## 3 Linear Optical Methods for Pulse Characterization
### Pulse Measurement  
As it may become obvious from the preceding sections, _SSRI_ offers a fast, simple and robust technique to measure the relative spectral phase of an unknown pulse to a well-described one. That is, unlike other pulse characterization techniques based on the (nonlinear) interferometry _SPIDER_[18, 21] and similar solutions, _SSRI_ is a linear, but not a self-referencing method [49]. Hence, the parameters of the reference pulse have to be well known at a certain point of the laser system (e.g., at the output of the oscillator, or measured by a different technique, which provides absolute measurements). Let us suppose that the phase derivates between the reference point and the input of the interferometer change by  
\[GD_{{}_{F}}=SGD^{{}^{\textit{med}}}\cdot L_{{}_{F}}+GD_{F}^{{}^{\textit{opt}}}, \tag{26}\]  
\[GDD_{{}_{F}}=SGDD^{{}^{\textit{med}}}\cdot L_{{}_{F}}+GDD_{F}^{{}^{\textit{ opt}}},\ \text{and} \tag{27}\]  
\[TOD_{{}_{F}}=STOD^{{}^{\textit{med}}}\cdot L_{{}_{F}}+TOD_{F}^{{}^{\textit{ opt}}} \tag{28}\]  
during the free-space propagation of the geometrical length of \(L_{F}\) and passing through certain optical elements with well-known dispersion (\(GD_{F}^{{}^{\textit{opt}}},GDD_{F}^{{}^{\textit{opt}}}\) and \(TOD_{F}^{{}^{\textit{opt}}}\)). If the absolute phase derivates at the reference point are \(GD_{RP}\), \(GDD_{RP}\) and \(TOD_{RP}\), the pulses of the sample arm at the output of the interferometer are characterized by  
\[GD_{{}_{P}}=\Delta GD+GD_{{}_{R}}+GD_{{}_{F}}+GD_{{}_{RP}}-m\cdot T, \tag{29}\]  
\[GDD_{{}_{P}}=\Delta GDD+GDD_{{}_{R}}+GDD_{{}_{F}}+GDD_{{}_{RP}}-SGDD^{{}^{ \textit{med}}}\cdot m\,\frac{T}{SGD^{{}^{\textit{med}}}}, \tag{30}\]  
Figure 8: Demonstration of separated effect of phase derivates on simulated interferograms. The values of the applied phase derivates are \(GD\) = 60 fs (**a**), \(GDD\) = 200 fs\({}^{2}\) (**b**) and \(TOD\) = 2400 fs\({}^{3}\) (**c**), respectively.  
\[TOD_{p}=\Delta TOD+TOD_{R}+TOD_{F}+TOD_{RP}-STOD^{med}\cdot m\frac{T}{SGD^{med}}, \tag{31}\]  
where \(\Delta GD\), \(\Delta GDD\) and \(\Delta TOD\) are measured from the recorded interferograms. This process gives a simple method to determine absolute pulse parameters at basically any point of the system.  
Please note that this is basically a single-shot, linear method which describes the absolute spectrum and the relative spectral phase of the pulse to be characterized, so that the relative temporal shape can be calculated to great accuracy. Linear interferometric methods can be utilized in most of the practical applications like spectroscopy, linear and nonlinear dispersion measurements, time-resolved studies, as well as compressor alignment and beam monitoring along high intensity laser systems. On the one hand, these methods provide the experimentalists with more accurate information than the nonlinear measurements methods, which offer an absolute temporal shape, but with a factor of 2-5 higher error than the linear methods. On the other hand, the linear methods would never be able to provide the absolute pulse duration on their own, since the first-order auto- (and cross-) correlation function to be determined by linear interferometry offers coherence information of the interacting pulses only [19].