Non-diffracting multi-﻿electron vortex beams balancing their electron–electron interactions

The wave-like nature of electrons has been known for almost a century, but only in recent years has the ability to shape the wavefunction of EBeams (Electron-Beams) become experimentally accessible. Various EBeam wavefunctions have been demonstrated, such as vortex, self-accelerating, Bessel EBeams etc. However, none has attempted to manipulate multi-electron beams, because the repulsion between electrons rapidly alters the beam shape. Here, we show how interference effects of the quantum wavefunction describing multiple electrons can be used to exactly balance both the repulsion and diffraction-broadening. We propose non-diffracting wavepackets of multiple electrons, which can also carry orbital angular momentum. Such wavefunction shaping facilitates the use of multi-electron beams in electron microscopy with higher current without compromising on spatial resolution. Simulating the quantum evolution in three-dimensions and time, we show that imprinting such wavefunctions on electron pulses leads to shape-preserving multi-electrons ultrashort pulses. Our scheme applies to any beams of charged particles, such as protons and ion beams.

frame this magnetic field effect can be seen as length contraction. However, this effect is very small in the case analyzed here, due to low electron velocities.
Next, we assume that all the electrons have the same wave function, as reflected by Eq. 1 in the main text. We also assume that the self-interaction (when j=i) is negligible, which is the case when the beam is very dense (N is a large number).
Proceeding to substitute the wavefunction from Eq. 1 and the potential form from the main text, we recover Eqs. 6, 7 there, which in cylindrical coordinates are: It is now convenient to define (0) = 0 2 � 2 ℏ 2 − 2 �, which simplifies Supplementary Eq. (2) to Eq. 4 from the main text: The normalization requirement is: The initial conditions for the nonlinear set of equations are: where is determined from the normalization requirement.
To find that satisfies Supplementary Eq. (5), we use an iterative algorithm. In the first iteration, the algorithm guesses the value of and finds the wavefunction using the 'ode45' matlab procedure (the algorithm is an explicit Runge-Kutta formula). From there on, in each iteration , we update the value of to +1 as hence the only free parameter in Supplementary Eqs. (3) and (4) is -which can vary between 0 and infinity. At the vicinity of = 0 , Supplementary Eq. (4) gives the Bessel equation, whose solution is ( ) = 1 ( ) + 2 ( ). However, ( ), is unphysical because it diverges at = 0. As such, we are left with the first term only, which is the reason why is the only remaining degree of freedom. 3 The electron density on the axis of an EBeam (as appears in Eq. 5 in the main text) can be derived from the current , and the acceleration voltage : As a side note, we would like to add that, while the above treatment is nonrelativistic, it can be directly extended to a fully relativistic quantum formalism. In case the propagation is limited to small angles (paraxial EBeams), the Schrödinger Equation only needs to be changed by multiplying the mass by the relativistic gamma, and decreasing the interaction terms by the same factor. In any case, the nonrelativistic equation above is a very good approximation for the parameters we simulate in the paper.

2.1: Beam propagation code for spatio-temporal electron pulses
To simulate the evolution of the wavefunction in time and 3D space, which includes charge distribution in full 3D, we use a modified version of the commonlyused Beam Propagation Method (BPM). We begin with Eq. 4 from the main text.
Then, we define the diffraction operator and the non-linear operator, as The equation of motion becomes: For a small propagation step in time, dt , the solution of Eq. (10) is Note, that the diffraction operator � is now diagonal in momentum space, while the non-linear operator � is diagonal in 3D real space. Therefore, we cannot evolve both operators together as diagonal matrices. We therefore use the Baker-Campbell-Hausdorff theorem: where, for small enough dt we get ( � + �) ≈ � � , hence . The evolution step in a time element dt is: where the potential is calculated by solving numerically Eq. 5 from the main text: The calculation is performed in momentum space, as follows: This procedure is used to simulate the evolution in time and 3D space of a pulsed electron beam whose charge distribution is in full 3D.

2.2: Beam propagation code for continuous wave electron beams
Similar to the evolution of the electron pulse in time and 3D space, we simulate the evolution of continuous wave (CW) electron beam in time and 2D space. Again, we use a modified version of BPM. We begin with Eq. 4 from the main text.
Note, that here the wavefunction evolves in time in a non-harmonic fashion (i.e., it does not evolve with − ). Rather, the evolution in time depends on the initial condition. Next, we transform (17) to cylindrical coordinates The equation of motion obtains the following form: Where, ⊥ 2 is the Laplacian in the transverse plane. Then, we define the diffraction operator and the non-linear operator, as For a small propagation step in time, dt Again the diffraction operator − ℏ 2 ⊥ 2 , in the momentum space, is of the , where ⊥ 2 is the transverse wavenumber, squared. The evolution step in a time element dt is: where the potential is calculated by solving numerically Eq. 5 from the main text: Where, is the electron density on the z axis of an EBeam (as appears in Supplementary Eq. 7). The calculation is performed in momentum space, as follows:

Supplementary Note 3: Neglecting the spin-spin and spinorbit interaction
This approach is along the lines of previous work addressing related questions (see, reference [4]) that showed that, in most standard EBeam conditions, the coulomb interaction dominates over any spin-related effect. Here, the energy of the spin-spin and spin-orbit interaction is as follows: In a similar vein, previous work that compared the spin-spin interaction and the coulomb repulsion led to similar conclusions (see ref. [4]): in most standard EBeam conditions the spin-spin interaction is negligible relative to the space charge effect.

Supplementary Note 4: Stability to modifications in the current and energy-broadening effects
In this section, we show in simulations that our non-diffracting wavefunction is stable under variations in the density in the EBeam. Namely, we show in simulations that our non-diffracting wavefunction is robust to modifications in the current and against energy broadening. In a physical setting, it would be expected that our nondiffracting wavefunction would be stable against energy broadening and would be able to adjust to modifications in the current.
In next figure, we actually examine the evolution of our non-diffracting wavefunction with a current different from the current it was designed for. We take the wavefunction that solves Supplementary Eqs.

Supplementary Note 5: Non-diffracting range and effective current vs. beam width, for different BSS
The following figure presents a quantitative comparison in the performance between our shape-preserving multi-electron wavefunction and multi-electron Bessel and Gaussian beams. Similar to the compassion that was made in Fig. 4 in the paper but with lower beam current of 500 (recall the total beam current in Fig. 4 in the paper is = 5 ). We present the results for two BSS: 140 (as in the example in We can see, in Supplementary figure 2, tendencies similar to the results presented in Fig. 4. However, the Critical width here is much larger, showing strong dependence on the BSS, while only weak dependence on the total current. Additionally, as the BSS is increased, the EBeam density decreases (because of total probability normalization) and the EBeam experiences less repulsion, resulting in larger non-diffracting range. This figure explains the larger non-diffracting range of the wide BSS when compare to the smaller BSS (see cyan vs. blue curves), but also leads to the lower current inside the main lobe (green vs. red). Further comparison is shown in Fig. 4 in the main text, which is the zoom-in of the area marked by the magenta rectangle.