Topological aspects in nonlinear optical frequency conversion

Nonlinear optical frequency conversion, observed more than half a century ago, is a cornerstone in modern applications of nonlinear and quantum optics. It is well known that frequency conversion processes are constrained by conservation laws, such as momentum conservation that requires phase-matching conditions for efficient conversion. However, conservation laws alone could not fully capture the features of nonlinear frequency conversion. Here it is shown that topology can provide additional constraints in nonlinear multifrequency conversion processes. Unlike conservation laws, a topological constraint is concerned with the conserved properties under continuous deformation, and can be regarded as a new indispensable degree of freedom to describe multifrequency processes. We illustrate such a paradigm by considering sum frequency generation under a multifrequency pump wave, showing that, akin topological phases in topological insulators, topological phase transitions can be observed in the frequency conversion process both at the classical and quantum levels.