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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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
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
Daniel Sjoberg, Department of Electrical and Information Technology
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)
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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′
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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!
Daniel Sjoberg, Department of Electrical and Information Technology
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
)
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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!
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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|
Daniel Sjoberg, Department of Electrical and Information Technology
Linear polarization
The simplest case is given by E0i = 0, implying E(t) = E0r cosωt.
x
y
E0r
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
Circular polarization and propagation direction
Daniel Sjoberg, Department of Electrical and Information Technology
Elliptical polarization
The general polarization is elliptical
The direction e is parallel to the Poynting vector (the power flow).
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
Examples of lossy media
I Metals (high conductivity)
I Liquid solutions (ionic conductivity)
I Resonant media
I Just about anything!
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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 =
õc
εc
Daniel Sjoberg, Department of Electrical and Information Technology
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.
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ReIm
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ReIm
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
Poynting’s theorem
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)
Daniel Sjoberg, Department of Electrical and Information Technology
Poynting’s theorem
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
õ0
ε′cis proportional to conductivity and independent of frequency.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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).
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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+ · · ·
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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 →∞
Daniel Sjoberg, Department of Electrical and Information Technology
Beam width
The beam angle θb is characterized by
tan θb =B(z)z
=1kb
Small initial width compared to wavelength implies large beamangle.
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology
Negative material parameters
Passivity requires the material parameters εc and µc to be in thelower half plane,
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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?
Daniel Sjoberg, Department of Electrical and Information Technology
Negative refractive index
The product of two complex valued parameters, both in the lowerhalf plane, can be located anywhere in the complex plane.
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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)
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
Negative refraction
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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
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.
Daniel Sjoberg, Department of Electrical and Information Technology