formulation of SHG was initially described by N. Bloembergen and P. S. Pershan at Harvard in 1962. In their extensive evaluation of Maxwell's equations at the planar interface between a linear and nonlinear medium, several rules for the interaction of light in non-linear mediums were elucidated. Before the development of the first laser, Peter Franken made a $10 USD bet with his friend and colleague, Willis Lamb, that second-harmonic generation was possible. Lamb refused to believe such a development was possible. Upon the publication of Franken's results, he went to Lamb and demanded his payment. Lamb wrote him a check for $10 USD and Franken framed and hung it on his office wall, proudly showing it off to all visitors. Since his death, his wife Peg Nash Franken has maintained possession of the check. Types in crystals Critical phase-matching Second-harmonic generation occurs in three types for critical phase-matching, denoted 0, I and II. In Type 0 SHG two photons having extraordinary polarization with respect to the crystal will combine to form a single photon with double the frequency/energy and extraordinary polarization. In Type I SHG two photons having ordinary polarization with respect to the crystal will combine to form one photon with double the frequency and extraordinary polarization. In Type II SHG, two photons having orthogonal polarizations will combine to form one photon with double the frequency and ordinary polarization. For a given crystal orientation, only one of these types of SHG occurs. In general to utilise Type 0 interactions a quasi-phase-matching crystal type will be required, for example periodically poled lithium niobate (PPLN). Non-critical phase-matching Since phase-matching process basically means to adapt the optical indices n at ω and 2ω, it can also be done by a temperature control in some birefringent crystals, because n changes with the temperature. For instance, LBO presents a perfect phase-matching at 25 °C for a SHG excited at 1200 or 1400 nm, but needs to be elevated at 200 °C for SHG with the usual laser line of 1064 nm. It is called "non-critical" because it does not depend on the crystal orientation as usual phase-matching. Optical second-harmonic generation Since media with inversion symmetry are forbidden from generating second-harmonic light via the leading-order electric dipole contribution (unlike third harmonic generation), surfaces and interfaces make interesting subjects for study with SHG. In fact, second-harmonic generation and sum frequency generation discriminate against signals from the