Rigorous solution to second harmonic generation considering transmission and reflection of light at air–crystal interface

Abstract

Considering the transmission and reflection of TE-polarized pump light at the air–crystal interface, the second harmonic generation (SHG) in a lithium niobate (LN) crystal is investigated theoretically and systematically in this work. In previous studies, the theoretical analyses of reflection and transmission of incident wave in the process of nonlinear frequency conversion were not considered in LN crystal on account of the complicated calculations. First, we establish a physical picture describing that a beam of light in TE mode transports in the LN crystal considering transmission and reflection at the crystal surface and generates nonlinear second-order optical polarization in crystal. Then we analytically derive the reflection coefficient and transmission coefficient of pump light by using the dispersion relationships and electromagnetic boundary conditions. We construct the nonlinear coupled wave equations, derive and present the small signal approximation solution and the general large signal solution exactly. Under the transmission model and reflection model, we find that the conversion efficiency of the second-harmonic wave is obviously dependent on transmission coefficient and other general physical quantities such as the length of LN crystal and the amplitude of pump light. Our analytical theory and formulation can act as an accurate tool for the quantitative evaluation of the SHG energy conversion efficiency in an LN crystal under practical situations, and it can practically be used to treat other more complicated and general nonlinear optics problems.