Recover conventional optical tensor structure from finite-wavevector lattice coefficients

Derive the contraction of the finite-wavevector lattice nonlinearity coefficient \(\Lambda_{pqr}(\hat{q})\) with the input and output optical field polarizations to recover the conventional crystal-frame second-order susceptibility \(\chi^{(2)}_{abc}\) lobes-and-nodes structure.

Background

The paper computes the nonlocal nonlinear response using phonon-polarization channels p,q,r∈{L,T1,T2}p,q,r\in\{L,T_1,T_2\} relative to the propagation direction q^\hat q. The resulting Λpqr(q^)\Lambda_{pqr}(\hat q) coefficients are reported before projection onto the actual input and output optical field polarizations.

A complete optical interpretation requires an additional tensor contraction that converts these mode-projected finite-wavevector coefficients into the conventional crystal-frame susceptibility χabc(2)\chi^{(2)}_{abc}, including its polarization-dependent lobes and nodes. The paper explicitly leaves this contraction unresolved, making it the sole qualifying future-work problem identified in the provided text.

References

Recovering the conventional \chi{(2)}_{abc} lobes/nodes structure from \Lambda_{pqr}(\hat{q}) requires an additional contraction of the resulting susceptibility tensor with the actual input and output optical field polarizations; the present \Lambda-plot reports the finite-k lattice coefficient one step prior to that optical-geometry projection, and we leave this contraction to future work.

— Nonlocal-nonlinear phonon polaritons  (2608.24344 - Álvarez-Pérez et al., 25 Aug 2026) in Methods, subsection “Density-functional perturbation theory calculations”