Sum-frequency generation at molecule-nanostructure interfaces from diagrammatic theory of nonlinear optics

We apply the loop diagrammatic method for linear and nonlinear optics to the calculation of the sum-frequency response of a molecule-nanostructure composite system. The presence of the nanostructure modifies the molecular response through dipolar energy exchange, and the molecular hyperpolarizability is factorized by nanostructure response functions of increasing orders. We provide a general method to transform these functions into products of first-order nanostructure polarizabilities, accounting for enhancements of the molecular response by coupling to plasmonic or excitonic resonances. In particular, we show how the diagrams may be directly read to determine the response functions and their factorization without explicit calculation. The methodology provides a frame for various applications to other systems, interactions, and nonlinear optical processes.

Figure 1

List of the diagrams for bipartite systems made of a nanostructure (upper loop) and a molecule (lower loop), when $V=0$ and 1. The arrow of each loop indicates the position of the initial state, and that of each virtual boson gives the direction of the frequency (i.e., energy) exchange.

Figure 2

List of the bipartite diagrams for $V=2$, when the photon ${\omega }_1$ is interacting with the molecule (lower loop). See Fig. 1 for the meaning of loops and arrows.

Figure 3

List of the bipartite diagrams when $V=3$. See Fig. 1 for the meaning of loops and arrows.

Figure 4

Illustration of the method to directly compute the contribution of the 24 two-boson diagrams ($V=2$) without explicit calculations, based on diagram $[2,1,\downarrow \uparrow ]$ of Fig. 2. The quantity ${\alpha }^{-}$ is the resonant part of $\alpha$ resulting from the upper loop splitting (see Ref. [22], Sec. 3.1).