wavefront starts from the source, at a distance $h$ from the boundary. When the incident wave reaches the interface, the reflected and refracted waves start to propagate in media 1 and 2, respectively. The incident and reflected waves propagate with velocity $v_{1} = c / n_{1}$ , while the refracted wave propagates with velocity $v_{2} = c / n_{2}$ , where $c$ is the speed of light in a vacuum, and $n_{1}$ and $n_{2}$ are the refractive index of media 1 and 2, respectively. The reflected wave is centered at $- h$ because of the image principle. Therefore, the incident and reflected wavefronts are always coincident on the interface; see Fig. 5(a). When the geometrical-optics rays of the incident and reflected waves form an angle $θ > θ_{L}$ , total reflection begins and no transmitted wave will be excited. Since $v_{2} > v_{1}$ , the last refracted wavefront will reach a further distance from the origin on the interface, with respect to the incident and reflected waves. The lateral wave is needed to ensure the continuity of the wavefronts; see Fig. 5(a). Through geometric considerations, we can prove that $\sin θ_{L} = n_{2} / n_{1}$ . In fact, by considering Fig. 5(a), we see that the cylindrical wavefronts of the incident and reflected waves, $V_{i}$ and $V_{r}$ , respectively, at an angle $θ > θ_{L}$ , intercept the interface at point R, while the transmitted cylindrical wave, $V_{t}$ , centered at the origin intercepts the interface at point T. The segment passing to $T$ and tangent to the wavefront $V_{r}$ is the line that makes the fronts continuous at the interface. Therefore, the segment $\bar{ST}$ represents the plane front of the lateral wave. The propagation direction of this wave forms an angle $θ_{L}$ with the perpendicular direction to the interface. Now, we can recognize that after a time $t$ , from the starting time, the reflected wave has traveled a distance $\bar{CS} = \bar{CR} = c t / n_{1}$ . On the other hand, at the same time $t$ , the transmitted wave relevant to the incidence at $θ = θ_{L}$ has traveled a distance $\bar{OT} = c (t - t_{L}) / n_{2}$ , where $t_{L}$ is the delay time due to the fact that the transmitted wave starts when the incident wave impinges at an angle of $θ_{L} < θ$ . From Fig. 5(a), the following relation holds:

Fig. 5. (a) Representation of the cylindrical wavefronts of the incident, reflected, and transmitted waves, $V_{i}$ , $V_{r}$ , and $V_{t}$ , respectively, due to a line source centered on a point on the $z$ axis and placed at a distance $h$ from the interface. The segment $\bar{ST}$ represents the plane front of the lateral wave that makes the field at the interface continuous. (b) Representation of the elementary plane wave, relevant to the spectrum of the field excited by the line source, at the incident angle $θ_{L}$ .

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\bar{OT} = \bar{{OS}^{'}} + \bar{S^{'} T},

with

\bar{{OS}^{'}} = h \tan θ_{L},

\bar{S^{'} T} = \frac{\bar{S^{'} S}}{\sin θ_{L}} = \frac{\bar{CS} - \bar{{CS}^{'}}}{\sin θ_{L}} = \frac{1}{\sin θ_{L}} (\frac{c t}{n_{1}} - \frac{h}{\cos θ_{L}}) .

Hence,

\frac{c (t - t_{L})}{n_{2}} = h \tan θ_{L} + \frac{1}{\sin θ_{L}} (\frac{c t}{n_{1}} - \frac{h}{\cos θ_{L}}) .

Solving for

t_{L}

, we find the following expression:

\frac{c t_{L}}{n_{2}} = c t (\frac{1}{n_{2}} - \frac{1}{n_{1} \sin θ_{L}}) + h \cot θ_{L} .

At this point, we note that the delay time,

t_{L}

, must be a constant, independent of

t

. Imposing this condition, we find

\frac{1}{n_{2}} - \frac{1}{n_{1} \sin θ_{L}} = 0 \Rightarrow \sin θ_{L} = \frac{n_{2}}{n_{1}},

obtaining the well-known expression of the critical angle of total reflection. Moreover, inserting such expression in Eq. (18), we get the following relation for the delay time:

\frac{{c t}_{L}}{n_{1}} = h \cos θ_{L} .

This expression is the path traveled by the incident wave from the starting time in correspondence with the current line position to the incidence at

θ = θ_{L}

; see Fig. 5(b). This result allows us to relate the previous quasi-optical arguments to the plane-wave spectrum considerations of Section 1. In fact, the current line pulse can be decomposed into an angular spectrum, and Eq. (20) tells us that the only elementary wave of the spectrum that excites the lateral wave is the one incident at an angle

θ_{L}

, i.e., the plane wave relevant to the branch-point singularity

k_{2}

. Furthermore, we see that when the source is on the interface, i.e.,

h = 0

, then there is no delay between the three waves, i.e., the transmitted wave at the critical angle is excited at the starting time, and the distance between the fronts is due only to the speed difference.

Finally, we want to show an important connection between the lateral wave and the so-called Goos–Hänchen shift, an important effect in optics, involving the reflection of beams with a finite-width wavefront [41–43]. If we consider a prism and a beam incident perpendicularly to its edge, the incident beam will be reflected by the other edge of the prism; see Fig. 6. We call $2 w$ the beam width, and we suppose that the incident angle is equal to the critical angle $θ_{L}$ . Solving the problem geometrically, we expect that the reflected beam is centered on a line obtained by the reflection of the center ray of the incident beam. On the other hand, measurements show that the reflected beam is shifted by a certain distance $D$ with respect to such line.

Fig. 6. Representation of a ray with a finite plane front, of width $2 w$ , incident on a prism with refraction index $n_{1}$ immersed in a medium with index $n_{2} < n_{1}$ . The incident angle of the beam on the horizontal interface of the prism is $θ_{L}$ . The beam is totally reflected. The dashed arrows represent the geometrical reflected beam, while the solid arrows represent the actual reflected beam, shifted a distance $D$ . The gray arrows represent the lateral wave because of the total reflection.

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We consider the incident beam as a Gaussian beam. Defining a reference frame $(x_{i}, z_{i})$ , with $x_{i}$ parallel to the incident-beam propagation direction, and a reference frame $(x_{r}, z_{r})$ , with $x_{r}$ parallel to the reflected-beam propagation direction, the incident field perpendicular to the plane of incidence can be written as follows:

V_{i} = \frac{\exp [- j k_{1} z_{i} - {(x_{i} / w)}^{2}]}{\sqrt{π} w} .

The reflected beam at the horizontal interface can be obtained as a superposition of two fields:

V_{r} = V_{0} + V_{1}

. The expressions of these fields are quite complicated and can be written as follows [39]:

V_{0} = R (k_{x i}) \frac{\exp [- j k_{1} z_{r} - {(x_{r} / w)}^{2}]}{\sqrt{π} w},

V_{1} = A (θ_{i}) V_{0} {{(- δ)}^{1 / 2} - \frac{{[2 \exp (γ^{2} - j π)]}^{1 / 4}}{\sqrt{k_{1} w}} D_{1 / 2} (γ)},

where

D_{1 / 2}

is the parabolic cylinder function of order

1 / 2

in the Whitaker notation, related to the Weber–Hermite function [44]. (It must not be confused with the Weber function

E_{n}

related to the inhomogeneous Bessel differential equation and strongly related to the Struve function.) Moreover, we define

A (θ_{i}) = \frac{4 m \cos^{2} θ_{L} \sin θ_{i}}{[\cos^{2} θ_{i} + m^{2} (\sin^{2} θ_{i} - \sin^{2} θ_{L})] \sqrt{\cos θ_{i} (\sin θ_{i} + \sin θ_{L})}},

γ = \sqrt{2} (\frac{i k_{1} w δ}{2} - \frac{x_{r}}{w}),

δ = (\sin θ_{i} - \sin θ_{L}) \sec θ \approx θ_{i} - θ_{L},

where the approximation in Eq. (26) holds only if

θ_{i} \approx θ_{L}

. Looking at Eq. (22), we immediately recognize the geometric reflected beam (without shift) as represented by the dashed arrows in Fig. 6. On the other hand, the field in Eq. (23) is quite complicated, and it is not easy to recognize its behavior. However, we can see that for

x_{r} < 0

this field decays exponentially very fast, while for

x_{r} ≫ w

its behavior can be simplified as follows [39]:

V_{1} \approx \frac{k_{1}}{2 \sqrt{π}} R (k_{x}) A (θ_{i}) \exp [- {(k_{1} w δ / 2)}^{2}] \frac{\exp [- j (k_{2} x + π / 4)]}{{(k_{1} x \cos θ_{i})}^{3 / 2}} .

Now, we can recognize the typical decreasing behavior of a lateral wave

x^{3 / 2}

. Therefore, we can assume that the field

V_{1}

is a lateral wave, as shown with the gray arrows in Fig. 6. Summarizing, the reflection of a Gaussian beam can be considered as the superposition of two contributions: the geometric reflection

V_{0}

plus the lateral wave

V_{L}

. Analyzing the amplitude of these two fields, we find that

V_{0}

is a Gaussian bell centered at

x = 0

, while

V_{1}

is composed of two bells having a common zero in

x = Δ x

; see Fig. 7. (The phases of

V_{0}

and

V_{1}

are plotted in the case

z_{r} = 0

, because the difference between these phases is independent of

z_{r}

. In fact, as can be seen from Eqs. (22) and (23), the phase dependence on

z_{r}

is the same in the two fields.) The first bell, for

x < Δ x

, has a phase equal to

- 135 °

, and then is in a destructive superposition with

V_{0}

, while the second bell, for

x > Δ x

, has a phase equal to 45°, and then is in a constructive superposition with

V_{0}

. Now, we can understand the relation between the Goos–Hänchen shift and the lateral wave. In the zone between

x = 0

and

x = Δ x

, the two fields

V_{0}

and

V_{1}

subtract from each other and result in a zero field, while in the zone

x > Δ x / 2

, they sum to each other giving a beam shift of

Δ x

.

Fig. 7. (a) Magnitude, in a logarithmic scale, normalized with respect to its maximum value and (b) phase, in degrees, of the field $V_{0}$ in Eq. (22) (dashed line). Also, (a) magnitude, in a logarithmic scale, normalized with respect to the maximum value of $V_{0}$ and (b) phase, in degrees, of the field $V_{1}$ in Eq. (23) (solid line). The interface is considered between the medium with refractive index $n_{1} = 1.94$ and air, $n_{2} = 1$ , while the incident wave has a half-width of $w = 1000 λ_{2}$ . The plots are considered in the case $z_{r} = 0$ .

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3. SURFACE WAVE

A surface wave is a field propagating along a planar surface, without radiation. In this sense, surface waves strongly differ from lateral and leaky waves because such waves propagate either in the first or in the second medium, while the surface wave is confined to the interface; then it is a guided wave. This guiding behavior makes the surface wave very important in applications, and it is the reason for much of the efforts made in its study. Several specialized papers and books have been published on the topic, e.g., [6,7,9,12,13,15,45–50]. The nature of the surface wave as a guiding wave gives us an easy way to define it by simple considerations. In fact, we can ask, “Is it possible to obtain a guided wave at the interface between two homogeneous lossless dielectrics?” The answer can be easily obtained by the following argument. Let us consider the interface between two media stratified along the $z$ direction, and let us consider a propagating wave along the $x$ direction. Being a two-dimensional structure, we know that the electromagnetic field can be decomposed into two uncoupled parts: a TE field, having only the components $E_{y}$ , $H_{x}$ , and $H_{z}$ ; and a TM field, having the components $E_{x}$ , $E_{z}$ , and $H_{y}$ . In both cases, the problem can be solved by considering only one scalar function, i.e., the $E_{y}$ component in the TE field and the $H_{y}$ component in the TM field. Let us consider first the TE case. The electric field in the two media can be written as follows,

E_{1 y} = A_{1} \exp [- j (k_{x} x + k_{1 z} z)],

E_{2 y} = A_{2} \exp [- j (k_{x} x + k_{2 z} z)],

while the magnetic field components can be obtained by Maxwell’s equations. We are looking for a field that propagates along the surface. The wave-vector components orthogonal to the interface must be purely imaginary:

k_{1 z} = \sqrt{k_{1}^{2} - k_{x}^{2}} = - j α_{1 z}

and

k_{2 z} = \sqrt{k_{2}^{2} - k_{x}^{2}} = j α_{2 z}

, with

α_{1 z}

,

α_{2 z} > 0

. Here, the different signs of the components orthogonal to the interface are due to the radiation condition that the field in medium 1 must decay in the positive

z

direction, while the field in medium 2 must decay in the opposite direction. By imposing the continuity of the tangential components to the interface of the electric and magnetic fields, i.e.,

E_{y}

and

H_{x}

, we find the following conditions:

A_{1} = A_{2},

A_{1} k_{1 z} = A_{2} k_{2 z} .

Hypothesizing

A_{1} \neq 0

, from Eq. (31) we find

α_{1 z} + α_{2 z} = 0 .

This equation does not allow solutions since

α_{1 z}

and

α_{2 z}

are both positive real numbers. Therefore, a TE field cannot be a guided mode of a planar boundary between two homogeneous lossless media.

Let us now consider the TM polarization. In this case, the field is completely described by the following scalar components:

H_{1 y} = A_{1} \exp [- j (k_{x} x + k_{1 z} z)],

H_{2 y} = A_{2} \exp [- j (k_{x} x + k_{2 z} z)],

where the electric field components can be obtained by Maxwell’s equations. The same previous considerations on

k_{1 z}

and

k_{2 z}

apply. Imposing the continuity of the tangential field components, we find

A_{1} = A_{2},

A_{1} \frac{k_{1 z}}{ω ε_{0} ε_{1}} = A_{2} \frac{k_{2 z}}{ω ε_{0} ε_{2}} .

Hypothesizing

A_{1} \neq 0

, from Eq. (36) we find

ε_{2} α_{1 z} + ε_{1} α_{2 z} = 0 .

If we now suppose that the two permittivities

ε_{1}

and

ε_{2}

have different signs, then Eq. (37) can be solved. Therefore, we proved that a guided TM mode can propagate at the interface between two homogeneous lossless dielectrics. One may note that we are considering an idealized situation, with two lossless dielectrics, without surface roughness, and with a perfectly planar boundary; in real life, such a kind of wave cannot exist. However, what we did is exactly what is done in the derivation of the TEM-wave propagation associated with twin parallel conductors where a lossless guided wave is supposed, or with the guided modes inside smooth hollow metal tubes modeled by perfectly conducting walls. Such waves cannot exist in practice, but they are suitable models to describe practical structures [13].

If now we suppose $ε_{1} > 0$ and $ε_{2} < 0$ , the expression of the wave-vector component parallel to the interface can be obtained:

k_{x} = k_{0} \sqrt{\frac{ε_{1} ε_{2}}{ε_{1} + ε_{2}}} .

Here, we see that the wave-vector component is purely real only if

| ε_{2} | > ε_{1}

. If this condition is not fulfilled, the wave-vector component becomes purely imaginary and the mode does not propagate anymore. Before giving further details on the properties of the surface wave, we must understand why it exists only in TM polarization. The answer to this query can be found in solid-state physics. In fact, the propagation component in Eq. (38) corresponds to the propagation constant of a surface polariton mode. A polariton is a wave due to the coupling between electric polarization and a photon [51]. Actually, several different types of polaritons can be considered, for example, the polariton due to the interaction between an electric field and electrons in a plasma, called plasmon polariton, or the polariton due to the interaction between an electric field and the vibrational modes of the ions in a polar dielectric; this time the coupling of the electric polarization is with a phonon, so it is called a phonon polariton. If we consider the surface plasmon polariton, i.e., the coupling between the electric field and the electrons of a plasma on a surface, we can easily understand why the mode must be TM. In fact, the coupling must be along the surface, i.e., the electrons and the electric field interact along the boundary between the two media. Therefore, a component of the electric field along the interface must exist in order to guarantee the interaction.

Equation (38) is very important and allows us to consider the dispersive properties of a surface wave by knowing the dispersive behavior of the materials. However, before going through these properties, we want to show two alternate ways to reach such an expression. These two ways are based on totally different considerations, forgetting the idea of guiding waves and considering propagating solid waves. If we consider a wave propagating in medium 1, incident on the planar boundary with medium 2, then we can write the expression of the Fresnel reflection coefficients, for both TE and TM polarizations, as follows:

R_{TE} = \frac{k_{1 z} - k_{2 z}}{k_{1 z} - k_{2 z}},

R_{TM} = \frac{ε_{2} k_{1 z} - ε_{1} k_{2 z}}{ε_{2} k_{1 z} - ε_{1} k_{2 z}} .

If we look for the cancellation of these coefficients, as if we were looking for the total transmission of the incident wave, i.e., as if we were looking for the so-called Brewster angle, we obtain exactly Eqs. (31) and (36). Therefore, we find the same results, i.e., the effect occurs only in TM polarization, and the parallel component of the incident wave vector is Eq. (38). Actually, the wave found following this way is not, in general, a surface wave. In fact, we know that if

k_{1 z}

and

k_{2 z}

are two purely real quantities, then the solution corresponds to the incident and transmitted waves at the Brewster angle. However, if we impose that the incident wave is inhomogeneous, i.e., that it presents an attenuation in a direction perpendicular to the phase direction, then we find a surface wave. In other words, we can say that the surface wave is an inhomogeneous plane wave incident at the boundary between two media with

ε_{1} ε_{2} < 0

, at the Brewster angle. It can easily be proved that the Brewster angle in this case is 90°. The connection between the surface wave and the Brewster angle was identified in very early research on the subject [13].

The last way we want to show to reach the result in Eq. (38) is of a mathematical type. If we consider a generic cylindrical source in medium 1, near the boundary with a medium 2, then the plane-wave spectrum of the field due to the interaction between the incident field and the interface will be expressed by the integral in Eq. (1). The function $s (k_{x})$ , as said in Section 1, depends on the source properties and the interface. Actually, when the boundary between two homogeneous media is considered, then the spectrum can be written as follows:

s (k_{x}) = R (k_{x}) s^{'} (k_{x}),

where

R (k_{x})

is the Fresnel reflection coefficient of the relevant polarization, while

s^{'} (k_{x})

is the plane-wave spectrum of the field excited by the source. As explained in Section 1, the surface waves are related to the real polar singularities of the kernel in the spectral representation in Eq. (1). Independently of the behavior of the source, in the spectrum from Eq. (41), the polar singularities of the Fresnel coefficient must be considered. If we consider the TE polarization, we can easily see that the denominator of the coefficient in Eq. (39) cannot be canceled when the square-root determination is the same for the two components

k_{1 z}

and

k_{2 z}

. On the other hand, the denominator of the coefficient in Eq. (40) can be canceled when

ε_{1}

and

ε_{2}

have different signs, finding the expression for

k_{x}

in Eq. (38).

Here, we showed how the existence of a surface wave can be proven by three different arguments: by looking for a guided wave at the interface between two homogeneous materials, by looking for the total transmission of an inhomogeneous wave at the interface between two materials, and by looking for the polar-singularity contribution in the plane-wave spectrum of the reflected field by a boundary between two materials in the presence of an arbitrary source. In all three cases, we find the same characteristics: (1) the surface wave is a TM wave; (2) to support the surface wave, the sign of the permittivity must be different in the two media; (3) the component of the propagation vector parallel to the interface must have the expression in Eq. (38). It is important to point out that these considerations hold in the simple case of lossless materials. The consideration of dissipative materials makes the analysis much more complicated, and we will have only a look at this case.

At this point, we can analyze Eq. (38) in more detail. To understand the behavior of a surface wave, we must specify the kind of material that we are considering. We start from the case of a surface plasmon polariton, i.e., a surface wave at the interface between a lossless dielectric, medium 1, e.g., air, and a metal behaving as a plasma, medium 2, described with the Drude model [51]. If we neglect dissipation, the relative permittivity of a free-electron plasma can be written as follows:

ε_{2} = 1 - \frac{ω_{p}^{2}}{ω^{2}},

where

ω_{p} = \sqrt{\frac{{N e}^{2}}{ε_{0} m_{0}}}

is the plasma angular frequency,

e

is the electron charge,

m_{0}

is the electron rest mass, and

N

is the number of electrons per unit volume. Typical values of

ω_{p} / (2 π)

for metals are between 0.5 and

2.5 \times 10^{16} Hz

, i.e., at optical frequencies. The dispersive behavior in Eq. (42) is well known: the permittivity is negative below the critical value

ω_{p}

, and is positive above this value, tending to 1 when the frequency goes to infinity. Therefore, metals have a negative permittivity below the plasma angular frequency and the surface wave can propagate at the dielectric–metal interface. As previously noted, the surface wave can propagate only if the quantity under the square root in Eq. (38) is positive. If

ε_{2} < 0

, it happens when

| ε_{2} | > ε_{1}

. Therefore, by inserting the Drude permittivity, when the angular frequency is less than a critical value,

ω < ω_{sp} = \frac{ω_{p}}{\sqrt{1 + ε_{1}}},

if

ε_{2} < 0

and

ω > ω_{sp}

, the argument of the square root becomes negative and the parallel component of the propagation vector becomes purely imaginary. As a consequence, the surface wave becomes evanescent, being purely attenuated in both the

x

and

z

directions. When

ω > ω_{p}

,

ε_{2}

is larger than zero and the propagation starts again. However, for these values of permittivity the perpendicular component

k_{z}

cannot be purely imaginary, and the wave is not a surface wave anymore. In fact, in this case we come back to the total transmission of a homogeneous wave at the boundary between two dielectrics. It is interesting to note that the band

ω_{sp} < ω < ω_{p}

is a forbidden band for the propagation, i.e., it is a so-called bandgap.

The dispersive behavior of $k_{x}$ in the case of a surface plasmon polariton is shown in Fig. 8, using normalized quantities. We can see that for low frequencies, when $ε_{2} \to - \infty$ , the parallel component of the propagation vector tends to $k_{0}$ . On the other hand, for values $ω \to ω_{sp}$ , the parallel component grows indefinitely. This growth leads the wave to have a wavelength much smaller than the vacuum wavelength. This behavior, well known from the 1970s [52], has been reconsidered in recent years as a way to go beyond the diffraction limit [19,53,54]. During the 1990s, the possibility of subwavelength propagation was described, opening the way to nanophotonics [55,56], i.e., the propagation and manipulation of light at the nanoscale, impossible with conventional waves because of the diffraction limit. The analysis of nanophotonics features go behind the scope of this paper. Here, we just give a brief explanation on the origin of such subwavelength behavior. If we consider the component of the propagation vector parallel to the interface, Eq. (38), we can define the surface-wave wavelength as follows:

λ_{s} = \frac{2 π}{k_{x}} = λ_{0} \sqrt{\frac{ε_{1} + ε_{2}}{ε_{1} ε_{2}}} .

In the frequency band where the surface-wave propagation is allowed, i.e.,

ω < ω_{sp}

, in the case of plasmon polaritons,

k_{x}

is always larger than

k_{0}

. As a consequence, the surface-wave wavelength will always be lower than the vacuum wavelength, i.e.,

λ_{s} < λ_{0}

. Therefore, the wave interacts with distances much smaller than the wavelength. This behavior can be exploited to realize guiding structures at the nanoscale for waves at optical frequencies, or to achieve the focusing of an optical beam in a region of space much smaller than

λ_{0}

.

Fig. 8. Dispersion behavior (solid line) of the component of the propagation vector parallel to the interface, $k_{x}$ , of a surface wave, Eq. (38), for the interface between air and silver, where the permittivity of silver is obtained by the Drude model in Eq. (42). The variables are normalized with respect to the plasma angular frequency $ω_{p}$ , Eq. (43). The light speed line is represented by the dashed line. The two horizontal dotted lines correspond to the angular frequencies $ω_{sp}$ and $ω_{p}$ .

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A consequence of the subwavelength behavior of the surface waves is the fact that their phase velocity is slower than the speed of light:

v_{ϕ} = \frac{ω}{k_{x}} = \frac{ω λ_{s}}{2 π} = c \sqrt{\frac{ε_{1} + ε_{2}}{ε_{1} ε_{2}}} .

Therefore,

v_{ϕ} < c

. For this reason, surface waves are often called slow waves. Moreover, we can see that the dispersion curve of a surface wave is always below the light speed line, as can be seen in Fig. 8.

We discussed the propagation of a surface wave at the air–metal interface, due to the interaction between the electric field and the free electrons of the metal. The surface plasmon polaritons (SPPs) can be supported by more complicated structures such as multilayer ones, e.g., air–metal–air or metal–metal interfaces [50]. Moreover, SPPs are not the only polaritons that can be considered. As mentioned before, the electric field interacts with the vibrational modes in polar solids, resulting in phonon polaritons [51]. As is well known, the vibrational modes can be either longitudinal or transverse. When these modes interact with light, they are called longitudinal optic (LO) phonon modes and transverse optic (TO) phonon modes, respectively. If we consider a polar dielectric, i.e., a dielectric with bound ions, the interaction between the electric oscillating field and the ions can be treated by considering a linear chain: while the field wavelength is much larger than the ion dimension, the positive and negative ions oscillate out of phase relative to each other. Being the electromagnetic wave transversely polarized, the TO phonon modes are excited. In the differential equation, describing the interaction between the electric field and the ions, there will be second-order terms related to the acceleration, zero-order terms related to the restoring force due to the ion bounds, and the external force due to the electric field. As a consequence, the behavior is described by a dielectric constant similar to the one of a Lorentz model [51]:

ε_{2} = ε_{\infty} + (ε_{st} - ε_{\infty}) \frac{ω_{TO}^{2}}{ω_{TO}^{2} - ω^{2}},

where the damping has been neglected. In Eq. (47),

ε_{\infty}

is the high-frequency permittivity and

ε_{st}

is the low-frequency (static) one, while

ω_{TO}

is the characteristic resonance angular frequency of the TO phonon, related to the restoring force and the masses of the bounded ions. Since the losses have been neglected, the expression in Eq. (47) is discontinuous in

ω_{TO}

, tending to

+ \infty

in the left neighborhood and to

- \infty

in the right neighborhood of

ω_{TO}

. The interesting feature of Eq. (47) is that it can be zero for a particular angular frequency

ω_{LO}

:

ω_{LO} = ω_{TO} \sqrt{\frac{ε_{st}}{ε_{\infty}}} .

Since

ε_{st} > ε_{\infty}

, we can see that

ω_{LO} > ω_{TO}

. The relation in Eq. (48) is known as the Lyddane–Sachs–Teller relationship, and it has been experimentally validated in several polar solids, e.g., gallium arsenide, boron nitride, and zinc selenide [51,57]. When the permittivity is equal to zero, the propagation of longitudinal electromagnetic waves is allowed, and this is the reason for the subscript of the frequency. (We remind the reader that in the absence of free charges, Gauss’ law gives

\nabla \cdot \underset{̲}{D} = 0

. If we consider a plane wave with a spatial dependence of the type

\exp (- j \underset{̲}{k} \cdot \underset{̲}{r})

, such an equation becomes

- j \underset{̲}{k} \cdot (ε \underset{̲}{E}) = 0

. This equation requires that the electric field is always transverse to the propagation direction, except when the permittivity is zero. In this case, the propagation of longitudinal waves is allowed.) Therefore, at this frequency, the electromagnetic wave can excite LO phonon modes. It can be seen that for

ω < ω_{TO}

and

ω > ω_{LO}

, the permittivity (47) is always positive, while in the range of

ω_{TO} < ω < ω_{LO}

it is negative. This zone is called the reststrahlen band (reststrahlen in German means “residual rays” and indicates a zone where light cannot propagate), where the electromagnetic wave propagation is forbidden; in such band, the polar dielectric presents a reflectivity equal to unity. If losses are considered, the reflectivity decreases but measurements show reflectivities up to 90%. In this band, polariton propagation is allowed. The dispersion behavior of the component parallel to the interface of the propagation vector at the interface between air and a polar dielectric is shown in Fig. 9. In contrast to the plasmonic case, we see that this time there is no bandgap. At any frequency, the propagation is allowed. However, outside the reststrahlen band, the dispersion curve lies over the speed of light line, i.e.,

k_{x} < k_{0}

. On the other hand, into the reststrahlen band, the curve is below the light speed line, indicating the propagation of a phonon polariton mode. It must be noted that the dispersion curve in the reststrahlen band does not tend to the value

ω_{LO}

but to a smaller angular frequency

ω_{s}

. Such angular frequency can be obtained by imposing

ε_{2} = - ε_{1}

in Eq. (47), obtaining [58,59]

ω_{s} = ω_{TO} \sqrt{\frac{ε_{st} + ε_{1}}{ε_{\infty} + ε_{1}}} .

Fig. 9. Dispersion behavior (solid line) of the component of the propagation vector parallel to the interface, $k_{x}$ , of a surface wave, Eq. (38), for the interface between air and gallium arsenide, where the permittivity of the gallium arsenide is obtained by Eq. (47). The variables are normalized with respect to the TO angular frequency $ω_{TO}$ . The light speed line is represented by the dotted line. The two horizontal dashed lines represent the angular frequencies $ω_{TO}$ and $ω_{LO}$ .

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As a final remark, we want to share a few words about the surface waves at the interface with a dissipative medium. The possibility to define a surface wave at the interface between lossy media is an important task widely discussed in the literature. The crucial point here is that in an absorbing material, because of the dispersion equation, the constant-phase and constant-amplitude planes of an inhomogeneous plane wave cannot be perpendicular to each other. Therefore, if we require that the surface wave has the phase vector parallel to the interface, then it cannot have the attenuation vector perpendicular to it. As a consequence, either the attenuation or the phase is oblique with respect to the perpendicular direction to the interface. For this reason, in the case of dissipative media, the surface wave takes several names due to the different characteristics of the involved media. In Table 1, a possible classification of the different kinds of surface waves in lossy media is presented [50,60]. The classification is relevant to the case of a boundary between air and a dissipative material. When the second medium has a negative real part of the permittivity and is slightly dissipative, then the surface mode is a polariton mode, i.e., it is a “true” surface mode, being the wave propagating parallel to the surface and closely confined on it. On the other hand, when the real part of the permittivity is $- 1 < ε_{2}^{'} < 0$ , the wave cannot be confined to the interface because the amplitudes of the phase and attenuation vectors are almost equal, making the wave strongly attenuated. When the real part of the permittivity is positive and the losses are small, the transmitted wave can be considered similar to the one in the lossless case as in the total-transmission phenomenon. Finally, when the real part of the permittivity is positive, but the losses are high, we find the so-called Zenneck waves propagating in the lossy medium closely confined to the boundary.

Table 1. Summary of Surface Waves

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Until now, we considered the boundary between two homogeneous media, and we proved that in such a circumstance the surface-wave propagation requires that one of the two media presents a negative permittivity. However, if we suppose that one of the two media be inhomogeneous, e.g., a stratified medium, then the constraints disappear and we find a surface wave considering only ordinary dielectrics. In this case, the structure can be considered a dielectric-based waveguide, widely studied in the electromagnetic applications [17]. Obviously, a great variety of structures might be considered, increasing the number of dielectrics involved. The simplest structures that can be considered are those with three dielectrics, i.e., a dielectric slab bounded by two half-spaces. Two notable cases can be cited: the symmetric slab, where a dielectric with a finite thickness is bounded by two half-spaces filled with the same material, and the asymmetric slab, where the dielectric is bounded by two different materials. Furthermore, the case of a grounded slab can be considered: a dielectric of finite thickness is bounded on one side by a half-space filled by a homogeneous material and on the other side by a perfect electric conducting plane. This structure is important for its applications at microwave frequencies. The symmetric slab, as a symmetric structure, can be bisected into two different structures: the grounded slab and the dielectric slab over a perfect magnetic conducting plane. Moreover, the methods applied to study the grounded slab are applicable to any stratified structure. Therefore, here we study the grounded slab as an example of a stratified structure. The simplest way to obtain the dispersion curves of a stratified structure is the transverse resonance method (TRM) [17].

Let us consider a perfect electric plane in $z = 0$ , a dielectric slab over such plane, with thickness $t$ , relative electric permittivity $ε_{2}$ , and a half-space $z > 0$ filled by air; see Fig. 10(a). The TRM consists of establishing a transmission line in the transverse direction, in this case the $z$ direction. The perfect electric plane corresponds to a short circuit, the dielectric slab to a transmission line of length $t$ , with propagation constant $k_{2 z}$ and characteristic impedance $Z_{2}$ , dependent on the polarization, and the air half-space corresponds to a matched impedance $Z_{0}$ , equal to the characteristic impedance of air, Fig. 10(b). The line impedance has the following expressions in the two polarizations [17]:

Z_{0} = \frac{ω μ_{0}}{k_{0 z}}, Z_{2} = \frac{ω μ_{0}}{k_{2 z}} in TE polarization;

Z_{0} = \frac{k_{0 z}}{ω ε_{0}}, Z_{2} = \frac{k_{2 z}}{ω ε_{0} ε_{2}} in TM polarization;

where

k_{0 z}

is the component of the wave vector perpendicular to the interface in air. The method requires the choice of a generic transverse section of the transmission line and equating the sum of the impedances seen upward and downward to zero:

\overset{↑}{Z} + \overset{↓}{Z} = 0 .

Choosing the section on the boundary between the dielectric and air, the up-impedance is

Z_{0}

, while the down-impedance is

\overset{↓}{Z} = j Z_{2} \tan (k_{2 z} t) .

Therefore, the resonance condition becomes

Z_{0} \cos (k_{2 z} t) + j Z_{2} \sin (k_{2 z} t) = 0 .

Equation (11) represents the dispersion equation of a grounded slab. The solutions of this equation return all the possible modes of the structure, i.e., the guided modes and the radiative modes. This is a transcendental complex equation, so it has in general complex solutions. In order to study the solutions, we can write the dispersion equation in the two polarizations. Inserting Eqs. (50) and (51) into Eq. (54), we get the following equations:

k_{2 z} \cos (k_{2 z} t) + j k_{0 z} \sin (k_{2 z} t) = 0 in TE polarization,

k_{0 z} \cos (k_{2 z} t) + j \frac{k_{2 z}}{ε_{2}} \sin (k_{2 z} t) = 0 in TM polarization .

Now, we can see that, for a surface wave,

k_{0 z}

is purely imaginary and

k_{2 z}

is purely real. Then these equations become real, and they have infinite countable real solutions, corresponding to the discrete guided modes of the structure. These real solutions correspond to the pole singularities on the real axis of the spectrum in Eq. (1). Of course, such equations do not admit solutions with both

k_{0 z}

and

k_{2 z}

real. On the other hand, several complex solutions can be considered, corresponding to radiation modes and to the complex pole singularities of the spectrum in Eq. (1): they are the leaky modes of the structure.

Fig. 10. (a) Sketch of a grounded dielectric slab with thickness $t$ and relative permittivity $ε_{2}$ . (b) The equivalent transmission line of the grounded slab in the direction transverse to the propagation.

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Being transcendental equations, Eqs. (55) and (56) must be solved numerically. The dispersion curves for both TE and TM modes are shown in Fig. 11, as a function of normalized quantities. We can see that the surface waves propagate in the interval $k_{0} < k_{x} < k_{2}$ . In the zone $k_{x} < k_{0}$ , we find the continuous spectrum of the structure, i.e., the radiative modes. Finally, we can see in Fig. 11 that TE modes are labeled with odd numbers, while TM modes are labeled with even numbers. This is because the modes of a grounded dielectric slab represent half of the modes of a symmetric dielectric slab. As previously pointed out, the other half of the modes are those of a dielectric slab on a perfect magnetic conductor. It can be seen that for such a structure, at the same cutoff frequency, a complementary mode starts to propagate, so it shows the even TE modes and the odd TM modes.

Fig. 11. Dispersion curves of a grounded dielectric slab, with $n = 1.5$ , as a function of normalized quantities. The curves correspond to the real solutions of Eqs. (55) and (56), indicated with (dashed line) ${TE}_{n}$ and (solid line) ${TM}_{n}$ , respectively, with $n = 0,1, 2 \dots$ .

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4. LEAKY WAVE

Leaky waves are a discrete set of radiated modes by a guiding structure. Early research on leaky waves considers the metallic waveguide with some kind of aperture along the propagation direction. From such aperture, the electromagnetic field is leaked and the radiation can arise [61–63]. Dielectric guiding structures, being open waveguides, are naturally suited to lose energy by radiation, then are particularly useful in the generation of leaky waves. However, if one considers the simple planar boundary between two homogeneous media, then one can note that a leaky wave is not supported by this structure. For this structure, no complex poles are present in the plane-wave spectrum. Therefore, to study leaky waves, one must consider stratified structures. As an example, we can consider the grounded dielectric slab already studied in Section 3 with reference to the surface waves. Let us consider a wave propagating in the grounded dielectric slab of Fig. 10(a) in the positive $x$ direction, with a dependence $\exp (- j k_{x} x)$ . If the wave is guided, i.e., it is a surface wave, the $z$ component of the propagation vector will be purely real in the dielectric and purely imaginary in air. Then the dispersion equation, either Eq. (55) or Eq. (56) depending on the polarization, becomes purely real. However, if one supposes that $k_{0 z}$ and $k_{2 z}$ may be complex, then several other solutions of the dispersion equation are possible. These solutions are relevant to the leaky wave. In this case, $k_{0 z}$ presents both a real and an imaginary part. As a consequence, $k_{x}$ must present both a real and an imaginary part, too. We can write the propagation vector as a sum of the real and imaginary parts, $\underset{̲}{k} = \underset{̲}{β} - j \underset{̲}{α}$ , both of them with components along the coordinate axes. The dispersion equation in free space, $\underset{̲}{k} \cdot \underset{̲}{k} = k_{0}^{2}$ , leads to

{| \underset{̲}{β} |}^{2} - {| \underset{̲}{α} |}^{2} = β^{2} - α^{2} = k_{0}^{2},

\underset{̲}{β} \cdot \underset{̲}{α} = β_{x} α_{x} + β_{z} α_{z} = 0 .

These equations give the following information: (1) the magnitude of the phase vector must always be larger than the amplitude of the attenuation vector, and the magnitudes of the phase and attenuation vectors are dependent on each other; (2) the constant-phase planes and constant-amplitude planes are perpendicular; and (3) the four components of the phase and attenuation vectors are not independent, and particularly, fixing the sign of three components, the fourth must be determined by Eq. (58). From these considerations, we can conclude that the leaky wave is an inhomogeneous wave propagating in air. Let us now consider the signs of the components of the phase and attenuation vectors. The sign of

α_{x}

indicates the direction in which the guiding structure is losing energy. If

α_{x} > 0

, the energy is lost during the propagation in the positive

x

direction, and vice versa. Therefore, we can pose

α_{x} > 0

by supposing a propagation inside the guiding structure in the positive

x

direction. At this point, we can consider the phase vector. In particular, we can say that the leaky wave rises from the guiding structure to the air, so the energy flow is upward. The Poynting vector of an inhomogeneous wave in lossless media is parallel to the phase vector, so the component

β_{z}

must be positive. These two constraints are the only physical conditions on the sign of the two vectors. Then we have one free sign. From Eq. (58), since

α_{x}

,

β_{z} > 0

, if

β_{x} > 0

, then

α_{z}

must be negative, and if

β_{x} < 0

, then

α_{z}

must be positive. From this argument, we find that two different leaky waves can radiate from the structure: one with

β_{x} > 0

, called a forward leaky wave because the phase variation along

x

is concordant with the energy propagation direction; and another with

β_{x} < 0

, called a backward leaky wave because the phase variation along

x

is opposite to the energy propagation direction.

The forward leaky wave has an amplitude decreasing with $x$ , but increasing with $z$ . Therefore, it exists only in an angular sector $θ > θ_{0}$ ; see Fig. 12(a). This limitation is required to respect the radiation condition. If we consider a dielectric slab as a guiding structure, it is interesting to note that if the slab is denser than the upper medium, then there are no leaky-wave solutions with small values of $α_{x}$ . In fact, in this case we find some solutions with $α_{x} = 0$ , i.e., surface-wave solutions, and the first solution with nonzero attenuation is strongly attenuated. It means that for a dense dielectric slab, the leaky-wave solution is not dominant, being strongly attenuated. On the other hand, if we consider a dielectric slab rarer than the upper medium, then we find some dominant leaky waves, i.e., leaky waves with small values of $α_{x}$ [16].

Fig. 12. Sketch of the (a) forward and (b) backward leaky waves. The dashed lines represent the constant-amplitude planes. The direction of the components of the phase and attenuation vectors are specified.

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Turning to the backward leaky wave, its principal properties are the following: (1) their domain of existence is for an angular sector with $θ_{0} < 0$ , i.e., in the negative $x$ direction, see Fig 12(b); (2) the attenuation is positive in both the positive $x$ and $z$ directions; (3) the propagation direction of energy, parallel to $\underset{̲}{β}$ , is opposite, along $x$ , to the propagation direction in the guiding structure. The backward leaky waves are supported by structures with negative permittivity, such as plasmas, or by nonuniform structures, such as periodic planar structures, that are widely used in the generation of leaky waves.

We can see that both the surface and the leaky waves are solutions of the dispersion Eqs. (55) and (56). Therefore, it is possible to analyze the transition from a solution to the other one [64]. Let us consider the dispersion diagram in Fig. 11, and let us zoom in on one of the cutoff values; see Fig. 13. If we call $C$ the starting point of the dispersion curve of the surface wave, then before this value we find the curves from point A to point B and from point B to point C. Before point A, being $β_{0 x} < k_{0}$ , the components of the propagation vector must be complex, then we find a leaky wave, Fig. 14(a). From A to B, $k_{x}$ becomes purely real, while $k_{0 y}$ is purely imaginary, directed in the negative $z$ direction; see Fig. 14(b). It is as if the leaky wave was rotated from point A to point B, changing the radiation angle from $θ_{0}$ to $π / 2$ . The fact that $k_{x}$ is purely real, but $k_{x} > k_{0}$ , depends on the dispersion equation: $k_{0}^{2} = k_{x}^{2} + k_{y}^{2} = k_{x}^{2} - α_{y}^{2}$ . This wave is not physically acceptable because it does not respect the radiation condition in air in any angular sector. Therefore, it is a mathematical solution of the dispersion equation that does not correspond to a wave in the real structure. From point B, the dispersion equation presents two solutions: one is the dotted line in Fig. 13, for which the attenuation along $z$ keeps the same direction and grows in magnitude, and the other one is the solid line from point B to point C, for which the attenuation along $z$ goes to zero. The dotted line is the improper solution that does not correspond to a physical wave, while the solid line corresponds to a wave with a $k_{2 y}$ purely imaginary, that goes to zero. This solution does not exist, but it brings one to the proper solution. In C, we find the wave in Fig. 14(c), i.e., a surface wave at the cutoff. From point C, the wave becomes a proper surface wave and the dispersion curve becomes the same as Fig. 11. The propagation vector of the wave is the one shown in Fig. 14(d). In this way, we see that from point A to point C we studied the transition from the leaky wave to the surface wave through a curve that does not correspond to a physical wave. The steps of the transition in terms of the propagation vector can be seen in Fig. 14.

Fig. 13. Dispersion curve in the transition region from a leaky wave to a surface wave.

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Fig. 14. Complex propagation vector of (a) a leaky wave, (b) a nonphysical wave in the transition region, (c) a surface wave at the cutoff, and (d) a surface wave above the cutoff.

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The leaky wave generated by the guiding structure propagates in air with an angle $θ_{0} = \arctan (β_{x} / β_{z})$ . If the attenuation in the longitudinal direction is small, i.e., the guiding structure radiates a small quantity of energy per unit length, $α_{x} / k_{0} ≪ 1$ , then $β_{x} \approx k_{0} \sin θ_{0}$ . From this relation it is possible to obtain an approximate relation for the radiation angle:

\sin θ_{0} \approx \frac{β_{x}}{k_{0}} .

From the dispersion diagram, we know that the ratio

β_{x} / k_{0}

varies with frequency; then the radiation angle

θ_{0}

depends on the frequency. This dependence is one of the most important properties of a leaky-wave antenna: it can generate a wave at different angles simply by changing the working frequency. On the other hand, a disadvantage of this kind of antennas is their physical length. Since the longitudinal attenuation is small, in order to radiate almost all the guided energy, the antenna must have a length several times the wavelength. It is possible to obtain the following approximate relation:

\frac{L}{λ_{0}} \approx \frac{0.183 k_{0}}{α_{x}},

where

L

is the longitudinal length of the antenna. Another crucial point about leaky-wave antennas concerns the beam width. In fact, from a theoretical point of view, the radiated wave should be a plane wave. However, several poles contribute to the radiation, then the radiated wave is the superposition of several plane waves at different angles. Therefore, the leaky wave presents a divergence of the beam

Δ θ

. This divergence is proportional to the longitudinal attenuation:

Δ θ \propto \frac{α_{x}}{k_{0}} .

This is another reason why a small value of

α_{x} / k_{0}

is often required: in order to enhance the directivity of the antenna.

Now, we want to give an optical explanation of the leaky-wave excitation [16]. Let us consider a dielectric slab on a ground plane. We suppose that the slab terminates on the left-hand side with an interface with air; see Fig. 15. The normal to this interface forms an angle $θ_{1}$ with $z$ . Now, we consider a plane wave coming from a point A, below the slab, incident at an angle $θ_{1}$ , and transmitted into the slab. This wave will be partially refracted and partially reflected at the point $O$ , the origin of the reference frame. After this first reflection, the ray inside the slab is reflected on the ground plane and it is again refracted and reflected at point $C$ . The field in air is composed of the sum of the refracted rays, where each of them is attenuated with respect to the previous one due to the several reflections; see Fig. 15. Let us consider the amplitude of the field incident on the origin equal to unity. Then the first refracted field will be equal to

E_{0} = T e^{- j k_{1} (x \sin θ_{0} + z \cos θ_{0})},

where

T (k_{x})

is the Fresnel transmission coefficient, while

θ_{0}

is the transmitted angle. To obtain the expression of the second refracted ray, we must consider the optics length traveled inside the slab from the first refraction to the second one, i.e., from

O

to

C

. This length is equal to

k_{2} d / \cos (θ_{1})

. Therefore, the

r

th refracted ray in air will assume the following expression:

E_{r} = (- R)^{r} T e^{\frac{- 2 r j k_{2} d}{\cos θ_{1}}} e^{- j k_{1} (x \sin θ_{0} + z \cos θ_{0})} .

Here,

R

is the Fresnel reflection coefficient and the minus sign in the first set of brackets on the right-hand side is due to the reflections on the ground plane. At this point, we can note that the equation of the line of the

r

th ray is the following:

x - z \tan θ_{0} = 2 r d \tan θ_{1} .

The right-hand side of this equation represents the interception between the

r

th ray and the

x

axis. In order to give Eq. (63) a more readable form, we must make some assumptions about the reflection coefficient. In particular, we must suppose that the reflection coefficient is almost equal to unity, i.e., we must suppose that the leaky wave is slightly attenuated in the

x

direction. In this case, we can write the reflection coefficient as follows:

- R = (1 - δ) e^{- j Δ},

where

δ

is a positive real number. If the leaky wave is slightly attenuated, then

δ ≪ 1

, and we can write

- R \approx e^{- δ} e^{- j Δ} .

Substituting Eqs. (64) and (66) into Eq. (63), we obtain

E_{r} \approx T \exp {- j [x (k_{1} \sin θ_{0} + \frac{k_{2}}{\sin θ_{1}} + \frac{Δ - j δ}{2 d \tan θ_{1}}) + z (k_{1} \cos θ_{0} - \frac{k_{2} \tan θ_{0}}{\sin θ_{1}} - \frac{Δ - j δ}{2 d \tan θ_{1}} \tan θ_{0})]} .

In this expression, we recognize the

x

and

z

components of the propagation vector:

k_{x} = k_{1} \sin θ_{0} + \frac{k_{2}}{\sin θ_{1}} + \frac{Δ - j δ}{2 d \tan θ_{1}},

k_{z} = k_{1} \cos θ_{0} - \frac{k_{2} \tan θ_{0}}{\sin θ_{1}} - \frac{Δ - j δ}{2 d \tan θ_{1}} \tan θ_{0} .

Both of these components are complex, and in particular the imaginary part, i.e., the attenuation, is negative in the

x

component and positive in the

z

component, indicating that the wave is attenuated in the positive

x

direction and amplified in the positive

z

direction. In this way, we reconstructed a plane-wave expression for the leaky wave and we found several characteristics: (1) the complex nature of the components of the propagation vector, (2) the field in the upper space is produced by a leakage of energy from the guiding structure, (3) along the

z

direction the field is amplified (improper wave), and (4) the leaky wave exists only in a wedge-shaped region with an angle

θ_{0}

. It is important to note that the propagation vector obtained from the components in Eqs. (68) and (69) does not comply with the dispersion relations in Eqs. (57) and (58) because of the approximations made in the procedure.

Fig. 15. Geometrical-optics interpretation of a leaky wave: an optic ray is generated in A, refracted inside a dielectric slab, partially reflected at the points O, C, E, and G, and totally reflected on the ground plane at the points B, D, and F. The wave that emerges in the upper space ( $z > 0$ ) occupies a wedge-shaped region, is attenuated in the $x$ direction, and is amplified in the $z$ direction.

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Wave	Dielectric Function $ε_{2} = ε_{2}^{'} + i ε_{2}^{''}$	Parallel Wave-Vector Component	Perpendicular Wave-Vector Component	Phase Velocity	Comment
Fano	$ε_{2}^{'} < - 1$ $ε_{2}^{''} ≪ \| ε_{2}^{'} \|$	$k_{x}^{'} ≫ k_{x}^{''}$	$k_{z}^{'} ≪ k_{z}^{''}$	$v_{ϕ} < c$	polariton mode
Evanescent	$- 1 < ε_{2}^{'} < 0$ $ε_{2}^{''} \approx \| ε_{2}^{'} \|$	$k_{x}^{'} \approx k_{x}^{''}$	$k_{z}^{'} \approx k_{z}^{''}$		rapidly attenuating
Brewster	$ε_{2}^{'} > 0$ $ε_{2}^{''} ≪ \| ε_{2}^{'} \|$	$k_{x}^{'} ≫ k_{x}^{''}$	$k_{z}^{'} ≫ k_{z}^{''}$	$v_{ϕ} > c$	bound to surface only in the presence of damping
Zenneck	$ε_{2}^{'} > 0$ $ε_{2}^{''} ≫ \| ε_{2}^{'} \|$	$k_{x}^{'} ≫ k_{x}^{''}$	$k_{z}^{'} \approx k_{z}^{''}$	$v_{ϕ} > c$	propagates in the conducting medium

Electromagnetic inhomogeneous waves at planar boundaries: tutorial

Abstract

1. INTRODUCTION

2. LATERAL WAVE

3. SURFACE WAVE

4. LEAKY WAVE

REFERENCES

Cited By

Figures (15)

Tables (1)

Equations (69)

Journal of the Optical Society of America A