Second-Harmonic Generation: Phase Matching and Conversion

An interactive view of second-harmonic generation (SHG) in a $\chi^{(2)}$ crystal under the plane-wave slowly varying envelope approximation. In the undepleted-pump limit the second-harmonic intensity is $I_{2\omega}(z) = (\gamma z)^2 \operatorname{sinc}^2(\Delta k z/2)$: at perfect phase matching ($\Delta k = 0$) it grows exactly as $z^2$, while for $\Delta k \neq 0$ it is a pure $\sin^2$ oscillation of fixed amplitude $(2\gamma/\Delta k)^2$ with coherence length $L_c = \pi/|\Delta k|$, so the energy cycles between the fundamental and the harmonic every $2 L_c$ and never accumulates. At perfect phase matching with pump depletion the exact closed form is $I_\omega = \operatorname{sech}^2(z/L_{NL})$, $I_{2\omega} = \tanh^2(z/L_{NL})$ with $L_{NL} = 1/\gamma$, so $I_\omega + I_{2\omega} = 1$ identically (energy/Manley-Rowe conservation) and the conversion efficiency rises monotonically toward but never reaches 100% ($\tanh^2 < 1$). The phase-matching acceptance is the $\operatorname{sinc}^2(\Delta k L/2)$ curve, and the dispersion panel uses the $\beta$-BBO Sellmeier equations to show why birefringent angle tuning is needed and to compute the type-I phase-matching angle (about 22.8 degrees for 1064 to 532 nm). Manley-Rowe power conservation ($I_\omega + I_{2\omega} = 1$) holds identically and the conversion efficiency rises monotonically toward but never reaches unity.