Adjoint Approach to Optimization and Sensitivity Analysis of Beam Wave Interaction in Vacuum Electronic Devices

We demonstrate a new approach to optimization and sensitivity analysis of beam-wave interaction in any vacuum electronic device (VED) that consists of a circuit interacting with a linear beam through a series of gaps. The basis of the method is a consequence of the Hamiltonian form of the equations that govern the beam-wave interaction, which implies the conservation of symplectic area for two perturbed solutions. Using this property of the system we have derived a relationship between the perturbed solution and an adjoint solution to the linearized equations. We show that proper selection of the adjoint solution allows obtaining compact symplectic equations. Consequently when the adjoint solution is obtained using a simulation code, it may be used to evaluate the multi-dimensional derivatives needed for efficient optimization and for sensitivity analysis. For sensitivity analysis the single adjoint solution allows to construct a sensitivity function what acts like a type of Green's function for many variable parameters of the system. The approach applies to small or large signal operation of standing and traveling wave devices using either single or multiple round beams.