In this Letter we introduce a systematic perturbation method for analyzing the eeect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schrr odinger equation (PNLS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpliied, depending on whether PNLS is power conserving...

Achieving tunable focus of electromagnetic field energy at multiple target points is a critical challenge in the wireless power transfer (WPT) domain. Although techniques such as optimal constrained focusing (OCPF) and time reversal (TR) have been proposed. The former presents limited practical applicability while latter noteworthy for its adaptive spatio-temporal synchronous characteristics. H...