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We derive the $\mathcal{T}$-matrix formalism tailored for the numericalanalysis of second-harmonic (SH) generation from arbitrarily-shaped particlesmade of centrosymmetric optical materials. First, the transfer matrix of asingle particle is computed \textit{via} the extended boundary conditionmethod, in which the electromagnetic fields both at fundamental frequency andSH are expanded in vector spherical wave functions and the integral formulationis satisfied away from the surface of the scatterer. We allow for the accuratephysical description of the SH sources by taking into account both localsurface and nonlocal bulk polarization contributions to the nonlinearpolarization density source responsible for the generation of the SH signal ina particle. This single-particle formalism is then extended to arbitrarydistributions of particles by incorporating into the formalism linear andnonlinear electromagnetic wave scattering from the particles in the cluster.Importantly from a practical point of view, our method can be applied toparticles of arbitrary shape made of optical materials characterized by generalfrequency-dispersion relations, so that it can describe the linear andnonlinear optical response of clusters of metallic, semiconductor, orpolaritonic particles, as well as mixtures of such particles. The approachproposed here is faster and more memory-efficient than well-establishednumerical techniques, especially in the analysis of spheroidal particles, dueto the favourable symmetries of spherical wave basis functions used in the wavescattering analysis.

$\mathcal{T}$-matrix method for computation of second-harmonic generation upon optical wave scattering from clusters of arbitrary particles