Mode matching method 111
V H
Ω Ω
A (z)+ B (z)+
B (z)
-
-A (z) z
z
0
Figure 5.16: The geometry for the mode matching method.
and is called the Green dyadic, which is a vectorial analogue of the Green function for scalar fields.
extract the transverse components of the fields from these sums
EVT(r) = X
ν=TM,TE,TEMn
a+nνeikz nV z+ a−nνe−ikz nV z
EVT nν(ρ)
HVT(r) = X
ν=TM,TE,TEMn
a+nνeikz nV z− a−nνe−ikz nV z
HVT nν(ρ)
z < z0
EHT (r) = X
ν=TM,TE,TEMn
b+nνeikz nHz+ b−nνe−ikHz nz
EHT nν(ρ)
HHT(r) = X
ν=TM,TE,TEMn
b+nνeikHz nz− b−nνe−ikHz nz
HHT nν(ρ)
z > z0
Note that transverse mode functions are different in the two waveguides since the trans-verse wavenumber ktn depends on the cross section of the waveguide. We denote the mode functions in the waveguides EVT nν and EHT nν. The same indices are used for the mode functions for the magnetic field and for the longitudinal wavenumbers, i.e., kH,Vz n . At z = z0 the transverse components of the electric and magnetic fields are continuous on ΩV, while the transverse electric field is zero over the remaining part of ΩH, i.e., in the part ρ∈ ΩH but ρ /∈ ΩV. This leads to
EHT(ρ, z0) =
(0, ρ∈ ΩH and ρ /∈ ΩV
EVT(ρ, z0), ρ∈ ΩV
HHT(ρ, z0) = HVT(ρ, z0), ρ∈ ΩV
(5.60)
We now evaluate the normal surface integral over ΩH of the vector product between HHT n∗0ν0 and the upper equation in (5.60). We utilize the normalization integral (5.37) on
page 97 ZZ
ΩH
zˆ· EHT nν(ρ)× HHT n∗0ν0(ρ)
dS = 2PnνHδnn0δνν0
where the mode powers in the lossless waveguide are given by, see (5.38),
PnνH,V =
0ωkzH,Vn
2 ktH,Vn 2, ν = TM µ0µωkzH,Vn ∗
2 ktH,Vn 2 , ν = TE 1
2η0η, ν = TEM
(5.61)
The following relation is obtained
P(B+(z0) + B−(z0)) = Qt(A+(z0) + A−(z0)) (5.62) where A±(z0) and B±(z0) are column vectors A±nν(z) = a±nνe±ikzVnzand Bnν±(z) = b±nνe±ikzHnz wheret denotes transpose. The matrix Q is given by
Qnν,n0ν0 = 1 2
ZZ
ΩV
ˆ
z· EVT nν(ρ)× HHT n∗0ν0(ρ) dS
Mode matching method 113
while P is the diagonal matrix
Pnν,n0ν0 = PnνHδnn0δνν0
Next we evaluate the normal surface integral over ΩV of the vector product of EVT n∗0ν0 and the lower of the equations in (5.60) and get the relation
Q∗(B+(z0)− B−(z0)) = R∗(A+(z0)− A−(z0)) (5.63) where R is the diagonal matrix
Rnν,n0ν0 = PnνV δnn0δνν0 The mode power PnνV is given by (5.61).
The system of equations (5.62) and (5.63) gives the relations between the amplitudes of the incident modes, A+(z0), B−(z0), and the amplitudes of the outgoing modes A−(z0), B+(z0):
A−(z0) B+(z0)
!
=
S11 S12 S21 S22
A+(z0) B−(z0)
!
= S A+(z0) B−(z0)
!
where S is the scattering matrix with elements given by
S11= (Q∗P−1Qt+ R∗)−1(R∗− Q∗P−1Qt) S12= 2(Q∗P−1Qt+ R∗)−1Q∗
S21= 2(QtR∗−1Q∗+ P )−1Qt
S22= (QtR∗−1Q∗+ P )−1(QtR∗−1Q∗− P )
We have so far considered waveguides for which ΩV <ΩH. The case when ΩV >ΩH leads to similar expressions for the relations between the amplitudes
A−(z0) B+(z0)
!
= Se11 Se12 Se21 Se22
! A+(z0) B−(z0)
!
where the scattering matrix is
Se11= (R + eQtP∗−1Qe∗)−1( eQtP∗−1Qe∗− R) Se12= 2(R + eQtP∗−1Qe∗)−1Qet
Se21= 2( eQ∗R−1Qet+ P∗)−1Qe∗
Se22= ( eQ∗R−1Qet+ P∗)−1(P∗− eQ∗R−1Qet) and where
Qenν,n0ν0 = 1 2
ZZ
ΩH
ˆ
z· EHT nν(ρ)× HVT n∗0ν0(ρ) dS
Example 5.14
Consider a transition between the two planar waveguides depicted in figure 5.17. An incident TEM wave propagates in the positive z-direction and gives rise to a reflected and transmittet TEM-wave at the transition z = 0. The frequency is low enough such
a = 7.5 mm b = 15 mm 0
z
Figure 5.17: Transition between two planar waveguides.
(T-line theory) T
) T
R R
(T-line theory)
Frekvens/GHz (mode matching)
(mode matching) 1.2
0.8
0
0 5 10 15
0.2 0.4 0.6 1 1.4
Figure 5.18: Reflection and transmission coefficients for the TEM-mode.
that no other modes than the TEM-mode can propagate. For low enough frequencies the transition can be treated by transmission line theory. This gives the reflection coefficient Γ = (Z2 − Z1)/(Z2 + Z1) and the transmission coefficient T = 2Z2/(Z2 + Z1). For a planar waveguide the characteristic impedance is given by Z = dη/w, where d is the distance between the plates and w is the width of the two conductors. The reflection and transmission coefficients are then given by
R= b− a b+ a T = 2b
b+ a
(5.64)
These coefficients can be compared with the corresponding coefficients obtained from the mode matching method. In this case B−(z0) = 0. The scattering matrix element S11
acts as a reflection matrix and the matrix element S21
rZ2
Z1 as a transmission matrix, where
rZ2 Z1 =
rb
a is the quotient of the characteristic impedances of the transmission lines at port 1 and 2, c.f., (3.39). In figure 5.18 the reflection coefficient for the TEM wave calculated from the mode matching method is compared with the corresponding coefficients calculated from transmission line theory. In this case a = 7.5 mm and b = 15 mm. We see that transmission line theory is only accurate up to approximately 1 GHz.
Mode matching method 115 5 GHz
15 GHz
25 GHz
Figure 5.19: The magnetic field distribution for the junction in figure 5.17 generated by COMSOL at 5, 15, and 25 GHz. The incident TEM-wave enters from the left. The total length of the waveguide is 32 cm. At 5 GHz only the TEM wave propagates. At 15 GHz only the TEM wave propagates in the left waveguide and both the TEM and TM1 modes propagate in the right part. At 25 GHz TEM and TM1 propagate in the left part and TEM, TM1, TM2 propagate in the right part.
This despite that the cut-off frequency for the next propagating modes TE1 and TM1 is 10 GHz.
Example 5.15
To analyze the transition in figure 5.17 with COMSOL we choose 2D>Electromagnetic waves>Frequency domain. We draw the geometry and define the boundary conditions.
All of the surfaces are perfect conductors except the vertical surface to the left, which is the input port, and the vertical surface to the right, which is the output port. In boundary conditions we specify the vertical surface to the left to be the input port and the vertical surface to the right to be the output port. Now we can specify the frequency in Study>Frequency domain and let COMSOL calculate the field in the waveguide.
Figure 5.19 shows the magnetic field at 5 GHz, 15 GHz and 25 GHz. The cut-off frequencies for the TM1 mode is 10 GHz in the right part of the waveguide and 20 GHz in the left part. This is in accordance with the three figures.
It seems that we can handle junctions with FEM. FEM is more flexible than the mode matching technique since it does not rely on analytical expressions for the waveguide modes. We might then get the impression that the mode matching method is redundant, but this is not all true. There are a number of cases where the mode-matching technique is superior to FEM. If there are long distances between junctions then the mode matching technique is very efficient. The mode matching technique decomposes the waves in their mode sums and that is not as straightforward with FEM.
5.11.1 Cascading
A waveguide with several discontinuities can be treated by a cascading method. Assume the geometry depicted in figure 5.20. The relation between A±(z0) and B±(z0) can be
z
z0 z1
A+(z)
A−(z)
B+(z)
B−(z)
C+(z)
C−(z)
Figure 5.20: Three cascaded waveguides.
written, see equations (5.62) and (5.63)
P0 P0
Q∗0 −Q∗0
B+(z0) B−(z0)
=
Qt0 Qt0 R∗0 −R∗0
A+(z0) A−(z0)
The relation between B±(z1) and C±(z1) is given by
P1 P1 Q∗1 −Q∗1
C+(z1) C−(z1)
=
Qt1 Qt1 R∗1 −R∗1
B+(z1) B−(z1)
The relation between B±(z1) and B±(z0) is
B+(z1) B−(z1)
=
E+(z1− z0) 0 0 E−(z1− z0)
B+(z0) B−(z0)
where Enν,n± 0ν0(z) = δnn0δνν0exp(±ikz nz). By matrix multiplication we obtain the relations between A±(z0) and C±(z1). This is straightforward to generalize to a waveguide with an arbitrary number of transitions.
A continuous (tapered) transition from one waveguide to another can be treated by cascading a large number of waveguides with constant cross sections.
5.11.2 Waveguides with bends
With FEM we can analyze waveguides that are not straight. In the example in figure 5.21 a TE10mode enters the left port and exits at the upper port. It is straightforward to draw the geometry in COMSOL. All of the surfaces are perfect conductors except the ports. At the ports we specify that the mode is the first TE-mode, i.e., the TE10mode. One can let COMSOL calculate the reflection and transmission coefficient as a function of frequency.