Adaptations for Modes with Azimuthal Variations

For n > 0, the procedure is the same as for Eρ and Hφ. For the other components one

It can be observed that, except for a different sign, the ratio ofE_φtoH_ρ is the same as the ratio E_ρ and H_φ. Therefore, the transfer matrix for these field components can be derived in an analogous way and can be written as

−E_φ(z_e,i) concluded that the matrices express the same conditions. The difference in sign of the electric field in (B.47) leads (by construction of the wave solution) to an orientation of the transverse component of the wave vector which is the same as in (B.42).

For mode order index n = 0 and with PEC boundary conditions at the outermost layer interfaces it is thereby shown that the same transfer matrix can be used for the determi-nation of the wave numbers of the radial waveguides. The following cases can be shown in an analogous way:

• Cases with n >0 and with the complementary sinusoidal azimuthal dependence.

• Non-PEC boundary conditions with decaying fields away from the waveguide analog to the method in Section 4.3.3.

B.4 Analytical Derivatives for the TRM B.3.4 Approximate Solution for a PEC-bounded Single Layer

For the single layer it follows that

k_ρ=k_i =ω√

ε_iµ_i (B.48)

If a loss tangent tanδ of the dielectric medium is defined by ε_i =ε₀ε_r,i(1−jtanδ), then for small loss tangents, using the series expansion of the square root, the wave number can be given as

This is the same result for the attenuation due to dielectric loss (in an unlayered medium) as the one derived through considerations of power in [91, Eq. A2.8].

B.4 Analytical Derivatives for the TRM

The numerical efficiency and robustness can be improved by making use of analytical derivatives of the function of which the roots need to be determined when using one of the root-search techniques that involve derivatives of the characteristic function. Consider the following form of the transfer matrix for layer i

T_i = cos(θ_i) −jε_r,isin(θ_i)/k_i θ_i =k_it_i. Note that the sign chosen for the k_i of the intermediate layers is irrelevant. This is because cos(k_it_i), sin(k_it_i)/k_i, and k_isin(k_it_i) are all even functions for their complex argument k_i (see also Appendix F.2) and therefore invariant to its sign.

The transfer function for two consecutive layer i and j is given by the matrix product

T_{i,j} =T_iT_j (B.51)

and for all N layers analogously by T =T{1,2,···,N} =

where the product is defined recursively by

The characteristic functionf(k_z)for the general case (arbitrary non-magnetic materials at outermost layers) is then given by

f(kz) =−jksT1,1/εr,s−jkcT2,2/εr,c−T2,1 +kskcT1,2/(εr,sεr,c), (B.54) where for the space below the first layer (“substrate”) k_s = ±q

k²_z−k_0,s² and ε_r,s = ε_s/ε₀. For the space above the uppermost layer (“cladding”) k_c = ±q

k²_z−k²_0,c and ε_r,c = ε_c/ε₀. In both cases the sign for the wavenumber has to be chosen such that the real part is positive which corresponds to a wave that decays in the transverse direction(s) away from the waveguide.

B.4.1 First Derivative of the Transfer Matrix

The first derivative of the transfer matrix of layeri with respect to the longitudinal wave number k_x can be given as

The first derivative of the transfer matrix for two consecutive layer iand j is given by the sum of matrix products involving the transfer matrices and their derivatives:

∂T{i,j}

∂k_x = ∂(T_iT_j)

∂k_x = ∂T_i

∂k_xT_j+T_i∂T_j

∂k_x (B.56)

The first derivative of the overall transfer matrix for all N layers can be formulated in a recursive way as

B.5 Characteristic Impedance and Wave Impedances of Parallel Plate Waveguides B.4.2 First Derivative of the Characteristic Function

The derivative of the characteristic function is given by differentiation of (4.26) as

∂F(k_z)

B.5 Characteristic Impedance and Wave Impedances of Parallel Plate Waveguides

B.5.1 Impedances of the Homogeneously Filled Waveguide

Consider a parallel plate waveguide with perfectly conducting plates, width w and plate separation d. The homogeneous filling has a permittivity ε and a permeability µ0. The characteristic impedance can be expressed as a function of the per-unit-length capacitance and inductance which are easily derived as

L⁰ =εw/d, (B.60)

C⁰ =µ₀d/w. (B.61)

The characteristic impedance of the (fundamental) TEM mode is then given as Z0 =

Equation (B.62) can applied, e.g., when using the characteristic impedance Z₀ from port solutions of FEM full-wave simulations or the network parameter of a transmission line segment to determine the wave impedance η of the material.

B.5.2 Impedances of the Layered Waveguide

For the ith layer of the multilayer waveguide, a TM-wave exists for which the longitudinal impedance can be related to the longitudinal wave number as

η_TMx = k_x

ωε_i. (B.63)

I.e., the wave impedance is related to the permittivity of the respective layer, but the longitudinal wave number is the same for all layers and is determined with one of the methods discussed before.

Next, the characteristic impedance of the layered waveguide is considered for the case where it can be related to capacitances and inductances. For the silicon structures that are of interest in this thesis this has been found to be applicable for the slow-wave and quasi-TEM regimes where the (effective) wave number has been with good accuracy approximated as

k_x =ω√

Using L⁰ which is identical to the empty guide and C⁰ for the characteristic impedances, one obtains

In the last step of (B.66) an effective wave impedance η_eff is defined that is analog to the one of the homogeneously filled guide and is related to the characteristic impedance by only geometrical parameters. It is different in that it cannot in general relate the transverse electric and magnetic field components of the mode. For the effective wave impedance, also (B.63) can be used together with(B.64) as

η_eff= kx

From the last result in (B.67) it can be seen that with these assumptions, the TM- and TE-mode impedances coincide.

Note that these results are equivalent to those used in other publications. For the modeling

B.5 Characteristic Impedance and Wave Impedances of Parallel Plate Waveguides of the multi-conductor lines in [25], a characteristic impedance

Z_c= 1

√ε_effc₀C₀ (B.68)

is used where c₀ is the speed of light, C₀ is the capacitance of the structure with all dielectrics replaced by vacuum, and

ε_eff = k_zc₀

ω 2

. (B.69)

This formulation can be traced back to [131–133]. It should be noted that, while [132]

relates the characteristic impedance to the (real) phase velocity, it is related to the com-plex propagation constant by [131]. Applicability of the latter is assumed with the above formulas.