Nonlinear optical phenomena have been extensively studied theoretically, experimentally, and practically after second-harmonic generation (SHG) was first observed by Franken et al. and the theoretical framework had been established, mainly by Bloembergen and his colleagues. Most nonlinear optical phenomena are successfully described by nonlinear optical tensors with adequate rank, whether the involved processes are resonant or nonresonant, parametric or dissipative, slow or fast, temporally dependent or not. The magnitudes of nonlinearities are determined by complex physical processes, including real or virtual electronic transitions, electron-phonon interactions, and even the thermal properties of materials. Therefore, the quantitative studies of optical nonlinearities require precise experimental techniques to extract intrinsic data to be compared by theoretical values obtained from large amounts of numerical computation. However, there are simple symmetry relationships among the tensor components of the nonlinear optical coefficients determined only by the point group to which materials belong. For example, it is well known that the second-order optical nonlinearity vanishes in any material with spatial inversion but does not vanish in noncentrosymmetric systems. Understanding the symmetry properties of the relevant tensors is critical to distinguishing and assigning the observed nonlinear effects of various physical origins, especially when small signals from magnetic effects are investigated.
Regarding atomic positions, all crystals can be classified into 32 point groups and 230 space groups. According to von Neumann's principle, the symmetry properties of nonlinear optical phenomena, as well as any other physical property, are determined by the types of symmetries each point group respects. These classifications are, however, not complete when magnetism affects the properties of materials. Since angular momentum vectors, such as spin vectors, are defined as the vector product of two conventional vectors, the behavior under a coordinate transformation is different from that of the usual vectors. For example, spatial inversion reverses the direction of conventional vectors, while it does not change the direction of spins. Instead, the time-reversal operation flips the direction of the spin vector. If we consider a material with spins aligned in an antiferromagnetic order on a lattice having centrosymmetry, the conventional spatial inversion cannot be a symmetry element; however, the combination of spatial inversion and temporal reversal reproduces the equivalent alignment. In order to describe the correct symmetry of magnetic materials, we need to employ the time-reversal operation as a transformation element.
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