Nonlinear Optics Through the Field Tensor Formalism

The “field tensor” is the tensor product of the electric fields of the interacting waves during a sum- or difference-frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light-matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non-coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time-space multi-convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all-optical and purely electric poling, and also bulk single crystals.