New ducting model for analyzing the Gaussian beam propagation in nonlinear Kerr media and its application to spatial self-phase modulations

Knowing the Gaussian beam parameters, such as its radius of curvature and spot size during propagation in nonlinear Kerr media, is of paramount importance in describing the observable far-field diffraction ring patterns as well as in design and stability analysis of Kerr-lens mode-locked resonators. Specifically, the sign of the beam radius of curvature after exiting these media has been proposed to be of assistance in recognizing their optical nonlinearity sign through determining the type of diffraction ring pattern in the far field. In order to be able to trace the evolution of the beam parameters in the Gaussian beam formalism, we have used the common aberration-free theory. We have shown that the nonlinear propagation problem of a fundamental Gaussian beam in a Kerr medium with an intensity-dependent index of refraction can be handled by assuming a ducting index profile along the propagation direction. Knowing the familiar ABCD matrix of a duct, the evolution of the mentioned beam parameters can be traced during propagation using the ABCD law in Gaussian beam theory. We have validated our ducting model by comparing its results with the outcomes of one widely used and accepted model which has been known to yield consistent results when electronic optical nonlinearity prevails. We have shown that when thermal optical nonlinearity is dominant, as in diffraction ring observation experiments, our ducting model yields sensible results and should be used. Our model predicts that when the sign of the thermal nonlinearity and the beam radius of curvature on the entrance plane of the medium are positive, the sign of the beam radius of curvature on the exit plane may have either sign, depending on the medium thickness used in the experiment. Hence, two types of diffraction ring pattern may be obtained using the same medium with two different thicknesses and this may cast doubt on the validity of the methods proposing the detection of the optical nonlinearity signs by observing these patterns. We have proposed a simple procedure for experimentally obtaining the two different types of diffraction pattern from the same medium.