Statistical Analysis of Crossed Undulator for Polarization Control in a SASE FEL

There is a growing interest in producing intense, coherent x-ray radiation with an adjustable and arbitrary polarization state. In this paper, we study the crossed undulator scheme (K.-J. Kim, Nucl. Instrum. Methods A {\bf 445}, 329 (2000)) for rapid polarization control in a self-amplified spontaneous emission (SASE) free electron laser (FEL). Because a SASE source is a temporally chaotic light, we perform a statistical analysis on the state of polarization using FEL theory and simulations. We show that by adding a small phase shifter and a short (about 1.3 times the FEL power gain length), $90^\circ$ rotated planar undulator after the main SASE planar undulator, one can obtain circularly polarized light -- with over 80% polarization -- near the FEL saturation.