# Higher-Order Dispersion

The dispersion discussed in the last section was for the lowest order dispersion, which is sufficient is most cases of interest. However, for operation near the zero-dispersion wavelength or for pulses shorter than approximately 100 fs, inclusion of $\beta_3$, the third order dispersion (TOD) term, is necessary. This section discusses how to do that and the effects of TOD on the pulse.

INTRODUCTION

This section will consider the effects of both SOD and TOD on pulses while still neglecting the nonlinear effects. In a similar manner to the differential equation we found for the SOD, the normalized amplitude satisfies:

$$i\frac{\partial{U}}{\partial{z}} = \frac{1}{2}\beta_2\frac{\partial^2U}{\partial{T^2}} + \frac{i}{6}\beta_3\frac{\partial^3U}{\partial{T^3}}$$

The equation can again be solved using the Fourier transform of a frequency domain solution, and would be given as:

$$U(z,T) = \frac{1}{2\pi}\int_{-\infty}^{\infty}U(0,\omega)e^{\frac{i}{2}\beta_2\omega^2z+\frac{i}{6}\beta_3\omega^3z-i\omega{T}}d\omega$$

CHANGES IN PULSE SHAPE

Pulse evolution and changes in pulse shape along a fiber depends on the relative strengths of $\beta_2$ and $\beta_3$, as well as the initial conditions of the pulse. It is first useful to introduce a length parameter for the TOD, much like we did for SOD:

$$L_D’ = T_0^3/|\beta_3|$$

where $T_0$ denotes the initial pulse width. For pulses longer than approximately 100 fs, the third order dispersion term only becomes non-negligible when the wavelength approaches or approximates the zero-order dispersion term, or when the effects of third order dispersion in a material are greater than the second order dispersion (i.e. when $L_D’ \leq L_D$. For pulses less than or equal to 100 fs, though, the situation changes completely because the SOD term would need to be on the order of 1 ps$^2$/km for a 100 fs pulse before the $\beta_3$ contribution would be negligible.

The effect of higher order dispersion terms is to distort the pulse shape such that an initially Gaussian pulse loses its shape in the presence of TOD. This differs from the case of second order dispersion, where a Gaussian pulse would maintain its shape on propagation but would just broaden. The origin of this can be seen in the fact that the third order dispersion term depends on a higher harmonic of the frequency.