Symmetry-based identification and enumeration of independent tensor properties in nonlinear and chiral optics

For many laser-based methods of material characterisation and analysis, a tensor formulation of theory is necessary, especially techniques that exploit nonlinear or chiral optics. The fundamental interactions that underpin such methods offer various levels of approach to theory, but the most rigorous often lead to equations of considerable complexity. To compute the values for individual material parameters frequently demands making assumptions of extreme simplicity, overly dependent on calculational method, yet still providing unsatisfactory results. A pragmatic and entirely rigorous symmetry-based approach to irreducible tensorial structures circumvents many of these problems, securing reliable results and guiding the pathway to applications. Instead of focusing on individual tensor components, such an approach can rapidly determine the number of linearly independent quantities - and hence the number of operationally different setups necessary for full characterisation. Hence, one can directly ascertain how variations of optical polarisation and beam geometry reliably capture the response of any material system. The use of an irreducible tensor method operates independently of any means to calculate material properties. It removes the need for common simplifying assumptions such as the approximation of tensorial structure by a scalar representation, adoption of a two-state model, or disregarding near-resonance damping. It also obviates any dependence on a choice of simulation package, or quantum-calculational software. In this paper, the principles are set down and illustrated by application to experiments of varying degrees of complexity, including interactions of growing significance in the realm of chiral nonlinear optics. Limitations of this approach are also critically assessed.