## 3 Linear Optical Methods for Pulse Characterization
### Dispersion Measurement of Optical Elements  
Although the _SSRI_ measures only the relative spectral phase of a light pulse, even this can be used for _absolute_ measurement of dispersion of any optical element. The scheme is simple: let us measure the relative spectral phase of a light pulse to a reference pulse before and after the optical element. The difference of spectral phases arises purely from the dispersion of the optical element. In detail, as the first step of the measurement, differences of phase derivatives are measured in case of the empty interferometer, so the phase derivatives of the arms are revealed as:  
\[GD_{X}^{(0)}=SGD^{med}\cdot L_{X}^{(0)}+GD_{X}^{opt}, \tag{32}\]  
\[GDD_{X}^{(0)}=SGD^{med}\cdot L_{X}^{(0)}+GDD_{X}^{opt}\text{ and } \tag{33}\]  
\[TOD_{X}^{(0)}=STOD^{med}\cdot L_{X}^{(0)}+TOD_{X}^{opt},\text{ where }X\in S,R. \tag{34}\]  
After an identical group delay has been adjusted, the left side of Equation (24) equals to zero, therefore  
\[\Delta GD^{(0)}\equiv GD_{S}^{(0)}-GD_{R}^{(0)}=SGD^{med}\cdot\left(L_{S}^{(0)}-L_{R}^{(0)} \right)+GD_{S}^{opt}-GD_{R}^{opt}=m\cdot T\hskip 28.452756 pt(m\in\mathbf{Z}) \tag{35}\]  
can be written, where \(m\) equals zero if the same pulses interfere at the output, which were split at the input of the interferometer. In the second step, the sample having a geometrical length of \(L_{sample}\) is inserted in the sample arm of the empty interferometer. This leads to the fact that Equations (32)-(35) will be applied with the following modifications: on one hand, the laser pulses will travel in the sample arm a length shorter by \(L_{sample}\) compared to the empty interferometer, thus  
\[L_{S}=L_{S}^{(0)}-L_{sample}, \tag{36}\]  
and on the other hand, the length of the reference arm must be modified by \(\Delta L_{R}\) to compensate the group delays, thereby  
\[L_{R}=L_{R}^{(0)}+\Delta L_{R}. \tag{37}\]After this adjustment, the equation \(GD_{S}-GD_{R}=m\,\dot{T}\) can be used, where \(m\,\dot{}\) is an integer as well, though not necessarily the same as \(m\). For the group delay of the sample  
\[GD_{\text{\emph{sample}}}=SGD^{med}\cdot\left(L_{\text{\emph{sample}}}+\Delta L _{R}\right)+(m^{\prime}-m)\cdot T \tag{38}\]  
can be deducted. The group delay dispersion is given by  
\[GDD_{\text{\emph{sample}}}=\Delta GDD-\Delta GDD^{(0)}+SGDD^{med}\cdot\left(L _{\text{\emph{sample}}}+\Delta L_{R}+(m^{\prime}-m)\cdot T\,/\,SGD^{med} \right), \tag{39}\]  
where \(\Delta GDD^{(0)}\) and \(\Delta GDD\) are obtained from the evaluation of the interferograms captured during the two measurements. Similarly, third-order dispersion can be acquired from  
\[TOD_{\text{\emph{sample}}}=\Delta TOD-\Delta TOD^{(0)}+STDD^{med}\cdot\left(L _{\text{\emph{sample}}}+\Delta L_{R}+(m^{\prime}-m)\cdot T\,/\,SGD^{med} \right). \tag{40}\]  
Depending on the nature of the sample, further measurements are also possible if the spectral phase shift of the sample can be tuned in such a way that neither the sample length nor the phase derivatives of the other components of the interferometer change (for example, changing the temperature of an optical element, the concentration of a solution, or the pressure of a gas). Besides such tuning of a state variable of the sample, equal group delays should be adjusted before each measurement if necessary, and the differences of the phase derivatives will finally be determined. If the dispersions of all the other elements within the system remain constant during the series of measurement, the correlation between the object's variable quantity and the phase derivatives of the sample's material can be represented by a fitted function. After the first experimental demonstration of pulse measurement by the _SSRI_[49], its precision and reliability were investigated in an extensive study in Reference [57]. Simulations showed that one of the most dominant sources of inaccuracy originates from the noise of the spectrograph's detector.