Sum- and Difference-Frequency Generation

Two waves of comparable intensity and frequency ω₁ and ω₂ are initially present, and a wave at the third frequency ω₃ is created through the nonlinear interaction
P(ω₃) = ∈₀ χ⁽²⁾E(ω₁)E(ω₂), .........(1)
where ω3 = ω1+ω2. Similarly, a wave at the difference frequency ω₃−ω₂ = ω₁ may be generated (ω₃ > ω₂ > ω₁). A very common technique to produce the third harmonic of a neodymium laser is, first, the generation of the second harmonic and then sum frequency generation by mixing the second harmonic (ω₂) with the unconverted portion of the fundamental beam (ω₁).