Non-linear crystals and frequency-mixing processes

In addition to the various tuneable laser sources described in the previous section, sources of tuneable 

only a short summary of the basic principles behind the non-linear effects is given, and some practical experimental realizations are discussed. 

All non-linear processes in crystalline materials rely on the fact that the macroscopic polarization of the medium, with non-linear susceptibility x, which is subjected to a strong electromagnetic field E, can be approximated as a power series expansion in terms of that applied field. The basic formalism for this, associated with the three-wave mixing processes of relevance here (i.e. two incident waves generate a new outgoing wave), is given in Box 4.1. 

In essence, the polarization induced in the nonlinear material by two incident waves E and Ei acts as a source of new waves at frequencies 

The new wave propagates through the non-linear medium with phase velocity 

---