Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization

Birgin, Ernesto G.; Chambouleyron, I.; Martínez, José Mário

doi:10.1006/jcph.1999.6224

Cited by 254 publications

(198 citation statements)

References 10 publications

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“…Glunt et al (1993) present a direct application of the BB method in chemistry. Birgin et al (1999) use a globalized BB method to estimate the optical constants and the thickness of thin films, while in Birgin et al (2000) further extensions are given, leading to more efficient projected gradient methods. Liu & Dai (2001) provide a powerful scheme for solving noisy unconstrained optimization problems by combining the BB method and a stochastic approximation method.…”

Section: )mentioning

confidence: 99%

The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

174

163

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In the cyclic Barzilai-Borwein (CBB) method, the same Barzilai-Borwein (BB) stepsize is reused for m consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when m > n/2 3, where n is the problem dimension, CBB is locally superlinearly convergent. In the special case m = 3 and n = 2, it is proved that the convergence rate is no better than linear, in general. An implementation of the CBB method, called adaptive cyclic Barzilai-Borwein (ACBB), combines a non-monotone line search and an adaptive choice for the cycle length m. In numerical experiments using the CUTEr test problem library, ACBB performs better than the existing BB gradient algorithm, while it is competitive with the well-known PRP+ conjugate gradient algorithm.

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Section: )mentioning

confidence: 99%

The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

174

163

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“…The retrieval process of α fails when a break occurs in the smooth α vs E curve. It is followed by an almost constant value of α.Summarizing, a numerical method to extract the optical constants and the thickness of thin dielectric films from transmission data only [6] has been tested with a−Si:H …”

mentioning

confidence: 99%

“…Summarizing, a numerical method to extract the optical constants and the thickness of thin dielectric films from transmission data only [6] has been tested with a−Si:H…”

mentioning

confidence: 99%

Determination of thickness and optical constants of amorphous silicon films from transmittance data

Mulato

Chambouleyron

Birgin

et al. 2000

Applied Physics Letters

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This work presents the application of a recently developed numerical method to determine the thickness and the optical constants of thin films using experimental transmittance data only. This method may be applied to films not displaying a fringe pattern and is shown to work for a−Si:H (hydrogenated amorphous silicon) layers as thin as 100 nm. The performance and limitations of the method are discussed on the basis of experiments performed on a series of six a−Si:H samples grown under identical conditions, but with thickness varying from 98 nm to 1.2 µm. 2Modern electronic devices, such as thin−film transistors, solar cells, active matrix displays and image sensors, possess thin semiconductor layers of hydrogenated amorphous silicon (a−Si:H). For most electronic applications, the optical properties and the thickness t of these films play an important role, in the sense that they govern the device performance. The quality of the as−deposited material can be monitored in production lines through the in−situ determination of its optical constants (refractive index n and extinction coefficient k) and the thickness homogeneity. For that aim, ellipsometry is the most appropriate tool [1] due to the fact that it is not influenced by the adopted substrate. Alternatively, for the ex−situ analysis of samples grown on top of transparent substrates like glass, the use of optical transmittance is the most attractive method because optical transmission is a very easy, accurate and non−destructive measure.The problem of estimating the thickness and the optical constants of thin films using transmission data only represents a very ill−conditioned inverse problem with many local A set of experimental data [λ i , T meas (λ i )], λ min ≤ λ i ≤ λ i+1 ≤ λ max , for i = 1,…,N, is given, and we want to estimate t, n(λ), and k(λ). The problem seems highly underdetermined. In fact, for known t and given λ, the following must hold [5]: T meas (λ) = T theor (λ, s(λ), t, n(λ)where T theor is the calculated transmission of the film+substrate [3] and s the refractive index of the transparent substrate. This equation has two unknowns n(λ) and k(λ) and, in general, its set of solutions (n,k) is a curve in the two−dimensional (n(λ), k(λ)) space. Therefore, the set of functions (n,k) satisfying T meas = T theor for a given t is infinite and, roughly speaking, is represented by a nonlinear manifold of dimension N in R 2n . However, physical constraints (PC) drastically reduce the range of variability of the unknowns n(λ), k(λ). The optimization process looks for a thickness that, subject to the physical input of the problem, minimizes the difference between the measured and the theoretical spectra, i. e.,(1)The minimization process starts sweeping a thickness range ∆t R divided into thickness steps ∆t S and proceeds decreasing ∆t R and ∆t S until the optimized thickness t opt is found. In the examples to follow, the starting ∆t R and ∆t S were 5 µm and 100 nm, respectively.As seen, the most important issue of the present method is the ret...

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“…On the other hand, the direction d k could be obtained by solving the system (4). If the order of the system were small it could be solved using the LU factorization with partial pivoting or the QR factorization.…”

Section: Implementation Featuresmentioning

confidence: 99%

“…They were introduced by Barzilai and Borwein [1], and later analyzed by Raydan [21]. They have been applied to find local minimizers of large scale problems [4,5,22], and also to explore faces of large dimensions in box-constrained optimization (see [3] and [12]). More recently, spectral gradient methods were extended to minimize general smooth functions on convex sets.…”

Section: Introductionmentioning

confidence: 99%

Spectral Gradient Methods for Linearly Constrained Optimization

Martínez¹,

Pilotta²,

Raydan³

2005

J Optim Theory Appl

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Linearly constrained optimization problems with simple bounds are considered in the present work. First, a preconditioned spectral gradient method is defined for the case in which no simple bounds are present. This algorithm can be viewed as a quasiNewton method in which the approximate Hessians satisfy a weak secant equation. The spectral choice of steplength is embedded into the Hessian approximation, and the whole process is combined with a nonmonotone line search strategy. The simple bounds are then taken into account by placing them in an exponential penalty term that modifies the objective function. The exponential penalty scheme defines the outer iterations of the process. Each outer iteration involves the application of the previously defined preconditioned spectral gradient method for linear equality constrained problems. Therefore, an equality constrained convex quadratic programming problem needs to be solved at every inner iteration. The associated extended KKT matrix remains constant unless the process is reinitiated. In ordinary inner iterations, only the right hand side of the KKT system changes. Therefore, suitable sparse factorization techniques can be effectively applied and exploited. Encouraging numerical experiments are presented.

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Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization

Cited by 254 publications

References 10 publications

The cyclic Barzilai-–Borwein method for unconstrained optimization

The cyclic Barzilai-–Borwein method for unconstrained optimization

Determination of thickness and optical constants of amorphous silicon films from transmittance data

Spectral Gradient Methods for Linearly Constrained Optimization

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