Spatial-Domain Methods. The first attempts were based on the solution of the Helmholtz equation in an arbitrarily shaped, inhomogeneous medium with the help of the finite-element method together with the boundary-element method. Toshiharu Matsuszwa et al. applied this approach to study photoresist bleaching on a stepped, perfectly conducting substrate . H. Paul Urbach and Douglas Bernard improved this method by extending it to more general domains and partial coherence [153,154,155,156]. Eytan Barouch et al. adopted the related spectral element method  for lithographic applications [71,158]. This method is a combination of spectral and finite-element methods in which relatively few spatial elements are needed since the local basis functions inside each element are tensor products of high-order polynomials, e.g., Chebyshev or Legendre polynomials. Hence an exponential decrease of the numerical error with the degree of the polynomial is achieved. Based on this method the first comprehensive three-dimensional notching simulation on nonplanar substrate was performed .
Time-Domain Methods. The above spatial-domain methods calculate the steady-state field distribution. Employing a time-domain finite-difference discretization scheme also transient phenomena can be studied. The foundations of this approach go back to Kane Yee, who was one of the first to replace the Maxwell equations by a set of finite-difference equations . Allen Taflove and Morris Browding successfully applied this method to two-dimensional steady-state scattering investigations of uniform and circular dielectric cylinders . However, because of its computation-intensive nature time-domain solutions of EM problems in photolithography were impractical until the advent of supercomputers. Gregory Wojcik et al. introduced them into lithography simulation by first studying light scatter from silicon surfaces  and later laser alignment mark signals . Concurrently, Roberto Guerrieri  formulated and John Gamelin  implemented the time-domain finite-difference method on a massively parallel supercomputer. The work was performed at the University of California at Berkeley under the guidance of Professor Andrew Neureuther and resulted in the program TEMPESTc . The simulator TEMPEST was applied to a wide variety of difficult lithographic problems such as to the study of reflective notching , metrology of polysilicon gates structures , mask material and coating effects on image quality , alignment mark signal integrity , and mask topography effects in projection printing of phase-shifting masks [68,167]. The algorithm was extended to three-dimensions  and the latest versions also include features to account for partially coherent imaging .
Frequency-Domain Methods. A totally different approach is pursued in case of frequency-domain methods as the EM field is expressed by a superposition of some basis functions. Generally two different discretization techniques can be applied to the Maxwell equations. Either the partial differential equations are directly transformed to a linear algebraic equation system by solving repeatedly an eigenvalue problem or, alternatively, an ordinary differential equation system is set up first and is then discretized also resulting in an algebraic system. If harmonic basis functions are chosen, the two techniques are called modal approach [169,170,171,172,173,174,175] and coupled-wave approach [176,177,178,179,180,181,182], respectively. Both methods describe the EM field in a rigorous manner without approximations. As such they are closely related and have been shown to be exact and equivalent .
The foundations of frequency-domain methods are due to Lord Rayleigh, when he made the first attempts to rigorously analyze the EM scattering problem from a periodic grating structure by assuming that the field can be expressed as a linear superposition of propagation and evanescent waves [184, p. 124]. His approach is commonly referred to as the Rayleigh expansion. Based on this work the modal expansion technique as well as the coupled-wave approach were developed.
For lithographic applications such as edge detection Diana Nyyssonen and Chris Kirk adopted the modal approach . In the lithography community this method is better known as the waveguide method. Chi-Min Yuan et al. extended it to transverse-magnetic polarization  and Hiroyoshi Tanabe presented a three-dimensional formulation  that was implemented by Kevin Lucas [10,187]. Most of this work was performed at the Carnegie-Mellon University at Pittsburgh under the guidance of Professor Andrzej Strojwas and was incorporated into the lithography simulator METROPOLE. The three basic problems of light scattering in lithography, namely alignment, metrology, and resist exposure were successfully simulated with METROPOLE in two as well as three dimensions [105,73,188,69,8,189,190].
At the same time Michael Yeung used the coupled-wave approach to model high numerical aperture optical lithography . For lithography simulation this method is also known as the differential method. It was implemented in the two-dimensional lithography simulator iPHOTO developed at INTEL. The extension to three-dimensions has been performed in this thesis and a detailed discussion is presented in the following chapter. Simulation results for relevant lithographic scattering problems are given in Chapter 8.