Using the Panofsky-Wenzel theorem in the analysis of radio-frequency deflectors

In a 1956 paper Panofsky and Wenzel considered the transverse momentum imparted to a fast particle moving parallel to the-axis of a cavity excited in either a TE (no component of the electric field parallel to the axis) or TM mode (no component of the magnetic field parallel to the axis). One conclusion of this paper was that in a TE mode the deflecting impulse of the electric field exactly cancels the impulse of the magnetic field. This result is sometimes misinterpreted as concluding that if the electric field acting on a particle is purely transverse, the deflection impulses from the electric and magnetic fields must cancel one another. This conclusion is false. Instead p⊥=(e/ω₀)∫^d₀ (-i)∇⊥E_zdz, implicitly derived in the 1956 paper, is a more useful form of the theorem for deflecting cavities. In this equation, p⊥ is the transverse momentum imparted to the particle, d is the length of the cavity, e is the charge of the particle, ω₀ is the angular frequency of the cavity, and ∇⊥E_z, is the transverse gradient of the z component of the electric field along the path of the particle. The -i represents a 90° phase advance of the integrand with respect to the electric field. In other words, the integrand has the same phase as the magnetic field. Eq. (1) is not restricted to TE or TM modes. In particular, it applies to two similar-looking modes in a high-energy deflector that was studied for the Accelerator Transmutation of Waste (ATW) project at Los Alamos National Laboratory