## 3 Prism Pair CEP Shifter
### Theory
#### 3.2.2 Minimal Deviation Configuration  
We now suppose that the incident angle corresponds to the prism minimal deviation at the carrier frequency, \(\omega_{0}\). This condition, which gives the highest angular dispersion, can be written as:  
\[\theta_{1}=Arc\sin\left\{n_{0}\sin\left(\beta/2\right)\right\} \tag{20}\]  
As these parameters are more relevant from an experimental point of view, instead of using the parameters, \(a\) and \(b\), we now switch to the parameters, \(d_{2}\) and \(d_{3}\), corresponding, respectively, to the distances covered between prism 1 and 2 in air and to the total path inside the prisms for the ray at the carrier frequency (Figure 6). The spectral phase takes the form (Appendix III):  
\[\phi\left(\omega,E\right)=\frac{\omega}{c}\left(d_{2}\cos\left[\theta_{1}- \theta_{4}\left(\omega,E\right)\right]+d_{3}\frac{\sin\left[\theta_{4}\left( \omega,E\right)\right]}{2\sin\left(\beta/2\right)}\right) \tag{21}\]  
We will now express the results in the particular case of central wavelength, \(\lambda_{0}\), corresponding to the central angular frequency, \(\omega_{0}=2\pi\,c/\lambda_{0}\), and using \(\lambda\) instead of \(\omega\). Equation 21 becomes (Appendix IV):  
\[\Delta\phi\left(\lambda_{0},E\right)\approx\frac{2\pi d_{3}n_{0}\varepsilon}{ \lambda_{0}}\,E \tag{22}\]  
This shows that, at central wavelength, \(\lambda_{0}\) the variation of the spectral phase with respect to the applied electric field does not depend on the distance, \(d_{2}\), nor on the apex angle, \(\beta\). **Figure 6.** Geometrical arrangement of the EO prism pair CEP shifter at prism minimal deviation. The group delay variation with respect to the applied electric field is written:  
\[\Delta\tau_{g}\left(\lambda_{0}\right)=\frac{E}{c}\left[4\varepsilon n_{0} \lambda_{0}d_{2}\frac{dn_{0}}{d\lambda}\frac{\sin^{2}\left(\beta/2\right)}{ \cos^{2}\theta_{1}}+\right.\right. \tag{23}\]  
where \(\lambda\) is the wavelength corresponding to the angular frequency, \(\omega\), and where we use the derivatives taken with respect to the wavelength, \(\lambda\), instead of \(\omega\). Contrary to the case of the variation of the spectral phase with the electric field, the variation of the group delay with the field depends on \(d_{3}\) and \(d_{2}\). Analytical expression of the group dispersion with respect to the electric field can also be derived, but, as in the general configuration, is not given here for conciseness. Analytical expressions of the ratio, \(d_{2}/d_{3}\), at \(\lambda=\lambda_{0}\) for isochronous CEP shifter and PGD generator configuration are, respectively, given below:  
\[\frac{d_{2}}{d_{3}}=\frac{-\lambda_{0}e\bigg{(}\frac{dn_{0}}{d \lambda}\bigg{)}\tan^{2}\bigg{(}\frac{\beta}{2}\bigg{)}+\lambda_{0}\bigg{(} \frac{d(\varepsilon n_{0})}{d\lambda}\bigg{)}-\varepsilon n_{0}}{4\lambda_{0} \varepsilon n_{0}\bigg{(}\frac{dn_{0}}{d\lambda}\bigg{)}\sin^{2}\big{(}\beta \big{)}}\cos^{2}\theta_{1}\quad\text{(Isochronous CEP shifter)} \tag{24}\]  
\[\frac{d_{2}}{d_{3}}=\frac{\bigg{(}\frac{d(\varepsilon n_{0})}{d \lambda}\bigg{)}-\varepsilon\bigg{(}\frac{dn_{0}}{d\lambda}\bigg{)}\tan^{2} \big{(}\beta/2\big{)}}{4\varepsilon n_{0}\bigg{(}\frac{dn_{0}}{d\lambda} \bigg{)}\sin^{2}\big{(}\beta/2\big{)}}\cos^{2}\theta_{1}\qquad\text{(PGD)} \tag{25}\]  
When the isochronous CEP shifter configuration is chosen, the variation of the CEP is equal to the variation of the spectral phase with the electric field, which is given by Equation 22.  
Another interesting parameter to evaluate is the group-delay dispersion (GDD) induced on the beam by the two prism compressor for \(E=0\) at \(\lambda=\lambda_{0}\).