Transverse Magnetic ($\text{TM}_{mn}$) Mode in Rectangular Waveguide
Interactive $\text{TM}_{mn}$ Mode Wave Visualizer
Switch between the 3D Longitudinal Traveling Wave View along the $z$-axis and the 2D Cross-Sectional View ($x$-$y$ plane). The lowest existing TM mode is $\text{TM}_{11}$.
Longitudinal $E_z$ Heatmap
⚠️ For TM modes, both m and n MUST be greater than zero ($m \ge 1, n \ge 1$). Setting $m=0$ or $n=0$ forces all TM field components to vanish completely!
1. TM Mode Definition & Field Components
In the Transverse Magnetic (TM) mode, the magnetic field vector $\mathbf{H}$ is entirely transverse (perpendicular) to the direction of wave propagation along the $z$-axis. Consequently, there is no longitudinal component of magnetic field ($H_z = 0$), while the longitudinal electric field component is non-zero ($E_z \neq 0$).
$$H_z = 0, \quad E_z \neq 0$$
The longitudinal electric field distribution $E_z^\circ$ inside a waveguide of width $a$ ($x$-direction) and height $b$ ($y$-direction) is given by:
$$E_z^\circ = C \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right)$$
Using Maxwell's equations, the transverse field amplitude components expressed in terms of $E_z^\circ$ are:
$$E_x^\circ = -\frac{\gamma}{h^2} C \left(\frac{m\pi}{a}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right)$$
$$E_y^\circ = -\frac{\gamma}{h^2} C \left(\frac{n\pi}{b}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right)$$
$$H_x^\circ = \frac{j\omega\varepsilon}{h^2} C \left(\frac{n\pi}{b}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right)$$
$$H_y^\circ = -\frac{j\omega\varepsilon}{h^2} C \left(\frac{m\pi}{a}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right)$$
Including the propagation factor $e^{-\gamma z}$ along the length of the guide:
$$E_z = C \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$
$$E_x = -\frac{\gamma}{h^2} C \left(\frac{m\pi}{a}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$
$$E_y = -\frac{\gamma}{h^2} C \left(\frac{n\pi}{b}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$
$$H_x = \frac{j\omega\varepsilon}{h^2} C \left(\frac{n\pi}{b}\right) \sin\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$
$$H_y = -\frac{j\omega\varepsilon}{h^2} C \left(\frac{m\pi}{a}\right) \cos\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{-\gamma z}$$
Where $h^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2$ and mode indices $m, n = 1, 2, 3\dots$
2. Cutoff Parameters & Lowest TM Mode
For $\text{TM}_{mn}$ modes, neither $m$ nor $n$ can be zero. If $m=0$ or $n=0$, $\sin(0) = 0$, which forces $E_z = 0$ everywhere and causes all field components ($E_x, E_y, H_x, H_y$) to vanish identically. Thus, modes such as $\text{TM}_{00}$, $\text{TM}_{10}$, and $\text{TM}_{01}$ do not exist.
$$f_c = \frac{u'}{2} \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}$$
Lowest TM Mode ($\text{TM}_{11}$):
The lowest order mode in Transverse Magnetic propagation is $\mathbf{\text{TM}_{11}}$. Its cutoff frequency is given by:
$$f_{c_{\text{TM}_{11}}} = \frac{u'}{2} \sqrt{\frac{1}{a^2} + \frac{1}{b^2}}$$
Note: While $\text{TE}_{10}$ is the absolute dominant mode of a rectangular waveguide overall, $\text{TM}_{11}$ is the lowest existing mode for Transverse Magnetic propagation.
3. Wave Impedance
The wave impedance for the TM mode represents the ratio of transverse electric field to transverse magnetic field and is given by:
$$\eta_{\text{TM}} = \frac{E_x}{H_y} = -\frac{E_y}{H_x} = \sqrt{\frac{\mu}{\varepsilon}} \sqrt{1 - \left(\frac{f_c}{f}\right)^2} = \eta' \sqrt{1 - \left(\frac{f_c}{f}\right)^2}$$
Where $\eta' = \sqrt{\frac{\mu}{\varepsilon}}$ is the intrinsic impedance of the unbounded medium.