The purpose of the last section was to add context behind how a grating compressor may work to compress pulses with normal dispersion. A fiber grating compressor uses a short length of single mode fiber to provide normal dispersion and SPM, and then a grating to counteract the effects of the normal dispersion by adding anomalous dispersion.

**Contents:**

**HISTORY**

Before diving into the theory of fiber-grating compressors, it is interesting to know some of the history behind how fibers have come to be used to add SPM-broadening, and where the present state of the research is.

**THEORY**

The two most relevant questions when designing a fiber-grating compressor are (i) is there an optimum fiber length for given values of input pulse parameters and (ii) is there an optimum grating separation to obtain high-quality output pulses with maximum compression. To answer this, we need to first consider how a pulse may evolve inside a fiber in the presence of both GVD and SPM. This was considered in some detail in the fourth subsection of this “Fiber Optics” section, but it is useful to repeat the relevant equation here. The normalized propagation equation including the effects of both nonlinearity and dispersion are:

\begin{equation} i\frac{2}{\pi}\frac{\partial{U}}{\partial(z/z_0)} – \frac{1}{2}\frac{\partial^2U}{\partial\tau^2}+N^2e^{-\alpha{z}}|U|^2U = 0 \end{equation}

where $\tau = T/T_0$,

\begin{equation} z_0 = \frac{\pi}{2}L_D = \frac{\pi{T_0^2}}{2|\beta_2|} \end{equation}

and the parameter N is given as:

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

To analyze the situation at hand, we will first start with just the case of pure SPM in the absence of GVD. As we saw in section 4, the effect of SPM in the temporal domain is to add a linear frequency up-chirp, but only over the central part of the pulse (in the example of a Gaussian pulse). When such a pulse is then passed through a grating pair, the anomalous dispersion would only compress the central part. Since this would mean then that a significant amount of energy remains in the wings, the compressed pulse would not be of good quality.

It turns out that normal GVD can improve the pulse quality immensely. The effect of the GVD would be to add a nearly linear chirp across the entire pulse width and as a result, a grating pair can compress most of the pulse energy into a narrow pulse. An unfortunate effect of the presence of GVD, though, is to reduce the total possible compression amount at a given input peak power.

\begin{equation} F_c = T_{FWHM}/T_{comp} \end{equation}

\begin{equation} Q_c = |U_{out}(0)|^2/F_c \end{equation}

where $T_{comp}$ is the FWHM of the compressed pulse, such that $F_c$ is the compression factor. The parameter $Q_c$ is a measure of the quality of the compressed pulse.

**EXPERIMENTAL RESULTS**