Analysis of a Chiral Dielectric Supported Broadband Helix Slow-Wave Structure for Millimeter-Wave TWTs

Abstract

A novel technique of broadbanding a helical slow-wave structure through negative dispersion shaping is proposed. The model considers a simple continuous chiral dielectric support for the helix inside a metallic barrel, unlike conventional helix slow-wave structures with three discrete dielectric supports at 120⁰ apart. The dispersion relation of the slow-wave structure was derived following sheath-helix abstraction, suitably benchmarked for special cases, and was used for analyzing the dispersion behavior of a typical slow-wave structure. Chiral dielectric loading could easily provide negative dispersion characteristics (required for broadband operation) by merely controlling the chirality parameter alone. The scheme with its simple geometric configuration is expected to be useful for millimeter-wave devices providing better thermal management.

Appendix

In region-II, the propagation is characterized by the chiral constitutive relations, in which the electric and magnetic fields are coupled through the chirality parameter, given, following the notations of Lindell et al. [4], as:

$$ {\overrightarrow{D}} = \varepsilon {\overrightarrow{E}} - j\kappa {\sqrt {\varepsilon \mu } }{\overrightarrow{H}} ,\;and\;{\overrightarrow{B}} = \mu {\overrightarrow{H}} + j\kappa {\sqrt {\varepsilon \mu } }{\overrightarrow{E}} $$

(A1)

Here, ɛ and μ are the permittivity and permeability of the medium, respectively, and κ is the chirality parameter. Putting these constitutive relations in source free Maxwell’s equations, one may easily get the Helmholtz’s equations in terms of the decomposed longitudinal electric field given as:

$$ {\left( {\nabla ^{2}_{t} + {\left( {k^{\,2}_{ \pm } - \beta ^{2} } \right)}} \right)}E_{{z \pm }} = 0 $$

(A2)

with $k_{ \pm } = k_{0} n{\left( {1 \pm \kappa } \right)}$ being the axial wave numbers of the two circularly polarized waves, n being the refractive index of the medium, and $k_{0} = \omega {\sqrt {\varepsilon _{0} \mu _{0} } }$ the free space propagation contrast. The total solutions of the longitudinal components of the electric and magnetic fields are given as:

$$ E_{z} = E_{{z + }} + E_{{z - }} ,\;and\;H_{z} = \frac{j} {\eta }{\left( {E_{{z + }} - E_{{z - }} } \right)} $$

(A3)

where E _z+ and E _z- are the solutions of (A2), and for slow-wave propagation with azimuthal symmetry are expressed as:

$$ \begin{array}{*{20}c} {{E_{{z + }} = {\text{A}}_{2} I_{{0r}} + {\text{C}}_{2} K_{{0r}} }} \\ {{E_{{z - }} = {\text{B}}_{2} I_{{0r}} + {\text{D}}_{2} K_{{0r}} }} \\ \end{array} $$

(A4)

Here, we have considered the slow-wave assumption of transverse propagation constants in both region-I and region-II being same as that of γ. This is viable under slow-wave assumption, where the axial wave-numbers corresponding to unbounded waves (k _±) are very small compared to the axial phase-propagation constant β of the guided slow-wave. Here, $\gamma = {\sqrt {\beta ^{2} - k^{2}_{ \pm } } }$; A₂, B₂, C₂, and D₂ are constants to be evaluated by enforcing the boundary conditions. The relations between the other field components for azimuthally symmetric propagating modes are given as:

$$ {\left( {\begin{array}{*{20}c} {{E_{r} }} \\ {{E_{\theta } }} \\ {{H_{r} }} \\ {{H_{\theta } }} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{j\frac{\beta } {{\gamma ^{2} }}}}{{j\frac{\beta } {{\gamma ^{2} }}}} \\ {{\frac{{k_{ + } }} {{\gamma ^{2} }}}}{{ - \frac{{k_{ - } }} {{\gamma ^{2} }}}} \\ {{ - \frac{\beta } {{\eta \gamma ^{2} }}}}{{\frac{\beta } {{\eta \gamma ^{2} }}}} \\ {{j\frac{{k_{ + } }} {{\eta \gamma ^{2} }}}}{{j\frac{{k_{ - } }} {{\eta \gamma ^{2} }}}} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{\frac{{\partial E_{{z + }} }} {{\partial r}}}} \\ {{\frac{{\partial E_{{z - }} }} {{\partial r}}}} \\ \end{array} } \right)} $$

(A5)

The required field components in region-II may now be given with the help of the equations (A3), (A4) and (A5) as:

$$ \begin{array}{*{20}l} {{E_{{z2}} = {\text{A}}_{2} I_{{0r}} + {\text{C}}_{2} K_{{0r}} + {\text{B}}_{2} I_{{0r}} + {\text{D}}_{2} K_{{0r}} } \hfill} \\ {{H_{{z2}} = \frac{j} {\eta }{\left( {{\text{A}}_{2} I_{{0r}} + {\text{C}}_{2} K_{{0r}} - {\text{B}}_{2} I_{{0r}} - {\text{D}}_{2} K_{{0r}} } \right)}} \hfill} \\ {{E_{{\theta 2}} = {\left( {\frac{{k_{0} }} {\gamma }} \right)}{\left( {\xi _{1} - \xi _{2} } \right)}} \hfill} \\ {{H_{{\theta 2}} = {\left( {\frac{{jk_{0} }} {{\eta \gamma }}} \right)}{\left( {\xi _{1} + \xi _{2} } \right)}} \hfill} \\ {{\xi _{1} = {\left( {{k_{ + } } \mathord{\left/ {\vphantom {{k_{ + } } {k_{0} }}} \right. \kern-\nulldelimiterspace} {k_{0} }} \right)}{\left( {{\text{A}}_{2} I_{{1r}} - {\text{C}}_{2} K_{{1r}} } \right)}} \hfill} \\ {{\xi _{2} = {\left( {{k_{ - } } \mathord{\left/ {\vphantom {{k_{ - } } {k_{0} }}} \right. \kern-\nulldelimiterspace} {k_{0} }} \right)}{\left( {{\text{B}}_{2} I_{{1r}} - {\text{D}}_{2} K_{{1r}} } \right)}} \hfill} \\ \end{array} $$

(A6)

These field components are reproduced in (2), which have been subsequently used in the analysis.