ETEN05 Electromagnetic Wave Propagation Lecture 3: Time ...

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





6 Conclusions


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.


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


Outline

1 Introduction





6 Conclusions


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}


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 —


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.


Outline

1 Introduction





6 Conclusions


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}).


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)


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.


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:


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:


Example: permittivity of water

Microwave properties (one Debye model):

Light properties (many Lorentz resonances):


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′


Outline

1 Introduction





6 Conclusions


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)∗}.


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)


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.


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.


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)


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.


Outline

1 Introduction





6 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{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|


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)


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:

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.


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.


Outline

1 Introduction





6 Conclusions


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.


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.


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