Z-scan. In this problem we develop a mathematical description of the zscan procedure for measuring the nonlinear refractive index. The basis of this procedure is that a sample is translated longitudinally through the beam-waist region of a focused gaussian laser beam, and the variation of the on-axis intensity in the far field is measured as a function of sample position. The on-axis intensity in the far field is usually determined by measuring the power transmitted through a small aperture placed on the system axis. The variation of measured power with sample position is found to be proportional to the nonlinear phase shift experienced in passing through the sample, and from this measured phase shift the value of n2 of the sample can be determined from the known sample thickness L and laser intensity at the sample In detail, you are to derive an expression for the dependence of the fractional change in on-axis intensity ?I/I on the sample position z relative to the position z0 of the beam waist of the incident laser beam. This expression will also depend on the values of n2, of L, and on the parameters of the incident laser beam. For simplicity, assume that the sample is thin both in the sense that L is much smaller than the confocal parameter b of the incident laser beam and in that the maximum nonlinear phase shift acquired in passing through the sample is much smaller than unity. Here are some suggestions on how to proceed. Begin with the expression for a gaussian laser beam. Determine first how the beam will be modified in passing through the nonlinear material. Note that under the assumed conditions the beam diameter at the sample will be unchanged, but that the wave front curvature will be modified by nonlinear refraction. Note that the modified beam itself approximates a gaussian beam, but with a different value of the wavefront radius of curvature. By assuming that this beam propagates according to the same laws that govern the propagation of a gaussian beam, determine its on-axis intensity in the far field as a function of z - z0. Note that a more detailed analysis of the z-scan procedure (SheikBahae et al., 1990) leads to the somewhat different result

Where max is the nonlinear phase shift on-axis when the sample is at the beam waist of the incident laser beam. Plot the functional dependence of both your result and the literature result