Electromagnetic Wave Propagation
Lecture 4: Propagation in lossy media,
complex waves
Daniel Sjöberg
Department of Electrical and Information Technology
September 13, 2012
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
2 / 46
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
3 / 46
Lossy media
We study lossy isotropic media, where
D = d E, J = σE, B = µH
The conductivity is incorporated in the permittivity,
σ
J tot = J + jωD = (σ + jωd )E = jω d + E
jω
which implies a complex permittivity
σ
c = d − j
ω
Often, the dielectric permittivity d is itself complex, d = 0d − j00d ,
due to molecular interactions.
4 / 46
Examples of lossy media
I Metals (high conductivity)
I Liquid solutions (ionic conductivity)
I Resonant media
I Just about anything!
5 / 46
Characterization of lossy media
In a previous lecture, we have shown that a passive material is
characterized by
ξ ξ
Re jω = −ω Im ≥0
ζ µ ζ µ
For isotropic media with = c I, ξ = ζ = 0 and µ = µc I, this
boils down to
c = 0c − j00c 00c ≥ 0
⇒
µc = µ0c − jµ00c µ00c ≥ 0
6 / 46
Maxwell’s equations in lossy media
Assuming dependence only on z we obtain
∂
ẑ × E = −jωµc H
(
∇ × E = −jωµc H
⇒ ∂z
∇ × H = jωc E ∂ ẑ × H = jω E
c
∂z
Nothing really changes compared to the lossless case, for instance
it is seen that the fields do not have a z-component. This can be
written as a system
∂ E 0 −jkc E
=
∂z ηc H × ẑ −jkc 0 ηc H × ẑ
where the complex wave number kc and the complex wave
impedance ηc are
√
r
µc
kc = ω c µc , and ηc =
c
7 / 46
The parameters in the complex plane
√
For passive media, the parameters c , µc , and kc = ω c µc take
p
their values in the complex lower half plane, whereas ηc = µc /c
is restricted to the right half plane.
Im Im
11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
11111111111111
00000000000000
Re
00000000
11111111 Re
00000000000000
11111111111111 00000000
11111111
00000000000000
11111111111111 00000000
11111111
00000000000000
11111111111111 00000000
11111111
00000000000000
11111111111111 00000000
11111111
00000000000000
11111111111111 00000000
11111111
00000000000000
11111111111111 00000000
11111111
00000000
11111111
Equivalently, all parameters (jωc , jωµc , jkc , ηc ) take their values in
the right half plane.
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Solutions
The solution to the system
∂ E 0 −jkc E
=
∂z ηc H × ẑ −jkc 0 ηc H × ẑ
can be written (no z-components in the amplitudes E + and E − )
E(z) = E + e−jkc z + E − ejkc z
1
H(z) = ẑ × E + e−jkc z − E − ejkc z
ηc
Thus, the solutions are the same as in the lossless case, as long as
we “complexify” the coefficients.
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Exponential attenuation
The dominating effect of wave propagation in lossy media is
exponential decrease of the amplitude of the wave:
kc = β − jα ⇒ e−jkc z = e−jβz e−αz
Thus, α = − Im(kc ) represents the attenuation of the wave,
whereas β = Re(kc ) represents the oscillations.
The exponential is sometimes written in terms of γ = jkc = α + jβ
as
e−γz = e−jβz e−αz
where γ can be seen as a spatial Laplace transform variable, in the
same way that the temporal Laplace variable is s = ν + jω.
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Power flow
The power flow is given by the Poynting vector
∗
1 −jβz−αz 1 −jβz−αz
P(z) = Re E 0 e × ẑ × E 0 e
2 ηc
1 1
= ẑ Re |E 0 |2 e−2αz = P(0)e−2αz
2 ηc∗
11 / 46
Characterization of attenuation
The power is damped by a factor e−2αz . The attenuation is often
expressed 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 per
meter as
αdB = 8.686α
Instead of the attenuation coefficient, often the skin depth (also
called penetration depth)
δ = 1/α
is used. When the wave propagates the distance δ, its power is
attenuated a factor e2 ≈ 7.4, or 8.686 dB ≈ 9 dB.
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Characterization of losses
A common way to characterize losses is by the loss tangent
(sometimes denoted tan δ)
00c 00d + σ/ω
tan θ = =
0c 0d
which usually depends on frequency. In spite of this, it is often
seen that the loss tangent is given for only one frequency. This is
acceptable if the material properties vary only little with frequency.
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Example of material properties
From D. M. Pozar, Microwave Engineering:
Material Frequency 0r tan θ
Beeswax 10 GHz 2.35 0.005
Fused quartz 10 GHz 6.4 0.0003
Gallium arsenide 10 GHz 13. 0.006
Glass (pyrex) 3 GHz 4.82 0.0054
Plexiglass 3 GHz 2.60 0.0057
Silicon 10 GHz 11.9 0.004
Styrofoam 3 GHz 1.03 0.0001
Water (distilled) 3 GHz 76.7 0.157
The imaginary part of the relative permittivity is given by
00r = 0r tan θ.
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Approximations for weak losses
In weakly lossy dielectrics, the material parameters are (where
00c 0c )
c = 0c − j00c = 0c (1 − j tan θ)
µc = µ0
The wave parameters can then be approximated as
√ p
0
1
kc = ω c µc ≈ ω c µ0 1 − j tan θ
2
r r
µc µ0 1
ηc = ≈ 1 + j tan θ
c 0c 2
If the losses are caused mainly by a small conductivity, we have
00c = σ/ω, tan θ = σ/(ω0c ), and the attenuation constant
r
1 p 0 σ σ µ0
α = − Im(kc ) = ω c µ0 0 =
2 ωc 2 0c
is proportional to conductivity and independent of frequency.
15 / 46
Example: propagation in sea water
A simple model of the dielectric properties of sea water is
4 S/m
c = 0 81 − j
ω0
that is, it has a relative permittivity of 81 and a conductivity of
σ = 4 S/m. The imaginary part is much smaller than the real part
for frequencies
4 S/m
f = 888 MHz
81 · 2π0
for which we have α = 728 dB/m. For lower frequencies, the exact
calculations give
f = 50 Hz α = 0.028 dB/m δ = 35.6 m
f = 1 kHz α = 1.09 dB/m δ = 7.96 m
f = 1 MHz α = 34.49 dB/m δ = 25.18 cm
f = 1 GHz α = 672.69 dB/m δ = 1.29 cm
16 / 46
Approximations for good conductors
In good conductors, the material parameters are (where σ ω)
σ
c = − jσ/ω = 1 − j
ω
µc = µ
The wave parameters can then be approximated as
r r
√ σ ωµσ
kc = ω c µc ≈ ω −j µ = (1 − j)
ω 2
r r r
µc µ ωµ
ηc = ≈ = (1 + j)
c −jσ/ω 2σ
√
This demonstrates that the wave number is proportional to ω
rather than ω in a good conductor, and that the real and
imaginary part have equal amplitude.
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Skin depth
The skin depth of a good conductor is
r
1 2 1
δ= = =√
α ωµσ πf µσ
For copper, we have σ = 5.8 · 107 S/m. This implies
f = 50 Hz δ = 9.35 mm
f = 1 kHz δ = 2.09 mm
f = 1 MHz δ = 0.07 mm
f = 1 GHz δ = 2.09 µm
This effectively confines all fields in a metal to a thin region near
the surface.
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Surface impedance
Integrating the currents near the surface z = 0 implies (with
γ = α + jβ)
Z ∞ Z ∞
σ
Js = J (z) dz = σE 0 e−γz dz = E 0
0 0 γ
Thus, the surface current can be expressed as
air E0
1
Js = E0 metal
Zs J(z) = σE0 e−γz
z
where the surface impedance is
r
γ α + jβ α 1 ωµ
Zs = = = (1 + j) = (1 + j) = (1 + j) = ηc
σ σ σ σδ 2σ
19 / 46
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
20 / 46
Generalized propagation factor
For a wave propagating in an arbitrary direction, the propagation
factor is generalized as
e−jkz → e−jk·r
Assuming this as the only spatial dependence, the nabla operator
can be replaced by −jk since
∇(e−jk·r ) = −jk(e−jk·r )
Writing the fields as E(r) = E 0 e−jk·r , Maxwell’s equations for
isotropic media can then be written
( (
−jk × E 0 = −jωµH 0 k × E 0 = ωµH 0
⇒
−jk × H 0 = jωE 0 k × H 0 = −ωE 0
21 / 46
Properties of the solutions
Eliminating the magnetic field, we find
k × (k × E 0 ) = −ω 2 µE 0
This shows that E 0 does not have any components parallel to k,
and the BAC-CAB rule implies k × (k × E 0 ) = −E 0 (k · k). Thus,
the total wave number is given by
k 2 = k · k = ω 2 µ
It is further clear that E 0 , H 0 and k constitute a right-handed
triple since k × E 0 = ωµH 0 , or
k k 1
H0 = × E 0 = k̂ × E 0
ωµ k η
22 / 46
Preferred direction
What happens when k̂ is not along the z-direction (which could be
the normal to a plane surface)?
I There are then two preferred directions, k̂ and ẑ.
I These span a plane, the plane of incidence.
I It is natural to specify the polarizations with respect to that
plane.
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, we
have transverse electric polarization (TE).
23 / 46
TM and TE polarization
From these figures it is clear that the transverse impedance is
Ex A cos θ
ηTM = = 1 = η cos θ
Hy ηA
Ey B η
ηTE = = 1 =
−Hx η B cos θ
cos θ
24 / 46
Transverse wave impedance
The transverse wave vector kt = kx x̂ corresponds to the angle of
incidence θ as
kx = k sin θ
The transverse impedance is
η
E t = Zt · (H t × ẑ), Zt = η cos θx̂x̂ + ŷ ŷ
| {z cos θ }
isotropic case
The transverse wave impedance can be generalized to bianisotropic
materials by solving the eigenvalue problem from last lecture
kz Et 0 −ẑ × I tt ξ tt I 0 Et
= · − A(kt ) · ·
ω H t × ẑ I 0 ζ tt µtt 0 ẑ × I H t × ẑ
and studying the eigenvectors [E t , H t × ẑ]. The eigenvalue
kz /ω = n/c0 corresponds to the refractive index.
25 / 46
Complex waves
When the material parameters are complexified, we still have
kc2 = k · k = ω 2 c µc
with a complex wave vector
k = β − jα ⇒ e−jk·r = e−jβ·r e−α·r
The real vectors α and β do not need to be parallel.
26 / 46
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
27 / 46
The plane wave monster
So far we have treated plane waves, which have a serious
drawback:
I Due to the infinite extent of e−jkz z in the xy-plane, the plane
wave has infinite energy.
However, the plane wave is a useful object with which we can build
other, more physically reasonable, solutions.
28 / 46
Finite extent in the xy-plane
We can represent a field distribution with finite extent in the
xy-plane using a Fourier transform (where kt = kx x̂ + ky ŷ):
∞
ZZ
1
E t (kx , ky ; z)e−jkt ·r dkx dky
E t (x, y; z) =
(2π)2
−∞
∞
ZZ
E t (x, y; z)ejkt ·r dx dy
E (k , k ; z) =
t x y
−∞
The z dependence in E t (kx , ky ; z) corresponds to a plane wave
E t (kx , ky ; z)e−jkt ·r = E t (kx , ky ; 0)e−jkt ·r e−jkz z
The total wavenumber for each kt is given by k 2 = ω 2 µ and
k 2 = |kt |2 + kz2 = kx2 + ky2 + kz2 ⇒ kz (kt ) = (k 2 − |kt |2 )1/2
29 / 46
Initial distribution
Assume a Gaussian distribution in the plane z = 0
2 +y 2 )/(2b2 )
E t (x, y; 0) = Ae−(x
The transform is itself a Gaussian
2 2 2 /2
E t (kx , ky ; 0) = A2πb2 e−(kx +ky )b
30 / 46
Paraxial approximation
The field in z ≥ 0 is then
∞
ZZ
1 2 2 2 /2−j(k x+k y)−jk (k )z
E t (x, y; z) = Ab2 e−(kx +ky )b x y z t
dkx dky
2π
−∞
The exponential makes the main contribution to come from a
region close to kt ≈ 0. This justifies the paraxial approximation
kz (kt ) = (k 2 − |kt |2 )1/2 = k(1 − |kt |2 /k 2 )1/2
1 |kt |2 |kt |2
4 4
=k 1− + O(|k t | /k ) = k − + ···
2 k2 2k
31 / 46
Computing the field
Inserting the paraxial approximation in the Fourier integral implies
∞
ZZ
1 2 2 b2 z
E t (x, y; z) ≈ Ab2 e−(kx +ky )( 2 −j 2k )−j(kx x+ky y)−jkz dkx dky
2π
−∞
Ab2 −(x2 +y2 )/(2F 2 ) −jkz
= e e
F2
1
where F 2 = b2 − jz/k = jk (z + jkb2 ) = q(z)/(jk).
I q(z) = z + jz0 is known as the q-parameter of the beam.
I z0 = kb2 is known as the Rayleigh range.
The final expression for the beam distribution is then
2 2
A − x +y
E t (x, y; z) ≈ e 2b2 (1−jz/z0 ) e−jkz
1 − jz/z0
32 / 46
Beam width
The power density of the beam is proportional to
2 +y 2
−x Re 1
e 2b2 1−jz/z0
and the beam width is then
b b p
B(z) = q =q = b 1 + (z/z0 )2
1 1+jz/z0
Re 1−jz/z Re 1+(z/z 2
0 0)
where z0 = kb2 . For large z, the beam width is
z
B(z) → bz/z0 = , z→∞
kb
33 / 46
Beam width
The beam angle θb is characterized by
B(z) 1
tan θb = =
z kb
Small initial width compared to wavelength implies large beam
angle.
34 / 46
How can beams be used?
Beams can be an efficient representation of fields, determined by
three parameters:
I Propagation direction ẑ
I Polarization A(ω)
I Initial beam width b(ω)
High frequency propagation in office Raytracing in optics, FRED
spaces (Timchenko, Heyman, Boag,
EMTS Berlin 2010).
35 / 46
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
36 / 46
The Doppler effect
Classical formulas:
c c
fb = = fa
λb c − va
c − vb c − vb
fb = = fa
λa c
The relativistically correct formula is
r
c−v vb − va
v
fb = fa ≈ fa 1 − where v=
c+v c0 1 − va vb /c20
Lots more on relativistic Doppler effect in Orfanidis. Do not dive
too deep into this, it is not central material in the course.
37 / 46
Negative material parameters
Passivity requires the material parameters c and µc to be in the
lower half plane,
Im
11111111111111
00000000000000
Re
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
This means we could very well have c /0 ≈ −1 and µc /µ0 ≈ −1
for some frequency.
p What is then the appropriate value for
√ p
k = ω c µc = k0 (c /0 )(µc /µ0 ) = k0 (−1)(−1),
k = +k0 or k = −k0 ?
38 / 46
Negative refractive index
Simple solution: consider all parameters in the right half plane and
approach the negative axis from inside the half plane, using the
standard square root (with branch cut along negative real axis):
p
jkc = (jωc )(jωµc )
Im Im Im Im Im
jk
ǫ jωǫ √ n=
µ jωµ jωǫ · jωµ jk = jωǫ · jωµ jk0
Re Re Re Re Re
The refractive index is then
jkc p
n= = −j (jc /0 )(jµc /µ0 ) = −1
jk0
39 / 46
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, even
though the Poynting vector 12 Re{E × H ∗ } = ẑ 12 Re( η1∗ )|E 0 |2 is
c
still pointing in the positive z-direction.
I The power flow is in the opposite direction of the phase
velocity!
I Snel’s law has to be “inverted”, the rays are refracted in the
wrong direction.
First investigated by Veselago in 1967. Enormous scientific interest
since about a decade, since the materials can now (to some
extent) be fabricated.
40 / 46
Negative refraction
41 / 46
Realization of negative refractive index
I Artificial materials,
“metamaterials”
I Periodic structures
I Resonant inclusions
I Small losses
required
I Theoretical and
practical challenges
I Very hot topic since
about 10 years
To describe the structure as a material usually requires
microstructure a λ, which is not easily achieved.
42 / 46
Band limitations
If the negative properties are realized with passive, causal materials,
they must satisfy Kramers-Kronig’s relations (∞ = lim (ω))
ω→∞
Z ∞ 00 0
1 (ω )
0 (ω) − ∞ = p.v. 0−ω
dω 0
π −∞ ω
Z ∞ 0 0
00 1 (ω ) − ∞
(ω) = − p.v. dω 0
π −∞ ω0 − ω
These relations represent restriction on the possible frequency
behavior, and can be used to derive bounds on the bandwidth
where the material parameters can be negative.
43 / 46
Example: two Lorentz models
²(!)
4
²s
2 ²m =1.5
Re ²1
Im !
0
0.1 1 10
²m =-1
-2
j²(!)-²m j
1
0.8
0.6
0.4
0.2 j²(!)+1j
j²(!)+1j j²(!)-1.5j
!
0
0.1 1 10
An m between s and ∞ is easily realized for a large bandwidth,
whereas an m < ∞ is not. With fractional bandwidth B:
(
B 1/2 lossy case
max |(ω) − m | ≥ (∞ − m )
ω∈B 1 + B/2 1 lossless case
44 / 46
Outline
1 Propagation in lossy media
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
45 / 46
Conclusions
I Lossy media leads to complex material parameters, but plane
wave formalism is the same as in lossless media.
I At oblique propagation, the transverse fields are most
important.
I The paraxial approximation can be used to describe beams.
The beam angle depends on the original beam width in terms
of wavelengths.
I The Doppler effect can be used to detect motion.
I Negative refractive index is possible, but only for very narrow
frequency band.
46 / 46
