ABSTRACT
Figure 20.1. The ξ -dependence of the plane wave function As a result of the additional degree of freedom arising from the transverse orientation of the field vectors, electromagnetic waves exhibit polarization properties, which are not manifested by scalar waves. Consider the plane wave solutions to the vector electromagnetic wave equations (14.7) in linear, homogenous and isotropic media which contain no free charge distribution. As illustrated in Figure 20.1, a plane wave
function is conveniently written in terms of the distance ner GG ⋅=ξ travelled along the
direction of propagation rather than in terms of the position vector ,ne G rG of an arbitrary
point on the wavefront. We have
ξ ξξξ
∂ ∂
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂+∂ ∂=∂
∂+∂ ∂+∂
∂=∇ z
e y
e x
e z
e y
e x
e zyxzyx GGGGGG
( ) ( ) ( )x n y n z ne e r e e r e e rx y z ξ ⎡ ⎤∂ ∂ ∂= ⋅ + ⋅ + ⋅⎢ ⎥∂ ∂ ∂⎣ ⎦ G G G G G G G G G ∂
∂
( ) ( ) ( )x n x y n y z n z ne e e e e e e e e eξ ξ∂ ∂⎡ ⎤= ⋅ + ⋅ + ⋅ =⎣ ⎦ ∂ ∂G G G G G G G G G G (20.1) so that Eqs.(14.7) can be rewritten in terms of ξ as
t
HEen ∂ ∂−=∂
∂× GGG µξ (20.2)
t EEHen ∂
∂+=∂ ∂×
GGGG εσξ (20.3)
0=∂ ∂⋅ ξ Hen
GG (20.4)
0=∂ ∂⋅ ξ Een
GG (20.5)
where ( ) ( ).,and, tHHtEE ξξ GGGG == Equation (20.5) also reads ( ) ,0// =∂∂=∂⋅∂ ξξ nn EEe GGG and this implies that the longitudinal component of the electric field might be a function of time only, which can be obtained from Eq.(20.3) giving
( ) nnnn EtEetHee ⎟⎠⎞⎜⎝⎛ ∂∂+=⋅⎟⎠⎞⎜⎝⎛ ∂∂+==⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂×⋅ εσεσξ
GGGGG 0 (20.6)
The solution
in conductors, as 0=nE τ is very small. In nonconducting media, where ,0=σ Eq.(20.6) reduces to thus the longitudinal component of the electric field is independent of both time and position. A similar result holds for the longitudinal component of the magnetic field, since Eq.(20.4) and Eq.(20.2) can be reduced to
,0/ =∂∂ tEn
/ 0, and /n nH H 0,tξ∂ ∂ = ∂ ∂ = respectively. Therefore the field vectors of plane electromagnetic waves, defined as functions of both position and time, are not allowed to have components along the propagation direction .ne
G This type of plane wave, where both HE
GG and are contained in the wavefront plane, is called a transverse
electromagnetic wave. If we assume that the field variation with both position and time is harmonic and has the form (16.25)
where nekk GG = is the wave vector, as defined by Eq.(16.23), it is a straightforward matter
to derive, by direct substitution of Eq.(20.8), the following properties
EkiE GGG ⋅=⋅∇ HkiH GGG ⋅=⋅∇
EkiE GGG ×=×∇ HkiH GGG ×=×∇ (20.9)
Ei t E GG ω−=∂
∂ Hi t
H GG ω−=∂ ∂
Thus, for a plane harmonic wave representation (20.8) of the field vectors, two operator relations hold as
ki G→∇ , ωi
t −→∂
∂ (20.10) and this allow Maxwell's equations (14.7), in nonconducting media ( ),0=σ to be reduced to the form
k E Hµω× =G G G k H Eεω× = −G G G (20.11)
0=⋅ Hk GG 0=⋅ Ek GG Equations (20.11) show that not only both HE
GG and are perpendicular to the direction
of propagation ,k G
but also
( ) ( EeEekHHEek nnn )GGGGGGGG ×=×==× µεµωµω or (20.12) where we have used the definitions (16.24) and (15.54) for the wave number k and velocity v. It follows that the three vectors kHE
GGG and, constitute a right-handed
orthogonal set, as illustrated in Figure 20.2. Since the cross product ,HE GG × which defines
the Poynting vector (14.13), has the same direction as the wave vector ,k G
the direction of energy flow in isotropic media coincides with the propagation direction.
