Interface Algorithm

The interface algorithm mixes exact algebra with engineering intuition. The derivations of the interface formulas are exact within the stated assumptions. The passivity and truncation arguments are intentionally more heuristic: they are useful for reasoning about numerical behavior, but they should not be read as formal proofs.

Problem Setup

We consider a step discontinuity between two rib waveguides, labeled left (L) and right (R), with propagation along z.

In a z-invariant waveguide, the transverse fields are expanded in eigenmodes:

\[ \mathbf{E}_t(x,y,z) = \sum_p \left[a_p^{+} e^{-j \beta_p z} + a_p^{-} e^{+j \beta_p z}\right] \mathbf{e}_p(x,y) \]

\[ \mathbf{H}_t(x,y,z) = \sum_p \left[a_p^{+} e^{-j \beta_p z} - a_p^{-} e^{+j \beta_p z}\right] \mathbf{h}_p(x,y) \]

The sign flip on H for backward-propagating modes follows from Maxwell's curl equation.

At the interface z = 0, tangential field continuity gives:

\[ \sum_p (a_p^{+} + a_p^{-}) \mathbf{e}_p^L = \sum_q (b_q^{+} + b_q^{-}) \mathbf{e}_q^R \]

\[ \sum_p (a_p^{+} - a_p^{-}) \mathbf{h}_p^L = \sum_q (b_q^{+} - b_q^{-}) \mathbf{h}_q^R \]

Inner Product

To convert the vector continuity equations into matrix equations, we project onto test functions using the asymmetric modal overlap:

\[ \langle \mathbf{e}_a, \mathbf{h}_b \rangle = \frac{1}{2}\int_S (\mathbf{e}_a \times \mathbf{h}_b)\cdot\hat{z}\, dA \]

This overlap is not symmetric across different waveguides.

Define:

• O_LR[i,j] = <e_i^L, h_j^R>
• O_RL[i,j] = <e_i^R, h_j^L>
• G_LL[i,j] = <e_i^L, h_j^L>
• G_RR[i,j] = <e_i^R, h_j^R>

For lossless waveguides with the unconjugated product:

• reciprocity within one waveguide gives G_LL = G_LL^T and G_RR = G_RR^T;
• orthogonal modes with distinct propagation constants have vanishing off-diagonal self-overlaps;
• in general O_LR != O_RL^T.

If modes are orthonormalized in this same inner product, then G_LL = G_RR = I.

Why The Symmetric Product Fails For Reflection

The symmetric product is

\[ \langle \mathbf{e}_a, \mathbf{h}_b \rangle_{\mathrm{sym}} = \frac{1}{4}\int_S \left[ (\mathbf{e}_a \times \mathbf{h}_b) + (\mathbf{e}_b \times \mathbf{h}_a) \right]\cdot\hat{z}\, dA \]

By construction it is commutative, so it forces O_LR = O_RL^T even for modes of different waveguides. That makes the reflection term vanish algebraically.

This is the main reason the asymmetric product is used for interface scattering.

Interface Formulas

Left Incidence

For left incidence, set b^- = 0. Project E continuity with h_k^L and H continuity with e_k^L:

\[ G_{LL}(a^+ + a^-)=O_{RL}^T b^+ \]

\[ G_{LL}(a^+ - a^-)=O_{LR} b^+ \]

Adding:

\[ 2 G_{LL} a^+ = (O_{LR} + O_{RL}^T) b^+ \]

\[ T_{LR} = 2 (O_{LR} + O_{RL}^T)^{-1} G_{LL} \]

Subtracting:

\[ R_{LL} = \tfrac{1}{2} G_{LL}^{-1} (O_{RL}^T - O_{LR}) T_{LR} \]

For orthonormal modes, G_LL = I, so:

\[ T_{LR} = 2 (O_{LR} + O_{RL}^T)^{-1} \]

\[ R_{LL} = \tfrac{1}{2} (O_{RL}^T - O_{LR}) T_{LR} \]

Two equivalent continuity forms are also useful:

\[ R_{LL} = O_{RL}^T T_{LR} - I = I - O_{LR} T_{LR} \]

Averaging those two recovers the standard difference form:

\[ \tfrac{1}{2} \left[ (O_{RL}^T T_{LR} - I) + (I - O_{LR} T_{LR}) \right] = \tfrac{1}{2}(O_{RL}^T - O_{LR}) T_{LR} \]

In practice, the continuity forms are often numerically preferable because they avoid explicitly subtracting two nearly equal overlap matrices before multiplying by T.

Right Incidence

For right incidence, set a^+ = 0:

\[ G_{RR}(b^+ + b^-) = O_{LR}^T a^- \]

\[ G_{RR}(b^+ - b^-) = -O_{RL} a^- \]

Hence:

\[ T_{RL} = 2 (O_{RL} + O_{LR}^T)^{-1} G_{RR} \]

\[ R_{RR} = \tfrac{1}{2} G_{RR}^{-1} (O_{LR}^T - O_{RL}) T_{RL} \]

And for orthonormal modes:

\[ R_{RR} = O_{LR}^T T_{RL} - I = I - O_{RL} T_{RL} \]

Full S-Matrix

Assemble the interface two-port as

\[ \begin{pmatrix} a^- \\ b^+ \end{pmatrix} = S \begin{pmatrix} a^+ \\ b^- \end{pmatrix} \]

with block structure

\[ S= \begin{pmatrix} R_{LL} & T_{RL} \\ T_{LR} & R_{RR} \end{pmatrix} \]

The Metric

The self-overlap matrices G_LL and G_RR are metrics on modal amplitude space.

If modes are orthonormalized in the same inner product used for the interface solve, those metrics become identity matrices and the simple formulas above apply directly.

If the basis is not orthonormal in that metric, then the generalized formulas must be used. In that setting, passivity is also naturally expressed in the metric-weighted form

\[ S^\dagger G_{\mathrm{out}} S \le G_{\mathrm{in}} \]

rather than in plain Euclidean coordinates.

Analytical Sanity Checks

Same Medium

If the two sides are identical, then O_LR = O_RL = G.

For orthonormal modes, G = I, so

\[ T = 2 (I + I)^{-1} = I \]

\[ R = \tfrac{1}{2}(I - I) I = 0 \]

This is the basic self-consistency check for the interface formulas.

Fresnel Limit

In a single-mode homogeneous limit, the modal overlap formulas reduce to the usual normal-incidence Fresnel coefficients.

With normalized overlaps

\[ O_{LR} = \sqrt{n_R / n_L}, \qquad O_{RL} = \sqrt{n_L / n_R} \]

one obtains

\[ T = \frac{2}{\sqrt{n_R/n_L} + \sqrt{n_L/n_R}} = \frac{2 \sqrt{n_L n_R}}{n_L + n_R} \]

\[ R = \frac{\sqrt{n_L/n_R} - \sqrt{n_R/n_L}} {\sqrt{n_R/n_L} + \sqrt{n_L/n_R}} = \frac{n_L - n_R}{n_L + n_R} \]

which matches the Fresnel result in the chosen mode-amplitude convention.

Practical Numerical Pipeline

The practical pipeline used by the library is:

1. compute modes;
2. post-process them into a filtered, orthonormal basis;
3. build the overlap matrices;
4. solve for the transmission blocks with a TSVD-regularized inverse;
5. reconstruct the reflection blocks from continuity;
6. assemble the interface S-matrix;
7. enforce passivity on the singular values of S;
8. optionally enforce reciprocity by symmetrization.

The key sharp edge is that steps 2 and 3 must use compatible inner products.

TSVD Regularization

The raw interface solve is most sensitive to small singular values of

\[ A_{LR} = O_{LR} + O_{RL}^T \]

and its right-incidence counterpart.

Rather than using a direct inverse, the library solves with a truncated SVD:

\[ T_{LR} = 2 A_{LR,\mathrm{TSVD}}^{+} \]

where singular values below

\[ s_{\mathrm{cut}} = rcond \cdot s_{\max} \]

are discarded.

This is a pragmatic regularization. It is not exact physics; it is a way of preventing a truncated basis from letting tiny singular directions dominate the low-order scattering coefficients.

Passivity And Truncation

With a complete basis, the lossless interface solve is unitary.

With a truncated basis, the computed S-matrix can violate passivity. A common engineering interpretation is that the missing channels are mostly radiation or continuum-like modes that should carry power away from the retained guided subspace.

The following argument is heuristic, not a proof.

In the scalar case, if truncation makes a transmission coefficient appear as

\[ T = 1 + \varepsilon \]

with eps > 0, one may interpret the excess as power that should have gone into omitted channels. That motivates correcting overly large singular values of the assembled S.

Common corrections are:

• clip: sigma -> min(sigma, 1)
• invert: sigma -> 1 / sigma for sigma > 1
• subtract: sigma -> max(0, 2 - sigma) for sigma > 1

clip is the simplest algebraic projection.

invert and subtract both map mild gain above one to mild loss below one, which can be a useful model when a truncated solve is interpreted as missing radiation loss.

These corrections are practical modeling choices. They should not be confused with proof of convergence.

What The Algorithm Does Well

• captures reflection that is destroyed by the symmetric product;
• provides a consistent overlap-based interface operator;
• exposes regularization and passivity controls explicitly;
• works naturally with unequal numbers of modes on each side.

What Still Requires Care

• reflection is often a small difference of nearly equal overlap terms, so it is more sensitive than transmission to mesh, ordering, and truncation;
• passivity correction can improve robustness, but it does not replace basis convergence;
• if you override the default inner product, you should usually pass the same callable both to mode post-processing and to interface construction.