Nonlinear optics and harmonic generation

Every material responds to light by having its electrons pushed around by the light's electric field, and that motion is what re-radiates the light as it passes through — the basis of refraction itself. At everyday intensities, the response is proportional to the field: double the field strength and you double the response, which is exactly what "linear optics" means, and it's why ordinary light passing through glass or water comes out the same colour it went in.

Push the intensity high enough — the kind of intensity you only get by focusing a laser to a tiny spot — and that proportionality stops being exact. The material's response picks up extra terms that go as the square of the field strength, the cube, and so on; these are small correction terms at ordinary intensities, which is why nonlinear effects go unnoticed in daily life, but they become significant when the field is strong enough. The second-order term is responsible for second-harmonic generation: expand out what a field oscillating at frequency ω looks like when squared, and you get not just a term at frequency 2ω but a genuine new oscillation the material can radiate at double the original frequency.

None of this violates energy conservation, because it isn't really about one wave changing colour — it's two photons at the original frequency being absorbed together and one photon at twice the frequency (and twice the energy) being emitted, a process only some crystals support at any useful efficiency, and only a small fraction of the beam typically converts. This is exactly how a common green laser pointer works: a diode laser producing invisible 1064 nm infrared light passes through a nonlinear crystal that converts a fraction of it to visible 532 nm green light, and a filter blocks the unconverted infrared before it reaches your eye.