OPTICAL PROPERTIES OF MATERIALS Summer 2002

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OPTICAL PROPERTIES OF MATERIALS - Summer
2002 Wednesday, May 15 Some reminders about free
space propagation of plane waves
(vacuum)
Plane wave solution
(as P 0)
For non-magnetic media,
Poynting vector
2
Take time average
Now,
Hence,
0
in vacuum.
For vacuum / isotropic media
3
For anisotropic media,
We will not be concerned with the details of
anisotropic propagation in this course.
Waves normally incident on a dielectric
interface We are mainly going to be interested
in the material properties, not in the details of
the electromagnetics, so we will just look at the
result for normal incidence
Usually in this course, we are interested in the
case where one of the materials is air or vacuum,
e.g. ?i1.
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- Now that we know how R, T and A relate to n and
?, and how these relate to ?. - We need a
physical model for ?(?).
Classical Lorentz oscillator model for absorption
dispersion
Atom
Atom in E-field
Force on electron
Assume electrons bound to atom by a linear
restoring force
(Hooke-s law)
Equation of motion
To solve, take Fourier transform
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Now the induced dipole moment is defined as
Now,
Atomic or molecular polarizability - generally a
tensor. However, we will treat as a scalar.
(Careful not to confuse with absorption
coefficient.)
With N atoms per unit volume, the net dipole
moment per unit volume is
Hence, in scalar terms,
- We have assumed local and macroscopic fields
are equal, and have ignored spatial averaging -
assumed all dipoles are free to point in
direction of field.
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Note that ? is dimensionless, so that the term
has dimensions of ?2. ? we set
For reasons that will become apparent later,
?p is known as the plasma frequency.
See fig 3.1 in Wooten for plots of these
quantities.
Resonance Approximation We can simplify these
expressions near resonance, under the
approximation ?0 - ? ltlt ?0. In this case, ?0
? ? 2 ?0 , therefore
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Hence,
- Resonance Approximation ? is the Full-
width at half maximum (FWHM) of the imaginary
part of ?, which has a Lorentzian functional
form. Note the symmetry of the real imaginary
parts. - ? is symmetric about ?0, while ? is
antisymmetric about ?0. Notice that in the
resonance approximation that ? appears
antisymmetric (i.e. about ? 0) while ? appears
symmetric. The converse is actually true, as
seen before from the reality condition. - This
serves to highlight that the resonance
approximation is strongly invalid far from
resonance. Real Atoms TRK Sum Rule In general,
atoms molecules have several resonances, not
just a single one as in our model. - Quantum
mechanically, there are several resonances
electronic resonances, and molecules may exhibit
vibrational and rotational resonances also. It
turns out that perturbation methods in quantum
mechanics yield a result very similar to the
classical one. We find
?j is the frequency for a transition between two
electronic states with energy difference
. ?j is the decay rate for the final state and
fj is known as the oscillator strength, which
obeys the Thomas-Reich-Kuhn sum rule
for an atom with Z electrons. This tells us that
the total absorption, integrated over all
frequencies is dependent only on Z. Usually one
resonance dominates all others.
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j4
j3
j2
j1
f4
f3
f2
f1
Ground State
?
?
The naturallinewidth is dictated by ?. Recall
? is a damping or decay rate, which corresponds
to (1/lifetime) of a state. This could range
from kHz to GHz. Often the electronic states are
split into many sub states. In molecules, each
electronic state can exist for many possible
vibrational or rotational states of the molecule.
Which strongly broadens the electronic states
into bands. In solids, the electronic levels
are broadened into very broad electronic energy
bands. All of these broaden out the optical
resonances far in excess of ?. In addition to
several electronic resonances, other degrees of
freedom, such as atomic motions, including
molecular vibrations, lattice vibrations,
molecular rotations, can interact with the
electromagnetic field, producing many resonances
over the electromagnetic spectrum. This is
illustrated in Wooten, Fic 3.2.
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Our calssical, or quantum, model gives the real
and imaginary parts of the susceptibility, from
which it is easy to obtain the dielectric
function, ?r (?) 1 ?(?). Here ?r ? / ?0
is the relative permittivity. It is less
straightforward, but still not difficult to find
expressions for n(?) and ?(?). We start from
From which we obtain
Armed with n and ?, we can find the absorption
and reflectance of the material. Wooten
illustrates this in Figs. 3.3 and 3.4. It is
often illustrative to plot n, ?, absorption ? and
R for different values of ?0, ?p and ?. On the
next page is a example, using values of 4, 8 and
1, for each of these parameters, respectively. -
These seem like strange, unrealistic numbers to
use, but our expression for ? contains only
frequencies, so the scale is relative. However,
it is realistic to think of these numbers as
photon energies, , in electron volts
(eV). Note the shapes of the curves are as we
would expect from Kramers-Kronig relations.
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Lorentz model for ?0 4, ?p 8, ? 1.
Note that ? drops below zero between ?0 and ?p .
We can calculate n and ? from the above
relations. n is small over the region where ? is
negative. Above ? p , n gradually rises to 1.
Noting that ? 2? ?/c, we have also plotted a
scaled version of ? , by plotting 2??, scaled to
? p. - Note how ? is skewed to higher
frequencies.
R is large in the region where n is small, which
makes sense, as in the limt of n?0, R ?1.
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Lorentz model for ?0 4, ?p 8, ? 0.3.
As expected, for a smaller damping, ? 0.3, the
curves are narrower and have larger maximum
values.
Above ?0, n is well below unity. Above ?p, n
rises, but only in the high-frequency limit does
n approach unity. This is a good predictor of
actual materials. - Above the highest frequency
resonance, usually in the deep uv/soft x-ray
region, n is indeed less than unity. Causality
is not violated, though. Note that near ?p, ? 0
and hence n ?.
The reflectance is much higher and has sharper
edges. Note the high R starts around ?0 and falls
near ?p. This is because ? gtgt n in the
reflecting band. Note ? is large above ?0 even
where ? has dropped to a small value.

• T-A-R-T
• As described in Wooten, the material has four
  distinct regions of optical properties,
• Transmissive, ? lt ?0 - ?/2,
• Absorptive, ?0 - ?/2 lt ? lt ?0
• Reflective, ?0 ?/2 lt ? lt ?p
• Transmissive ? gt ?p
• These frequency ranges are approximate. The
  regions are more distinct for smaller ? and
  larger ?p.

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Addendum to handout from Thursday, Sept
4 Solids Real solids are very different from
single atoms, but most of the above principles
are still useful. As individual atoms are
brought together to forma solid, electron
interactions cause levels to broaden out into
bands. But these bands still retain some of the
characteristics of their atomic origins. For
example, in GaAs, the valence bands are mainly
p-like in nature, while the conduction band is
s-like.
Energy
Atom Solid
In covalent bonded solids, (e.g. GaAs, Si, Ge)
outer electrons are quite evenly shared between
different atoms, so the electron orbitals are
extended throughout the crystal, and the energies
bands corresponding to these orbitals are very
broad. However in ionic bonded solids, (e.g.
NaCl, MgF2, KCL) electrons are much more
localized and the bands are narrower. The
absorption spectra of these materials retain more
of the characteristics of atoms. (See figures
3.6 and 3.7 in Wooten.) Density of States and
Fermi Golden Rule An important property of
continuous energy bands relevant to their optical
properties is the Density of States, ?(E). The
quantity ?(E)dE number of states (per atom) in
the energy range (E, EdE). This is used in the
quantum mechanical transition rate from state k
to state n, Wk?n , upon illumination with photons
of energy
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Where E0 is the applied electric field and ?kn
is the transtion matrix element for states k
and n. - This tells us how strong the transition
is. For some states, ?kn 0 (forbidden
transition). Relation of W to absorption
coefficient Wk?n rate of photons absorbed
per atom
So the absorption coefficient depends on the
dipole transition matrix element and on the
density of states.