Time-and-frequency domains approach to data processing in multiwavelength optical scatterometry of dielectric gratings

Abstract

This paper focuses on scatterometry problems arising in lithography production of periodic gratings. Namely, the paper introduces a theoretical and numerical-modeling-oriented approach to scatterometry problems and discusses its capabilities. The approach allows for reliable detection of deviations in gratings' critical dimensions (CDs) during the manufacturing process. The core of the approach is the one-to-one correspondence between the electromagnetic (EM) characteristics and the geometric/material properties of gratings. The approach is based on highly accurate solutions of initial boundary-value problems describing EM waves' interaction on periodic gratings. The advantage of the approach is the ability to perform simultaneously and interactively both in frequency and time domains under conditions of possible resonant scattering of EM waves by infinite or finite gratings. This allows a detection of CDs for a wide range of gratings, and, thus is beneficial for the applied scatterometry.

Rigorous 2D Model for Study of Pulsed and Monochromatic Waves Propagation Near the Earth’s Surface

“…All the processes, the necessity of which has been discussed above, are described by the following 2D initial boundary value problem [6,7]: An exception has been made for the "time" which represents a product of the true time and the free-space velocity of light and hence is expressed in meters.…”

“…The problems (1) and (2) are equivalent in Ω int [7,8] as exact absorbing conditions do not distort the simulated physical processes. The outgoing from domain Ω int wave ( , ) seems to be absorbed by domain Ω ext = Ω \ Ω int or by its boundary Γ, and there is no reflection in domain Ω int .…”

“…The solution ( , ) of the problem (2) which is constructed for the points ∈ Ω int and values of from the observation time interval [0, ] using the standard computational schemes of the finite-difference method [4] and which continued, if it is necessary, from the domain Ω int into the domain Ω ext by the operator method (by the method of transport operators which determine space-andtime deformations of pulses along finite segments of their propagation in regular structures [6][7][8][9]) can be converted, using the integral transformatioñ…”

“…The algorithms and computer codes presented in the paper, similar to other algorithms of the exact absorbing condition method [6][7][8], are thoroughly tested. The principal tasks for computer codes generalization and unification that we are about to resolve are the following: (i) the automation of the routine for trace "prolongation" by means of introduction of the new instant sources ( ) and ( ) in the problem (2) with new computational domain Ω int and new current sources ( , ) = 0; (ii) the construction of more efficient computational schemas for the solution to the problem (2) that can essentially reduce required computer resources (the usage of acceleration scheme based on the blocked fast Fourier transform [12] for calculation of the temporal convolutions in exact absorbing conditions of the problem (2), changing from computational schemes of the finitedifference method to schemas of the discontinuous Galerkin finite-element method [13]); (iii) the straightforward comparison (basing on criteria of universality, accuracy, and consumption of computer resources) of our approach with approaches of other authors, with those ones which are already considered to be classic [1,2,14,15] and with relatively novel [16][17][18][19][20][21][22][23][24].…”

“…In other words, it is necessary to reduce the open problem to an equivalent closed one. In the paper, this rather complex procedure is performed using the method of exact absorbing conditions [6,7], which made it possible over the recent several years to comprehensively study a number of important problems of radio physics.…”