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

Abstract

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.

Describing the propagation of intense laser pulses in nonlinear Kerr media using the ducting model

“…In addition, for the linear refractive index n 0 and absorption coefficient α of the Kerr medium, typical values of 1 [19] and 10 m −1 [20] are set in our simulations. It is assumed that the laser beam waist is equal to 0.033 mm and the medium is placed 1 cm after the beam waist position where R 1 is positive and equal to 0.014 m and w 1 has a value of 0.061 mm [12].…”

“…This function can have any appropriate shape to describe the different kinds of laser pulses, like Gaussian, super-Gaussian, rectangular, hyperbolic secant, etc. From this point forward, the model will closely follow the ducting model relations [12], apart from the fact that now all the relations should contain the pulse shape function f (t). Therefore, the time-dependent Gaussian beam parameters, namely its spot size and radius of curvature, can be found at each z position inside the Kerr medium using the following relations [12]:…”

“…where a is a correction factor that may take on values between 3.77 and 6.4 [12,15] and P denotes the laser pulse peak power.…”

“…In order to be able to know the values of w(z, t) and R(z, t) of the spatially Gaussian laser pulse during its propagation in the Kerr medium, the same numerical procedure as described in our previous report [12] can be used. At each instant of time, w(z, t) can be obtained by numerically solving the equation which is accessible by inserting the expressions A(z, t) and B(z, t) into equation (3).…”

“…Recently, a ducting model has been developed for predicting the Gaussian beam parameters during propagation in nonlinear Kerr media [12]. This is a stationary model which is based on the aberration-free theory and the assumption that nonlinear Kerr media are of a ducting nature.…”