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We propose and investigate the quasi-phase-matched (QPM) nonlinear optical frequency conversion of optical vortices in periodically poled lithium niobate. Laguerre-Gaussian (LG) modes are used to represent the orbital angular momentum (OAM) states, characterized with the azimuthal and radial indices. Typical three-wave nonlinear interactions among the involved OAM modes are studied with the help of coupling wave equations. Being different from normal QPM process where the energy and quasimomentum conservations are satisfied, both of the azimuthal and radial indices of the OAM states stay constant in most of the cases. However, abnormal change of the radial index is observed when there is asynchronous nonlinear conversion in different parts of the beams. The QPM nonlinear evolution of fractional OAM states is also discussed showing some interesting properties. In comparison with the traditional birefringent phase matching, the QPM technique avoids the undesired walk-off effect to reserve high-quality LG modes. We believe the QPM is an effective way to convert, amplify, and switch OAM states in various optical vortex related applications. [ABSTRACT FROM AUTHOR]