Lectromagnetic waves and antennas combined

When the vector J represents the energy flux of a propagating electromagnetic wave andρthe corresponding energy per unit volume, then because the speed of propagation is the velocity of l

1 Maxwell’s Equations

The first is Faraday’s law of induction, the second is Amp`ere’s law as amended by

Maxwell to include the displacement current∂D/∂t, the third and fourth are Gauss’ laws

for the electric and magnetic fields

The displacement current term∂D/∂tin Amp`ere’s law is essential in predicting the

existence of propagating electromagnetic waves Its role in establishing charge

conser-vation is discussed in Sec 1.7

Eqs (1.1.1) are in SI units The quantities E and H are the electric and magnetic

field intensities and are measured in units of [volt/m] and [ampere/m], respectively

The quantities D and B are the electric and magnetic flux densities and are in units of

[coulomb/m2] and [weber/m2], or [tesla] D is also called the electric displacement, and

B, the magnetic induction.

The quantitiesρand J are the volume charge density and electric current density

(charge flux) of any external charges (that is, not including any induced polarization

charges and currents.) They are measured in units of [coulomb/m3] and [ampere/m2]

The right-hand side of the fourth equation is zero because there are no magnetic

mono-pole charges Eqs (1.3.17)–(1.3.19) display the induced polarization terms explicitly

The charge and current densitiesρ,J may be thought of as the sources of the

electro-magnetic fields For wave propagation problems, these densities are localized in space;

for example, they are restricted to flow on an antenna The generated electric and

mag-netic fields are radiated away from these sources and can propagate to large distances to

(source-free Maxwell’s equations) (1.1.2)

The qualitative mechanism by which Maxwell’s equations give rise to propagatingelectromagnetic fields is shown in the figure below

For example, a time-varying current J on a linear antenna generates a circulating and time-varying magnetic field H, which through Faraday’s law generates a circulating electric field E, which through Amp`ere’s law generates a magnetic field, and so on Thecross-linked electric and magnetic fields propagate away from the current source Amore precise discussion of the fields radiated by a localized current distribution is given

mdv

dt =F= q(E+v×B) (1.2.2)wheremis the mass of the charge The force F will increase the kinetic energy of the

charge at a rate that is equal to the rate of work done by the Lorentz force on the charge,

that is, v·F Indeed, the time-derivative of the kinetic energy is:

Wkin=12mv·v ⇒ dWkin

dt = mv·dv

dt =v·F= qv·E (1.2.3)

We note that only the electric force contributes to the increase of the kinetic energy—

the magnetic force remains perpendicular to v, that is, v· (v×B)=0

1.3 Constitutive Relations 3

Volume charge and current distributions ρ,J are also subjected to forces in the

presence of fields The Lorentz force per unit volume acting onρ,J is given by:

f= ρE+J×B (Lorentz force per unit volume) (1.2.4)

where f is measured in units of [newton/m3] If J arises from the motion of charges

within the distributionρ, then J= ρv (as explained in Sec 1.6.) In this case,

f= ρ(E+v×B) (1.2.5)

By analogy with Eq (1.2.3), the quantity v·f= ρv·E=J·E represents the power

per unit volume of the forces acting on the moving charges, that is, the power expended

by (or lost from) the fields and converted into kinetic energy of the charges, or heat It

has units of [watts/m3] We will denote it by:

dPloss

dV =J·E (ohmic power losses per unit volume) (1.2.6)

In Sec 1.8, we discuss its role in the conservation of energy We will find that

elec-tromagnetic energy flowing into a region will partially increase the stored energy in that

region and partially dissipate into heat according to Eq (1.2.6)

1.3 Constitutive Relations

The electric and magnetic flux densities D,B are related to the field intensities E,H via

the so-called constitutive relations, whose precise form depends on the material in which

the fields exist In vacuum, they take their simplest form:

ampere·m=henry

mFrom the two quantities0, μ0, we can define two other physical constants, namely,

the speed of light and the characteristic impedance of vacuum:

These are typically valid at low frequencies The permittivityand permeabilityμ

are related to the electric and magnetic susceptibilities of the material as follows:

where the quantity P= 0χE represents the dielectric polarization of the material, that

is, the average electric dipole moment per unit volume In a magnetic material, we have

B= μ0(H+M)= μ0(H+ χmH)= μ0(1+ χm)H= μH (1.3.7)

where M= χmH is the magnetization, that is, the average magnetic moment per unit

volume The speed of light in the material and the characteristic impedance are:

c=√1μ, η=

μ

n

η=

μ

D(r, t)= (r)E(r, t)

1.3 Constitutive Relations 5

In anisotropic materials,depends on thex, y, zdirection and the constitutive

rela-tions may be written component-wise in matrix (or tensor) form:

⎡

⎢ DDxy

Ez

⎤

Anisotropy is an inherent property of the atomic/molecular structure of the

dielec-tric It may also be caused by the application of external fields For example, conductors

and plasmas in the presence of a constant magnetic field—such as the ionosphere in the

presence of the Earth’s magnetic field—become anisotropic (see for example, Problem

1.10 on the Hall effect.)

In nonlinear materials,may depend on the magnitudeEof the applied electric field

in the form:

D= (E)E, where (E)=  + 2E+ 3E2+ · · · (1.3.12)

Nonlinear effects are desirable in some applications, such as various types of

electro-optic effects used in light phase modulators and phase retarders for altering

polariza-tion In other applications, however, they are undesirable For example, in optical fibers

nonlinear effects become important if the transmitted power is increased beyond a few

milliwatts A typical consequence of nonlinearity is to cause the generation of higher

harmonics, for example, ifE= E0ejωt, then Eq (1.3.12) gives:

D= (E)E = E + 2E2+ 3E3+ · · · = E0ejωt+ 2E2e2jωt+ 3E3e3jωt+ · · ·

Thus the input frequencyω is replaced by ω,2ω,3ω, and so on In a

multi-wavelength transmission system, such as a multi-wavelength division multiplexed (WDM)

op-tical fiber system carrying signals at closely-spaced carrier frequencies, such

nonlinear-ities will cause the appearance of new frequencies which may be viewed as crosstalk

among the original channels For example, if the system carries frequenciesωi, i=

1,2, , then the presence of a cubic nonlinearityE3will cause the appearance of the

frequenciesωi± ωj± ωk In particular, the frequenciesωi+ ωj− ωkare most likely

to be confused as crosstalk because of the close spacing of the carrier frequencies

Materials with a frequency-dependent dielectric constant(ω)are referred to as

dispersive The frequency dependence comes about because when a time-varying

elec-tric field is applied, the polarization response of the material cannot be instantaneous

Such dynamic response can be described by the convolutional (and causal) constitutive

relationship:

D(r, t)= t

−∞(t− t)E(r, t) dt (1.3.13)which becomes multiplicative in the frequency domain:

All materials are, in fact, dispersive However,(ω)typically exhibits strong

depen-dence onωonly for certain frequencies For example, water at optical frequencies has

refractive indexn=√rel=1.33, but at RF down to dc, it hasn=9

or waveguide dispersion introduced by the confining walls of a waveguide

There exist materials that are both nonlinear and dispersive that support certaintypes of non-linear waves called solitons, in which the spreading effect of dispersion isexactly canceled by the nonlinearity Therefore, soliton pulses maintain their shape asthey propagate in such media [1177,874,875]

More complicated forms of constitutive relationships arise in chiral and gyrotropicmedia and are discussed in Chap 4 The more general bi-isotropic and bi-anisotropicmedia are discussed in [30,95]; see also [57]

In Eqs (1.1.1), the densitiesρ,J represent the external or free charges and currents

in a material medium The induced polarization P and magnetization M may be made

explicit in Maxwell’s equations by using the constitutive relations:

D= 0E+P, B= μ0(H+M) (1.3.16)Inserting these in Eq (1.1.1), for example, by writing∇∇ ×B = μ0∇∇ × (H+M)=

μ0(J+D˙+ ∇∇∇ ×M)= μ0(0˙+J+˙+ ∇∇∇ ×M), we may express Maxwell’s equations in

terms of the fields E and B :

∂t, ρpol= −∇∇∇ ·P (polarization densities) (1.3.18)

Similarly, the quantity Jmag= ∇∇∇ ×M may be identified as the magnetization current

density (note thatρmag=0.) The total current and charge densities are:

Jtot=J+Jpol+Jmag=J+∂P

∂t + ∇∇∇ ×M

ρ = ρ + ρ = ρ − ∇∇∇ ·P

(1.3.19)

1.4 Negative Index Media 7

and may be thought of as the sources of the fields in Eq (1.3.17) In Sec 14.6, we examine

this interpretation further and show how it leads to the Ewald-Oseen extinction theorem

and to a microscopic explanation of the origin of the refractive index

1.4 Negative Index Media

Maxwell’s equations do not preclude the possibility that one or both of the quantities

, μbe negative For example, plasmas below their plasma frequency, and metals up to

optical frequencies, have <0 andμ >0, with interesting applications such as surface

plasmons (see Sec 8.5)

Isotropic media withμ <0 and >0 are more difficult to come by [153], although

examples of such media have been fabricated [381]

Negative-index media, also known as left-handed media, have, μthat are

simulta-neously negative, <0 andμ <0 Veselago [376] was the first to study their unusual

electromagnetic properties, such as having a negative index of refraction and the

rever-sal of Snel’s law

The novel properties of such media and their potential applications have generated

a lot of research interest [376–457] Examples of such media, termed “metamaterials”,

have been constructed using periodic arrays of wires and split-ring resonators, [382]

and by transmission line elements [415–417,437,450], and have been shown to exhibit

the properties predicted by Veselago

Whenrel<0 andμrel<0, the refractive index,n2= relμrel, must be defined by

the negative square rootn = −√relμrel Because thenn <0 andμrel <0 will imply

that the characteristic impedance of the mediumη= η0μrel/nwill be positive, which

as we will see later implies that the energy flux of a wave is in the same direction as the

direction of propagation We discuss such media in Sections 2.12, 7.16, and 8.6

where ˆn is a unit vector normal to the boundary pointing from medium-2 into medium-1.

The quantitiesρs,Jsare any external surface charge and surface current densities on

the boundary surface and are measured in units of [coulomb/m2] and [ampere/m]

In words, the tangential components of the E-field are continuous across the

inter-face; the difference of the tangential components of the H-field are equal to the surface

current density; the difference of the normal components of the flux density D are equal

to the surface charge density; and the normal components of the magnetic flux density

B are continuous.

TheDnboundary condition may also be written a form that brings out the dence on the polarization surface charges:

depen-(0E1n+ P1n)−(0E2n+ P2n)= ρs ⇒ 0(E1n− E2n)= ρs− P1n+ P2n= ρs,totThe total surface charge density will beρs,tot= ρs+ρ1s,pol+ρ2s,pol, where the surfacecharge density of polarization charges accumulating at the surface of a dielectric is seen

to be (ˆn is the outward normal from the dielectric):

The relative directions of the field vectors are shown in Fig 1.5.1 Each vector may

be decomposed as the sum of a part tangential to the surface and a part perpendicular

to it, that is, E=Et+En Using the vector identity,

E=ˆn× (E׈n)+ˆn(ˆn·E)=Et+En (1.5.3)

we identify these two parts as:

Et=ˆn× (E×nˆ) , En=nˆ(nˆ·E)=nˆEn

Fig 1.5.1 Field directions at boundary.

Using these results, we can write the first two boundary conditions in the followingvectorial forms, where the second form is obtained by taking the cross product of thefirst with ˆn and noting that Jsis purely tangential:

ˆ

n× (E1׈n)−nˆ× (E2×nˆ)=0ˆ

n× (H1×nˆ)−ˆn× (H2×nˆ)=Js×nˆ or,

ˆ

n× (E1−E2)=0ˆ

n× (H1−H2)=Js

(1.5.4)

The boundary conditions (1.5.1) can be derived from the integrated form of Maxwell’sequations if we make some additional regularity assumptions about the fields at theinterfaces

1.6 Currents, Fluxes, and Conservation Laws 9

In many interface problems, there are no externally applied surface charges or

cur-rents on the boundary In such cases, the boundary conditions may be stated as:

E1t=E2t

H1t=H2t

D1n= D2n

B1n= B2n

(source-free boundary conditions) (1.5.5)

1.6 Currents, Fluxes, and Conservation Laws

The electric current density J is an example of a flux vector representing the flow of the

electric charge The concept of flux is more general and applies to any quantity that

flows.† It could, for example, apply to energy flux, momentum flux (which translates

into pressure force), mass flux, and so on

In general, the flux of a quantityQ is defined as the amount of the quantity that

flows (perpendicularly) through a unit surface in unit time Thus, if the amountΔQ

flows through the surfaceΔSin timeΔt, then:

J=ΔSΔtΔQ (definition of flux) (1.6.1)When the flowing quantityQis the electric charge, the amount of current through

the surfaceΔSwill beΔI= ΔQ/Δt, and therefore, we can writeJ= ΔI/ΔS, with units

of [ampere/m2]

The flux is a vectorial quantity whose direction points in the direction of flow There

is a fundamental relationship that relates the flux vector J to the transport velocity v

and the volume densityρof the flowing quantity:

This can be derived with the help of Fig 1.6.1 Consider a surfaceΔSoriented

per-pendicularly to the flow velocity In timeΔt, the entire amount of the quantity contained

in the cylindrical volume of heightvΔtwill manage to flow throughΔS This amount is

equal to the density of the material times the cylindrical volumeΔV= ΔS(vΔt), that

is,ΔQ= ρΔV = ρ ΔS vΔt Thus, by definition:

ΔSΔt=ρ ΔS vΔt

When J represents electric current density, we will see in Sec 1.12 that Eq (1.6.2)

implies Ohm’s law J= σE When the vector J represents the energy flux of a propagating

electromagnetic wave andρthe corresponding energy per unit volume, then because the

speed of propagation is the velocity of light, we expect that Eq (1.6.2) will take the form:

†In this sense, the terms electric and magnetic “flux densities” for the quantities D,B are somewhat of a

Fig 1.6.1 Flux of a quantity.

Similarly, whenJrepresents momentum flux, we expect to haveJmom = cρmom.Momentum flux is defined asJmom= Δp/(ΔSΔt)= ΔF/ΔS, wherepdenotes momen-tum andΔF= Δp/Δtis the rate of change of momentum, or the force, exerted on thesurfaceΔS Thus,Jmomrepresents force per unit area, or pressure

Electromagnetic waves incident on material surfaces exert pressure (known as diation pressure), which can be calculated from the momentum flux vector It can beshown that the momentum flux is numerically equal to the energy density of a wave, that

ra-is,Jmom= ρen, which implies thatρen= ρmomc This is consistent with the theory ofrelativity, which states that the energy-momentum relationship for a photon isE= pc

1.7 Charge Conservation

Maxwell added the displacement current term to Amp`ere’s law in order to guaranteecharge conservation Indeed, taking the divergence of both sides of Amp`ere’s law andusing Gauss’s law∇∇ ·D= ρ, we get:

Another consequence of Eq (1.7.1) is that in good conductors, there cannot be anyaccumulated volume charge Any such charge will quickly move to the conductor’ssurface and distribute itself such that to make the surface into an equipotential surface

1.8 Energy Flux and Energy Conservation 11

Fig 1.7.1 Flux outwards through surface.

Assuming that inside the conductor we have D= E and J= σE, we obtain

whereρ0(r)is the initial volume charge distribution The solution shows that the

vol-ume charge disappears from inside and therefore it must accumulate on the surface of

the conductor The “relaxation” time constantτrel= /σis extremely short for good

conductors For example, in copper,

τrel= 

σ =8.85×10−12

5.7×107 =1.6×10−19sec

By contrast,τrelis of the order of days in a good dielectric For good conductors, the

above argument is not quite correct because it is based on the steady-state version of

Ohm’s law, J= σE, which must be modified to take into account the transient dynamics

of the conduction charges

It turns out that the relaxation timeτrelis of the order of the collision time, which

is typically 10−14sec We discuss this further in Sec 1.13 See also Refs [138–141].

1.8 Energy Flux and Energy Conservation

Because energy can be converted into different forms, the corresponding conservation

equation (1.7.1) should have a non-zero term in the right-hand side corresponding to

the rate by which energy is being lost from the fields into other forms, such as heat

Thus, we expect Eq (1.7.1) to have the form:

∂ρen

∂t + ∇∇∇ ·Jen=rate of energy loss (1.8.1)

Assuming the ordinary constitutive relations D= E and B = μH, the quantities

ρen,Jendescribing the energy density and energy flux of the fields are defined as follows,

where we introduce a change in notation:

ρen= w =12|E|2+12μ|H|2=energy per unit volume

Jen= PPP =E×H=energy flux or Poynting vector

As we discussed in Eq (1.2.6), the quantity J·E represents the ohmic losses, that

is, the power per unit volume lost into heat from the fields The integrated form of

Eq (1.8.3) is as follows, relative to the volume and surface of Fig 1.7.1:

capaci-The energy density of the electric field between the plates isw= E2/2 Multiplying this

by the volume between the plates,A·l, will give the total energy stored in the capacitor.Equating this to the circuit expressionCV2/2, will yield the capacitanceC:

W=1

2E2· Al =1

2CV2=1

2CE2l2 ⇒ C = Al

Next, consider a solenoid withnturns wound around a cylindrical iron core of length

l, cross-sectional areaA, and permeabilityμ The current through the solenoid wire isrelated to the magnetic field in the core through Amp`ere’s lawHl= nI It follows that thestored magnetic energy in the solenoid will be:

1.9 Harmonic Time Dependence 13

current is assumed to be uniformly distributed over the cross-sectionAand will have

densityJ= σE

The power dissipated into heat per unit volume isJE = σE2 Multiplying this by the

resistor volumeAland equating it to the circuit expressionV2/R= RI2will give:

(J· E)(Al)= σE2(Al)=V2

R =E2l2

R ⇒ R = 1

σ

lA

The same circuit expressions can, of course, be derived more directly usingQ= CV, the

magnetic flux Φ= LI, andV= RI 

Conservation laws may also be derived for the momentum carried by electromagnetic

fields [41,1140] It can be shown (see Problem 1.6) that the momentum per unit volume

carried by the fields is given by:

1.9 Harmonic Time Dependence

Maxwell’s equations simplify considerably in the case of harmonic time dependence

Through the inverse Fourier transform, general solutions of Maxwell’s equation can be

built as linear combinations of single-frequency solutions:†

where the phasor amplitudes E(r),H(r)are complex-valued Replacing time derivatives

by∂t→ jω, we may rewrite Eq (1.1.1) in the form:

In this book, we will consider the solutions of Eqs (1.9.2) in three different contexts:

(a) uniform plane waves propagating in dielectrics, conductors, and birefringent

me-dia, (b) guided waves propagating in hollow waveguides, transmission lines, and optical

fibers, and (c) propagating waves generated by antennas and apertures

†Theejωt convention is used in the engineering literature, ande−iωtin the physics literature One can

Next, we review some conventions regarding phasors and time averages A valued sinusoid has the complex phasor representation:

whereA= |A|ejθ Thus, we haveA(t)=Re

0 A2(t) dt=1

2Re

AA∗]=1

2|A|2 (1.9.5)Some interesting time averages in electromagnetic wave problems are the time av-erages of the energy density, the Poynting vector (energy flux), and the ohmic powerlosses per unit volume Using the definition (1.8.2) and the result (1.9.4), we have forthese time averages:

The expression (1.9.6) for the energy densitywwas derived under the assumptionthat bothandμwere constants independent of frequency In a dispersive medium,, μ

become functions of frequency In frequency bands where(ω), μ(ω)are essentiallyreal-valued, that is, where the medium is lossless, it can be shown [153] that the time-averaged energy density generalizes to:

∇∇ ·E×H= −E·D˙−H·B˙−J·E (1.9.8)

As in Eq (1.8.3), we would like to interpret the first two terms in the right-hand side

as the time derivative of the energy density, that is,

dw

dt =E·D˙+H·B˙

1.9 Harmonic Time Dependence 15

Anticipating a phasor-like representation, we may assume complex-valued fields and

derive also the following relationship from Maxwell’s equations:

2Re

H∗·B˙

(1.9.10)

In a dispersive dielectric, the constitutive relation between D and E can be written

as follows in the time and frequency domains:†

D(t)= ∞

−∞(t− t)E(t)dt  D(ω)= (ω)E(ω) (1.9.11)where the Fourier transforms are defined by

Following [153], we assume a quasi-harmonic representation for the electric field,

E(t)=E0(t)ejω 0 t, where E0(t)is a slowly-varying function of time Equivalently, in the

frequency domain we have E(ω)=E0(ω− ω0), assumed to be concentrated in a small

neighborhood ofω0, say,|ω − ω0| ≤ Δω Because(ω)multiplies the narrowband

function E(ω), we may expandω(ω)in a Taylor series aroundω0and keep only the

linear terms, that is, inside the integral (1.9.14), we may replace:

ω(ω)= a0+ b0(ω− ω0) , a0= ω0(ω0) , b0= d

ω(ω) dω





ω0

(1.9.15)Inserting this into Eq (1.9.14), we obtain the approximation

used inside Eq (1.9.13) Inserting (1.9.16) into Eq (1.9.13), we find

2Re

E∗·D˙

= ddt

dω(ω) 0

dω |E0(t)|2



(1.9.18)Dropping the subscript 0, we see that the quantity under the time derivative in theright-hand side may be interpreted as a time-averaged energy density for the electricfield A similar argument can be given for the magnetic energy term of Eq (1.9.7)

We will see in the next section that the energy density (1.9.7) consists of two parts:one part is the same as that in the vacuum case; the other part arises from the kineticand potential energy stored in the polarizable molecules of the dielectric medium.When Eq (1.9.7) is applied to a plane wave propagating in a dielectric medium, onecan show that (in the lossless case) the energy velocity coincides with the group velocity.The generalization of these results to the case of a lossy medium has been studiedextensively [153–167] Eq (1.9.7) has also been applied to the case of a “left-handed”medium in which both(ω)andμ(ω)are negative over certain frequency ranges Asargued by Veselago [376], such media must necessarily be dispersive in order to make

Eq (1.9.7) a positive quantity even though individuallyandμare negative

Analogous expressions to (1.9.7) may also be derived for the momentum density of

a wave in a dispersive medium In vacuum, the time-averaged momentum density isgiven by Eq (1.8.5), that is,

1.10 Simple Models of Dielectrics, Conductors, and Plasmas

A simple model for the dielectric properties of a material is obtained by considering themotion of a bound electron in the presence of an applied electric field As the electricfield tries to separate the electron from the positively charged nucleus, it creates anelectric dipole moment Averaging this dipole moment over the volume of the materialgives rise to a macroscopic dipole moment per unit volume

A simple model for the dynamics of the displacementxof the bound electron is asfollows (with ˙x= dx/dt):

where we assumed that the electric field is acting in thex-direction and that there is

a spring-like restoring force due to the binding of the electron to the nucleus, and afriction-type force proportional to the velocity of the electron

The spring constantkis related to the resonance frequency of the spring via therelationshipω0=√k/m, or,k= mω2 Therefore, we may rewrite Eq (1.10.1) as

¨

x+ γx˙+ ω2x= e

1.11 Dielectrics 17

The limitω0=0 corresponds to unbound electrons and describes the case of good

conductors The frictional termγ˙xarises from collisions that tend to slow down the

electron The parameterγis a measure of the rate of collisions per unit time, and

therefore,τ=1/γwill represent the mean-time between collisions

In a typical conductor,τis of the order of 10−14seconds, for example, for copper,

τ=2.4×10−14sec andγ =4.1×1013 sec−1 The case of a tenuous, collisionless,

plasma can be obtained in the limitγ=0 Thus, the above simple model can describe

the following cases:

a Dielectrics,ω0 0, γ 0

b Conductors,ω0=0, γ 0

c Collisionless Plasmas,ω0=0, γ=0

The basic idea of this model is that the applied electric field tends to separate positive

from negative charges, thus, creating an electric dipole moment In this sense, the

model contains the basic features of other types of polarization in materials, such as

ionic/molecular polarization arising from the separation of positive and negative ions

by the applied field, or polar materials that have a permanent dipole moment

1.11 Dielectrics

The applied electric fieldE(t)in Eq (1.10.2) can have any time dependence In particular,

if we assume it is sinusoidal with frequencyω,E(t)= Eejωt, then, Eq (1.10.2) will have

the solutionx(t)= xejωt, where the phasorxmust satisfy:

From Eqs (1.11.1) and (1.11.2), we can find the polarization per unit volume P

We assume that there areNsuch elementary dipoles per unit volume The individual

electric dipole moment isp= ex Therefore, the polarization per unit volume will be:

This can be written in a more convenient form, as follows:

(ω)= 0+ 0ω

2 p

For a dielectric, we may assumeω0 0 Then, the low-frequency limit (ω=0) of

Eq (1.11.5), gives the nominal dielectric constant:

(0)= 0+ 0

ω2 p

ω2 = 0+ Ne2

The real and imaginary parts of (ω)characterize the refractive and absorptiveproperties of the material By convention, we define the imaginary part with the negativesign (because we useejωttime dependence):

2

pωγ(ω2− ω2)2+γ2ω2 (1.11.10)Fig 1.11.1 shows a plot of(ω)and (ω) Around the resonant frequencyω0,the real part(ω)behaves in an anomalous manner, that is, it drops rapidly withfrequency to values less than0and the material exhibits strong absorption The term

“normal dispersion” refers to an(ω)that is an increasing function ofω, as is thecase to the far left and right of the resonant frequency

1.11 Dielectrics 19

Fig 1.11.1 Real and imaginary parts of the effective permittivity(ω)

Real dielectric materials exhibit, of course, several such resonant frequencies

cor-responding to various vibrational modes and polarization mechanisms (e.g., electronic,

ionic, etc.) The permittivity becomes the sum of such terms:

(ω)= 0+ 0

i

fji(Ni− Nj)e2/m0

ω2

ji− ω2+ jωγji

(1.11.12)

whereωjiare transition frequencies between energy levels, that is,ωji= (Ej− Ei)/,

andNi, Njare the populations of the lower,Ei, and upper,Ej, energy levels The

quan-titiesfjiare called “oscillator strengths.” For example, for a two-level atom we have:

(ω)= 0+ 0

f ω2 p

as a classical absorbing dielectric However, if there is population inversion,Ni< Nj,

then the corresponding permittivity term changes sign This leads to a negative

imag-inary part, (ω), representing a gain Fig 1.11.2 shows the real and imaginary parts

of Eq (1.11.13) for the case of a negative effective oscillator strengthf= −1

The normal and anomalous dispersion bands still correspond to the bands where

the real part(ω)is an increasing or decreasing, respectively, function of frequency

But now the normal behavior is only in the neighborhood of the resonant frequency,

whereas far from it, the behavior is anomalous

Settingn(ω)=(ω)/0for the refractive index, Eq (1.11.11) can be written in the

following form, known as the Sellmeier equation (where theBiare constants):

Fig 1.11.2 Effective permittivity in a two-level gain medium withf= −1

In practice, Eq (1.11.14) is applied in frequency ranges that are far from any nance so that one can effectively setγi=0:

reso-n2(ω)=1+

i

Biω2 i

whereλ, λi denote the corresponding free-space wavelengths (e.g.,λ = 2πc/ω) Inpractice, refractive index data are fitted to Eq (1.11.15) using 2–4 terms over a desiredfrequency range For example, fused silica (SiO2) is very accurately represented over therange 0.2≤ λ ≤3.7μm by the following formula [147], whereλandλiare in units of

ω2− ω2+ jωγ=

jω0ω2 p

We note thatσ(ω)/jωis essentially the electric susceptibility considered above.Indeed, we haveJ= Nev = Nejωx = jωP, and thus,P= J/jω = (σ(ω)/jω)E Itfollows that(ω)−0= σ(ω)/jω, and

(ω)= 0+ 0ω

2 p

ω2− ω2+ jωγ= 0+

σ(ω)

1.12 Conductors 21

Since in a metal the conduction charges are unbound, we may take ω0 = 0 in

Eq (1.12.1) After canceling a common factor ofjω, we obtain:

σ(ω)= 0ω

2 p

The model defined by (1.12.3) is know as the “Drude model.” The nominal

conduc-tivity is obtained at the low-frequency limit,ω=0:

σ=0ω

2 p

γ =Ne2

mγ (nominal conductivity) (1.12.4)

Example 1.12.1: Copper has a mass density of 8.9×106gr/m3and atomic weight of 63.54

(grams per mole.) Using Avogadro’s number of 6×1023atoms per mole, and assuming

one conduction electron per atom, we find for the volume densityN:

N=

6×1023 atoms

mole

63.54 grmole

8.9×106 gr

m3 1electronatom

of copper can be calculated by

which lies in the ultraviolet range For frequencies such thatω γ, the conductivity

(1.12.3) may be considered to be independent of frequency and equal to the dc value of

Eq (1.12.4) This frequency range covers most present-day RF applications For example,

assumingω≤0.1γ, we findf≤0.1γ/2π=653 GHz 

So far, we assumed sinusoidal time dependence and worked with the steady-state

responses Next, we discuss the transient dynamical response of a conductor subject to

an arbitrary time-varying electric fieldE(t)

Ohm’s law can be expressed either in the frequency-domain or in the time-domain

with the help of the Fourier transform pair of equations:

−∞σ(t− t)E(t)dt (1.12.5)whereσ(t)is the causal inverse Fourier transform ofσ(ω) For the simple model of

J(t)= t0

0ω2pe−γ(t−t)Edt=0ω

2 p

γ E 1− e−γt

= σE 1− e−γt

whereσ= 0ω2

p/γis the nominal conductivity of the material

Thus, the current starts out at zero and builds up to the steady-state value ofJ= σE,which is the conventional form of Ohm’s law The rise time constant isτ=1/γ Wesaw above thatτis extremely small—of the order of 10−14sec—for good conductors.The building up of the current can also be understood in terms of the equation ofmotion of the conducting charges Writing Eq (1.10.2) in terms of the velocity of thecharge, we have:

J= Nev∞=Ne2

mγE= σE

1.13 Charge Relaxation in Conductors

Next, we discuss the issue of charge relaxation in good conductors [138–141] Writing(1.12.5) three-dimensionally and using (1.12.6), Ohm’s law reads in the time domain:

J(r, t)= ω2

p t

−∞e

−γ(t−t )

0E(r, t) dt (1.13.1)Taking the divergence of both sides and using charge conservation,∇∇ ·J+ρ˙=0,and Gauss’s law,0∇∇ ·E= ρ, we obtain the following integro-differential equation forthe charge densityρ(r, t):

−ρ(˙ r, t)= ∇∇∇ ·J(r, t)= ω2

p t

−∞e

−γ(t−t )

0∇∇ ·E(r, t)dt= ω2

p t

−∞e

−γ(t−t )ρ(r, t)dt

Differentiating both sides with respect tot, we find thatρsatisfies the second-orderdifferential equation:

¨

ρ(r, t)+γρ(˙r, t)+ω2

pρ(r, t)=0 (1.13.2)whose solution is easily verified to be a linear combination of:

e−γt/2cos(ωrelaxt) , e−γt/2sin(ωrelaxt) , where ωrelax=



ω2p−γ2

4

1.14 Power Losses 23

Thus, the charge density is an exponentially decaying sinusoid with a relaxation time

constant that is twice the collision timeτ=1/γ:

τrelax= 2

γ=2τ (relaxation time constant) (1.13.3)Typically,ωp  γ, so thatωrelaxis practically equal toωp For example, using

the numerical data of Example 1.12.1, we find for copperτrelax =2τ=5×10−14sec.

We calculate also: frelax = ωrelax/2π =2.6×1015Hz In the limitγ→ ∞, orτ→ 0,

Eq (1.13.2) reduces to the naive relaxation equation (1.7.3) (see Problem 1.9)

In addition to charge relaxation, the total relaxation time depends on the time it takes

for the electric and magnetic fields to be extinguished from the inside of the conductor,

as well as the time it takes for the accumulated surface charge densities to settle, the

motion of the surface charges being damped because of ohmic losses Both of these

times depend on the geometry and size of the conductor [140]

1.14 Power Losses

To describe a material with both dielectric and conductivity properties, we may take the

susceptibility to be the sum of two terms, one describing bound polarized charges and

the other unbound conduction charges Assuming different parameters{ω0, ωp, γ}for

each term, we obtain the total permittivity:

(ω)= 0+ 0ω

2 dp

ω2d0− ω2+ jωγd+ 0ω

2 cp

Denoting the first two terms byd(ω)and the third byσc(ω)/jω, we obtain the

total effective permittivity of such a material:

(ω)= d(ω)+σc(ω)

jω (effective permittivity) (1.14.2)

In the low-frequency limit, ω= 0, the quantitiesd(0)andσc(0)represent the

nominal dielectric constant and conductivity of the material We note also that we can

write Eq (1.14.2) in the form:

These two terms characterize the relative importance of the conduction current and

the displacement (polarization) current The right-hand side in Amp`ere’s law gives the

total effective current:

Jtot= J +∂D

∂t = J + jωD = σc(ω)E+ jωd(ω)E= jω(ω)E

where the termJdisp= ∂D/∂t = jωd(ω)Erepresents the displacement current The

relative strength between conduction and displacement currents is the ratio:

Example 1.14.1: This ratio can take a very wide range of values For example, assuming afrequency of 1 GHz and using (for illustration purposes) the dc-values of the dielectricconstants and conductivities, we find:

109 for copper withσ=5.8×107S/m and= 0

1 for seawater withσ=4 S/m and=720

10−9 for a glass withσ=10−10S/m and=20

Thus, the ratio varies over 18 orders of magnitude! If the frequency is reduced by a factor

of ten to 100 MHz, then all the ratios get multiplied by 10 In this case, seawater acts like

The time-averaged ohmic power losses per unit volume within a lossy material aregiven by Eq (1.9.6) Writing(ω)= (ω)−j(ω), we have:

Jtot= jω(ω)E= jω(ω)E+ ω(ω)EDenotingE2

dV =12ωd(ω)tanθE2

(ohmic losses) (1.14.10)

1.15 Plasmas

To describe a collisionless plasma, such as the ionosphere, the simple model

consid-ered in the previous sections can be specialized by choosingω0 = γ =0 Thus, the

conductivity given by Eq (1.12.3) becomes pure imaginary:

σ(ω)=0ω

2 pjω

The corresponding effective permittivity of Eq (1.12.2) becomes purely real:

We will see in Sec 2.6 that the propagation wavenumber of an electromagnetic wave

propagating in a dielectric/conducting medium is given in terms of the effective

Ifω > ωp, the electromagnetic wave propagates without attenuation within the

plasma But ifω < ωp, the wavenumberk becomes imaginary and the wave gets

attenuated At such frequencies, a wave incident (normally) on the ionosphere from the

ground cannot penetrate and gets reflected back

1.16 Energy Density in Lossless Dispersive Dielectrics

The lossless case is obtained from Eq (1.11.5) by settingγ=0, which is equivalent to

assuming thatωis far from the resonanceω0 In this case the permittivity is:

(ω)= 0



1+ ω

2 p



(1.16.1)Thus, the electric part of the energy density (1.9.7) will be:



(1.16.2)This expression can be given a nice interpretation: The first term on the right is theenergy density in vacuum and the second corresponds to the mechanical (kinetic andpotential) energy of the polarization charges [154,177] Indeed, the displacementxandvelocityv=˙xof the polarization charges are in this case:



ω2

p(ω2+ ω2)(ω2− ω2)2

40|E|2+w¯mech=w¯vac+w¯mech (1.16.3)

1.17 Kramers-Kronig Dispersion Relations

The convolutional form of Eq (1.3.13) implies causality, that is, the value of D(r, t)atthe present timetdepends only on the past values of E(r, t),t≤ t

This condition is equivalent to requiring that the dielectric response(t)be a sided (causal) function of time, that is,(t)=0 fort < 0 Then, Eq (1.3.13) may bewritten as ordinary convolution by extending the integration range over all times:

1.17 Kramers-Kronig Dispersion Relations 27

In the frequency domain, this becomes multiplicative: P(r, ω)= 0χ(ω)E(r, ω)

The Kramers-Kronig relations are the frequency-domain expression of causality and

re-late the real and imaginary parts of the susceptibility functionχ(ω) Here, the functions

χ(t)andχ(ω)are Fourier transform pairs:

functionu(t)in the equivalent manner:

Using the property that the Fourier transform of a product of two time functions is

the convolution of their Fourier transforms, it follows that Eq (1.17.4) can be written in

the equivalent frequency-domain form:

whereU(ω)is the Fourier transform of the unit-step Eq (1.17.5) is essentially the

Kramers-Kronig relation The functionU(ω)is given by the well-known expression:

Rearranging terms and canceling a factor of 1/2, we obtain the Kramers-Kronig

re-lation in its complex-valued form:†

sufficiently fast for largeωto make the integral in (1.17.5) convergent, whereas(ω)

tends to the constant0

Settingχ(ω)= χr(ω)−jχi(ω)and separating (1.17.7) into its real and imaginary

parts, we obtain the conventional form of the Kramers-Kronig dispersion relations:

generating single-sideband communications signals.

Because the time-responseχ(t)is real-valued, its Fourier transformχ(ω)will isfy the Hermitian symmetry propertyχ(−ω)= χ∗(ω), which is equivalent to the evensymmetry of its real part,χr(−ω)= χr(ω), and the odd symmetry of its imaginary part,

sat-χi(−ω)= −χi(ω) Taking advantage of these symmetries, the range of integration in(1.17.8) can be folded in half resulting in:

There are several other ways to prove the Kramers-Kronig relations For example,

a more direct way is to state the causality condition in terms of the signum functionsign(t) Indeed, because u(t)= 1+sign(t)

/2, Eq (1.17.4) may be written in theequivalent formχ(t)= χ(t)sign(t) Then, Eq (1.17.7) follows by applying the samefrequency-domain convolution argument using the Fourier transform pair:

sign(t)  Pjω2 (1.17.10)Alternatively, the causality condition can be expressed asu(−t)χ(t)=0 This ap-proach is explored in Problem 1.12 Another proof is based on the analyticity properties

ofχ(ω) Because of the causality condition, the Fourier integral in (1.17.3) can be stricted to the time range 0< t <∞:

This implies thatχ(ω)can be analytically continued into the lower halfω-plane,

so that replacingωbyw= ω − jαwithα≥0 still gives a convergent Fourier integral

in Eq (1.17.11) Any singularities inχ(ω)lie in the upper-half plane For example, thesimple model of Eq (1.11.7) has poles atω= ±ω¯0+ jγ/2, where ¯ω0=ω2− γ2/4.Next, we consider a clockwise closed contourC= C+C∞consisting of the real axis

Cand an infinite semicircleC∞in the lower half-plane Becauseχ(ω)is analytic in theregion enclosed byC, Cauchy’s integral theorem implies that for any pointwenclosed

byC, that is, lying in the lower half-plane, we must have:

χ(w)= − 1

2πj

C

χ(w)

where the overall minus sign arises becauseCwas taken to be clockwise Assuming that

χ(ω)falls off sufficiently fast for largeω, the contribution of the infinite semicirclecan be ignored, thus leaving only the integral over the real axis Settingw= ω − jandtaking the limit→0+, we obtain the identical relationship to Eq (1.17.5):

1.18 Group Velocity, Energy Velocity 29

An interesting consequence of the Kramers-Kronig relations is that there cannot

exist a dielectric medium that is purely lossless, that is, such thatχi(ω)=0 for allω,

because this would also require thatχr(ω)=0 for allω

However, in all materials,χi(ω)is significantly non-zero only in the neighborhoods

of the medium’s resonant frequencies, as for example in Fig 1.11.1 In the frequency

bands that are sufficiently far from the resonant bands,χi(ω)may be assumed to be

essentially zero Such frequency bands are called transparency bands [153]

1.18 Group Velocity, Energy Velocity

Assuming a nonmagnetic material (μ= μ0), a complex-valued refractive index may be

nr− jni=1+ χr− jχi Upon squaring, this splits into the two real-valued equations

i + (1+ χr)

2

⎤

⎦1/2

ni=sign(χi)

⎡

⎣

(1+ χr)2+χ2

This form preserves the sign ofχi, that is,niandχiare both positive for absorbing

media, or both negative for gain media The following approximate solution is often

used, which can be justified whenever|χ| 1 (for example, in gases):

nr(ω)−jnr(ω)=1+ χ(ω) 1+χ2 ⇒ nr=1+12χr, ni=12χi (1.18.3)

We will see in Chap 2 that a single-frequency uniform plane wave propagating, say,

in the positivez-direction, has a wavenumberk= ωn/c = ω(nr− jni)/c≡ β − jα,

wherecis the speed of light in vacuum Therefore, the wave will have a space-time

dependence:

ej(ωt−kz)= ej(ωt−(β−jα)z)= e−αzej(ωt−βz)= e−ωn i z/cejω(t−nr z/c) (1.18.4)

The real partnrdefines the phase velocity of the wave,vp= ω/β = c/nr, whereas

the imaginary partni, orα= ωni/c, corresponds to attenuation or gain depending on

the sign ofniorχi

When several such plane waves are superimposed to form a propagating pulse, we

will see in Sec 3.5 that the peak of the pulse (i.e., the point on the pulse where all the

individual frequency components add up in phase), propagates with the so-called group

velocity defined by:

nr+ ωdnrdω

Within an anomalous dispersion region, nr is decreasing rapidly withω, that is,

dnr/dω <0, as in Fig 1.11.1 This results in a group velocityvg, given by Eq (1.18.5),that may be larger thancor even negative Such velocities are called “superluminal.”Light pulses propagating at superluminal group velocities are referred to as “fast light”and we discuss them further in Sec 3.9

Within a normal dispersion region (e.g., to the far left and far right of the resonantfrequencyω0in Fig 1.11.1),nris an increasing function ofω, dnr/dω > 0, whichresults invg < c In specially engineered materials such as those exhibiting “electro-magnetically induced transparency,” the slopednr/dωmay be made so steep that theresulting group velocityvgbecomes extremely small,vg  c This is referred to as

“slow light.”

We close this section by showing that for lossless dispersive media, the energy locity of a plane wave is equal to the group velocity defined by (1.18.5) This result isquite general, regardless of the frequency dependence of(ω)andμ(ω)(as long asthese quantities are real.)

ve-We will see in the next chapter that a plane wave propagating along thez-directionhas electric and magnetic fields that are transverse to thez-direction and are related by:

|H| =1

η|E| , η =

μ

η2

d(ωμ)dω



|E|2The energy velocity is defined byven=P¯z/w¯ Thus, we have:

d(ωμ)dω

d(ωμ)dω

Eq (1.18.7) is also valid for the case of lossless negative-index media and implies thatthe group velocity, and hence the group refractive indexng= c0/vg, will be positive,

1.19 Problems 31

even though the refractive indexnis negative Writing= −||andμ= −|μ|in this

case and noting thatη=|μ|/||andn= −|μ|/√0μ0, andk= ωn/c0, we have:

from which we also obtain the usual relationshipng= d(ωn)/dω The positivity of

vgandngfollows from the positivity of the derivativesd(ω)/dωandd(ωμ)/dω, as

required to keepvenpositive in negative-index media [376]

A=ˆn×A×nˆ+ (ˆn·A)ˆn (ˆn is any unit vector)

In the last identity, does it a make a difference whether ˆn×A×n is taken to mean ˆˆ n×(A×nˆ)

1.3 Consider the infinitesimal volume elementΔxΔyΔzshown below, such that its upper half

lies in medium1and its lower half in medium2 The axes are oriented such that ˆn=ˆz.

Applying the integrated form of Amp`ere’s law to the infinitesimal faceabcd, show that

H2y− H1y= JxΔz+∂Dx

∂t Δz

In the limitΔz→0, the second term in the right-hand side may be assumed to go to zero,

whereas the first term will be non-zero and may be set equal to a surface current density,

that is,Jsx≡limΔz→0(JxΔz) Show that this leads to the boundary conditionH1y− H2y=

−Jsx Similarly, show thatH1x− H2x= Jsy, and that these two boundary conditions can be

combined vectorially into Eq (1.5.4)

2Re

A×B∗ , AA(t)·BBB(t) =1

2Re

A·B∗

1.5 Assuming that B= μH, show that Maxwell’s equations (1.9.2) imply the following

complex-valued version of Poynting’s theorem:

∇∇ · (E×H∗)= −jωμH·H∗−E·Jtot∗, where Jtot=J+ jωD

Extracting the real-parts of both sides and integrating over a volumeVbounded by a closedsurfaceS, show the time-averaged form of energy conservation:

x):

1.19 Problems 33

Tx= ExE+ μHxH−ˆx1

2(E2+ μH2)

Write similar equations of they, zcomponents The quantityGxis interpreted as the field

momentum (in thex-direction) per unit volume, that is, the momentum density

1.7 Show that the causal and stable time-domain dielectric response corresponding to Eq (1.11.5)

is given as follows:

(t)= 0δ(t)+0χ(t) , χ(t)=ω

2 p

¯

ω0

e−γt/2sin(ω¯0t)u(t) (1.19.2)

whereu(t)is the unit-step function and ¯ω0 =ω2− γ2/4, and we must assume that

γ <2ω0, as is typically the case in practice Discuss the solution for the caseγ/2> ω0

1.8 Show that the plasma frequency for electrons can be expressed in the simple numerical form:

fp=9√

N, wherefpis in Hz andNis the electron density in electrons/m3 What isfpfor

the ionosphere ifN=1012? [Ans 9 MHz.]

1.9 Show that the relaxation equation (1.13.2) can be written in the following form in terms of

the dc-conductivityσdefined by Eq (1.12.4):

1

γρ(¨r, t)+ρ(˙r, t)+σ

0

ρ(r, t)=0Then, show that it reduces to the naive relaxation equation (1.7.3) in the limitτ=1/γ→0

Show also that in this limit, Ohm’s law (1.13.1) takes the instantaneous form J= σE, from

which the naive relaxation constantτrelax= 0/σwas derived

1.10 Conductors and plasmas exhibit anisotropic and birefringent behavior when they are in the

presence of an external magnetic field The equation of motion of conduction electrons in

a constant external magnetic field ism˙= e(E+v×B)−mγv, with the collisional term

included Assume the magnetic field is in thez-direction, B=ˆzB, and that E=ˆxEx+ˆyEy

What is the cyclotron frequency in Hz for electrons in the Earth’s magnetic fieldB=

0.4 gauss=0.4×10−4Tesla? [Ans 1.12 MHz.]

b To solve this system, work with the combinationsvx± jvy Assuming harmonic

time-dependence, show that the solution is:

whereσ±(t)= γσ0e−γte∓jωB tu(t)is the inverse Fourier transform ofσ±(ω)and

u(t)is the unit-step function

e Rewrite part (d) in component form:

and identify the quantitiesσxx(t), σxy(t), σyx(t), σyy(t)

f Evaluate part (e) in the special caseEx(t)= Exu(t)andEy(t)= Eyu(t), whereEx,Ey

are constants, and show that after a long time the steady-state version of part (e) willbe:

What is the numerical value ofbfor electrons in copper ifBis 1 gauss? [Ans 4.3×10−7.]

g For a collisionless plasma (γ=0), show that its dielectric behavior is determined from

ω(ω± ωB)



whereωpis the plasma frequency Thus, the plasma exhibits birefringence.1.11 This problem deals with various properties of the Kramers-Kronig dispersion relations forthe electric susceptibility, given by Eq (1.17.8)

a Using the symmetry propertiesχr(ω)= χr(−ω)andχi(ω)= −χi(−ω), show that(1.17.8) can be written in the folded form of Eq (1.17.9)

b Using the definition of principal-value integrals, show the following integral:

P ∞

0

dω

ω2− ω2=0 (1.19.3)Hint : You may use the following indefinite integral: dx

a2− x2= 1

2aln



aa+ x− x

ω2− ω2 dω

(1.19.4)

Hint : You will need to argue that the integrands have no singularity atω= ω

d For a simple oscillator model of dielectric polarization, the susceptibility is given by:

χ(ω)= χr(ω)−jχi(ω)= ω

2 p

ω2− ω2+ jγω

2

p(ω2− ω2)(ω2− ω2)2+γ2ω2− j γωω

2 p

(ω2− ω2)2+γ2ω2

(1.19.5)

Show that for this model the quantitiesχr(ω)andχi(ω)satisfy the modified Kronig relationships (1.19.4) Hint : You may use the following definite integrals, forwhich you may assume that 0< γ <2ω0:

Kramers-2

π

∞ 0

dx(ω2− x2)2+γ2x2= 1

γω2, 2

π

∞ 0

x2dx(ω2− x2)2+γ2x2=1γ

Indeed, show that these integrals may be reduced to the following ones, which can befound in standard tables of integrals:

2

π

∞ 0

e Consider the limit of Eq (1.19.5) asγ→0 Show that in this case the functionsχr, χi

are given as follows, and that they still satisfy the Kramers-Kronig relations:

χr(ω)=P ω2p

ω0− ω+P ω2p

ω0+ ω, χi(ω)=πω

2 p

2ω0

δ(ω− ω0)−δ(ω + ω0)

1.12 Derive the Kramers-Kronig relationship of Eq (1.17.7) by starting with the causality tionχ(t)u(−t)=0 and translating it to the frequency domain, that is, expressing it as theconvolution of the Fourier transforms ofχ(t)andu(−t)

condi-1.13 An isotropic homogeneous lossless dielectric medium is moving with uniform velocity v with

respect to a fixed coordinate frameS In the frameSmoving with dielectric, the constitutive

relations are assumed to be the usual ones, that is, D= Eand B= μH Using the Lorentztransformations given in Eq (H.30) of Appendix H, show that the constitutive relations takethe following form in the fixed frameS:

D= E+ av×(H− v×E) , B= μH− av×(E+ μv×H) , a=μ− 0μ0

1− μv2

2 Uniform Plane Waves

2.1 Uniform Plane Waves in Lossless Media

The simplest electromagnetic waves are uniform plane waves propagating along some

fixed direction, say thez-direction, in a lossless medium{, μ}

The assumption of uniformity means that the fields have no dependence on the

transverse coordinatesx, yand are functions only ofz, t Thus, we look for solutions

of Maxwell’s equations of the form: E(x, y, z, t)=E(z, t)and H(x, y, z, t)=H(z, t)

Because there is no dependence onx, y, we set the partial derivatives†∂x=0 and

∂y=0 Then, the gradient, divergence, and curl operations take the simplified forms:

An immediate consequence of uniformity is that E and H do not have components

along thez-direction, that is,Ez = Hz =0 Taking the dot-product of Amp`ere’s law

with the unit vector ˆz, and using the identity ˆ z· (ˆz×A)=0, we have:

Because also∂zEz =0, it follows thatEzmust be a constant, independent ofz, t.Excluding static solutions, we may take this constant to be zero Similarly, we have

Hz=0 Thus, the fields have components only along thex, ydirections:

Now all the terms have the same dimension Eqs (2.1.5) imply that both E and H

satisfy the one-dimensional wave equation Indeed, differentiating the first equationwith respect tozand using the second, we have:

∂2E

∂z2 = −1c

and similarly for H Rather than solving the wave equation, we prefer to work directly

with the coupled system (2.1.5) The system can be decoupled by introducing the called forward and backward electric fields defined as the linear combinations:

so-E+=12(E+ ηH׈z)

E−=12(E− ηH׈z)

(forward and backward fields) (2.1.7)

38 2 Uniform Plane Waves

Component-wise, these are:

Ex ±=12(Ex± ηHy) , Ey ±=12(Ey∓ ηHx) (2.1.8)

We show next that E+(z, t)corresponds to a forward-moving wave, that is, moving

towards the positivez-direction, and E−(z, t), to a backward-moving wave Eqs (2.1.7)

can be inverted to express E,H in terms of E+,E− Adding and subtracting them, and

using the BAC-CAB rule and the orthogonality conditions ˆz·E±=0, we obtain:

E(z, t)=E+(z, t)+E−(z, t)

H(z, t)= 1

ηˆz×E+(z, t)−E−(z, t) (2.1.9)

In terms of the forward and backward fields E±, the system of Eqs (2.1.5) decouples

into two separate equations:

∂E+

∂z = −1c

∂E+

∂t

∂E−

∂z = +1c

∂

∂t(E± ηH׈z)

Eqs (2.1.10) can be solved by noting that the forward field E+(z, t)must depend

onz, tonly through the combinationz− ct(for a proof, see Problem 2.1.) If we set

E+(z, t)=F(z− ct), where F(ζ)is an arbitrary function of its argumentζ= z − ct,

then we will have:

∂E+

∂t

Vectorially, F must have onlyx, ycomponents, F=ˆxFx+yˆFy, that is, it must be

transverse to the propagation direction, ˆz·F=0

Similarly, we find from the second of Eqs (2.1.10) that E−(z, t)must depend onz, t

through the combinationz+ ct, so that E−(z, t)=G(z+ ct), where G(ξ)is an arbitrary

(transverse) function ofξ= z + ct In conclusion, the most general solutions for the

forward and backward fields of Eqs (2.1.10) are:

E+(z, t)=F(z− ct)

with arbitrary functions F and G, such that ˆ z·F=ˆz·G=0

Inserting these into the inverse formula (2.1.9), we obtain the most general solution

of (2.1.5), expressed as a linear combination of forward and backward waves:

To see this, consider the forward field at a later timet+ Δt During the time interval

Δt, the wave moves in the positivez-direction by a distanceΔz= cΔt Indeed, we have:

t− Δt = t − Δz/c, that is,

E+(z+ Δz, t)=E+(z, t− Δt)

Similarly, we find that E−(z, t+ Δt)=E−(z+ Δz, t), which states that the backwardfield at timet+ Δtis the same as the field at timet, translated to the left by a distance

Δz Fig 2.1.1 depicts these two cases

Fig 2.1.1 Forward and backward waves.

The two special cases corresponding to forward waves only(G=0), or to backwardones(F=0), are of particular interest For the forward case, we have:

E(z, t)=F(z− ct)

H(z, t)=η1ˆz×F(z− ct)=η1ˆz×E(z, t)

(2.1.13)

40 2 Uniform Plane Waves

This solution has the following properties: (a) The field vectors E and H are

perpen-dicular to each other, E·H=0, while they are transverse to thez-direction, (b) The

three vectors{E,H,ˆz}form a right-handed vector system as shown in the figure, in the

sense that E×H points in the direction of ˆ z, (c) The ratio of E to H׈z is independent

ofz, tand equals the characteristic impedanceηof the propagation medium; indeed:

H(z, t)= 1

ηˆz×E(z, t) ⇒ E(z, t)= ηH(z, t)׈z (2.1.14)The electromagnetic energy of such forward wave flows in the positivez-direction

With the help of the BAC-CAB rule, we find for the Poynting vector:

P

P =E×H=ˆz1

η|F|2= cˆz|F|2 (2.1.15)where we denoted|F|2=F·F and replaced 1/η= c The electric and magnetic energy

densities (per unit volume) turn out to be equal to each other Because ˆz and F are

mutually orthogonal, we have for the cross product|ˆz×F| = |ˆz||F| = |F| Then,

w= we+ wm=2we= |F|2 (2.1.16)

In accordance with the flux/density relationship of Eq (1.6.2), the transport velocity

of the electromagnetic energy is found to be:

v=P

w =cˆz|F|2

|F|2 = cˆz

As expected, the energy of the forward-moving wave is being transported at a speed

calong the positivez-direction Similar results can be derived for the backward-moving

solution that has F=0 and G=0 The fields are now:

The Poynting vector becomesPP =E×H= −cˆz|G|2and points in the negative

z-direction, that is, the propagation direction The energy transport velocity is v= −cˆz.

Now, the vectors{E,H,−ˆz}form a right-handed system, as shown The ratio ofEtoH

is still equal toη, provided we replace ˆz with−ˆz:

H(z, t)=1

η(−ˆz)×E(z, t) ⇒ E(z, t)= ηH(z, t)×(−ˆz)

In the general case of Eq (2.1.12), theE/Hratio does not remain constant ThePoynting vector and energy density consist of a part due to the forward wave and a partdue to the backward one:

Example 2.1.1: A source located atz=0 generates an electric field E(0, t)=ˆxE0u(t), where

u(t)is the unit-step function, andE0, a constant The field is launched towards the positive

z-direction Determine expressions for E(z, t)and H(z, t)

Solution: For a forward-moving wave, we have E(z, t)=F(z− ct)=F

Because of the unit-step, the non-zero values of the fields are restricted tot− z/c ≥0, or,

z≤ ct, that is, at timetthe wavefront has propagated only up to positionz= ct Thefigure shows the expanding wavefronts at timetandt+ Δt

Example 2.1.2: Consider the following three examples of electric fields specified att=0, anddescribing forward or backward fields as indicated:

E(z,0)=ˆxE0cos(kz) (forward-moving)

E(z,0)=yˆE0cos(kz) (backward-moving)

E(z,0)=ˆxE1cos(k1z)+yˆE2cos(k2z) (forward-moving)wherek, k1, k2are given wavenumbers (measured in units of radians/m.) Determine the

corresponding fields E(z, t)and H(z, t)

Solution: For the forward-moving cases, we replacezbyz− ct, and for the backward-movingcase, byz+ ct We find in the three cases:

42 2 Uniform Plane Waves

The first two cases are single-frequency waves, and are discussed in more detail in the

next section The third case is a linear superposition of two waves with two different

2.2 Monochromatic Waves

Uniform, single-frequency, plane waves propagating in a lossless medium are obtained

as a special case of the previous section by assuming the harmonic time-dependence:

E(x, y, z, t)=E(z)ejωt

where E(z)and H(z)are transverse with respect to thez-direction

Maxwell’s equations (2.1.5), or those of the decoupled system (2.1.10), may be solved

very easily by replacing time derivatives by∂t → jω Then, Eqs (2.1.10) become the

first-order differential equations (see also Problem 2.3):

∂E±(z)

∂z = ∓jkE±(z) , where k=ω

c = ω√μ (2.2.2)with solutions:

E+(z)=E0+ −jkz (forward)

E−(z)=E0 − jkz (backward)

(2.2.3)

where E0±are arbitrary (complex-valued) constant vectors such that ˆz·E0±=0 The

corresponding magnetic fields are:

Inserting (2.2.3) into (2.1.9), we obtain the general solution for single-frequency

waves, expressed as a superposition of forward and backward components:

E(z)=E0 + −jkz+E0 − jkz

H(z)=1

ηˆz×E0 + −jkz−E0 − jkz (forward+backward waves) (2.2.6)

Setting E0 ±=ˆxA±+ˆyB±, and noting that ˆz×E0 ±=ˆz×(ˆxA±+ˆyB±)=ˆyA±−ˆxB±,

we may rewrite (2.2.6) in terms of its cartesian components:

Ex(z)= A+ −jkz+ A− jkz, Ey(z)= B+ −jkz+ B− jkz

Hy(z)=η1A+ −jkz− A− jkz

, Hx(z)= −η1B+ −jkz− B− jkz (2.2.7)

Wavefronts are defined, in general, to be the surfaces of constant phase A forward

moving wave E(z)=E0e−jkzcorresponds to the time-varying field:

E(z, t)=E0ejωt−jkz=E0e−jϕ(z,t), where ϕ(z, t)= kz − ωt

A surface of constant phase is obtained by settingϕ(z, t)=const Denoting thisconstant byφ0= kz0and using the propertyc= ω/k, we obtain the condition:

ϕ(z, t)= ϕ0 ⇒ kz − ωt = kz0 ⇒ z = ct + z0Thus, the wavefront is thexy-plane intersecting thez-axis at the pointz= ct + z0,moving forward with velocityc This justifies the term “plane wave.”

A backward-moving wave will have planar wavefronts parametrized byz= −ct+z0,that is, moving backwards A wave that is a linear combination of forward and backwardcomponents, may be thought of as having two planar wavefronts, one moving forward,and the other backward

The relationships (2.2.5) imply that the vectors{E0 +,H0 +,ˆz}and{E0 −,H0 −,−ˆz}will

form right-handed orthogonal systems The magnetic field H0±is perpendicular to the

electric field E0 ±and the cross-product E0 ±×H0 ±points towards the direction of agation, that is,±ˆz Fig 2.2.1 depicts the case of a forward propagating wave.

prop-Fig 2.2.1 Forward uniform plane wave.

The wavelengthλis the distance by which the phase of the sinusoidal wave changes

by 2πradians Since the propagation factore−jkzaccumulates a phase ofkradians permeter, we have by definition thatkλ=2π The wavelengthλcan be expressed via thefrequency of the wave in Hertz,f= ω/2π, as follows:

44 2 Uniform Plane Waves

scale factorncompared to the free-space values, whereas the wavenumberkis increased

by a factor ofn Indeed, using the definitionsc=1/√μ

Example 2.2.1: A microwave transmitter operating at the carrier frequency of 6 GHz is

pro-tected by a Plexiglas radome whose permittivity is=30

The refractive index of the radome isn=/0=√3=1.73 The free-space wavelength

and the wavelength inside the radome material are:

We will see later that if the radome is to be transparent to the wave, its thickness must be

chosen to be equal to one-half wavelength,l= λ/2 Thus,l=2.9/2=1.45 cm

Example 2.2.2: The nominal speed of light in vacuum isc0=3×108m/s Because of the

rela-tionshipc0= λf, it may be expressed in the following suggestive units that are appropriate

in different application contexts:

500 nm×600 THz (visible spectrum)

100 nm×3000 THz (UV)Similarly, in terms of length/time of propagation:

c0 = 36 000 km/120 msec (geosynchronous satellites)

300 km/msec (power lines)

300 m/μsec (transmission lines)

30 cm/nsec (circuit boards)The typical half-wave monopole antenna (half of a half-wave dipole over a ground plane)

has lengthλ/4 and is used in many applications, such as AM, FM, and cell phones Thus,

one can predict that the lengths of AM radio, FM radio, and cell phone antennas will be of

the order of 75 m, 0.75 m, and 7.5 cm, respectively

A more detailed list of electromagnetic frequency bands is given in Appendix B The precise

value ofc0and the values of other physical constants are given in Appendix A

Wave propagation effects become important, and cannot be ignored, whenever the

physical length of propagation is comparable to the wavelengthλ It follows from

Eqs (2.2.2) that the incremental change of a forward-moving electric field in propagating

Thus, the change in the electric field can be ignored only ifΔz λ, otherwise, gation effects must be taken into account

propa-For example, for an integrated circuit operating at 10 GHz, we haveλ=3 cm, which

is comparable to the physical dimensions of the circuit

Similarly, a cellular base station antenna is connected to the transmitter circuits byseveral meters of coaxial cable For a 1-GHz system, the wavelength is 0.3 m, whichimplies that a 30-meter cable will be equivalent to 100 wavelengths

2.3 Energy Density and Flux

The time-averaged energy density and flux of a uniform plane wave can be determined

by Eq (1.9.6) As in the previous section, the energy is shared equally by the electricand magnetic fields (in the forward or backward cases.) This is a general result for mostwave propagation and waveguide problems

The energy flux will be in the direction of propagation For either a forward- or abackward-moving wave, we have from Eqs (1.9.6) and (2.2.5):

w= we+ wm=2we=1

2|E0 ±|2 (2.3.1)For the time-averaged Poynting vector, we have similarly:

ing energy velocity is, as in the previous section:

com-46 2 Uniform Plane Waves

2.4 Wave Impedance

For forward or backward fields, the ratio of E(z)to H(z)׈z is constant and equal to

the characteristic impedance of the medium Indeed, it follows from Eq (2.2.4) that

E±(z)= ±ηH±(z)׈z

However, this property is not true for the more general solution given by Eqs (2.2.6)

In general, the ratio of E(z)to H(z)׈z is called the wave impedance Because of the

vectorial character of the fields, we must define the ratio in terms of the corresponding

x- andy-components:

Zx(z)=



E(z)x

Thus, the wave impedances are nontrivial functions ofz For forward waves (that is,

withA−= B−=0), we haveZx(z)= Zy(z)= η For backward waves (A+= B+=0), we

haveZx(z)= Zy(z)= −η

The wave impedance is a very useful concept in the subject of multiple dielectric

interfaces and the matching of transmission lines We will explore its use later on

2.5 Polarization

Consider a forward-moving wave and let E0=ˆxA++yˆB+be its complex-valued

pha-sor amplitude, so that E(z)=E0e−jkz= (ˆxA++ˆyB+)e−jkz The time-varying field is

obtained by restoring the factorejωt:

E(z, t)= (ˆxA++yˆB+)ejωt−jkz

The polarization of a plane wave is defined to be the direction of the electric field

For example, ifB+=0, theE-field is along thex-direction and the wave will be linearly

polarized

More precisely, polarization is the direction of the time-varying real-valued field

EE(z, t)= Re

E(z, t)] At any fixed pointz, the vectorEE(z, t)may be along a fixed

linear direction or it may be rotating as a function oft, tracing a circle or an ellipse

The polarization properties of the plane wave are determined by the relative tudes and phases of the complex-valued constantsA+, B+ Writing them in their polarformsA+= Aejφ aandB+= Bejφ b, whereA, Bare positive magnitudes, we obtain:

magni-E(z, t)= ˆxAejφa+ˆyBejφb ejωt−jkz=ˆxAej(ωt−kz+φa )+ˆyBej(ωt−kz+φb ) (2.5.1)Extracting real parts and setting EE(z, t)=Re

E(z, t)

=ˆxEx(z, t)+ˆyEy(z, t), wefind the corresponding real-valuedx, ycomponents:

Ex(t)= Acosωtcosφa−sinωtsinφa

Ey(t)= Bcosωtcosφb−sinωtsinφb

Solving for cosωtand sinωtin terms ofEx(t),Ey(t), we find:

 Ey(t)

B sinφa−Ex(t)

A sinφb

2+ Ey(t)

A2 +E

2 y

B2 −2 cosφExEy

AB =sin2φ (polarization ellipse) (2.5.4)Depending on the values of the three quantities{A, B, φ}this polarization ellipsemay be an ellipse, a circle, or a straight line The electric field is accordingly calledelliptically, circularly, or linearly polarized

48 2 Uniform Plane Waves

To get linear polarization, we setφ=0 orφ= π, corresponding toφa= φb=0,

orφa=0, φb= −π, so that the phasor amplitudes are E0=ˆxA±ˆyB Then, Eq (2.5.4)

B2 ∓2ExEy

A ∓EyB

To get circular polarization, we setA= Bandφ= ±π/2 In this case, the

polariza-tion ellipse becomes the equapolariza-tion of a circle:

E2 x

A2 +E

2 y

A2 =1The sense of rotation, in conjunction with the direction of propagation, defines left-

circular versus right-circular polarization For the case,φa=0 andφb = −π/2, we

haveφ= φa− φb= π/2 and complex amplitude E0= A(ˆx− jˆy) Then,

Ex(t)= Acosωt

Ey(t)= Acos(ωt− π/2)= Asinωt

Thus, the tip of the electric field vector rotates counterclockwise on thexy-plane

To decide whether this represents right or left circular polarization, we use the IEEE

convention [115], which is as follows

Curl the fingers of your left and right hands into a fist and point both thumbs towards

the direction of propagation If the fingers of your right (left) hand are curling in the

direction of rotation of the electric field, then the polarization is right (left) polarized.†

Thus, in the present example, because we had a forward-moving field and the field is

turning counterclockwise, the polarization will be right-circular If the field were moving

backwards, then it would be left-circular For the case,φ= −π/2, arising fromφa=0

andφb= π/2, we have complex amplitude E0= A(ˆx+ jyˆ) Then, Eq (2.5.3) becomes:

Ex(t)= Acosωt

Ey(t)= Acos(ωt+ π/2)= −Asinωt

The tip of the electric field vector rotates clockwise on thexy-plane Since the wave

is moving forward, this will represent left-circular polarization Fig 2.5.1 depicts thefour cases of left/right polarization with forward/backward waves

Fig 2.5.1 Left and right circular polarizations.

To summarize, the electric field of a circularly polarized uniform plane wave will be,

in its phasor form:

A2 +E

2 y

B2 =1

Finally, ifA= Bandφis arbitrary, then the major/minor axes of the ellipse (2.5.4)will be rotated relative to thex, ydirections Fig 2.5.2 illustrates the general case

50 2 Uniform Plane Waves

Fig 2.5.2 General polarization ellipse.

It can be shown (see Problem 2.15) that the tilt angleθis given by:

tan 2θ= 2AB

A2− B2 cosφ (2.5.5)The ellipse semi-axesA, B, that is, the lengths OC and OD, are given by:

(2.5.6)

wheres=sign(A− B) These results are obtained by defining the rotated coordinate

system of the ellipse axes:

A2+E2y

The polarization ellipse is bounded by the rectangle with sides at the end-points

±A, ±B, as shown in the figure To decide whether the elliptic polarization is left- or

right-handed, we may use the same rules depicted in Fig 2.5.1

The angleχsubtended by the major to minor ellipse axes shown in Fig 2.5.2 is given

as follows and is discussed further in Problem 2.15:

52 2 Uniform Plane Waves

type are given below:

case A B φ A B θ rotation polarization

In the linear case (b), the polarization ellipse collapses along itsA-axis (A = 0) and

becomes a straight line along itsB-axis The tilt angleθstill measures the angle of theA

axis from thex-axis The actual direction of the electric field will be 90o−36.87o=53.13o,

which is equal to the slope angle, atan(B/A)=atan(4/3)=53.13o

In case (c), the ellipse collapses along itsB-axis Therefore,θcoincides with the angle of

the slope of the electric field vector, that is, atan(−B/A)=atan(−3/4)= −36.87o

With the understanding thatθalways represents the slope of theA-axis (whether

collapsed or not, major or minor), Eqs (2.5.5) and (2.5.6) correctly calculate all the special

cases, except whenA= B, which has tilt angle and semi-axes:

θ=45o, A= A1+cosφ , B= A1−cosφ (2.5.10)The MATLAB functionellipse.m calculates the ellipse semi-axes and tilt angle,A,

B,θ, given the parametersA,B,φ It has usage:

[a,b,th] = ellipse(A,B,phi) % polarization ellipse parameters

For example, the function will return the values of theA, B, θcolumns of the

pre-vious example, if it is called with the inputs:

phi = [-90, 0, 180, 60, 45, -45, 135, -135]’;

To determine quickly the sense of rotation around the polarization ellipse, we use

the rule that the rotation will be counterclockwise if the phase differenceφ= φa− φb

is such that sinφ >0, and clockwise, if sinφ <0 This can be seen by considering the

electric field at timet=0 and at a neighboring timet Using Eq (2.5.3), we have:

E

E(0)=ˆxAcosφa+yˆBcosφb

E

E(t) =ˆxAcos(ωt+ φa)+ˆyBcos(ωt+ φb)

The sense of rotation may be determined from the cross-product EE(0)×EEE(t) If

the rotation is counterclockwise, this vector will point towards the positivez-direction,

and otherwise, it will point towards the negativez-direction It follows easily that:

EE(0)×EEE(t)=ˆzABsinφsinωt (2.5.11)

Thus, fortsmall and positive (such that sinωt > 0), the direction of the vector

EE(0)×EEE(t)is determined by the sign of sinφ

2.6 Uniform Plane Waves in Lossy Media

We saw in Sec 1.14 that power losses may arise because of conduction and/or materialpolarization A wave propagating in a lossy medium will set up a conduction current

Jcond = σE and a displacement (polarization) current Jdisp = jωD = jωdE Both

currents will cause ohmic losses The total current is the sum:

Jtot=Jcond+Jdisp= (σ + jωd)E= jωcE

wherecis the effective complex dielectric constant introduced in Eq (1.14.2):

jωc= σ + jωd ⇒ c= d− j σ

The quantitiesσ, dmay be complex-valued and frequency-dependent However, wewill assume that over the desired frequency band of interest, the conductivityσis real-valued; the permittivity of the dielectric may be assumed to be complex,d= 

d− j

d.Thus, the effectivechas real and imaginary parts:

The assumption of uniformity (∂x= ∂y=0), will imply again that the fields E,H are

transverse to the direction ˆz Then, Faraday’s and Amp`ere’s equations become:

c

(2.6.5)

They correspond to the usual definitionsk = ω/c = ωμandη =μ/ withthe replacement→ c Noting thatωμ= kcηcandωc = kc/ηc, Eqs (2.6.4) may

54 2 Uniform Plane Waves

be written in the following form (using the orthogonality property ˆz·E= 0 and the

BAC-CAB rule on the first equation):

and backward components:

andη→ ηc The lossless case is obtained in the limit of a purely real-valuedc

Becausekcis complex-valued, we define the phase and attenuation constantsβand

αas the real and imaginary parts ofkc, that is,

kc= β − jα = ωμ(− j) (2.6.12)

We may also define a complex refractive indexnc= kc/k0that measureskcrelative

to its free-space valuek0= ω/c0= ω√μ00 For a non-magnetic medium, we have:

tinction coefficient andnr, the refractive index Another commonly used notation is the

propagation constantγdefined by:

It follows fromγ = α + jβ = jkc = jk0nc = jk0(nr− jni)thatβ = k0nrand

α= k0ni The nomenclature about phase and attenuation constants has its origins inthe propagation factore−jkc z We can write it in the alternative forms:

e−jkc z= e−γz= e−αze−jβz= e−k 0 n i ze−jk0 n r z (2.6.15)Thus, the wave amplitudes attenuate exponentially with the factore−αz, and oscillatewith the phase factore−jβz The energy of the wave attenuates by the factore−2αz, ascan be seen by computing the Poynting vector Becausee−jkc zis no longer a pure phasefactor andηcis not real, we have for the forward-moving wave of Eq (2.6.11):

Eq (2.6.16), we have for the dB attenuation atz, relative toz=0:

AdB(z)= −10 log10

... data-page="19">

2 Uniform Plane Waves< /b>

2.1 Uniform Plane Waves in Lossless Media

The simplest electromagnetic waves are uniform plane waves propagating along some... frequency

bands that are sufficiently far from the resonant bands,χi(ω)may be assumed to be

essentially zero Such frequency bands are called transparency bands [153]

1.18... indexnis negative Writing= −| |and? ?= −|μ|in this

case and noting thatη=|μ|/||andn= −|μ|/√0μ0, andk= ωn/c0,