Higher-Order Waveguide Modes

Before advanced electromagnetic (EM) simulation tools became commonplace, it was necessary to analyze and model waveguides by applying Maxwell’s equations and solving for EM wave behavior within waveguide structures. For specific geometries, such practices can mathematically describe the EM fields and predict the electrical response of various microwave components. In modern practice, the application of classical EM analysis has become less common. However many engineers can still benefit from being familiar with the nature of EM fields in guided wave structures.

EM wave propagation occurs in a wide variety of waveguide structures. Waveguides are commonly understood to be metallic tubes with rectangular or circular cross-sections. For any waveguide, there are an infinite number of waveguide “modes” that describe the possible electric and magnetic field patterns within the waveguide. For rectangular and circular waveguides, the modes can be determined mathematically by solving Maxwell’s equations with appropriate boundary conditions defined by the regular shape of the waveguide walls. Waveguide modes are often depicted graphically using lines that indicate the direction and relative magnitude of the electric (E) and magnetic (H) fields (Figs. 1-2).

The conducting walls of a waveguide impose boundary conditions on the electric fields, which are vector quantities having both direction and magnitude. All E-fields terminating on a waveguide wall must do so at right angles to the wall. That is, the tangential component of the E-field vector must be zero (or nearly zero) at the waveguide wall. The finite conductivity of the wall prevents the tangential E-field from being exactly zero at the conductor surface, resulting in some ohmic loss but not affecting the EM fields appreciably.

Fig. 1 – Rectangular waveguide modes are periodic in the horizontal (x) and vertical (y) directions.

Fig. 2 – Circular waveguide modes are periodic in the angular (phi) and radial (r) directions.

Tangential H-fields are permitted to be non-zero at the waveguide walls. They induce surface currents on the waveguide walls, with currents flowing at right angles to the direction of the tangential H-fields. In corrugated waveguides, longitudinal currents are intentionally disrupted or modulated by circumferential grooves.  The grooves can induce coupling between waveguide modes, or suppress modes that have tangential H-fields at the waveguide walls.

The transverse components of the E and H fields are perpendicular to the direction of propagation. Wave propagation is normally taken to be flowing in the positive-z direction in rectangular or cylindrical coordinates. The transverse E and H fields are at right angles to each other, and their magnitudes are proportional to each other at any given point in the waveguide. Their ratio (E/H) at any point defines the wave impedance of the waveguide mode.

Basic waveguide modes are described as either Transverse Electric (TE) or Transverse Magnetic (TM). A TE mode has no longitudinal electric field while a TM mode has no longitudinal magnetic field. Some waveguides, such as coaxial, coplanar, microstrip and strip-line structures, support Transverse Electric and Magnetic (TEM) modes in addition to TE and TM modes. No longitudinal fields exist in TEM modes.

Non-TEM modes are denoted using a pair of integer subscripts that are related to periodic variations in the transverse fields across the waveguide cross-section. For example in a rectangular waveguide, the TE₁₀ mode has transverse fields that vary by a half cycle in the horizontal (x) direction and are constant in magnitude in the vertical (y) direction. Conversely, the TE₀₁ mode has transverse fields that are constant in the x direction and vary by a half cycle in the y direction. For circular waveguides, the first subscript indicates periodic variations in the angular (phi) direction while the second subscript is related to variations in the radial (r) direction.

Waveguide Cutoff Frequencies

Non-TEM waveguide modes have a lower cut-off frequency below which wave propagation is highly attenuated and reactive in nature. Below the cutoff frequency, wave propagation along the waveguide axis is “evanescent,” meaning it diminishes quickly in magnitude over distance. The E and H fields are reactive because they are 90 degrees out of phase. When an input signal is applied, most of the incident energy is stored in the EM fields and reflected back to the source. A small fraction is dissipated by ohmic losses. A short section of waveguide below the cutoff may behave like a reactive load that can be used for impedance matching purposes, or it may act as a resonator that can be useful in waveguide filters. At frequencies just above the cutoff, signal attenuation and dispersion can be excessive. As a result, the lowest operating frequency in a waveguide is typically several percent above the lowest cutoff frequency.

Different waveguide modes tend to have different cutoff frequencies as well as different propagation constants  (phase vs distance relationships). Modes with larger subscripts also tend to have higher cut-off frequencies although many modes have the same cutoff frequencies. Such modes are considered  “degenerate” in the jargon of mathematicians and microwave engineers. The “dominant” mode has the lowest cutoff frequency. With some notable exceptions, waveguides are operated at frequencies between the lowest and next-lowest cutoff frequencies. The reason for this is that when two or more propagating modes are possible, coupling between the modes can result in unwanted signal loss or distortion. Mode coupling can occur at any discontinuity, bend or taper.

Rectangular waveguides with a 2:1 width-to-height ratio provide a 2:1 margin between the lowest and next-lowest cutoff frequencies (Fig. 3). Square waveguides provide a margin of just 1.4:1, while circular waveguides have only a 1.3:1 margin between the two lowest cutoff frequencies. In square and circular waveguides, modes that are not radially symmetric may be duplicated orthogonally (the fields are at right angles to each other). In the case of square waveguides, this duplication may be viewed as the merging of cutoff frequencies for the TE₁₀ and TE₀₁ modes as the width-to-height ratio in rectangular waveguides approaches unity.

Fig. 3 – Rectangular waveguides with a 2:1 width-to-height ratio provide a 2:1 margin between the lowest and next-lowest cutoff frequencies. Circular waveguides provide only a 1.3:1 margin between the lowest and next-lowest cutoff frequencies.

Hybrid Waveguide Modes

For many waveguide structures and operating conditions, wave propagation is described by “hybrid” modes that are combinations of TE and TM modes. One of the most commonly encountered hybrid modes is HE₁₁ (Fig. 4).

Fig. 4 – The HE₁₁ hybrid waveguide mode is commonly used for antenna aperture illumination and low-loss transmission of RF power. The HE₂₁ mode is often used to generate different signals in monopulse radar antennas.

HE₁₁ is the principal mode appearing at the radiating aperture of corrugated waveguide horn antennas (Fig. 5). Such horns are typically fed with a circular waveguide carrying the dominant TE₁₁ mode. The tapered and corrugated waveguide wall provides coupling between a series of higher-order modes that combine to form the HE₁₁ mode at the antenna aperture. Other modes are usually present to a lesser degree. The E and H fields are both nearly zero at the conducting edges of the aperture, resulting in low edge diffraction and low sidelobes in the antenna beam. The HE₁₁ mode also exhibits a high degree of symmetry, producing a symmetric antenna pattern with low cross-polarization over a wide range of off-boresight angles.

Fig. 5 – Corrugated horn antennas provide symmetric beams and low cross-polarization responses.

Another notable application for the HE₁₁ mode is the illumination of steered antenna arrays in compact satellites or “CubeSats” as they are commonly known. A patch array operates as either a reflective or transmissive steering element that requires circularly polarized illumination across the entire aperture. Because the illuminating antenna is in close proximity to the steering array, the antenna must produce circular polarization over a wide beam width. This is achieved using a dual-polarized corrugated horn antenna with the HE₁₁ mode appearing predominantly at the antenna aperture.

Low-loss transmission of RF power at microwave and millimeter-wave frequencies is another common use for the HE₁₁ mode. In a typical application, a gyrotron or a similar high-power device generates signals in a circular waveguide structure with the principal output mode being TE₀₁. The output signal is fed to a mode converter consisting of a series of resonators or curved waveguide sections. The mode converter provides strong coupling between the input TE₀₁ mode and the output TM₁₁ mode. The TM₁₁ signal is fed to another mode converter, usually a tapered circular waveguide with internal grooves similar to those found in corrugated waveguide horns but tuned to the gyrotron frequency. The resulting signal is carried to its final destination through corrugated waveguides that are designed to support the HE₁₁ mode exclusively. Attenuation is extremely low due to the near absence of electromagnetic fields at the conducting surfaces. To minimize power loss caused by coupling between HE₁₁ and various other modes, the corrugated waveguide must be as symmetric and straight as possible. To preserve mode purity, changes in direction are achieved using mirrors mounted inside water-cooled miter bends.

Additional Uses For Higher-Order Waveguide Modes

The TE₀₁ mode in circular waveguides is often used because it does not induce longitudinal currents on the waveguide wall. This phenomenon results from the absence of any angular (phi) component in the magnetic field. The TE₀₁ mode appears in many rotary joint designs because no current flows across the junction between stationary and rotating waveguides. The waveguide sections can be separated by a small gap with negligible signal loss or distortion. The TE₀₁ mode is often generated using a network of rectangular waveguides surrounding a circular waveguide (Fig. 6).

Fig. 6 – In many rotary joints the TE₀₁ waveguide mode is generated by splitting input and output channels eight ways and recombining the signals around the circumference of a cylindrical waveguide.

The TE₀₁ mode is also used in cylindrical waveguide cavity resonators that employ piston tuners to vary the cavity length. A small gap between the piston and the cavity wall does not appreciably affect the resonator response.

Many monopulse radar antennas employ higher-order waveguide modes at the antenna feed. For example, a corrugated conical horn antenna can use the HE₁₁ mode to transmit and receive on-axis Sum signals while the HE₂₁ mode receives off-axis Difference signals. Another type of monopulse antenna, realized with a rectangular waveguide, uses four waveguide channels operating in the dominant TE₁₀ mode. They feed a set of mode converters that drive a pyramidal horn antenna (Fig. 6).  The rectangular modes used by the antenna include TE₁₀, TE₃₀, HE₁₂, TE₂₀, HE₂₂, HE₁₁, HE₁₃ and HE₃₁ which combine variously to form the Sum and Difference channels in both azimuth and elevation.

Fig. 7 – A monopulse radar antenna generates a number of higher-order waveguide modes inside a pyramidal waveguide horn.

Evanescent waveguide modes operating at or below their cutoff frequency also have many uses. A notable characteristic of a waveguide segment operated well below its cutoff frequency is that the output amplitude and phase are largely independent of frequency. This behavior is useful in directional couplers, where arrays of holes (short lengths of waveguide) transfer signals between adjacent waveguide sections. Evanescent modes are also generated in a variety of other waveguide structures, including step discontinuities, tuning elements, junctions, and waveguide-to-coaxial transitions.

Coaxial transmission lines can support many waveguide modes in addition to the dominant TEM mode, which has no lower cutoff frequency. The upper-frequency limit of coaxial connectors and cables is generally dictated by the diameter of the outer conductor. The presence of the inner conductor raises the waveguide cutoff frequency of a coaxial transmission line slightly above that of a hollow waveguide with the same diameter. The upper-frequency limit can be estimated as the lowest cutoff frequency of a hollow waveguide with the same diameter as the coaxial component. From waveguide theory, this frequency may be calculated as c/(πb)/(ϵ_r)^1/2 where c is the speed of light in vacuum, ϵ_r is the relative dielectric constant within the connector or cable, and b is the radius of the outer conductor. For example, a coaxial connector with a diameter of 1 mm and a relative dielectric constant of 3 would have an upper-frequency limit around 110 GHz. Above this frequency, signals are likely to exhibit increased levels of reflection and distortion as they excite one or more waveguide modes.

In conclusion, there are many graphical representations of waveguide modes available in the literature. They can provide excellent insight into how various waveguide components work and the ways in which they can fail when used outside of their designed frequency range. Microwave and millimeter-wave engineers who become more familiar with EM field patterns in various waveguide structures can gain a better understanding of their work overall.