Theory of free electron vortices.

The recent creation of electron vortex beams and their first practical application motivates a better understanding of their properties. Here, we develop the theory of free electron vortices with quantized angular momentum, based on solutions of the Schrödinger equation for cylindrical boundary conditions. The principle of transformation of a plane wave into vortices with quantized angular momentum, their paraxial propagation through round magnetic lenses, and the effect of partial coherence are discussed.

Introduction

Electron vortices are free electrons carrying orbital angular momentum. They are characterized by a spiraling wavefront with a screw dislocation along the propagation axis. With the publication of two seminal papers on the creation of electron vortices in the electron microscope [1,2] the matter was transformed from a theoretical possibility [3] to reality. The first practical application was a filter for magnetic transitions in the ferromagnetic 3d metals, thus facilitating experiments in the field of energy loss magnetic chiral dichroism (EMCD) [4]. The potential of vortices is much wider, ranging from probing chiral structures to manipulation of nanoparticles, clusters and molecules, exploiting the magnetic interaction [5].

For a review of vortex applications in optics, see [6]. The theory of field vortices was first described in the 1970s [7] in the context of dislocations in sound waves; it took almost one decade before such vortices in the optical frequency range were produced [8]. It was then realized that optical vortices were related to angular momentum of light, which already Poynting in 1909 speculated about [9]. The two formerly rather separated fields of research began to merge. The first experimental demonstration of laser light carrying quantized orbital angular momentum came in 1992 [10].

In the context of orbital angular momentum optical vortices are similar to electron vortices. In a sense, the latter are simpler than optical ones because the electron is described by a scalar wave equation1 (the Schrödinger equation) whereas optical vortices are vector fields. But for unpolarized light the description of vortex modes can be simplified, and in the paraxial approximation they have in fact been described by a Schrödinger type equation [11]. One of the surprising and original features of electron vortices is that, according to their rotational component of probability current they carry also a magnetic moment, even for beams without spin polarization. This, together with the much smaller scale of electron vortices compared to light, makes them so attractive.

We present here a theory of electron vortices based on solutions of the Schrödinger equation. We assume cylindrical geometry, starting with the general case allowing external scalar and vector potentials. Then we concentrate on potential-free systems, having in mind vortices propagating freely in space.2 It turns out that the solutions fall into families with quantized angular momentum along the optic axis.

The Schrödinger equation in cylindrical geometry

We assume an electron propagating along the z axis of the microscope and defined within a circle of radius r (the aperture). There may or may not be a magnetic field present in the space where the free electron propagates. We look for the general solution of the Schrödinger equation. A natural coordinate system is cylindrical for this geometry. The Hamilton operator in SI units iswhere M is the electron mass, e its charge, and are the vector and scalar potential operators, and is the momentum operator. The first term can be written as With and (in Coulomb gauge) , so thatWe write the time-independent Schrödinger equation in cylindrical coordinates . In these coordinates the nabla operator reads Now that we have For the special case of a constant magnetic field along z and a vanishing electrostatic potential V=0 (which will be important as an approximation to magnetic lenses in a follow-up paper) we can conveniently use a gauge where , so The Schrödinger equation with Hamiltonian Eq. (1) readsSeparation of variablesinserted into Eq. (3) transforms the partial derivatives into simple derivativesThe separation of variables is consistent with the fact that the Hamiltonian of a rotationally invariant system (free space or a radially symmetric vector potential) commutes with and with ; as a consequence, the eigenfunctions of :and of :where m is integer because it must be in the azimuth due to periodic boundary conditions are also eigen functions of the Hamiltonian.

Electrons in field free space

We consider the case of an electron limited by a circular aperture. We also assume that no potentials exist. The case of non-vanishing magnetic flux will be considered elsewhere.

Boundary conditions

The boundary condition at the rim of an aperture of radius r centered at the origin isWe look for solutions inside the aperture consistent with Eq. (8) and match them with the trivial solution outside the aperture . Since we deal with free electrons there are no scalar or vector potentials. Expressing the energy E of the electron by the wave number of its plane wave solutionmultiplying Eq. (5) with r2, filling in solutions Eqs. (6) and (7), and rearranging terms we obtain the differential equationfor the radial part of the wave function. We have used the abbreviationThe substitution and yieldsThis is Bessel's differential equation. The solutions are Bessel functions of the first kind J and the second kind Y with index m. The Y are divergent at x=0 and can be excluded for round apertures centered on the optic axis. The radial solutionsare indexed according to quantum numbers k and m. The factor Z(z) depends on the quantum number k. The energy spectrum is given by Eq. (9) as Note that the energy is always positive because the particle is free. States are degenerate in the magnetic quantum number m. Note that without magnetic field m is integer.3 It is important to note that the parameter k is real since .

Special case

The special case k=0 gives a particular solution In order to avoid singularities at r=0 the exponent must be positive; it must also vanish at the rim of the beam defining aperture with radius which is impossible for . For m=0 the solutions read (going back to Eq. (12) for this special case) That is to say that k=0 does not allow solutions with , nor a constant intensity within the aperture That means that solutions inevitably have and . In the next section we discuss properties of beams with discrete angular momentum, including .

Solutions with quantized angular momentum

The general solution of the present problem for , Eq. (4), is It is convenient to classify the solutions according to the eigenvalue m of Eq. (7), aswith arbitrary coefficients . The reason is that is an eigenfunction of with eigenvalue . To see this, one may calculate the expectation value directly: the angular momentum of the electron is In cylindrical coordinates the angular momentum operator reads For solutions belonging to the J family, Eq. (14) we findThe boundary condition Eq. (8) reduces the allowed values of k to a discrete set where the are the zeros of the Bessel function J. For m=0 a frequent situation is given by a wave function that is constant within the round aperture and vanishes outside, where is the radial step function

Expanding into eigenfunctions of this geometry we obtainwith which by definition of the zeros vanishes at the rim . This gives a set of

As a demonstration example we assume an aperture of in Fig. 1; the first 10 coefficients give already a good approximation. This solution is of type J0; from Eq. (7), its angular momentum is .

Fig. 1

Plane wave filling a round aperture of . First 10 basis functions.

The relative deviations of k from k0 for a 200 kV incident beam are of the order of 10−10 which translates into 10−5 eV, so the splitting of the k levels is unobservable in practice. The set of different k creates the oscillations within the aperture, visible in Fig. 1. In combination with the set of k they propagate the solution into z direction.

For m=1, the solutions are Bessel functions J1. By analogy to the case m=0 we find the expansion for a wave with constant amplitude in the aperture in the family J, : The result is given in Fig. 2. Contrary to the m=0 case, we see here the amplitude dropping to zero at the origin despite the fact that we assumed a constant amplitude. Expansion to higher order will make the slope steeper, eventually leaving a ‘pinhole’ at the origin. This is a consequence of the central vortex line where the phase is undefined, also known as topological charge.

Fig. 2

Amplitude of a helical wave with m=1 filling a round aperture of . First 10 basis functions.

Focussed vortices

The solutions discussed in the previous section had constant amplitude over the aperture. Can we focus such extended beams? The case m=0 is trivial—it is exactly what happens when one illuminates an aperture in the electron microscope with a plane wave. An ideal lens will create an Airy disk in the diffraction (back focal) plane. The action of the lens is described by a Fourier transform.

We can perform such transforms for beams with any quantum number . According to the Fourier–Bessel transform of a function with azimuth angle , the Fourier transform of isWe may calculate the far field solution via Eq. (17), equivalent to the diffraction pattern produced by an ideal lens. The intensity is isotropic. The radial intensity profiles are shown in Fig. 3. Only the m=0 beam has a maximum at the center; it is the well known Airy disk. The vortex beams are characterized by a ringlike structure with a maximum position depending on the helicity m. Note that contrary to intuition, according to Plancheret's theorem, the total intensity is equal for each m provided the are equally normalized. The helicity of a beam can then be determined by measuring the intensity as a function of momentum . This is the basis for further analysis of vortex beams. It should be noted that the focussed vortex has the same angular momentum as the original one since the azimuthal part is still of type . More generally, after passage through the mask, the electron remains in the same eigenstate of the angular momentum operator because the angular momentum is a constant of motion in free space.

Fig. 3

Radial intensity distribution scaled to the maximum value in the back focal (i.e. diffraction) plane of a lens for vortex waves filling a round aperture of diameter, with m=0 (full line) m=1 (dashed), m=2 (dash-dotted), and m=3 (dotted). C=0, 200 kV. Abscissa: scattering angle in . Zero denotes the vortex centers.

Arbitrary wave forms

Any wave function can be expanded into the eigenfunctions of the cylindrical geometry described here, Its FT isFor amplitudes f(r) defined within a finite aperture one may use a series expansion similar to Eq. (16). with Evaluating the transform Eq. (17) at positions , one can apply the orthogonality relations for Bessel functions. The result isas can be verified by inserting the c. Eventually,Note that each m has different zeroes .

Creation of vortex electrons

Vortex electrons can be created with spiral phase plates [1] or with holographic masks [2]. We illustrate the working principle underlying the creation of electron vortices, a plane wave passing through a holographic mask. Since the subject of this paper is the theory of electron vortices and not any technical aspect we restrict the discussion to the elementary form of a fork dislocation in the following.4

A plane wave is by definition infinitely extended and can therefore not be a solution of the present case. We ask for solutions that match a plane wave within the aperture and vanish outside. I.e. we simulate a plane wave impinging onto a round aperture with absorptive structure given by the transmission functionwhere is the azimuthal angle in the aperture plane, and with the lattice distance d. The exit wave function iswhere k is the wave number of the fast electron that propagates in the z direction. Matching this wave function to the boundary conditions of cylindrical geometry means a decomposition of into eigenfunctions of a round aperture. We have already encountered the first term in brackets—expression 16 for a plane wave. The other terms belong to the family of vortex solutions with m=±1,The phase factor describes propagation of the partial waves under the angle with respect to the optic axis, in direction k, caused by diffraction on the periodic mask.5 The transmission function of the holographic mask is shown in Fig. 4.

Fig. 4

Transmission function Eq. (21) for creating vortex beams. The aperture has diametre. Scale in .

The diffraction pattern of the holographic mask is shown in Fig. 5. The phase of the wave function is coded as hue, the range corresponding to colours continuously changing from blue to red (rainbow chart). The central Airy disk and its first two concentric rings are well visible. The m=±1 side bands show the characteristic volcano shape of vortices.6 Whereas the central Airy disk has a cylinder symmetric phase, the vortices show azimuthal phase variation over . Compare also the radial phase jumps at the central pattern (blue to red to blue) with the more gradual phase variations at the vortices. A trace through the diffraction pattern is shown in Fig. 6. Here the typical volcano profile of the side bands is clearly visible.

Fig. 5

Diffraction pattern of the mask in Fig. 4 obtained by Fourier transform, showing the central Airy disk and two focussed vortices. The Bragg angle at 200 kV is . The phase is coded as hue. The helical structure of the vortices is well visible (Color coding: Rainbow chart from 0 to ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Horizontal trace through the diffraction pattern shows the volcano-like profiles of the vortices. Scale in .

Spherical aberration

The focussed vortices of Fig. 5 with wave functions given by Eq. (17) are in the far field, or – equivalently – in the diffraction (back focal) plane of an ideal lens. Spherical aberration and defocus distort this ideal wave function via the wave transfer function W describing the wave front aberrations of the lens as This can be written as a product of the Fourier transformed functions with the wave front aberration and the aperture defining function : Note that is the variable after Fourier transform from the q plane. The beam defining aperture appears under a half angle (i.e. the half convergence angle seen from a point in the q plane). Taking into account only defocus df and spherical aberration C the phase imposed by the lens is radially symmetric, e.g. [13], Note that the scattering angle relates to the coordinate in the aperture as for a lens of focal length f (provided that ). Eq. (17) for the perfect lens is now replaced bywhich we evaluate at the Scherzer defocus . The azimuth angle is now in the diffraction plane where the vortex is focussed. In Figs. 7–9 are shown the radial traces through the intensity patterns for different aperture/mask diameters. For an ideal lens the profiles would be identical with the profile over the right or left vortex in Fig. 5. The angle is measured with respect to the center of the vortex (which is at in Fig. 5).

The maxima of the vortices move to higher radii with increasing aberration. For the largest mask the change is well visible; less for the mask, and negligible for the small mask. The side maxima in the profiles increase with m and with the aberration constant. Interestingly, the central zero for the vortices with remains always zero independent of the aberration. The vortex diameter scales with the reciprocal aperture size. Even for the smallest mask it is in the range of . Such structures are difficult to observe.7

Incoherent illumination

In reality, a perfectly coherent probe does not exist. Rather we have illumination with extended sources, which reduces the spatial (lateral) coherence of the beam. The grade of coherence is given by the angle subtended by the source on the aperture, i.e. the convergence angle. An exact treatment of partial coherence would involve calculation of the density matrix of the electron. Although only diagonal terms of this matrix are observable, the off-diagonal terms which are closely related to the mixed dynamic form factor [14,15] may lead to visible effects when the electron interacts with matter [16-18]. The incoherent superposition of waves coming from different points in the source reduces the size of the coherence patch on the aperture. This gives rise to a broadening of the diffraction spots. The resulting vortex profiles are calculated from a convolution of Eq. (17) with the incoherent intensity distribution of the source, projected on the diffraction plane. We assume here a Gaussian source shape With the charge density in the back focal plane from a perfectly coherent (point) source which is a function of radial distance q only, we calculate the convolutionas The Gaussian can be separated into Only the last factor depends on the azimuth, and the integral over the variable is with the modified Bessel function of first kind and order zero, I0. So, the convolution can be written as

Fig. 10 shows the effect of partial coherence on the m=1 vortex. The central dip rises rapidly with increasing angular width of the source subtended on the aperture. This constitutes a very sensitive monitor of coherence. On the other hand, it renders the creation of pure vortices difficult. Very small sources are needed, the critical source size being inversely proportional to the mask diameter. For a mask, the angular width of the source should not exceed in order to see a drop of the central intensity to 50%.

Fig. 10

Effect of a Gaussian source distribution on the m=1 vortex. Radial intensity profile for a mask with diameter. All intensities scaled to 1. The abscissa is relative to the vortex center. The parameter is of the source subtended on the aperture: (full line), 0.05 (dashed), 0.10 (dash-dotted), 0.15 (dotted) .

Conclusions

We have developed the theory of electrons carrying quantized orbital angular momentum. To make connection to realistic situations, we considered a plane wave moving along the optic axis of a lens system, intercepted by a round, centered aperture.8 It turns out that the movement along the optic axis can be separated off; the reduced Schrödinger equation operating in the plane of the aperture can be mapped onto Bessel's differential equation. The ensuing eigenfunctions fall into families with discrete orbital angular momentum along the optic axis where m is a magnetic quantum number. Those vortices can be produced by matching a plane wave after passage through a holographic mask with a fork dislocation to the eigenfunctions of the cylindrical problem. Vortices can be focussed by magnetic lenses into volcano-like charge distributions with very narrow angular divergence, resembling loop currents in the diffraction plane. Inclusion of spherical aberration changes the ringlike shape but does not destroy the central zero intensity of vortices with . Partial coherence of the incident wave leads to a rise of the central intensity minimum. It is shown that a very small source angle (i.e. a very high coherence) is necessary so as to keep the volcano structure intact. Their small angular width in the far field may allow the creation of nm-sized or smaller electron vortices but the demand for extremely high coherence of the source poses a serious difficulty.