## 9 Dispersion Compensation
### Prism Pair  
In contrast to material dispersion (where only the refractive index, \(n\), is wavelength-dependent), one can also introduce geometric dispersion in which the geometrical path, \(L(\lambda)\) (Equation (15)), is also wavelength-dependent. Geometric dispersion can be introduced using a prism pair, illustrated in Figure 9c, in either one of the cavity arms. The general concept of a prism pair is to manipulate the optical path of different frequencies in such a way that all the frequencies will experience the same cavity round-trip time. The relative time delay between the frequencies caused by the prism pair compensates for the time delay caused by other dispersive material in the cavity (e.g., the Ti:sapphire crystal). The result is that a prism pair can generate both negative and positive GDD in a controlled, tuned manner [64,65]. The GDD that the pulse experiences as it passes through the prism pair can be calculated using Equation (15), and the optical path, \(\mathrm{OP}(\lambda)\), in the entire prism pair system (air and material) is calculated using geometrical optics [66, 67, 68], resulting in:  
\[\mathrm{GDD}=-\frac{2\lambda}{\pi c^{2}}\left(\lambda\frac{dn}{d\lambda} \right)^{2}R_{p}+\left(\frac{\lambda^{3}}{\pi c^{2}n^{2}}\frac{d^{2}n}{d\lambda ^{2}}\right)H \tag{16}\]  
where \(R_{p}\) is the separation between the prisms tips, \(H\) is the second prism penetration and \(n\) is the refractive index of the prisms. Therefore, the net GDD that the pulse experiences as it passes through a prism pair can be continuously controlled by changing \(R_{p}\) or \(H\). Note that increasing the separation between the prisms always introduces negative GDD, while increasing the prism penetration can produce both positive and negative GDD. For the Ti:sapphire crystal in the NIR spectrum, increasing the prism penetration corresponds to positive GDD. Optimization guidelines for a prism pair setup can be found in [69], using a dispersion map in which the second and third order dispersions are plotted as orthogonal coordinates. The resulting dispersion vector can be compensated for by optimizing the prism pair parameters (separation, penetration and prism material) to reduce the dispersion vector to zero. Similarly, a grating pair can be used for larger values of controlled negative dispersion [70] at the expense of increasing losses. The combination of a prism and a grating in a single device (termed: \"grism\") was demonstrated for higher order dispersion management [71]. One can also use a single prism and a wedged mirror to compensate for dispersion [72].  
In this regard, one should be aware of the additional linear astigmatism introduced by prisms (or wedge windows): because a prism pair is effectively a Brewster window separated in two parts, it produces astigmatism (see Section 5.4), which is generally much smaller than that of the crystal. This added astigmatism was not included while solving Equations (9) and (17) and will require fine-tuning of the folding angles of the curved mirrors to re-compensate for the overall astigmatism in the cavity. Note, however, that astigmatism from a prism pair exists only if the beam passing through the pair is not collimated (_i.e._, the pair is placed in the long arm while mode-locking near \(\delta_{1}\) or in the short arm while mode-locking near \(\delta_{2}\)). This is because, for a collimated beam, the pair astigmatism, \(\Delta L\), is completely negligible compared to the long Rayleigh range of the beam.