Hollow rectangular waveguides

A hollow rectangular guide confines the field between two pairs of walls. Its aspect ratio and polarisation affect both dispersion and loss. The equations below use the large-core approximation; the field is truncated at the walls and the wall response enters the complex effective index.

RectMode describes a gas-filled hollow guide of rectangular cross-section, half-widths \(a\) along x and \(b\) along y, with dielectric or metal walls. Like the capillary it is a grazing-incidence model of a guide many wavelengths wide, and it shares the capillary’s gas fill and loss policy. The mode index m counts along x and n along y. Each positive integer pair has two distinct polarisation partners, polarisation="x" (the default) and "y": their fields occupy different electric-field components and their propagation constants usually differ, so they are separate mode rows.

from solprop.modes import RectMode, ModeCollection
from solprop.physdata import silver

mode = RectMode(50e-6, 12.5e-6, "Ar", 5.0, cladding=silver())  # x-polarised (1, 1)
pair = ModeCollection.with_polarisations([mode])  # the x and y partners

The Laakmann–Steier wall-reflection model gives

\[ n_{\mathrm{eff}}=\sqrt{1+\chi-X-Y},\qquad X=\left(\frac{m\lambda}{4a}\right)^2 \left(1-i\frac{\lambda a_c}{2\pi a}\right)^2, \quad Y=\left(\frac{n\lambda}{4b}\right)^2 \left(1-i\frac{\lambda b_c}{2\pi b}\right)^2. \]

Write \(s=\sqrt{\epsilon_{\mathrm{cl}}-\epsilon_{\mathrm{co}}}\) with \(\epsilon_{\mathrm{co}}=1+\chi\). For x polarisation, \((a_c,b_c)=(\epsilon_{\mathrm{cl}}/(\epsilon_{\mathrm{co}}s),1/s)\); exchange these for y polarisation. Both the contrast and the TM-like coefficient’s permittivity ratio use the filled-core permittivity. The reason is the wall’s reflection at grazing incidence. With the wall normal along x, the cladding’s normal wavenumber is \(\sqrt{k^2(\epsilon_{\rm cl}-\epsilon_{\rm co})+\kappa_x^2}\simeq ks\), and expanding the two Fresnel coefficients about grazing incidence gives \(r_s\simeq-\exp(-2\kappa_x/(ks))\) and \(r_p\simeq-\exp(-2(\epsilon_{\rm cl}/\epsilon_{\rm co})\kappa_x/(ks))\); the paired-wall round trip \(\exp(4ia\kappa_x)r^2=1\) then gives \(\kappa_x\) as in \(X\) above with \(q_{\rm TE}=1/s\) and \(q_{\rm TM}=\epsilon_{\rm cl}/(\epsilon_{\rm co}s)\). The printed paper, which treats an evacuated core, writes \(\epsilon_{\rm cl}/s\) with \(s=\sqrt{\epsilon_{\rm cl}-1}\); the two agree only when the core is evacuated. At 800 nm, argon and the dimensions above, the loss changes from that evacuated-core form are

wall	x, 1 bar	x, 10 bar	y, 1 bar	y, 10 bar
constant 1.45	+0.0216%	+0.219%	−0.0276%	−0.276%
silver	−0.0193%	−0.194%	−0.0533%	−0.534%
aluminium	−0.0267%	−0.268%	−0.0545%	−0.546%

and the real refractivity moves by up to 1.1e-4 relative (aluminium, y, 1 bar).

Complex wall corrections affect phase as well as attenuation. Each paired wall’s TE- or TM-like boundary condition gives a complex transverse wavenumber: for a complex \(q=A+iB\) the correction factor is \(1+B/(ka)-iA/(ka)\), so keeping \(A\) for the attenuation while dropping \(B\) from the phase would be inconsistent with the same expansion. solprop retains the whole complex \(\kappa_x^2+\kappa_y^2\) in \(\beta^2=k^2\epsilon_{\mathrm{co}}-\kappa_x^2-\kappa_y^2\), so the wall’s contribution to \(\operatorname{Re}n_{\rm eff}\) is kept, as a principled continuation of the first-order boundary approximation. The squared factors retain some higher orders without being a complete higher-order solution, and the model is not exact at arbitrary wavelength or aspect ratio. The paper’s equations were checked through a text extraction, not the typeset page, and the content of its published erratum (doi:10.1364/AO.15.002029) has not been checked; the coefficients above rest on the derivation given here.

Validity, and the warning. The model is first order in each wall pair’s grazing angle, \(m\lambda/4a\) (and \(n\lambda/4b\)), and in its wall correction, \(\lambda|q|/2\pi a\) (and \(\lambda|q|/2\pi b\)), with \(\lambda\) the vacuum wavelength (Laakmann and Steier’s conditions 1–2: small grazing angles, well past Brewster’s). Wherever either ratio exceeds 0.1 over the evaluated band, the mode raises a GrazingIncidenceWarning naming the ratio, its value and the wavelength — once per mode, its frozen copies along z included. The neglected terms are second order in these ratios, about 1 % of the mode’s decrement \(X\) at 0.1 and growing quadratically beyond it, and a grazing angle of 0.1 rad is still well short of glass’s Brewster grazing angle \(\arctan(1/n)\) (0.6 rad for silica). A metal’s TM-like coefficient is large (\(|q|\approx15\) for silver at 2 µm), so on a metal-walled guide the wall correction usually reaches the threshold first. The warning does not refuse: narrow the band towards shorter wavelengths, widen the guide or keep lower orders.

The analytic derivative chains through both material slopes and the wall corrections. Refractivity is formed directly, by the exactly rationalised \(z/(\sqrt{1+z}+1)\) with \(z=\chi-X-Y\), which is accurate to rounding in the large-guide limit where the spelling matters. The GasFillMixin shares gas resolution, mixtures, species absorption, parameter freezing, the four-way loss policy, gain clamp and evanescent floor with Marcatili and Vincetti modes.

The field is a product of cosines for odd indices and sines for even indices, supported inside the rectangle. Its closed forms are

\[N=\frac{\epsilon_0 c}{2}ab,\qquad A_{\mathrm{eff}}=\frac{16ab}{9}.\]

The basis normalisation uses the vacuum admittance \(\tfrac12\sqrt{\epsilon_0/\mu_0}\), as for every mode: assembly and output convert between this internal basis and the true-power amplitude with \(b=\sqrt{\operatorname{Re}n_{\rm eff}}a\), and run statistics evaluate the Poynting flux (power and intensity).

Cartesian Gauss quadrature integrates these trigonometric products, which have no finite polynomial exactness degree, so the bound is measured: cartesian_nodes() is \(6h+6\) nodes per axis for highest order \(h\), covering the measured 12, 18, 22, 26 and 35 nodes that reach a relative \(10^{-12}\) at orders 1, 2, 3, 4 and 6. For (n, m) = (3, 2) 22 nodes per axis are enough and 20 leave \(6.8\times10^{-11}\). A rule below the bound warns, as a coarse azimuthal rule does, and both Cartesian defaults of prop_waveguide take it.

Gas-filled rectangular guides run through prop_waveguide() and from_waveguide(), which take a supplied mode set rather than a radius and a mode count; n_r and n_theta are then the x and y node counts. A Cartesian product rule has no Kronrod extension, so those runs record no quadrature_error and the quadrature alarm is off; the node bound above is what stands in its place.

Both widths may be profiles. Uniform scaling is available when both half-widths are constant or both axes share the same profile object; independent tapers support the linear operator and mode-averaged area, while a modal transform refuses a non-scale-invariant taper. Two width paths and a density path — three parameters — are carried on the linear operator’s z table rather than by its expansion in the parameters, which every CPU backend runs (the native backend’s GPU builds do not). Constant widths and a constant fill report z_constant=True, so the fixed-quadrature transform evaluates the fields once rather than at every step.

Reference: K. D. Laakmann and W. H. Steier, Waveguides: characteristic modes of hollow rectangular dielectric waveguides, Applied Optics 15, 1334–1340 (1976), doi:10.1364/AO.15.001334.