Consider a linear dielectric medium. Then 
and 
are related
by:
where 
is a 
matrix. The elements will be complex if
there are losses in the material; for physically real materials, there is
the constraint:
We assume the crystal is lossless, so (5.2) implies is a real symmetric matrix (hence reciprocity holds).
Now, every symmetric matrix can be diagonalized by rotating the coordinate
axes to a certain reference frame, called the principal axes. In this
coordinate system, has the form:
so:
The scalars 
, i=1,2,3, are called the principal
permittivities. The dielectric energy density U associated with the field
is:
If we normalize the field via:
and we define:
(which assumes 
) then:
This is the equation of an ellipsoid, illustrated in Figure 5. It is denoted by various names, including the index ellipsoid, index of wave normals, optical indicatrix and reciprocal ellipsoid.
Define the effective index of refraction 
via:
This is motivated by the property that:
which corresponds to the formula for isotropic media. The value of
can be read from the index ellipsoid as follows; consider a ray emanating
from the origin in the direction of the field (i.e., in the
direction of polarization of the field). The length of the ray from the
origin to the intercepted point on the index ellipsoid is the value of the
effective index of refraction, and this motivations the name given to the
ellipsoid.
Crystals are classified according to certain structural symmetries which, in turn, impose symmetries upon the index ellipsoid.
;
the ellipsoid is a sphere; the material is isotropic.
. The ellipsoid is an ellipse with one axis
along the 3-direction, rotated around the 3-axis. That is, the ellipsoid
exhibits the same symmetry as the crystal. Such crystals are called uniaxial.
and so are call
ed biaxial. The structure is orthorhombic, monoclinic or triclinic. All three
principal axes of the ellipsoid are different.
For a plane wave characterized by wavenumber vector 
(that is, 
, etc.) in a
linear homogeneous isotropic medium, we have , and in the
same direction, all are 
to 
, and all are to the
Poynting vector 
. The Poynting vector, computed as:
is in the direction of power flow and its flux through a surface is the
power flow through the surface; it has units of 
. In optics, the
Poynting vector is called the ray vector.
For anisotropic materials, the orthogonal relation between and , and the coline
ar relation between and is not
apparent. The following results, however, do hold:
, and are all in the plane
to . , we can find two
linearly polarized waves with orthogonal ; that is, we can find
valid two linearly polarized waves, 
and 
, respectively, such
that , 
and are mutually perpendicular.
To prove the first statement, we write:
and consider Maxwell's equations in phasor form:
Here, 
is the phasor displacement
current (in the time domain, it is 
). The plane wave formulation leads to:
and hence is to and . Similarly:
leads to:
and hence is to and . Also (5.11) implies 
and 
.
As we continue the analysis, let 
denote the unit vector in the
direction of propagation, that is:
Also, is related to the phase velocity 
of the wave via:
We can define the (effective) index of refraction as:
Note that for anisotropic media the value of n is not constant but may
depend on . We can combine equations (5.14) and (5.16)
to obtain:
and hence:
Since is a unit vector, we can interpret 
as the negative of the component of
that is
to . For isotropic media, 
; for
anisotropic media, has a component colinear with . However,
(5.21) implies 
and in fact is in the
direction of the component of which is to .
The magnitude of is 
times this component of . We can make a further statement. , 
, and all lie in the same plane; 
and 
.
That is, 
is the component of in the
direction of , so 
is the component of
which is to . Hence:
In the principal coordinate system, defined by the unit vectors 
, i=1,2,3, let:
and similarly for . Then:
and hence:
Equation (5.25) leads to the following result:
We also have:
since is a unit vector. Define the principal velocities of
propagation 
as:
Then we can combine (5.26) and (5.27) to obtain:
which is called Fresnel's equation of wave normals. We correspondingly can write:
Note that the principal velocities are constants of the material,
whereas the phase velocity of a plane wave depends on its direction of
propagation , this relation being governed by Fresnel's equation (5.29). Indeed, given , it can be shown that we obtain a
quadratic equation for , which implies there are two phase
velocities and hence two indices of refraction associated with every
direction of propagation. Let us examine the details a bit more carefully.
Clearing the denominators of (5.29) yields:
which is a quadratic equation in 
. The discriminant can be shown to
be positive, so there are two real roots, and the sign variation indicates
both roots are positive; that is, there are two possible positive real
values for . Each yields a positive and a negative value for ;
we reject negative values for , and conclude there will be two positive
real values for .
Once we solve for , we can compute the components 
, i=1,2,3, up
to a scalar multiple ( 
), and in particular there is a fixed direction
for corresponding to each choice of . From the direction of we determine the direction of .
We conclude that there are two possible directions for for
each . Closer examination indicates that all components of
are real, and hence these two fields are each linearly polarized. We can
visualize this from the index ellipsoid as possible. Given , we cut
the ellipsoid in the center with a plane to . This
plane cuts the ellipsoid on a curve which is an ellipse. The lengths of the
semiaxes of this ellipse gives n (and hence, indirectly, ) and the
direction of these two choices for .
Let us specialize this result to the case of uniaxial crystals. The axis of
symmetry of the index ellipsoid is called the optic axis, and let us
take the 3-direction as the optic axis. This is called the extraordinary direction, and a plane wave with linearly polarized
in this direction has index of refraction 
and phase velocity 
. Note that is the intercept of the index ellipsoid on the optic axis.
The index ellipsoid intersects the 12-plane in a circle of radius 
,
corresponding to the ordinary directions, and the principal velocities
in this plane are 
.
Let 
denote the angle between the direction of propagation
and the optic axis. Then the two possible phase velocities can be computed
to be:
Thus, one wave propagates as it would in an isotropic medium, with phase
velocity independent of direction of propagation; this is called an ordinary wave. The other has a velocity (and hence effective n) that
depends on the direction of propagation, and is called an extraordinary
wave. The two phase velocities are equal (that is, no extraordinary wave
exists) if and only if 
, corresponding to propagation along the
optic axis.
Refer to Figure 5. Consider a plane wave incident from
medium1 into medium 2, and re-emergent into medium 1, where
the two boundaries are parallel. The incident wave will have a deflected
direction of propagation in medium 2, according to the angle of incidence
and the index of refraction in both media, and will re-emerge parallel to
but displaced from the original ray. If medium 2 is a uniaxial crystal,
then the incident wave will be split upon emergence into an extraordinary
and an ordinary wave; this is called double refraction or birefringence. Within the anisotropic medium, both waves have the same and so their wavef
ronts are parallel, but their ray vectors are in different directions; upon emergence, the 
vectors are realigned so as to be parallel. If we properly polarize the
incident wave, we can eliminate either the ordinary or the extraordinary
wave that emerges.
