Sum-frequency generation processes with minimal crystal length

Using the Pontryagin maximum principle (PMP), we address the crystal length minimization problem in sum-frequency generation (SFG) under the undepleted pump approximation. Our results show that the minimal crystal length is inversely proportional to the coupling coefficient, and that the optimal evolution trajectory follows a geodesic on the Bloch sphere along which the optical field propagates without energy loss or absorption. The crystal polarization period is derived from the optimal phase-matching function. These findings offer valuable guidance for the design of compact optical chips and nonlinear optical devices.