Much of nonlinear optics relates to the coupling of waves and changes in frequency components in the weakly nonlinear regime. The total field of a nonlinear optical waveguide can be expanded in terms of the modal fields of the linear waveguide, with the nonlinearity acting to couple power between the modes. For lossless systems there are at least two constants of the motion, one always being the conserved total power. The second constant has been constructed in various ways in specific problems and has sometimes been identified as a Hamiltonian. We show that a second constant can always be constructed by deriving a general formula for it in terms of the electromagnetic-field variables. Further, the second constant can then be used to write the coupled amplitude equations in Hamiltonian form. Specific examples are given.
[Optical Society of America ]
Colin Pask, David R. Rowland, and Wagdy Samir, "Constant of motion for modal interactions in nonlinear dielectric waveguides," J. Opt. Soc. Am. B 15, 1871-1879 (1998)