Length dependence of second-harmonic generation in periodically poled lithium niobate under quasi-phase matching

Second-harmonic generation (SHG) in periodically poled lithium niobate (PPLN) enables efficient χ2 frequency conversion via quasi-phase matching (QPM), yet the attainable external conversion efficiency is not a monotonic function of the crystal length due to the interplay among residual phase mismatch, pump depletion, linear loss, and finite-beam propagation. Here, we present a unified analytical–numerical study of SHG length scaling in PPLN. Starting from Maxwell’s equations with a second-order nonlinear polarization, we derive coupled-wave equations under the slowly varying envelope approximation (SVEA) with a consistent envelope normalization that directly links field amplitudes to optical powers. In the small-signal regime, we obtain closed-form expressions for SH power, including both QPM-induced effective phase mismatch and linear absorption, which naturally define an effective interaction length Leff. Beyond the undepleted-pump limit, we introduce a dimensionless formulation that identifies the governing parameters controlling the transition from quadratic growth to saturation and determines the optimal interaction length Lopt. Numerical integrations incorporating realistic dispersion (Sellmeier), absorption, and a reproducible Gaussian-beam overlap model validate the analytical trends and quantify how Lopt depends on pump power, focusing, and QPM design (order and duty cycle). These results provide practical design rules for high-efficiency QPM-based frequency converters in bulk and waveguide LN platforms.