“…The angle θ is defined as cos −1 (m/ χ' m,n ), where χ' m,n is the n -th zero of the derivative of the m -th order Bessel function, J m ' (z), and the eigenvalue of the TE m,n mode. The quantity m/χ' m,n is an important factor that determines the direction and extension of a beam emitted from a Vlasov launcher [7]. In geometrical optics, the radius of the caustic surface of the EM wave is represented by R w cosθ in the waveguide [23,24], where R w is the waveguide radius.…”