Cutoff and dispersion

For a homogeneous guide filled by a nondispersive medium, k=ω√(με). The transverse boundary problem supplies one positive eigenvalue k_c for each non-TEM mode. Longitudinal propagation then satisfies β²=ω²με−k_c².

Define the cutoff angular frequency by ω_c=k_c/√(με). If ω>ω_c, β is real; the +z traveling solution uses β>0 in exp(+jωt−jβz). If ω<ω_c, β² is negative and the mode is evanescent rather than propagating.

Cutoff is therefore not disappearance of the eigenfunction. The transverse field pattern still exists mathematically below cutoff; its longitudinal dependence changes from oscillatory propagation to exponential decay.

Below cutoff define α_z=√(k_c²−ω²με)>0. With AoN's exp(+jωt−jβz) convention, the forward-decaying analytic branch is β=−jα_z because exp[−j(−jα_z)z]=exp(−α_z z).

A single decaying evanescent mode in an ideal uniform lossless guide does not transport nonzero average power arbitrarily far in +z; its longitudinal electric and magnetic power-flow contributions are in quadrature. Nevertheless finite below-cutoff sections can participate in reactive coupling or tunnelling when both boundaries are present.

Do not call α_z a conductor/dielectric attenuation constant. Here the decay is geometric cutoff even though the walls and filling medium are ideal.

Above cutoff define λ_g=2π/β. Let v=1/√(με) be the bulk wave speed and λ=2π/k the wavelength in the unbounded filling medium. Eliminating β gives λ_g=λ/√(1−(ω_c/ω)²).

Thus λ_g exceeds the bulk-medium wavelength and diverges as the frequency approaches cutoff from above. Far above cutoff, k_c becomes a small correction and λ_g approaches λ.

The guide wavelength describes longitudinal phase repetition. It is not the transverse size of the mode and it is not the free-space wavelength unless the filling medium is vacuum and the mode is far above cutoff.

The phase velocity along z is v_p=ω/β=v/√(1−(ω_c/ω)²), so v_p>v for every finite frequency above cutoff. Near cutoff it diverges because β tends to zero.

This does not imply that energy, information or a causal modulation travels at v_p. A monochromatic phase front can move faster than the bulk wave speed without violating causality.

The meaningful envelope velocity for a narrowband packet is obtained from the dispersion slope, not from ω/β.

Differentiate β²=ω²/v²−k_c² while treating k_c and the nondispersive material parameters as frequency-independent. This gives dβ/dω=ω/(v²β), so v_g=(dβ/dω)⁻¹=v²β/ω.

Hence v_g=v√(1−(ω_c/ω)²), which is smaller than v and tends to zero at cutoff. Multiplying by v_p gives the exact nondispersive-guide identity v_p v_g=v².

For a length L, the narrowband group delay is τ_g=L/v_g. Near cutoff this delay grows rapidly, and the strong curvature of β(ω) also makes broadband pulses distort.

Chapter 02 found Z_TE=ωμ/β and Z_TM=β/(ωε). Introduce the intrinsic impedance η=√(μ/ε). Using the dispersion relation gives Z_TE=η/√(1−(ω_c/ω)²) and Z_TM=η√(1−(ω_c/ω)²) for propagating modes.

Therefore Z_TE rises without bound as cutoff is approached from above, while Z_TM falls toward zero. Far above cutoff both approach η because the guide mode becomes increasingly plane-wave-like along z.

These are ratios of modal transverse fields. They are not two-conductor characteristic impedances and should not be inserted blindly into circuit formulas that require a uniquely defined terminal voltage/current pair.

For a lossless nondispersive guide carrying one propagating mode, the longitudinal time-average Poynting flux divided by the stored electromagnetic energy per unit length yields an energy velocity equal to v_g. Thus the energy transport tends to zero as cutoff is approached even while v_p diverges.

The physical picture is not that photons or signals accelerate and then stop. The guided field contains increasing transverse phase structure relative to its longitudinal advance; the same transverse eigenvalue that subtracts k_c² from k² reduces β and the longitudinal energy transport.

In dispersive filling media the stored-energy density requires the appropriate frequency derivatives of material response, and v_pv_g=v² no longer follows in this simple form.

Take a vacuum-filled mode with f_c=10 GHz and operate at f=15 GHz. The normalized propagation factor is √(1−(10/15)²)=√5/3≈0.74536.

Hence v_p≈1.34164c, v_g≈0.74536c and v_pv_g=c². The guide wavelength is λ_g=λ/0.74536. Since λ at 15 GHz is about 19.986 mm, λ_g≈26.814 mm.

With η₀≈376.73 Ω, Z_TE≈505.5 Ω while Z_TM≈280.8 Ω. At f=8 GHz, below cutoff, α_z/k_c=√(1−(8/10)²)=0.6, so the amplitude decays by exp(−0.6 k_c z). Chapter 04 will obtain k_c from rectangular geometry instead of prescribing f_c.

1. Starting from β²=k²−k_c² with k=ω/v, derive ω_c and the propagating condition.
2. For the exp(+jωt−jβz) convention, show which imaginary branch of β produces decay toward +z below cutoff.
3. Derive λ_g=λ/√(1−(ω_c/ω)²). What happens at cutoff and far above cutoff?
4. Derive v_p and v_g from the exact dispersion relation and prove v_pv_g=v² under the stated assumptions.
5. A vacuum mode has f_c=6 GHz and operates at 10 GHz. Compute v_p/c, v_g/c and λ_g/λ.
6. Using Z_TE=ωμ/β and Z_TM=β/(ωε), derive their expressions in terms of η and f_c/f.
7. For a vacuum mode with f_c=10 GHz at 15 GHz, compute Z_TE and Z_TM using η₀=376.73 Ω.
8. At f=0.8f_c, compute α_z/k_c and the amplitude ratio after a distance z=2/k_c.
9. Explain why v_p>c in a vacuum waveguide does not imply superluminal information transfer.
10. A narrowband pulse traverses L=0.30 m in a vacuum guide with f_c=8 GHz at f=10 GHz. Compute v_g and group delay; compare with L/c.

Chapter 04 will now supply the missing geometry: solve the rectangular cross-section exactly, obtain the TE_mn and TM_mn eigenfunctions, identify the dominant TE_10 mode, and connect the dimensions a,b to cutoff frequencies and field patterns.