## ABSTRACT

This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book presents methods for numerically solving James Clerk Maxwell equations or equations derived from them such as the wave or Helmholtz equations. It shows how a very general propagation equation may be derived, specialized to the case of a singlemode waveguide, and eventually reduced to the nonlinear Schrodinger equation through various approximations. Maxwell presented his well-known set of equations governing electromagnetic fields in 1861. These equations describe the interrelationship between electric and magnetic fields and electric charges and currents. All derivatives in Maxwell’s equations are approximated numerically by finite-difference schemes reducing differential equations to algebraic ones. The field inside or on the surface of the scattering object is obtained by solving self-consistent equations. However, many generalizations of the equation are possible, accounting for effects of higher-order dispersion, delayed nonlinear response, mode profile variations, vectorial effects, and multimode behaviour.