Discussion and analytical test for inclusion of advanced field and boundary condition in theory of free electron lasers

By the covariant statement of the distance in space-time separating transmitter and receivers, the emission and absorption of the retarded and advanced waves are all simultaneous. In other words, for signals carried on electromagnetic waves (advanced or retarded) the invariant interval (cdt)2−dr2 between the emission of a wave and it's absorption at the non-reflecting boundary is always identically zero. Utilizing this principle, we have previously explained the advantages of including the coherent radiation reaction force as a part of the solution to the boundary value problem for FELs that radiate into “free space” (Self Amplified Spontaneous Emission (SASE) FELs) and discussed how the advanced field of the absorber can interact with the radiating particles at the time of emission. Here we present an analytical test which verifies that a multilayer mirror can act as a band pass filter and can contribute to microbunching in the electron beam. Here we will discuss motivation, conditions and requirements, and method for testing this effect.