Upper limit for the accelerating gradient in the collinear wakefield accelerator as a function of the transformer ratio

The interrelation between the accelerating gradient and the transformer ratio in the collinear wakefield accelerator has been analyzed. It has been shown that the high transformer ratio and the high efficiency of the energy transfer from the drive bunch to the witness bunch can only be achieved at the expense of the accelerating gradient. Rigorous proof is given that in best cases of meticulously shaped charge density distributions in the drive bunch, the maximum accelerating gradient falls proportionally to the gain in the transformer ratio. Conclusions are verified using several representative examples.