Waveguide ports and S-parameter extraction#

The port machinery is Magnelio’s largest methodological block. It combines standard 2D mode analysis with a family of exact discrete transparent boundary conditions (DTBC) that were derived in-house for the FIT leapfrog lattice; the published antecedents of each layer are listed per section.

2D port-mode solvers#

Curl-curl eigenvalue problem (TE/TM and hybrid modes)#

Port cross-section modes are computed from the generalised eigenvalue problem \(\mathbf K \hat e = \omega_c^2 \mathbf M \hat e\) with \(\mathbf K = \mathbf C_{2D}^{\mathsf T}\,\mathrm{diag}(M_\mu^{-1})\, \mathbf C_{2D}\), where \(\mathbf C_{2D}\) is the row/column slice of the 3D FIT curl matrix at the port plane (ports/modal/curl_curl_2d.py). Discretising the 2D problem in the same metric as the 3D problem (rather than with an independent 2D solver) is the property that makes the discrete mode an exact eigenvector of the 3D-restricted transversal operator; the FIT curl-curl eigenformulation itself is standard [2, 3]. The TM variant is the exact restriction of the same operator, an in-house derivation (DD-055).

TEM / quasi-TEM Laplace solver#

Multi-conductor cross-sections use the electrostatic route (ports/modal/tem_laplace.py): a 2D FIT Laplace problem \(\nabla\!\cdot(\varepsilon\nabla\varphi_k) = 0\) per signal conductor with unit-potential boundary conditions, \(\hat e_k = -\nabla\varphi_k\), line capacitance from the discrete energy. For inhomogeneous filling the quasi-TEM effective parameters follow the classic two-capacitance construction \(\varepsilon_{\text{eff}} = C'/C'_0\), \(Z_0 = 1/(c\sqrt{C' C'_0})\) — standard quasi-static transmission-line theory as found in the microwave-engineering literature [16, 17] (the specific formulation is textbook material; no single originating paper is claimed). The line-impedance definition used for TEM modes is the power–current impedance \(Z_{\pi}\) (DD-025); the coexistence of \(Z_{PI}\), \(Z_{PV}\), \(Z_{VI}\) definitions is classical waveguide theory [16, 18]. Analytical reference modes for coaxial and rectangular-waveguide ports are closed-form textbook solutions [16].

Mode classification and multi-mode merge#

TE/TM and TEM/QTEM branches are merged into one unified multi-mode port; degenerate multi-TEM subspaces are orthonormalised through a Gram eigenbasis, and multi-channel projections use the dual basis (Gram inverse) (DD-066). Standard linear algebra.

Exact discrete transparent boundary conditions (DTBC)#

Scalar DTBC on uniform feed lines (TEM: DD-054, TE/TM: DD-055)#

On a uniform feed line, the longitudinal dynamics of one modal amplitude in the FIT leapfrog is exactly a 1D leapfrog discrete Klein–Gordon chain with modal Courant number \(r\) (read off the co-located pair product \(M_\varepsilon M_\mu\)) and mass \(q = \hat\omega_c \Delta t\) from the 2D eigenvalue (\(q=0\) for TEM). The exact transparent termination of the semi-infinite continuation is obtained by \(\mathcal Z\)-transform: the outgoing characteristic root \(\lambda(z)\) yields the ghost relation \(\hat u_{K+1} = \lambda(z)\,\hat u_K\), realised in time domain as a causal convolution over the boundary history whose kernel is computed by contour integration and auto-extended past the run length, making the boundary exact (reflection-free to machine precision) within any finite run (ports/modal/dtbc.py). Incident waves are prescribed at the ghost plane through the same kernel.

This construction was derived in-house for the FIT/FDTD leapfrog lattice (derivation in DD-054/DD-055; measured port floors −124 to −250 dB). The general concept of exact discrete transparent boundary conditions — deriving the boundary kernel from the discretised interior scheme rather than discretising a continuous ABC — is established in the numerical-analysis literature, e.g. by Arnold, Ehrhardt and Sofronov [19] and Ehrhardt’s work on discrete TBCs for Schrödinger-type equations [20]; discretely nonreflecting boundary closures for finite-difference schemes of linear hyperbolic systems are constructed by Rowley and Colonius [21]. None of these antecedents treats modal FIT/FDTD waveguide ports; that application appears to be original to Magnelio.

CW true-mode ports for inhomogeneous lines (DD-056)#

For inhomogeneous cross-sections measured at a single frequency, the true discrete modes of the z-uniform feed section are computed as eigenpairs of the quadratic ζ-pencil

\[ \left[\zeta^2 \mathbf D_{+1} + \zeta(\mathbf D_0 - \hat\sigma) + \mathbf D_{-1}\right]\varphi = 0 \]

built from the production system matrices at the port (ports/modal/zeta_pencil.py), solved by sparse shift-invert ARPACK on the linearised pencil. Quadratic eigenvalue problems and their linearisations are covered by the survey of Tisseur and Meerbergen [22]; computing waveguide Bloch/propagation modes from a transfer/period formulation is established practice in computational photonics and microwave theory (no single originating publication is claimed). The frequency-local closed-form \((r_\text{eff}, q_\text{eff})\) chain fit (Hellmann–Feynman derivative matching so the scalar DTBC is exact at the drive frequency) is an in-house derivation.

Galerkin band-subspace DTBC for broadband runs (DD-057)#

For pulsed broadband runs on inhomogeneous lines, the tracked mode-family traces over the band span a low-rank W-orthonormal subspace; the exterior half-line is Galerkin-projected onto it, inheriting the palindromic symmetry that makes the projected lattice lossless, and is closed by the exact small-system DTBC kernel (ports/modal/band_dtbc.py). Projection-based model order reduction (Galerkin projection onto an SVD/POD subspace) is standard numerical practice [23] (survey reference; the specific passive-by-construction boundary closure is in-house).

Excitation and recording#

Port excitation prescribes the incident modal amplitude at the ghost plane (DTBC branch) or injects on the port plane (Mur branch); V/I are recorded on a single co-located plane (DD-041). Time signals are smooth pulses (Gaussian and derived shapes, signals/waveforms.py) — standard practice [5].

S-parameters#

S-parameters are computed as power waves with \(\sqrt{\mathrm W}\) normalisation, \(a,b = (V \pm Z_0 I)/(2\sqrt{Z_0})\) per mode, from the recorded modal V/I after Fourier transform (postprocessing/modal_sparameters.py, DD-042/DD-078). The power-wave formalism is Kurokawa’s [26]. Two discrete-exactness refinements are in-house (DD-063, DD-056): the a/b split de-staggers E and H with the exact discrete factor \(\lambda^{1/2}(z)\) and uses the exact discrete wave impedance of the leapfrog chain (rather than the continuum \(Z_0\)), which removes the \(O(\beta\Delta z)\) staggering leak; and CW measurements solve the exact 2×2 phasor system per port (lock-in demodulation of incident and reflected discrete waves).

Excitation amplitudes are pinned to physical units at the source (C = 1 convention, DD-085), so recorded V/I and monitor fields are in SI units — an in-house calibration convention.