## 1 Introduction
Figure 1: Measurement setup for the problem to identify GNP-related dispersion parameters and plasmonic resonances. Figure 2: Hollow waveguide with a dielectric layer. The dashed curve illustrates qualitatively the effective relative permittivity \(\varepsilon_{r}\) as it changes from \(\varepsilon_{\text{G}}\) to \(\varepsilon_{\text{L}}\) and back. The general exact analytical fomulas for transmission, reflection, and absorption coefficients in waveguide settings with inserted GNP-substrates are known from our previous work [20; 21]. These general fomulas are further developed here to be readily applicable to the calculation of absorption coefficient within the inserted lossy layer only, quantifying the absorption in the GNP-fed cancer tissue. In doing so, a theoretical scaling factor is added as an extension to the theory developed in [20]. The scaling factor is valuable for both the analytical solution and the numerical results when one wants to investigate losses within the thin layer only and eliminate the smaller and unessential absorption in the surrounding lossy medium. This is particularly handy for numerical methods that use scattering parameters, where distinguishing between absorption regions is difficult when the ambient medium is lossy. Although the general analytical results are known, it is of interest to study these analytical solutions in more detail and to compare them to state-of-the-art numerical simulations to confirm their usefulness and prediction power. We therefore compare the numerical simulation results for power transmission, reflection and absorption coefficients, obtained using COMSOL Multiphysics software, with the corresponding results obtained from the modified exact analytic fomulas. The study includes both a simple example of constant complex permittivities, and a more realistic example where a dispersive model of permittivity is used to describe human tissue and the electrophoretic motion of charged gold nanoparticles. Thus, in this paper, we developed a numerical approach to the graded waveguide model using COMSOL Multiphysics, a modeling software based on the finite element method. The numerical examples in previous work on graded waveguides [20; 21] used constant permittivities with respect to frequency, and their magnitudes were chosen with no particular application in mind. In [18; 19], material values were chosen based on metal particles immersed in saline water, and with no dispersion. In the present paper, we also include a more realistic numerical example based on dispersive permittivity describing actual human tissue, as well as the dispersive permittivity describing the thin layer with a concentration of GNPs absorbed in such a tissue where their electrophoretic particle acceleration is described by the Drude model. Due to the similar mathematical notation and descriptions, it is important to particularly emphasize the novelty of the present work, as compared to our previous publications [20; 21; 22]. In [20], the general theory of TE-wave propagation in the waveguide setting was developed for the first time. It constitutes an important mathematical basis for the present work. However, the work reported in [20] did not include any energy considerations such as, e.g., energy transmission, reflection and absorption, and did not consider the problem of extracting the energy absorption in the GNP-fed thin layer from the absorption in the rest of the infinite waveguide. It does not consider the more realistic finite waveguide either. Furthermore, no particular dispersion models for the GNP-fed target layer and the surrounding tissue were considered, and there were no attempts to numerical verification. These issues are essential for any future simulations and measurements. The work reported in [21] addressed some of these issues. It generalized the results reported in [20] to both transverse electric (TE) and transverse magnetic (TM) polarization, and included a first attempt to provide energy considerations.