Electromagnetic mode laboratory
Explore Gaussian-beam coupling, cutoff, dispersion and copper-wall loss in an air-filled circular waveguide—directly in your browser.
Perfect-conductor circular eigenfields with first-order copper surface-resistance loss. Centered Gaussian overlap is evaluated numerically. Flanges, roughness, offsets and lens aberrations are not included.
Reference notes
The model combines ideal circular-waveguide eigenmodes, Gaussian beam optics, overlap-integral coupling, and a first-order copper surface-loss correction.
In a uniform, air-filled circular guide, the longitudinal field satisfies the scalar Helmholtz equation. Separation in cylindrical coordinates gives radial solutions proportional to Bessel functions:
TE modes have Ez = 0 and use roots χ′mn of J′m. TM modes have Hz = 0 and use roots χmn of Jm. The dominant circular-guide mode is TE11, with χ′11 = 1.84118.
For guide radius a, the transverse wavenumber is kc = χ/a. Each mode propagates only when the operating frequency exceeds its cutoff:
Above cutoff, β is real and power travels along the tube. Below cutoff, β is imaginary and the field is evanescent. Waveguide dispersion also makes the group velocity vg = c√(1 − (fc/f)²), producing delay relative to free space.
A collimated Gaussian THz beam illuminating a thin lens is focused to an approximate 1/e electric-field waist:
Here λ is the free-space wavelength, f is the lens focal length, and wL is the Gaussian radius on the lens. The simulator lets wL vary as a fraction of the lens radius. A shorter focal length or a more fully illuminated lens produces a smaller focused waist.
Coupling is evaluated from the normalized transverse-field overlap between the centered, x-polarized Gaussian beam and the selected waveguide eigenmode:
The result includes the fraction of Gaussian power entering the circular aperture. Symmetry can make the overlap of a centered beam with some modes zero even when those modes are above cutoff.
In interference mode the two complex output fields are added coherently. Their output phase is φ0 + (βTM11 − βTE11)L, and the axial beat length is Lb = 2π/|βTE11 − βTM11|. Orthogonality makes total integrated modal power additive, while the local field pattern depends strongly on relative phase.
Finite copper conductivity creates a surface resistance that increases with the square root of frequency:
The wall-loss power is obtained from the tangential magnetic field at r = a. The resulting attenuation α gives conductor loss in dB m⁻¹ and output power Pout/Pin = η exp(−2αL).
This is a rapid analytical design model, not a full 3D FEM or FDTD calculation. It assumes a straight, uniform circular tube, an ideal thin lens, an aligned centered beam, and smooth copper walls.
Use the results for parameter exploration, then validate a final design with measurement or a calibrated full-wave solver.