## 3 Linear Optical Methods for Pulse Characterization
### Spectrally Resolved Interferometry  
One of the most common methods used for the measurement of spectral phase shift is the so-called spectrally resolved interferometry, _SRI_[63; 64; 65; 66; 67; 68; 69; 70; 71; 72], which can be interpreted as a special case of _SSRI_ when \(\alpha=0\). The beams here leave the Michelson or Mach-Zehnder interferometer in a collinear manner, and an interference is established due to the temporal delay \(t^{\bullet}\) within the time scale of the coherence time, kept to be constant between the beams. The beams are directed in a spectrograph and reveal a modulated spectrum as represented in Figure 9a. The larger \(t^{\bullet}\) delay is set the narrower is the period of the modulation. Since in most cases spatial modulation is not present along the slit, a spectrograph with array detector is sufficient. One of the most widely used Fourier transform-based evaluation methods of _SRI_ interferograms was developed by [PERSON] _et al._[64]. According to his method, the measured spectral interferogram (Figure 9a) is first inverse Fourier-transformed, and then a filter is applied on it at the time difference corresponding to the time interval between the pulses, \(t^{\bullet}\), using a window function having optimal width (Figure 9b). In the final step, the filtered signal is transformed back into the spectral domain. The angle of the complex values of this spectrum reveals the spectral phase difference of the pulses (Figure 9c). The slope of the phase at \(\omega_{0}\) corresponds to \(\omega_{0}\)\(t^{\bullet}\), and the phase derivatives of higher orders can also be determined by a polynomial fitting. This method can be used with broadband laser pulses as well as with continuously operating white light sources, since the only precondition of the measurement is that the spectral bandwidth of the light source should be broad enough. _SRI_ is thereby often used to determine the dispersion of optical material, using an interferometer illuminated by continuously operating white-light lamps [43; 44; 73; 74]. When overwhelming difference in dispersion of the sample and reference arms is present, _SSRI_ and _SRI_ might be able to resolve the interference pattern only on a small wavelength range of spectrum. Stationary phase point method, _SPP_[75; 76] is an implementation of _SRI_ (_i.e._, \(\alpha=0\) is still a necessary condition), which can be applied in this case. The plane-wave approximation can no longer be applied because the different curvatures of the phase fronts play an important role in the position-dependence of the phase difference occurring between the pulses that meet on the detector  
Figure 9: Steps of the spectral phase calculation by Fourier-transform _SRI_. The recorded interferogram (**a**) is inverse Fourier-transformed to time domain, and filtered (**b**). Finally, the remainder is transformed back to spectral domain. The complex angle corresponds to the spectral phase difference between the arms of the interferometer (**c**). surface. _SPP_ method is based on the determination of the spectral position of the stationary phase point as a function of the delay between the sample ad the reference pulse. This technique is particularly useful when one wants to compare stretched pulses of a _CPA_ system with nearly transform-limited seed pulses from the oscillator [48]. It has been also proved to be useful during the measurements of the dispersion of optical fibers [77, 78, 79]. Let us suppose that the two beams are coaxial, and they reach the detector at the position \(y_{0}\). The stationary phase point is located at the central frequency \(\omega_{0}\), if the group delays of the arms are carefully aligned to satisfy the condition in Equation (30). In this case, the temporal delay \(t^{\ast}\) is zero. When one introduces the delay \(t^{\ast}\) of the pulses of the sample arm, the stationary phase point will be shifted along the \(\omega\)-axis.