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Transverse Electric ($\text{TE}_{mn}$) Mode in Rectangular Waveguide

Interactive $\text{TE}_{mn}$ Mode Wave Visualizer

Switch between the 3D Longitudinal Traveling Wave View along the $z$-axis and the 2D Cross-Sectional View ($x$-$y$ plane).

Electric Field ($E$)
Magnetic Field ($H$)

1. TE Mode Definition & Field Components

In the Transverse Electric (TE) mode, the electric field is transverse (normal) to the direction of wave propagation along the $z$-axis.

$$E_z = 0, \quad H_z \neq 0$$

The longitudinal magnetic field distribution $H_z^\circ$ is defined with $B = \frac{m\pi}{a}$ and $A = \frac{n\pi}{b}$:

$$H_z^\circ = C \cos(Bx) \cdot \cos(Ay)$$

The transverse field amplitude components expressed in terms of $H_z^\circ$ are:

$$H_x^\circ = \frac{j\beta}{h^2} C B \sin(Bx) \cos(Ay)$$ $$H_y^\circ = \frac{j\beta}{h^2} C A \cos(Bx) \sin(Ay)$$ $$E_x^\circ = \frac{j\omega\mu}{h^2} C A \cos(Bx) \sin(Ay)$$ $$E_y^\circ = -\frac{j\omega\mu}{h^2} C B \sin(Bx) \cos(Ay)$$

Substituting $A$ and $B$, the full field components with propagation factor $e^{-\gamma z}$ are:

$$E_x = \frac{j\omega\mu}{h^2} C \left(\frac{n\pi}{b}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$ $$E_y = -\frac{j\omega\mu}{h^2} C \left(\frac{m\pi}{a}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$ $$H_x = \frac{j\beta}{h^2} C \left(\frac{m\pi}{a}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$ $$H_y = \frac{j\beta}{h^2} C \left(\frac{n\pi}{b}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$

Where $h^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2$ and mode indices $m, n = 0, 1, 2, \dots$

2. Cutoff Parameters & Dominant Mode

For the $\text{TE}_{mn}$ mode, mode indices can be $(0,1)$ or $(1,0)$, but $\text{TE}_{00}$ does not exist. Both $m$ and $n$ cannot be zero simultaneously because it forces all field components to vanish.

$$f_c = \frac{u'}{2} \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$

Dominant Mode ($\text{TE}_{10}$):

The Dominant mode is the mode with the lowest cutoff frequency (or longest cutoff wavelength).

By standard convention, rectangular waveguides are designed with $a > b$ (width greater than height), so $\frac{1}{a^2} < \frac{1}{b^2}$. Thus, $\text{TE}_{10}$ is the lowest mode because its cutoff frequency is lower than $\text{TE}_{01}$:

$$f_{c_{\text{TE}_{10}}} = \frac{u'}{2a} < f_{c_{\text{TE}_{01}}} = \frac{u'}{2b}$$

Therefore, $\text{TE}_{10}$ is the dominant mode of the rectangular waveguide.

3. Wave Impedance

The wave impedance for the TE mode is derived as:

$$\eta_{\text{TE}} = \frac{\sqrt{\frac{\mu}{\varepsilon}}}{\sqrt{1 - \left(\frac{f_c}{f}\right)^2}} = \frac{\eta'}{\sqrt{1 - \left(\frac{f_c}{f}\right)^2}}$$

Where $\eta' = \sqrt{\frac{\mu}{\varepsilon}}$ is the intrinsic impedance of the unbounded medium.