## 2 Propagation of Ultrashort Pulses
The derivatives of spectral phase, the so-called _phase derivatives_, have the following conventional notations:  
\[GD\equiv\varphi^{(1)}=\frac{\mathrm{d}\varphi(\omega,z)}{\mathrm{d}\omega} \Biggl{|}_{z;\omega=\omega_{0}}, \tag{6}\]  
\[GDD\equiv\varphi^{(2)}=\frac{\mathrm{d}^{2}\varphi(\omega,z)}{\mathrm{d} \omega^{2}}\Biggl{|}_{z;\omega=\omega_{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
The definitions above give _group delay_ (_GD_), _group-delay dispersion_ (_GDD_) and _third order dispersion_ (_TOD_). It could be very useful in some cases, when material dispersion is given for a unity geometrical length of propagation. For this purpose, we can use _specific_ terms like _SGD_, _SGDD_ and _STOD_ for the derivatives of (6)-(8) by parameter \(z\).  
Phase derivatives considerably influence the temporal shape of the envelope. For example, let us consider a medium with first and second order dispersion. The evolution of the electric field will be given by  
\[E(t,z)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}\tilde{E}(\omega) \exp\biggl{[}iot-i\biggl{(}\varphi(\omega_{0})+GD\cdot(\omega-\omega_{0})+ \frac{1}{2}GDD\cdot(\omega-\omega_{0})^{2}\biggr{)}\biggr{]}\mathrm{d}\omega, \tag{9}\]  
which provides a mathematically similar solution to the formalism of the Gaussian beam approximation. A pulse with an initial shape described by Equation (1) will have a form of  
\[E(t,z)=E^{*}\exp\biggl{[}-2\ln(2)\frac{(t-GD)^{2}}{\tau^{2}}\biggr{]}\cdot\exp \bigl{[}i\Phi(t,z)\bigr{]}, \tag{10}\]  
where  
\[E^{*}=E_{0}\frac{\tau_{0}}{\sqrt[4]{\tau_{0}^{4}+\bigl{(}4\ln(2)\cdot GDD \bigr{)}^{2}}} \tag{11}\]  
is the peak of the envelope of the electric field  
\[\tau=\tau_{0}\sqrt[4]{1+\biggl{(}\frac{GDD\cdot 4\ln(2)}{\tau_{0}^{2}} \biggr{)}^{2}} \tag{12}\]  
is the pulse duration, and  
\[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]0  
is the temporal phase. It can be seen, that _GDD_ affects \(E*\) and \(\tau\) considerably, while _GD_ determines only the temporal position of the pulse. Figure 1 shows the effects of the first three derivatives on the carrier wave and the envelope. The shape of the pulse in the case of _TOD_ is calculated numerically. \[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]1 Schematic temporal effect of the first three phase derivatives. For few-cycle pulses, the position of the carrier wave under the envelope plays a significant role since the outcome of many experiments could be drastically influenced if the electric field exceeds the ionization threshold by more or less one time during the course of a pulse relative to the previous ones. The quantity for the characterization of this property is the phase at \(t-GD=0\) (_i.e._, at the peak of the pulse) and is the so-called _carrier-envelope phase_ (\(CEP\), see Figure 2), which can be approximated by  
\[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]2  
The different spectral components of an ultrafast laser pulse may propagate not only at different velocities but also into different directions. In this latter case, the propagation is affected by the _angular dispersion_. Such phenomenon takes place when e.g., a beam of ultrashort laser pulses refracts at the boundary between media with different dispersions, passes through a prism, or diffracts on a grating. Angular dispersion can be defined in two ways. The most obvious one, based on geometrical optical considerations, is the wavelength dependence of the direction of propagation of the different spectral components, and is called _propagation direction angular dispersion,_ denoted by \(\gamma_{PD}\)[26].