A. La Rosa Lecture Notes APPLIED OPTICS ________________________________________________________________________ Lecture‐5: Light propagation in non‐
magnetic materials 5.1 Surface (pol). volume (pol), and current (jpol) densities produced by polarization charges The objective in this section is to derive expressions for the surface, volume and current densities contributed by charges bound to atoms (i.e. charges that do not travel freely in a medium, in contrast to the free charges in metals). Subsequently, in the next section, those expressions will be properly integrated into the Maxwell equations. That is, at this stage, we distinguish the bound charges (typical from dielectric materials) from the free mobile charges (encountered in metals). That way we will be able to identify the effects of bound charges to the propagation of light in such a non‐conducting medium 5.1.A The Polarization vector An electrically neutral atom acquires a dipole moment upon the application of an external electric field. 
Neutral
atom
qp
+
- qp -
Electric
field
E
p +
Electric
dipole
E
r
P
Fig.1 An electric field induces electric dipoles on a neutral atom. A collection of small dipoles in a volume 
The induced electric dipole p is expressed as 

p  q p
In a given material subjected to the application of an external electric filed, the vector POLARIZATION is defined as 
P  total dipole moment per unit volume
(1)
Let N be the number of molecules per unit volume in the material. Assuming that in a given unit volume each atom acquires the same dipole moment, then, 

PN p
Notice, in general   
P  P(r )

That is, P may vary from point to point inside the material. (2)
5.1.B Polarization‐charge surface density  pol
If the polarization of a material is not uniform, the application of an external electric field will, in general, create accumulation of charges non uniformly across the material volume as well as on the boundary surface. A special case occurs when the material is 
isotropic, i.e. P is uniform throughout the material, where only net accumulation of polarized charge occurs at the materials boundary. Case: Isotropic materials The uniformity of the polarization P throughout the volume of the slab, implies that there is not accumulation of charge at the interior of the slab. Notice that in the dashed volume #1, shown in the figure below, the amount of induced charges getting inside that volume is the same as the one leaving out. Accumulation of charge happens only at the boundaries of the dielectric slab. In volume #2, only negative charges leave that volume, thus leaving uncompensated positive charges; the latter constitutes the surface charge density.) -Q
-Q
2 + + + + + + + +
1 Qpol
+ + + + + + + +
- - - - - - - -
-Qpol
P
- - - - - - - A
+Q
pol = Qpol /A
+Q
Fig.2 Absence of net charge accumulation inside the bulk of an isotropic material under 
a uniform external electric filed. Still, notice there exists a polarization
that is constant P

(  0 ) throughout the interior of the material. Let’s find out a relationship between the surface density of the polarization charge and the polarization. To that effect, consider one of the two layers, the top layer for example (see Fig.3): 
+ + + + + + + +
Qpol
A
Fig.3 Zoom‐in of the top section of Fig.2 in order to visualize better the charge distribution and relate it to the polarization vector. Total number of molecules in the top layer is N x AEach molecule, having a dipole p=qecontributes with an amount of charge qe to the surface charge.Hence, the total amount of charge in the layer is 
Qpol = qe Nx A and the surface charge density is then given by,  pol
= Qpol / A = qe N (i) On the other hand, the total dipole moment of the layer is: (total number of molecules in the layer) x p = (N Ax p
Since the volume of the layer is A, the dipole per unit volume (in the layer) is then given by P = (NAx p) / (A)
= N p = N qe  (ii) From (i) and (ii), P   pol (2) The polarization surface charge density is numerically equal to the polarization inside the material. Notice, although the analysis above considered an extended area A, the argument is valid for the case of a more localized area on the surface. That is, if P varied along the sides of the slab (but still uniform along the vertical direction), one would obtain, 

P( r )   pol ( r ) that is, the local surface charge density depends on the local value of the polarization vector. Polarization‐charge surface density along an arbitrary surface In the previous case we found the surface density of the polarization‐charge along a 
surface perpendicular to the polarization vector P . I many occasions we will encounter boundaries which do not align perpendicular to P and still we have to find the correspondent surface charge densities. Here we find such a relationship 
First, notice that when the surface is perpendicular to P , then the total polarization‐

charge crossing the surface is P times the area, or  pol  P . But if P were tangential to the surface, no net charge crosses the surface  pol  0 . p +
p
-
p
+
- +
-
Fig.4 Upon the application of an electric field, the net amount of charge crossing a hypothetical surface (dashed lines) depends on the relative orientation of the induced dipoles 
c Qp
b a 
n

P Fig.5 Another view of the same situation depicted in Fig.4, this time to quantify the surface density of the polarization charge established along local surfaces of different orientations relative to the 
polarization vector P . In figure 5, Qp crosses the surface area ab and establishes a surface density Qp /(ab),
which, we know, it is equal to P,
Qpol /(ab) = P
However, the same charge establishes a charge density along the surface cb
n =Qpol /(bc) Since a= c Cos()


n = [Qpol /(ba) ] Cos()


 = P Cos()


n = P  n Generalizing this result,  
 pol  P  n
Net surface charge density along an hypothetical 
surface perpendicular to the unit vector n .
(3) P
n
Surface
r
  

Fig.6 A local surface charge density  pol (r )  P(r )  n crosses an hypothetical surface upon the application of an electric filed. 5.1.C Polarization volume charge density  pol 
In anisotropic materials, the non‐uniformity of P can give rise to a net accumulation of charges inside the dielectric. For example, the diagram below shows two different dielectric slabs. Because the polarizations are different, the volume of the dashed boundary comprises a net polarization charge -Q
P1
P2
+Q
Fig.7 Accumulation of charges occurs when the polarization is not uniform
  
The diagram below displays a more general case, where it is assumed that P  P(r ) .  
P(r )
n
Qpol
Volume V
Surface S
Fig.8 Diagram to evaluate the volumetric polarization‐charge density
 
The amount of charge crossing the boundary of the volume V would be  P  n dA . An S
equal excess of charge, of opposite sign, is left behind inside the volume. Thus,  
 Q pol    P  n dA S
The charge in the volume V can be considered as a charge distributed according to a charge density pol.
 Q pol    pol dV
V
Hence,  
  pol dV    P  n dA
S
V
The latter surface integral can be converted to a volume integral using the Gauss  

mathematical theorem  P  n dA     P dV ,
S

V
Hence, pol
V

dV      P dV
V

 pol    P
(4)
r
pol
 
P(r )
 
Fig. In a material of non‐uniform polarization P(r ) , its divergence gives the net density of charge in the material. 
5.1.D Polarization current density j pol Expression (4) is also valid when the driving electric filed vary with time. As  pol 
varies with time, it will generate a corresponding polarization current density j pol . Since the polarization charges are real charges (and no fictitious charges) the conservation of charges should apply. Accordingly,  pol
t

   j pol  0 (statement of charge conservation)





  P
P
P 
Using (4), 
   j pol   
   j pol    ( 
 j pol )  0 . This expression t
t
t
is satisfied if, 

P
Polarization current density
(5)
j pol 
t