Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics,...

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´ Contents Simple Model for (ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern´ andez Professor: Jorge Alfaro Instituto de F´ ısica Pontificia Universidad Cat´ olica de Chile 1/23

Transcript of Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics,...

Page 1: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Frequency Dispersion: Dielectrics, Conductors, and Plasmas

Carlos Felipe Espinoza Hernandez

Professor: Jorge Alfaro

Instituto de FısicaPontificia Universidad Catolica de Chile

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Page 2: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Contents

1 Simple Model for ε(ω)

2 Anomolous Dispersion and Resonant Absorption

3 Low-Frequency Behavior, Electric Conductivity

4 High-Frequency Limit, Plasma Frequency

5 Example: Liquid Water

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Page 3: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Simple Model for ε(ω)

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Page 4: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Simple Model for ε(ω)

Electron of charge −e bound by a harmonic force to an atom, and acted onby an electric field

Equation of motion:

m[~x + γ~x + ω2

0~x]

= −e~E (~x , t) (1)

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Page 5: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

If the field varies harmonically in time with frequency ω.

~E ∼ ~Ee−iωt (2)

~x ∼ ~x0e−iωt (3)

Dipole moment contributed by one electron:

~p = −e~x =e2

m(ω20 − ω2 − iωγ)

~E =e2

m

(ω20 − ω2) + iωγ

(ω20 − ω2)2 + γ2ω2

~E (4)

Difference of phase between the electric field and the dipole moment:

φ = tan−1

(ωγ

ω20 − ω2

)(5)

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Page 6: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Atomic contribution to the dielectric constant for N molecules per unitvolume, fj electrons per molecule with binding frequency ωj and dampingconstan γj :

ε(ω)

ε0= 1 + χe = 1 + N

e2

ε0m

∑j

fj(ω2

j − ω2 − iωγj)(6)

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Page 7: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Anomalous Dispersion and ResonantAbsorption

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Page 8: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Anomalous Dispersion and Resonant Absorption

In a dispersive medium, the wave equation for the electric field reads

∇2~E = εµ0∂2~E

∂t2(7)

it admits plane wave solutions

~E (z , t) = ~E0ei(kz−ωt) (8)

with the complex wave number

k =√µ0εω (9)

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Page 9: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

We can write the wave number in terms of this real and imaginary parts

k = β + iα

2(10)

and the electric field becomes

~E (z , t) = ~E0e−α

2ze i(βz−ωt) (11)

Intensity proportional to E 2 (to e−αz), α is called the absorptioncoefficient.

Index of refraction: n =cβ

ω

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Page 10: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

ε(ω)

ε0= 1 + N

e2

ε0m

∑j

fj(ω2

j − ω2 − iωγj)

The damping constants γj aregenerally small compared withthe resonant frequencies ωj .

Normal dispersion region isassociated with the increase ofRe(ε) with ω.

Anomalous dispersion (resonantabsorption) is where theimaginary part of ε isappreciable, and that representsdissipation of energy.

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Page 11: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Low-Frequency Behavior, ElectricConductivity

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Page 12: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Low-Frequency Behavior, Electric Conductivity

ε(ω)

ε0= 1 + N

e2

ε0m

∑j

fj(ω2

j − ω2 − iωγj)

Fraction f0 of electrons per molecule free (ω0 = 0)

ε(ω) = εb(ω) +Ne2

m

f0−ω2 − iωγ0

= εb(ω) + iNe2

m

f0ω(γ0 − iω)

(12)

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Page 13: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Maxwell-Ampere equation: ~∇× ~H = ~J + ∂D∂t

Assuming medium obbeys Ohm’s law: ~J = σ~E , ~D = εb~E

~∇× ~H = −iω(εb + i

σ

ω

)~E (13)

Assuming all the properties due the medium: ~D = ε(ω)~E

~∇× ~H = −iωε(ω)~E (14)

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Page 14: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

By comparison we get the electric conductivity (model of Drude)

σ =f0Ne

2

m(γ0 − iω)(15)

The dispersive properties of the medium can be attributed as well to acomplez dielectric constant, as to a frequency-dependent conductivity and adielectric constant.

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Page 15: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

High-Frequency Limit, PlasmaFrequency

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Page 16: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

High-Frequency Limit, Plasma Frequency

ε(ω)

ε0= 1 + N

e2

ε0m

∑j

fj(ω2

j − ω2 − iωγj)

At frequencies far above the highest resonant frequency

ε(ω)

ε0≈ 1−

ω2p

ω2(16)

where ωp =NZe2

ε0mis called the plasma frequency of the medium.

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Page 17: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

The wave number is given by

ck =√ω2 − ω2

p (17)

In dielectric media, the approximation holds only for ω2 >> ω2p.

In plasmas, the electrons are free and the damping force is negligible, sothe approximation holds over a wide range of requencies, includingω < ωp.

In conductors, the approximation holds for frequencies ω >> γ0 and thebehavior of incident waves for ω << ωp is similar to the behavior forplasma

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Page 18: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Example: Liquid Water

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Page 19: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

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Page 20: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

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Page 21: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

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Page 22: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Bibliography

1 J. Jackson, Classical Electrodynamics. Chapter 7.5

2 D. Griffiths, Indtroduction to Electrodynamics. Chapter 9.4

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Page 23: Frequency Dispersion: Dielectrics, Conductors, and Plasmas · Frequency Dispersion: Dielectrics, Conductors, and Plasmas Carlos Felipe Espinoza Hern andez Professor: Jorge Alfaro

Contents Simple Model for ε(ω) Anomolous Dispersion and Resonant Absorption Low-Frequency Behavior, Electric Conductivity High-Frequency Limit, Plasma Frequency Example: Liquid Water

Thank You

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