12—Tensors 306The second root hasE 0 y= 0, and an eigenvector computed as
(
k^2 cos^2 α−μ 0 ω^2 11
)
E 0 x−
(
k^2 sinαcosα
)
E 0 z= 0
(
k^2 cos^2 α−k^2
(
11 sin^2 α+ 33 cos^2 α
)
/ 33
)
E 0 x−
(
k^2 sinαcosα
)
E 0 z= 0
11 sinαE 0 x+ 33 cosαE 0 z= 0
If the two’s are equal, this says thatE~is perpendicular to~k. If they aren’t equal then~k.E~ 6 = 0and
you can write this equation for the direction ofE~ as
E 0 x=E 0 cosβ, E 0 z=−sinβ, then
11 sinαcosβ− 33 cosαsinβ= 0
and so tanβ=
11
33
tanα
x
z ~k
E~
α
β
In calcite, the ratio 11 / 33 = 1. 056 , makingβa little bigger thanα.
The magnetic field is in the~k×E~direction, and the energy flow of the light is along the Poynting
vector, in the directionE~×B~. In this picture, that putsB~along theyˆ-direction (into the page), and
then the energy of the light moves in the direction perpendicular toE~andB~. That isnotalong~k. This
means that the propagation of the wave is not along the perpendicular to the wave fronts. Instead the
wave skitters off at an angle to the front. The “extraordinary ray.” Snell’s law says that the wavefronts
bend at a surface according to a simple trigonometric relation,and they still do,but the energy flow of
the light doesnotfollow the direction normal to the wavefronts. The light ray does not obey Snell’s
law
ordinary ray extraordinary ray~k S~ ~k S~
In calcite as it naturally grows, the face of the crystal is not parallel to any of thex-yorx-zor
y-zplanes. When a light ray enters the crystal perpendicular to the surface, the wave fronts are in the
plane of the surface and what happens then depends on polarization. For the light polarized along one
of the principle axes of the crystal (they-axis in this sketch) the light behaves normally and the energy
goes in an unbroken straight line. For the other polarization, the electric field has components along
two different axes of the crystal and the energy flows in an unexpected direction — disobeying Snell’s