Waveguide mode

A waveguide mode is a specific transverse field pattern E(x,y) e^{i(βz − ωt)} that propagates along a waveguide without changing its transverse shape, characterised by a propagation constant β and by its transverse polarisation structure. Each waveguide has a discrete set of guided modes, labelled by integer indices (e.g., TE_mn, TM_mn in rectangular metal waveguides, or LP_mn in optical fibres), and each mode has a cutoff frequency ω_c below which β becomes imaginary and the mode no longer propagates — it becomes evanescent, decaying exponentially along z.

The mode structure comes from solving the transverse Helmholtz equation (∇_⊥² + (k² − β²))E_⊥ = 0 subject to the waveguide's boundary conditions (metal walls for microwave guides, dielectric discontinuity for optical fibres). In a rectangular metal waveguide of dimensions a × b, the dominant TE₁₀ mode has cutoff frequency f_c = c/(2a); the common WR-90 X-band guide (22.9 × 10.2 mm) cuts off at 6.6 GHz and is used up to 12.4 GHz before higher modes start propagating. In a single-mode optical fibre, the V-parameter V = (2π/λ) a · NA must be less than 2.405 for only the fundamental LP₀₁ mode to propagate — the condition used to choose core diameter for a given wavelength. Every microwave filter, every optical-fibre design, every integrated-photonic circuit is built on top of the mode spectrum of its underlying waveguide geometry.