A Schrödinger Formulation for Paraxial Light Beam Propagation and Its Application to Propagation through Nonlinear Parabolic-Index Media

The Helmholtz equation is reduced to the Schrödinger-like equation and then the quantities representing the gross features for a paraxial optical beam, such as the width, divergence, radius of curvature of the wave front, complex beam parameter, beam quality factor, and the potential function representing beam propagation stability, are studied by using the quantum mechanical methods. The results derived in other ways previously are rederived by our formulation in a more systematical and explicit fashion analytically, and some new results are demonstrated. The general equations for the evolution of these quantities, i.e., the first- and second-order differential equations with respect to the propagation distance, such as the universal formula for the width and curvature radius, the general formula for the first derivative of the complex beam parameter with respect to the axial coordinate, the general formula for the second derivative of the width with respect to the axial coordinate, and some general criteria for the conservation of the beam quality factor and the existence of a potential well of the potential function, are derived. We also discuss the application of our formulation to nonlinear parabolic-index media.