Effective parameters for dielectric waveguides and resonators in linear and nonlinear regimes

Modern waveguiding structures contain various materials that extend their properties beyond mere waveguiding. Here, the most common cases are considered: materials with linear loss or gain and Kerr-type nonlinearity. A convenient and insightful description of how guided waves are affected by these materials is to recast their properties into material parameters and electromagnetic modal parameters, giving rise to the concept of effective parameters, e.g., geometrical loss factor, gain confinement factor, effective modal area, and effective nonlinear refractive index. The literature contains abundant formulas for the effective parameters. At times, they are applied beyond their range of validity, scope of applicability, or are even incorrect. Here, key effective parameters are consistently derived and corresponding formulas from the literature are discussed. This Tutorial serves two purposes. First, it guides the reader step-by-step through the derivation of waveguide and resonator characteristics, introduces effective parameters, and discusses related issues. Second, it provides formulas for many practical situations for the reader more interested in using them directly for specific calculations, such as analyzing experimental data or making theoretical predictions, without delving into mathematical details. This also applies to choosing correct parameters in commercial software packages.