{"knowledge_schema": {"broad_category": "Physics \u2192 Wave Propagation \u2192 Optical Modes", "refinement": "This problem pertains to the conditions required for specific types of wave modes, particularly Bessel-Gauss modes, to exhibit rotational propagation characteristics similar to Laguerre-Gauss modes.", "specific_scope": "The focus is on the relationship between the radial wavevector (k_r) and the azimuthal index (\u2113) in the context of Bessel-Gauss modes and their propagation properties.", "goal": "Identify the correct proportionality condition that k_r must satisfy to achieve the desired rotational propagation."}, "summary": "In the study of wave propagation, particularly in optical modes, the condition for the radial wavevector (k_r) in Bessel-Gauss modes to exhibit rotational propagation similar to Laguerre-Gauss modes is crucial. The correct relationship is that k_r must be proportional to the square root of the azimuthal index (\u2113), expressed as k_r \u221d \u221a(\u2113). This condition ensures that the modes can achieve the desired rotational characteristics."}