Abstract

Ellipsometry is an optical technique that is widely used for determining optical and geometrical properties of optical thin films. These properties are in general extracted from the ellipsometric measurement by solving an inverse problem. Classical methods like the Levenberg–Marquardt algorithm are generally too long, depending on direct calculation and are very sensitive to local minima. In this way, the neural network has proved to be an efficient tool for solving these kinds of problems in a very short time. Indeed, it is rapid and less sensitive to local minima than the classical inversion method. We suggest a complete neural ellipsometric characterization method for determining the index dispersion law and the thickness of a simple SiO2 or photoresist thin layer on Si, SiO2, and BK7 substrates. The influence of the training couples on the artificial neural network performance is also discussed.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  2. O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
    [CrossRef]
  3. K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).
  4. S. A. Alterovitz and B. Johs, “Multiple minima in the ellipsometric error function,” Thin Solid Films 313-314, 124-127(1998).
    [CrossRef]
  5. A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
    [CrossRef]
  6. Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
    [CrossRef]
  7. O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
    [CrossRef]
  8. C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1996).
  9. S. Robert, A. Mure-Ravaud, and D. Lacour, “Characterization of optical diffraction gratings by use of a neural method,” J. Opt. Soc. Am. A 19, 24-32 (2002).
    [CrossRef]
  10. I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
    [CrossRef]
  11. M. F. Tabet and W. A. McGahan, “Use of artificial neural networks to predict thickness and optical constants of thin films from reflectance data,” Thin Solid Films 370, 122-127 (2000).
    [CrossRef]
  12. M. Fried and L. Rédei, “Non-destructive optical depth profiling and real-time evaluation of spectroscopic data,” Thin Solid Films 364, 64-74 (2000).
    [CrossRef]
  13. L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
    [CrossRef]
  14. F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
    [CrossRef]
  15. F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
    [CrossRef]

2007 (1)

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

2006 (1)

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

2004 (2)

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[CrossRef]

2003 (1)

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[CrossRef]

2002 (1)

2000 (2)

M. F. Tabet and W. A. McGahan, “Use of artificial neural networks to predict thickness and optical constants of thin films from reflectance data,” Thin Solid Films 370, 122-127 (2000).
[CrossRef]

M. Fried and L. Rédei, “Non-destructive optical depth profiling and real-time evaluation of spectroscopic data,” Thin Solid Films 364, 64-74 (2000).
[CrossRef]

1998 (3)

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

S. A. Alterovitz and B. Johs, “Multiple minima in the ellipsometric error function,” Thin Solid Films 313-314, 124-127(1998).
[CrossRef]

F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
[CrossRef]

1992 (1)

F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
[CrossRef]

1944 (1)

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

Alterovitz, S. A.

S. A. Alterovitz and B. Johs, “Multiple minima in the ellipsometric error function,” Thin Solid Films 313-314, 124-127(1998).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bársony, I.

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

Barton, D.

F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1996).

Boubani, N. I.

F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
[CrossRef]

Dang, M.

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[CrossRef]

Fried, M.

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

M. Fried and L. Rédei, “Non-destructive optical depth profiling and real-time evaluation of spectroscopic data,” Thin Solid Films 364, 64-74 (2000).
[CrossRef]

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

Gereige, I.

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Granet, G.

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Jamon, D.

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Johs, B.

S. A. Alterovitz and B. Johs, “Multiple minima in the ellipsometric error function,” Thin Solid Films 313-314, 124-127(1998).
[CrossRef]

Kudla, A.

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[CrossRef]

Lacour, D.

Levenberg, K.

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

Lohner, T.

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

McGahan, W. A.

M. F. Tabet and W. A. McGahan, “Use of artificial neural networks to predict thickness and optical constants of thin films from reflectance data,” Thin Solid Films 370, 122-127 (2000).
[CrossRef]

Mure-Ravaud, A.

Park, D. C.

F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
[CrossRef]

Petrik, P.

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

Polgár, O.

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

Rédei, L.

M. Fried and L. Rédei, “Non-destructive optical depth profiling and real-time evaluation of spectroscopic data,” Thin Solid Films 364, 64-74 (2000).
[CrossRef]

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

Robert, S.

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

S. Robert, A. Mure-Ravaud, and D. Lacour, “Characterization of optical diffraction gratings by use of a neural method,” J. Opt. Soc. Am. A 19, 24-32 (2002).
[CrossRef]

Rousseau, J. J.

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Tabet, M. F.

M. F. Tabet and W. A. McGahan, “Use of artificial neural networks to predict thickness and optical constants of thin films from reflectance data,” Thin Solid Films 370, 122-127 (2000).
[CrossRef]

F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
[CrossRef]

Urban, F. K.

F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
[CrossRef]

F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
[CrossRef]

Wallinga, H.

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

Zhaoxian, Y.

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[CrossRef]

Appl. Surf. Sci. (1)

O. Polgár, P. Petrik, T. Lohner, and M. Fried, “Evaluation strategies for multi-layer, multi-material ellipsometric measurements,” Appl. Surf. Sci. 253, 57-64 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

I. Gereige, S. Robert, D. Jamon, J. J. Rousseau, and G. Granet, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Q. Appl. Math. (1)

K. Levenberg, “A method for the solution of certain problems in least squares,” Q. Appl. Math. 2, 164-168 (1944).

Thin Solid Films (9)

S. A. Alterovitz and B. Johs, “Multiple minima in the ellipsometric error function,” Thin Solid Films 313-314, 124-127(1998).
[CrossRef]

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[CrossRef]

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[CrossRef]

O. Polgár, M. Fried, T. Lohner, and I. Bársony, “Evaluation of ellipsometric measurements using complex strategies,” Thin Solid Films 455-456, 95-100 (2004).
[CrossRef]

M. F. Tabet and W. A. McGahan, “Use of artificial neural networks to predict thickness and optical constants of thin films from reflectance data,” Thin Solid Films 370, 122-127 (2000).
[CrossRef]

M. Fried and L. Rédei, “Non-destructive optical depth profiling and real-time evaluation of spectroscopic data,” Thin Solid Films 364, 64-74 (2000).
[CrossRef]

L. Rédei, M. Fried, I. Bársony, and H. Wallinga, “A modified learning strategy for neural networks to support spectroscopic ellipsometric data evaluation,” Thin Solid Films 313-314, 149-155 (1998).
[CrossRef]

F. K. Urban III, D. C. Park, and M. F. Tabet, “Development of artificial neural networks for real time, in-situ ellipsometry data reduction,” Thin Solid Films 220, 247-253 (1992).
[CrossRef]

F. K. Urban III, D. Barton, and N. I. Boubani, “Extremely fast ellipsometry solutions using cascaded neural networks alone,” Thin Solid Films 332, 50-55 (1998).
[CrossRef]

Other (2)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford University, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Graph of a three layer ANN with P inputs, N outputs, and N s neurons in the hidden layer. w j , i is the connection weight between the jth neuron in the input layer and the ith in the hidden layer.

Fig. 2
Fig. 2

Structure of a thin film deposited on a substrate.

Fig. 3
Fig. 3

Dispersion laws estimated by a classical optimization method (LM) and by the ANN regression from a simulated ellipsometric signature corresponding to a SiO 2 thin film on a Si substrate.

Fig. 4
Fig. 4

Influence of the number of samples on the ANN performance for different numbers of hidden neurons.

Fig. 5
Fig. 5

Influence of the number of training couples and the number of hidden neurons (N) on the training time.

Fig. 6
Fig. 6

Mapping of the mse function for a theoretical sample defined by ε s = 2 , ω t = 15 eV , and d = 80 nm for different presumed values of ε s and d. ω t is fixed at 15 eV .

Fig. 7
Fig. 7

Influence of the local minima on the algorithm convergence thickness resulting from a neural characterization for different training widths and from a LM optimization for different initial guesses.

Fig. 8
Fig. 8

Dispersion law of a SiO 2 thin film on a Si substrate resulting from the ANN and the LM optimization.

Fig. 9
Fig. 9

Dispersion laws of five photoresist films on SiO 2 or BK7 substrates resulting from the ANN optimization.

Tables (2)

Tables Icon

Table 1 Evaluation of the Global Performances of Four ANNs during the Characterization of 100 Simulated Photoresist Samples

Tables Icon

Table 2 Comparison between the Thicknesses of Five Photoresist Films Determined by an ANN Inversion and an LM Optimization

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

ρ = r p r s = tan ψ e i Δ ,
I s = sin 2 ψ sin Δ , I c = sin 2 ψ cos Δ .
o j = f ( i = 0 P w j , i x i ) with x 0 = 1.
f ( α ) = 1 1 + exp ( α ) .
s m = j = 0 N z m , j f ( i = 0 P w j , i x i ) ,
E = M ( s s sim ) 2 .
rmse = [ 1 N N ( p t p s ) 2 ] 1 / 2 = mse 1 / 2 ,
n ( ω ) = 1 + ( ε s 1 ) ω t 2 ω t 2 ω 2 ,
n ( λ ) = A + B × 10 4 λ 2 + C × 10 9 λ 4 ,
rmse = [ 1 N N ( I mes I sim ) 2 ] 1 / 2 .
1.58 < A < 1.62 , 0.1 nm 2 < B < 1.1 nm 2 , 0.49 nm 4 < C < 1.6 nm 4 .

Metrics