In a nonlinear crystal (χ⁽²⁾ material), a fundamental wave at ω drives a second-harmonic at 2ω. The coupled equations are:
dA₁/dz = −iκ A₂ A₁* e^{+iΔkz}
dA₂/dz = −iκ A₁² e^{−iΔkz}
The phase matching condition Δk = k_{2ω} − 2k_ω = 0 is critical: if unmatched, energy flows back from SHG to fundamental periodically over the coherence length L_c = π/Δk. At Δk=0 (phase matched), efficiency grows monotonically as sin²(κL).
Quasi-phase matching (QPM): a periodically poled crystal flips the sign of χ⁽²⁾ every coherence length, preventing back-conversion. This achieves high efficiency without birefringent phase matching. Used in PPLN (periodically poled lithium niobate).