Fig. 2. Bremsstrahlung enhancement beyond the free electron spatial probability distribution.

We show that non-paraxial electron wavefunctions with identical spatial probability distribution, but different phases, can emit distinctly different radiation patterns. We consider 20 keV shaped electron wavefunctions ${\mathrm{\Psi }}_p$ with two momentum components (each with an incident angle of 15°), having two different superpositions: $\left(\left|p_x\right.⟩-2\left|-p_x\right.⟩\right)/\sqrt{5}$ and $\left(2\left|p_x\right.⟩-\left|-p_x\right.⟩\right)/\sqrt{5}$. Their phases $\mathrm{Arg}\left(u_z^{†}{\mathrm{\Psi }}_p\right)$ (at $z$ = 0) are shown in (a(i), a(ii)), respectively, where both of them form identical spatial probability distributions as shown in (b). Their corresponding bremsstrahlung emission differential cross sections $\mathrm{d}\sigma /\left(\mathrm{d}{\omega }_{k^{\prime }}\mathrm{d}{\mathrm{\Omega }}_{k^{\prime }}\right)$, for a 15 keV photon, are shown in (c(i), c(ii)), respectively. These two examples indicate that the shaping of output radiation does not simply depend on the electron transverse spatial probability distribution and substantially deviate from the prediction under the non-recoil and paraxial approximation