Analytical solutions for the half-loaded waveguide#

The analytical solutions for the half-loaded waveguide with perfect electric conducting walls are described in Harrington’s Time-harmonic electromagnetic fields. We will skip the full derivation, and we just mention that the problem can be decoupled into \(\mathrm{TE}_x\) and \(\mathrm{TM}_x\) modes, and the possible \(k_z\) can be found by solving a set of transcendental equations, which is shown here below:

\[\begin{split} \begin{aligned} \textrm{For TE}_x \textrm{ modes}: \begin{cases} &k_{x d}^{2}+\left(\frac{n \pi}{b}\right)^{2}+k_{z}^{2}=k_0^{2} \varepsilon_{d} \\ &k_{x v}^{2}+\left(\frac{n \pi}{b}\right)^{2}+k_{z}^{2}=k_0^{2} \varepsilon_{v} \\ & k_{x d} \cot k_{x d} d + k_{x v} \cot \left[k_{x v}(h-d)\right] = 0 \end{cases} \end{aligned} \end{split}\]
\[\begin{split} \begin{aligned} \textrm{For TM}_x \textrm{ modes}: \begin{cases} &k_{x d}^{2}+\left(\frac{n \pi}{b}\right)^{2}+k_{z}^{2}= k_0^{2} \varepsilon_{d} \\ &k_{x v}^{2}+\left(\frac{n \pi}{b}\right)^{2}+k_{z}^{2}= k_0^{2} \varepsilon_{v} \\ & \frac{k_{x d}}{\varepsilon_{d}} \tan k_{x d} d + \frac{k_{x v}}{\varepsilon_{v}} \tan \left[k_{x v}(h-d)\right] = 0 \end{cases} \end{aligned} \end{split}\]

with:

  • \(\varepsilon_d\rightarrow\) dielectric permittivity

  • \(\varepsilon_v\rightarrow\) vacuum permittivity

  • \(h\rightarrow\) height of the waveguide

  • \(d\rightarrow\) height of the dielectric

  • \(k_0\rightarrow\) vacuum wavevector

  • \(k_{xd}\rightarrow\) \(x\) component of the wavevector in the dielectric

  • \(k_{xv}\rightarrow\) \(x\) component of the wavevector in the vacuum

  • \(\frac{n \pi}{b} = k_y\rightarrow\) \(y\) component of the wavevector for different \(n\) harmonic numbers

Let’s define the set of equations with the \(\tan\) and \(\cot\) function:

import numpy as np

def TMx_condition(kx_d, kx_v, eps_d, eps_v, d, h):
    return (kx_d / eps_d * np.tan(kx_d * d)
            + kx_v / eps_v * np.tan(kx_v * (h - d)))

def TEx_condition(kx_d, kx_v, d, h):
    return kx_d / np.tan(kx_d * d) + kx_v / np.tan(kx_v * (h - d))

Then, we can define the verify_mode function, to check whether a certain \(k_z\) satisfy the equations (below a certain threshold):

def verify_mode(
        kz, w, h, d, lmbd0, eps_d, eps_v, threshold):

    k0 = 2 * np.pi / lmbd0

    n = 1
    ky = n * np.pi / w

    kx_d_target = np.sqrt(k0**2 * eps_d - ky**2 + - kz**2 + 0j)

    alpha = kx_d_target**2

    beta = alpha - k0**2 * (eps_d - eps_v)

    kx_v = np.sqrt(beta)
    kx_d = np.sqrt(alpha)

    f_tm = TMx_condition(kx_d, kx_v, eps_d, eps_v, d, h)
    f_te = TEx_condition(kx_d, kx_v, d, h)

    return np.isclose(
        f_tm, 0, atol=threshold) or np.isclose(
        f_te, 0, atol=threshold)