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In this chapter, the problems concerning with the transmission line model and performance are presented. The subjects include transmission line models, transmission line voltage regulation, transmission line compensation, and features of transmission matrix. In this chapter, the problems are categorized in different levels based on their difficulty levels (easy, normal, and hard) and calculation amounts (small, normal, and large). Additionally, the problems are ordered from the easiest problem with the smallest computations to the most difficult problems with the largest calculations.

In this chapter, criteria for existence of propagating optical modes which are transversely bound at the interface of two materials are studied. In particular, quite general cases are considered, where the materials involved are assumed to be anisotropic, but also demonstrating magneto-electric effects. Moreover, surface states of two-dimensional materials like topological insulators and graphene are also modeled via consideration of a conductivity sheet existing at the interface. A characteristic equation for obtaining the propagation constant of generalized interface modes is presented. Furthermore, optical modes sustained by a thin film of anisotropic materials with magneto-electric effect and topological surface states are also investigated. It is shown that interface modes supported by such a system are hybrid in nature, and can be further decomposed into the well-known classes of transverse magnetic and electric modes, only at the absence of magneto-electric effect. Although the formulations driven here are mathematically abstract, they can be used to investigate polaritons in van der Waal materials, hyperbolic materials, and topological insulators.

In this chapter, the fundamentals of the nodal finite element method (FEM) are presented, including the first-order element and second-order element. The nodal FEM is introduced for the scalar concept of the propagation constant of 2D waveguide cross section. Then, it is extended to include the time domain analysis under perfectly matched layer absorbing boundary conditions. A simple sensor based on optical grating is thereafter simulated using the time domain FEM. Also, the full vectorial analysis is discussed through the application of the penalty function method on the nodal FEM and the vector finite element method (VFEM). For the penalty function method, a global weighting factor is used to incorporate the effect of the divergence-free equation. In the VFEM, nodes are used to represent the orthogonal component of the field while the edges are used to represent the tangential component for accurate application of the boundary conditions. Finally, surface plasmon resonance photonic crystal fiber biosensor is introduced as an example of the full vectorial analysis using the VFEM.

In this last chapter we apply spectral techniques to the wave equation and transmission lines. Transmission lines in a way supersede diffusion ones (last chapter) and in fact if we set the inductance to zero we fall back on the diffusion RC line. The wave equation governs the evolution of voltage and current along the T-line, both in space and time. It is second order both in space and time. To solve it we need initial conditions and boundary ones. We start the chapter with the un-driven line with either Neumann (fixed) or Dirichlet (tangential) boundary conditions and solve for various initial conditions. Then we move to the more general case with forced boundary conditions. The connecting link here is stimulating the line with a complex exponential and finding the solution which comes out in the form of forward propagating wave and backwards propagating one. Once that is known we can swap it with a general excitation and then use Fourier/Laplace transforms and superposition to find corresponding solution. In the chapter we examine physical parameters such as line input impedance, propagation constant, termination impedance, wave propagation, wave speed, wave reflection, and transmission. We also put emphasis on the impact of line dissipation (resistance) on wave propagation. In most cases we plot in 3D the wave propagation along the line for various time points and explain results. We also compare the two propagation phenomena of waves propagation and diffusion. We wrap the chapter with some notes on T-line input impedance and explain the meaning of zero or infinite impedance.

In this chapter, the graphene monolayer is modeled as surface current density without thickness in order to explore the properties of the plasmons propagating along it. The proposed model is verified analytically in terms of the propagation constant of the plasmons. By using this zero-thickness model, the nanofocusing properties of the mid infrared electromagnetic wave and the slow-light system of infrared wave are investigated. A focus width of 1.6 nm for a wavelength of 56 THz is achieved on a gradient chemical potential distribution graphene. Variant frequency components of the plasmons are stopped at the different locations of the gradient chemical potential distributed graphene monolayer forming “rainbow” capture effect. Additionally, the tunable whispering-gallery mode properties of the graphene monolayer coated semiconductor nano wire are also studied. An effective mode area smaller than \( 3.75\times {\displaystyle {10}^{-5}}{\displaystyle {\left({\lambda}_0\right)}^2} \), and a quality factor of 195 are obtained in a nanowire with a diameter of 5 nm. The proposed model and the plasmonic structures may find broad implications in the fields of high density plasmonic integrated circuits technique and transformation plasmonics.

Among the components needed in photonic integrated circuits, dielectric waveguides and small footprint ring resonators play a key role for many applications and require sophisticated electromagnetic analysis and design. In this work, we present an accurate vectorial mode solver based on the finite element method. Considering a general nonreciprocal permittivity tensor, the proposed method allows us to investigate important cases of practical interest. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node elements and the penalty function is added to remove the spurious solutions. Although the use of the penalty function is well known for the waveguide problem, it has been introduced for the first time (to the best of our knowledge) in the ring resonator modal analysis. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ-formulation). Unlike the earlier developed mode solvers, our approach allows us to precisely compute both forward and backward propagating modes in the nonreciprocal case. Moreover, it avoids time-consuming iterations and preserves matrix sparsity, ensuring high accuracy and computational efficiency.

We present a new method for determination of dielectric permittivity constant using measurements of fundamental mode of propagation constant in optical fiber’s. We first solve the forward spectral problem to compute the dispersion curve for the fundamental mode. Then using this curve we present an effective and accurate spline-collocation method for calculation of permittivity.

M.Yu. Kokurin and A. Bakushinsky, Interatively regularized Gauss-Newton methods under random noise.- L. Beilina, N.T. Thanh, M.V. Klibanov, and J.B. Malmberg, Methods of quantitative reconstruction of shapes and refractive indices from experimental data.- J.B. Malmberg, A posteriori error estimate in the Lagrangian setting for an inverse problem based on a new formulation of Maxwell's system.- E. Karchevskii, A. Spiridonov, and L. Beilina, Determination of permittivity from propagation constant measurements in optical fibers.- L. Angermann, Yu.V. Shestopalov, and V.V. Yatsyk, Eigenmodes of linearised problems of scattering and generation of oscillations on cubically polarisable layers.- S. Soltani, R. Andersson, and B. Andersson, Time resolution in transient kinetics.- L. Beilina and A. Eriksson, Reconstruction of dielectric constants in a cylindrical waveguide.- L. Beilina and I. Gainova, Time-adaptive FEM for distributed parameter identification in mathematical model of HIV infection with drug therapy.- L. Beilina and E. Karchevskii, The layer-stripping algorithm for reconstruction of dielectrics in an optical fiber.- L. Beilina, M. Cristofol, and K. Niinimaki, Simultaneous reconstruction of Maxwell's coefficients from backscattering data.

In the last decade a new research area based on the study of Metamaterials has emerged. There is not an absolute definition of Metamaterials but the most commonly used is that they are artificial materials made of a periodic arrangement of unit cells much smaller than the wavelength with some unique properties [1, 2, 3].

A plane multilayered waveguide structure is considered. The layers are located between two half-spaces with constant permittivities. The permittivity in each layer can be a constant or nonlinear (depends arbitrarily on modulus of the electric field intensity). We consider propagation of polarized electromagnetic waves in such a structure. The physical problem is reduced to (nonlinear) boundary eigenvalue problem in a multiply-connected domain. We suggest a numerical approach to calculate propagation constants (eigenvalues) for (nonlinear) layered waveguide structures based on numerical solution of a Cauchy problem in each layer. By means of transmission conditions on the layer boundaries we can define initial data for each Cauchy problem. When all Cauchy problems are solved we construct a function that depends on the spectral parameter. The zeros of this function which can be effectively calculated are the sought-for propagation constants (eigenvalues).