nonlinear-optics Physics

Second-harmonic generation is a nonlinear process in which intense light at frequency ω passing through a nonlinear crystal drives a polarization component proportional to E², radiating new light at 2ω. At weak fields the quadratic polarization term is negligible compared with the linear one, and although the generated power scales with the square of the fundamental intensity it remains vanishingly small. Only when a laser supplies sufficiently high field strength and the crystal satisfies phase matching—so that fundamental and harmonic waves stay in step—can conversion efficiencies reach sizable tens of percent.

Phase matching demands that the fundamental and the second harmonic propagate at equal phase velocity so harmonic contributions everywhere add in phase; otherwise the coherence length is only tens of micrometers and conversion oscillates rather than accumulates. Mathematically it is k(2ω) = 2k(ω), and at zero mismatch Δk the output power grows quadratically with crystal length. Birefringent crystals exploit the direction- and polarization-dependent refractive index of ordinary and extraordinary waves, tuning angle or temperature until some polarization combination satisfies n(ω) = n(2ω).

The second-harmonic intensity grows as I₂ ∝ I₁² d_eff² L² because it originates from the second-order nonlinear polarization, so even small increases in the fundamental power boost the output quadratically. Efficient buildup additionally requires the fundamental and harmonic waves to propagate with the same phase velocity — i.e. n(ω) = n(2ω) — so that contributions generated along the crystal add in phase; when Δk ≠ 0 the output oscillates as sin²(ΔkL/2) and is capped by the coherence length. In practice phase matching is achieved by temperature or angle tuning of birefringent crystals.