# Understanding Modal Propagation in Rectangular Waveguides

### Key Takeaways

• Rectangular waveguides do not support the transverse electromagnetic (TEM) mode of wave propagation.

• Only TE and TM modes of wave propagation are supported by rectangular waveguides.

• The mode of propagation with the lowest cut-off frequency is called the dominant mode. TE10 is the dominant mode in rectangular waveguides.

The alignment of the electric and magnetic fields in the direction of propagation defines the different modes of propagation

In a waveguide, electromagnetic waves propagate in different modes. The alignment of the electric and magnetic fields in the direction of propagation defines these modes. The geometry of the waveguide is also an influencing factor in defining the modes of propagation through it. The correct mode of propagation needs to be excited and other modes must be suppressed. The modal propagation in a rectangular waveguide can either be the transverse electric (TE) mode or transverse magnetic (TM) mode.

## Modal Propagation in Rectangular Waveguides

Rectangular waveguides are the earliest type of waveguiding structure. A hollow metallic pipe of a rectangular cross-section forms a rectangular waveguide. Usually, the broader dimension of the rectangular waveguide is represented by ‘a’ (x-direction) and the breadth is represented by ‘b’ (y-direction). The direction of propagation of the waves is in the z-direction.

The electromagnetic fields are confined within the rectangular waveguide walls and they guide the electromagnetic waves to their destination. The distribution of electric and magnetic fields inside the rectangular waveguide varies with the frequency of electromagnetic waves.

## Maxwell’s Equations and the Phase Constant in Rectangular Waveguides

The electromagnetic fields inside a rectangular waveguide obey Maxwell’s equations. For instance, the electric field can be described using the wave equation:

In a lossless rectangular waveguide, the propagation constant γ is equal to the phase constant  β. By applying boundary conditions, equation (1) can be solved and the relationship given in equation (2) can be obtained, where βz is the phase constant in the direction of propagation (in this case, the z-direction) and β is a constant. The symbol ω gives the frequency of the electromagnetic signal,  μ is the permeability of the medium inside the rectangular waveguide, and ε is the permittivity of the medium:

Waveguide dimensions and the mode of propagation in the waveguide determine the value β. In a rectangular waveguide β , can be given by equation (3):

### Mode Indexes

The numbers (m, n) describe the eigenvalues or characteristic values in a rectangular waveguide. The numbers m and n correspond to the number of wave halves present in the waveguide walls of dimensions a and b, respectively. The mode index is applicable to the two modes of wave propagation possible in a rectangular waveguide (TE and TM mode). Every combination of m and n defines a mode in a rectangular waveguide except (0, 0). Each of the TEmn or TMmn modes are characterized by the velocity of propagation and field distribution.

## The Different Modes of Propagation in Rectangular Waveguides

In a rectangular waveguide, electromagnetic waves are reflected from the walls. Since there is only one conductor present in a rectangular waveguide, it does not support the transverse electromagnetic (TEM) mode of propagation. Only TE and TM modes are supported by rectangular waveguides.

### The TE Mode of Propagation

In this mode, the electric field is transverse to the direction of propagation, and only magnetic fields exist in this direction. There is an infinite number of solutions for the wave equation corresponding to the magnetic fields. The solutions are distinguished using the mode indexes, and the modes are represented as TEmn or Hmn. The m and n values can be 0, 1, 2, etc., and must satisfy the relationship m≠n. Each TEmn mode is associated with a cut-off frequency. Below this frequency, no TE propagation occurs in rectangular waveguides. The mode of propagation with the lowest cut-off frequency is called the dominant mode. TE10 is the dominant mode in rectangular waveguides.

### The TM  Mode of Propagation

There is no magnetic field component in the direction of propagation in TM mode. Maxwell’s equation or wave equation for the electric fields in the rectangular waveguide is solved, and infinite solutions can be distinguished using the mode indexes. In a rectangular waveguide, neither m nor n can be equal to zero in TM waveguide mode. Therefore, the dominant mode in the rectangular waveguide is TM11.

Geometric dimensions and electromagnetic signal frequency heavily influence modal propagation in rectangular waveguides. To excite a particular TE or TM mode in a rectangular waveguide with the given electromagnetic signal, the geometry of the waveguide should be designed carefully. Cadence’s software can help you design rectangular waveguides that satisfy mode propagation requirements.