Electromagnetic wave propagation in nonlinear kerr media

Electromagnetic elds within a nonlinear medium satisfy Maxwell's equations. In Kerr-type media the nonlinearity is created through an intensity dependent refractive index. We examine the Maxwell equations with a Kerr nonlinearity using four di erent approaches; rigorous vector theory of volume gratings, Hamilton's canonical perturbation theory of classical mechanics, variational calculus that approximates the exact nonlinear solution, and analytical methods that produce, for example, the nonlinear solutions in the form of self-guided waves. The latter three techniques make use of a mathematical construction called the action. We consider models that consist of several in nite layers of (non)linear media. Structures of this kind present simple and widely used geometries in optics, photonics, and laser technology. In the nonlinear wave theory the KerrMaxwell equations simplify signi cantly due to the layered structure. From the experimental point of view the fabrication of such samples is highly developed. Most (non)linear phenomena, such as guided waves, bistability, and solitons, are readily demonstrated in layered media. Our results show that the exact electromagnetic theory of volume gratings predicts similar nonlinear phenomena that are found in experiments. However, we make evident that in many cases simple analytical approximations for the nonlinear elds are still more useful and accurate enough to give a qualitatively correct nonlinear behavior. We also show that the action integral formalism is an advanced and e ective technique when it is applied to layered nonlinear problems. In particular, our approach gives new insight into optical systems with Kerr-type media and it elucidates the complex theory of electromagnetic nonlinear waves.