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Ground, Chapman & Hall, London, 1993.

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24. H. Bremmer, Terrestrial Radio Waves, Elsevier, New York,1949.

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Problems, Pergamon Press, Oxford, 1965.

ELECTROMAGNETIC WAVE PROPAGATION

FERNANDO L. TEIXEIRA

The Ohio State UniversityColumbus, Ohio

FERNANDO J. S. MOREIRA

ODILON M. C. PEREIRA-FILHO

Federal University of MinasGerais

Belo Horizonte, Brazil

1. INTRODUCTION

1.1. Historical Perspective

Electromagnetic waves play a major role in remote sen-sing and communication systems. The equations govern-ing electromagnetic wave propagation were firstestablished by J. C. Maxwell (1831–1879) in 1864. How-ever, one should not overlook previous contributions ofmany scientists over the centuries, especially in optics.Speculations regarding the nature of light date sinceancient Greece, when philosophers were already ac-quainted with the rectilinear propagation, reflection, andrefraction of light [1]. The major contributions to the field,however, initiated during the Renaissance, on the founda-tions of the experimental method introduced by Galileo(1564–1642). Some examples are the law of refractiondiscovered by Snell (1580–1626) in 1621, the principle ofleast time established by Fermat (1601–1665) in 1657, theobservation of light diffraction by Grimaldi (1618–1663)and Hooke (1635–1703) circa 1665, and Huygen’s (1629–1695) envelope construction (leading to the principlenamed after him) in 1678 [1]. Such observations andexperiments lead to the development of a wave theory toexplain the nature of light as luminous sources vibratingin adjacent portions of an ethereal medium, an interpreta-tion similar to that of acoustic waves. The wave theory oflight was, however, later rejected by Newton (1642–1727),who proposed a corpuscular interpretation instead [1].

Many years passed until new experiments reinforcedthe wave theory of light. Among the leading scientistsresponsible for that is Fresnel (1788–1827), who developedthe theory of light reflection and refraction in a morequantitative way and as well as a firmer mathematicalbasis for the interpretation of light as a transverse wave.Fresnel’s work temporarily obscured the corpuscular the-ory of light, which regained strength only after Planck(1858–1947) and Einstein (1879–1955) started to unveilthe quantum aspects of light in the beginning of the 1900s.The nineteenth century also witnessed the efforts of many

1280 ELECTROMAGNETIC WAVE PROPAGATION

physicists to experimentally determine the speed of light,such as Michelson (1852–1931).

Apart from the developments in optics, many scientistsdedicated efforts to establish the physical nature ofmagnetism and electricity. Among them, it is worthmentioning Coulomb (1736–1806), who demonstratedthe inverse square law for electrical forces in 1785;Oersted (1777–1851), who was perhaps the first to observeexperimentally the effect of electrical currents on themagnetic field in 1820; and Ampere (1775–1836), whoformulated the circuit force law and postulated magnet-ism as an electrical phenomenon. The milestone experi-ment for the unification of magnetism and electricity,however, was carried out by Faraday (1791–1867), whodiscovered electromagnetic induction in 1831. Theelectromagnetic theory known at that time was puton analytical ground by Maxwell (1831–1879) in 1855/56.He further developed a model to explain electromagne-tics as a mechanical phenomenon, which lead into theconcept of displacement currents and the consequentgeneralization of Ampere’s law in 1861. The mechanicalmodel was finally abandoned and Maxwell, in 1864,published his third paper on the subject, establishingthe basis of the electromagnetic theory [2]. Besidespostulating the displacement current, Maxwell was thefirst to predict the propagation of electromagnetic wavesand to postulate light itself as an electromagnetic radia-tion. This provoked strong opposition from scientists atthat time, as no evidence of such waves had been observedby experiments.

Maxwell did not survive to see his ideas been acceptedby the majority of the scientists, which came just after thefirst experimental observation of electromagnetic wavepropagation by Hertz, published in 1888. After that, theuse of electromagnetic waves for practical purposes wasjust a matter of time, and in 1901 Marconi (1874–1937)achieved the first transmission of radio signals across theAtlantic. Today, electromagnetic waves permeate most ofmodern technologies.

In this article, we will briefly discuss some fundamen-tal aspects of the electromagnetic wave propagation.Further details on this vast subject can be found in severalbooks, such as Refs. 1 and 3–11.

1.2. The Electromagnetic Wave Spectrum

Electromagnetic waves can propagate at different fre-quencies. It is common to classify electromagnetic wavesaccording to their frequency range as 3–30 Hz, ELF (extra-low-frequency or extremely-LF) waves; 30–300 Hz, SLF(super-low-frequency) waves; 300 Hz–3 kHz, ULF (ultra-low-frequency) waves; 3–30 kHz, VLF (very-low-fre-quency) waves; 30–300 kHz, LF (low-frequency or long)waves; 300 kHz–3 MHz, MF (medium-frequency or med-ium) waves, which include most AM radiowaves; 3–30 MHz, HF (high-frequency or short) waves, which in-clude most of shortwave radio; 30–300 MHz, VHF waves,which include FM radio and TV signals; 300 MHz–3 GHz,UHF waves, which include TV signals, radar waves at Land S bands, and microwave oven radiation; 3–30 GHz,SHF (centimeter) waves, which include radars at C, X, Ku,

and K bands, satellite communication links, and aircraftlanding systems; and 30–300 GHz, EHF (millimeter)waves, which include radars at Ka band.

Electromagnetic waves can also be classified by theirwavelength. For an electromagnetic wave propagating inair or vacuum, the wavelength l and frequency f arerelated by l¼ c/f, where cE3�10� 8 m/s is the speed oflight in vacuum. Microwaves correspond to electromag-netic waves with l around 1 cm–1 m (300 MHz–30 GHz).Millimeter and submillimeter waves have l around 1 mm–1 cm (30–300 GHz) and just below it, respectively. Visiblelight is a form of electromagnetic wave with l¼ 0.38–0.72 mm. Wavelengths just below visible light correspondto ultraviolet waves, while wavelengths just above visiblelight correspond to near-infrared waves (0.72–1.3mm) andthermal infrared waves (7–15 mm). At even smaller wave-lengths (higher frequencies) one encounters X and gammarays [10].

2. MAXWELL’S EQUATIONS AND THE WAVE EQUATION

The propagation of electromagnetic waves is governed byMaxwell’s equations, which in SI units and for macro-scopic field quantities are

r�E¼ �@B

@tr .D¼q

r�H¼Jþ@D

@tr .B¼ 0

ð1Þ

These four equations are supplemented by constitutiverelations that relate D and B to E and H. We shallassume propagation in simple media, with vacuum as aspecial case. The term simple media in this contextdenotes linear, homogeneous, and isotropic materials,which will be assumed lossless for the time being. Theconstitutive relations in this case are expressed as (seeMAXWELL’S EQUATIONS article)

D¼ eE and B¼ mH ð2Þ

where e is the permittivity (e¼ e0E8.85418782� 10�12

F/m in vacuum) and m is the permeability (m¼ m0¼ 4p�10�7 H/m in vacuum), constant scalar numbers for simplemedia. The electric current J and charge q densities canbe interpreted as the sources of the electromagnetic fieldand are interrelated by the continuity equation:

r .J¼ �@q

@tð3Þ

which is implicit in (1) (see MAXWELL’S EQUATIONS article).From (1) one can also derive a law for the conservation ofelectromagnetic energy, the Poynting theorem (see MAX-

WELL’S EQUATIONS article). For our purposes, we just need toemphasize the Poynting vector, defined as

S¼E�H ð4Þ

ELECTROMAGNETIC WAVE PROPAGATION 1281

This vector gives the magnitude and direction of powerflow density (watts per square meter).

We will discuss the relations between sources ðJÞ andfields (E andH) later. For now, we shall only investigatesimple characteristics of the electromagnetic propagation.In regions of a simple media where no source is present(i.e., J¼ q¼ 0) the (homogeneous) wave equations belowcan be derived from (1) (see MAXWELL’S EQUATIONS article):

r2E�1

c2

@2E

@t2¼ 0

r2H�1

c2

@2H

@t2¼ 0

ð5Þ

where c¼ 1=ffiffiffiffiffimep

is the speed of light in the medium (c¼c0¼ 299,792,458 m/s in vacuum, by definition). From (5)one can readily show that each Cartesian component of Eand H also satisfies the following homogeneous (scalar)wave equation

r2c�1

c2

@2c@t2¼ 0 ð6Þ

where c represents one of such components. This waveequation governs the behavior of c with respect to bothposition (x,y,z) and time (t). For simplicity, let us consider aone-dimensional problem (i.e., with no variation in x or y),such that (6) simplifies to

@2c@z2�

1

c2

@c@t¼ 0 ð7Þ

It can be easily verified that a solution to (7) is

c¼ f ðz� ctÞþ gðzþ ctÞ ð8Þ

where f and g are arbitrary real functions [12]. These arecalled D’Alembert’s solutions, where f(z� ct) represents anarbitrary waveform propagating in the positive z directionwith speed equal to c, while g(zþ ct) represents anotherwaveform with the same speed but propagating in thenegative z direction.

Although (8) represents the solution of an idealizedproblem, it illustrates one fundamental property of thesolutions of (6); namely, for unbounded simple media theyrepresent a wave traveling with the speed of light. In real-life situations, obstacles (ground, vegetation, human-pro-duced constructions, etc.) are present and impose addi-tional constraints (boundary conditions) on the solutionsof (6). In general, the presence of obstacles complicates theproblem and simple analytical solutions can be found onlyfor simple geometries.

2.1. Time-Harmonic Regime

Most RF and microwave applications deal with time-invariant linear media [6]. In this case, the use of atime-harmonic representation is often more convenientfor dealing with electromagnetic wave phenomena. As-suming a ejot time variation, where o is the angular

frequency, Maxwell’s equations (1) are rewritten as [6]

r�E¼ � joB r .D¼ r

r�H¼Jþ joD r .B¼ 0ð9Þ

where the functions (of space only) in (9) are now complexphasor representations of the corresponding quantities in(1), according to

E¼ReðEejotÞ ð10Þ

Here Re denotes the real part. A similar relation holds forthe other fields (and sources). For simple media and from(2), the constitutive relations are then

D¼ eE and B¼ mH ð11Þ

The wave equations (5) also assume a simpler represen-tation [6]:

r2Eþ k2E¼ 0

r2Hþ k2H¼ 0ð12Þ

known as the (homogeneous) Helmholtz equations, wherek¼o

ffiffiffiffiffimep

is the wave number.It is useful to define time-average powers when dealing

with time-harmonic fields. For instance, from (4) and thedefinition in (10) one can define the complex Poyntingvector

S¼1

2E�H� ð13Þ

The real part of the complex Poynting vector is the timeaverage of S [4]. Equation (13) stresses an importantcharacteristic of electromagnetic wave propagation. In-phase components of E and H contribute to ReðSÞ, repre-senting the power density propagating in the direction ofReðSÞ [6]. On the other hand, components of E and H inphase quadrature contribute to the imaginary part of S,representing a (stationary) reactive power density.

In the time-harmonic regime, (7) becomes

d2Cdz2þ k2C¼ 0 ð14Þ

where C is the complex representation of c according to(10). The solution of (14) is

C¼p1ejðkzþf1Þ þp2e�jðkzþf2Þ ð15Þ

Consequently, from (10) and observing that for simplemedia k¼o/c, we obtain

c¼p1 cos½kðzþ ctÞþf1� þp2 cos½kðz� ctÞþf2� ð16Þ

which is a particular time-harmonic solution of (7) and (8)known as a plane wave.


3. PLANE, TEM, AND STATIONARY WAVES

The plane wave is the simplest solution of the homoge-neous wave equation. Many important properties of elec-tromagnetic wave propagation can be inferred from thisfundamental solution. Moreover, an arbitrary electromag-netic wave in sourceless linear media can be decomposedin terms of plane waves (e7jkz in one dimension) in amanner similar to the decomposition of a time-domainsignal in terms of its spectral components ejot [13].

The complex representation of a plane wave in (15)specifies an amplitude (p) and phase (kzþf). Moreover,since C represents a Cartesian component of E or H, italso indicates the orientation (i.e., the polarization) of thevector field.

The plane-wave solution of C in three dimensions iswritten as e�jðkxxþ kyyþ kzzÞ, where kx,y,z/k can be understoodas the cosine directors of the wavefront. By defining thepropagation vector k¼ kxxxþ kyyyþ kzzz and putting thecomponents of E and H back together, we arrive at amore general complex representation for plane waves [6]

E¼E0e�jk . r and H¼H0e�jk . r ð17Þ

where r¼ xxxþ yyyþ zzz denotes the obervation point, andamplitudes and phases of each component are incorpo-rated in the constant complex vectors E0 and H0 (whichalso determine the wave polarization). Substituting (17)into (12), one obtains the characteristic equation or dis-persion relation (for simple media):

k2¼o2me¼ k2x þ k2

y þk2z ð18Þ

The plane-wave relations for E and H in simple media areobtained by substituting (11), (17), and (18) into (9) withno sources present

k�E¼omH k .E¼ 0

k�H¼ � oeE k .H¼ 0ð19Þ

from which one observes that for sourceless simple mediaAmpere’s and Faraday’s laws are independent equations.Actually, this is also true for any wave solution in suchmedia, which can be inferred from (9) and (11) [6].

Note that kx,y,z can be complex and still satisfy (18).Consequently, it is useful to regard k as a complex vector

k¼ b–ja ð20Þ

where a and b are real vectors [6]. Since both a and b arereal, they can be interpreted geometrically. Substituting(20) into (17), one verifies that the plane-wave spatialvariation in complex notation is of the form e�a

. re�jb . r,representing the variations of the amplitude (first term)and phase (second one) of the wave as it propagatesthrough the medium. Consequently, the constant ampli-tude and phase surfaces are planes (hence the name planewaves), with a and b pointing to their normal directions,respectively. If such planes coincide (i.e., ajjb), then theplane wave is denoted as a uniform plane wave. Other-

wise, it is denoted as nonuniform. Furthermore, bb (i.e.,unit normal vector to the equiphase surface) denotes thedirection of propagation of the plane wave.

For uniform plane waves, one can show from (18) and(20) that k¼ kkk, where kk¼ bb is the direction of propaga-tion. Now k has a geometrically defined direction and onecan inspect from (19) that Ampere’s and Faraday’s lawsreduce to

H¼k

omkk�E¼

1

Zkk�E

E¼�k

oekk�H¼ � Zkk�H

ð21Þ

where Z¼ffiffiffiffiffiffiffim=e

pis the intrinsic impedance of the medium

(Z¼ Z0E376.730313O in vacuum). So, a uniform planewave has E, H, and kk mutually orthogonal to each other(which is not the case for nonuniform plane waves). Waveswith such characteristics, that is, obeying the relations in(21), are called transverse electromagnetic (TEM) waves,the simplest example of which is the plane wave.

3.1. Wavelength and Phase Velocity

For simplicity, let us consider a uniform plane wavepropagating in the kk¼ zz direction. Consequently,k . r¼ kz. Since kk .E¼ 0, let us further considerE¼E0e�jkzxx with E0¼pejf. Consequently, from (21),H¼ ðE0=ZÞe�jkzyy. Note that this is a one-dimensional pro-blem; specifically, the components of E and H satisfy (14),with constant amplitude and phase planes perpendicularto zz.

To obtain corresponding expressions in time domain,we apply the definition in (10):

E¼p cos ðot� kzþfÞxx

H¼p

Zcos ðot� kzþfÞyy

ð22Þ

So, for a certain instant of time, the spatial variation of Eand H is sinusoidal with a period equal to l¼ 2p/k, thewavelength of the electromagnetic wave. Since k¼o/c,then

l¼c

fð23Þ

where f is the frequency. For such sinusoidal variation,o/k¼ c is called the phase velocity (vp) of the TEM wave.

3.2. Polarization

In the previous example we have assumed Ejjxx, but anycombination between x and y components also satisfieskk .E¼ 0 for kk¼ zz. In this situation, such components aresaid to be orthogonal to each other and their phase andamplitude relationships describe the nature of the wavepolarization. Wave polarization plays an important role inRF and microwave systems, as the interaction betweenfields and obstacles can strongly depend on it. In principle,


it is possible to transmit and receive orthogonal polariza-tions in a communication channel independently, thusdoubling the capacity of the channel. In practice, depolar-ization effects (partial conversion of energy in one polar-ization to another) often occur. This effect can cause, forinstance, cochannel interference in wireless communica-tions. Orthogonal polarizations can also be used in closelyspaced radiolinks to minimize cochannel interference.

According to (21), TEM wave polarization can befully characterized by E and kk. So, let us assumeE¼ ðpxejfx xxþpyejfy yyÞe�jkz. Then, from (10), we obtain

E¼px cosðot� kzþfxÞ xxþpy cosðot� kzþfyÞ yy ð24Þ

If the x and y components are in phase (i.e., fx¼fy), then(24) can be rewritten as

E¼ ðpxxxþpyyyÞ cosðot� kzþfxÞ ð25Þ

and, for any position and time, E is always parallel to theplane containing zzð¼ kkÞ and pxxxþpyyy. Then, the wave issaid to have a linear polarization, as depicted in Fig. 1.Note that fx¼fy7p also provides a linear polarization.Now, let us assume that px¼py¼p and fy�fx¼7p/2.Then, from (24), we have

E¼p½cosðot� kzþfxÞxx sinðot� kzþfxÞyy� ð26Þ

and the wave has a circular polarization, as the tip of Edescribes a circular helix in space with an axis in thedirection of zz. This is depicted in Fig. 2. For kk¼ zz, fy�

fx¼ p/2 (¼ �p/2) defines a left (right)-hand circularpolarization. For any other combination between fx,fy

and px,py, the wave polarization is elliptical. Finally, fromthe discussion conducted here it should be clear that anycircular or elliptical polarization can be decomposed intotwo orthogonal linear polarizations with appropriate am-plitude and phase relations. For pictorial depictions of thedifferent wave polarizations, the reader is referred to, forinstance, the text by Balanis [14].

3.3. Complex Poynting Vector and Stationary Waves

We will now investigate in further detail the behavior of Sfor uniform plane waves.

From (13) and (21) we can verify that for a TEM planewave

S¼jEj2

2Zkk¼

ZjHj2

2kk ð27Þ

indicating that (in a lossless simple media) S is real (i.e., apure active power density) and, consequently, the direc-tion of propagation of a uniform plane coincides with thedirection of energy flux. For instance, in the examples ofSections 3.1 and 3.2, one immediately observes thatReðSÞjjzz.

So, let us now extend the example of Section 3.1 toinvestigate the case of a stationary wave, which is de-scribed here as a superposition of two TEM plane wavespropagating in opposite directions (i.e., kk¼ � zz)

E¼ Eþ0 e�jkzþE�0 ejkz� �

xx

H¼1

ZEþ0 e�jkz � E�0 ejkz� �

yyð28Þ

where H was directly obtained from (21), noting that eachindividual wave has its kk pointing in the opposite directionof the other. Substituting (28) into (13), we obtain

S¼jEþ0 j

2

2Z�jE�0 j

2

2Zþ jjEþ0 jjE

�0 j

Zsinð2kz� fþ þf�Þ

� �zz

ð29Þ

where f7 are the corresponding phases of E�0 . So, (29)indicates that S has real (corresponding to the net flux ofpower density) and imaginary (corresponding to the sta-tionary reactive power density) parts. This is the case, forinstance, of fields on a transmission line terminated by aload that does not match the line impedance. If jEþ0 j ¼ jE

�0 j

the wave of (28) is purely stationary and there is noaverage power flux [i.e., Re(S)¼0, that occurs when theline is terminated, for example, by a short circuit].

py

px

y

z

x

�

�

Figure 1. Snapshot of the electric field vector of a linearlypolarized wave traveling in the z direction.

py

z

x

�

�

Figure 2. Snapshot of the electric field vector of a circularlypolarized wave traveling in the z direction.


4. WAVES AND SOURCES

Until this point, we have focused on wave propagationthrough regions of simple media where no source ispresent. We shall now present relations describing fieldsproduced by elementary electric current sources in simplemedia. The most common way to accomplish this objectiveis by obtaining the so-called Green’s functions. For asystematic treatment, the reader is referred to Collin’stext [7] (for guided-wave examples) and Silver’s treatise[9] (for radiation examples with applications to antennatheory). The approach adopted here is more restricted andcan be found in most of the Bibliography.

Observing from (1) that r .B¼ 0, one can define themagnetic vector potential A such that r�A¼B. Insimple media, and after applying the Lorenz gauge condi-tion given by c2

r �A¼ � @F/@t, where F is the electricscalar potential, it can be shown that A satisfies a waveequation as follows [4]:

r2A�1

c2

@2A

@t2¼ � mJ ð30Þ

For time-harmonic fields, this equation can be representedin complex notation as

r2Aþ k2A¼ � mJ ð31Þ

A solution of these equations can be written in a genericform as [4]

A¼ mZZZ

V

Jðr0ÞGðr; r0Þdv0 ð32Þ

where V denotes the volumetric region encompassingJ; r0 and r denote source and observation points, respec-tively; and Gðr; r 0Þ is the Green’s function. Gðr; r0Þ can beinterpreted physically as the field produced by a pointsource, namely, the solution of

r2Gþ k2G¼ � dðr� r0Þ ð33Þ

satisfying the appropriate boundary conditions [4]. Forinstance, for bounded sources in unbounded simple media(free space), it can be shown that

Gðr; r0Þ ¼e�jkjr�r 0 j

4pjr� r0jð34Þ

which is known as the free-space Green’s function.Furthermore, the time-harmonic electromagnetic fieldcan be written in terms of A as [4]

H¼1

mr�A

E¼ � jo Aþ1

k2rðr .AÞ

� � ð35Þ

Use of the Lorenz condition in the derivation of (30) hasan important consequence—one does not need to explicitlyconsider the charge densities to obtain the field. Thatshould come as no surprise, since charges are related tocurrents by the continuity equation (3). In the time-harmonic regime their effects become implicit in (35). Inany event, it is important to mention that (in a classicalsense) gauge conditions other than those due to Lorenzcan be applied [4].

As a simple example, we consider a uniform time-harmonic electric current distribution over a small seg-ment (wire) ‘ along the z axis, centered at the origin(z¼ 0). The length ‘ is assumed very small, such that‘5l. The electric current flows along zz , with a constantphasor I0 for jzj � ‘=2 (I0¼ 0 for jzj > ‘=2). This sourcedistribution is called infinitesimal electric dipole. If thedipole radiates in free space, then the resulting vectorpotential A is given by (32) and (34). Since r0 ¼ z0zz withjz0j � ‘=25l, then kjr� r0j � kr and, consequently,Gðr; r0Þ � e�jkr=ð4prÞ. The integral in (32) can then bereadily evaluated to give [9]

A¼mI0‘

4pe�jkr

rzz ð36Þ

where I0‘ is the electric dipole moment. The expressions forthe associated electric and magnetic fields are obtained bysubstituting (36) into (35). The result can be easily ex-tended to arbitrary dipole’s locations and orientations [9].

Note from (35) and (36) that the dipole radiationdepends on the length ‘ multiplied by k, that is, on theratio ‘=l (also called the electrical length). Any boundedsource distribution in a simple medium can be written as asuperposition of infinitesimal dipoles. As a result, thefree-space radiation properties of any bounded sourcedistribution depend on its electrical dimension, namely,its dimension relative to the wavelength. This is also thecase for scattering by bounded obstacles immersed insimple media.

We note that the integral (32) is generally difficult toevaluate except for simple current distributions and sim-ple Green’s functions (such as the free-space Green’sfunction described above).

4.1. Far-Field (Radiation) Region

In free space and for observation points located sufficientlyfar away from (bounded) sources, (34) and, consequently,(32) can be simplified by means of a Taylor expansion onjr� r0j for small values of jr0j with respect to |r|. Bykeeping just the first few terms on this expansion, oneends up with a simplified relation valid for the so-calledradiation (or far-field) region [9]

A �m4p

e�jkr

r

ZZZ

V

J ðr0Þe jkrr . r 0dv0 ð37Þ


where r¼ jrj, in which case relations (35) simplify to

E � �jo½A� ðA . rrÞrr�

H �1

Zrr�E

ð38Þ

4.2. Spherical and Cylindrical TEM Wavefronts

A closer look at (37) and (38) reveals some interestingproperties of the far-field radiation of bounded sources insimple media. The field dependence on the radial distancer is of the form e–jkr/r. This indicates that the surfaces ofconstant phase are concentric spheres and that the fieldintensity decays with 1/r. Also, the propagation direction(normal to the equiphase surface) is rr. Furthermore, E, H,and rr are mutually orthogonal. This characterizes aspherical TEM wave, for which the basic TEM relationsof Section 3 still hold, but now with kk¼ rr. For instance, acomplex E-field notation of the form E¼pðy;fÞðyy� jffÞ expð�jkrÞ=r characterizes a spherical TEM wavewith a circular polarization, according to the basic defini-tions of Section 3.2. For such a wave, the complex Poyntingvector is given by S¼ kkjEj2=ð2ZÞ¼ rrjpðy;fÞj2=ðZr2Þ, accord-ing to (27).

Such observations show that the average power densityof a spherical TEM wave decays with 1/r2 in losslesssimple media. This result is also expected from simpleconsiderations about energy conservation. Since totalradiated power in a lossless medium is conserved andthe spherical area of the wavefront increases with r2, thepower density of the spherical wavefront must decay with1/r2. Furthermore, note that a spherical TEM wave be-haves locally as a plane wave when krb1, so that, in freespace, the field close to a receiver antenna located suffi-ciently far away from the transmitter antenna can belocally approximated as a plane wave.

Apart from plane and spherical waves, another simplekind of wavefront is the cylindrical one. It is useful topicture a cylindrical wave as the field produced by aninfinitely long line current [6]. For observation points faraway from the current axis, the corresponding wavefrontscan be approximated as cylindrical surfaces with a cross-sectional area that increases linearly with distance (thecylindrical r-coordinate). Consequently, from considera-tions on energy conservation, the radiating field decayswith r�1/2 [6]. It can also be shown that such cylindricalwave is also a TEM field obeying those basic relations ofSection 3, now with kk¼ qq. Furthermore, for krb1 thecylindrical wavefront can also be locally approximated asa plane wave.

4.3. Ray Optics Limit

The discussion in the previous section leads toward thepicture of TEM wavefronts propagating far away from theradiating sources, with directions of propagation normalto the corresponding equiphase surfaces. As stressedpreviously, distances and dimensions are to be consideredlarge or small vis-a-vis the wavelength of operation. In thelimit of l-0 or, alternatively, k-N, the electromagnetic

field is expected to behave locally as a TEM wave (i.e., faraway from sources). This is often called the geometricoptics (GO) limit, where diffraction effects are neglectedand a number of simplifications of the nature of electro-magnetic wave propagation can be made, such as those in(21) [15].

The GO principles are directly related to the classicaltreatment of light. The developments date from ancientGreece and are intrinsically related to those of Euclideangeometry [1]. Obviously, such studies were not based onMaxwell’s equations, but, as expected, the GO principlescan be derived from Maxwell’s equations. The usual start-ing point for this derivation is to assume a monochromaticwave (i.e., a time-harmonic field) governed by (12) incomplex notation. Adopting the notation of Section 2.1,we let C be a Cartesian component of E or H (a function ofposition only). Consequently

r2Cþ k2C¼ 0 ð39Þ

From the TEM waves discussed up to here, it can beassumed, as a first-order ansatz, that [1]

C � pðrÞe�jk0FðrÞ ð40Þ

where p represents the (complex) amplitude variation, Frepresents the phase variation with position only, and k0 isthe wavenumber for vacuum. For other simple media onecan define the index of refraction

n¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðmeÞ=ðm0e0Þ

pð41Þ

such that k¼nk0. In (40), n is accounted for by FðrÞ.Furthermore, the wavefront is given by the surfaces ofconstant FðrÞ (equiphase surfaces). Consequently, thedirection of propagation is kk¼rF=jrFj. The equationthat establishes the optical path (trajectory) in terms ofthe wavefront properties is accomplished by substituting(40) into (39) and, in the k0-N limit, assuming that anyvariation of p(r) with position is negligible with respect tothat of k0F(r). After some algebraic manipulations, onearrives at the so-called eikonal equation [1]:

jrFj2¼n2 ð42Þ

This equation can be used to derive Fermat’s principle [1]

Fðr2Þ � Fðr1Þ¼

Z r2

r1

n d‘¼

Z tðr2Þ

tðr1Þ

c dt ð43Þ

where Fðr2Þ � Fðr1Þ is the optical arc length (along theoptical path ‘) between points r1 and r2, as illustrated inFig. 3. This principle states that light (or, more precisely,electromagnetic waves in the GO limit) follows the pathcorresponding to the shortest travel time [1]. For simplemedia, both n and c are constants, but (43) holds forinhomogeneous media as well. From (43) one can alsoderive the laws of reflection and refraction [1], or establishapproximate trajectories of radiowave propagationthrough Earth’s atmosphere [16].


GO principles are useful (although approximate) toolsfor the characterization of electromagnetic wave propaga-tion not only at optical frequencies but also often at RFand microwave frequencies. For instance, such approxi-mation can be used (possibly augmented by small correc-tions) for characterization of some radio channels at UHFor higher frequencies [17]. The range of validity of suchapproximation can depend on many factors, but it is fair tosay that GO can be used as long as the characteristiclengths of the problem (obstacle feature sizes, mutualdistances, etc.) are much larger than l.

5. WAVES IN LOSSY AND DISPERSIVE MEDIA

So far, we have considered the propagation of electromag-netic waves in simple media without losses or dispersion.This assumption implies that the permittivity and perme-ability in the constitutive relations (2) and (11) are con-stant scalar parameters. In general, however, constitutiverelations are not that simple. For example, ferroelectricand ferromagnetic materials are an example of nonlinearmedia, where the constitutive parameters at a pointdepend on the field strength at that point [4]. Evendielectrics can exhibit nonlinear effects under sufficientlylarge field strengths (e.g., breakdowns inside capacitors ortransmission lines). On the other hand, crystals present awell-organized atomic structure, where the field responseis highly dependent on wave polarization. Crystals are anexample of anisotropic materials, where the permittivity(or permeability) has a tensorial nature [18]. The iono-sphere is another example of anisotropic medium at radio-frequencies, although due to very different reasons [4].

Wave propagation on nonlinear or anisotropic mediawill not be considered here; The interested reader mayconsult Refs. 4 and 18 for further details. We shall con-tinue to assume simple (linear, homogeneous, and isotro-pic) media, but now accounting for the presence of losses(lossy simple media).

In simple media and for a time-harmonic regime, lossesin the medium cause a phase delay between the electricdisplacement ðDÞ and the electric field ðEÞ, such that, incomplex notation, we obtain

D¼ eðoÞE¼ ½e0ðoÞ � je00ðoÞ�E ð44Þ

where e0 and � e00 are the real and imaginary parts of thecomplex permittivity e, respectively. The negative imagin-ary part of e indicates the phase delay on D. Here, we stillassume that (11) applies for the magnetic relation, sincefor conductors and dielectrics mEm0 at radiofrequencies,although, in general, a complex permeability of the formm(o)¼ m0(o)� jm00(o) can also be defined [6].

For dielectrics, the ratio e00/e0 defines the loss tangent,where tan� 1(e00/e0) is the phase difference between E and Ddue to the polarization inertia of the atomic structure(macroscopic interpretation). For conductors, net freecharges are present and generate a conduction currentJc whenever an external field is applied. For most con-ductors, Jc is given by Ohm’s law [4]

Jc¼ sE ð45Þ

where s is the material conductivity of the medium(approximately constant for frequencies below the infra-red region). For conductive media, Jc and the correspond-ing net charge density rc can be subtracted from J and r in(9), respectively, and Maxwell’s equations in complexnotation can be rearranged in the same form as in (9),but now with the permittivity replaced by the complexpermittivity of (44), where e0 is the real permittivity ande00 ¼ s/o [6]. After that, J and r in (9) correspond to theimpressed (external) sources only, as Jc and rc are im-plicitly taken into account by the complex e.

It is important to note that, once the complex permit-tivity in (44) is adopted, k¼o

ffiffiffiffiffimep

and Z¼ffiffiffiffiffiffiffim=e

pare

complex quantities as well. These quantities now dependon o but are still spatially uniform (for homogeneousmedia). Therefore, all the results derived from Maxwell’sequations (in complex notation) in previous sections stillhold true, except for those where one had to deal withconjugate or absolute values (i.e., relations regardingaverage energy and power densities). For instance, (4) isstill valid, since it is a general definition. However, for aTEM wave propagating in a lossy simple medium, (27)must account to the fact that Z is now complex and hencegeneralizes to

S¼jEj2

2Z�kk¼

ZjHj2

2kk ð46Þ

5.1. Wave Attenuation and Frequency Dispersion

One of the important consequences of losses is the at-tenuation of electromagnetic waves inside the medium. Toillustrate this, we revisit the problem of Section 3.1, wherea uniform plane wave propagates in the zz direction, butnow in a lossy simple medium. From (20), and as the waveis considered uniform, then akbkzz and, consequently,k¼ kzz¼ kkk, where k is a complex-valued quantity. So, ifone defines

k¼ b� ja ð47Þ

WavefrontOpticalpath

r2

r1

k

Origin

∧

Figure 3. Wavefront and optical path in the geometric opticsapproximation.


where a and b are nonnegative real quantities, then a¼ akkand b¼ bkk. Such definition is valid for any TEM wavepropagating through a lossy simple medium in the direc-tion kk. The expressions for a and b are obtained from thedefinition k¼o

ffiffiffiffiffimep

and (44):

a

b

( )¼o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffime0

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þe0 0

e0

� �2s

� 1

0@

1A

vuuut ð48Þ

The plane-wave complex representation of Section 3.1 nowbecomes

E¼E0e�aze�jbzxx

H¼E0

Ze�aze�jbzyy

ð49Þ

Note that Z is now complex. We observe from (49) that thephase factor b controls the phase variation with distance,whereas the attenuation factor a controls the amplitudeattenuation as the wave propagates (i.e., as z increases).Despite the attenuation effect, the surfaces of constantamplitude and phase are still planes perpendicular to zz,characterizing a uniform plane wave.

Note that the oscillatory behavior is controlled by e� jbz,and this factor defines the wavelength and phase velocity,now represented as

l¼2pb

and vp¼ob

ð50Þ

To have a better picture of the wave behavior, we apply thedefinition in (10) to (49) in order to obtain E in timedomain. Assuming E0¼pe� jf, we obtain

E¼pe�az cosðot� bzþfÞxx ð51Þ

So, the picture here is that of a sinusoidal function (withspatial period l) multiplied by an envelope variation givenby |pe� az|. The field behavior is sketched in Fig. 4 for afixed instant of time.

By extending the present analysis to the spherical wavesolution of Section 4.1, one arrives at a field dependencewith the radial distance r according to e� are� jbr/r. Thedefinitions in (50) still apply in this case. Besides theoriginal (algebraic) amplitude attenuation caused by thespherical spreading factor 1/r, the presence of losses pro-duce an additional (exponential) attenuation factor e� ar.

Equation (48) indicates that both b and a have a non-linear variation with o, due to e0(o) and e00(o). Conse-quently, one observes from (50) that vp varies with o, thatis, different frequency components travel at differentphase velocities. This causes the dispersion of an electro-magnetic wave composed by several frequencies. In RFand microwave applications, electromagnetic waves oftenhave a certain frequency bandwidth, and dispersion maycause waveform distortion as the wave propagates. Ingeneral, losses can significantly decrease the signal-to-noise ratio (SNR) over RF and microwave links, and need

to be minimized and/or compensated (by using, e.g.,repeaters through long-distance links).

5.2. Group Velocity

In a dispersive (simple) medium, the phase velocity vp

depends on o. In this case, it becomes necessary to definethe velocity of a wavepacket (i.e., a wave composed bydifferent spectral components) for vp is no longer ade-quate. To better understand this aspect, let us assume awavepacket composed of two time-harmonic TEM compo-nents, propagating in a lossy simple medium in the zzdirection and with the same linear polarization, such that

E¼ ½p1e�a1z cosðo1t� b1zþf1Þ

þp2e�a2z cosðo2t� b2zþf2Þ�xxð52Þ

For the sake of simplicity, we assume p1¼p2¼p, f1¼f2¼

0, and small losses, so that a1Ea2E0. Furthermore, wedefine oc¼ (o1þo2)/2, such that o1,2¼oc8do, wheredo¼ (o2�o1)/2. By so doing, (52) can be rewritten as

E¼ 2p cos oct� ðb2þ b1Þz

2

h icos dot� ðb2 � b1Þ

z

2

h ixx ð53Þ

which corresponds to the envelope distribution depicted inFig. 5. The first cosine (representing, e.g., a carrier withangular frequency oc) propagates with a velocity equal to2oc/(b2þ b1). In the limit do-0, (b2þ b1)/2Ebc (where bc isthe phase constant at oc) and, consequently, the carrier’svelocity tends to the phase velocity at oc. The secondcosine in (53) modulates the amplitude of the carrier (seeFig. 5) and propagates with a velocity equal to do/db, wheredb¼ (b2� b1)/2 corresponds to the variation of b around oc.We then define the group velocity

vg¼ limdo!0

dodb¼

@b@o

� ��1��o¼oc

ð54Þ

as the velocity of the wavepacket envelope, which can alsobe interpreted as the velocity of the signal (information)

Vector Field

z

vp = �/�

|pe−�z|

�

Figure 4. Uniform plane wave propagating in a lossy medium.


that modulates the carrier. A spectral analysis will de-monstrate that (54) still holds for more arbitrary butbandlimited wavepackets propagating through a mediumwith little dispersion (i.e., with relatively small variationsof b) [4]. As the wave energy is intrinsically related to thefield strength (amplitude), for bandlimited signals vg isalso defined as the velocity of energy flow [4]. Obviously,vg¼ vp¼ c for a TEM propagating through a losslesssimple medium. More detailed discussions on wave velo-cities can be found in Refs. 4 and 5.

6. ELECTROMAGNETIC WAVE INTERACTION WITHOBSTACLES

6.1. Boundary Conditions

Boundary conditions are needed for the proper solution ofMaxwell’s equations at the interface between two dissim-ilar media (due to the discontinuity on their physicalproperties) or at discontinuous source distributions.

One boundary condition is associated with each one of(1) [6]. For a time-invariant interface between two regions1 and 2, the four boundary conditions read as

nn� ðE2 �E1Þ¼ 0 nn . ðD2 �D1Þ¼ qs

nn� ðH2 �H1Þ¼Js nn . ðB2 �B1Þ¼ 0ð55Þ

where nn is the unit normal to the interface pointingtoward region 2 and Js and qs denote surface (or line)current and charge distributions over the interface, re-spectively. In (55), the field components are evaluatedright next to each side of the interface, at region 1 or 2,respectively. In the time-harmonic regime, the complexrepresentation of the boundary conditions have the sameform as (55). If both regions represent simple media, then

(11), (44), and the complex representation of (55) gives

nn� ðE2 � E1Þ¼ 0 nn . ðe2E2 � e1E1Þ¼ rs

nn� ðH2 �H1Þ ¼Js nn . ðm1H2 � m1H1Þ¼ 0ð56Þ

where mj and ej refer to region j (j¼ 1, 2).In simple media and if no surface currents or charges

are present at the interface, both (55) and (56) indicatethat the tangential (to the interface) components of theelectric and magnetic fields must be continuous across theinterface, and these are the only boundary conditions thatmust be imposed for the proper wave solution, as atregions of a simple medium without sources only Fara-day’s and Ampere’s laws are needed [6]. A particular caseof interest is that at the interface of a perfect electricconductor, which will be discussed in Section 6.3.

6.2. Reflection and Refraction of Plane Waves

The plane-wave reflection and refraction at a planarinterface is a canonical problem in electromagnetic wavetheory and provides many useful insights into the beha-vior of waves in the presence of more general obstacles[19–22]. Consider a plane interface between two simplemedia (lossy or not), characterized by their (complex)

z

Vector Field Envelope

vp

vg

Figure 5. Group and phase velocities for a wave with twospectral components with distinct frequencies.

(a)

Hi

y

Htkt

Et

Hrki

EiEr

kr

i r

t

∧

x∧

z∧

(b)

Hi

y

Htkt

Et

Hr

ki

Ei

Er

kr

i r

t

∧

x∧

z∧

�1,�1�2,�2

�1,�1�2,�2

Figure 6. Plane-wave incidence on a plane interface: (a) perpen-dicular and (b) parallel polarizations.


permeabilities (m1 and m2) and permittivities (e1 and e2), asshown in Fig. 6. An incident plane wave (index i) impingeson the interface from medium 1, generating a reflectedplane wave (index r), propagating back into medium 1,and a transmitted plane wave (index t) into medium 2. Theexact reflected and transmitted energies of the corre-sponding waves depend on the media properties and onthe direction of propagation and polarization of theincident wave.

Any arbitrary polarization can be decomposed into twolinear and orthogonal ones in simple media (see Section3.2). For the present analysis, the most commonly used arethe perpendicular (no E-field component normal to theinterface) and parallel (no H-field component normal tothe interface) linear polarizations, as depicted in Figs. 6aand 6b, respectively. The formulas to be presented arevalid for any incident plane wave, uniform or not. Accord-ing to the discussion in Section 3, the geometric inter-pretation illustrated in Fig. 6 (i.e., with real-valued yangles) is valid only for uniform waves. Depending onthe case, some y angles may come out complex, indicatingthat the corresponding plane wave is nonuniform. In anyevent, Fig. 6 is indeed useful for establishing the tangen-tial (and normal) components of the fields at the interface,necessary for the application of the pertinent boundaryconditions and, consequently, the solution of the problem.For uniform waves, Fig. 6 also provides a nice picture ofthe problem.

We will assume a time-harmonic regime (i.e., mono-chromatic waves) and use the complex phasor notation.According to (19), we can assume, without loss of general-ity, that the incident propagation vector ðkiÞ has nocomponent in the yy direction. The boundary conditionswill further prove that the same occurs for the reflectedðkrÞ and transmitted ðktÞ propagation vectors. With thatwe can define the xz plane in Fig. 6 as the plane ofincidence. For uniform plane waves, that is in conformitywith Fermat’s principle; specifically, the trajectories arerectilinear (in simple media) and corresponding to theleast travel time (see Section 4.3).

Starting with the perpendicular polarization and withthe help of (17) and Fig. 6a, we define the electric fields ofthe plane waves as

Ei¼E0e�jki. ryy

Er¼G?E0e�jkr. ryy

Et¼T?E0e�jkt. ryy

ð57Þ

where G> and T> are Fresnel’s reflection and transmis-sion coefficients for the perpendicular polarization, respec-tively, relating the amplitude and phase of thecorresponding field to those of the incident one. Thepropagation vectors are written as

ki¼ k1ðsin yi xxþ cos yi zzÞ

kr¼ k1ðsin yr xx� cos yr zzÞ

kt¼ k2ðsin yt xxþ cos yt zzÞ

ð58Þ

where kj¼o ffiffiffiffiffiffiffiffimjejp

(with j¼1,2). Note that (58) is in agree-ment with (18). From (19), (57), and (58) one immediatelyobtains the H-field expressions, observing thatZ¼ Zj¼

ffiffiffiffiffiffiffiffiffiffimj=ej

qwith j¼ 1 for the incident and reflected

waves and¼ 2 for the transmitted wave.Enforcing the boundary conditions (i.e., the continuity

of the tangential E- and H-field components) at the inter-face plane z¼ 0, one comes up with Snell’s law

k1 sin yi¼ k1 sin yr¼ k2 sin yt ð59Þ

and also with

G? ¼Z2 cos yi � Z1 cos yt

Z2 cos yiþ Z1 cos yt

T? ¼2Z2 cos yi

Z2 cos yiþ Z1 cos yt

ð60Þ

Snell’s law (59) comes from the matching of the phasevariation of the fields at the interface. For a uniformincident wave (with a real-valued yi), (59) imposes thatyr¼ yi (i.e., the reflected wave is also uniform), which is thelaw of reflection known for centuries in optics and inconformity with Fermat’s principle [1]. The Snell law forrefraction is given by the second equality in (59), whichcan be rewritten with the help of (41)

n1 sin yi¼n2 sin yt ð61Þ

where n1,2 are the indices of refraction at regions 1 and 2,respectively. Equations (59) and (61) provide the value ofyt with respect to yi and the physical properties of themedium. If one obtains a complex-valued yt, this indicatessimply that the transmitted wave is nonuniform. Thatmay happen even for a uniform incident wave if losses arepresent in one of the media (resulting in a complex indexof refraction) or at total reflection.

The analysis of the parallel polarization follows along aline similar to that of its perpendicular counterpart and acomparison between Figs. 6a and 6b provides the neces-sary insights. After enforcement of the boundary condi-tions at the interface, one comes up with (59) and (61) oncemore (i.e., Snell’s law is valid for any plane-wave polari-zation) and

Gjj ¼Z2 cos yt � Z1 cos yi

Z2 cos ytþ Z1 cos yi

Tjj ¼2Z2 cos yi

Z2 cos ytþ Z1 cos yi

ð62Þ

which are the Fresnel’s reflection and transmission coeffi-cients for the parallel polarization, respectively.

For simplicity, we shall assume next that both mediaare lossless simple dielectrics (i.e., m1Em2Em0 and real-valued e1 and e2) and that the incident plane wave isuniform (yi is real). The incidence is defined external ife24e1 (e.g., the incidence of a radiowave on ground in a RFlink). For real-valued yt, (61) provides that ytoyi for


external incidences. Otherwise, the incidence is internal(e.g., a refraction from water into air).

The behavior of |G| and |T| with respect to yi issketched in Figs. 7a and 7b for external and internalincidences, respectively. From the figures one observestwo particular angles of incidence: the Brewster (or polar-izing) angle yB, for which jGkj ¼ 0, and the critical angle yC,such that, at internal incidence and for yiZyC, |G|¼ 1 andtotal reflection occurs. For lossless and simple media, theBrewster angle just occurs for the parallel polarization [1].Its value can be determined from (61) and (62) by settingGk ¼ 0:

tan yB¼n2

n1ð63Þ

If losses are present, then Gk is complex and jGkj ¼ 0 isnever met for any yi. Instead, |G8| passes through aminimum, and that defines an effective Brewster angle.In any event, one can infer from Fig. 7 that |G8|r|G>|

for 0oyiop/2. For this reason, polarizing glasses aredesigned to block horizontally polarized light reflected bythe ground. For this reason also, vertically polarized RFwaves are preferred when attempting to obtain a uniformcoverage of a urban cell in mobile communications at UHF(due to reflections from vertical buildings).

In lossless simple media, the critical angle yC occursjust at internal incidence. This is established from (61)and from the fact that n14n2 for internal incidence,yielding yt4yi for yiryC with yt¼p/2 when yi¼ yC bydefinition. Besides, yt becomes complex for yCoyirp/2,indicating that the transmitted wave is nonuniform forincidence angles greater than yC. The primary conse-quence is |G|¼ 1 for yiZyC, as depicted in Fig. 7b.

To better understand the behavior of the transmittedwave at total reflection, let us recall the propagationvector kt of (58). For the present scenario and from (59)one will verify that the x component of kt is real-valuedwhile its z component is purely imaginary, such that thetransmitted wave propagates along the xx direction whileits intensity dies off exponentially from the interface. So,the transmitted wave is strongly localized near the inter-face and propagates parallel to it, characterizing a surfacewave. Also, if one obtains the E- and H-field compo-nents—with the help of (21) and (58)—and substitutesthem into (13), it will be shown that the complex Poyntingvector has a purely imaginary zz component and, conse-quently, there is no average flux of active (real) powerthrough the interface (i.e., the incident power is totallyreflected by the interface). In practical situations, losseswill prevent such idealized conditions from occuring and a(small) amount of energy will eventually be transmittedthrough the interface [4]. The critical angle and theconsequent total reflection helps in providing a nicepicture of how light is guided throughout an opticalfiber [23].

6.3. Waves on Good Electric Conductors

The equations of Section 6.2 can also be used to investi-gate the behavior of a plane wave incident on the surfaceof a good electric conductor. Let us assume that region 1 inFig. 6 is a lossless simple medium (i.e., m1 and e1 are real-valued constants), while region 2 is a good conductor, inwhich case m2Em0 and e2 is complex and according to (44).By definition, for good conductors sboe, such thate2E� js/o for our purposes, where s is the conducti-vity of region 2. Consequently, k2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jm0s=o

pand

Z2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijom0=s

p.

Applying such definitions to (59), one should observethat yi¼ yr, as expected, and that ytEp/2 as s-N. Be-sides, from (58) it is verified that kt � k2zz �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jm0s=o

pzz.

So, we come to the conclusion that the transmitted planewave tends to behave as a uniform one inside the goodconductor, propagating in the direction normal to theconductor’s surface. The corresponding amplitude factora2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm0sÞ=ð2oÞ

phas a considerably large value, so that

the field intensity dies off very rapidly away from thesurface (tending to zero when s-N, as expected). Thus, askin depth (d) is defined as the propagation distance

Perpendicular Pol.Parallel Pol.

1

0 B

B C i

i�

�

/2

2�2�2+�1

�2+�1

�2−�1

2�2�2+�1

�2+�1

�2−�1

|T|

|Γ|

(a)

(b)

Perpendicular Pol.Parallel Pol.

2

1

0 /2

2�2�1

|T|

|Γ |

Figure 7. |G| and |T| as a function of yi: (a) external and (b)internal incidences.


needed for the field intensity to decay by e�1, namely,d¼1=a2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2oÞ=ðm0sÞ

p.

As a consequence, Et is highly concentrated nearby andparallel to the surface. So, from (45) we end up concludingthat there will be a volumetric conduction current highlyconcentrated nearby and flowing parallel to the conduc-tor’s surface. In the limit of a perfect electric conductor(i.e., s-N), such a current behaves as a surface one ðJsÞ,in which case (55) and (56) apply with null fields insideregion 2 (note that nn¼ zz in Fig. 6).

It is interesting to stress that the results presentedhere are valid for any angle of incidence yi and for anywave polarization. So, according to the discussion in thebeginning of Section 3, the results are valid for anyelectromagnetic wave impinging on the surface of a goodelectric conductor.

The limiting case of a perfect electric conductor is notuseful just for good conductors. As an example, let usassume an horizontally polarized radiowave propagatingover ground and at grazing incidence (i.e., with yi-p/2,which is generally the case in a long-distance UHF radi-olink). This corresponds to a perpendicular polarizationand, according to (60), G>-� 1. Note also from (60) thatG>-� 1 as well for incidences on a good electric conduc-tor, as |Z2|-0 in this case. So, it is often used toapproximate the ground as a perfect (or good) electricconductor to simplify the analysis [16]. For vertical polar-ization the approximation to be adopted also depends onother factors (such as the value of yB) [16].

6.4. Scattering and Diffraction

The scattering of an electromagnetic wave by an arbitraryobstacle is a difficult problem to solve using purely analy-tical techniques. Usually, numerical methods are em-ployed. Among these, one can cite the method ofmoments [24], the finite-element method [25], and thefinite-difference time-domain method [26].

However, if the incidence wave is locally TEM and if theobstacle is immersed in a simple medium and has asmooth surface, the concepts discussed in Sections 4.3and 6.2 can be adopted (as approximations), to the extentthat the obstacle’s dimensions are large compared to thewavelength. Corrections to account for the curvatures ofthe incident wavefront (in case it is not a plane wave) andof the obstacle surface can be derived from GO principlesand included in G of (60) and (62), according to the wavepolarization [15,27]. The procedure is then conducted bytracing rays from the transmitter point to the receiver,such that any reflection on the obstacle must obey Fer-mat’s principle, according to which the trajectories arerectilinear (in simple media) and yi¼ yr with respect to thesurface’s normal. The difficulty of such technique gener-ally appears in the determination of the specular points(where reflection occurs) over the surface of the obstacle.

If the obstacle presents curvature discontinuities (suchas, e.g., at the edge of a wedge or at the border of a reflectorantenna), then the propagation mechanisms associatedwith the incidence on such regions is classified as adiffraction (as the diffraction of a laser beam by a metallicslit). Asymptotic (in the sense of k-N) techniques based

on the geometric theory of diffraction (GTD) can be appliedto account for diffraction [27]. Such techniques are alsobased on ray tracing and can be accommodated togetherwith GO principles to characterize the wave propagationthrough regions with several obstacles, such as a urbanscenario in mobile communications [18].

7. GUIDED WAVES

Electromagnetic waves can propagate either in open spaceor through guiding structures such as transmission linesand waveguides. The choice of transmission lines dependson characteristics such as frequency of operation, band-width, power-handling capability, and losses. Some of themost usual transmission lines are two-wire lines, coaxialcables, rectangular and circular waveguides, microstrips,and striplines. These structures are discussed in moredetail elsewhere in this encyclopedia in the article titledHIGH FREQUENCY TRANSMISSION LINES.

Some authors employ the waveguide denomination forguiding structures that allow propagation only of trans-verse electric (TE) and/or transverse magnetic (TM) waves,as described below, while the term transmission lines isused for guiding structures that allow propagation of TEMwaves as well. Other authors use these terms interchange-ably, as we will do here. We discuss TE and TM wavesnext.

7.1. TE and TM Waves

We will assume that the waves are guided in thez direction. TE and TM waves are a class of solutions ofMaxwell’s equations. As we are looking for propagatingfields, their z dependence will be assumed on the forme�jkzz. In this case, the transverse components of theelectric and magnetic fields can be written as [20]

Et¼1

k2 � k2z

½�jomðrtHzÞ� zz� jkzrtEz�

Ht¼1

k2 � k2z

½joeðrtEzÞ� zz� jkzrtHz�

ð64Þ

where rt stands for the transverse (to the z direction)portion of the r operator. The electric field of the TEcomponent is entirely transverse (Ez¼ 0), while the mag-netic field has a longitudinal component (Hza0). From(64), the TE field is given by Hz and the correspondingtransverse components Et and Ht. On the other hand, themagnetic field of the TM component is entirely transverse(Hz¼ 0), while the electric field has a longitudinal compo-nent (Eza0). From (64) the TM field is given by thelongitudinal electric field Ez and the corresponding trans-verse components.

TE and TM waves are supported by waveguides con-taining one or more perfect conductors and a homoge-neous dielectric. In this case, each wave (or mode) satisfiesthe boundary conditions at the waveguide walls, and thesewaves are decoupled from each other. A useful example ofsuch structure is the rectangular waveguide, formed byfour metallic walls (perfect conductors), as shown in Fig. 8.


In general, a rectangular waveguide may be empty insideor loaded, that is, partially or completely filled with adielectric. We will consider homogeneous (i.e., empty orcompletely filled with a uniform dielectric) rectangularwaveguides next.

For TE modes (also known as H modes), the long-itudinal component Hz(x,y,z) is given by the solution ofthe scalar wave equation that satisfies the appropriateboundary conditions at the four metallic walls (in thiscase, @Hz/@n¼ 0)

Hz¼H0 cosmpx

a

� �cos

npy

b

� �e�jkzz ð65Þ

where kz¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � ðmp=aÞ2 � ðnp=bÞ2

q, m¼ 0,1,2y, n¼

0,1,2y (but not m¼n¼ 0). The transverse componentsof the electric and magnetic fields can be obtained directlyfrom (64). These modes are denoted TEmn, in reference tothe indices m and n of (65). According to the frequency ofoperation, the TEmn modes may propagate or not insidethe waveguide. A propagating mode is characterized by areal kz. The parameter kz is real only if k24(mp/a)2

þ

(np/b)2. Otherwise, the mode is evanescent (exponentiallydecaying). The threshold frequency (where kz¼ 0) is calledcutoff frequency and is given by

fmn¼1

2ffiffiffiffiffimep

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim

a

� �2þ

n

b

� �2r

ð66Þ

For TM modes (also known as E modes), the longitudi-nal component of the electric field is obtained from thesolution of the scalar wave equation satisfying the bound-ary conditions at the four metallic walls (in this case,Ez¼0)

Ez¼E0 sinmpx

a

� �sin

npy

b

� �e�jkzz ð67Þ

where m¼ 1,2y, n¼ 1,2y, and kz is the same as above.The transverse components of the fields are obtained bysubstituting (67) into (64). Again the TMmn modes propa-gate only for frequencies above the cutoff frequency fmn

given by (66).

Assuming that a4b, the first mode that propagates(i.e., the one with smallest cutoff frequency) is the TE10,which is called the dominant (or fundamental) mode.Usually, waveguides are designed to operate in a fre-quency range where only the dominant mode can propa-gate. This avoids intermodal dispersion [7] that resultsfrom the different phase velocities of two or more propa-gating modes. Note that if the waveguide is filled with alossy dielectric, the modes are attenuated even abovecutoff.

Mathematically, TE and TM modes are orthogonalto each other and form a complete set. This means thatany field distribution inside the waveguide can be repre-sented as a superposition of TE and TM modes[7]. However, when the finite conductivity of the metallicwalls are considered, this orthogonality is no longer valid,and the modes become coupled. Also, when the dielectricis not homogeneous, the propagating modes becomea combination of TE and TM fields, also called hybridmodes [7].

7.2. Surface and Leaky Waves

Surface waves were introduced in Section 6.2, underthe condition of total internal reflection. This type ofwave exhibits an exponential decay away from a guidinginterface, while propagating in a direction parallel to it.Such a wave is also supported by dielectric waveguides,where no conductor is needed to guide the fields. A simpleexample of this kind of waveguide is the dielectric slabwaveguide, formed by a dielectric layer (infinite in xx and zzdirections), surrounded by air, as shown in Fig. 9. In theair, surface-wave modes are evanescent, and there is noaverage power flow from the dielectric to the air. TE andTM modes can be obtained for dielectric waveguide byfollowing a procedure similar to the one described inSection 7.1 [6]. For TE and TM modes, there is also apossibility of even and odd modes. For example, even TMmodes are given by [6]

Ez¼

Ed cosðbyyÞe�jkzz for jyj �d

2

E0e�ayjyje�jkzz for jyj d

2

8>><

>>:ð68Þ

The associated transverse field components are obtainedby substituting (68) into (64). The characteristic equations

a

Metallic wall

b

Z

Y

X

Figure 8. Rectangular waveguide geometry.

Dielectric

d2

d2

X

Y

Z

�o

�o

�d

Figure 9. Dielectric slab waveguide.


in the dielectric and in air are given by k2z þ b2

y ¼o2m0ed

and k2z � a2

y ¼o2m0e0. The boundary conditions at thedielectric interfaces provides the relation betweenthe amplitudes ½Ed cosðbyd=2Þ¼E0e�ayd=2� and requirethat [6]

by cotbyd

2¼�ayed

e0ð69Þ

which is the transcendental equation for kz, and conse-quently by and ay. Multiple solutions of this transcenden-tal equation are often identified by a subscript n (TMn

mode).These modes present surface waves properties when ay

is real and positive. The cutoff frequency of the TMn modeis the lowest frequency for which it propagates with noattenuation ½ay¼ 0; by¼o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim0ðed � e0Þ

p�. In this case, (69)

results in

fn¼n

2dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim0ðed � e0Þ

p ;with n¼ 1; 3; 5; . . . ð70Þ

For frequencies below the cutoff, the power is no longerconfined within the dielectric, and part of it is radiatedinto air. As a consequence of this radiation loss, the modespropagate with attenuation, having complex kz¼ bz� jaz.This kind of wave is called a leaky wave [7]. A simpleinterpretation of these waves comes from the plane-waveexpansion of a field within the dielectric. As mentionedearlier, any field can be written as a superposition ofplane waves in a sourceless linear medium. From thedielectric interface problem, it is known that the planewaves with incidence angles above the critical angle willbe totally reflected back to the dielectric, forming a sur-face wave. But those plane waves with incidence anglesbelow the critical angle will be refracted into the air(radiation), forming a leaky wave. Odd TM modes andboth even and odd TE modes can be obtained similarly,and present the same properties as described for even TMmodes.

An important example of dielectric waveguide is theoptical fiber [23], which is extensively used in long-dis-tance and high-bandwidth communications. An opticalfiber is formed by a dielectric rod, called core, and one ormore surrounding cylindrical dielectric layers, calledcladdings. In the simplest format, only one cladding isused. Usually the refraction index of the core is slightlyabove that of the cladding, allowing the propagationof surface waves. When the refraction index of the coreis constant, the configuration is known as a stepped-indexfiber. Otherwise, it is known as a graded-index fiber.The modes that propagate in the fiber are, in general,hybrid (combination of TE and TM), and the domi-nant mode is called the HE11 mode, with no cutofffrequency [23].

8. COHERENT AND INCOHERENT WAVES

Electromagnetic waves can also be classified as coherentor incoherent [1]. Coherent waves have definite phase

fronts (for each wavelength), while incoherent waves donot. Coherent waves are produced by sources that emitenergy through a collectively dependent process such asantennas. Most electromagnetic waves produced by syn-thetic devices at RF and microwave frequencies are co-herent, as we have been considering here. Lasers are alsoan example of coherent wave sources.

On the other hand, incoherent waves result from theradiation of many collectively independent sources. In thiscase, the wavefront is not well defined and random phasefluctuations occur across space (spatial incoherence) andwith varying wavelengths at random intervals (temporalincoherence). Incoherent sources include most naturalsources of radiation such as the sun, light from fluorescentlamps and lightbulbs, and LEDs (light-emitting diodes). Ingeneral, electromagnetic waves exhibit some partial de-gree of coherence. As such, this classification correspondsto two ideal extremes. In reality, electromagnetic wavesare at most highly incoherent (in one extreme) or mostlycoherent (in the other extreme).

9. FURTHER REMARKS

In this article, we have focused mainly on classicalaspects of electromagnetic wave propagation in simplemedia. For the vast majority of RF and microwave appli-cations, this classical description is sufficient. However,the detailed interaction and propagation of electromag-netic waves in material media, including molecular andatomic effects, depends on quantum aspects that are notcovered by a strictly classical description. The theoryof electromagnetic interaction that takes into accountthe laws of quantum mechanics is called quantumelectrodynamics (QED) [28,29]. The developmentof QED was the basis for the 1965 Nobel Prize in Physics,shared by Tomonaga (1906–1979), Schwinger (1918–1994), and Feynmann (1918–1988), who followed earlierdevelopments by Dirac (1902–1984). When averagedat a macroscopic level, QED reduces to Maxwell’s equa-tions augmented by phenomenological equations thatcan be expressed in terms of macroscopic constitutivelaws.

As a final side remark, it should also be noted here thatMaxwell’s equations in their classical form are invariantto Lorentz transformations [30] and already incorporatespecial relativity. Indeed, the need to preserve the form ofMaxwell’s equations in any nonaccelerated frames ofreference (special relativity principle) was a major moti-vation to the development of special relativity theory byEinstein [31].

BIBLIOGRAPHY

1. M. Born and E. Wolf, Principles of Optics, 6th ed., PergamonPress, Oxford, 1980.

2. J. C. Maxwell, Electricity and Magnetism, Academic Press,New York, 1935.

3. R. S. Elliott, Electromagnetics: History, Theory, and Applica-

tions, IEEE Press, Piscataway, NJ, 1993.


4. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, NewYork, 1999.

5. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, NewYork, 1941.

6. R. F. Harrington, Time-Harmonic Electromagnetic Fields,McGraw-Hill, New York, 1961.

7. R. E. Collin, Field Theory of Guided Waves, 2nd ed., IEEEPress, Piscataway, NJ, 1991.

8. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of

Waves, IEEE Press, Piscataway, NJ, 1994.

9. S. Silver, ed., Microwave Antenna Theory and Design,McGraw-Hill, New York, 1949.

10. J. A. Kong, Electromagnetic Wave Theory, EMW Publishing,Cambridge, MA, 2000.

11. W. Chew, Waves and Fields in Inhomogeneous Media, IEEEPress, Piscataway, NJ, 1995.

12. P. M. Morse and H. Feshbach, Methods of Theoretical Physics,Vol. 1, New York: McGraw-Hill Book Co., 1953.

13. P. C. Clemmow and J. Wait, The Plane Wave Spectrum

Representation of Electromagnetic Fields, Oxford Univ. Press,1996.

14. C. A. Balanis, Antenna Theory Analysis and Design, 2nd ed.,Wiley, New York, 1997.

15. G. A. Deschamps, Ray techniques in electromagnetics, Proc.

IEEE 60(9):1022–1035 (Sept. 1972).

16. D. E. Kerr, ed., Propagation of Short Radio Waves, McGraw-Hill, New York, 1949.

17. J. D. Parsons, The Mobile Radio Propagation Channel, 2nded., Wiley, New York, 2000.

18. A. Yariv and P. Yeh, Optical Waves in Crystals, Wiley, NewYork, 1984.

19. C. A. Balanis, Advanced Engineering Electromagnetics, Wi-ley, New York, 1989.

20. D. K. Cheng, Field and Wave Electromagnetics, 2nd ed.,Addison-Wesley, Reading, MA, 1992.

21. D. M. Pozar, Microwave Engineering, 2nd ed., Wiley, NewYork, 1998.

22. S. Ramo, J. R. Whinnery, and T. Van Duzer, Field and Waves

in Communication Electronics, 2nd ed., Wiley, New York,1984.

23. J. A. Buck, Fundamentals of Optical Fibers, Wiley, New York,1995.

24. J. J.-H. Wang, Generalized Moment Methods in Electromag-

netics, Wiley, New York, 1991.

25. J. Jin, The Finite Element Method in Electromagnetics, 2nded., Wiley, New York, 2002.

26. A. Taflove and S. C. Hagness, Computational Electrody-namics: The Finite-Difference Time-Domain Method, 2nded., Artech House, Boston, 2000.

27. D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe,Introduction to the Uniform Geometrical Theory of Diffrac-tion, Artech House, Boston, 1990.

28. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,Quantum Electrodynamics, 2nd ed., Pergamon Press, Oxford,1982.

29. J. Schwinger, ed., Selected Papers on Quantum Electrody-namics, Dover, New York, 1958.

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31. A. Einstein and L. Infeld, Evolution of Physics, Simon &Schuster, New York, 1966.

ELECTROMAGNETIC WAVE SCATTERING

RANDALL L. MUSSELMAN

USAF AcademyColorado Springs, Colorado

1. TYPES OF ELECTROMAGNETIC SCATTERING

Fundamentally, there are three types of electromagneticscattering mechanisms: reflection, refraction, and diffrac-tion. These scattering mechanisms can radiate specularlyor diffusely. Specular scattering, caused by electricallylarge flat objects, means that electromagnetic reradiationtravels in parallel rays. Diffuse scattering, caused by ir-regular or electrically small objects, means that the elec-tromagnetic reradiation spreads as it propagates awayfrom the scattering object.

1.1. Specular Scattering

Of the three fundamental scattering mechanisms, themost familiar are specular reflection and refraction. Ifany corners or bends that exist at the boundary are verygradual compared to the wavelength of the incident field,then the boundary tends to cause specular scattering. Op-tical scattering is often assumed to be specular becausemost obstructing bodies that are smooth are electricallylarge compared to optical wavelengths. Specular scatter-ing can be modeled with the specular law of reflection andSnell’s law of refraction.

1.1.1. Reflection. A familiar example of specular reflec-tion is the common reflection of a visible image in a mirror,since the dimensions of the mirror would be huge com-pared to the wavelength of visible light. Other examplesare the radar return from a large flat target, cellular tele-phone multipath from a water tower, or a parabolic dishused to reflect microwave energy to the antenna at its fo-cal point. The ratio of the reflected electric field Er, to theincident electric field Ei, is called the reflection coefficient:

G¼Er

Eið1Þ

If the electric field in Fig. 1 is parallel to the plane ofincidence containing all three propagation paths (i.e., in-cident, reflected, and transmitted paths), then it is hasparallel polarization. When the electric field is perpendic-ular to the plane of incidence (the x–z plane in Fig. 1), thenit has perpendicular polarization. The reflection coeffi-cients for parallel and perpendicular polarizations are

Gjj ¼Z2 cos yt � Z1 cos yi

Z2 cos ytþ Z1 cos yið2aÞ

G? ¼Z2 cos yi � Z1 cos yt

Z2 cos yiþ Z1 cos ytð2bÞ

ELECTROMAGNETIC WAVE SCATTERING 1295

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