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    We minimized crystal length for sum-frequency generation (SFG) using the Pontryagin maximum principle. Optimal designs are inversely proportional to coupling strength, guiding compact nonlinear optical device creation.

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    Area of Science:

    • Nonlinear Optics
    • Quantum Optics
    • Materials Science

    Background:

    • Sum-frequency generation (SFG) is crucial for nonlinear optical processes.
    • Minimizing crystal length is key for developing compact photonic devices.
    • The undepleted pump approximation simplifies theoretical analysis of SFG.

    Purpose of the Study:

    • To determine the minimal crystal length required for efficient SFG.
    • To identify optimal conditions for crystal length minimization.
    • To provide design guidelines for compact nonlinear optical devices.

    Main Methods:

    • Application of the Pontryagin maximum principle (PMP) for optimization.
    • Analysis under the undepleted pump approximation.
    • Derivation of optimal phase-matching functions and trajectories.

    Main Results:

    • Minimal crystal length is inversely proportional to the coupling coefficient.
    • Optimal evolution follows a geodesic on the Bloch sphere.
    • The optical field propagates without energy loss or absorption along the optimal path.

    Conclusions:

    • The study provides a theoretical framework for minimizing crystal length in SFG.
    • Findings guide the design of efficient and compact nonlinear optical devices.
    • The derived optimal phase-matching function is critical for device engineering.