On limitations of Schwinger formulae for coherent synchrotron radiation produced by an electron bunch moving along an arc of a circle

Re-examination of dogmatic “truths” can sometimes yield surprises. For years we were led to believe that famous Schwingerʹs formulas are directly applicable to describe synchrotron radiation from dipole magnet and even now no attention is usually paid to the region of applicability of these expressions. While such formulas are valid in order to describe radiation from a dipole in the X-ray range, their long-wavelength asymptote are not valid, in general. In the long-wavelength region, Schwingerʹs formulas must be analyzed from a critical viewpoint, and corrections must be discussed when one is looking for an application to CSR-based diagnostics. In this paper, we perform such a task by means of a consistent use of similarity techniques, discussing the limits of validity of Schwingerʹs formulas which arise from a finite magnet length, from a finite distance of the detector to the sources and from diffraction effects (due to the presence of vacuum pipe and aperture limitations).