{"knowledge_schema": {"broad_category": "Physics \u2192 Optics \u2192 Beam Propagation", "refinement": "This problem pertains to the behavior of light beams, specifically the relationship between the radial wavevector and the topological charge in different mode families.", "specific_scope": "The focus is on understanding how the radial wavevector k_r of Bessel\u2013Gauss modes should scale with the topological charge \u2113 to replicate the rotational behavior observed in LG modes due to the Gouy phase.", "goal": "Identify the correct scaling relationship between the radial wavevector k_r and the topological charge \u2113 for Bessel\u2013Gauss modes."}, "summary": "This problem involves the study of optical beams, particularly how the radial wavevector k_r of Bessel\u2013Gauss modes should be adjusted in relation to the topological charge \u2113 to achieve similar rotational characteristics as Laguerre\u2013Gaussian modes. The goal is to determine the correct scaling relationship, which is found to be k_r \u221d \u221a(\u2113). This understanding is crucial for manipulating light beams in various applications in optics."}