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ETEN05 Electromagnetic Wave PropagationLecture 3: Time harmonic fields, material and
polarization classification
Daniel Sjoberg
Department of Electrical and Information Technology
September 8, 2009
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Scope
I The theory given in this lecture (and the entire course) isapplicable to the whole electromagnetic spectrum.
I However, different processes are dominant in different bands,making the material models different.
I Here, you learn what restrictions are imposed by therequirements
1. Linearity2. Causality3. Time translational invariance4. Passivity
Requirements 1–3 have already been enforced in the time domain.We will now transform to the frequency domain and add thepassivity requirement.
Daniel Sjoberg, Department of Electrical and Information Technology
Electromagnetic spectrum, c0 = fλ = 3 · 108 m/s
Band Frequency WavelengthELF Extremely Low Frequency 30–300 Hz 1–10 MmVF Voice Frequency 300–3000 Hz 100–1000 kmVLF Very Low Frequency 3–30 kHz 10-100 kmLF Low Frequency 30–300 kHz 1–10 kmMF Medium Frequency 300–3000 kHz 100–1000 mHF High Frequency 3–30 MHz 10–100 mVHF Very High Frequency 30–300 MHz 1–10 mUHF Ultra High Frequency 300–3000 MHz 10–100 cmSHF Super High Frequency 3–30 GHz 1–10 cmEHF Extremely High Frequency 30–300 GHz 1–10 mm
Submillimeter 300–3000 GHz 100–1000µmInfrared 3–300 THz 1–100µmVisible 385–789 THz 380–780 nmUltraviolet 750 THz–30 PHz 10–400 nmX-ray 30 PHz–3 EHz 10 nm–100 pmγ-ray >3 EHz <100 pm
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Three ways of introducing time harmonic fields
I Fourier transform (finite energy fields)
E(r, ω) =∫ ∞−∞
E(r, t)eiωt dt
E(r, t) =12π
∫ ∞−∞
E(r, ω)e−iωt dt
I Laplace transform (causal fields, zero for t < 0)
E(r, s) =∫ ∞
0E(r, t)e−st dt
E(r, t) =1
2πi
∫ α+i∞
α−i∞E(r, s)est ds
I Real-value convention (purely harmonic cosωt, preservesunits)
E(r, t) = Re{E(r, ω)e−iωt}
Daniel Sjoberg, Department of Electrical and Information Technology
Some examples
E(r, t) Fourier Laplace Real-value
cos(ω0t) π(δ(ω − ω0) + δ(ω + ω0)) — 1
sin(ω0t) iπ(δ(ω − ω0)− δ(ω + ω0)) — i
e−atu(t) 1−iω+a
1s+a —
e−at cos(ω0t)u(t) −iω+a(−iω+a)2+ω2
0
s+a(s+a)2+ω2
0—
e−at sin(ω0t)u(t) ω0
(−iω+a)2+ω20
ω0
(s+a)2+ω20
—
δ(t) 1 1 —
Daniel Sjoberg, Department of Electrical and Information Technology
Different time conventions
Different traditions:
Physics: Time dependence e−iωt, plane wave factor ei(k·r−ωt).
Systems: Time dependence ejωt, plane wave factor ej(ωt−k·r).
If you use i and j consistently, all results can be translated betweenconventions using the simple rule
j = −i
Kristensson’s book uses e−iωt, whereas Orfanidis uses ejωt.
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Constitutive relations in the time domain
Convolutions model materials with transient processes:
D(t) = ε0
{ε·E(t)+
t∫−∞
χee(t−t′)·E(t′) dt′+
t∫−∞
χem(t−t′)·η0H(t′) dt′}
= ε0
{ε ·E(t) + [χee ∗E](t) + [χem ∗ η0H](t)
}
B(t) =1c0
{ t∫−∞
χme(t−t′)·E(t′) dt′+µ·η0H(t)+
t∫−∞
χmm(t−t′)·η0H(t′) dt′}
=1c0
{[χme ∗E](t) + µ · η0H(t) + [χmm ∗ η0H](t)
}Now use the general relation F{f ∗ g} = (F{f}) (F{g}).
Daniel Sjoberg, Department of Electrical and Information Technology
Constitutive relations in the frequency domain
Applying a Fourier transform implies
D(ω) = ε0
{ε ·E(ω) + χee(ω) ·E(ω) + χem(ω) · η0H(ω)
}
B(ω) =1c0
{χme(ω) ·E(ω) + µ · η0H(ω) + χmm(ω) · η0H(ω)
}or (
ε−10 D(ω)c0B(ω)
)=(ε(ω) ξ(ω)ζ(ω) µ(ω)
)·(E(ω)η0H(ω)
)where
ε(ω) = ε+ χee(ω) ξ(ω) = χem(ω)ζ(ω) = χme(ω) µ(ω) = µ+ χmm(ω)
Daniel Sjoberg, Department of Electrical and Information Technology
Modeling arbitriness
In frequency domain, the arbitrariness in modeling is more obviousthan in the time domain. Assume the models
J(ω) = σ(ω)E(ω), D(ω) = ε(ω)E(ω)
The total current in Maxwell’s equations can then be written
J(ω)− iωD(ω) = [σ(ω)− iωε(ω)]︸ ︷︷ ︸=σ′(ω)
E(ω) = J ′(ω)
= −iω[σ(ω)−iω
+ ε(ω)]
︸ ︷︷ ︸=ε′(ω)
E(ω) = −iωD′(ω)
where σ′(ω) and ε′(ω) are equivalent models for the material.
Daniel Sjoberg, Department of Electrical and Information Technology
Debye material in frequency domain
The susceptibility kernel is χ(t) = αe−t/τu(t), with the Fouriertransform
χ(ω) =∫ ∞
0αe−t/τeiωt dt =
α
−iω + 1/τ=
ατ
1− iωτ
The frequency dependent permittivity is ε(ω) = 1 + χ(ω), withtypical behavior as below:
Daniel Sjoberg, Department of Electrical and Information Technology
Lorentz material in frequency domain
The susceptibility kernel is χ(t) = ω2p
ν0e−νt/2 sin(ν0t)u(t), with the
Fourier transform
χ(ω) =∫ ∞
0
ω2p
ν0e−νt/2 sin(ν0t) dt = −
ω2p
ω2 − ω20 + iων
The frequency dependent permittivity is ε(ω) = 1 + χ(ω), withtypical behavior as below:
Daniel Sjoberg, Department of Electrical and Information Technology
Example: permittivity of water
Microwave properties (one Debye model):
Light properties (many Lorentz resonances):
Daniel Sjoberg, Department of Electrical and Information Technology
Some observations
I The imaginary part is positive.
I The susceptibility is an analytic function of the frequency inthe upper ω-plane.
The latter property is subtle, and is linked to the causality. Itimplies the Kramers-Kronig relations (writingχ(ω) = χr(ω) + iχi(ω) for the real and imaginary part)
χr(ω) =1π
p. v.∫ ∞−∞
χi(ω′)ω′ − ω
dω′
χi(ω) = − 1π
p. v.∫ ∞−∞
χr(ω′)ω′ − ω
dω′
See Orfanidis Section 1.10. In particular, we have the sum rule
χr(0) =1π
p. v.∫ ∞−∞
χi(ω′)ω′
dω′
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Poynting’s theorem
In the time domain we had
∇· (E(t)×H(t))+H(t) · ∂B(t)∂t
+E(t) · ∂D(t)∂t
+E(t) ·J(t) = 0
For time harmonic fields, we consider the time average over oneperiod:
∇·〈E(t)×H(t)〉+⟨H(t) · ∂B(t)
∂t
⟩+⟨E(t) · ∂D(t)
∂t
⟩+〈E(t) · J(t)〉 = 0
The time average of a product of two harmonic signals is〈f(t)g(t)〉 = 1
2 Re{f(ω)g(ω)∗}.
Daniel Sjoberg, Department of Electrical and Information Technology
Poynting’s theorem, continued
The different terms are
〈E(t)×H(t)〉 =12
Re{E(ω)×H(ω)∗}⟨H(t) · ∂B(t)
∂t
⟩=
12
Re {iωH(ω) ·B(ω)∗}⟨E(t) · ∂D(t)
∂t
⟩=
12
Re {iωE(ω) ·D(ω)∗}
〈E(t) · J(t)〉 =12
Re {E(ω) · J(ω)∗}
For a purely dielectric material, we have D(ω) = ε(ω) ·E(ω) and
2 Re {iωE(ω) ·D(ω)∗} = iωE(ω)·[ε(ω)·E(ω)]∗−iωE(ω)∗·ε(ω)·E(ω)
= −iωE(ω)∗ · [ε(ω)− ε(ω)†] ·E(ω)
Daniel Sjoberg, Department of Electrical and Information Technology
Poynting’s theorem, final version
In source free regions, we have
∇·〈S(t)〉 =iωε04
(E(ω)η0H(ω)
)†·(ε(ω)− ε(ω)† ξ(ω)− ζ(ω)†
ζ(ω)− ξ(ω)† µ(ω)− µ(ω)†
)·(E(ω)η0H(ω)
)Passive material: ∇ · 〈S(t)〉 ≤ 0Active material: ∇ · 〈S(t)〉 > 0Lossless material: ∇ · 〈S(t)〉 = 0This boils down to conditions on the material matrix
−iω(ε(ω)− ε(ω)† ξ(ω)− ζ(ω)†
ζ(ω)− ξ(ω)† µ(ω)− µ(ω)†
)= 2ω Im
{(ε(ω) ξ(ω)ζ(ω) µ(ω)
)}which is positive for a passive material, and zero for lossless media.
Daniel Sjoberg, Department of Electrical and Information Technology
Lossless media
The condition on lossless media can also be written(ε(ω) ξ(ω)ζ(ω) µ(ω)
)=(ε(ω) ξ(ω)ζ(ω) µ(ω)
)†That is, the matrix should be hermitian symmetric.
Daniel Sjoberg, Department of Electrical and Information Technology
Isotropic materials
A bi-isotropic material is described by(ε(ω) ξ(ω)ζ(ω) µ(ω)
)=(ε(ω)I ξ(ω)Iζ(ω)I µ(ω)I
)The passivity requirement implies that all eigenvalues of the matrix
−iω(ε(ω)− ε(ω)∗ ξ(ω)− ζ(ω)∗
ζ(ω)− ξ(ω)∗ µ(ω)− µ(ω)∗
)are positive. If ξ = ζ = 0 it is seen that this requires
ω Im ε(ω) > 0, ω Imµ(ω) > 0
and if ξ and ζ are nonzero we also require
|ξ(ω)− ζ(ω)∗|2 < 4 Im ε(ω) Imµ(ω)
Daniel Sjoberg, Department of Electrical and Information Technology
Classification of materials
The same classification as in the time domain can be made
with the addition that in the frequency domain we can also classifythe passivity of the material.
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 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{E0e−iωt}
where
E0 = xE0x + yE0y + zE0z = x|E0x|eiα + y|E0y|eiβ + z|E0z|eiγ
= E0r + iE0i
where E0r and E0i are real-valued vectors. We then have
E(t) = Re{(E0r + iE0i)e−iωt} = 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)
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:
ie · (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+ iy (right handed) or E0 = x− iy (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+ iy and x− iy
Sometimes the linear basis is preferrable, sometimes the circular.
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Introduction
2 Time harmonic fields
3 Constitutive relations
4 Classification of materials
5 Classification of polarization
6 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
Conclusions
I Constitutive relations can be represented by dyadics(matrices).
I Passivity requires the matrix to have positive imaginary part.
I A lossless material has hermitian symmetric material matrix.
I Polarization is in general elliptic, and can be left- orright-handed.
I The complex amplitude of the electric field has all informationon the polarization.
Daniel Sjoberg, Department of Electrical and Information Technology
Next lecture
I We “solve” the wave equation for a completely arbitrarymaterial.
I The fundamental parameters of wave propagation are defined:wave speed and wave impedance.
I Lecture notes available on the home page for the arbitrarymaterial part, rest in Orfanidis Ch2.1–5.
Daniel Sjoberg, Department of Electrical and Information Technology