## 3 Linear Optical Methods for Pulse Characterization
### Spectrally and Spatially Resolved Interferometry
Beside these factors, a further phase shift must be considered, induced by certain optical elements such as beam splitters, output couplers, filters, dielectric and chirped mirrors. Those physical quantities which summarize their effects will be denoted by a superscript index \"_opt_\". Since phase derivatives are additive quantities according to the theory of laser pulse propagation discussed before [19] (this can easily be shown if their linear dependence of \(z\) is considered), the total phase derivatives of the sample arm are  
\[GD_{S}=SGD^{med}\cdot L_{S}+GD_{S}^{opt}+GD_{sample}, \tag{19}\]  
\[GDD_{S}=SGD^{med}\cdot L_{S}+GDD_{S}^{opt}+GDD_{sample}, \tag{20}\]  
\[TOD_{S}=STOD^{med}\cdot L_{S}+TOD_{S}^{opt}+TOD_{sample}, \tag{21}\]  
Figure 6: Formation of interference in the case of crossed monochromatic beams. while that of the reference arm are  
\[GD_{R} =SGD^{med}\cdot L_{R}+GD_{R}^{opt}\,, \tag{22}\] \[GDD_{R} =SGD^{med}\cdot L_{R}+GDD_{R}^{opt}\,,\] (23) \[TOD_{R} =STOD^{med}\cdot L_{R}+TOD_{R}^{opt}\,. \tag{24}\]  
A precondition of interference is that the components with identical frequency should coincide within the coherence time and the coherence length. Interference can emerge not only between such laser pulses, which were generated at the same time. Those interferometers which establish interference between consecutive laser pulses at their outputs are called asymmetric interferometers. In this case, the optical path difference of the arms must equal to the multiple of distance between subsequent pulses of the train. Based on Equation (10), pulses propagating in different pathways will overlap each other in time if the difference of the total group delays between the arms and the delay corresponding to \(m\) periods of the temporal separation between the pulses (the latter originates from the asymmetry of the interferometer) are together within the coherence time \(t_{coh}\), _i.e._  
\[\left|GD_{S}-GD_{R}+m\cdot T\right|<t_{coh}.\quad(m\in\mathbf{Z}) \tag{25}\]  
In the frames of the spectral resolution of ultrashort laser pulses, the meaning of coherence time slightly differs from its common definition, since here the coherence of the individual modes of the frequency comb must be considered. This falls in the picosecond range, taking into consideration that the line width of a mode is approx. 150 Hz [56]. In this case, the observation of interference is (beside the noise resulted from the mechanical vibrations of the interferometer) only limited by the resolution of the spectrograph, which may correspond to several millimeters in terms of the difference between the arms. Figure 7 demonstrates a typical interference pattern resulted in a _SSRI_, when the dispersion in the arms is identical. The interferogram presented here is centered along the vertical axis to the position of \(y_{0}\), and the interference fringe corresponding to this height is set to be horizontal. In the interferogram, interference fringes diverge in a fan-shaped manner from left to right toward increasing frequencies, thereby increasing also the periodicity of the \(y\)-directed modulation. Figure 7: Simulated _SSRI_ interferogram with arms of equal dispersions. With increasing distance measured from the position \(y_{0}\) along the spatial axis \(y\), the slope of the interference fringes increases in absolute value, which can be explained by the group delay between the phase fronts corresponding to the position \(y\). If the spectral phase difference between the two arms is not zero, then the pattern of the interferogram exactly follows the shape of the spectral phase difference function between the sample arm and the reference arm, apart from the extra group delay caused by the distance measured from the position \(y_{0}\). Figure 8 demonstrates how the effect of the phase derivatives of different orders are reflected in the interferograms.