Undulator radiation brightness and coherence near the diffraction limit

The brightness and transverse coherence of undulator radiation in synchrotron light sources is investigated using the Wigncr formulation in a full 4D treatment. New results are presented on the impact of energy spread on the brightness and coherence as a function of the emittance. In contrast to other work on this topic, it is found that as emittance decreases, the effect of energy spread increases, but as the diffraction limit is approached, the behavior of brightness and coherence diverge. The accuracy of the commonly used Gaussian approximation for brightness and coherence is also explored in detail, resulting in a new conclusion that the commonly used projected source sizes and divergences are not the appropriate quantities to use in the Gaussian approximation to obtain the most accurate results. New expressions are derived for the rms source size and divergence as a function of the energy spread for use in the Gaussian approximation.