= Relativistic solenoid magnetic lens
For a near-axis beam of equal incoming speed $v$ through a uniform solenoid of length $l$, define $\omega=|qB|/(2\gamma m)$. Neglecting the small axial-speed changes, the internal radius is $\rho=\rho_0\cos(\omega z/v)$. The exit fringe removes the azimuthal velocity for zero incident canonical angular momentum. The field-free radial trajectory then reaches the axis a distance
$$
f=\frac v\omega\cot\left(\frac{\omega l}v\right)
$$
beyond the exit, when $0<\omega l/v<\pi/2$. In the weak-lens limit, $f\simeq4(\gamma mv)^2/(q^2B^2l)$. Without the paraxial approximation, the radius-dependent axial speed produces aberration; the ideal common focus is then only approximate.
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