Standard Parameters for Photonic Hopfion Stability

This technical dataset and reference document provides the analytical bounds required for the independent tuning of poloidal ($p$) and toroidal ($q$) winding numbers in high-order photonic hopfions. Current experimental approaches utilizing Laguerre-Gaussian superpositions face a fundamental unresolved challenge: the decoupled tuning of $(p,q)$ indices is strictly limited by Orbital Angular Momentum (OAM) phase coupling. To overcome this limit, we present strict stability parameters obtained from analytical modeling of Hopf fibration geometry ($S^3 \to S^2$) in free-space manifolds. The document defines the Critical Phase Limit ($\Phi_{crit}$) to prevent topological tearing during propagation, utilizing the geometric threshold constant $A \approx 137.036$ rad. Additionally, it establishes the Lattice Stiffness Coefficient ($Z_B$) based on the topological rigidity constant $B \approx 3283.515$ for multi-beam array synchronization. Numerical Validation: These parameters have been numerically validated against multi-beam Laguerre-Gaussian interference models, demonstrating structural locking with >99.9% precision. Experimental Protocol: Parameters are provided as vacuum ideal values ($n=1$). For practical Spatial Light Modulator (SLM) phase mask generation in dielectric media, a necessary dielectric screening calibration ($1/n^2$ scaling) is detailed. Adherence to these analytical thresholds ensures phase-gradient stability and suppresses topological decay in free-space transport.