Orientational Polarizability
• No restoring force: analogous to conductivity
O
- p
H
+
θ
O
O
p=0
Analogous to conductivity, the
molecules collide after a certain
time t, giving:
α
CD
= oα
τωi−1
+q
C
H
-q
For a group of many molecules at some temperature:
f =e
−U
k bT
pE cosθ
k bT
= e
After averaging over the polarization of the
ensemble molecules (valid for low E-fields):
α DC
p2
~
3kbT
1
©1999 E.A. Fitzgerald
Dielectric Loss
•
•
•
For convenience, imagine a low density of molecules in the gas phase
C-M can be ignored for simplicity
There will be only electronic and orientational polarizability
ε r = 1+ χ e + χ o = n 2 +
Nα DC
3ε o (1− iωτ )
ωτ << 1, , ε r = ε so = n 2 +
Nα DC
3εo
ε so − n 2
∴ εr = n +
1− iωτ
2
ε’,ε’’
αe+αi
εso
We can write this in terms of a real and imaginary dielectric constant if we choose:
ε r = ε '+iε ''
ε so − n 2
ε so − n 2
; ε '' =
ε'= n +
ωτ
2 2
2 2
1+ ω τ
1+ ω τ
2
©1999 E.A. Fitzgerald
αe
n2
-2
0
logωτ
Water molecule: τ=9.5x10-11 sec, ω~1010
microwave oven, transmission of E-M waves
+2
2
Dielectric Constant vs. Frequency
•
Completely general ε due to the localized charge in materials
ε
molecules
αo
ions
αi
n2
1
electrons
αe
1/τ
ωT
ωoe
ω
Dispersion-free regions, vg=vp
3
©1999 E.A. Fitzgerald
Dispersion
•
Dispersion can be defined a couple of ways (same, just different way)
– when the group velocity ceases to be equal to the phase velocity
– when the dielectric constant has a frequency dependence (i.e. when dε/dω not 0)
ω
Dispersion-free
Dispersion
ω =
vp =
vp =
ω
k
ω
k
=
=
c
ε
r
c
εr
k
=
∂ω
= vg
∂k
∂ω
c
≠
= vg
ε r (ω ) ∂k
k
4
©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
•
Snell’s Law
θ1
n1
Boundary conditions for E-M wave gives
Snell’s Law:
n2
Refraction
n1 sin θ1 = n2 sin θ 2
θ2
θ1
θ2
n1
n2
Internal Reflection: θ1=90°
θ 2 = θ c = sin −1
n1
n2
Glass/air, θc=42°
5
©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
•
•
Attenuation
– Absorption
• OH- dominant, SiO2 tetrahedral mode
– Scattering
• Raleigh scattering (density fluctuations) αR~const./λ4 (<0.8 μm
not very useful!)
Dispersion
– material dispersion (see slide i13)
x
•Light source always has Δλ
•parts of pulse with different l propagate at
– modal dispersion
different speeds
Black wave arrives later than red wave
y
n1
Solution: grade index
©1999 E.A. Fitzgerald
n2
n
6
Characteristics of Optical Fiber
Egyptian
106
Venetian
Optical loss dB/km
105
Bohemian
104
Window Glass
103
100
10
1
0.1
3000 BC
1000 AD
1964
1974
1988
Figure by MIT OpenCourseWare.
7
©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
Image removed due to copyright restrictions.
Please see any diagram of the typical optical attenuation curve, such as
http://www.nikhef.nl/~nooren/dispersie3_files/image010.jpg
8
©1999 E.A. Fitzgerald
Characteristics of Optical Fiber
Pluse spreading (ns/km)
-8
-6
LED ∆λ = 30nm
-4
-2
-0
2
Laser ∆λ = 3nm
Narrow Width
Laser ∆λ = 0.1nm
0.7
0.8
0.9
1.0
1.1
Wavelength (µm)
1.2
1.3
1.4
Figure by MIT OpenCourseWare.
9
©1999 E.A. Fitzgerald
Ferroelectrics
• ‘Confused’ atom structure creates metastable relative positions of
positive and negative ions
Barium titanate (BaTiO3)
Titanium (Ti4+)
Barium (Ba2+)
Oxygen (O2-)
Figure by MIT OpenCourseWare.
10
©1999 E.A. Fitzgerald
Ferroelectrics
• Each unit cell a dipole!
• Very large Ps (saturated polarization, P(E=0)
• No iron involved; ‘Ferro’ since hysterisis loop analogous to magnetic
materials
E
Two equal-energy atom positions
Can flip cell polarization by applying
large enough reverse E-field to get over
barrier
P
Ps
ΔE
‘normal’ dielectric
Ro
Ec
E
R
11
©1999 E.A. Fitzgerald