Operator Formalism for Laser–Plasma Mode Coupling

In this paper, we develop an operator-based framework for laser–plasma interactions, providing a compact and systematic description of the coupled dynamics of laser fields and plasma response, with particular relevance to laser wakefield acceleration (LWFA). The formalism employs key operators: the transverse modal operator K^, the nonlinear plasma operator N^[Ψ], the plasma oscillation operator Ω^p2, and the ponderomotive source operator α^, which together describe mode coupling, plasma oscillations, and nonlinear feedback induced by the ponderomotive force. In the linear regime, the system is characterized by invariant subspaces associated with stable modal structures, while nonlinear interactions break these invariances, leading to mode mixing. The operators are defined on explicit Hilbert spaces. We establish the unitarity of linear propagation, the conservation of modal power under the nonlinear flow, the equivalence between the operator and PDE descriptions, and an explicit error bound for finite-mode truncations. We also characterize invariant subspaces through selection rules. Local well-posedness of the nonlinear problem is established under stated regularity assumptions. A reduced two-mode model is validated against direct numerical integration of the paraxial equation, and its accuracy and computational cost are quantified, the validation does not cover wakefield generation (transient plasma wave, pump depletion, injection or acceleration). The formalism provides a unified operator-based framework for the cold-fluid Maxwell–plasma model.