  Femtosecond optical parametric oscillators (OPOs) are widely used in
ultrafast nonlinear frequency conversion and quantum information. However,
conventional OPOs based on quasi-phase-matching (QPM) crystals have many
parasitic non-phase-matched processes which decreases the conversion
efficiency. Here, we propose nine-wave coupled equations (NWCEs) to simulate
all phase-matched and non-phase-matched interactions in QPM crystals to improve
conventional three-wave coupled equations (TWCEs), especially for the situation
of high intensity ultrashort pulses and complexly structured crystals. We
discuss how to design the poling period of QPM crystal to maximize the
conversion efficiency of signal light for a given OPO system. The simulation
reveals that the OPO based on chirped periodically poled lithium niobate
(CPPLN) with certain chirp rate has higher signal wave conversion efficiency
than that of a PPLN, and demonstrates that NWCEs illustrates more details of
the pulse evolution in the OPO cavity. Starting from a CPPLN, an aperiodically
poled lithium niobate (APPLN) design is available by modifying the domain
lengths of the crystal and optimizing the OPO output power via dynamical
optimization algorithm. The results show that by using a properly designed
APPLN crystal, a 1600nm OPO, when pumped by a femtosecond laser with 1030nm
central wavelength, 150 femtosecond pulse duration and 5 GW/cm2 power intensity
at the focus, can achieve very efficient output with a signal light conversion
efficiency of 50.6%, which is higher than that of PPLN (25.2%) and CPPLN
(40.2%). The scheme in this paper will provide a reference for the design of
nonlinear QPM crystals of OPOs and will help to understand the complex
nonlinear dynamical behavior in OPO cavities.
