2. Linear crystal optics for monochromatic plane waves
D =
1
2
De
−i(ωt−k·r) + D∗
e
i(ωt−k·r)
. (2.3)
The actual fields are represented by the real quantities P, E, and D, but
for many calculations it is more convenient to use the complex envelope
functions D, E, and P . These complex vectors are in general functions of
(x, y, z, t). However, for monochromatic plane waves, they are independent
of space and time so we omit these arguments in this chapter, and treat
the envelope functions as simple complex vectors.
We substitute the expansions of Eqs. (2.1)-(2.3) in the wave equation,
Eq. (1.13)
∇ × ∇ × E = −µ◦
∂
2
∂t2 D = −µ◦
∂
2
∂t2
ǫ◦E + P
. (2.4)
The operator ∇ becomes (±ik) when it operates on the exponent (±ik ·
r). Similarly the operator (∂/∂t) becomes (±iω) when it operates on the
exponent (±iωt). Making these substitutions in Eq. (2.4) and equating the
positive (or negative) frequency components on each side of the equation
yields
k × k × E = −µ◦ω
2D. (2.5)
In deriving this equation we assumed that E and D do not change on prop-
agation, so this is a wave equation for eigenpolarized light. This expression
implies that D must be normal to k, but D is not necessarily parallel to
E. However, k, D, and E must lie in a single plane.
We use a similar procedure to rewrite the Poynting vector equation to find
the energy flow for monochromatic plane waves. We start with Eq. (1.16),
the general Poynting vector equation,
S =
1
µ◦
E × B = E × H. (2.6)
For monochromatic plane waves the third Maxwell equation, Eq. (1.3),
relates H to E for eigenpolarized light in a nonmagnetic material by
H =
k × E
µ◦ω
. (2.7)
Substituting the expansions for E and H in Eq. (2.6), equating equal fre-
quency components, and using Eq. (2.7) plus
ǫ◦µ◦c
2 = 1, (2.8)
we arrive at
S =
nǫ◦c
2
|E|
2 eˆ× kˆ × eˆ,
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