## 1 Introduction  
The electric field of a laser pulse is generally described by the product of a wave envelope and a carrier wave. In a dispersive medium, phase and group velocities are different, inducing a slippage of the carrier frequency wave inside the envelope. For ultra-short pulses containing only a few optical cycles, laser-matter interactions can drastically depend on the electric field and not only on its envelope. In this case, controlling the carrier-envelope-phase (CEP) is of prime importance [1, 2, 3].  
Because of the difference between effective group and phase velocities in laser cavities and of environmental effects, such as vibrations and thermal drift, the generated pulses from ultra-short chirped pulse amplification (CPA) lasers do not have the same CEP. Various methods have been developed in order to obtain, through the use of a fast control loop, a train of CEP stabilized pulses from mode-locked oscillators [4, 5, 6]. Different ways also exist to stabilize the CEP of the amplified pulses of a (CPA) laser system seeded by a CEP stabilized mode-locked oscillator. They are mainly based on a slow feedback loop containing a f-2f interferometer [7], a proportional-integral-derivative controller (PID) and a specific CEP correction technique [8].  
Examples of those techniques are the use of a pair of wedges to modify the optical path in the dispersive element composing these wedges [9, 10], the modification of one parameter of the compressor or of the stretcher (this parameter can be the distance between the gratings) [11, 12], the use of an Acousto-Optic Programmable Dispersive Filter (AOPDF) [13, 14, 15] or of a 4f system with an adaptive phase modulator device [16].  
We recently proposed an original method based on the linear electro-optic (EO) effect in a bulk material (LiNbO\({}_{3}\)--lithium niobate) [17, 18] and successfully applied it to the CEP control of a titanium-sapphire CPA laser [19] with stabilization performances of the order of those obtained with classical methods. Compared to the main equivalent CEP shifters (as will be detailed at the end of this paper), the major advantages of EO CEP shifters are related to the fact that they do not need mechanical displacements and especially to their high correction bandwidth (\(>\)10 kHz). In this paper, we introduce another original method, still based on the use of the EO effect, but in a more complex optical scheme and which relies on the use of an EO dispersive prism pair. While experimental demonstration of CEP shifts and CEP control [17, 18, 19] of CPA amplified pulses with a \"longitudinal\" shifter (simple rubidium titanyl phosphate (RTP) or LiNbO\({}_{3}\) rectangular slab) has been proven, no experimental results will be given here concerning the prism pair EO shifter, which will be the topic of a future paper.  
The structure of the paper is the following. We first recall analytical results explaining how a CEP shift can be induced in an EO crystal, the former being linked to the static electric field dependence of the difference between the induced group and phase delays in the crystal. We then give briefly the main experimental results already obtained with CEP shifters consisting of an EO material rectangular slab (that will be called the \"longitudinal configuration\" of the CEP shifter in the rest of the paper) on which a transverse electric field is applied. The new prism pair configuration is then theoretically detailed, and it is shown that it has many advantages compared to the first set-up, as, in particular, making it possible to shift the CEP without changing the group delay. This may be essential (typically in high resolution pump-probe experiments and especially in experiments making use of attosecond laser pulses) when anychange in timing between pulses is to be avoided. In the last part of the paper, we provide some elements to compare the different solutions for CEP stabilization.