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Electron Bessel Beam
The electron Bessel beam is a solution of Schroedinger equation for electrons.

Electron Bessel beams are among the most interesting beams for theory developement and for experiments. Unfortunately Bessel beams are not $$L^2$$ normalizable so only approximations can be produced in a laboratory. Bessel beams are characterized by two quantum numbers $$K_\rho$$, \ell. The second number \ell is related to the Orbita Angular Momentum since Bessel beam are a particular cathegory of Electron vortex beam.

Derivation


 * $$\frac{1}{\rho}\frac{\partial}{\partial{\rho}}\Big(\rho\frac{\partial\Psi}{\partial{\rho}}\Big)+\frac{1}{\rho^{2}}\frac{\partial^{2}\Psi}{\partial\varphi^{2}}+\frac{\partial^{2}\Psi}{\partial{z}^{2}}+\Big(E-\frac{2mc}{\rho}\Big)=0$$

It is obtained by variable separation
 * $$\Psi(\rho,\phi,z)=R(\rho)\Phi(\phi)Z(z).$$

Obtaining an equation for the radial part as
 * $$\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\frac{\rho\partial R}{\partial\rho}\right)+{\ell}^2\frac{1}{\rho^2}R(\rho)=(\rho^2(K^2-K_z^2)+\ell^2)R(\rho)$$

From this it follows
 * $$R(\rho)=J_{|\ell|}(K_\rho \rho)$$

Therefore
 * $$\Psi(\rho,\phi,z)=J_{|\ell|}(K_\rho \rho)e^{iK_z+il\phi}$$

Where
 * $$K_\rho=\sqrt{K^2-K_z^2}$$

Notice that the radial number $$K_\rho$$ does not depend on the OAM number $$\ell$$.

Relativistic version

Whereas a relativistic extension can be derived for the Bessel wave based on the Dirac equation the process of genralisation is not univocal. The most important characteristic is that Bessel beams are eigenstates of the OAM that is not conserved in the Dirac equation. Conversely the Dirac equation conserves the total angular momentum
 * $$\mathbf{J} = \mathbf{L} + \mathbf{S}$$

and in particular
 * $$\mathbf{J_z} = \mathbf{L_z} + \mathbf{S_z}$$

In particular a beam eigenstate of OAM e.g. L=0 should slowly oscillate to another state L=+/-1 with and associated spin variation. However in a paraxial and non-relativistic approximation this oscillation is negligible. The main parameter governing the transition is $$\Delta=\sqrt{1-\frac{m}{E}}sin^2(\theta)$$

Properties

The main properties of Bessel beams in general vali also for electrons is property of being diffraction invariant and self healing.

Generation

Bessel beams are also generated in electron microscopy with hollow cone illumination and electron holograms. The general form of the electron hologram in polar coordinates is
 * $$\Phi(\rho,\theta,z)=a\rho sin(\theta)+b\rho +\ell \theta $$

Applications

Bessel beams have an application in electron microscopy for strain measurement, for spectroscopy for dynamic diffraction and tomography.