Adjoint Optimization of Self-Consistent Particle Trajectories Due to External Fields

Given many parameters to be varied, the optimization of an objective function can be hastened by computing the gradient (the change in the parameters that maximally improves the objective function). Unfortunately, finding the gradient can be prohibitively expensive when each evaluation of the objective function involves running a separate particle-in-cell simulation. We derive a general (so called “adjoint”) method to find both the value and gradient of an objective function using only two simulations, a substantial savings. To do this, we consider the general case where particle motion is described by a Hamiltonian with arbitrary dependence on external parameters. The structure of the perturbed Hamilton's equations yields a simple recipe for the evaluation of the adjoint equation. Our work offers a new perspective on the method of Antonsen et al. (2019) for optimization of trajectories in an electron gun [1]. We generalize their method to accommodate field-dependent particle emission, and use this new formulation to compute geometric optimizations of nanoscale vacuum-channel transistors.