A. La Rosa
Portland State University
Lecture Notes
A microscopic view of the index of refraction
Analysis from a DIFFERENTIAL EQUATION perspective
• Introduction
• Dielectric material response to a STATIC electric field
• Hypothesis about the relationship between the
polarization vector P and the electric field E
• Material's response to an ALTERNATING electric field E = Eo COS(ωt)
• The Maxwell Equations. The wave equation
Introduction
We know the speed of light appears to slow down when it travels
through a medium
Light
We also know that when ordinary light (the one from the sun , for
example) passes through a prism, it splits into its different color
components
Since different colors have different frequencies, the observations
indicate that the atoms in the prism interact differently with light
of different frequencies.
One way to describe the two phenomena mentioned above is to resort to the
Maxwell equations (ME), which describes the propagation of electromagnetic
waves inside a material medium.
The ME predicts that the light waves in the medium propagate with a speed
(v) smaller than the speed of light in vacuum(c). The ratio of these speeds is
called the index of refraction n of the medium.
n = c/v.
It turns out n depends on the frequency,
n = n(ω)
and typically
n>1
But this approach needs information about how specifically the material
responds to incident light.
In particular we know that electromagnetic fields induce polarization on the
material (i.e. induces every atom to become an individual dipole).
The induced polarization is characterized by the "Polarization vector P" of the
material.
In this section, we derive an expression for the polarization vector P using a
method where the atomic excitations (in response to an alternating electric
field) are modeled by simple harmonic oscillators.
This expression for P is then inserted in to the wave equation (an equation
derived by from the ME) and hence an expression for n(ω)
is obtained.
In short, we present in this section a mathematical model to calculate the
Polarization vector P of the medium, where we take into account the dynamic
response from individual atoms to t he external incident light. Thus is what
we call the MICROSCOPIC VIEW. P is then plugged into the wave equation (a
differential equation that result from the Maxwell equation), in which the
index of refraction n is identified. The latter constitute a DIFFERENTIAL
EQUATION VIEW of the index of refraction.
How can light interact with atoms?
It is because
light is a travelling wave of electric fields and atoms contain charges.
We can foresee then that the index of refraction of a medium has its origin
in the peculiar response of the charges (contained in the atoms) to the
electric fields (from the incident light).
E
Incident
Light
Atom
Electron cloud
oscillating back and
forth
Dielectric material response to a STATIC electric
field
-e
Charge distribution
under no external
electric field.
The "center of mass"
of the positive and
negative charges
coincide; hence, no
electric dipole exists)
An electrostatic field Eo is
applied, which causes a redistribution of charges
inside the atom
In this particular simplified
model p and Eo point in the
same direction
-e
Simplified model of
charge distribution
So, an external
electric field Eo
induces an "electric
dipole"
+e
(1)
What do a collection of electric dipoles do?
To grasp some ideas about the effect of a collection of dipoles, let's consider a
couple of parallel plates (see figure on the right side below), charged with
charges Q and -Q respectively. As a consequence, there exists an electric field Eo
between the plates.
Subsequently, we fill the region between the plates with some plastic material.
The net effect is the induction of electric dipoles (as shown below, in the figure
on the left) caused by the presence of the electric field Eo.
Since the charges between contiguous dipoles essentially compensate, the net
charge distribution in the plastic material is shown in the graph at the bottom:
negative charge - qind on the left side and qind on the right side.
Notice, the induced charge produce collectively an additional electric field
that point opposite to the external field Eo, which causes to have a net
electric field E of smaller magnitude than Eo.
Charge
+Q
Charge
-Q
Charge
+Q
Eo
Charge
-Q
Metallic plate
of area A
Metallic plate
of area A
The presence of a
material between
the plates gives rise
to the formation of
dipoles
The electric field between the
plates is given by,
Eo
An over-simplified version of the
above dipole charge distribution is
shown below:
The net electric field is given by
E
Charge
+Q
Charge
-Q
-qind
+qind
E
qind
What is the relationship between Q and qind?
qind
Charge
+Q
Let's take a unit volume
and add-up all the dipoles
contained in that volume.
l
Charge
-Q
A
The resulting total dipole
is called the
(2)
Polarization vector P
P is therefore a dipole moment density
Total dipole contained
between the plates
= PAl
On the other hand, notice in the figure above that one way to evaluate the
total dipole moment inside the plates is to add all the individual dipoles.
Adding dipoles in a row means adding qindd1+ qindd2 + qindd3 + .... (where all
the di are equal). Hence the result is qind(d1 + d2 + d3 + ...) =qind(l)
Total dipole contained
between the plates
Charge
+Q
-qind
l
Charge
-Q
qind
= qind
You can ignore the
following two pages
From the last two expression we obtain,
qind
=σind
The Polarization P of the
material (dipole moment per
unit volume) is equal to the
induced surface charge σind
(3)
The net electric field between the plate, can then be expressed
now in terms of the polarization
Charge - Q
Charge
+Q
qind
E
qind
-qind
qind
(4)
We can not go further, unless we know how the
polarization P depends on the electric field E
Also, still we do not have a relationship between
the free charge Q on the plates
and
the induced charge qindon the surface of the dielectric.
So, we can not find the net
electric field between the
plates
Q - qind
A εο
E=
Because we do not know
qind yet.
E
-qind
qind
E
Hypothesis about the relationship between P and E:
The polarizability χ
E
E
Notice,
the stronger E
the bigger the separation d
E value of the dipole,
the greater the
the larger the Polarization vector
E
Accordingly, it is plausible to think then that
the Polarization P
is proportional to
the Electric field
E being the net electric filed inside the material (no the externally applied
field)
The constant of proportionality is chosen to be of the form
(5)
called "Polarizability"
Polarization
vector
(It is determined experimentally
for each material)
Replacing (5) in (4),
Charge
+Q
Charge
-Q
E
E
(6)
which can be re-written as follows:
Since the electric field between the plates
when no material is present it given by
Eo=σ/εo, therefore expression (7) indicates:
(7)
E
E
K
(8)
That is, the presence of the material reduces the electric field by a factor of K;
K = ε/εo .
(9)
K is called the "dielectric constant" of the dielectric material
Material's response to an
ALTERNATING electric field E = Eo COS(ωt)
Electron attracted to the nucleus by electrical
forces.
This electrical centripetal force
prevents the electron from going
away from the nucleus.
Thus the electron ends up orbiting around the
nucleus in an equilibrium orbit.
We ae of course overlooking the fact that the the
accelerated charge emits radiation, hence making the
atom unstable. A QM view would be necessary to
explain why the atom is stable)
Under an external perturbation Eo cos (wt) we expect the electron
cloud to oscillate accordingly around its equilibrium orbit
This view makes appealing to use a
mechanical model to describe the
motion of the electron cloud
Spring model
e is a positive quantity
E = Eo cos(ωt)
In the previous sections of these NOTES we showed also
explicitly how to select the "radiation damping constant b " in
the mechanical model (b was given in terms of the atomic
quantities involved in the description of the electromagnetic
radiation emmited by a accelerated charge).
Assuming, then, that we have properly selected the spring
constant k and the damping constant b, the equaton of
motion for the electron's motion is,
Force "felt" by
by the atom inside
insi
the dielectric
!
(10)
e is a positive quantity
Electric field "felt" by the atom
inside the dielectric
We are already familiar with this equation. In the previous
section we found the following solution:
The electric dipole of a single atom will be given by,
=
=
(11)
!
If there are N atoms per unit volume, the Polarization is given
by,
P= Np
Eo
P(t) =
(12)
Thus, we find that the Polarization is out of phase
with the electric field "felt" by the atoms in the material
=
Eo
(13)
For the sake of keeping the analysis using real variables (omitting
for the moment a complex variable analysis) let's express the
Polarization vector P(t) into its Sin (ωt) and cos(ωt) components
=
=
= =
+
(14)
The first term on the right side of the equality in expression (14)
above is,
The second term on the right side of the equality in expression
(14) above is therefore responsible for the energy absorption of
the oscillator
=
Eo
+
(15)
Eo
Interpretation
Since the first term in (15) is proportional to the electric field
inside the material (both vary as COS(ωt) ), we will use the
analog expression P= χE used in electrostatic to define the
frequency dependent polarizability. E
Accordingly, notice in (15) that we can identify,
χ
(16)
Accordingly, the electrical permitivity will be given by,
=
(17)
=
E = Eo COS(ωt)
(18)
What does all this Polarization vector and Force oscillations
stuff has to do with the index of refraction n of a material?
n
Why is v c ?
=
(19)
In any region in space where does exists electric and magnetic
fields, the total field (the one resulting from adding
the field from the incident light,
plus the FIELD produced by the oscillating dipoles that
constitute the Polarization vector,
plus the contribution from any other source)
must satisfy the MAXWELL EQUATIONS (ME)
Maxwell Equations
Gauss' law
No magnetic
monopole
Faraday's
law
Ampere's
law
E
Equation of motion
for the electric
field travelling in
vacuum
E
(20)
This is nothing but the wave equation
E
E
(21)
E is a wave that travels with speed "c"
In expression (20) we can identify then the speed of light as,
(22)
Using the tabulated values for εo and μo, one obtains,
which happens to coincide with
the speed of light in vacuum
E
E
P
P