## Abstract

It is commonly believed that the Hamiltonian approach applies only to linear systems, or nonlinear systems with proper linearization procedures. In this Letter, we show that the Hamiltonian approach based on linear coupled-mode theory applies very well for cross-mode modulation, i.e., field variation of the probe light induced by a strong co-propagating pump light in a degenerate mode group. By deriving eigenvalues and eigenvectors of the nonlinear system, evolution of the probe light can be obtained, which agrees with numerical simulations of multimode nonlinear Schrödinger equations. Using the Hamiltonian approach, a simple approach for determining the pump mode for realizing arbitrary mode conversion is discussed. Two concrete scenarios of the optically induced vector mode conversions within the ${\mathrm{LP}}_{11}$-mode group are further exemplified.

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