# Optics: Lecture 2 Light in Matter:... Dispersive Prism: Dependence of the index of refraction, n(),...

```Optics: Lecture 2
Light in Matter: The response of dielectric materials
Dispersive Prism: Dependence of the index of refraction, n(), on frequency or
wavelength of light.
Sir Isaac Newton used prisms to disperse white light into its
constituent colors over 300 years ago.
When white light
passes through a
prism, the blue
constituent
experiences a larger
index of refraction
than the red
component and
therefore it deviates
at a larger angle, as
we shall see.
The effect of introducing a homogenous, isotropic dielectric changes Maxwell’s
equations to the extent that o   and o  .
The phase speed in the medium becomes:
v  1 / 
The ratio of the speed of an E-M wave in vacuum to that
in matter is defined as the index of refraction n:
c

n 
 KE KM
v
 o o

KE 
o

KM 
o
Relative Permittivity and Relative Permeability
For most dielectrics of interest that are transparent in the visible, these
are essentially non-magnetic and to a good approximation KM  1.
To a good approximation also known as Maxwell’s Relation:
n  KE
KE is presumed to be the static dielectric constant (and works well only for
some simple gases, as shown on next slide).
In reality, KE and n are actually frequency-dependent, n(), known as
dispersion.
Resonant process h  E
n
Scattering and Absorption:
h
m n
•Non-resonant scattering:
•Energy is lower than the resonant
frequencies.
•E-M field drives the electron cloud into
oscillation
•The oscillating cloud relative to the
positive nucleus creates an oscillating
dipole that will re-radiate at the same
frequency.
 Em
Gas/Solids
Excitation energy can be
transferred via collisions
before a photon is re-emitted.
Works well
Doesn’t work so well
A small displacement x from
equilibrium causes a restoring force F.
F = -kEx
and results in resonant frequency:
o  k E / me
E(t)
x-axis
+
-
Light  E(t) and produces a
classical forced oscillator.
Amplitude ~ 10-17 m for bright
sun light.
The result can be modeled like a classical forced oscillator with
FE = qeEocos(t) = qeE(t). Using Newton’s 2nd law:
Driving Force – Restoring Force = ma, where Rest. Force = -kEx
2
d
x
2
 qe Eo cos t  meo x  me 2
dt
To solve, let x(t) = xocost
 qe Eo cos t  meo xo cos t  me 2 xo cos t
2
qe Eo
qe Eo
qe E (t )
 xo 
; x
cos t ; x 
2
2
2
2
2
me o  
me o  
me o   2




*Note that the phase of the displacement x depends on
 &gt; o or  &lt; o which gives x   qE(t).


The electric polarization or density of dipole moments (dipole
moment/vol.) is given by:
qe2 NE / me
P  qe xN  2
o   2
Where N = number
electrons per volume.
 
We learn from the dielectric properties of solids that (   o ) E  P.
2
q
P
(
t
)
Therefore
e N / me
  o 
 o  2
E (t )
o   2
Since n2 = KE = /o it follows that we obtain the following
dispersion equation:
*Note that  &gt; o  n &lt; 1 (above resonance)
2
 1  (Displacement is 180 out-of-phase with
q
2
e N
 2

n ( )  1 
2  driving force.)
 o me  o   
and  &lt; o  n &gt;1 (below resonance)
(Displacement is in-phase with driving force.)
For light  = ck = 2c/, we can write the dispersion relation as
(n 2  1) 1  C / 2  C / 2o ; C  4 2 c 2 o me /( Nqe2 )
Thus, if we plot (n2-1)-1 versus -2 we should arrive at a straight line.
•In reality, there are several transitions in which n &gt; 1 and n &lt; 1 for
increasing , i.e., there are several oi resonant frequencies
corresponding to the complexity of the material.
•Therefore, we generalize the above result for N molecules/vol. with fj
different oscillators having natural frequencies oj, where j = 1, 2, 3..
2
qe N
2
n ( )  1 
 o me

fj 
j   2   2 
 oj

A quantum mechanical treatment shows further that
f
j
1
j
where fj are weighting factors known as Q.M. oscillator strengths
and represent the transition probability for each mode j.
The energy  oj is the energy of absorption or emission for a
given electronic, atomic, or molecular transition.
•When  = oj then n is discontinuous (and blows up). Actual
observations show continuity and finite n.
•The conclusion is that a damping force which is proportional to the
speed medx/dt should generally be included when there are strong
interactions occurring between atoms and molecules, such as in
liquids and solids.
•With damping, (1) energy is lost when oscillators re-radiate and (2)
heat is generated as a result of friction between neighboring atoms
and molecules.
•The corrected dispersion, including damping effects, is as follows:
2
q N
n 2 ( )  1  e
 o me


fj

j   2   2  i  
j
 oj

This expression often works fine for gases.
•In a dense solid material, the atoms/molecules may experience an additional
field that is induced by the surrounding medium and is given by P(t)/3o.
•With this induced field, the dispersion relation becomes
Nqe2
n2 1

2
n  2 3 o me

j
fj
2
oj
  2  i j 
•We will see that a complex index of refraction will lead to absorption.
•Presently, we will consider regions of negligible absorption in which n
is real and oj2   2   j.
•Thus
Nqe2
n2 1

2
n  2 3 o me
For various glasses,

j
fj
2
oj
2
oj2   2 , n2 ()  Const.
Since oj ~ 100 nm in the ultra-violet (UV).
•Note that as   oj, n() gradually increases and the behavior is called
“Normal Dispersion.”
•Again, at  = oj, n is complex and leads to an absorption band.
•Also, when dn/d &lt; 0, the behavior is called “Anomalous Dispersion.”
When white light
passes through a
glass prism, the
blue constituent
experiences a larger
index of refraction
than the red
component and
therefore it deviates
at a larger angle, as
seen in the first
slide.
•Note the rise of n in the
UV and the fall of n in the
IR, consistent with
“Normal Dispersion.”
•At even lower frequencies
materials become again
transparent with n &gt; ~1.
•Transparency occurs
when  &lt;&lt; o or  &gt;&gt; o.
•When  ~ o, dissipation,
friction and therefore
absorption occurs, causing
the observed opacity.
Propagation of Light, Fermat’s Principle (1657)
Involves the principle of least time: The path between two points that is taken
by a beam of light is the one that is transversed in the least amount of time.
t SP
SO OP
t 

vi
vt
b 2  (a  x) 2
h2  x2


vi
vt
To find the path of least time,
set dt/dx=0.

dt
x
(a  x)


dx vi h 2  x 2 vt b 2  (a  x) 2
0
sin  i sin  t


 ni Sin  i  nt Sin  t
vi
vt
Since ni = c/vi and nt = c/vt .
Snell’s Law of Refraction
c
1 c o
where o is the
Note that   v    

n
n
n vacuum wavelength.
In general, for many layers having different n, we can write
t SP
m
sm
s
s1 s2
 t    ...   i
v1 v2
vm i 1 vi
1 m
c
t   ni si ; ni 
c i 1
vi
Note that if the layers are very thin, we
can write
m
n s
i 1
i i
P
  n( s)ds  OPL
S
= OPL (Optical Path Length)
We can compute t as simply
OPL
t
.
c
P
m
Note that the spatial path length is
 s   ds
i 1
i
and for a medium possesing a fixed index
n1,
S
OPL
o

s

.
•Fermat’s principle can be re-stated: Light in going from SP traverses the
route having the smallest OPL.
•We will begin next with the E-M approach to light waves incident at an
interface and derive the Fresnel Equations describing transmission and
reflection.
```