## 3 Prism Pair CEP Shifter
### Theory
#### 3.2.1 General Configuration  
We first consider the general case where the angle of incidence can take any value. As the optical path is obviously invariant for parallel incident rays, we calculate the path of a ray going through the apex of the first prism (Figure 5). For a single pass in the system, it can be written as:  
\[\Lambda_{op}=A_{\mathrm{1}}B+BC+CD \tag{5}\]  
Surprisingly, following the method given in [24], a rather simple calculation shows that it can be written in an elegant mathematical form. We consider (Figure 5) a ray NA\({}_{2}\) parallel to A\({}_{1}\)B (NA\({}_{1}\) normal to A\({}_{1}\)B) intercepting the apex A\({}_{2}\) of prism 2. This ray is deviated by prism 2 along the path, A\({}_{2}\)A\({}_{3}\). As A\({}_{1}\)B and NA\({}_{2}\) are parallel incident rays, the light paths, \(\Lambda_{opt}=\) A\({}_{1}\)BCD and NA\({}_{2}\)A\({}_{3}\), are identical. Let \(l\) be the distance between the apices of the two prisms and \(\rho\) the angle, (\(NA_{2}A_{1}\)) (Figure 5), the optical pass, \(\Lambda_{opt}\), can be written as:  
\[\Lambda_{op}=l\cos\rho+A_{\mathrm{2}}A_{\mathrm{3}} \tag{6}\]  
If \(\big{(}\varphi=\rho+\theta_{4}\big{)}\) is the angle, (\(A_{1}A_{2}O\)), between A\({}_{1}\)A\({}_{2}\) and A\({}_{1}\)O, one obtains the following relation:  
\[\Lambda_{op}=l\cos\big{(}\varphi-\theta_{4}\big{)}+A_{\mathrm{2}}A_{\mathrm{3 }}=l\cos\varphi\cos\theta_{4}+l\sin\varphi\sin\theta_{4}+A_{\mathrm{2}}A_{ \mathrm{3}} \tag{7}\]  
Figure 4: EO prism pair CEP shifter. Yellow parts correspond to gold coating. This last equation leads finally to the result:  
\[\Lambda_{ap}=a\cos\theta_{4}+b\sin\theta_{4}+A_{2}A_{3} \tag{8}\]  
where (Figure 5) \(\theta_{4}\) is the external refraction angle at the exit of prism 1 in air, \(a\) the slant distance, A\({}_{1}\)O, between the output face of prism 1 and input face of prism 2, \(b\) the distance, OA\({}_{2}\), and A\({}_{2}\)A\({}_{3}\) the distance between the apex of the second prism and a reflecting mirror (used in a two pass configuration to remove spatial chirp). A\({}_{2}\)A\({}_{3}\) being a constant, we can remove it, and the spectral phase can thus be written:  
\[\phi\left(\omega\right)=\left(a\cos\theta_{4}+b\sin\theta_{4}\right)\omega/c \tag{9}\]  
This expression is equivalent to that given in [24]. An analogous calculation was also done in [25], but the above expression was not given. The group delay, \(\tau_{g}\), the CEP shift, \(\Delta\phi_{CEP}\), and second order dispersion, \(\phi^{\left(2\right)}\), have, respectively, the following expression:  
\[\tau_{g}\left(\omega\right)=\frac{\partial\phi}{\partial\omega} \left(10a\right) \tag{10}\] \[\phi_{CEP}\left(\omega\right)=\phi\left(\omega\right)-\omega \tau\left(\omega\right) \left(10b\right)\] \[\phi^{\left(2\right)}\left(\omega\right)=\frac{\partial^{2}\phi }{\partial\omega^{2}} \left(10c\right)\]  
Because only the variation of these parameters with the applied static electric field, \(E\), is relevant, we define:  
\[\Delta\tau_{g}\left(\omega,E\right)=\tau_{g}\left(\omega,E\right)- \tau_{g}\left(\omega,0\right) \tag{11a}\] \[\Lambda_{op}=l\cos\rho+A_{\mathrm{2}}A_{\mathrm{3}} \tag{6}\]0 (11b) \[\Lambda_{op}=l\cos\rho+A_{\mathrm{2}}A_{\mathrm{3}} \tag{6}\]1  
The isochronous CEP shifter condition at carrier angular frequency, \(\omega_{0}\) (CEP shift without induced group delay), corresponds to:  
\[\Lambda_{op}=l\cos\rho+A_{\mathrm{2}}A_{\mathrm{3}} \tag{6}\]2 Geometrical arrangement of the EO prism pair. \[\Lambda_{op}=l\cos\rho+A_{\mathrm{2}}A_{\mathrm{3}} \tag{6}\]3  
This is to be fulfilled for any value of \(E\).