# Extended Field Interactions in [PERSON]'s Equation Revision
## 3 Potential Applications
### Generation of Electron Beams Carrying Orbital Angular Momentum  
The generation of electron beams carrying orbital angular momentum (OAM) [53; 54; 55; 56; 57; 58; 59] refers to the creation of electron beams that possess a well-defined helical phase front, similar to a spiral staircase. This helical structure of the electron beam arises due to the presence of OAM, which is a property of a wave that describes the rotation of the wave's phase around its axis of propagation. One way to generate electron beams carrying OAM is by using electron holography, which is a technique that allows for the measurement and manipulation of electron wavefronts [62; 63; 64; 65; 66]. In electron holography, a beam of electrons is split into two paths, with one path being used as a reference wave and the other passing through a sample. The two beams are then recombined, resulting in an interference pattern that contains information about the electron wavefront. By manipulating the electron wavefront using holography, it is possible to introduce a helical phase shift into the electron beam, thereby generating an electron beam carrying OAM. Such electron beams have been used in a variety of applications, such as electron microscopy and nanofabrication, and have the potential to revolutionize fields such as quantum computing and communication. This formalism's application to the issue will be illustrated. The phase shift associated with an electron beam carrying OAM can be described mathematically using the following equation:  
\[\Psi(r,\theta,z)=A(r,\theta,z)\exp(il\theta), \tag{19}\]  
where \(\Psi\) is the electron wavefunction, \(A\) is the amplitude of the wavefunction, \(r\), \(\theta\), and \(z\) are the cylindrical coordinates, and \(l\) is the azimuthal quantum number that determines the amount of OAM carried by the electron beam. The total angular momentum \(J\) of an electron beam carrying OAM can be expressed as \(J=\hbar l\), where \(\hbar\) is the reduced Planck constant. Observing that the axial vector \(\vec{\omega}\) possesses a dimensionality of \(1/L\), it prompts the proposal that the axial vector can be articulated as a product of a spherical harmonic function \(Y_{lm}(\theta,\phi)\) and a radial unit vector \(\mathbf{r}/r\). Based on this postulate, expressions for the divergence of the vector \(\Phi(r)\)\(\vec{\omega}\) are derived using the properties of spherical harmonics. Specifically, it uses the expressions for the divergence of \(\Phi(r)\mathbf{Y}_{l,l+1,m}(\theta,\phi)\), \(\Phi(r)\mathbf{Y}_{l,l,m}(\theta,\phi)\), and \(\Phi(r)\mathbf{Y}_{l,l-1,m}(\theta,\phi)\), which are derived using the differential operators for spherical coordinates. These expressions involve derivatives of the radial function \(\Phi(r)\) and the spherical harmonic functions, and they provide a way to relate the divergence of \(\Phi(r)\)\(\vec{\omega}\) to the function \(\Phi(r)\) and its derivatives (see also Ref. [70] with further information on the properties and use of spherical harmonics in physics; and Refs. [71; 72; 73] for applications). Hence, the Axial vector \(\vec{\omega}\) is defined by  
\[\vec{\omega}=\frac{\mathbf{r}}{r}Y_{lm}(\theta,\phi). \tag{20}\]A more extended expression for \(\mathbf{\widetilde{\omega}}\) in terms of vector spherical harmonics is  
\[\mathbf{\widetilde{\omega}}=-\bigg{[}\frac{l+1}{2l+1}\bigg{]}^{1/2}\mathbf{Y}l,l+1,m +\bigg{[}\frac{l}{2l+1}\bigg{]}^{1/2}\mathbf{Y}l,l-1,m \tag{21}\]  
and for the divergence of \(\Phi(r)\mathbf{\widetilde{\omega}}\), it is obtained that  
\[ \begin{split}\mathbf{\
abla}\cdot[\Phi(r)\mathbf{\widetilde{\omega}}]=- \bigg{[}\frac{l+1}{2l+1}\bigg{]}^{1/2}\mathbf{\
abla}\cdot[\Phi(r)Y_{l,l-1,m}( \theta,\phi)]\\ +\bigg{[}\frac{l}{2l+1}\bigg{]}^{1/2}\mathbf{\
abla}\cdot[\Phi(r) \mathbf{Y}_{l,l-1,m}(\theta,\phi)].\end{split}  \tag{22}\]  
Since the equations discussed thus far are based on the mathematics of vector spherical harmonics and the characteristics of angular momentum, they offer an effective foundation for explaining helical waves and OAM in electron beams.