Analytical solution to incident angle quasi-phase-matching engineering for second harmonic generation in a periodic-poled lithium niobate crystal

Phase matching or quasi-phase matching (QPM) is of significant importance to the conversion efficiency of second harmonic generation (SHG) in artificial nonlinear crystals like lithium niobate (LN) crystal or microstructured nonlinear crystals like periodic-poled lithium niobate (PPLN) crystals. In this paper, we propose and show that the incident angle of pump laser light can be harnessed as an alternative versatile tool to engineer QPM for high-efficiency SHG in a PPLN crystal, in addition to conventional means of period adjusting or temperature tuning. A rigorous model is established and analytical solution of the nonlinear conversion efficiency under the small and large signal approximation theory is obtained at different incident angles. The variation of phase mismatching and walk-off length with incident angle or incident wavelength are also explored. Numerical simulations for a PPLN crystal with first order QPM structure are used to confirm our theoretical predictions based on the exact analytical solution of the general large-signal theory. The results show that the narrow-band tunable SHG output covers a range of 532 nm-552.8 nm at the ideal incident angle from 0 degrees to 90 degrees. This theoretical scheme, fully considering the reflection and transmission at the air-crystal interface, would offer an efficient theoretical system to evaluate the nonlinear frequency conversion and help to obtain the maximum SHG conversion efficiency by selecting an optimum incident wavelength and incident angle in a specially designed PPLN crystal, which would be very helpful for the design of tunable narrow-band pulse nanosecond, picosecond, and femtosecond laser devices via PPLN and other microstructured LN crystals.