Quantification of Laser Polarization by Position Dependent Refractive Indices

The electro-optic (EO) crystals used in the generation of tera-hertz radiation [Shan et al, 2000; Wen & Lindenberg, 2009; Shen et al., 2008] can be applied to non-invasive measurements of the electron bunch in the accelerator. The bunch length and arrival time of the electron bunch with respect to the laser can be measured with this technique. The electric field generated by the electron bunch changes the optical properties of the EO crystal. The laser passing through the EO crystal will thus experience a modulation in the polarization state. The amount of optical power modulation, which can be detected by detectors such as photodiodes and cameras, can be calculated by Jone's matrices [Jones, 1941; Hecht, 2002]. In recent years, the EO technique has been implemented to measure femtosecond electron bunches with high energy at several facilities such as the free-electron laser for infrared experiments (FELIX), the sub-picosecond pulse source (SPPS), and the free-electron laser in Hamburg (FLASH) [Yan et al, 2000; Wilke et al., 2002; Berden et al., 2004; Cavalieri et al., 2005; Casalbuoni et al., 2008; Steffen et al., 2009]. In these applications, the electric field from the electron bunch makes a finite angle with the crystallographic axis, because the electric field is generated radially from the electron bunch. In a real diagnostics setup, the principal axes at the passing position of the laser in the EO crystal are no longer 45o with respect to the crystallographic axes. However, the polarization vector of the laser is usually fixed as parallel to an crystallographic axis of the EO crystal in the measurement. When an electric field is applied to the EO crystal, it shows a birefringence which is described by an ellipsoid equation of refractive indices. If the ellipsoid equation can be expressed without any cross term between the three axes in a coordinate system, they are called the principal axes. If a linearly polarized laser is passing through the EO crystal with an external electric field applied to it, the polarization of the laser is modulated elliptically or even circularly. Conventional theories to describe the measurement result of an electron bunch with EO crystal have only used the principal axes and the principal refractive indices. To quantify the polarization state, we need to define amplitudes and phases of the electric field components of the laser. If the amplitudes of the decomposed electric field are the same, we need the phase information only to quantify the polarization state. To decompose the electric field with the same amplitude, we need to choose decomposition axes at 45o with respect to the polarization vector. The spatial