Abstract

Diffraction gratings have a wide array of applications in optics, diagnostics, food science, sensing, and process inspection. Scattering effects from defects can severely degrade the performance of such gratings. In this paper, we consider three classes of defects: Two classes introduced at the grating/air interface, as a change in line heights, and one class introduced as a sinusoidal variation of the grating/substrate interface. The scattering properties of the gratings are modelled using rigorous coupled wave analysis, and defects are approximated with a new semi-analytical model and a neural network. The new methods make it possible to avoid the time consuming library generation/search strategy commonly used in scatterometry. The method does not introduce new numerical parameters, and therefore no new parameter correlations. This work enables improved grating reconstruction, especially of non-diffracting short pitch gratings. It is found that two of the defect classes can be adequately described by the semi-analytical model, while the third defect is accurately reconstructed by a neural network. The network is demonstrated to be faster than a library search and more versatile for related structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

C. Pan, Z. Liu, Y. Pang, X. Zheng, H. Cai, Y. Zhang, and Z. Huang, “Design of a high-performance in-coupling grating using differential evolution algorithm for waveguide display,” Opt. Express 26(20), 26646–26662 (2018).
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J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

2017 (4)

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

M. G. S. Bernd, S. R. Bragança, N. Heck, and L. C. P. S. Filho, “Synthesis of carbon nanostructures by the pyrolysis of wood sawdust in a tubular reactor,” J. Mater. Res. Technol. 6(2), 171–177 (2017).
[Crossref]

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

P.-E. Hansen, M. H. Madsen, J. Lehtolahti, and L. Nielsen, “Traceable Mueller polarimetry and scatterometry for shape reconstruction of grating structures,” Appl. Surf. Sci. 421, 471–479 (2017).
[Crossref]

2016 (2)

M. H. Madsen and P.-E. Hansen, “Scatterometry—fast and robust measurements of nano-textured surfaces,” Surf. Topogr. 4(2), 023003 (2016).
[Crossref]

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

2014 (1)

W. Lee and S.-J. Park, “Porous anodic aluminum oxide: anodization and templated synthesis of functional nanostructures,” Chem. Rev. 114(15), 7487–7556 (2014).
[Crossref] [PubMed]

2012 (1)

J. E. Harvey, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51(1), 013402 (2012).
[Crossref]

2011 (1)

2008 (3)

2005 (2)

B. G. Kermani, S. S. Schiffman, and H. T. Nagle, “Performance of the Levenberg–Marquardt neural network training method in electronic nose applications,” Sens. Actuators B Chem. 110(1), 13–22 (2005).
[Crossref]

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing, and other techniques,” Chem. Rev. 105(4), 1171–1196 (2005).
[Crossref] [PubMed]

2002 (1)

1999 (1)

1998 (1)

M. Gardner and S. Dorling, “Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences,” Atmos. Environ. 32(14-15), 2627–2636 (1998).
[Crossref]

1993 (1)

1984 (1)

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23(4), 234406 (1984).
[Crossref]

1981 (1)

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[Crossref]

1961 (1)

1944 (1)

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

Adad, A.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Badran, F.

Barde, K. S.

D. Ramakrishnan, T. N. Singh, N. Purwar, K. S. Barde, A. Gulati, and S. Gupta, “Artificial neural network and liquefaction susceptibility assessment: a case study using the 2001 Bhuj earthquake data, Gujarat, India,” Comput. Geosci. 12(4), 491–501 (2008).
[Crossref]

Bennett, H. E.

Bernd, M. G. S.

M. G. S. Bernd, S. R. Bragança, N. Heck, and L. C. P. S. Filho, “Synthesis of carbon nanostructures by the pyrolysis of wood sawdust in a tubular reactor,” J. Mater. Res. Technol. 6(2), 171–177 (2017).
[Crossref]

Bilenberg, B.

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

Boher, P.

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

Bouachrine, M.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Bragança, S. R.

M. G. S. Bernd, S. R. Bragança, N. Heck, and L. C. P. S. Filho, “Synthesis of carbon nanostructures by the pyrolysis of wood sawdust in a tubular reactor,” J. Mater. Res. Technol. 6(2), 171–177 (2017).
[Crossref]

Cai, H.

Chtita, S.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Coriand, L.

Czolkos, I.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

Delort, T.

Dorling, S.

M. Gardner and S. Dorling, “Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences,” Atmos. Environ. 32(14-15), 2627–2636 (1998).
[Crossref]

Duparré, A.

Dwarakanath, D.

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

Filho, L. C. P. S.

M. G. S. Bernd, S. R. Bragança, N. Heck, and L. C. P. S. Filho, “Synthesis of carbon nanostructures by the pyrolysis of wood sawdust in a tubular reactor,” J. Mater. Res. Technol. 6(2), 171–177 (2017).
[Crossref]

Gardner, M.

M. Gardner and S. Dorling, “Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences,” Atmos. Environ. 32(14-15), 2627–2636 (1998).
[Crossref]

Garnæs, J.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

Garnaes, J.

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

Gates, B. D.

B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing, and other techniques,” Chem. Rev. 105(4), 1171–1196 (2005).
[Crossref] [PubMed]

Gaylord, T. K.

Gereige, I.

Gillespie, C. H.

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23(4), 234406 (1984).
[Crossref]

Granet, G.

Gulati, A.

D. Ramakrishnan, T. N. Singh, N. Purwar, K. S. Barde, A. Gulati, and S. Gupta, “Artificial neural network and liquefaction susceptibility assessment: a case study using the 2001 Bhuj earthquake data, Gujarat, India,” Comput. Geosci. 12(4), 491–501 (2008).
[Crossref]

Gupta, S.

D. Ramakrishnan, T. N. Singh, N. Purwar, K. S. Barde, A. Gulati, and S. Gupta, “Artificial neural network and liquefaction susceptibility assessment: a case study using the 2001 Bhuj earthquake data, Gujarat, India,” Comput. Geosci. 12(4), 491–501 (2008).
[Crossref]

Hansen, P. E.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

Hansen, P.-E.

P.-E. Hansen, M. H. Madsen, J. Lehtolahti, and L. Nielsen, “Traceable Mueller polarimetry and scatterometry for shape reconstruction of grating structures,” Appl. Surf. Sci. 421, 471–479 (2017).
[Crossref]

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

M. H. Madsen and P.-E. Hansen, “Scatterometry—fast and robust measurements of nano-textured surfaces,” Surf. Topogr. 4(2), 023003 (2016).
[Crossref]

Harvey, J. E.

J. E. Harvey, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51(1), 013402 (2012).
[Crossref]

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19(10), 9820–9835 (2011).
[Crossref] [PubMed]

Heck, N.

M. G. S. Bernd, S. R. Bragança, N. Heck, and L. C. P. S. Filho, “Synthesis of carbon nanostructures by the pyrolysis of wood sawdust in a tubular reactor,” J. Mater. Res. Technol. 6(2), 171–177 (2017).
[Crossref]

Hmamouchi, R.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Huang, Z.

Jensen, S. A.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

Johansson, A.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

Jørgensen, J. F.

J. S. Madsen, P. E. Hansen, P. Boher, D. Dwarakanath, J. F. Jørgensen, B. Bilenberg, J. Nygård, and M. H. Madsen, “Study on microgratings using imaging, spectroscopic, and fourier lens scatterometry,” J. Micro Nano-Manuf. 5(3), 031005 (2017).
[Crossref]

Kallioniemi, I.

Kermani, B. G.

B. G. Kermani, S. S. Schiffman, and H. T. Nagle, “Performance of the Levenberg–Marquardt neural network training method in electronic nose applications,” Sens. Actuators B Chem. 110(1), 13–22 (2005).
[Crossref]

Lacour, D.

Lakhlifi, T.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Larif, M.

R. Hmamouchi, M. Larif, S. Chtita, A. Adad, M. Bouachrine, and T. Lakhlifi, “Predictive modelling of the LD 50 activities of coumarin derivatives using neural statistical approaches: Electronic descriptor-based DFT,” J. Taibah Univ. Sci. 10(4), 451–461 (2016).
[Crossref]

Lee, W.

W. Lee and S.-J. Park, “Porous anodic aluminum oxide: anodization and templated synthesis of functional nanostructures,” Chem. Rev. 114(15), 7487–7556 (2014).
[Crossref] [PubMed]

Lehtolahti, J.

P.-E. Hansen, M. H. Madsen, J. Lehtolahti, and L. Nielsen, “Traceable Mueller polarimetry and scatterometry for shape reconstruction of grating structures,” Appl. Surf. Sci. 421, 471–479 (2017).
[Crossref]

Levenberg, K.

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

Li, L.

Liu, Z.

Madsen, J. S.

J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
[Crossref]

J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
[Crossref]

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J. S. Madsen, L. H. Thamdrup, I. Czolkos, P.-E. Hansen, A. Johansson, J. Garnaes, J. Nygard, and M. H. Madsen, “In-line characterization of nanostructured mass-produced polymer components using scatterometry,” J. Micromech. Microeng. 27(8), 085004 (2017).
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P.-E. Hansen, M. H. Madsen, J. Lehtolahti, and L. Nielsen, “Traceable Mueller polarimetry and scatterometry for shape reconstruction of grating structures,” Appl. Surf. Sci. 421, 471–479 (2017).
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J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
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J. S. Madsen, S. A. Jensen, L. Nakotte, A. Vogelsang, L. H. Thamdrup, I. Czolkos, A. Johansson, J. Garnæs, T. Nielsen, J. Nygård, and P. E. Hansen, “Scatterometry for optimization of injection molded nanostructures at the fabrication line,” Int. J. Adv. Manuf. Technol. 99(9-12), 2669–2676 (2018).
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P.-E. Hansen, M. H. Madsen, J. Lehtolahti, and L. Nielsen, “Traceable Mueller polarimetry and scatterometry for shape reconstruction of grating structures,” Appl. Surf. Sci. 421, 471–479 (2017).
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Figures (7)

Fig. 1
Fig. 1 Sketches of the different grating imperfections examined. The denoted parameters are the height of the perfect grating, h, the width of the grating, w, and the size of the imperfection, d, the period of the simple grating, Δ, and the period of the supercell, Γ. The marked volume shows where the defect is introduced and the plot to the left shows the profile of the defect used in the semi-analytical model. (a) Simple grating defect. (b) Sinusoidal grating defect. (c) A perfect grating on a sinusoidal substrate. All imperfections have been exaggerated for clarity.
Fig. 2
Fig. 2 Defect class 1 (top), class 2 (middle) and class 3 (bottom). Fully drawn lines show the δ η Num while the slightly darker crosses show δ η SA . d is illustrated in Fig. 1. Note the different y-axes.
Fig. 3
Fig. 3 Mean square error as a function of defect size used in the semi-analytical model for defect class 2 using pure RCWA (a) and RCWA with added white Gaussian noise (b). The legends and the dashed lines indicate the defect size d used in the RCWA calculations, while the fully drawn lines indicate the defect sizes used in the semi-analytical model.
Fig. 4
Fig. 4 Same approach as Fig. 3 but using the simplified (first order Taylor) TIS model for defect class 2. The legends and the dashed lines indicate the defect size d used in the RCWA calculations, while the fully drawn lines indicate the defect sizes used in the semi-analytical model. We see that this model already falls off for defect values above 28 nm (corresponding to 4% of the grating height).
Fig. 5
Fig. 5 Sketch of the neural network. The input layer has a node for each wavelength simulated (121). The hidden layer has a total of 10 nodes, and the output layer has a single node finding the defect magnitude d. The nodes from the input layer are connected to the hidden layer through a weighted Tansig transfer function. In the same manner, all nodes in the hidden layer are connected to the output node through a Purelin transfer function.
Fig. 6
Fig. 6 Performance of the Neural network. The defect found by the neural network, dF, is plotted vs the targeted defect, dT, from the simulated models with added noise. The black dashed line shows the best linear fit to the data.
Fig. 7
Fig. 7 Evaluation of the developed network using different substrate defects. Circles shows the magnitude found by the network, and the dashed lines show the best linear fit. The diagonal line shows the fit obtained from the uncut sinusoidal in Fig. 6 to guide the eye. Upper insert shows the simulated structures from uncut to 50% cut. The curves have been displaced for clarity. The grating lines on top of the substrate has been omitted in the illustration. The Arrows mark 2d for the different cut values, where d is the target value for the given structure. Lower insert: table showing the linear coefficient for the fit: dF = a dT.

Equations (7)

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

TIS= R T R S R T =1exp( ( 4π σcos( θ I ) λ ) 2 )
σ= 1 Γ 0 Γ f(x) 2 dx
η SA = I S I I = η Grat (1TIS)
MSE(δ,d)= 1 N i=1 N ( η SA ( λ i ,δ) η Num ( λ i ,d) ) 2
TIS= R T R S R T =1exp( ( 4π σcos( θ I ) λ ) 2 ) ( 4π σcos( θ I ) λ ) 2
g=4π σcos( θ I ) λ <<1
d F =0.9959 d T

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