**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 [152]. 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* [157] 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 [158].

**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 [159].
Allen Taflove and Morris Browding successfully applied this method to
two-dimensional steady-state
scattering investigations of uniform and circular dielectric
cylinders [160]. 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 [161] and later
laser alignment mark signals [162]. Concurrently, Roberto
Guerrieri [5] formulated and John Gamelin [70] 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
TEMPEST^{c} [7].
The simulator TEMPEST was
applied to a wide variety of difficult lithographic problems such as
to the study of reflective notching [163], metrology of polysilicon gates
structures [164], mask material and coating effects on image
quality [165], alignment mark signal integrity [166], and mask
topography effects in projection printing of phase-shifting
masks [68,167]. The algorithm was extended
to three-dimensions [6] and the latest versions also include features
to account for partially coherent imaging [168].

**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 [183].

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 [185]. In the lithography
community this method is better known as the *waveguide method*.
Chi-Min Yuan *et al.* extended it to transverse-magnetic
polarization [186] and Hiroyoshi Tanabe presented a three-dimensional
formulation [9] 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 [114].
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.

- ...
TEMPEST
^{c} - For detailed information on the simulation program TEMPEST including availability see the WWW at http://tanqueray.eecs.berkeley.edu/tcad/tempest/tempest.html.

1998-04-17