ω + ω → 2ω: converting red laser light to blue/green requires momentum conservation
SHG converts pump photons at ω to 2ω via χ⁽²⁾ susceptibility. Efficiency requires phase matching: Δk = k(2ω) − 2k(ω) = 0.
Mismatch → sinc² dependence:
I₂ω ∝ χ²I²L² sinc²(ΔkL/2)
Perfect phase matching (Δk=0): I₂ω ∝ L² — quadratic growth. With mismatch: coherence length L_c = π/Δk limits conversion.
Birefringent phase matching: crystal angle θ tunes the extraordinary refractive index n_e(θ), finding the angle where n_e(2ω) = n_o(ω).
Applications: green laser pointers (1064→532 nm), OPO, blue lasers (BBO, KDP crystals).