#### 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:

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, Λ

_{opt} = A

_{1}BCD and NA

_{2}A

_{3}, are identical. Let

l be the distance between the apices of the two prisms and ρ the angle, (

NA_{2}A_{1}) (

Figure 5), the optical pass, Λ

_{opt}, can be written as:

If

is the angle, (

A_{1}A_{2}O), between A

_{1}A

_{2} and A

_{1}O, one obtains the following relation:

This last equation leads finally to the result:

where (

Figure 5) θ

_{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:

This expression is equivalent to that given in [

24]. An analogous calculation was also done in [

25], but the above expression was not given.

**Figure 5.**
Geometrical arrangement of the EO prism pair.

**Figure 5.**
Geometrical arrangement of the EO prism pair.

The group delay, τ

_{g}, the CEP shift,

, and second order dispersion,

, have, respectively, the following expression:

Because only the variation of these parameters with the applied static electric field,

E, is relevant, we define:

The isochronous CEP shifter condition at carrier angular frequency, ω

_{0} (CEP shift without induced group delay), corresponds to:

This is to be fulfilled for any value of

E. In order to obtain analytical results, we rewrite Equation 2 as:

By derivation with respect to ω, one gets:

with the following relations:

Using for the refractive index, a first order Taylor expansion as a function of the static electric field,

E, the isochronous condition leads to the following relation (Appendix I) between

a and

b:

where θ

_{20} and θ

_{40} are respectively the values of θ

_{2} (internal refraction angle in first prism) and θ

_{4} (external refraction angle in air at the exit of first prism) for

E = 0 (

Figure 5) and β is the apex angle of the prisms. Similarly, generation of a group delay without variation of the CEP leads to the condition:

This situation, that will be called “Pure Group Delay” (PGD) generation in the rest of this paper, implies that

a and

b are such that (Appendix II):

Finally, the iso-dispersive condition being:

This leads again to a similar relation. In this case, however, the analytical result is more complex and is not given here for conciseness.

As a first conclusion, we see that for a particular set of the ratio, b/a, that is to say, a specific geometry, it is possible to make the system behave like an isochronous CEP shifter, a PGD generator or an iso-dispersive CEP shifter.

#### 3.2.2. Minimal Deviation Configuration

We now suppose that the incident angle corresponds to the prism minimal deviation at the carrier frequency, ω

_{0}. This condition, which gives the highest angular dispersion, can be written as:

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):

We will now express the results in the particular case of central wavelength, λ

_{0}, corresponding to the central angular frequency,

, and using λ instead of ω. Equation 21 becomes (Appendix IV):

This shows that, at central wavelength, λ_{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, β.

**Figure 6.**
Geometrical arrangement of the EO prism pair CEP shifter at prism minimal deviation.

**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:

where λ is the wavelength corresponding to the angular frequency, ω, and where we use the derivatives taken with respect to the wavelength, λ, instead of ω.

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

for isochronous CEP shifter and PGD generator configuration are, respectively, given below:

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

. It can be written in our notations as:

From this expression, we deduce the ratio,

d_{2}/

d_{3}, for which the system introduces no dispersion (

i.e.,

) at

:

Analytical results were checked numerically with a ray tracing program.