Statistical Optics approach to the design of beamlines for Synchrotron Radiation

摘要

In this paper we analyze the image formation problem for undulator radiationthrough an optical system, accounting for the influence of the electron beamemittance. On the one hand, image formation with Synchrotron Radiation isgoverned by the laws of Statistical Optics. On the other hand, the widely usedGaussian-Shell model cannot be applied to describe the coherence properties ofX-ray beams from third generation Synchrotron Radiation sources. As a result, amore rigorous analysis of coherence properties is required. We propose atechnique to explicitly calculate the cross-spectral density of an undulatorsource, that we subsequently propagate through an optical imaging system. Atfirst we focus on the case of an ideal lens with a non-limiting pupil aperture.Our theory, which makes consistent use of dimensionless analysis, also allowstreatment and physical understanding of many asymptotes of the parameter space,together with their applicability region. Particular emphasis is given to theasymptotic situation when the horizontal emittance is much larger than theradiation wavelength, which is relevant for third generation SynchrotronRadiation sources. First principle calculations of undulator radiationcharacteristics (i.e. ten-dimensional integrals) are then reduced toone-dimensional convolutions of analytical functions with universal functionsspecific for undulator radiation sources. We also consider the imaging problemfor a non-ideal lens in presence of aberrations and a limiting pupil aperture,which increases the dimension of the convolution from one to three. Inparticular we give emphasis to cases when the intensity at the observationplane can be presented as a convolution of an impulse response function and theintensity from an ideal lens.