Adding: combining layers

When two layers are combined — layer 1 (top, interfaces \(a\)–\(b\)) above layer 2 (bottom, interfaces \(b\)–\(c\)) — the composite properties are obtained by summing the geometric series of reflections in the gap at interface \(b\).

Because a composite of inhomogeneous sub-layers is itself inhomogeneous, all four directional matrices are required: \(\mat{R}_{ac}, \mat{T}_{ac}\) (illumination from above) and \(\mat{R}_{ca}, \mat{T}_{ca}\) (illumination from below).

Resolvents

Two resolvent matrices sum the multiple reflections between the facing surfaces of the two layers:

\[\mat{D}_1 = (\mat{I} - \mat{R}_{bc}\mat{R}_{ba})^{-1}, \qquad \mat{D}_2 = (\mat{I} - \mat{R}_{ba}\mat{R}_{bc})^{-1}.\]

Composite reflection and transmission

\[\begin{split}\mat{R}_{ac} &= \mat{R}_{ab} + \mat{T}_{ba}\mat{D}_1\mat{R}_{bc}\mat{T}_{ab}, \\ \mat{R}_{ca} &= \mat{R}_{cb} + \mat{T}_{bc}\mat{D}_2\mat{R}_{ba}\mat{T}_{cb}, \\ \mat{T}_{ac} &= \mat{T}_{bc}\mat{D}_2\mat{T}_{ab}, \\ \mat{T}_{ca} &= \mat{T}_{ba}\mat{D}_1\mat{T}_{cb}.\end{split}\]

Composite source vectors

The same combination law applies to both the thermal and the solar source vectors:

(3)\[\begin{split}\vect{s}^{+} &= \vect{s}^{+}_1 + \mat{T}_{ba}\mat{D}_1 (\vect{s}^{+}_2 + \mat{R}_{bc}\vect{s}^{-}_1), \\ \vect{s}^{-} &= \vect{s}^{-}_2 + \mat{T}_{bc}\mat{D}_2 (\vect{s}^{-}_1 + \mat{R}_{ba}\vect{s}^{+}_2).\end{split}\]

The upward source emerging from the composite is layer 1’s own upward source plus the light that layer 2 emits (and that layer 1 reflects back down and up) transmitted up through layer 1, and symmetrically for the downward source. Because these expressions are linear in the source vectors with temperature-independent operators, differentiating them with respect to the Planck sources gives the analytic temperature Jacobian at essentially no extra cost (Analytic temperature Jacobians).

Non-scattering layer optimisation

When one of the two layers has \(\omega = 0\), its reflection matrices vanish and its transmission is diagonal,

\[\mat{R} = \mat{0}, \qquad T_{ij} = \ee^{-\tau/\mu_i}\,\delta_{ij}.\]

Then \(\mat{D}_1 = \mat{D}_2 = \mat{I}\), both matrix inversions disappear, and the adding step collapses to \(O(M^2)\) diagonal scaling operations. The solver detects this case automatically and takes the fast path.