## 3 Linear Optical Methods for Pulse Characterization
### Spectrally and Spatially Resolved Interferometry  
The experimental implementation of _SSRI_ is based on the combination of a two-beam interferometer and a two-dimensional imaging spectrograph. For the sake of simplicity, a Mach-Zehnder interferometer is considered, illuminated by a broadband light source (Figure 5). The arm of the interferometer containing the dispersive object is called the sample arm; the other one is the reference arm. The spatial and spectral intensity distribution of the laser pulses travelling through the corresponding arms are denoted by \(I_{S}(y,\omega)\) and \(I_{R}(y,\omega)\), respectively. The pattern is finally detected by a spectrograph, which is equipped with a two-dimensional _CCD_ detector. For the pulses propagating in the reference arm, the shape of the spectral phase function at the output generally is not influenced by the sample arm and remains unaltered; therefore, these pulses can be considered a reference for the measurement of the spectral phase change caused by the dispersive sample. However, a delay stage is inserted usually in the reference arm in order to adjust the timing of the reference pulses. In order to establish a spatial resolution, the angle between the crossed beams must also be adjusted. The outcoupling mirrors of the individual arms are tilted vertically so that the beams cross each other at an angle \(\alpha\), and overlap spatially along the vertically aligned slit. After the correct adjustment of the spatial and temporal overlapping of the pulses travelling through the sample arm, a vertically modulated interference pattern emerges on the spectrograph slit as illustrated in Figure 6.  
The interference pattern on the two-dimensional detector surface of the imaging spectrograph is described by  
\[I(y,\omega)=I_{S}(y,\omega)+I_{R}(y,\omega)+2\sqrt{I_{S}(y,\omega)\cdot I_{R}( y,\omega)}\cdot\cos(\varphi_{i\%}(y,\omega)), \tag{16}\]  
where the phase of the interferogram can be calculated as  
\[\varphi_{i\%}(y,\omega)=\varphi_{S}(\omega)-\varphi_{R}(\omega)+(y-y_{0}) \cdot\alpha\omega n^{med}\ /\ c. \tag{17}\]  
Figure 5: Schematic experimental layout of the _SSRI_ technique. In the formula above, \(\varphi_{S}(\omega)\) and \(\varphi_{R}(\omega)\) denote the spectral phase of the pulses arriving from the sample arm and the reference arm, respectively. These quantities are supposed to be independent of the spatial coordinate \(y\). The third term of Equation (17) describes a phase modulation in direction \(y\), caused by the crossed arrangement of the beams, assuming that \(\alpha\) is small (see Figure 6). Here \(y_{0}\) denotes a reference point, where the phase difference arising from the tilted phase fronts is zero. At this height, the total group delay at the central frequency of \(\omega_{0}\) is exactly the same for the pulses travelling along the two different pathways when an empty interferometer is considered. If the group delay in any of the arms changes (e.g., the length of the reference arm changes through the operation of the delaying unit, or extra material is inserted into the sample arm), the equal \(GD\) level will be shifted along the \(y\) axis from the position of \(y_{0}\). The spectral phase shift in the arms of the interferometer can be approximated by Taylor expansion, similarly to Equation (5). If \(L_{S}\) and \(L_{R}\) distances are supposed to be travelled by the laser pulses in the arms filled by a medium (usually air) having an index of refraction \(n^{med}\), then their contributions to the total spectral phase shifts can be written as  
\[ \begin{split}\varphi_{X}^{med}(\omega)=&\frac{n^{ mod}(\omega_{0})\omega_{0}\cdot L_{X}}{c}+SGD^{med.}L_{X}\cdot\left(\omega-\omega_{0}\right)+\frac{SGD^{med}L_{X}}{2}\cdot\left(\omega-\omega_{0} \right)^{2}+\\ &+\frac{STOD^{med}L_{X}}{6}\cdot\left(\omega-\omega_{0}\right)^{ 3}+\ldots,\qquad\qquad\qquad X\in S,R\end{split}  \tag{18}\]  
where \(X\) can be replaced by either \(S\) or \(R\), depending on whether the sample arm or the reference arm is considered.