Electromagnetic Wave Propagation Lecture 4: Propagation in lossy media, complex wavescomplex waves
Daniel Sjoberg
September 2016
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
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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
The conductivity is incorporated in the permittivity,
J tot = J + jωD = (σ + jωεd)E = jω
( εd +
σ
jω
) E
εc = εd − j σ
ω
Often, the dielectric permittivity εd is itself complex, εd = ε′d − jε′′d, due to molecular interactions.
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I Resonant media
Characterization of lossy media
In a previous lecture, we have shown that a passive material is characterized by
Re
{ jω
)} ≥ 0
For isotropic media with ε = εcI, ξ = ζ = 0 and µ = µcI, this boils down to
εc = ε′c − jε′′c
µc = µ′c − jµ′′c ⇒ ε′′c ≥ 0
µ′′c ≥ 0
Maxwell’s equations in lossy media
⇒
∂
∂z z ×H = jωεcE
∂
ηcH × z
) where the complex wave number kc and the complex wave impedance ηc are
kc = ω √ εcµc, and ηc =
√ µc εc
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The parameters in the complex plane
For passive media, 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.


Re Im
Equivalently, all parameters (jωεc, jωµc, jkc, ηc) take their values in the right half plane, since multiplication with j is a rotation 90 degrees counter-clockwise.
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∂
ηcH × z
) can be written (no z-components in the amplitudes E+ and E−)
E(z) = E+e −jkcz +E−e
jkcz
−jkcz −E−ejkcz )
Thus, the solutions are the same as in the lossless case, as long as we “complexify” the parameters.
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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−jkcz = e−jβze−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−jβze−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
P(z) = 1
2 Re
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Characterization of attenuation
The power is decreased by a factor e−2αz. The attenuation is often expressed in logarithmic scale, decibel (dB).
A = P(0) P(z) = e2αz ⇒ 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 defined in the green box below (sometimes denoted tan δ)
tan θ = ε′′c ε′c
= ε′′d + σ/ω
ε′d
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.
103 106 109 1012 1015 f (Hz)
ε = ε′ − jε′′
From D. M. Pozar, Microwave Engineering:
Material Frequency ε′r tan θ
Beeswax 10GHz 2.35 0.005 Fused quartz 10GHz 6.4 0.0003 Gallium arsenide 10GHz 13. 0.006 Glass (pyrex) 3GHz 4.82 0.0054 Plexiglass 3GHz 2.60 0.0057 Silicon 10GHz 11.9 0.004 Styrofoam 3GHz 1.03 0.0001 Water (distilled) 3GHz 76.7 0.157
The imaginary part of the relative permittivity is given by ε′′r = ε′r tan θ.
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Approximations for weak losses
In weakly lossy dielectrics, the material parameters are (where ε′′c ε′c)
εc = ε′c − jε′′c = ε′c(1− j tan θ)
µc = µ0
kc = ω √ εcµc ≈ ω
( 1 + j
2 tan θ
) If the losses are caused mainly by a small conductivity, we have ε′′c = σ/ω, tan θ = σ/(ωε′c), and the attenuation constant
α = − Im(kc) = 1
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
εc = ε0
( 81− j
4 S/m
ωε0
) that is, it has a relative permittivity of 81 and a conductivity of σ = 4S/m. The imaginary part is much smaller than the real part for frequencies
f 4 S/m
81 · 2πε0 = 888MHz
for which we have α = 728 dB/m. For lower frequencies, the exact calculations give
f = 50Hz α = 0.028 dB/m δ = 35.6m
f = 1kHz α = 1.09 dB/m δ = 7.96m
f = 1MHz α = 34.49 dB/m δ = 25.18 cm
f = 1GHz α = 672.69 dB/m δ = 1.29 cm
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In good conductors, the material parameters are (where σ ωε)
εc = ε− jσ/ω = ε ( 1− j
σ
ωε
kc = ω √ εcµc ≈ ω
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
δ = 1
For copper, we have σ = 5.8 · 107 S/m. This implies
f = 50Hz δ = 9.35mm
f = 1kHz δ = 2.09mm
f = 1MHz δ = 0.07mm
f = 1GHz δ = 2.09µm
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)
J s =
J s = 1
Zs = γ
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
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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) = E0e −jk·r, Maxwell’s equations for
isotropic media can then be written{ −jk ×E0 = −jωµH0
−jk ×H0 = jωεE0 ⇒
{ k ×E0 = ωµH0
k ×H0 = −ωεE0
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, the total wave number is given by
k2 = k · k = ω2εµ
It is further clear that E0, H0 and k constitute a right-handed triple since k ×E0 = ωµH0, or
H0 = k
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Preferred direction
What happens when k is not along the z-direction (which could be the 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 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).
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kz
TE
From these figures it is clear that the transverse impedance is
ηTM = Ex Hy
= A cos θ
Transverse wave impedance
The transverse wave vector kt = kxx corresponds to the angle of incidence θ as
kx = k sin θ
The transverse impedance is
cos θ yy
isotropic case
The transverse wave impedance can be generalized to bianisotropic materials by solving the eigenvalue problem from last lecture
kz ω
) · [( εtt ξtt ζtt µtt
) · (
)
and studying the eigenvectors [Et,Ht × z]. The eigenvalue kz/ω = n/c0 corresponds to the refractive index.
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Complex waves
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.
α β αz αx βz βx complex wave type
0 0 0 + − oblique incidence ↑ → 0 + + 0 evanescent surface wave + + + − Zenneck surface wave − + + + leaky wave
x
z
α
β
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
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The plane wave monster
So far we have treated plane waves, which have a serious drawback:
I Due to the infinite extent of e−jkzz 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.
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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 = kxx+ kyy):
Et(x, y; z) = 1
Et(kx, ky; z) =
Et(x, y; z)e jkt·r dx dy
The z dependence in Et(kx, ky; z) corresponds to a plane wave
Et(kx, ky; z)e −jkt·r = Et(kx, ky; 0)e
−jkt·re−jkzz
The total wavenumber for each kt is given by k2 = ω2εµ and
k2 = |kt|2 + k2z = k2x + k2y + k2z ⇒ kz(kt) = (k2 − |kt|2)1/2
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Initial distribution
Et(x, y; 0) = Ae−(x 2+y2)/(2b2)
∼ b
∼ b
Et(kx, ky; 0) = A2πb2e−(k 2 x+k
2 y)b
Et(x, y; z) = 1
2 y)b
2/2−j(kxx+kyy)−jkz(kt)z dkx dky
The exponential makes the main contribution to come from a region close to kt ≈ 0. This justifies the paraxial approximation
kz(kt) = (k2 − |kt|2)1/2 = k(1− |kt|2/k2)1/2
= k
Et(x, y; z) ≈ 1
2 y)(
= Ab2
where F 2 = b2 − jz/k = 1 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
Et(x, y; z) ≈ A
2b2(1−jz/z0) e−jkz
e −x2+y2
2b2 Re 1
and the beam width as function of position is then
B(z) = b√
= b √ 1 + (z/z0)2
where z0 = kb2. For large z, the beam width is
B(z)→ bz/z0 = z
tan θb = B(z)
z =
1
kb Small initial width compared to wavelength implies large beam angle.
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How can beams be used?
Beams can be an efficient representation of fields, determined by three parameters:
I Propagation direction z I Polarization A(ω) I Initial beam width b(ω)
High frequency propagation in office spaces (Timchenko, Heyman, Boag, EMTS Berlin 2010).
Raytracing in optics, FRED
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
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fb = fa
vb − va 1− vavb/c20
Lots more on relativistic Doppler effect in Orfanidis. Do not dive too deep into this, it is not central material in the course.
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Negative material parameters


Re Im
This means we could very well have εc/ε0 ≈ −1 and µc/µ0 ≈ −1 for some frequency. What is then the appropriate value for k = ω
√ εcµc = k0
√ (εc/ε0)(µc/µ0) = k0
√ (−1)(−1),
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Negative refractive index
Simple solution: consider all parameters in the right half plane and approach the imaginary axis from inside the half plane, using the standard square root (with branch cut along negative real axis):
jkc = √ (jωεc)(jωµc)
Reµ
Im Re
n = jk
n = jkc jk0
= −j √ (jεc/ε0)(jµc/µ0) = −1
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 1
2 Re{E ×H∗} = z 1 2 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 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 more than a decade, since the materials can now (to some extent) be fabricated.
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Negative refraction
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I Artificial materials, “metamaterials”
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.
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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∞ ω′ − ω dω′
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.
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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. In this case, where B is the fractional bandwidth:
max ω∈B |ε(ω)− εm| ≥
B
Outline
2 Oblique propagation and complex waves
3 Paraxial approximation: beams (not in Orfanidis)
4 Doppler effect and negative index media
5 Conclusions
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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.
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Paraxial approximation: beams (not in Orfanidis)
Doppler effect and negative index media
Conclusions