Page 356 - Electromagnetics

Carrying out the integral we replace k x with κ n = κ + 2nπ/L, giving

∞ e − jk y,n |y| − jκ n x
e

˜
˜
E z (x, y,ω) =−ω ˜µI 0 (ω)
2Lk y,n
n=−∞
˜
˜
=− jω ˜µI 0 (ω)G ∞ (x, y|0, 0,ω) (4.430)

2
2
where k y,n = k − κ , and where
n
∞ − jk y,n |y−y | − jκ n (x−x )

e e
˜
G ∞ (x, y|x , y ,ω) = (4.431)
2 jLk y,n
n=−∞
is called the periodic Green’s function.
We may also find the field produced by an infinite array of line sources in terms of
the Hankel function representation of a single line source (4.345). Using the current
representation (4.428) and summing over the sources, we obtain
∞
ω ˜µ
˜ ˜ − jκnL (2) ˜ ˜
E z (ρ, ω) =− I 0 (ω)e H 0 (k|ρ − ρ n |) =− jω ˜µI 0 (ω)G ∞ (x, y|0, 0,ω)
4
n=−∞
where

2
|ρ − ρ n |=|ˆ yy + ˆ x(x − nL)|= y + (x − nL) 2
˜
and where G ∞ is an alternative form of the periodic Green’s function
∞
1 (2)
˜ − jκnL 2 2
G ∞ (x, y|x , y ,ω) = e H 0 k (y − y ) + (x − nL − x ) . (4.432)
4 j
n=−∞
The periodic Green’s functions (4.431) and (4.432) produce identical results, but are
each appropriate for certain applications. For example, (4.431) is useful for situations
in which boundary conditions at constant values of y are to be applied. Both forms are
difficult to compute under certain circumstances, and variants of these forms have been
introduced in the literature [203].

4.15 Problems

4.1 Beginning with the Kronig–Kramers formulas (4.35)–(4.36), use the even–odd be-
c
havior of the real and imaginary parts of ˜ to derive the alternative relations (4.37)–
(4.38).

4.2 Consider the complex permittivity dyadic of a magnetized plasma given by (4.88)–
c
(4.91). Show that we may decompose [ ˜ ¯ ] as the sum of two matrices
[ ˜ ¯σ]
c
[ ˜ ¯ ] = [ ˜ ¯ ] +
jω
where [ ˜ ¯ ] and [ ˜ ¯σ] are hermitian.

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