From now on we let the z−axis be parallel to the guiding structures that are treated in this book. We treat fields with the z−dependence exp(ikzz). If Ez and Hz have this z-dependence it follows from (4.3) that the transverse components of the fields have the same z−dependence and thus all fields can be written as
E(r, ω) = E(ρ, kz, ω)eikzz
The coefficient kz is referred to as the longitudinal wave number. The Maxwell equations now simplifies to partial differential equations in the transverse coordinates x and y
(zˆ· (∇T × ET(ρ, kz, ω)) = iωµ0µ(ω)Hz(ρ, kz, ω) ˆ
z· (∇T × HT(ρ, kz, ω)) =−iω0(ω)Ez(ρ, kz, ω) (4.5)
and (
ikzzˆ× ET(ρ, kz, ω)− iωµ0µ(ω)HT(ρ, kz, ω) = ˆz× ∇TEz(ρ, kz, ω)
ikzzˆ× HT(ρ, kz, ω) + iω0(ω)ET(ρ, kz, ω) = ˆz× ∇THz(ρ, kz, ω) (4.6) We observe that the transverse components of the vectors E and H, i.e. ET and HT can be expressed in terms of the longitudinal components Ez and Hz. We see this by operating with ˆz× on the first of the transverse equations in (4.6), utilizing A×(B ×C) = B(A· C) − C(A · B) (BAC-CAB-rule), and by eliminating ˆz × HT(ρ, kz, ω) using the second equation in (4.6). If we treat the second equation in (4.6) in the same manner we
get
− ikzET(ρ, kz, ω)−ωµ0µ(ω) kz
hzˆ× ∇THz(ρ, kz, ω)
− iω0(ω)ET(ρ, kz, ω)i
=−∇TEz(ρ, kz, ω)
− ikzHT(ρ, kz, ω) +ω0(ω) kz
hzˆ× ∇TEz(ρ, kz, ω)
+ iωµ0µ(ω)HT(ρ, kz, ω)i
=−∇THz(ρ, kz, ω)
or
ET(ρ, kz, ω) = ikz∇TEz(ρ, kz, ω)− ωµ0µ(ω)ˆz× ∇THz(ρ, kz, ω)
ω2
c20(ω)µ(ω)− kz2
HT(ρ, kz, ω) = ikz∇THz(ρ, kz, ω) + ω0(ω)ˆz× ∇TEz(ρ, kz, ω)
ω2
c20(ω)µ(ω)− kz2
We rewrite these equations
ET(ρ, kz, ω) = i
kt2 {kz∇TEz(ρ, kz, ω)− ωµ0µ(ω)ˆz× ∇THz(ρ, kz, ω)} HT(ρ, kz, ω) = i
k2t {kz∇THz(ρ, kz, ω) + ω0(ω)ˆz× ∇TEz(ρ, kz, ω)}
(4.7)
where we introduced the transverse wave number kt kt2 = ω2
c20(ω)µ(ω)− kz2 (4.8)
The relation between the wave numbers k(ω), kz, and kt is k2 = k2t + k2z
Since the transverse components of the the electric and magnetic fields are determined by the z−components of the fields, it is sufficient to determine the z-components of the electric and magnetic fields in order to construct the transverse parts. Thus the full vector problem that we started with has been reduced to much simpler scalar problems.
Each of the longitudinal components, Ez(ρ, kz, ω) och Hz(ρ, kz, ω), satisfies a partial differential equation in the variables x och y. We easily get these equations from (2.8) and (2.9) on page 15. The result is
(∇2TEz(ρ, kz, ω) + kt2Ez(ρ, kz, ω) = 0
∇2THz(ρ, kz, ω) + k2tHz(ρ, kz, ω) = 0 (4.9) where the transverse wave number kt is defined in (4.8). The transverse components also satisfy a system of partial differential equations. They are less useful since they are vectorial.
Problems in Chapter 4
∗4.1 Let A = ∇ × (∇ × F ). Determine AT och Az expressed in FT, Fz, ∇T och
∂
∂z.
4.2 Show that one can relate ET and HT in the following way:
If Ez= 0 then ET =−ωµ0µ(ω)
kz zˆ× HT
and
If Hz = 0 then HT = ω0(ω)
kz zˆ× ET
We have here assumed that all fields have the specific z-dependence exp(ikzz).
Summary 69
Summary of chapter 4
Decomposition E(r, ω) = E(ρ, k
z, ω)e
ikzzMaxwell equtions
(zˆ· (∇T × ET(ρ, kz, ω)) = iωµ0µ(ω)Hz(ρ, kz, ω) ˆ
z· (∇T × HT(ρ, kz, ω)) =−iω0(ω)Ez(ρ, kz, ω)
(ikzzˆ× ET(ρ, kz, ω)− iωµ0µ(ω)HT(ρ, kz, ω) = ˆz× ∇TEz(ρ, kz, ω) ikzzˆ× HT(ρ, kz, ω) + iω0(ω)ET(ρ, kz, ω) = ˆz× ∇THz(ρ, kz, ω)
Transverse components
ET(ρ, kz, ω) = i
kt2{kz∇TEz(ρ, kz, ω)− ωµ0µ(ω)ˆz× ∇THz(ρ, kz, ω)} HT(ρ, kz, ω) = i
kt2{kz∇THz(ρ, kz, ω) + ω0(ω)ˆz× ∇TEz(ρ, kz, ω)} kt2= ω2
c20(ω)µ(ω)− k2z
Equations for the longitudinal components
∇2TEz(ρ, kz, ω) + kt2Ez(ρ, kz, ω) = 0
∇2THz(ρ, kz, ω) + kt2Hz(ρ, kz, ω) = 0
Chapter 5
Waveguides at fix frequency
Waveguides are structures that guide waves along a given direction. Figure 5.1 gives an example of geometry for a waveguide. The surface of the waveguide is denoted S and the normal to the surface ˆn. Note that the normal ˆn is a function of the coordinates x and y, but not of the coordinate z. The cross section of the waveguide is denoted Ω and it is circumscribed by the curve Γ, c.f., figure 5.4. Figure 5.4a shows a waveguide with a simply connected cross section Ω, while the figure 5.4b shows a waveguide with an inner surface (the curve Γ consists of two non-connected parts). The analysis in this chapter is valid for waveguides with general cross section.
Two types of waveguides that are studied in this book. The first type is referred to as closed waveguide, or hollow waveguide and has metallic walls that enclose the region. The other type is the open waveguide, for which parts of the enclosing surface is not metallic.
Resonance cavities and dielectric resonators are related to hollow waveguides and dielectric waveguides, respectively, and these are also analyzed in this book.
Figure 5.2 shows two different hollow waveguides and an optical fiber, which is an open waveguide. The figure also shows reflector antennas that are fed by circular and rectangular horn antennas. In such antenna systems hollow waveguides are crucial. Figure 5.3 shows another application for hollow waveguides and hollow cavities. It is a klystron that generates electromagnetic fields with high power and a waveguide that leads this power to a cavity in a linear accelerator.
This chapter treats the hollow waveguides. A special type of open waveguides referred to as dielectric waveguides are treated in chapter 8. To analyse the hollow waveguides mathematically we need boundary conditions and wave equations. The boundary condi-tions for the metallic walls are treated in section 5.1. The derivation of the wave equation from the Maxwell equations in a source free region is given in section 5.2 and 5.3. In the same sections the solutions to the equations are discussed. In section 5.4 the solutions are expressed in terms of expansions in orthogonal and complete sets of basis functions. Some very important examples of waveguides are presented in 5.5. The normalizations for the sets of basis functions are given in section 5.7. Based upon these normalizations we derive expressions for the power flow and the losses in the walls and present them in sections 5.8 and 5.9.
71
z
S Ω
Γ
n
Figure 5.1: Geometry for waveguide.
5.1 Boundary conditions
We now analyze the boundary conditions for the electric and magnetic fields on the metallic surface S of a hollow waveguide. We assume isotropic material in the interior, i.e., the constitutive relations are given by
(D(r, ω) = 0(r, ω)E(r, ω) B(r, ω) = µ0µ(r, ω)H(r, ω)
The sufficient boundary conditions on a perfectly conducting surface are, c.f., (1.17) on
page 7, (
ˆ
n× E(r, ω) = 0
nˆ· H(r, ω) = 0 r on S since B = µ0µH for an isotropic material.
We express the boundary conditions in terms of the decomposed fields in section 4.1 (nˆ × (ET(r, ω) + ˆzEz(r, ω)) = 0
ˆ
n· (HT(r, ω) + ˆzHz(r, ω)) = 0 r on S
The unit normal vector ˆn to the surface S has no z-component, i.e., ˆn· ˆz = 0. Since ˆ
n× ET only has a component along the z-axis, while ˆn× ˆz is perpendicular to the z−axis (directed tangential to Γ), each term in the first equation has to be zero. The second term in the second equation is zero since ˆn and ˆz are perpendicular. The conditions are equivalent to
Ez(r, ω) = 0 ˆ
z· (ˆn× ET(r, ω)) = ˆn· (ET × ˆz) = 0 nˆ · HT(r, ω) = 0
r on S (5.1)
These equations are valid on the entire surface S, which implies that also the following z-derivative is zero (remember that ˆn is independent of z):
ˆ n· ∂
∂zHT(r, ω) = 0 r on S We utilize (4.4)
∂
∂zHT(r, ω) =∇THz(r, ω) + iω0(r, ω)ˆz× ET(r, ω)
TM- and TE-modes 73
reflector
horn rectangular waveguide
rectangular waveguide circular waveguide
coaxial cable
reflector
horn circular waveguide
coaxial cable coaxial cable optical fiber
Figure 5.2: Examples of waveguides. Reflector antenna with a rectangular and circular feed horn.
in order to eliminate the z-derivative, and get ˆ
n· (∇THz(r, ω) + iω0(r, ω)ˆz× ET(r, ω)) = 0 r on S
By utilizing the original boundary condition (5.1), the boundary conditions on the surface
S are reduced to
Ez(r, ω) = 0
∂
∂nHz(r, ω) = 0 r on S (5.2)
where ∂n∂ Hz(r, ω) = ˆn·∇THz(r, ω). These boundary conditions only contain the z−components of the fields and they are sufficient for determining the waves that can exist in a hollow waveguide.