# Extended Field Interactions in [PERSON]'s Equation Revision
## 3 Potential Applications
### Application of the Axial Vector and Its Divergence to Plasmonics  
To illustrate the potential use of equations for the axial vector and its divergence, let us consider an example in the field of plasmonics. The interaction of a twisted light beam with a metallic nanostructure, such as a metallic nanorod, will be analyzed to determine how the nanostructure influences the OAM properties of the incident electromagnetic wave. Initially, the axial vector \(\mathbf{\widetilde{\omega}}\), as defined in Equation (20), will be utilized. Following this, the expression for the divergence of \(\Phi(r)\mathbf{\widetilde{\omega}}\), presented in Equation (22), will be employed. The interaction between the twisted light beam and the metallic nanorod, as depicted in Figure 4, will be examined. This interaction modifies the amplitude \(\Phi(r)\) of the EM wave. In this case, we can calculate the new amplitude \(\Phi^{\prime}(r)\) after the interaction with the nanorod by solving [PERSON]'s equations or using numerical methods like finite-difference time-domain (FDTD) simulations. In this scenario, the new amplitude \(\Phi^{\prime}(r)\) following the interaction with the nanorod can be determined by solving [PERSON]'s equations or employing numerical methods such as finite-difference time-domain (FDTD) simulations. With the new amplitude \(\Phi^{\prime}(r)\), the new axial vector \(\vec{\mathbf{\omega}}^{\prime}\) can be computed, as well as its divergence. By comparing the divergence of \(\vec{\mathbf{\omega}}\) and \(\vec{\mathbf{\omega}}^{\prime}\), the impact of the metallic nanorod on the OAM properties of the incident EM wave can be analyzed. A simple model will be further developed, utilizing the axial vector and its divergence, to understand the impact of a metallic nanorod on the OAM properties of an incident electromagnetic wave. The analysis will focus on the changes in the axial vector and its divergence before and after interaction with the nanorod. Let us assume the following: The incident twisted light beam has a helical wavefront described by the wave function \(\Psi(r,\theta,z)=A(r,\theta,z)\exp(il\theta)\), where \(l\) is the azimuthal quantum number that determines the OAM. The metallic nanorod has a length \(L\) and a radius \(a\). It is positioned along the z-axis and interacts with the incident EM wave (see Figure 4). * Step 1: Calculating the initial axial vector and its divergence for the incident EM wave. The initial axial vector can be calculated using the equations provided for \(\vec{\mathbf{\omega}}\) and its divergence \(\mathbf{\
abla}\cdot(\Phi(r)\,\vec{\mathbf{\omega}})\) before the interaction with the nanorod. * Step 2: Defining the interaction between the twisted light beam and the metallic nanorod. For simplicity, it is assumed that the interaction between the twisted light beam and the nanorod can be represented by an interaction factor \(\alpha\). This factor depends on the size, shape, and material properties of the nanorod as well as the wavelength of the incident EM wave. The new amplitude after the interaction can be represented as \(\Phi^{\prime}(r)=\alpha\Phi(r)\). Figure 4: Illustration of the interaction between a twisted light beam and a metallic nanorod, demonstrating the principles of electron holography in generating electron beams with orbital angular momentum (OAM). The figure depicts the helical phase front of the electron beam, induced by the OAM, as it interacts with the nanorod, highlighting the potential for manipulation and analysis of the OAM properties in various applications, including electron microscopy and optical communications. * Step 3: Calculating the new axial vector and its divergence after the interaction. Using the new amplitude \(\Phi^{\prime}(r)\), the new axial vector \(\vec{\mathbf{\omega}}^{\prime}\) and its divergence \(\mathbf{\
abla}\cdot(\Phi^{\prime}(r)\ \vec{\mathbf{\omega}}^{\prime})\) after interaction with the nanorod can be calculated. * Step 4: Comparing the initial and final axial vectors and their divergences.