Creating electron vortex beams with light

Recently, experiments have shown that free electrons can carry orbital angular momentum (OAM), by passing electrons through nanofabricated diffraction holograms [1], [2]. While electrons are classically forbidden from carrying OAM (without being in the presence of external fields), quantum mechanics does allow free electrons, as well as other massive particles to carry OAM. To date OAM beams of electrons have been produced by diffracting them from nanofabricated material gratings. We propose an alternative method using the Kapitza-Dirac (KD) effect to transfer OAM directly from photons to free electrons [3]. In the KD effect, both linear momentum and energy are conserved in the electron/photon interaction. In this interaction orbital angular momentum must also be conserved, resulting in the transfer of quantized amounts of OAM from photons to free electrons. We show that this experiment can be completed using a low energy ultrafast electron microscope and current femtosecond lasers. The resulting electron pulses carrying OAM will have femtosecond durations and could be used to study a variety of systems include the ultrafast behavior of chiral plasmonic nanostructures.

Recently, experiments have shown that free electrons can carry orbital angular momentum (OAM), by passing electrons through nanofabricated diffraction holograms [1], [2].While electrons are classically forbidden from carrying OAM (without being in the presence of external fields), quantum mechanics does allow free electrons, as well as other massive particles to carry OAM.To date OAM beams of electrons have been produced by diffracting them from nanofabricated material gratings.We propose an alternative method using the Kapitza-Dirac (KD) effect to transfer OAM directly from photons to free electrons [3].In the KD effect, both linear momentum and energy are conserved in the electron/photon interaction.In this interaction orbital angular momentum must also be conserved, resulting in the transfer of quantized amounts of OAM from photons to free electrons.We show that this experiment can be completed using a low energy ultrafast electron microscope and current femtosecond lasers.The resulting electron pulses carrying OAM will have femtosecond durations and could be used to study a variety of systems include the ultrafast behavior of chiral plasmonic nanostructures.The net change in the momentum of the electron is equal to 2ħk.Right) The exchange of linear momentum in this situation is the same as the standard KD effect, however in this case one photon also has a non-zero OAM of nħ.After the scattering event the electron has gained the OAM from the photon.Both the standard and OAM KD effect conserve linear momentum, orbital angular momentum and energy.
The Kapitza-Dirac (KD) effect is the diffraction of electrons from a standing wave of light.It can be described in several ways, but the simplest is to think of it as the interaction of a free electron with two counterpropagating photons.In the interaction an electron first absorbs a photon from one of the counterpropagating light beams, and then a second photon from the other direction causes the electron to undergo stimulated emission.After the interaction there are two photons traveling in the same direction and the electron beam has been given a 2ħk transverse momentum kick in the direction opposite to the motion of the two photons (see Fig. 1).An important aspect of this interaction is that it simultaneously conserves both momentum and energy.When these momentum kicks of ±nħk are propagated to the far field, an electron diffraction pattern is observed.
If OAM is added to one of the counterpropagating light beams, conservation of OAM must also be considered.For OAM to be conserved through the stimulated emission process, the OAM originally carried by the photon must be transferred to the electron (see Fig. 1).To study this effect in more detail we used a quantum path integral model to predict the resulting diffraction patterns in an experimentally realistic setup.We model the ponderomotive potential created by interfering a laser beam with OAM=0 with another laser beam that has an of OAM=nħ.In the path integral model the wave function of an electron is propagated from a field emission tip to the overlapping laser beams, which creates the ponderomotive potential.The electron wave function then acquires a spatially dependent phase shift as it travels through the ponderomotive potential and finally the wave function is propagated to the far field where an electron detector would be placed.The model predicts that the far field diffraction pattern exhibits the 'donut' shaped peaks indicative of a beam carrying OAM.In addition, the model also predicts the helical phase fronts that would be present on each of the separate diffraction orders.
This method of creating electron OAM beams does not require the nanofabrication of material gratings, results in OAM electron pulses with femtosecond durations which are particularly suitable for use in ultrafast electron microscopes.In addition, by simply changing the photon OAM in the laser beams the electron beams can be given arbitrary amounts of OAM.These tailored OAM electron pulses could then be used to follow the ultrafast dynamics of chiral plasmonic structures and other ultrafast phenomena that involves OAM.

Figure 1 .
Figure 1.Particle picture for Kapitza-Dirac effect with and without OAM.Left) This figure shows the momentum before and after the scattering event.The net change in the momentum of the electron is equal to 2ħk.Right) The exchange of linear momentum in this situation is the same as the standard KD effect, however in this case one photon also has a non-zero OAM of nħ.After the scattering event the electron has gained the OAM from the photon.Both the standard and OAM KD effect conserve linear momentum, orbital angular momentum and energy.