Page 409 - Electromagnetics

Example of spherical TE–TM decomposition: a plane wave. Consider a uni-
form plane wave propagating in the z-direction in a lossless, homogeneous material of
permittivity and permeability µ, such that its electromagnetic field is
˜
˜
˜
E(r,ω) = ˆ xE 0 (ω)e − jkz = ˆ xE 0 (ω)e − jkr cos θ ,
˜
˜
E 0 (ω) − jkz E 0 (ω) − jkr cos θ
˜
H(r,ω) = ˆ y e = ˆ x e .
η η
We wish to represent this field in terms of the superposition of a field TE to r and a field
˜
˜
˜
˜
TM to r. We first find the potential functions A e = ˆ rA e and A h = ˆ rA h that represent
the field. Then we may use (5.166)–(5.170) and (5.175)–(5.179) to find the TE and TM
representations.
˜
˜
˜
From (5.166) we see that A e is related to E r , where E r is given by
˜
˜
˜
E 0 cos φ ∂
E r = E 0 sin θ cos φe − jkr cos θ = e − jkr cos θ .
jkr ∂θ
We can separate the r and θ dependences of the exponential function by using the identity
n
(E.101). Since j n (−z) = (−1) j n (z) = j −2n j n (z) we have
∞
" −n
e − jkr cos θ = j (2n + 1) j n (kr)P n (cos θ).
n=0
Using
0
∂ P n (cos θ) ∂ P (cos θ) 1
n
= = P (cos θ)
n
∂θ ∂θ
we thus have
˜
∞
j E 0 cos φ "
˜ −n 1
E r =− j (2n + 1) j n (kr)P (cos θ).
n
kr
n=1
1
Here we start the sum at n = 1 since P (x) = 0. We can nowidentify the vector potential
0
as
˜ ˜ ∞ −n
A e E 0 k " j (2n + 1) 1
= cos φ j n (kr)P (cos θ) (5.180)
n
r ω n(n + 1)
n=1
since by direct differentiation we have
1
∂ 2
˜ 2 ˜
E r = + k A e
jω ˜µ˜ c ∂r 2
˜
E 0 k " j −n (2n + 1) 1
∂ 2 2
∞
= cos φ P (cos θ) + k [rj n (kr)]
n
2
jω ˜µ˜ c n(n + 1) ∂r 2
n=1
˜
∞
j E 0 cos φ " −n 1
=− j (2n + 1) j n (kr)P (cos θ),
n
kr
n=1
which satisfies (5.166). Here we have used the defining equation of the spherical Bessel
functions (E.15) to showthat

2 2
∂ 2 ∂ ∂ 2
+ k [rj n (kr)] = r j n (kr) + 2 j n (kr) + k rj n (kr)
∂r 2 ∂r 2 ∂r
∂ 2 ∂
2
2
2
= k r + j n (kr) + k rj n (kr)
∂(kr) 2 kr ∂(kr)

n(n + 1) n(n + 1)
2 2
=−k r 1 − j n (kr) + k rj n (kr) = j n (kr).
(kr) 2 r
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