electromagnetic wave propagation lecture 2: uniform plane ...€¦ · electromagnetic wave...

Electromagnetic Wave PropagationLecture 2: Uniform plane waves

Daniel Sjoberg

Department of Electrical and Information Technology

March 25, 2010

Outline

1 Plane waves in lossless mediaGeneral time dependenceTime harmonic waves

2 Polarization

3 Propagation in lossy media

4 Oblique propagation and complex waves

5 Paraxial approximation: beams

6 Doppler effect and negative index media

7 Conclusions

Daniel Sjoberg, Department of Electrical and Information Technology

Plane electromagnetic waves

In this lecture, we dive a bit deeper into the familiar right-handrule.

This is the building block for most of the course.


Outline


2 Polarization





7 Conclusions


Propagation in source free, isotropic, non-dispersive media

The electromagnetic field is assumed to depend only on onecoordinate, z. This implies

E(x, y, z, t) = E(z, t)D(x, y, z, t) = εE(z, t)H(x, y, z, t) = H(z, t)B(x, y, z, t) = µH(z, t)

and by using ∇ → z ∂∂z we have

∇×E = −∂B∂t

z × ∂E

∂z= −µ∂H

∂t

∇×H =∂D

∂t=⇒ z × ∂H

∂z= ε

∂E

∂t

∇ ·D = 0∂Ez∂z

= 0 ⇒ Ez = 0

∇ ·B = 0∂Hz

∂z= 0 ⇒ Hz = 0


Rewriting the equations

The electromagnetic field has only x and y components, and satisfy

z × ∂E

∂z= −µ∂H

∂t

z × ∂H

∂z= ε

∂E

∂t

Using (z × F )× z = F for any vector F orthogonal to z, and

c = 1/√εµ and η =

√µ/ε , we can write this as

∂

∂zE = −1

c

∂

∂t(ηH × z)

∂

∂z(ηH × z) = −1

c

∂

∂tE

which is a symmetric hyperbolic system

∂

∂z

(E

ηH × z

)= −1

c

∂

∂t

(0 11 0

)(E

ηH × z

)Daniel Sjoberg, Department of Electrical and Information Technology

Wave splitting

We could eliminate the magnetic field to find the wave equation(∂2

∂z2− 1c2∂2

∂t2

)E(z, t) = 0

but it is often more convenient to keep the system formulation.The change of variables(

E+

E−

)=

12

(E + ηH × zE − ηH × z

)=

12

(1 11 −1

)(E

ηH × z

)is called a wave splitting and diagonalizes the system to

∂E+

∂z= −1

c

∂E+

∂t⇒ E+(z, t) = F (z − ct)

∂E−∂z

= +1c

∂E−∂t

⇒ E−(z, t) = G(z + ct)


Forward and backward waves

The total fields can now be written

E(z, t) = F (z − ct) +G(z + ct)

H(z, t) =1ηz ×

(F (z − ct)−G(z + ct)

)A graphical interpretation of the forward and backward waves is


Energy density in one single wave

A wave propagating in positive z direction satisfies H = 1η z ×E

x: Electric fieldy: Magnetic fieldz: Propagation direction

The Poynting vector and energy densities are (using the BAC-CABrule A× (B ×C) = B(A ·C)−C(A ·B))

P = E ×H =1ηE × (z ×E) =

1ηz|E|2

we =12ε|E|2

wm =12µ|H|2 =

12µ

1η2|z ×E|2 =

12ε|E|2 = we

The transport velocity of electromagnetic energy is then

v =P

we + wm=

1ηz

1ε

= cz


Energy density in forward and backward wave

If the wave propagates in negative z direction, we haveH = 1

η (−z)×E and

P = E ×H =1ηE × (−z ×E) = −1

ηz|E|2

Thus, when the wave consists of one forward wave F (z − ct) andone backward wave G(z + ct), we have

P = E ×H =1ηz(|F |2 − |G|2

)w =

12ε|E|2 +

12µ|H|2 = ε|F |2 + ε|G|2


Generalization for nonlinear media: simple waves

A generalization of the plane wave concept is to look for solutionsdepending only on a scalar function of space and time

E(x, y, z, t) = E(ϕ(x, y, z, t)) (linear media: ϕ = z − ct)

This can help analyze nonlinear material models with instantaneousresponse (for instance a Kerr model D = ε0(E + α|E|2E))

D(x, y, z, t) = D(E(x, y, z, t))

Maxwell’s equations can then be written

∇ϕ×E′ = −∂ϕ∂t

∂B

∂H·H ′

∇ϕ×H ′ = ∂ϕ

∂t

∂D

∂E·E′


Simple waves, continued

The wavefronts and wave slowness are defined by

ϕ(x, y, z, t) = constant

s = −∇ϕ∂ϕ∂t

(unit s/m)

Maxwell’s equations become a symmetric algebraic problem(0 −s× I

s× I 0

)·(E′

H ′

)=(∂D∂E 00 ∂B

∂H

)·(E′

H ′

)where ∂D

∂E = ε(E) is the differential permittivity and ∂B∂H = µ(H)

is the differential permeability. The “eigenvalue” s (and hencepropagation speed) depends on field strengths E and H.


Shock waves

Initial data propagate with a speed depending on field strength.

Rarefaction wave Shock waveFast wave outruns the slow wave Fast wave overtakes the slow wave


Outline


2 Polarization





7 Conclusions


Time harmonic waves

We assume harmonic time dependence

E(x, y, z, t) = E(z)ejωt

H(x, y, z, t) = H(z)ejωt

Using the same wave splitting as before implies

∂E±∂z

= ∓1c

∂E±∂t

= ∓ jωcE± ⇒ E±(z) = E0±e∓jkz

where k = ω/c is the wave number in the medium. Thus, thegeneral solution for time harmonic waves is

E(z) = E0+e−jkz +E0−ejkz

H(z) = H0+e−jkz +H0−ejkz =1ηz ×

(E0+e−jkz −E0−ejkz

)The triples {E0+,H0+, z} and {E0−,H0−,−z} are right-handedsystems.


Wavelength

A time harmonic wave propagating in the forward z direction hasthe space-time dependence E(z, t) = E0+ej(ωt−kz) andH(z, t) = 1

η z ×E(z, t).

The wavelength corresponds to the spatial periodicity according toe−jk(z+λ) = e−jkz, meaning kλ = 2π or

λ =2πk

=2πcω

=c

f


Refractive index

The wavelength is often compared to the correspondingwavelength in vacuum

λ0 =c0

f

The refractive index is

n =λ0

λ=

k

k0=

c0

c=√

εµ

ε0µ0

Important special case: non-magnetic media, where µ = µ0 andn =

√ε/ε0.

c =c0

n, η =

η0

n,︸︷︷︸

only for µ = µ0!

λ =λ0

n, k = nk0

Note: We use c0 for the speed of light in vacuum and c for thespeed of light in a medium, even though c is the standard for thespeed of light in vacuum!


Energy density and power flow

For the general time harmonic solution

E(z) = E0+e−jkz +E0−ejkz

H(z) =1ηz ×

(E0+e−jkz −E0−ejkz

)the energy density and Poynting vector are

w =14

Re[εE(z) ·E∗(z) + µH(z) ·H∗(z)

]=

12ε|E0+|2 +

12ε|E0−|2

P =12

Re[E(z)×H∗(z)

]= z

(12η|E0+|2 −

12η|E0−|2

)


Wave impedance

For forward and backward waves, the impedance η =√µ/ε relates

the electric and magnetic field strengths to each other.

E± = ±ηH± × z

When both forward and backward waves are present this isgeneralized as (in component form)

Zx(z) =[E(z)]x

[H(z)× z]x=Ex(z)Hy(z)

= ηE0+xe−jkz + E0−xejkz

E0+xe−jkz − E0−xe−jkz

Zy(z) =[E(z)]y

[H(z)× z]y= −Ey(z)

Hx(z)= η

E0+ye−jkz + E0−yejkz

E0+ye−jkz − E0−ye−jkz

Thus, the wave impedance is in general a non-trivial function of z.This will be used as a means of analysis and design in the course.


Material and wave parameters

We note that we have two material parameters

Permittivity ε, defined by D = εE.

Permeability µ, defined by B = µH.

But the waves are described by the wave parameters

Wave number k, defined by k = ω√εµ.

Wave impedance η, defined by η =√µ/ε.

This means that in a scattering experiment, we primarily getinformation on k and η, not ε and µ. In order to get material data,a theoretical material model must be applied.

Often, wave number is given by transmission data (phase delay),and wave impedance is given by reflection data (impedancemismatch). More on this in future lectures!


Outline


2 Polarization





7 Conclusions


Why care about different polarizations?

I Different materials react differently to different polarizations.

I Linear polarization is sometimes not the most natural.

I For propagation through the ionosphere (to satellites), orthrough magnetized media, often circular polarization isnatural.


Complex vectors

The time dependence of the electric field is

E(t) = Re{E0ejωt}

where the complex amplitude can be written

E0 = xE0x + yE0y + zE0z = x|E0x|ejα + y|E0y|ejβ + z|E0z|ejγ

= E0r + jE0i

where E0r and E0i are real-valued vectors. We then have

E(t) = Re{(E0r + jE0i)ejωt} = E0r cos(ωt)−E0i sin(ωt)

This lies in a plane with normal

n = ± E0r ×E0i

|E0r ×E0i|


Linear polarization

The simplest case is given by E0i = 0, implying E(t) = E0r cosωt.

x

y

E0r


Circular polarization

Assume E0r = x and E0i = −y. The total fieldE(t) = x cosωt+ y sinωt then rotates in the xy-plane:

x

y

E(t)

The rotation direction needs to be compared to the propagationdirection.


Circular polarization and propagation direction


Elliptical polarization

The general polarization is elliptical

The direction e is parallel to the Poynting vector (the power flow).


Classification of polarization

The following classification can be given:

−je · (E0 ×E∗0) Polarization= 0 Linear polarization> 0 Right handed elliptic polarization< 0 Left handed elliptic polarization

Further, circular polarization is characterized by E0 ·E0 = 0.Typical examples:

Linear: E0 = x or E0 = y.

Circular: E0 = x− jy (right handed) or E0 = x+ jy (lefthanded).

See the literature for more in depth descriptions.


Alternative bases in the plane

To describe an arbitrary vector in the xy-plane, the unit vectors

x and y

are usually used. However, we could just as well use the RCP andLCP vectors

x− jy and x+ jy

Sometimes the linear basis is preferrable, sometimes the circular.


Outline


2 Polarization





7 Conclusions


Lossy media

We study lossy isotropic media, where

D = εdE, J = σE, B = µH

The conductivity is incorporated in the permittivity,

J tot = J + jωD = (σ + jωεd)E = jω(εd +

σ

jω

)E

which implies a complex permittivity

εc = εd − jσ

ω

Often, the dielectric permittivity εd is itself complex, εd = ε′d − jε′′d.


Examples of lossy media

I Metals (high conductivity)

I Liquid solutions (ionic conductivity)

I Resonant media

I Just about anything!


Characterization of lossy media

In the previous lecture, we have shown that a passive material ischaracterized by

Re[jω(ε ξζ µ

)]≥ 0

For isotropic media with ε = εcI, ξ = ζ = 0 and µ = µcI, thisboils down to

εc = ε′c − jε′′cµc = µ′c − jµ′′c

⇒ε′′c ≥ 0µ′′c ≥ 0


Maxwell’s equations in lossy media

Assuming dependence only on z we obtain{∇×E = −jωµcH

∇×H = jωεcE⇒

∂

∂zz ×E = −jωµcH

∂

∂zz ×H = jωεcE

Nothing really changes compared to the lossless case, for instanceit is seen that the fields do not have a z-component. This can bewritten as a system

∂

∂z

(E

ηcH × z

)=(

0 −jkc

−jkc 0

)(E

ηcH × z

)where the complex wave number kc and the complex waveimpedance ηc are

kc = ω√εcµc, and ηc =

√µc

εc


The parameters in the complex plane

The parameters εc, µc, and kc = ω√εcµc take their values in the

complex lower half plane, whereas ηc =√µc/εc is restricted to the

right half plane.

��

��

ReIm

��

��

ReIm


Solutions

The solution to the system

∂

∂z

(E

ηcH × z

)=(

0 −jkc

−jkc 0

)(E

ηcH × z

)can be written (no z-components in the amplitudes E+ and E−)

E(z) = E+e−jkcz +E−ejkcz

H(z) =1ηcz ×

(E+e−jkcz −E−ejkcz

)Thus, the solutions are the same as in the lossless case, as long aswe “complexify” the coefficients.


Exponential damping

The dominating effect of wave propagation in lossy media isexponential damping of the amplitude of the wave:

kc = β − jα ⇒ e−jkcz = e−jβze−αz

Thus, α = − Im(kc) represents the damping of the wave, whereasβ = Re(kc) represents the oscillations.The exponential is sometimes written in terms of γ = jkc = α+ jβas

e−γz = e−jβze−αz

where γ corresponds to a spatial Laplace transform variable, in thesame way that the temporal Laplace variable is s = jω + ν.


Poynting’s theorem

Poynting’s theorem for isotropic media (εc = ε′c − jε′′c andµc = µ′c − jµ′′c ) is

∇ · 12

Re{E ×H∗} = −ω2(ε′′c |E|2 + µ′′c |H|2

)We will write out the left and right hand side explicitly to see ifthey are indeed equal.


The Poynting vector

Consider a wave propagating in the positive z direction,E(z) = E0e−jkcz

H(z) =1ηcz ×E0e−jkcz

The Poynting vector is

12

Re{E(z)×H(z)∗} =12

Re{E0 × (z ×E∗0)

1η∗c

e−jβz−αz+jβz−αz}

= z12

Re(1η∗c

)|E0|2e−2αz

with the divergence

∇·12

Re{E×H∗} =12∂

∂zRe(

1η∗c

)|E0|2e−2αz = −αRe(1η∗c

)|E0|2e−2αz


Material losses

For the plane wave under study we have H = 1ηcz ×E0e−jβz−αz,

which implies

|H|2 =1|η∗c |2

|E0|2e−2αz =|εc||µc||E0|2e−2αz

The right hand side of Poynting’s theorem is then

−ω2(ε′′c |E|2 + µ′′c |H|2

)= −ω

2

(ε′′c +

µ′′c |εc||µc|

)|E0|2e−2αz



In order for Poynting’s theorem to be satisfied, this requires

αRe(1η∗c

) =ω

2

(ε′′c +

µ′′c |εc||µc|

)This is not a trivial relation to prove. In the case of non-magneticmedia (µc = µ0, α = −ω Im(

√εcµ0), 1/ηc =

√εc/µ0), the

explicit form is

− Im(√εc) Re(

√εc) =

12ε′′c = −1

2Im(εc)

This is true due to the general relation (valid for any complexnumber w)

Im(w2) = 2Re(w) Im(w)



In conclusion, Poynting’s theorem

∇ · 12

Re{E ×H∗} = −ω2(ε′′c |E|2 + µ′′c |H|2

)is valid for waves propagating in lossy media, but the equality isobscured by some complex algebra.


Characterization of attenuation

The power is damped by a factor e−2αz. The attenuation is oftenexpressed in logarithmic scale, decibel (dB).

A = e−2αz ⇒ AdB = −10 log10(A) = 20 log10(e)αz = 8.686αz

Thus, the attenuation coefficient α can be expressed in dB permeter as

αdB = 8.686α

Instead of the attenuation coefficient, often the skin depth (alsocalled penetration depth)

δ = 1/α

is used. When the wave propagates the distance δ, its power isattenuated a factor e2 ≈ 7.4, or 8.686 dB ≈ 9 dB.


Characterization of losses

A common way to characterize losses is by the loss tangent(sometimes denoted tan δ)

tan θ =ε′′cε′c

=ε′′d + σ/ω

ε′d

which usually depends on frequency. In spite of this, it is often seenthat the loss tangent is given for only one frequency. This may beacceptable if the material properties vary little with frequency.


Example of material properties

From D. M. Pozar, Microwave Engineering:

Material Frequency εr tan θBeeswax 10 GHz 2.35 0.005Fused quartz 10 GHz 6.4 0.0003Gallium arsenide 10 GHz 13. 0.006Glass (pyrex) 3 GHz 4.82 0.0054Plexiglass 3 GHz 2.60 0.0057Silicon 10 GHz 11.9 0.004Styrofoam 3 GHz 1.03 0.0001Water (distilled) 3 GHz 76.7 0.157


Approximations for weak losses

In weakly lossy dielectrics, the material parameters are (whereε′′c � ε′c)

εc = ε′c − jε′′c = ε′c(1− j tan θ)µc = µ0

The wave parameters can then be approximated as

kc = ω√εcµc ≈ ω

√ε′cµ0

(1− j

12

tan θ)

ηc =√µc

εc≈√µ0

ε′c

(1 + j

12

tan θ)

If the losses are caused mainly by a small conductivity, we haveε′′c = σ/ω, tan θ = σ/(ωε′c), and the attenuation constant

α = − Im(kc) =12ω√ε′cµ0

σ

ωε′c=σ

2

√µ0

ε′cis proportional to conductivity and independent of frequency.


Example: propagation in sea water

A simple model of the dielectric properties of sea water is

εc = ε0

(81− j

4 S/mωε0

)that is, it has a relative permittivity of 81 and a conductivity ofσ = 4S/m. The imaginary part is much smaller than the real partfor frequencies

f � 4 S/m81 · 2πε0

= 888MHz

for which we have α = 728 dB/m. For lower frequencies, the exactcalculations give

f = 50Hz α = 0.028 dB/m δ = 35.6 mf = 1kHz α = 1.09 dB/m δ = 7.96 mf = 1MHz α = 34.49 dB/m δ = 25.18 cmf = 1GHz α = 672.69 dB/m δ = 1.29 cm


Approximations for good conductors

In good conductors, the material parameters are (where σ � ωε)

εc = ε− jσ/ω = ε(1− j

σ

ωε

)µc = µ

The wave parameters can then be approximated as

kc = ω√εcµc ≈ ω

√−jσ

ωµ =

√ωµσ

2(1− j)

ηc =√µc

εc≈√

µ

−jσ/ω=√ωµ

2σ(1 + j)

This demonstrates that the wave number is proportional to√ω

rather than ω in a good conductor, and that the real andimaginary part have equal amplitude.


Skin depth

The skin depth of a good conductor is

δ =1α

=√

2ωµσ

=1√πfµσ

For copper, we have σ = 5.8 · 107 S/m. This implies

f = 50Hz δ = 9.35 mmf = 1kHz δ = 2.09 mmf = 1MHz δ = 0.07 mmf = 1GHz δ = 2.09µm

This effectively confines all fields in a metal to a thin region nearthe surface.


Surface impedance

Integrating the currents near the surface z = 0 implies (withγ = α+ jβ)

J s =∫ ∞

0J(z) dz =

∫ ∞0

σE0e−γz dz =σ

γE0

Thus, the surface current can be expressed as

J s =1ZsE0

airmetalJ(z) = σE0e

−γz

z

E0

where the surface impedance is

Zs =γ

σ=α+ jβσ

=α

σ(1 + j) =

1σδ

(1 + j) =√ωµ

2σ(1 + j) = ηc

Since H0 × z = E0/ηc, this can also be writtenJ s = H0 × z = −z ×H0 = n×H0.


Outline


2 Polarization





7 Conclusions


Generalized propagation factor

For a wave propagating in an arbitrary direction, the propagationfactor is generalized as

e−jkz → e−jk·r

Assuming this as the only spatial dependence, the nabla operatorcan be replaced by −jk since

∇(e−jk·r) = −jk(e−jk·r)

Writing the fields as E(r) = E0e−jk·r, Maxwell’s equations forisotropic media can then be written{

−jk ×E0 = −jωµH0

−jk ×H0 = jωεE0⇒

{k ×E0 = ωµH0

k ×H0 = −ωεE0


Properties of the solutions

Eliminating the magnetic field, we find

k × (k ×E0) = −ω2εµE0

This shows that E0 does not have any components parallel to k,and the BAC-CAB rule implies k× (k×E0) = −E0(k · k). Thus,

k2 = k · k = ω2εµ

It is further clear that E0, H0 and k constitute a right-handedtriple since k ×E0 = ωµH0, or

H0 =k

ωµ

k

k×E0 =

1ηk ×E0


Preferred direction

What happens when k is not along the z-direction (which could bethe normal to a plane surface)?

I There are then two preferred directions, k and z.

I These span a plane, the plane of incidence.

I It is natural to specify the polarizations with respect to thatplane.

I When the H-vector is orthogonal to the plane of incidence,we have transverse magnetic polarization (TM).

I When the E-vector is orthogonal to the plane of incidence, wehave transverse electric polarization (TE).


TM and TE polarization

From these figures it is clear that the transverse impedance is

ηTM =ExHy

=A cos θ

1ηA

= η cos θ

ηTE = −EyHx

=B

1ηB cos θ

=η

cos θ


Interpretation of transverse wave vector

The transverse wave vector kt = kxx corresponds to the angle ofincidence θ as

kx = k sin θ

The transverse impedance is

Et = Zr · (Ht × z), Zr = η cos θxx+η

cos θyy︸︷︷︸

isotropic case

In a coming lecture, we will see how to generalize the transversewave impedance for oblique propagation in bianisotropic materials.


Complex waves

When the material parameters are complexified, we still have

k2c = k · k = ω2εcµc

with a complex wave vector

k = β − jα ⇒ e−jk·r = e−jβ·re−α·r

The real vectors α and β do not need to be parallel.


Outline


2 Polarization





7 Conclusions


The plane wave monster

So far we have treated plane waves, which have a seriousdrawback:

I Due to the infinite extent of e−jβz in the xy-plane, the planewave has infinite energy.

However, the plane wave is a useful object with which we can buildother, more physically reasonable, solutions.


Finite extent in the xy-plane

We can represent a field distribution with finite extent in thexy-plane using a Fourier transform:

Et(r, ω) =1

(2π)2

∞∫∫−∞

Et(z,kt, ω)e−jkt·r dkx dky

Et(z,kt, ω) =

∞∫∫−∞

Et(r, ω)ejkt·r dx dy

The z dependence in Et(z,kt, ω) corresponds to a plane wave

Et(z,kt, ω)e−jkt·r = Et(0,kt, ω)e−jkt·re−jβz

The total wavenumber for each kt is given by k2 = ω2εµ and

k2 = |kt|2 + β2 = k2x + k2

y + β2 ⇒ β(kt) = (k2 − |kt|2)1/2


Initial distribution

Assume a Gaussian distribution in the plane z = 0

Et(x, y, z = 0, ω) = A(ω)e−(x2+y2)/(2b2),

The transform is itself a Gaussian

Et(z = 0,kt, ω) = A(ω)2πb2e−(k2x+k2

y)b2/2


Paraxial approximation

The field in z ≥ 0 is then

Et(r, ω) =12πAb2

∞∫∫−∞

e−(k2x+k2

y)b2/2−j(kxx+kyy)−jβ(kt)z dkx dky

The exponential makes the main contribution to come from aregion close to kt ≈ 0. This justifies the paraxial approximation

β(kt) = (k2 − |kt|2)1/2 = k(1− |kt|2/k2)1/2

= k

(1− 1

2|kt|2

k2+ O(|kt|4/k4)

)= k − |kt|2

2k+ · · ·


Computing the field

Inserting the paraxial approximation in the Fourier integral implies(details can be found in Kristensson 5.3)

Et(r) ≈12πAb2

∞∫∫−∞

e−(k2x+k2

y)( b2

2−j z

2k)−j(kxx+kyy)−jkz dkx dky

= · · · = A

1− jξ(z, ω)e−(x2+y2)/(2F 2)−jkz

where F 2(z, ω) = b2 − jz/k = b2(1− jξ(z, ω)) and ξ = z/(kb2) isa real quantity.


Beam width

The power density of the beam is proportional to

e−(x2+y2)Re(1/F 2)

and the beam width is then

B(z) =1√

Re 1F 2

= · · · = bξ√

1 + ξ−2

where ξ = z/(kb2). For large z, the beam width is

B(z)→ bξ(z) =z

kb, z →∞


Beam width

The beam angle θb is characterized by

tan θb =B(z)z

=1kb

Small initial width compared to wavelength implies large beamangle.


Outline


2 Polarization





7 Conclusions


Classical Doppler effect

fb =c

λb= fa

c

c− va

fb =c− vbλa

= fac− vbc

fb = fac− vbc− va

≈ fa(

1 +va − vbc

)Lots more on relativistic Doppler effect in Orfanidis.


Negative material parameters

Passivity requires the material parameters εc and µc to be in thelower half plane,

��

��

ReIm

This means we could very well have εc/ε0 ≈ −1 and µc/µ0 ≈ −1for some frequency. What is then the appropriate value fork = ω

√εcµc = k0

√(εc/ε0)(µc/µ0) = k0

√(−1)(−1),

k = +k0 or k = −k0?


Negative refractive index

The product of two complex valued parameters, both in the lowerhalf plane, can be located anywhere in the complex plane.

��

��

��

��

��

��

ReIm

ReIm

ReIm

�arg

By defining the argument as negative, the square root operationbrings back the lower half plane (the square root reduces theargument of a complex number by a factor of 2). Thus, therefractive index is

n =√

(εc/ε0)(µc/µ0) =√

(−1)(−1) = −1

Alternatively, use the standard square root and

jωn =√

(jωεc/ε0)(jωµc/µ0)


Consequences of negative refractive index

With a negative refractive index, the exponential factor

ej(ωt−kz) = ej(ωt+|k|z)

represents a phase traveling in the negative z-direction, eventhough the Poynting vector 1

2 Re{E ×H∗} = z 12 Re( 1

η∗c)|E0|2 is

still pointing in the positive z-direction.

I The power flow is in the opposite direction of the phasevelocity!

I Snel’s law has to be “inverted”, the rays are refracted in thewrong direction.

First investigated by Veselago in 1967. Enormous scientific interestsince about a decade, since the materials can now (to someextent) be fabricated.


Negative refraction


Realization of negative refractive index

In order to obtain εc(ω)/ε0 = −1 and µc(ω)/µ0 = −1, artificialmaterials are necessary. Common strategy:

I Periodic structure with inclusions small compared towavelength.

I To obtain negative properties, the inclusions are usuallyresonant.

Drawbacks:

I The inclusions are not always small, putting the material pointof view in doubt.

I The interesting applications require very small losses.

I The negative properties require frequency variation, making itdifficult to maintain negative properties in a large frequencyband.


Band limitations

If the negative properties are realized with passive, causal materials,they must satisfy Kramers-Kronig’s relations (ε∞ = lim

ω→∞ ε(ω))

ε′(ω)− ε∞ =1πP∫ ∞−∞

ε′′(ω′)ω′ − ω

dω′

ε′′(ω) = − 1πP∫ ∞−∞

ε′(ω′)− ε∞ω′ − ω

dω′

These relations represent restriction on the possible frequencybehavior, and can be used to derive bounds on the bandwidthwhere the material parameters can be negative.


Example: two Lorentz models

-2

0

2

4

0

0.2

0.4

0.6

0.8

1

0.1 1 10

!

²(!)

j²(!)-² jm

ReIm !

j²(!)+1jj²(!)-1.5jj²(!)+1j

0.1 1 10

² =-1m

² =1.5m

² 1

² s

An εm between εs and ε∞ is easily realized for a large bandwidth,whereas an εm < ε∞ is not. With fractional bandwidth B:

maxω∈B|ε(ω)− εm| ≥

B

1 +B/2(ε∞ − εm)

{1/2 lossy case

1 lossless case


Outline


2 Polarization





7 Conclusions


Conclusions

I Plane waves are characterized by wave vector k and waveimpedance η.

I Polarization is deduced from time domain electric field.

I Lossy media leads to complex material parameters, but planewave formalism remains the same.

I At oblique propagation, the transverse fields are mostimportant.

I The paraxial approximation can be used to describe beams.The beam angle depends on the original beam width in termsof wavelengths.

I The Doppler effect can be used to detect motion.

I Negative refractive index is possible, but only for very narrowfrequency band.


electromagnetic wave propagation lecture 2: uniform plane ...€¦ · electromagnetic wave...

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