Abstract

Characterization of gratings with small period-to-wavelength ratios is difficult to perform but is very helpful in improving the fabrication process. We experimentally tested an inverse-scattering method using a neural network on silicon etched gratings. We also characterized the gratings by using two popular microscopic methods. The validity of each method was determined by comparing measured diffracted intensities with calculated ones obtained from measured profiles. An estimation of accuracy and repeatability was deduced from a scan along a grating sample. This method was thus well validated for nondestructive and noninvasive measurements under experimental conditions that were close conditions of actual usage. This method is easy to implement and requires the measurement of only a few diffracted intensities.

© 2002 Optical Society of America

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References

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  1. A. Roberts, “Probe–sample interaction in near-field microscopy: analysis of the interaction between a metal diffraction grating and a plane conductor,” Opt. Commun. 98, 225–230 (1993).
    [CrossRef]
  2. O. J. F. Martin, C. Girard, A. Dereux, “Dielectric versus topographic contrast in near-field microscopy,” J. Opt. Soc. Am. A 13, 1801–1809 (1996).
    [CrossRef]
  3. J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
    [CrossRef]
  4. A. Roger, D. Maystre, “Inverse scattering method in electromagnetic optics: application to diffraction gratings,” J. Opt. Soc. Am. 70, 1483–1495 (1980).
    [CrossRef]
  5. N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
    [CrossRef]
  6. M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
    [CrossRef]
  7. J. N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
    [CrossRef]
  8. 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]
  9. R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
    [CrossRef]
  10. 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]
  11. J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” presented at IJCNN’90, the 2nd IEEE International Joint Conference on Neural Networks, Washington, D.C., January 15–19, 1990.
  12. I. Kallioniemi, J. Saarinen, E. Oja, “Optical scatterometry of subwavelength diffraction gratings: neural network approach,” Appl. Opt. 37, 5830–5835 (1998).
    [CrossRef]
  13. I. Kallioniemi, J. Saarinen, E. Oja, “Characterization of diffraction gratings in a rigorous domain with optical scatterometry: hierarchical neural-network model,” Appl. Opt. 38, 5920–5930 (1999).
    [CrossRef]
  14. J. Bischoff, H. Truckenbrodt, J. Bauer, “Diffraction analysis based characterization of very thin gratings,” in Micro-optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. Parriaux, B. Culshaw, M. Breidne, E. B. Kley, eds., Proc. SPIE3099, 212–222 (1997).
    [CrossRef]
  15. S. Robert, A. Mure-Ravaud, D. Lacour, “Optical diffraction grating characterization using a neural method,” J. Opt. Soc. Am. A 19, 24–32 (2002).
    [CrossRef]
  16. L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
    [CrossRef]
  17. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  18. C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, New York, 1995).
  19. M. T. Hagan, M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989–993 (1994).
    [CrossRef] [PubMed]

2002 (2)

M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

S. Robert, A. Mure-Ravaud, D. Lacour, “Optical diffraction grating characterization using a neural method,” J. Opt. Soc. Am. A 19, 24–32 (2002).
[CrossRef]

2001 (1)

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

2000 (2)

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

J. N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

1999 (1)

1998 (1)

1996 (2)

O. J. F. Martin, C. Girard, A. Dereux, “Dielectric versus topographic contrast in near-field microscopy,” J. Opt. Soc. Am. A 13, 1801–1809 (1996).
[CrossRef]

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[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]

A. Roberts, “Probe–sample interaction in near-field microscopy: analysis of the interaction between a metal diffraction grating and a plane conductor,” Opt. Commun. 98, 225–230 (1993).
[CrossRef]

1980 (1)

Achtenhagen, M.

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[CrossRef]

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]

J. Bischoff, H. Truckenbrodt, J. Bauer, “Diffraction analysis based characterization of very thin gratings,” in Micro-optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. Parriaux, B. Culshaw, M. Breidne, E. B. Kley, eds., Proc. SPIE3099, 212–222 (1997).
[CrossRef]

Bischoff, J.

J. Bischoff, H. Truckenbrodt, J. Bauer, “Diffraction analysis based characterization of very thin gratings,” in Micro-optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. Parriaux, B. Culshaw, M. Breidne, E. B. Kley, eds., Proc. SPIE3099, 212–222 (1997).
[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]

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, New York, 1995).

Bonard, J. M.

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[CrossRef]

Byrne, D. M.

M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[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 IJCNN’90, the 2nd IEEE International Joint Conference on Neural Networks, Washington, D.C., January 15–19, 1990.

Dereux, A.

Destouches, N.

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Dongmo, L. S.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Drege, M.

M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Franke, J. E.

Ganière, J. D.

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[CrossRef]

Gillet, J. N.

Giovannini, H.

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Girard, C.

Gottscho, R. A.

Guérin, C.-A.

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[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]

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 IJCNN’90, the 2nd IEEE International Joint Conference on Neural Networks, Washington, D.C., January 15–19, 1990.

Jones, S. N.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Kallioniemi, I.

Kornblit, A.

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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[CrossRef]

Lacour, D.

Lequime, M.

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[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 IJCNN’90, the 2nd IEEE International Joint Conference on Neural Networks, Washington, D.C., January 15–19, 1990.

Martin, O. J. F.

Maystre, D.

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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (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]

Morier-Genoud, F.

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[CrossRef]

Mure-Ravaud, A.

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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[CrossRef]

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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[CrossRef]

Postek, M. T.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[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 scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[CrossRef]

Reed, J. A.

M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Renegar, T. B.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Robert, S.

Roberts, A.

A. Roberts, “Probe–sample interaction in near-field microscopy: analysis of the interaction between a metal diffraction grating and a plane conductor,” Opt. Commun. 98, 225–230 (1993).
[CrossRef]

Roger, A.

Saarinen, J.

Sheng, Y.

Song, J. F.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Truckenbrodt, H.

J. Bischoff, H. Truckenbrodt, J. Bauer, “Diffraction analysis based characterization of very thin gratings,” in Micro-optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. Parriaux, B. Culshaw, M. Breidne, E. B. Kley, eds., Proc. SPIE3099, 212–222 (1997).
[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]

Villarrubia, J. S.

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Appl. Opt. (3)

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. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

A. Roberts, “Probe–sample interaction in near-field microscopy: analysis of the interaction between a metal diffraction grating and a plane conductor,” Opt. Commun. 98, 225–230 (1993).
[CrossRef]

N. Destouches, C.-A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Opt. Eng. (1)

M. Drege, J. A. Reed, D. M. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. 41, 225–236 (2002).
[CrossRef]

Semicond. Sci. Technol. (1)

J. M. Bonard, J. D. Ganière, F. Morier-Genoud, M. Achtenhagen, “Characterization of semi-conductor sub-micron gratings: is there an alternative to scanning microscopy?” Semicond. Sci. Technol. 11, 410–414 (1996).
[CrossRef]

Ultramicroscopy (1)

L. S. Dongmo, J. S. Villarrubia, S. N. Jones, T. B. Renegar, M. T. Postek, J. F. Song, “Experimental test of blind tip reconstruction for scanning probe microscopy,” Ultramicroscopy 85, 141–453 (2000).
[CrossRef]

Other (5)

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, New York, 1995).

J. Bischoff, H. Truckenbrodt, J. Bauer, “Diffraction analysis based characterization of very thin gratings,” in Micro-optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. Parriaux, B. Culshaw, M. Breidne, E. B. Kley, eds., Proc. SPIE3099, 212–222 (1997).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, S. S. H. Naqvi, “Using scat-tered light modeling for semiconductor critical dimension metrology and calibration,” in Integrated Circuits Metrology, Inspection, and Process Control VII, M. T. Postek, ed., Proc. SPIE1926, 60–70 (1993).
[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]

J. N. Hwang, C. H. Chan, R. J. Marks, “Frequency selective surface design based on iterative inversion of neural networks,” presented at IJCNN’90, the 2nd IEEE International Joint Conference on Neural Networks, Washington, D.C., January 15–19, 1990.

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Figures (11)

Fig. 1
Fig. 1

(a) Image and (b) profile measured by an AFM characterization of grating number 1.

Fig. 2
Fig. 2

Images of grating number 1 by an SEM characterization. (a) top view of the sample, (b) cross-sectional view.

Fig. 3
Fig. 3

Model of the grating etched in silicon. The period Λ is fixed to 1 μm. The three parameters characterizing the trapezoidal shape are the sidewall projection b1, the linewidth b2, and the groove depth h.

Fig. 4
Fig. 4

Representation of a multilayer perception 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 neuron in the hidden layer. A neuron i (or k) calculates its output by application of the transfer function f (or g) on the sum of weighted inputs. f is a sigmoidal function and g a linear one.

Fig. 5
Fig. 5

Experimental setup that allows automatic measurement of the reflected diffracted efficiencies from the silicon grating. The laser beam (λ=670 nm) is split into two parts: the reference collected by a fixed photodetector and the incident laser beam on the grating. The sample can rotate about the vertical axis to set the incident angle θ. The diffracted efficiencies are collected by a detector moving automatically around the grating.

Fig. 6
Fig. 6

Comparison of grating number 1 profiles measured by microscopic methods (AFM and SEM) and reconstructed by a NN (30 neurons). The SEM profile is extracted from the image by an image treatment. The grating period is 1 μm.

Fig. 7
Fig. 7

Comparison of the measured efficiencies and the different calculated efficiencies from the three profiles reconstructed by the NN and by the SEM and AFM measurements. The profiles are assumed to be trapezoidal. The calculation is performed for the diffraction orders r1 and r0 for both TE and TM polarization at incidence angle θ=40°.

Fig. 8
Fig. 8

Measured reflected efficiencies normalized to efficiencies in the center when the grating is scanned in TE polarization.

Fig. 9
Fig. 9

Measured reflected efficiencies normalized to efficiencies in the center when the grating is scanned in TM polarization.

Fig. 10
Fig. 10

Deviation of the grating parameters (b1, b2, and h) predicted by the NN from the center values versus the distance from center.

Fig. 11
Fig. 11

Variation of the error E. The variation represents the validity of the predicted parameters (b1, b2, and h) for each set of measurements along a vertical section in the grating.

Tables (3)

Tables Icon

Table 1 Grating Parameters (b1, b2 and h ) Obtained for Grating Number 1 by Three NNs (20, 30, and 40 Neurons in the Hidden layer)

Tables Icon

Table 2 Grating Parameters b1, b2, and h for Grating Number 1 for the Three Profiles Reconstructed by the NN Method and AFM and SEM Measurement

Tables Icon

Table 3 Grating Parameters Predicted for Grating Number 2 by Three NNs (20, 30, and 40 Neurons in the Hidden Layer) Trained in Four Domains a

Equations (5)

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

si=fjwi,jxj,
f(α)=11+exp(-α).
err=M(Scal-Starg)2,
-sin γ=sin θ+m λΛ.
E=150150Imes-IcalImes21/2.

Metrics