  We introduce a new family of paraxial asymptotic models that approximate the
Vlasov-Maxwell equations in non-relativistic cases. This formulation is $n$-th
order accurate in a parameter $\eta$, which denotes the ratio between the
characteristic velocity of the beam and the speed of light. This family of
models is interesting, first because it is simpler than the complete
Vlasov-Maxwell equation, then because it allows us to choose the model
complexity according to the expected accuracy.
