Light's symmetry, asymmetry, and their role in nonlinear optics and ultrafast phenomena

The analysis of symmetries is extremely useful across science. In Physics, symmetries are used to derive conservation laws and selection rules for transitions in interacting systems. In the early days of nonlinear optics (NLO), symmetries were used to formulate a set of rules for photonic processes according to the medium's symmetries that are reflected in the NLO coefficient tensor. While this approach was believed to be complete and closed, the field has recently reignited as multi-color ultrashort laser pulses with tailored polarization and spatiotemporal structures become standard in NLO. A more complete theory has been recently emerging, which aims to incorporate all possible dynamical degrees of freedom of light: spin and orbital angular momentum, spatial structure, time-dependent polarizations, temporal envelopes, etc., in addition to the symmetries of the medium. This theoretical development is also accompanied by experimental advances that rely on tailored light beams that can now be generated with ever-increasing complexity, including topologies in real and a variety of synthetic dimensions, carrying poly-chromatic carrier waves, time-dependent varying angular momenta, local-chirality, and more. The nonlinear interactions between light fields with unique symmetries (or asymmetries) and matter is especially appealing, since that holds the key for developing new ultrafast spectroscopies with sub-femtosecond resolution, for exerting exact control over matter, and improving our fundamental understanding of how light and matter interact. We review these recent advances in this expanding field, focusing on the theory, its implications, and seminal experiments. We aim to establish a comprehensive database of symmetries and selection rules governing NLO light-matter interactions within the emerging new formalism, and invite the scientific community to contribute to this effort.