Abstract

Optical scatterometry by use of a neural network is now recognized as an efficient method for retrieving dimensions of gratings in semiconductors or glasses. For an on-line control, a small number of measurements and a rapid data treatment are needed. We demonstrate that these requirements can be met by combining data preprocessing and a proper neural learning method. A good accuracy is attainable with the measurement of only a few orders, even in the presence of experimental errors, with a reduction in learning and computing time.

© 2002 Optical Society of America

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References

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  1. A. Roger, D. Maystre, “Inverse scattering method in electromagnetic optics: application to diffraction gratings,” J. Opt. Soc. Am. 70, 1483–1495 (1980).
    [CrossRef]
  2. K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
    [CrossRef]
  3. R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
    [CrossRef]
  4. S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
    [CrossRef]
  5. R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
    [CrossRef]
  6. J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
    [CrossRef]
  7. A. D. Mc Aulay, J. Wang, “Optical diffraction of periodic structures using neural networks,” Opt. Eng. 37, 884–888 (1998).
    [CrossRef]
  8. J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” Presented at International Joint Conference on Neural Networks, Washington, D.C., 1990.
  9. I. Kallioniemi, J. Saarinen, E. Oja, “Optical scatterometry of subwavelength diffraction gratings: neural network approach,” Appl. Opt. 37, 5830–5835 (1998).
    [CrossRef]
  10. I. Kallioniemi, J. Saarinen, E. Oja, “Characterization of diffraction gratings in a rigorous domain with optical scat-terometry: hierarchical neural-network model,” Appl. Opt. 38, 5920–5930 (1999).
    [CrossRef]
  11. J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
    [CrossRef]
  12. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  13. J. Herault, C. Jutten, Réseaux Neuronaux et Traitement du Signal (Editions Hermes, Paris, 1994).
  14. G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control Signal Syst. 2, 303–314 (1989).
    [CrossRef]
  15. K. I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks 2, 183–192 (1989).
    [CrossRef]
  16. D. Nguyen, B. Widrow, “The truck back-upper: an example of self-learning in neural networks,” in Neural Networks for Robotics and Control, W. T. Miller, R. Sutton, P. Werbos, eds. (MIT Press, Cambridge, Mass., 1990), Vol. 12, pp. 287–299.
  17. M. T. Hagan, M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989–993 (1994).
    [CrossRef] [PubMed]

1999 (1)

1998 (2)

A. D. Mc Aulay, J. Wang, “Optical diffraction of periodic structures using neural networks,” Opt. Eng. 37, 884–888 (1998).
[CrossRef]

I. Kallioniemi, J. Saarinen, E. Oja, “Optical scatterometry of subwavelength diffraction gratings: neural network approach,” Appl. Opt. 37, 5830–5835 (1998).
[CrossRef]

1994 (2)

1993 (2)

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

1991 (1)

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

1989 (2)

G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control Signal Syst. 2, 303–314 (1989).
[CrossRef]

K. I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks 2, 183–192 (1989).
[CrossRef]

1980 (1)

Bauer, J.

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

Bauer, J. J.

J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
[CrossRef]

Baumgart, J. W.

J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
[CrossRef]

Bischoff, J.

J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
[CrossRef]

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

Chan, C. H.

J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” Presented at International Joint Conference on Neural Networks, Washington, D.C., 1990.

Clark, L. A.

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Cybenko, G.

G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control Signal Syst. 2, 303–314 (1989).
[CrossRef]

Franke, J. E.

Funahashi, K. I.

K. I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks 2, 183–192 (1989).
[CrossRef]

Gaspar, S. M.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Giapas, K. P.

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Gottscho, R. A.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Haak, U.

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

Haaland, D. M.

Hagan, M. T.

M. T. Hagan, M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989–993 (1994).
[CrossRef] [PubMed]

Herault, J.

J. Herault, C. Jutten, Réseaux Neuronaux et Traitement du Signal (Editions Hermes, Paris, 1994).

Hutschenreuther, L.

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

Hwang, J. N.

J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” Presented at International Joint Conference on Neural Networks, Washington, D.C., 1990.

Jutten, C.

J. Herault, C. Jutten, Réseaux Neuronaux et Traitement du Signal (Editions Hermes, Paris, 1994).

Kallioniemi, I.

Kornblit, A.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Krukar, D. M.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Krukar, R.

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

Krukar, R. H.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Kruskal, J.

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Lambert, D.

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Li, L.

Marks, R. J.

J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” Presented at International Joint Conference on Neural Networks, Washington, D.C., 1990.

Maystre, D.

Mc Aulay, A. D.

A. D. Mc Aulay, J. Wang, “Optical diffraction of periodic structures using neural networks,” Opt. Eng. 37, 884–888 (1998).
[CrossRef]

McNeil, J. R.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Menhaj, M.

M. T. Hagan, M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989–993 (1994).
[CrossRef] [PubMed]

Naqvi, S. S. H.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch-depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Nguyen, D.

D. Nguyen, B. Widrow, “The truck back-upper: an example of self-learning in neural networks,” in Neural Networks for Robotics and Control, W. T. Miller, R. Sutton, P. Werbos, eds. (MIT Press, Cambridge, Mass., 1990), Vol. 12, pp. 287–299.

Niemczyk, T. M.

Oja, E.

Peterson, G. A.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Prins, S. L.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

Reitman, E. A.

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

Roger, A.

Saarinen, J.

Sinatore, D.

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Truckenbrodt, H.

J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
[CrossRef]

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

Wang, J.

A. D. Mc Aulay, J. Wang, “Optical diffraction of periodic structures using neural networks,” Opt. Eng. 37, 884–888 (1998).
[CrossRef]

Widrow, B.

D. Nguyen, B. Widrow, “The truck back-upper: an example of self-learning in neural networks,” in Neural Networks for Robotics and Control, W. T. Miller, R. Sutton, P. Werbos, eds. (MIT Press, Cambridge, Mass., 1990), Vol. 12, pp. 287–299.

Appl. Opt. (2)

IEEE Trans. Neural Netw. (1)

M. T. Hagan, M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989–993 (1994).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

R. Krukar, A. Kornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Reitman, R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698–3706 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol. A (1)

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Math. Control Signal Syst. (1)

G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control Signal Syst. 2, 303–314 (1989).
[CrossRef]

Neural Networks (1)

K. I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks 2, 183–192 (1989).
[CrossRef]

Opt. Eng. (1)

A. D. Mc Aulay, J. Wang, “Optical diffraction of periodic structures using neural networks,” Opt. Eng. 37, 884–888 (1998).
[CrossRef]

Other (6)

J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” Presented at International Joint Conference on Neural Networks, Washington, D.C., 1990.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration ,” in Integrated Circuit Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–71 (1993).
[CrossRef]

J. Bischoff, J. W. Baumgart, H. Truckenbrodt, J. J. Bauer, “Photoresist metrology based on light scattering,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 678–689 (1996).
[CrossRef]

D. Nguyen, B. Widrow, “The truck back-upper: an example of self-learning in neural networks,” in Neural Networks for Robotics and Control, W. T. Miller, R. Sutton, P. Werbos, eds. (MIT Press, Cambridge, Mass., 1990), Vol. 12, pp. 287–299.

J. Bischoff, J. Bauer, U. Haak, L. Hutschenreuther, H. Truckenbrodt, “Optical Scatterometry of quarter-micron patterns using neural regression,” in Metrology, Inspection, and Process Control for Microlithography XII, B. Singh, ed., Proc. SPIE3332, 526–537 (1998).
[CrossRef]

J. Herault, C. Jutten, Réseaux Neuronaux et Traitement du Signal (Editions Hermes, Paris, 1994).

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Figures (15)

Fig. 1
Fig. 1

Symmetric grating profile, defined by sidewall projection b1, linewidth b2, groove depth h, and period Λ.

Fig. 2
Fig. 2

Relative variations of diffracted efficiencies versus the groove depth. The grating parameters are b1=0.105 μm, b2=0.32 μm, and h=0.32 μm. Incident light is TE polarized at θ=30°.

Fig. 3
Fig. 3

Relative variations of diffracted efficiencies r-2, r-1, r0, t-2, t-1, and t0 of a grating with Λ=1 μm, b1=0.105 μm, b2=0.32 μm, and h=0.32 μm when the groove depth is varying with TE polarized light at θ=20°.

Fig. 4
Fig. 4

Relative variations of diffracted efficiencies r-2, r-1, r0, t-2, t-1, and t0 of a grating with Λ=1 μm, b1=0 μm, b2=0.5 μm, and h=0.32 μm when the groove depth is varying with TE polarized light at θ=30°.

Fig. 5
Fig. 5

Absolute variations of diffracted efficiencies r-2, r-1, r0, t-2, t-1, and t0 of a grating Λ=1 μm, b1=0.105 μm, b2=0.32 μm, and h=0.32 μm when the groove depth is varying with TE polarized light at θ=20°.

Fig. 6
Fig. 6

Graph of a three-layer network with six inputs, three outputs, and four neurons in the hidden layer. wi,j is the connection weight between the jth neuron in the input layer and the ith in the hidden layer.

Fig. 7
Fig. 7

Structure of a formal neuron. xj represents one input of the neuron, ai the weighting sum and oi the neuron output.

Fig. 8
Fig. 8

Typical behavior of the rms error on training and test data sets during the training process.

Fig. 9
Fig. 9

Experimental variation of the error during the training process, for a neural network composed of 18 inputs, 3 outputs, and 15 neurons in the hidden layer. The training requires 1300 couples. It is stopped when the error calculated on the validation set reaches a minimum (33 epochs).

Fig. 10
Fig. 10

Representation of each grating parameter calculated by a neural network with exact inputs versus its theoretical values. The training is performed with 1300 exact {input/target} calculated couples.

Fig. 11
Fig. 11

Representation of each grating parameter calculated by a neural network with experimental simulation inputs versus its theoretical values. The training is performed with 1300 exact {input/target} calculated couples.

Fig. 12
Fig. 12

Representation of each grating parameter calculated by a neural network with exact inputs versus its theoretical value. The training is performed with 1300 noisy {input/target} couples.

Fig. 13
Fig. 13

Representation of each grating parameter calculated by a neural network with experimental simulation inputs versus its theoretical value. The training is performed with 1300 noisy {input/target} couples.

Fig. 14
Fig. 14

Variation of the zero-order transmitted efficiency versus the incident angle at 670 nm wavelength calculated in a grating defined by b1=0.094 μm, b2=0.473 μm, and h=0.260 μm. The same efficiency variation calculated in the gratings reconstructed by the neural network for the two cases of inputs (theoretical and noisy inputs) is represented.

Fig. 15
Fig. 15

Variation of zero-order transmitted efficiency versus wavelength at θ=20° calculated in a grating defined by b1=0.094 μm, b2=0.473 μm and h=0.260 μm. The same efficiency variation calculated in the gratings reconstructed by the neural network for the two cases of inputs (theoretical and noisy inputs) is represented.

Tables (3)

Tables Icon

Table 1 Summary of the rms Errors Calculated for Different Training and Test Sets

Tables Icon

Table 2 Effects of the Preprocessing (PCA) in the Reduction of the Input Number and Training Duration Time for Two Cases

Tables Icon

Table 3 Grating Parameter Prediction for Two Kinds of Neural Inputs for a Randomly Chosen Grating

Equations (2)

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e(p)rms=(1/n)(pex-pcal)21/2,
ηn=ηex-N(0,0.03),

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