Optimization of {beta} functions through insertion devices

It is generally noted that at an undulator straight section, the horizontal beta function is made to be large while the vertical beta function is relatively small. On the other hand, at a wiggler straight section, both horizontal and vertical beta functions are made to be small. In this note we describe a procedure with which optimum settings of the beta functions in the insertion straight section are to be determined. For this we consider separately for the undulator radiation and the radiation from the wiggler device. Since the brilliance of radiation is a canonically conserved quantity, we use the brilliance as a figure of merit for the consideration. Then the optimization process is to find a set of horizontal and vertical beta functions which would give the maximum brilliance when the natural emittance, emittance coupling constant, length of insertion device and the photon energy to which the insertion device is optimized are specified. The next step of the study is to find sensitivities of the beta functions to the brilliance