Generation with phase‐and‐amplitude electron holograms of Laguerre‐Gauss beams with orbital angular momentum up to 200ħ

Phase‐and‐amplitude electron holograms provide a flexible way to encode an arbitrary wavefunction by modulating only the hologram phase [1] [2]. This is an innovative step in the direction of novel experiments with structured electron waves [3]. The most interesting application example is the generation of Laguerre‐Gaussian (LG) beams as they can be used to match exactly a single Landau state of electrons inside the objective lens of a transmission electron microscope (TEM). LG beams are a solution of the paraxial Schrödinger equation. They are mainly characterised by two indexes: l, the azimuthal index, representing the orbital angular momentum (OAM), and p, the radial index, where p+1 is the number of intensity radial nodes. They will be referred to with the contract notation LG(l,p). Landau states, on the other hand, are the quantized eigenstates of a charged particle with OAM in a magnetic field. Remarkably, LG beams have, at a given plane, the same form of quantized Landau states [4]. They only differ in the z evolution: while in a magnetic field Landau states are non‐diffractive, LG beams in vacuum expand with defocus but maintain the same intensity shape. We fabricated the holograms with Focused Ion Beam (FIB) on Si 3 N 4 membranes. As a first check of fabrication accuracy, the Energy Filtered‐TEM thickness map is taken. Thickness maps can be considered as a measure of the phase of the electron wavefunction after the hologram. In the first row of figure 1 thickness maps of the holograms LG(0,10), LG(10,0) and LG(10,10) are shown. The second row shows on top the experimentally acquired intensity at the Fraunhofer plane, and at the bottom the intensity and phase (represented by the colour hue) calculated with the software STEM_CELL starting from the thickness maps. The beams with l=10 show indeed the well‐known azimuthal phase ramp of vortex beams while the beams with p=10 show 11 intensity nodes in radial direction with alternating phase as prescribed for LG beams. This allows us to say that we were able to produce LG beams with arbitrary l, p indexes. If the LG beam was generated inside the magnetic field of the objective lens of a TEM, this would permit the visualization of exact Landau states. A more direct test of the LG character of these beams is obtained by visualising their shape invariance after propagation. To this aim, we investigated the propagation behaviour of the beam LG(10,0) with simulations, with w 0 is the beam waist and z R = π W 0 2 /λ is the Rayleigh range for electrons with wavelength λ. Simulations are shown in the first row of figure 2, reporting the beam intensity shape propagated over Δz distance. When propagating, the beam width increases but the circular intensity shape remains the same. This is not true in general for all vortex beams. In order to make a comparison, we generated a vortex beam with 10ħ OAM (named L=10). The second row of figure 2 shows the experimental intensity of the beam L=10, acquired at the Fraunhofer plane (Δz = 0), and with different defocuses. The external intensity ripples are due to the abrupt intensity profile of the hologram (which in this case was not made with the phase‐and‐amplitude technique) and have the same character at all defocuses. The shape of the vortex, which is the circle with maximum intensity, with increasing defocus develops some internal ripples: this beam is therefore not shape invariant after propagation. As a further instance of the improvements due to the phase‐and‐amplitude scheme with respect to the previous ones, in figure 3 a phase‐and‐amplitude LG(200,0) beam (a) and its radial profile (b) is compared with an ordinary hypergeometric‐Gauss beam with L=200ħ (c) and its profile (d). The striking feature is the transverse confinement of the intensity in the LG beam, with respect to L=200 that shows many external ripples. Generating beams with phase‐and‐amplitude holograms shows clear advantages, from the suppression of the unwanted beam intensity ripples to the control of radial and azimuthal degrees of freedom of LG beams, and is of great importance in order to generate shape‐invariant LG beams that can match exactly a single Landau state.

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