Explaining TE and TM Modes

What are TE and TM Modes - and Why do we Care?

I get a lot of inquiries about the meaning and definition of transverse electric (TE) and transverse magnetic (TM) modes in electromagnetics. A big reason this topic is confusing is because we use the TE and TM labels in very different ways for different situations, so they do not always mean the same thing.

The two most common ways the labels are used are:

1.

to describe the polarization of a wave incident onto a flat surface, and
2.

to identify two different types of modes supported by certain kinds of waveguides.

Understanding TE and TM modes is important in electromagnetics because the modes behave differently and sometimes we want to keep them separate. No matter the situation, it only makes sense to describe waves as TE or TM when they are interacting with a device or interface.

TE and TM Modes at a Flat Interface

When a wave encounters an interface between two mediums, some of that wave may reflect and some may transmit. The amplitudes of the reflected and transmitted waves depend on whether they are TE or TM polarized.

Figure 1 shows the geometry of scattering at a flat interface and how the TE and TM directions are defined for each of the waves. Let the interface lie in the $xy$ plane and the $+z$ direction is downward. The direction of the incident wave is described by its wave vector $\vec{k}_{\textrm{inc}}$ . The incident wave vector $\vec{k}_{\textrm{inc}}$ and surface normal $\hat{a}_z$ define a plane called the plane of incidence (POI).

The incident, reflected, and transmitted waves all lie within the POI.

A TE polarized wave has its electric field $\vec{E}$ purely in the direction perpendicular to the POI. This puts its magnetic field $\vec{H}$ purely parallel to the POI for the TE mode. A TM polarized wave has its magnetic field $\vec{H}$ in the direction perpendicular to the POI and its electric field $\vec{E}$ parallel to the POI. It should be noted that it is impossible to define TE and TM directions for waves at normal incidence because a POI cannot be defined. If the interface involves anisotropic materials and/or diffraction gratings, it may still be possible to define TE and TM directions for normal incidence, but that is outside the scope of what I want to cover here.

The angles of the reflected and transmitted waves are described by Snell’s laws of reflection and refraction, respectively. These angles do not depend on polarization (i.e. TE or TM) at all. Given the angle of incidence $\theta_{\textrm{inc}}$ , angle of reflection $\theta_{\textrm{ref}}$ , angle of transmission $\theta_{\textrm{trn}}$ , refractive index of the incident region $n_{\textrm{inc}}$ , and refractive index of the transmission region $n_{\textrm{trn}}$ , these equations are

$\theta_{\textrm{ref}} = \theta_{\textrm{inc}}$

$n_{\textrm{inc}}\theta_{\textrm{inc}} = n_{\textrm{trn}}\theta_{\textrm{trn}}$

Snell’s Law of Reflection

Snell’s Law of Refraction

The amplitudes of the reflected and transmitted waves are different for TE and TM polarizations and are calculated using the Fresnel equations. Given the angles defined above along with the impedance of the incident region inc and impedance of the transmission region trn, the Fresnel equations are

$r_{\textrm{TE}} = \frac{E_{\textrm{TE,ref}}}{E_{\textrm{TE,inc}}} = \frac{\eta_{\textrm{trn}} \cos \theta_{\textrm{inc}} + \eta_{\textrm{inc}} \cos \theta_{\textrm{trn}}}{\eta_{\textrm{trn}} \cos \theta_{\textrm{inc}} - \eta_{\textrm{inc}} \cos \theta_{\textrm{trn}}}$

$t_{\textrm{TE}} = \frac{E_{\textrm{TE,trn}}}{E_{\textrm{TE,inc}}} = \frac{2\eta_{\textrm{trn}} \cos \theta_{\textrm{inc}}}{\eta_{\textrm{trn}} \cos \theta_{\textrm{inc}} - \eta_{\textrm{inc}} \cos \theta_{\textrm{trn}}}$

$r_{\textrm{TM}} = \frac{E_{\textrm{TM,ref}}}{E_{\textrm{TM,inc}}} = \frac{\eta_{\textrm{trn}} \cos \theta_{\textrm{trn}} + \eta_{\textrm{inc}} \cos \theta_{\textrm{inc}}}{\eta_{\textrm{trn}} \cos \theta_{\textrm{trn}} - \eta_{\textrm{inc}} \cos \theta_{\textrm{inc}}}$

$t_{\textrm{TM}} = \frac{E_{\textrm{TM,trn}}}{E_{\textrm{TM,inc}}} = \frac{2\eta_{\textrm{trn}} \cos \theta_{\textrm{inc}}}{\eta_{\textrm{trn}} \cos \theta_{\textrm{trn}} - \eta_{\textrm{inc}} \cos \theta_{\textrm{inc}}}$

Eq. (1a)

Eq. (1b)

Eq. (1c)

Eq. (1d)

For more details on what happens when a wave scatters from an interface, refer to the series of videos in Topic 7 of our free Electromagnetic Field Theory course.

TE and TM Modes in Waveguides

Some types of waveguides support modes that can be classified as TE or TM, but the definition and meaning of these labels are completely different here than for scattering at an interface. The most typical setup is a rectangular metal waveguide with a homogeneous fill, as illustrated in Figure 2, but some other types of waveguides also support TE and TM modes. All other types of waveguides support hybrid modes that are not purely TE or TM.

The electromagnetic fields inside of a waveguide must obey Maxwell’s equations. For this reason, only certain discrete configurations of fields are possible and these are called “modes.” The various modes that may be supported by a waveguide exist at the same time and propagate inside the waveguide independently. Each guided mode has its own unique electromagnetic field profile and unique propagation characteristics such as speed.

When waveguides have bends or discontinuities, the modes respond to these differences and may become coupled and exchange energy. Most often, waveguides are designed to support only a single mode. For all these reasons and more, it is important to identify and understand the different modes supported by a waveguide.

Figure 3 depicts a rectangular metal waveguide with a homogeneous fill, along with the electric and magnetic fields for the $\textrm{TE}_{11}$ mode. A TE mode has the electric field $\vec{E}$ completely transverse to the direction of propagation $+z$ . This places the electric field entirely in the $x$ and $y$ directions and the $z$ component is zero, $E_{\textrm{TE},z}=0$ .

A TM mode has the magnetic field $\vec{H}$ completely transverse to the direction of propagation $+z$ . This places the magnetic field entirely in the $x$ and $y$ direction and the $z$ component is zero, $H_{\textrm{TM},z}=0$ . The rectangular waveguide can support multiple TE modes and multiple TM modes. These are identified using the integer subscripts m and n as $\textrm{TE}_{mn}$ and $\textrm{TM}_{mn}$ . More about this can be found in the videos in Topic 9 of the Electromagnetic Field Theory course.

Origin of TE and TM Modes in Waveguides

Readers may be curious to understand in simple terms why some waveguides support TE and TM modes while others do not.

A homogeneous fill inside the wave guide eliminates coupling between the field components, giving rise to separate TE and TM modes. An inhomogeneous fill causes power to be coupled between the field components at the various material interfaces. In this general case, the modes are called “hybrid” modes.

The origin of TE and TM modes can also be understood mathematically. The electromagnetic fields in waveguides must obey Maxwell’s equations. To calculate the electromagnetic modes, Maxwell’s equations are combined to derive wave equations, usually written in terms of just the two longitudinal components $E_z$ and $H_z$ . The longitudinal components are chosen because it is easiest to apply boundary conditions since they are always tangential to the boundaries of the waveguide. For the general case, this gives two coupled partial differential equations containing both $E_z$ and $H_z$ . When a waveguide has a homogeneous dielectric, the two equations simplify and decouple to become two independent equations. These are

$\nabla^2 H_z + k^2_{\textrm{c}} H_z = 0$

$\nabla^2 E_z + k^2_{\textrm{c}} E_z = 0$

Eq. (2a)

Eq. (2b)

Since the two differential equations are independent, they represent unique solutions. Solving Eq. (2a) implies that $E_z = 0$ so its solutions are for the TE mode. Solving Eq. (2b) implies that $H_z = 0$ so its solutions are for the TM mode. To see all the details of the analysis, refer to the videos in Topic 9 of Electromagnetic Field Theory.