Rahul Dewan - Jacobs University

3. COMPUTATIONAL MODELING OF OPTICAL WAVE PROPAGATION
˛
˛
∇ · (E × H) dv =
V ol
˛
V ol
(E × H) · ds = − ∂ ˛
∂t
{
− ∂ ( 1
∂t 2 ɛE2 + 1 ) }
2 µH2 − σE 2 dv (3.18)
( 1
2 ɛE2 + 1 ) ˛
2 µH2 dv −
(
σE
2 ) dv
(3.19)
S
} {{ }
net power flow
out of V
V ol
} {{ }
decrease in the stored electric
and magnetic energy within V
V ol
} {{ }
ohmic losses
within V
Equation (3.19) establishes the conservation of energy in electromagnetics and is also
known as the Poynting theorem. It states that the power flow out of the surface S
equals to the decreasing rate of the stored electric and magnetic energies plus the
power supplied by the source. The quantity on the left hand side of equation is defined
as the Poynting vector. For an electromagnetic wave with an electric field E and
magnetic field H, the Poynting vector S is defined as
S = E × H (3.20)
S represents the power per unit area (power density) carried by the wave and its
direction is along the propagation direction of the wave. In practice, however, the
quantity of greater interest is the average power density of the wave. For example in
optics, when one refers to the amount of light illuminating a surface, it refers to the
average energy per unit area per unit time - called the irradiance (I). The time-average
value of the magnitude of S is a measure of I (I ≡ 〈S〉 T
). In the case of a harmonic and
linearly polarized plane wave traveling through free space, irradiance is proportional
to the square of the amplitude of the electric field.
I ≡ 〈S〉 T
= 1 2 cɛ 0E 2 (3.21)
where c is the speed of light and ɛ 0 is the electric permittivity of free space. It can be
noted from equation (3.21) that since I depends only on the magnitude of the electric
field, waves characterized by different polarization of light carry the same amount of
average power if their electric fields have the same magnitude. In this section, a very
brief review from the theories of electromagnetics were presented. Detailed description
on the topic can be found in several textbooks where it is discussed in depth, interested
readers are recommended to check references [68, 70–73].
3.3 Maxwell’s Equations Solver
Although the most accurate result for an electromagnetic field problem can be achieved
using mathematical analytical method, most of real-life electromagnetics problems
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