Modification of relativistic beam fields under the influence of external conducting and ferromagnetic flat boundaries

We derive analytical expressions for external fields of a relativistic bunch of charged particles with a circular and an elliptical cross section under different boundary conditions and interaction of the fields with an accelerator structural elements. The particle density in the bunch is assumed to be uniform as well as non-uniform. At distances far apart from the bunch, in free space the field reduces to the relativistic modified Coulomb form for a pointlike charge and at small distances the expressions reproduce the external fields of a continuous beam. In an ultra-relativistic limit the longitudinal components of the internal and external electric fields of the bunch are strongly suppressed by the Lorentz factor. If the bunch is surrounded by conducting surfaces, the bunch self-fields are modified. Image fields generated by a bunch between two parallel conducting plates are studied in detail. Exact summation of the electric, $E_y$, and magnetic, $B_x$, image field components allows the infinite series to be represented in terms of elementary trigonometric functions. The new expressions for modified fields are applied to study image forces acting on the bunch constituents and the bunch as a whole. The coherent and incoherent tune shifts for an arbitrary bunch displacement from the midplane are calculated in the framework of an improved linear theory, for both infinite and finite parallel flat surfaces. Moreover, the developed method allows us to generalize the Laslett image coefficients $\epsilon_1$, $\epsilon_2$, $\xi_1$, $\xi_2$ to the case of an arbitrary bunch offset and reveal relationships between these coefficients. Appendix C provides a brief historical background of the development of the method of electrical images.