Page 352 - Electromagnetics

4.14.1 Floquet’s theorem
Consider an environment having spatial periodicity along the z-direction. In this envi-
ronment the frequency-domain field may be represented in terms of a periodic function
˜
ψ p that obeys
˜
˜
ψ p (x, y, z ± mL,ω) = ψ p (x, y, z,ω)
˜
where m is an integer and L is the spatial period. According to Floquet’s theorem,if ψ
represents some vector component of the field, then the field obeys
˜
˜
ψ(x, y, z,ω) = e − jκz ψ p (x, y, z,ω). (4.422)
Here κ = β − jα is a complex wavenumber describing the phase shift and attenuation of
the field between the various cells of the environment. The phase shift and attenuation
may arise from a wave propagating through a lossy periodic medium (see example below)
or may be impressed by a plane wave as it scatters from a periodic surface, or may be
produced by the excitation of an antenna array by a distributed terminal voltage. It is
also possible to have κ = 0 as when, for example, a periodic antenna array is driven with
all elements in phase.
˜
Because ψ p is periodic we may expand it in a Fourier series
∞
˜ ˜ − j2πnz/L

ψ p (x, y, z,ω) = ψ n (x, y,ω)e
n=−∞
˜
where the ψ n are found by orthogonality:
1 L/2
˜ ˜ j2πnz/L
ψ n (x, y,ω) = ψ p (x, y, z,ω)e dz.
L −L/2
Substituting this into (4.422), we have a representation for the field as a Fourier series:
∞

˜ ˜ − jκ n z
ψ(x, y, z,ω) = ψ n (x, y,ω)e
n=−∞
where
κ n = β + 2πn/L + jα = β n − jα.
We see that within each cell the field consists of a number of constituents called space
harmonics or Hartree harmonics, each with the property of a propagating or evanescent
wave. Each has phase velocity
ω ω
v pn = = .
β n β + 2πn/L
A number of the space harmonics have phase velocities in the +z-direction while the re-
mainder have phase velocities in the −z-direction, depending on the value of β. However,
all of the space harmonics have the same group velocity
−1
−1

dω dβ n dβ

v gn = = = = v g .
dβ dω dω
Those space harmonics for which the group and phase velocities are in opposite directions
are referred to as backward waves, and form the basis of operation of microwave tubes
known as “backward wave oscillators.”

© 2001 by CRC Press LLC