The book brings together three key fields of physics: symmetry, magnetic or electric long-range ("ferroic") order, and nonlinear laser optics. In the first part, the fundamentals of these three fields are introduced with a focus on the details that are relevant for their combination. The second part discusses how nonlinear optical studies help revealing properties that are inaccessible with "standard characterization" techniques, followed by a systematic discussion of the unique degrees of freedom of nonlinear-optical probing of ferroics. The third part explores material classes of central interest to contemporary condensed-matter physics, including multiferroics with magnetoelectric correlations and oxide-electronic materials, along with application related to the optical properties of ferroic materials. The book concludes with an outlook towards future developments.

1 A preview of the subject of the book1.1 Symmetry considerations1.2 Ferroic materials1.3 Laser optics1.4 Creating the trinity1.5 Structure of this book2 Symmetry2.1 Describing interactions in condensed-matter systems2.2 Introduction to practical group theory2.3 Crystals2.4 Point groups and space groups2.5 From symmetries to properties3 Ferroic materials3.1 Ferroic phase transitions3.2 Ferroic states3.3 Antiferroic states3.4 Classification of ferroics4 Nonlinear optics4.1 Interaction of materials with the electromagnetic radiation field4.2 Wave equation in nonlinear optics4.3 Microscopic sources of nonlinear optical effects4.4 Important nonlinear optical processes4.5 Nonlinear spectroscopy of electronic states5 Experimental aspects5.1 Laser sources5.2 Experimental setups5.3 Temporal resolution6 Nonlinear optics on ferroics - an instructive example6.1 SHG contributions from antiferromagnetic Cr2O36.2 SHG spectroscopy6.3 Topography on antiferromagnetic domains6.4 Magnetic structure in the spin-flop phase7 The unique degrees of freedom of optical experiments7.1 Polarisation-dependent spectroscopy7.2 Spatial resolution - domains7.3 Temporal resolution - correlation dynamics8 Theoretical aspects8.1 Microscopic sources of SHG in ferromagnetic metals8.2 Microscopic sources of SHG in antiferromagnetic insulators9 SHG and multiferroics with magnetoelectric correlations9.1 Type-I multiferroics - the hexagonal manganites9.2 Type-I multiferroics - BiFeO39.3 Type-I multiferroics with strain-induced ferroelectricity9.4 Type-II multiferroics - MnWO49.5 Type-II multiferroics - TbMn2O59.6 Type-II multiferroics - TbMnO39.7 Type-II multiferroics with higher-order domain functionalities10 SHG and materials with novel types of primary ferroic order10.1 Ferrotoroidics10.2 Ferro-axial order - RbFe(MoO4)211 SHG and oxide electronics - thin films and heterostructures11.1 Growth techniques11.2 Thin epitaxial oxide films with magnetic order11.3 Thin epitaxial oxide films with ferroelectric order11.4 Poling dynamics in ferroelectric thin films11.5 Growth dynamics in oxide electronics by in-situ SHG probing12 Nonlinear optics on ordered states beyond ferroics12.1 Superconductors12.2 Metamaterials - photonic crystals12.3 Topological insulators13 A retrospect of the subject of the book