The phenomena discussed in the last section are really only applicable for relatively long pulses (i.e. greater than approximately 100 ps) such that the dispersion is negligible compared with the effects of nonlinearity. We did consider the effects of *initial* chirp, or dispersion, but we didn’t consider the effects that may occur as the pulse propagates. For most cases of interest in ultrafast optics, the pulse duration is much shorter than this limit, and thus the effects of both GVD and SPM on pulse evolution need to be considered.

**Table of Contents:**

- Pulse Evolution
- Broadening Factor
- Optical Wave Breaking
- Effect of Higher-Order Dispersion

**PULSE EVOLUTION**

Pulse evolution with SPM and GVD can be discussed using the normalized propagation equation:

\begin{equation} i\frac{\partial{U}}{\partial{\zeta}} = \pm\beta_2\frac{1}{2}\frac{\partial^2U}{\partial\tau^2}-N^2e^{-\alpha{z}}|U|^2U \end{equation}

where $\zeta$ and $\tau$ represent the normalized distance and time variables,

\begin{equation} \zeta = z/L_D \end{equation}

\begin{equation} \tau = T/T_0 \end{equation}

and where the parameter N is given as:

\begin{equation} N^2 = \frac{L_D}{L_{NL}} = \frac{\gamma{P_0}T_0^2}{|\beta_2|} \end{equation}

As can be seen from this equation for N, the value of N governs the relative importance of SPM vs GVD. For example, if N << 1, SPM would dominate while if N>>1, GVD would dominate. If N $\approx$ 1, then both SPM and GVD play an important role in the pulse propagation. Using the split step Fourier method, the normalized equation can be solved numerically.

For the normal dispersion regime, the pulse broadens much faster temporally than in the presence of dispersion alone, and the spectrum narrows much more slowly. This can be understood by noting that SPM adds red-shifted spectral components to the leading edge and blue-shifted spectral components to the trailing edge of the pulse due to the positive nonlinear index. Since the red-shifted components travel faster than the blue-shifted components in normal dispersion, the pulse broadens much more rapidly than without SPM.

The situation is much different for the anomalous dispersion regime. Basically, here, the SPM and GVD balance each other to produce a chirp-free pulse. This is soliton evolution. If the pulse was chosen to be a hyperbolic secant on input, both the pulse shape and spectrum would be unchanged during propagation.

**BROADENING FACTOR **

See methods to solve the nonlinear propagation equation analytically.

**OPTICAL WAVE BREAKING**

**EFFECTS OF HIGHER-ORDER DISPERSION**

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