 N2 is 3.2 x 10-20 m ²/W and it is therefore very small compared to the linear refractive index, N1, which for silica glass at 1550 nm is 1.444. The difference is 20 orders of magnitude, but the impact of N2 is still important because of the long length inter- action between material and radiation and the high intensity of the radiation. Over a fiber length L the total phase shift due to An(t) is given by (5.54) (5.55) To proceed, one needs an analytical expression of the electric field. For convenience let us assume that the input light pulse has a Gaussian envelope: (5.56) where is the peak pulse amplitude, to is the time position of the pulse, and T is the half- width at the 1/e point of the peak amplitude. Substituting equation 5.56 into equation 5.54, we obtain (5.57) A0max (5.58) Nonlinear Optical Effects in Fibers where the length L corresponds to the nonlinear interaction length, Leff. The maximum fre- quency change is obtained by substituting equation 5.57 into equation 5.55: Aw(t)=AQmax = 4(t - (5.59) At the center of the pulse the frequency shift is zero. To find the position of the maximum frequency shift we calculate [[sa(t)=0)] (5.60) One finds that the maximum frequency changes occur at (t - to) = +T/2. For example, consider the Gaussian pulse with T = 1 psec, max = 8. The results are summarized in Figure 5.23. Time (ps) Time (ps) Time (ps) Fig. 5.23 (a) Normalized input pulse amplitude; (b) phase shift due to nonlinearity; (c) frequency chirp. Chapter 5 Optical Fibers Fig. 5.24 SPM followed by negative and positive dispersion: (a) D < 0, additional pulse broadening; (b) D > 0, pulse compression. For the central position of the pulse the frequency chirp is linear between +T/2 and is given by (5.61) The impact of SPM on the performance of an optical system depends on the dispersion. If there were no dispersion in the fiber, SPM would have no impact on performance. First consider operation at 2>20. Then D < 0 and the pulse will broaden because the leading "red" frequencies travel faster and the "blue