12—Tensors 305The combination that really appears in Eq. (12.26) is 0 +α. The first term is a scalar, a multiple of
the identity, so the matrix for this is
() = 0
1 0 0
0 1 0
0 0 1
+ (α) =
11 0 0
0 22 0
0 0 33
(12.28)
I’m setting this up assuming that the crystal has three different directions with three different polariz-
abilities. When I get down to the details of the calculation I will take two of them equal — that’s the
case for calcite. The direction of propagation of the wave is~k, and Eq. (12.26) is
k^2
1 0 0
0 1 0
0 0 1
−μ 0 ω^2
11 0 0
0 22 0
0 0 33
−
k 1 k 1 k 1 k 2 k 1 k 3
k 2 k 1 k 2 k 2 k 2 k 3
k 3 k 1 k 3 k 2 k 3 k 3
E 0 x
E 0 y
E 0 z
= 0 (12.29)
In order to have a non-zero solution for the electric field, the determinant of this total 3 × 3
matrix must be zero. This is starting to get too messy, so I’ll now make the simplifying assumption that
two of the three directions in the crystal have identical properties: 11 = 22. This makes the system
cylindrically symmetric about thez-axis and I can then take the direction of the vector~kto be in the
x-zplane for convenience.
~k=k(~e 1 sinα+~e 3 cosα)
The matrix of coefficients in Eq. (12.29) is now
k^2
1 0 0
0 1 0
0 0 1
−μ 0 ω^2
11 0 0
0 11 0
0 0 33
−k^2
sin^2 α 0 sinαcosα
0 0 0
sinαcosα 0 cos^2 α
=
k^2 cos^2 α−μ 0 ω^2 11 0 −k^2 sinαcosα
0 k^2 −μ 0 ω^2 11 0
−k^2 sinαcosα 0 k^2 sin^2 α−μ 0 ω^2 33
(12.30)
The determinant of this matrix is
(
k^2 −μ 0 ω^2 11
)[(
k^2 cos^2 α−μ 0 ω^2 11
)(
k^2 sin^2 α−μ 0 ω^2 33
)
−
(
−k^2 sinαcosα
) 2 ]
=
(
k^2 −μ 0 ω^2 11
)[
−μ 0 ω^2 k^2
(
11 sin^2 α+ 33 cos^2 α
)
+μ^20 ω^4 11 33
]
= 0 (12.31)
This has a factorμ 0 ω^2 = 0, and the corresponding eigenvector is~kitself. In column matrix notation
that is( sinα 0 cosα). It is another of those static solutions that aren’t very interesting. For the
rest, there are factors
k^2 −μ 0 ω^2 11 = 0 and k^2
(
11 sin^2 α+ 33 cos^2 α
)
−μ 0 ω^2 11 33 = 0
The first of these roots,ω^2 /k^2 = 1/μ 0 11 , has an eigenvector~e 2 =yˆ, or( 0 1 0 ). ThenE~ =
E 0 yeˆikz−ωt and this is a normal sort of transverse wave that you commonly find in ordinary non-
crystalline materials. The electric vector is perpendicular to the direction of propagation. The energy
flow (the Poynting vector) is along the direction of~k, perpendicular to the wavefronts. It’s called the
“ordinary ray,” and at a surface it obeys Snell’s law. The reason it is ordinary is that the electric vector
in this case is along one of the principal axes of the crystal. The polarization is alongE~just as it is in
air or glass, so it behaves the same way as in those cases.