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

Abstract

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. Especially, 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.

Diagrammatic theory of magnetic and quadrupolar contributions to sum-frequency generation in composite systems

“…1a and b) and calculated in previous papers according to the Feynman rules dedicated to optical response functions. 11,12 They encompass two and three vertices, respectively, representing light-matter interactions at the electric dipole level, thus governed by the Hamiltonian H e = − p • E. For the polarizability α ee (ω), we have:…”

“…Still, the higher order terms involving energy exchange may dominate the response, as is well-known for a molecule-nanoparticle system under the electric dipole approximation: the (eee) molecular SFG response function indeed involves first-order terms driven by the nanoparticle polarizability, which becomes giant when the beam energy matches the surface plasmon resonance energy. 9,12,14 Below, we consider the SFG response function of the molecule and the way it is modi-fied by the presence of a partner when the whole system is excited by two light beams at frequencies ω 1 and ω 2 , generating a third beam at frequency ω 3 = ω 1 + ω 2 . As we have seen in Section II, beyond the electric dipole approximation, this response also encompasses magnetic and quadrupole vertices.…”

“…ω b = ω 1 , ω 2 or ω 3 ), which simplifies the frequency filling of the two-loop diagrams. 12 For magnetic couplings in particular, it is worth noting that the energies linked to a direct interaction between orbital or spin magnetic moments of the molecule and its partner lie far below the accessible optical frequencies at stake in an SFG process. Vector bosons carrying energy in the optical range are actually rather linked to interaction Hamiltonians involving electromagnetic fields (and their gradients) oscillating at optical frequencies.…”

“…This approach follows the same path as was done previously for all-electric dipole diagrams involving only one boson. 11,12 In a general way, the unperturbed molecular hyperpolarizability β [0] is modified by the presence of the partner when one of the three beams (incoming or emitted) interacts with the partner (through one of its polarizabilities α) and is conveyed to the molecule (by the boson propagator W ) where the nonlinear SFG process takes place. Here, the result is complicated by the higher rank of the tensors describing the elementary building blocks β [0] , W and α when they involve quadrupolar vertices.…”

“…This approach has been successfully applied to the molecule-nanoparticle systems to account for plasmon-enhanced vibrational SFG spectroscopy. 12 In this case, electric dipole-dipole coupling between molecules and particles was postulated, as usual in the description of the optical response of complex systems. 8,13,14 This corresponds to the first-order contribution to the multipolar perturbative expansion of the electromagnetic interaction.…”