&lt;title&gt;Numerical reference models for optical metrology simulation&lt;/title&gt;

Wojcik, G.; Mould, John; Marx, Egon; Davidson, Mark P.

doi:10.1117/12.59785

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…Unfortunately, many of the popular methods are only low order accurate and agree in test problems to only 1 digit of accuracy [45]. Finite difference approximation to the time-domain wave equation (FDTD) [16,42] is common; it has the advantage of being quite general, but has the disadvantage of requiring large regions of free space to be simulated, with significant dispersion error for electrically large domains.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

“…Moving to the frequency domain, finite element methods [5,30] require non-trivial grid generation, especially in three dimensions, and also demand the discretization of substantial regions of free space together with the imposition of non-local radiation conditions constructed from (1.6) and (1.7). The engineering community has developed many other methods-including coupled wave analysis, the waveguide method [45], the C method [24], and cylindrical (multipole) expansions [27,37]-which are often based on uncontrolled approximations.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

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A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green's function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green's function diverges for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.In this paper, we bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. Summing nearby images directly leaves auxiliary densities that are smooth, hence easily represented in the Fourier domain using Sommerfeld integrals. Wood's anomalies are handled analytically by deformation of the Sommerfeld contour. The resulting integral equation is of the second kind and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls are handled easily and automatically. We include an implementation and simple code example with a freely-available MATLAB toolbox.Communicated by Per Lötstedt.

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Section: Introduction and Problem Formulationmentioning

confidence: 99%

Section: Introduction and Problem Formulationmentioning

confidence: 99%

A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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“…Secondly, we exploit the fact that filling (rather than solving) the matrix blocks in (56) often accounts for the bulk of the solution time. Consider one of the integral operators used in the BIE matrix (defined in (19)),…”

Section: Accelerated Sweep Over Multiple Incident Angles At One Frequ...mentioning

confidence: 99%

“…There are many low-order numerical methods used to solve multilayer scattering problems, which in test problems may only agree to 1 digit of accuracy [56]. Finite difference time-domain (FDTD) [55,27] is easy to code but has dispersion errors, and requires artificial absorbing boundary conditions and arbitrarily long settling times near resonances.…”

Section: Introductionmentioning

confidence: 99%

Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers

Cho¹,

Barnett²

2015

Opt. Express

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We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green's function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green's function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 10⁵ unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.

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“…[1][2][3][4][5] However, the mark selection procedure using real process wafers was time consuming because of numerous experimental evaluations of the mark signal and overlay accuracy. Though mark selection is done with great care, issues concerning the mark frequently arise in production.…”

Section: Introductionmentioning

confidence: 99%

Alignment mark signal simulation system for the optimum mark feature selection

Satō

Endo

Higashiki

et al. 2005

J. Micro/Nanolith. MEMS MOEMS

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Recently, requirements concerning overlay accuracy have become much more restrictive. For the accurate overlay, signal intensity and wave form from the topographical alignment mark have been examined by signal simulation. However, even if the results were in good agreement with actual signal profiles, it would be difficult to select particular alignment marks at each mask level by the signal simulation. Therefore, many mark candidates are left in the kerf area after mass production. To facilitate the selection, we propose a mark TCAD system. It is a useful system for the mark selection with the signal simulation performed in advance. In our system, the alignment mark signal can be easily simulated after input of some process material parameters and process of record (POR). The POR is read into the system and a process simulator makes stacked films on a wafer. Topographical marks are simulated from the stacked films and the resist pattern. The topographical marks are illuminated and reflected beams are produced. Imaging of the reflected beams through inspection optics is simulated. In addition, we show two applications. This system is not only for predicting and showing a signal wave form, but is also helpful for finding the optimum marks.

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<title>Numerical reference models for optical metrology simulation</title>

Cited by 5 publications

References 13 publications

A new integral representation for quasi-periodic scattering problems in two dimensions

A new integral representation for quasi-periodic scattering problems in two dimensions

Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers

Alignment mark signal simulation system for the optimum mark feature selection

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