## 5 Methods for Angular Dispersion Detection
### Phase Front Angular Dispersion--An Absolute Measurement with SSRI  
The _SSRI_ technique can be applied also for the determination of either the _relative_ (_i.e._, occurring between the arms) or the _absolute_ phase front angular dispersion \(\gamma_{PF}\) along the direction of the slit of the spectrograph. Keeping in mind that when reflected on a mirror, angular dispersion changes sign along the plane of reflection, the method can be modified accordingly. When a Mach-Zehnder interferometer is used with the same number of reflections in the arms (or their difference is an even number), the angular dispersion of an object inserted in the sample arm can be measured as the difference in angular dispersion of the interfering pulses. For the measurement of the  
Figure 14: Typical layout for simple, two-dimensional detection of propagation dispersion angular dispersion (**a**). If the original beam is not angularly dispersed enough to separate the peaks, an additional grating or prism can be inserted to have a well-known constant bias in the angular dispersion (**b**). On the right side, a sample image is shown (**c**). absolute_ angular dispersion, an _empty_ Mach-Zehnder interferometer has to be altered so that the difference between the numbers of reflections from optical surfaces in the two arms is an odd number (e.g., one arm should contain an additional mirror). Therefore, at the output of the interferometer, the pulse will establish interference with its replica, mirrored to the direction of the propagation. The magnitude of the angular dispersion will be unaffected by this reflection; however, its direction will be inverted. This implementation of the interferometer is called _inverted Mach-Zehnder interferometer_[105, 106]. The accordingly modified scheme is shown on Figure 15.  
The angle between the phase fronts will then be measured by spectrally and spatially resolved interferometry, and will depend on the wavelength due to the angular dispersion. If a small angle is adjusted between the beams in the direction of polarization, then this (wavelength-independent) angle will also occur between the phase fronts, and thereby the interference pattern will be modulated in the direction of polarization. The frequency of this modulation is directly proportional to the angle and to the frequency of the light. The manually adjusted angle between the beams can be measured as the function of the wavelength by a two-dimensional imaging spectrograph having a slit oriented in the direction of polarization. The derivative of the measured angle with respect to the wavelength results in a value twice the phase front angular dispersion occurring in the direction of polarization. The angle between the phase fronts as defined by Equation (17) will depend on the wavelength. Using the parameters \(a_{i}\) of the cosine fit outlined earlier, \(\alpha\) values can easily be obtained for each wavelength. Phase front angular dispersion reveals then as follows:  
\[\gamma_{PF}=\frac{1}{2}\frac{\mathrm{d}\alpha(\lambda)}{\mathrm{d}\lambda} \Bigg{|}_{\lambda=\lambda_{0}} \tag{44}\]  
An inappropriate adjustment of the optical elements in a _CPA_ system may cause an angular dispersion not only in the horizontal, but also in the vertical plane, therefore beam rotational stage is required here as well. Accuracy of the phase front angular dispersion depends on the detector noise similarly as it was shown in the case of _SSRI_[57], since the phase extraction process is the very same in both cases. Experimental studies demonstrated 0.2 \(\upmu\)rad/nm precision [53, 106]. Spatial invertion of the beam profile can be effectively used in nonlinear autocorrelators as well. Since no spectral dimension is needed during the evaluation, both spatial directions can be inverted,  
Figure 15: Experimental set-up of the interferometric detection of phase front angular dispersion with an inverted Mach–Zehnder interferometer. hence pulse front tilt can be detected in 2D in a single-shot measurement [107].