Electrical Properties of Matter Fields and Waves EE 6316 Spring 2008 1-30-2008
Electrical Properties of Matter
Fields and WavesEE 6316
Spring 20081-30-2008
Topics:• Review of Dipole, Polarization ,Susceptibility etc in isotropic medium
• Classical Harmonic Oscillator Model
•Abraham Lorentz Equation
•Damping
•Dispersion plots •Application of the model : some examples
•Plasma
• Dielectric behavior in anisotropic medium
•Permittivity tensor
• Example: Modulator
Review+
+
External fieldNo field
EP
whereEEEPED
e
re
r
r
rrrrrr
0
000
1
)1(
εχ
εεεχεε
=
==+=+=
∑ Δ=
→Δ i
i
V VpP
0lim
r
Qlpi =l
Q
In terms of molecular polarizability,α
localENPrr
α=
α has contribution due to electronic, ionic and permanent dipole polarization
Classical Electron Oscillator (CEO)Model
• electron cloud is modeled as a spring mass system, with attractive electric force between nucleus and electron cloud as the spring providing the restoring force
-
+-
l
E field
Mechanical Equivalent model of CEO
Mathematical development of CEO Model
( ) ⎟⎠⎞
⎜⎝⎛+−
==
=
=
=++⇒
==++
mdj
eEmQ
ell
ms
md
where
eEmQl
dtdl
dtld
eEQtEQsldtdld
dtldm
tj
tjss
tj
tj
ωωω
ω
α
ωα
ω
ω
ω
ω
220
0
0
0
0202
2
02
2
2
2
)(
r
r
rr
In the presence of an applied electric field: Abraham Lorentz Equation
d: Damping coefficient
s: spring constant
:Resonant frequency
Steady state solution
CEO Model :Damping Conditions
Underdamped3) α<ω0
Critically damped2) α=ω0
overdamped1) α>ω0
Classification of SolutionCondition
Frequency dependent dielectric response: Dispersion
( )
( ))Im()Re(
220
2
00
220
2
εεωωω
εεε
ωωω
j
mdj
mQN
EP
mdj
EmQN
P
−=⎟⎠⎞
⎜⎝⎛+−
⎟⎠⎞
⎜⎝⎛
+=+=
⎟⎠⎞
⎜⎝⎛+−
⎟⎠⎞
⎜⎝⎛
=
r
r
r
rPolarization:
Permittivity:
Dispersion Relation of Permittivity
Similar relationship for relative permittivity hence refractive index can be derived.The real and imaginary parts of refractive index are related to each other through Kramers-Kronig relationship.
Dispersion Plots
Application of CEO Model: Example
Ref: APPLIED OPTICS / Vol. 27, No. 12 /pp.2549-2553/1998
Further Application of CEO Model
• This model can be extended to model optical processes in semiconductors– Spontaneous and Stimulated emission– Rabi Oscillation– Collision Broadening – Radiative lifetimes etc
Plasma• Plasma is a sea of free electrons in a
background of positive ions of same density.
• Due to existence of free electrons, they are very conductive.
• Plasma response to electrical field is very strong too
Dynamics of Plasma• Motion of free electrons is governed by collision frequency f
Efm
fnejfmnej
jfmEneEjJEjHX
jfmEnevneJ
jfmEev
fvmEedtvdm
r
rrrrr
rrr
rr
rrr
⎥⎦
⎤⎢⎣
⎡
+−⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
++=+=∇
+=−=
+−=⇒
−−=
)()(
)(
)(
)(
22
2
22
2
0
2
00
2
ωωωεω
ωωεωε
ω
ωFor sinusoidal variation:
Convection Current:
Inserting in Curl Eqn:
Real Part Imaginary part=0 as f 0
Plasma Frequency
mne
where
mne
p
p
0
2
2
2
02
2
0 1
εω
ωω
εω
εε
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
Plasma frequency
Permittivity is negative for frequencies below plasma frequency. Physically this means wave is reflected off of plasma and attenuated inside.
Anisotropic Media: Dielectric tensors
[ ] [ ][ ]EDrr
ε=
zzzyzyxzxz
zyzyyyxyxy
zxzyxyxxxx
z
y
x
zzzyzx
yzyyyx
xzxyxx
z
y
x
EEED
EEED
EEED
EEE
DDD
εεε
εεε
εεε
εεεεεεεεε
++=
++=
++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Electro-Optic Effect: Modulator
cEnlr me
m 2
333ω−
=ΔΦModulation Index:
Suggested Reading
• Chapter 2 :Advanced Engineering Electromagnetics by Constantine A. Balanis
• Chapter 13: Fields and Waves in Communication Electronics by Simon Ramo