## 3 Prism Pair CEP Shifter
### RTP Prism Pair CEP Shifter  
This paragraph presents numerical results obtained in RTP crystals. The geometry corresponds to Figure 7, with the path length of the ray at the carrier frequency chosen equal in the two crystals.  
This geometry is in practice to be preferred in order to obtain an homogeneous \"static\" electric field in the crystal (which would not be the case at the tip of a prism). The RTP Sellmeier formula for the refractive index is taken from [26] and the wavelength dispersion of the EO coefficients from [20].  
Figure 8 plots the ratio, \(d_{2}/d_{3}\), _versus_ the apex angle, \(\beta\), of the prisms when the isochronous (\(\Delta\tau_{g}\big{(}\omega_{0}\big{)}=\tau_{g}\big{(}\omega_{0},E\big{)}- \tau_{g}\big{(}\omega_{0},0\big{)}=0\), thick red), the PGD (\(\Delta\phi_{CEP}\big{(}\omega_{0}\big{)}=\theta_{CEP}\big{(}\omega_{0},E \big{)}-\phi_{CEP}\big{(}\omega_{0},0\big{)}=0\), thick  
blue) and the iso-dispersive conditions (\(\Delta\phi^{(2)}\left(\omega_{0}\right)=\phi^{(2)}\left(\omega_{0},E\right)-\phi^{( 2)}\left(\omega_{0},0\right)=0\), thin green) are imposed. This gives a means of comparing the performances of each configuration. The thick brown line corresponds to a zero dispersion condition when no electric field is applied (\(\phi^{(2)}\left(\omega_{0},E=0\right)=0\) ) to the set-up.  
It appears that it is not possible to obtain simultaneously an isochronous and iso-dispersive system, nor an isochronous system without dispersion. In the isochronous configuration, the set-up introduces a positive dispersion (as the red curve is on the left of the brown curve, which corresponds to a zero dispersive system at zero electric field). Figure 8 also shows that the PGD configuration (no CEP shift) is very close to the iso-dispersive configuration. This implies that the iso-dispersive configuration should be generally inefficient as a CEP shifter (_i.e.,_ will require a far higher voltage to generate the same CEP shift than the isochronous configuration). This is confirmed by looking at the CEP shift, which is plotted (for \(d_{3}=40\) mm and \(E=1\) kV/cm) on the same graph in the iso-dispersive configuration (dotted green line). Finally, the dotted red line curve (which corresponds to the isochronous CEP shift as a function of \(\beta\)) illustrates the fact that, in the isochronous configuration, the CEP shift (at central frequency \(\omega_{0}\) is independent of the apex angle of the prism--as can be seen from Equation 22.  
Figure 8: Ratio, \(d_{2}/d_{3}\), _versus_ apex angle for different configurations and corresponding CEP shift.  
Figure 7: Proposed practical geometry for the prism pair CEP shifter set-up.  
In order to have an idea of the practical values of \(d_{2}\) corresponding to each configuration, Figure 9 plots the induced CEP shift (red line), the induced group delay (_i.e._, \(\omega_{\rm V}\Delta\tau_{g}\)--blue line) and the induced GDD (brown line) at central frequency, \(\omega_{0}\), as a function of \(d_{2}\). The group delay dispersion, \(\phi^{(2)}\)(GDD) for \(E=0\) is also plotted (dotted brown line). These results are obtained in the case of an apex angle of the prisms, \(\beta=56.5^{\circ}\), a total path length, \(d_{3}=40\) mm, at \(\lambda=800\) nm and a static electric field, \(E=1\) kV/cm.  
This graph shows again that PGD and iso-dispersive configurations correspond to nearly the same value of \(d_{2}\) and that the CEP phase shifter is not efficient in the iso-dispersive configuration. One can also conclude that at fixed CEP shift and, for a given pair of prisms, increasing \(d_{2}\) (_i.e._, the distance between the prisms) can increase the CEP shift range or reduce the electric field. This, however, cannot be done while maintaining the isochronous condition. In any case, \(d_{2}\) is limited by the size of the beam on the second crystal, which depends on its spectral extent.