## 2 Propagation of Ultrashort Pulses
The other definition originates from wave optics and hence addresses the wavelength dependence of the angle between the phase fronts; this approach results in the so-called _phase front angular dispersion_, \(\gamma_{PF}\)[27]. Comparing the two definitions (Figure 3), one can see that they are equivalent for plane waves, but different for spherical, hence the Gaussian beams [28]. \[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]3 Comparison of the two types of angular dispersion in the case of plane wave and Gaussian beam approximation. \[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]4 Definition of the carrier-envelope phase. Figure 4 demonstrates the effect of angular dispersion on ultrashort laser pulses propagating in vacuum. The spectral components of the initial pulse are spatially and temporally overlapping, but their directions of propagation are slightly different due to the angular dispersion. In the case of plane waves propagating in vacuum (or in any isotropic media), the magnitude of angular dispersion remains constant; however, its effect on the pulse becomes more significant with increasing path length of the pulse travelling from the source of angular dispersion. Conversely, the magnitude of angular dispersion affecting Gaussian beams usually decreases during propagation [28]. In Figure 4, the spatial intensity distribution of the spectral components constituting the pulse is represented for different scenarios, after having propagated a certain distance of \(z\). If angular dispersion is present, with increasing distance measured from the middle of the pulse along the direction \(x\) perpendicular to the direction of propagation, pulse length also increases and the spectrum becomes shifted at the same time. The pulse length in the center (at \(x=0\)) will also differ from the original pulse length, since the pulse will be affected by a phase modulation [19] corresponding to the group delay dispersion  
\[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]5  
This can be converted easily into a stretch in pulse length based on Equation (12) [29]. Angular dispersion is practically always accompanied by some pulse front tilt and spatial chirp, both demonstrated in \[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]6 Pulse front tilt means a group delay along the cross-section of the beam, in the direction perpendicular to the propagation \(z\) and with a magnitude of \(2\pi c\gamma_{PD}/\omega_{0}\) per unit length along the cross sectional area [19, 30, 31]. Spatial chirp involves a spatial anisotropy in the spectral content across the beam [32]. This means that one part of the beam (the blue side of the pulse in Figure 4) is shifted toward the shorter wavelengths, while the other part (the red side of the pulse) is shifted toward the longer wavelengths. One can see easily that spatiotemporal pulse distortions usually arise simultaneously; moreover, they are coupled with each other. A general mathematical analysis of these coupling effects is discussed in Reference [33]. \[\tilde{E}(\omega)=\frac{E_{0}\tau_{0}}{2\sqrt{\ln(2)}}\exp\Biggl{[}-\frac{\tau_{ 0}^{2}}{8\ln(2)}\bigl{(}\omega-\omega_{0}\bigr{)}^{2}\,\Biggr{]}. \tag{2}\]7 Illustration of spatial effects of the angular dispersion and pulse front tilt.