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We propose an all-optical method of creating electron vortices utilizing the Kapitza-Dirac effect. This technique uses the transfer of orbital angular momentum from photons to free electrons creating electron vortex beams in the process. The laser intensities needed for this experiment can be obtained with available pulsed lasers and the resulting electron beams carrying orbital angular momentum will be particularly useful in the study of magnetic materials and chiral plasmonic structures in ultrafast electron microscopy.
© 2015 Optical Society of America
1. Introduction
Although classically forbidden, quantum mechanics predicts that free electrons, along with other massive particles can carry orbital angular momentum (OAM) [1]. Electrons carrying OAM have been demonstrated experimentally [2–4], and are often referred to as vortex beams. To experimentally realize electron vortex beams several methods have been used, which include passing electrons through nanofabricated material holographic gratings [2,3], through a naturally occurring graphite phase plate [4] or by scattering from magnetic whiskers [5]. Other proposed methods for creating vortex beams include utilizing magnetic lens aberrations [6]. These electron vortex beams can be used for measuring the magnetic response of nanomaterials [3], manipulating nanoparticles and measuring dichroism in plasmonic structures [7].
Here, we propose the generation of electron vortex beams by Kapitza-Dirac (KD) scattering [8] from photons carrying orbital angular momentum (OAM). In 1933, Kapitza and Dirac suggested that an electron beam sent through a standing light wave would experience scattering and the standing wave would behave similar to a material grating. The diffraction of electrons by a standing wave of light, which came to be known as the KD effect, was not experimentally demonstrated until 2001 [9].
To observe KD scattering, a standing optical wave is created by having two counterpropagating electromagnetic waves interfere. In the overlap region, a standing wave forms and after time averaging regions of high and low intensity are arranged in alternating lines that resemble a diffraction grating. We propose replacing the optical plane waves in the counterpropagating beams that form the standing wave with optical vortex beams, otherwise known as Laguerre-Gaussian (LG) beams. LG optical beams are created by passing an optical beam through a spiral phase plate [10], an amplitude grating with an appropriate hologram [11] or by reflecting the optical beam from a spatial light modulator with an appropriate phase hologram [12]. The hologram used to create an optical beam with OAM is a diffraction grating with a fork dislocation at its center. When the optical beam diffracts from this grating, the diffracted orders carry OAM. When two of these OAM optical beams replace the plane wave beams, the resulting standing wave formed in the overlap region contains a fork dislocation in its intensity distribution that looks similar to material holographic gratings used to generate both optical and electron vortices [2,11]. Just as in the typical KD effect the forked optical standing wave will cause the electron beam to diffract. However, in addition to the transfer of linear momentum from the photons to the electron, quantized amounts of OAM will also be transferred. The observation of electron vortices in the proposed experiment would be a demonstration of the transfer from photons to electrons of orbital angular momentum and the carrying of orbital angular momentum by individual electrons. It would also present new way of creating electron vortices without the need for a nanometer scale material diffraction grating for the electrons.
2. ‘Particle picture’ description of OAM transfer from photons to electron
The formation of electron vortices through OAM transfer from photons to electrons is supported through two separate descriptions of the KD effect. An intuitive way to describe the KD effect (often referred to as the ‘particle picture’ [13]) is to describe an electron as a particle that is scattering from two counterpropagating photons. In this description an electron first absorbs a photon from one of the counterpropagating light beams, after which 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(a)]. An important aspect of this interaction is that it simultaneously conserves both energy and linear momentum. When these transverse momentum kicks of ± 2nħk are propagated to the far field an electron diffraction pattern is observed.
Fig. 1 Cartoon of the conservation of linear and orbital angular momentum in Kapitza-Dirac scattering. The left figure shows the before and after for the scattering of two photons by an electron. Linear momentum ( ± ħk) is conserved for this standard KD scattering event, by transferring + 2ħk of momentum from the photons to the electrons. On the right hand figure the photons are carrying ± ħ of OAM, in addition to linear momentum. To conserve OAM, in the scattering event, + 2ħ of OAM must be transferred to the electron from the photons. In both scattering situations energy is conserved.
Like both linear momentum and energy, orbital angular momentum must also be conserved in the electron/photon interaction. The result is that quantized amounts of OAM can be exchanged between free electrons and photons in the KD effect. 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(b)]. The particle picture describing this interaction can be more rigorously described using adding/lowering operators and utilizing the second-quantization procedure (as described in [13]). Here for simplicity we choose to rely on a ‘wave picture’ to describe the interaction and predict the resulting diffraction patterns.
3. ‘Wave picture’ description of OAM transfer from photons to electrons
In an alternate description electrons are modeled as waves which pass through an optical standing wave produced by the counterpropagating laser pulses. The interaction of the electron wave with an intense light field can be described using the ponderomotive potential, Up. The cycle averaged ponderomotive potential is related to the optical intensity of the standing wave through [14],
where e is the elementary electric charge, λ is the wavelength of the light, me is the mass of an electron, ε0 is the permittivity of free space and I(r,t) is the space and time dependent intensity of light. An electron passing through the ponderomotive potential undergoes a phase shift [14],where t is the interaction time of the electron with the potential. Electrons interacting with regions of high intensity such as at the crests undergo large phase shifts while electrons in regions of low intensity, such as the troughs undergo minimal phase shifts. When two optical beams, each carrying ± lħ of OAM interfere they produce a standing wave that has an interference pattern that produces a spatial intensity pattern with a fork type dislocation in its center. Electrons that interact with this optical interference pattern obtain a spatially dependent phase shift that gives rise to a diffraction pattern in the far field. Depending on the diffraction order n, and the OAM of optical beam l, each diffracted electron beam has an orbital angular momentum of,transferred to it. Thus both the ‘wave’ and ‘particle’ pictures of the KD effect with OAM optical beams produce the same result of transferring quantized amounts of OAM from photons to free electrons.4. Results and discussion
To predict the resulting diffraction patterns of an electron beam undergoing diffraction from the optical standing wave described above, we use a path integral method [15] to describe the electrons propagation from a field emission tip, through the collimation slit, including its interaction with the ponderomotive potential (standing optical wave) and then its propagation to the detection screen. This path integral method relies on the calculation of the phase accumulated by the electron wave along the possible classical paths, which is represented by a Kernel, (see Eq. (6) in [15]). By summing up all the phases from the different paths the initial wavefunction, ψi can be propagated to a new positon by multiply it with the Kernel and integrating over the source dimensions (see Eq. (5) in [15]), resulting in the new wavefunction, ψf. This method while computationally intensive does not require solving a differential equation and has been shown to be very accurate in calculating low energy electron diffraction patterns [15]. All of our calculations were carried out in Matlab. The model we use assumes a field emission tip as the source of the electrons, which is held at a negative voltage. This negative voltage accelerates the electrons and determines the kinetic energy of the resulting electron pulse, KEe = -eV.
For all results presented an energy of 250 eV was used for the electrons, which is a reasonable energy for a field emission tip source [16]. The electron source is assumed to be fully coherent, which is a standard approximation [16]. Field emission tip sources can be triggered with femtosecond laser pulses [17–19], which would allow the electrons to arrive at the same time as the laser pulses that would create the standing optical wave, making them ideally suited for the proposed experiment. After emission from the tip, the electron is propagated a distance d = 0.25 m to a circular pinhole of diameter 10µm. This pinhole has two uses: the first is that it acts as an alignment tool for the spatial overlap of the two laser beams that will create the standing wave and second is to act as a collimation slit for the electron beam.
To model the standing light wave produced by the interference of two LG beams we start with the simplified expression for the electric field for a linearly polarized LG electromagnetic wave [20],
where w is the waist of the focus, l is an integer. We calculate the interference pattern of two beams with an angle of ± 45° with respect to the electron propagation direction (see Fig. 2), on a plane perpendicular to the electron beam propagation. The interference between a beam of + l and –l produces a pattern that includes a fork dislocation at its center (see Fig. 3). It should be noted that a circular region of reduced intensity is found in the middle of the interference pattern, which is a direct consequence of the zero electric field in the center of each of the LG beams, Eq. (3).Fig. 2 Schematic of proposed experimental setup. An ultrashort electron pulse is created by irradiating a field emission tip with a femtosecond pulse of light. This electron pulse travels a distance d to a collimation pinhole, where two intense femtosecond laser pulses are coincident with it. The laser pulses, carrying -lħ and + lħ amounts of OAM respectively interfere creating an optical standing wave with a fork dislocation at its center. The ultrashort electron pulse passes through the collimation slit, diffracts from the standing wave and travels a distance D to the two dimensional detector plane. The different diffraction orders carry 2nlħ of OAM, according to Eq. (3).
Fig. 3 Calculation results showing predicted electron beams carrying OAM. a) Optical intensity patterns of two l = 0 optical beams. The circle is 10 microns in diameter. b) Electron diffraction pattern on detector after interaction with standing optical wave. The energy of each of the laser pulses that create the standing wave were 3 µJ. c) Simulation of the electron diffraction pattern after interfering with the plane wave electron beam (phase φ = 0). The reason this produces the spiral pattern is due to the spherical wavefronts of the diffraction peaks, which is ultimately due to the point like source of the field emission tip. The central intensity in the zeroth order only depends on the relative phase factor between the electron wave and plane wave. d) Optical intensity patterns of an l = −1 and l = + 1 optical beams. The circle is 10 microns in diameter. e) Electron diffraction pattern on detector after interaction with standing optical wave. The energy of the laser pulses that create the standing wave were 5 µJ per pulse. f) Simulation of the electron diffraction pattern after interfering with plane wave electron beam. The first order diffraction peaks, after interference with the plane wave show two spiral arms. This indicates that the first orders carry ± 2ħ of OAM. g) Optical intensity patterns of an l = −2 and l = + 2 optical beams. The circle is 10 microns in diameter. h) Electron diffraction pattern on detector after interaction with standing optical wave. The energy of the laser pulses that create the standing wave were 16 µJ. i) Simulation of the electron diffraction pattern after interfering with plane wave electron beam. The first order diffraction peaks, after interference with the plane wave show four spiral arms. On all images white indicates higher intensity of electrons or light respectively and the width of the patterns are 300 µm. The parameters for the laser pulses were all 300 fs, 515 nm and all were focused to a waist of 5 µm. The slight oval shapes and non-uniform intensity distributions of the diffracted beams shown in e and h are due to the slight oval shape in the standing wave intensity and the spherical front of the electron beams. The oval shape in the standing wave is caused by the angled interference of the laser beams and the spherical wavefront of the electron beam, due to the field emission tip source used in the calculation.
To calculate the interference pattern of the two beams with angles of ± 45° with respect to the electron propagation (z) direction Eq. (4) is first multiplied by exp(I(kz-ωt)-((z-ct)/zp)2), which describes the propagation of the electromagnetic wave and the envelope of the femtosecond laser pulse where zp = ctp, with tp being the duration of the laser pulse. The y-direction was chosen as vertical and the x-direction as horizontal and the substitution r = (x2 + y2)1/2 is made. To find the average intensity the electron experiences on the collimation pinhole plane (xy plane) the following substitutions were made, x = xccos(45°) and z = ± xcsin(45°), where xc is the x position on the xy plane at z = 0. The ± in the z substitution are for the left and right titled beams respectively. Finally, the two fields are added together and a numerical integration over t is performed. From this result the average intensity the electron would experience at the plane of the collimation pinhole over the duration of the laser pulse is calculated. Figures 3(a), 3(d) and 3 (g) show the results of these calculated intensities. The average intensity is then used to calculate the spatially dependent phase acquired by the electron (described by Eq. (1) and 2) at the collimation plane, with tp used as the electron interaction duration in Eq. (1). This calculation for the phase shift is valid as long as the electron does not travel a significant fraction of the depth of the standing wave and as long as the electron experiences many oscillations of the laser field [14]. Both of these conditions are met and discussed in more detail below.
After the electrons pass through the pinhole and interact with the standing optical wave through the ponderomotive potential, they are propagated to a detection plane which is a distance D = 0.25m from the pinhole. At the detection plane the wave function of the electron is found, from which we can calculate the probability distribution, Fig. 3. The electron beams that have diffracted from the overlapping LG optical waves clearly show the ‘donut’ intensity profile that would be expected of an electron beam carrying OAM. In addition to the probability distribution at the detector, we calculate what would result if the diffracted electron beams were interfered with a plane electron wave. Because the electrons that diffract from standing optical wave have a slightly spherical wave front (resulting from being emitted by a ‘point source’ field emission tip) the resulting interference pattern with a plane reference wave is representative of the phase of the diffracted electron beams, Fig. 3. The predicted phase of the diffracted beams is of the expected spiral form that would be associated with a wavefront containing a phase singularity at its center. The number of spiral arms in the pattern is equal to the amount of OAM the electron beams are carrying. As the OAM, l, of the optical beams is changed the amount transferred to the electron beams (see Fig. 3) follows directly as predicted by Eq. (3). The energy of each of the two laser pulses that interfere to create the standing wave is included in the Fig. 3 caption, and all are on the order of 10 µJ per pulse. The periodicity of the standing wave created by the interfering optical pulses is determined by the laser wavelength λ, and angle θ between them and takes the form: ∆xp = λ/2(1/sin(θ/2)) [21]. In addition the interaction time of the electrons with the laser standing wave is determined by the duration of the laser pulses tlaser.
To get a significant amount of diffraction from the standing wave the pulsed electron bunches must arrive when the two laser pulses interfere. This can be done by using a beam splitter to use a small amount of the laser pulse that creates the standing, to initiate the femtosecond laser pulse from the cathode. In addition, the electrons should not travel all the way through the standing wave during the time it is present. To estimate this we consider how far the 250 eV electrons travel in 300 fs (duration that the standing wave is present). This distance is ~3 microns and is less than the depth of the standing wave which would be on the order of the laser focus of ~10 microns. Commercially available femtosecond lasers can be obtained with ~300 fs, ~500 µJ pulses at 1030 nm (which can be easily frequency doubled to 515 nm). These lasers also have repetition rates in the tens of kilohertz to the megahertz range. Because the same laser that is used to make the standing wave would be used to trigger the pulsed electron source the high repetition rates assure a large number of electron counts per second. The spatial overlap of the electron and laser pulses will be facilitated with the collimation pinhole, while the temporal overlap of the two laser pulses can be achieved by letting them hit a field emission tip, placed in the position where the collimation pinhole will be. When the two laser pulses are temporally overlapped, the photoelectron emission from this tip will be modulated as one laser pulse is delayed with respect to the other.
To create a standing wave, a pulse is split in two, and both of them are sent into dispersion compensated optical vortex generators [22]. An efficiency of 20% can be expected after creation of the OAM pulse [22], so starting with a pulse of ~250 µJ, an OAM optical pulse of ~50 µJ can be obtained. When overlapped and focused to a spot of 20 µm diameter an intensity on the order of 1011 W/cm2 can be achieved, which would be sufficient to cause appreciable electron diffraction in the KD effect. Due to the decreased intensity in the center of the standing wave patterns when the ‘donut’ modes are interfered, which can be seen Figs. 3(d) and 3(g), the diffraction efficiency is not maximized. To remedy the decreased intensity in the center of the pattern [25], slightly more complicated setups could be used to create vortex beams that have increased central intensity [26,27]. By using these techniques the experimental realization of the proposed method to create electron vortex beams can be attempted using lower power lasers. With the use of a high repetition rate fs fiber laser, electron count rates (103-105 electrons per second) would be sufficient to acquire electron diffraction patterns on the order of tens of seconds [23]. To resolve the diffraction peaks on the electron detector they must be separated more than the detector resolution. Typical multichannelplate/phosphor screen electron detectors have a spatial resolution in the order of 50 microns. The periodicity of the standing wave can be determined using the before mentioned equation and an angle of 90° between the beams and a wavelength of 515 nm gives a periodicity of ∆xp = 364nm. Electrons at 250 eV have a wavelength of 77 picometers. Using the diffraction equation and the distance from the grating to the detection screen of 0.25 m, the diffraction peaks are separated by ~50 microns, which matches the diffraction patterns shown in Fig. 3. Since this is of the same order as the detector resolution, a magnetic or electric electron lens will need to be placed before the detector to magnify the pattern appropriately. These types of lenses are commonly used in electron diffraction setups [24] and in low energy electron interferometers [25], so their use in the proposed experiment should not pose a problem.
5. Conclusion
In conclusion, we present a method of creating electron vortices using only light, which utilizes the KD effect. While these vortex beams may be of practical use for material science, observation of the predicted effect would be fundamental demonstration of the quantized exchange of orbital angular momentum between free electrons and photons. These pulsed electron vortex beams also hold the promise of extending vortex beam electron microscopy to the ultrafast time domain by their use in an ultrafast electron microscope [17].
Acknowledgements
This work was supported by a Connecticut Space Grant and partially supported by a Trinity College Faculty Research Committee Grant.
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