“…As a consequence, these fields cause instabilities such as bunch lengthening, head-tail turbulence, energy dispersion and emittance growth which result in deterioration of the beam quality and can even affect strongly the lasing process. The conventional treatment of the wakefield [9,10], energy dispersion [11,12], and emittance growth [13] considers an ultrarelativistic beam of uniform density traveling at the speed of light. Although it does not take into account any axial energy dispersion inside the pulse, this treatment greatly simplifies the calculation and leads to ignoring the self-field.…”