Second-harmonic generation in dielectric nanoparticles with different symmetries

Abstract

In this work we study second-harmonic generation a monocrystalline nanoparticle with a non-centrosymmetric crystalline lattice. It was shown that breaking the symmetry of the nanoparticle's shape can significantly affect the second harmonic radiation pattern. We propose a method for explaining and predicting the generated field for arbitrary nanoparticles and provide selection rules for nanoparticles with several different symmetries. IntroductionSecond-harmonic generation (SHG) by resonant nanoparticles has recently been actively studied both theoretically [1-5] and experimentally [6][7][8][9][10][11] in order to develop nanosized light sources. The absence of phase matching conditions at the subwavelength scales results in significant drop of the generation efficiency, making the exploitation of resonances in such nanoscale structures the only way to enhance SHG. During the last few decades, metallic nanostructures supporting localized surface plasmon esonances have been actively studied -while the lattice of typical plasmonic materials has a center of inversion, second harmonic (SH) generation is still possible by surface and nonlocal volume effects [12][13][14][15][16]. In more recent developments, nanoparticles from dielectrics and semiconductors with a non-centrosymmetric crystalline lattice and bulk second order non-linearity [17][18][19][20] have been extensively studied. Mie-resonances supported by these dielectric nanoparticles [21] can lead to several orders of magnitude increase in SHG efficiency compared to metal structures [22]. Despite the large number of studies in this area, the full physical picture explaining SHG enhancement in all-dielectric nanostructures has yet to be identified. Selection rules, i.e. rules that determine the correspondence between the modes excited at the fundamental frequency and the modes at the second harmonic, play one of the most important roles. Selection rules for SHG in nanoparticles from materials that do not have bulk nonlinearity, such as gold, silicon, etc., were studied in detail in [23] for spherical particles, and in [24] for particles of other shapes. Despite the many studies devoted to the investigation of SHG from nanoparticles with bulk nonlinearity, the nature of the multipole composition of the generated fields, as well as the effect of nanoparticle symmetry on it, has not been studied in detail. In most of the works, the results of the SH fields' multipole expansion numerical calculations are presented without an explanation of the physical nature of the appearance of certain modes in the spectrum. The goal of this work is to determine the selection rules for SHG by dielectric nanoparticles with arbitrary symmetries and with a nonlinear susceptibility tensor χ (2) . In particular, we will show that a violation of the symmetry can strongly affect the mode composition and, as a result, the radiation pattern of the SH. Derivation of selection rulesAccording to [21,[25][26][27][28], the field E, which satisfies the Helmholtz equation ∇ 2 E + ε ω c 2 E = 0 ...

High-harmonic generation at the nanoscale boosted by bound states in the continuum

“…One strategy is to employ optical cavities with resonances both at the fundamental and harmonic frequencies [2]. However, this approach has yet to lead to the demonstration of a large efficiency due to mode-symmetry selection rules [3][4][5]. A substantial progress in nonlinear optics at the nanoscale was achieved with the use of plasmonic structures [6].…”

Second-harmonic generation in Mie-resonant dielectric nanoparticles made of noncentrosymmetric materials

“…These two processes change the selection rules A-C. However, the formulation of the exact selection rules in this case is a matter of the future work, there will be a particular resemblance to the case of the SHG from a cylindrical nanoparticle with isotropic linear permittivity tensor [58].…”

Multipolar origin of bound states in the continuum

“…We introduce an additional notation W piprmn for the both types of VSHs, where inversion parity index p i = (−1) n+1 for M prmn (k, r), and p i = (−1) n for N prmn (k, r). Index n is the multipole order, m varies from 0 to n, and p r = 1 if W piprmn is even under reflection from y = 0 plane (ϕ → −ϕ in the spherical system), and p r = −1 if it is odd [46].…”