Transverse ElectricEdit

Transverse Electric (TE) is a fundamental concept in electromagnetism that describes a class of wave modes in which the electric field is entirely transverse to the direction of propagation. TE modes are most familiar in the context of waveguides and transmission lines, where boundaries and materials shape how electromagnetic energy can travel. They are typically discussed together with transverse magnetic (TM) modes and, in some cases, with transverse electromagnetic (TEM) modes, each defined by which field components survive along the direction of travel. TE modes are essential for understanding microwave engineering, photonics, and RF systems, where precise control of field distributions enables efficient energy transfer and device performance.

In practical terms, a TE mode propagating along a principal axis has no electric field component along that axis, while a magnetic field may have a longitudinal component. This simple rule of thumb hides rich mathematics and a family of discrete solutions dictated by geometry, material properties, and boundary conditions. TE modes exist only for structures that guide the waves, such as metal walls or dielectric boundaries that enforce specific tangential electric-field conditions. The existence and properties of TE modes depend on the cross-sectional shape of the guide (e.g., rectangular, circular) and on the operating frequency relative to a geometry-dependent cutoff frequency.

Definition and scope

TE stands for transverse electric mode. In a waveguide where the propagation direction is taken as z, TE modes satisfy E_z = 0, with electric-field components lying entirely in the transverse plane while the magnetic-field component can have a longitudinal part. See transverse electric mode for related terminology and conventions.
TE modes are contrasted with TM (transverse magnetic) modes, where B_z = 0 and E_z may be nonzero, and with TEM (transverse electromagnetic) modes, where both E and B have no longitudinal components. See transverse magnetic mode and transverse electromagnetic mode.
Common contexts for TE modes include waveguides in microwave engineering, rectangular waveguides, and cylindrical waveguides in which geometry imposes standing-field patterns and a discrete set of modes. See rectangular waveguide and cylindrical waveguide.
The mathematical treatment of TE modes rests on Maxwell's equations and appropriate boundary conditions, notably the requirement that the tangential electric field vanish on conducting walls (for ideal conductors) or meet the specified boundary on dielectrics. See Maxwell's equations and boundary conditions.

Mathematical formulation

TE modes arise as eigenfunctions of the cross-sectional (transverse) Laplacian operator in a guide. If the wave propagates along z with time-harmonic dependence e^{-iωt}, the fields satisfy Maxwell's equations in the given medium (often modeled as lossless and isotropic), leading to a set of coupled partial differential equations. A key result is that the longitudinal magnetic field Hz (or, equivalently, other transverse components) must satisfy a Helmholtz-type equation in the cross-section, with a separation constant that defines the mode’s cutoff.
The propagation constant β along the axis and the transverse wavenumber k_c are related by k^2 = k_c^2 + β^2, where k = ω√(με). TE modes require E_z = 0, which constrains the field relations between Ex, Ey, Hz, and the remaining components. See Poynting vector for how energy flows in a TE mode.
Boundary conditions at conducting walls require the tangential electric field to vanish: n × E = 0 on the surface. This yields discrete sets of allowable k_c values, which in turn determine the cutoff frequencies for each mode. See conductor and boundary conditions.

Modes in common waveguides

Rectangular waveguides: The dominant TE mode is TE10, characterized by a single half-wave variation of Ex in the wider dimension and no variation along the narrower one. The cutoff frequency for TE_mn in a rectangular guide with dimensions a × b is f_c_mn = (c/2) sqrt[(m/a)^2 + (n/b)^2], where c is the speed of light in the guide material. See rectangular waveguide and dominant mode.
Circular (cylindrical) waveguides: TE modes in circular geometry follow roots of Bessel function derivatives. The lowest-order TE mode is TE11, followed by higher-order TE_mn combinations. The cutoff condition involves the first zeros of J'_m(x), linking geometry to allowable frequencies. See cylindrical waveguide.
Propagation above cutoff: For frequencies above their respective cutoff, TE modes transport power along the guide with a longitudinal magnetic field component and transverse electric fields shaping the field distribution in cross-section. See Poynting vector for how axial power flow is computed.

TE polarization in optics and photonics

In planar and optical waveguides, TE-like polarization refers to the orientation of the electric field relative to the guide and propagation axis. In many optical systems, TE and TM designations help describe polarization states important for coupling, confinement, and dispersion. In single-mode planar structures and in optical fibers, the so-called LP modes approximate TE and TM character under weak guiding, while real fibers require more complete vector-mode descriptions. See planar waveguide and optical fiber.
TE modes in optical components—such as dielectric slabs, waveguides, and integrated photonic circuits—play a central role in polarization control, mode matching, and low-loss transmission. See waveguide and photonics.

Practical considerations and applications

TE modes are central to microwave transmission, radar, and communications systems, where rectangular and circular waveguides are employed to confine and direct energy with minimal loss. The selection of TE modes (e.g., TE10 in a rectangular guide) often maximizes bandwidth and minimizes dispersion for a given geometry. See microwave engineering and RF engineering.
In high-frequency systems, mode purity is important: higher-order TE modes can be excited unintentionally, leading to undesired coupling, insertion loss, or interference. Designers use geometry, mode filters, and termination strategies to suppress unwanted modes. See mode filter and impedance matching.
TE polarization effects are also relevant in satellite and terrestrial communications, where waveguides and transmission lines transition to free space or onto printed circuit boards, demanding careful control of field orientation and boundary conditions. See antenna and transmission line.

History and terminology

The study of TE and TM modes grew out of early investigations into wave propagation in waveguides and the application of Maxwell's equations to practical guiding structures. The terminology reflects whether the electric or magnetic field has a longitudinal component with respect to the propagation direction. See history of electromagnetism and Maxwell's equations.