Noncollinear phase matching and effective nonlinear coefficient calculations for biaxial crystal out of the principal plane

The essential factor in laser frequency conversion involves phase matching within nonlinear optical crystals. To our knowledge, few studies have investigated the noncollinear phase matching calculation for biaxial crystal out of the principal plane. In this paper, we propose an arbitrary direction phase matching model and a computational method based on gradient descent (GD) algorithm, which can be applied to noncollinear in the principal plane, collinear and noncollinear out of the principal plane. In the case of 1053 nm third harmonic generation (THG) in LiB₃O₅ (LBO) crystal, the phase matching conditions are converted into a system of nonlinear equations with six variables and six equations, which can be solved by iterative optimization search with the GD algorithm and includes type-I (ss-f) and type-II (fs-f). We reveal the relationship of phase matching angles and effective nonlinear coefficients ( \(d_{eff}\) $d_{\mathit{eff}}$ ) for various structures. Our method uncovers the existence of many solutions in the non-principal plane with γ > 8° and the \(d_{eff}\) $d_{\mathit{eff}}$ close to the maximum value 0.66834 pm/V at θ = 90°, φ = 141.84° and γ = 0. The resolution of the arbitrary direction phase matching problem holds significant importance, as it expands the possibilities for laser frequency conversion, especially for noncollinear structures.