82 unifying physics of accelerators, lasers and plasma
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FIGURE 5.12
TEM wave in free space.
FIGURE 5.13
Boundary conditions on perfectly conducting surfaces.
FIGURE 5.14
Two-wave interference.
A plane EM wave with transverse electric and magnetic fields
(TEM wave) propagating in free space in x-direction is shown
in Fig.5.12. The quantity
P = (E × B) /μ 0
(5.6)
is called the Poynting vector and equals to the local power flux.
5.2.2 Conducting surfaces
In the derivations presented in this section, we discuss the
behavior of EM waves bounded in metal boxes; therefore, we
need to recall the boundary conditions of a wave at a perfectly
conducting metallic surface.
On the surface of a perfect conductor, the tangential component of an electric field E || and the normal component of a
magnetic field B ⊥ will vanish, as illustrated in Fig.5.13.
A non-ideal surface with conductivity σ is characterized
by skin depth; an EM wave entering a conductor is dampened
to 1/e of its initial amplitude at the depth
1
δ S = .
(5.7)
πf μ 0 μ r σ
This allows us to introduce the notion of surface resistance
1
R surf =
(5.8)
σ δ S
which plays an important role in determining performance
of accelerating cavities.
5.2.3 Group velocity
In preparation to discuss dispersion properties of waveguides
and RF structures, let us recall the derivation of group velocity
— the propagation velocity of energy (and information) in an
EM wave.
Consider the interference between two continuous waves
of slightly different frequencies ω ± dω and wavenumbers k ±
dk (see Fig.5.14):
E = E 0 sin [(k + dk) x − (ω + dω) t]
+E 0 sin [(k − dk) x − (ω − dω) t]
(5.9)
= 2E 0 sin [kx − ωt] cos [dk x − dω t]
= 2E 0 f 1 (x, t) f 2 (x, t)
The last line of Eq.5.9 contains two functions, f 1 and f 2 .
The first function corresponds to a continuous wave with the
mean wavenumber and frequency: f 1 (x, t) = sin [kx − ωt]. In
this wave, any given phase is propagated such that kx − ωt
remains constant, which gives us the equation for the phase
velocity of the wave:
ω
v p =
(5.10)
k
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[
FIGURE 5.12
TEM wave in free space.
FIGURE 5.13
Boundary conditions on perfectly conducting surfaces.
FIGURE 5.14
Two-wave interference.
A plane EM wave with transverse electric and magnetic fields
(TEM wave) propagating in free space in x-direction is shown
in Fig.5.12. The quantity
P = (E × B) /μ 0
(5.6)
is called the Poynting vector and equals to the local power flux.
5.2.2 Conducting surfaces
In the derivations presented in this section, we discuss the
behavior of EM waves bounded in metal boxes; therefore, we
need to recall the boundary conditions of a wave at a perfectly
conducting metallic surface.
On the surface of a perfect conductor, the tangential component of an electric field E || and the normal component of a
magnetic field B ⊥ will vanish, as illustrated in Fig.5.13.
A non-ideal surface with conductivity σ is characterized
by skin depth; an EM wave entering a conductor is dampened
to 1/e of its initial amplitude at the depth
1
δ S = .
(5.7)
πf μ 0 μ r σ
This allows us to introduce the notion of surface resistance
1
R surf =
(5.8)
σ δ S
which plays an important role in determining performance
of accelerating cavities.
5.2.3 Group velocity
In preparation to discuss dispersion properties of waveguides
and RF structures, let us recall the derivation of group velocity
— the propagation velocity of energy (and information) in an
EM wave.
Consider the interference between two continuous waves
of slightly different frequencies ω ± dω and wavenumbers k ±
dk (see Fig.5.14):
E = E 0 sin [(k + dk) x − (ω + dω) t]
+E 0 sin [(k − dk) x − (ω − dω) t]
(5.9)
= 2E 0 sin [kx − ωt] cos [dk x − dω t]
= 2E 0 f 1 (x, t) f 2 (x, t)
The last line of Eq.5.9 contains two functions, f 1 and f 2 .
The first function corresponds to a continuous wave with the
mean wavenumber and frequency: f 1 (x, t) = sin [kx − ωt]. In
this wave, any given phase is propagated such that kx − ωt
remains constant, which gives us the equation for the phase
velocity of the wave:
ω
v p =
(5.10)
k