Phase-matched optical second-harmonic generation in helically twisted smectic-C*phase

Phase matching for optical second-harmonic generation in the helically twisted smectic-$C^{*}$ liquid-crystalline phase is analyzed theoretically. Ten distinctive phase-matched combinations of the fundamental and the second-harmonic eigenmodes exist. The effective second-order nonlinear optical susceptibility and the relation between the optical wave vectors and the wave vector of helical structure are determined for each of the combinations. The results obtained within the standard approximation of the slowly varying amplitude are compared with the exact numerical solutions of the nonlinear wave equation. The discrepancy between the approximative and exact results is crucial only for the two phase-matched combinations that are realized when the second-harmonic frequency is close to the selective reflection band. In this case an additional enhancement of the second-harmonic beam can appear.