Abstract

Diffuse scattering of optical one-dimensional gratings becomes increasingly critical as it constrains the performance, e.g., of grating spectrometers. In particular, stochastic disturbances of the ideal grating structure provoke straylight. In this paper, the straylight spectrum of stochastically disturbed gratings is examined. First, a 1D-method is presented that allows to calculate 2D-diffuse scattering of arbitrarily polarized light originating from stochastic disturbances of the grating geometry on the basis of standard optical simulation tools. Within the scope of this method an enormous reduction of computational effort is achieved compared to the full 2D-simulation approach, i.e., the computation time can be reduced by several orders of magnitude. Hence, the method also allows to address even large period gratings that are not possible to calculate within a full 2D-approach. In analogy to scattering theories for surface roughness the method relies on typical characteristics of straylight originating from small disturbances, that the angle resolved scattering (ARS) can be separated into a product of the power spectral density describing the 2D stochastic process and additional factors depending on the undisturbed 1D grating structure. In a second part, an analytical model within Fourier optics utilizing thin element approximation (TEA) describing the wide angle scattering of lamellar gratings disturbed by line edge roughness (LER) for TE-polarized light is derived and verified by applying the 1D-simulation method. For shallow gratings, we find an excellent agreement between simulation and TEA over the whole transmission half space. In addition, this model allows a descriptive understanding of the underlying physical effects and, accordingly, the influence of relevant parameters (grating geometry, refractive indices, illumination) onto the scattering spectra is discussed. Further, it is shown that LER-scattering can be described within a modified Rayleigh-Rice-ARS usually found within the frame of surface roughness.

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

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2017 (3)

M. Kroneberger and S. Fray, “Scattering from reflective diffraction gratings: the challenges of measurement and simulation,” Adv. Opt. Technol. 6, 379–386 (2017).

M. Heusinger, M. Banasch, and U. D. Zeitner, “Rowland ghost suppression in high efficiency spectrometer gratings fabricated by e-beam lithography,” Opt. Express 25, 6182–6191 (2017).
[Crossref] [PubMed]

H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017).
[Crossref]

2016 (2)

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

M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016).
[Crossref]

2015 (1)

B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015).
[Crossref]

2014 (3)

S. Schröder, D. Unglaub, M. Trost, X. Cheng, J. Zhang, and A. Duparré, “Spectral angle resolved scattering of thin film coatings,” Appl. Opt. 53, A35–A41 (2014).
[Crossref] [PubMed]

C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithography, MEMS MOEMS 13, 020501 (2014).
[Crossref]

T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014).
[Crossref]

2012 (3)

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, “Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,” Appl. Opt. 51, 7384–7394 (2012).
[Crossref] [PubMed]

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

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

2011 (2)

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, 9820–9835 (2011).
[Crossref] [PubMed]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011).
[Crossref] [PubMed]

2010 (2)

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010).
[Crossref] [PubMed]

A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010).
[Crossref]

2009 (1)

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

2004 (1)

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[Crossref]

2003 (1)

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

2002 (1)

E. N. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” JOSA A 19, 702–715 (2002).
[Crossref] [PubMed]

2001 (1)

B. Schnabel and E.-B. Kley, “On the influence of the e-beam writer address grid on the optical quality of high-frequency gratings,” Microelectron. Eng. 57, 327–333 (2001).
[Crossref]

2000 (2)

1995 (2)

M. D. Perry, R. D. Boyd, J. A. Britten, D. Decker, B. W. Shore, C. Shannon, and E. Shults, “High-efficiency multilayer dielectric diffraction gratings,” Opt. Lett. 20, 940–942 (1995).
[Crossref] [PubMed]

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

1994 (1)

D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994).
[Crossref]

1978 (1)

M. Sharpe and D. Irish, “Straylight in diffraction grating monochromators,” Opt. acta 25, 861–893 (1978).
[Crossref]

1965 (1)

Acheta, A.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Balintfy, J. L.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

Banasch, M.

M. Heusinger, M. Banasch, and U. D. Zeitner, “Rowland ghost suppression in high efficiency spectrometer gratings fabricated by e-beam lithography,” Opt. Express 25, 6182–6191 (2017).
[Crossref] [PubMed]

M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

Bär, M.

H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017).
[Crossref]

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, “Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,” Appl. Opt. 51, 7384–7394 (2012).
[Crossref] [PubMed]

Bell, S.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Belting, C.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Benkenstein, T.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Bergner, B. C.

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010).
[Crossref] [PubMed]

Bézy, J.-L.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Bhatti, I.

D. Lobb and I. Bhatti, “Applications of immersed diffraction gratings in earth observation from space,” in ICSO 2010, 10565 (SPIE, 2017), p. 105651M.

Bischoff, J.

J. Bischoff and K. Hehl, “Scatterometry modeling for gratings with roughness and irregularities,” in Metrology, Inspection, and Process Control for Microlithography XXX, 9778 (SPIE, 2016), p. 977804.

Bouvet, M.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Boyd, R. D.

Britten, J. A.

Burdick, D. S.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

Castleman, Z.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Cheng, X.

Choi, N.

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

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011).
[Crossref] [PubMed]

Chu, K.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

Coatantiec, C.

B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

Constantoudis, V.

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

E. Gogolides, V. Constantoudis, and G. Kokkoris, “Towards an integrated line edge roughness understanding: metrology, characterization, and plasma etching transfer,” in Advanced Etch Technology for Nanopatterning II, 8685 (SPIE, 2013), p. 868505.

Coriand, L.

Decker, D.

Deep, A.

B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015).
[Crossref]

B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

Del Bello, U.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Drake, G.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Drusch, M.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Duparré, A.

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[Crossref]

Espejo, J.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Flügel-Paul, T.

M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016).
[Crossref]

M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

Fray, S.

M. Kroneberger and S. Fray, “Scattering from reflective diffraction gratings: the challenges of measurement and simulation,” Adv. Opt. Technol. 6, 379–386 (2017).

Frenner, K.

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

Fuchs, F.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Gaylord, T.

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

Germer, T. A.

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010).
[Crossref] [PubMed]

E. Marx, T. A. Germer, T. V. Vorburger, and B. C. Park, “Angular distribution of light scattered from a sinusoidal grating,” Appl. Opt. 39, 4473–4485 (2000).
[Crossref]

T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” in Advanced Lithography, (SPIE, 2007), pp. 65180Z.
[Crossref]

Glytsis, E. N.

E. N. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” JOSA A 19, 702–715 (2002).
[Crossref] [PubMed]

Gogolides, E.

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

E. Gogolides, V. Constantoudis, and G. Kokkoris, “Towards an integrated line edge roughness understanding: metrology, characterization, and plasma etching transfer,” in Advanced Etch Technology for Nanopatterning II, 8685 (SPIE, 2013), p. 868505.

Grann, E. B.

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994).
[Crossref]

Gross, H.

H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017).
[Crossref]

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, “Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,” Appl. Opt. 51, 7384–7394 (2012).
[Crossref] [PubMed]

Guérin, C.-A.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[Crossref]

Guldimann, B.

B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015).
[Crossref]

Hagen, C. W.

T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014).
[Crossref]

Hansen, P.-E.

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

Harnisch, B.

B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Harvey, J. E.

J. E. Harvey, N. Choi, S. Schroeder, and A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 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, 9820–9835 (2011).
[Crossref] [PubMed]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011).
[Crossref] [PubMed]

Harzendorf, T.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Hehl, K.

J. Bischoff and K. Hehl, “Scatterometry modeling for gratings with roughness and irregularities,” in Metrology, Inspection, and Process Control for Microlithography XXX, 9778 (SPIE, 2016), p. 977804.

Heidenreich, S.

H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017).
[Crossref]

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, and M. Bär, “Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,” Appl. Opt. 51, 7384–7394 (2012).
[Crossref] [PubMed]

Henn, M.-A.

Hessel, A.

Heuerman, K.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Heusinger, M.

M. Heusinger, M. Banasch, and U. D. Zeitner, “Rowland ghost suppression in high efficiency spectrometer gratings fabricated by e-beam lithography,” Opt. Express 25, 6182–6191 (2017).
[Crossref] [PubMed]

M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016).
[Crossref]

M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

Irish, D.

M. Sharpe and D. Irish, “Straylight in diffraction grating monochromators,” Opt. acta 25, 861–893 (1978).
[Crossref]

Kato, A.

Kley, E.-B.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

B. Schnabel and E.-B. Kley, “On the influence of the e-beam writer address grid on the optical quality of high-frequency gratings,” Microelectron. Eng. 57, 327–333 (2001).
[Crossref]

Kokkoris, G.

E. Gogolides, V. Constantoudis, and G. Kokkoris, “Towards an integrated line edge roughness understanding: metrology, characterization, and plasma etching transfer,” in Advanced Etch Technology for Nanopatterning II, 8685 (SPIE, 2013), p. 868505.

Kopp, G.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Kraft, S.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

Kroneberger, M.

M. Kroneberger and S. Fray, “Scattering from reflective diffraction gratings: the challenges of measurement and simulation,” Adv. Opt. Technol. 6, 379–386 (2017).

Kruit, P.

T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014).
[Crossref]

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011).
[Crossref] [PubMed]

La Fontaine, B.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Lamprecht, B.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Lemarchand, F.

H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” JOSA A 17, 1048–1058 (2000).
[Crossref] [PubMed]

Levinson, H. J.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Lobb, D.

D. Lobb and I. Bhatti, “Applications of immersed diffraction gratings in earth observation from space,” in ICSO 2010, 10565 (SPIE, 2017), p. 105651M.

Loewen, E. G.

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Lu, T.-M.

Y. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface – Principles and Applications (Elsevier, 2000).

Mack, C. A.

C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithography, MEMS MOEMS 13, 020501 (2014).
[Crossref]

Madsen, M. H.

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

Marx, E.

Michaelis, D.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Moharam, M.

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994).
[Crossref]

Naylor, T. H.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

Oliner, A.

Oliva, M.

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Osten, W.

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

Palmer, C. A.

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

Park, B. C.

Patsis, G.

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

Pawloski, A. R.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Paz, V. F.

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

Penalver, D. H.

Perry, M. D.

Pilewskie, P.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Pommet, D. A.

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994).
[Crossref]

Rafler, S.

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

Rathsfeld, A.

Rigneault, H.

H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” JOSA A 17, 1048–1058 (2000).
[Crossref] [PubMed]

Schnabel, B.

B. Schnabel and E.-B. Kley, “On the influence of the e-beam writer address grid on the optical quality of high-frequency gratings,” Microelectron. Eng. 57, 327–333 (2001).
[Crossref]

Scholze, F.

Schröder, S.

Schroeder, S.

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

Schuster, T.

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

Sentenac, A.

H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” JOSA A 17, 1048–1058 (2000).
[Crossref] [PubMed]

Shannon, C.

Sharpe, M.

M. Sharpe and D. Irish, “Straylight in diffraction grating monochromators,” Opt. acta 25, 861–893 (1978).
[Crossref]

Shore, B. W.

Shults, E.

Smith, P.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering, 1995).
[Crossref]

Suleski, T. J.

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010).
[Crossref] [PubMed]

Trost, M.

Tserepi, A.

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

Tünnermann, A.

Unglaub, D.

Verduin, T.

T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014).
[Crossref]

Vermeer, B.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

Vink, R.

B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015).
[Crossref]

B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

Vorburger, T. V.

Wallow, T.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

Wang, G.-C.

Y. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface – Principles and Applications (Elsevier, 2000).

Zeitner, U. D.

M. Heusinger, M. Banasch, and U. D. Zeitner, “Rowland ghost suppression in high efficiency spectrometer gratings fabricated by e-beam lithography,” Opt. Express 25, 6182–6191 (2017).
[Crossref] [PubMed]

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

Zeitner, U.-D.

M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016).
[Crossref]

Zhang, J.

Zhao, Y.

Y. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface – Principles and Applications (Elsevier, 2000).

Adv. Opt. Technol. (1)

M. Kroneberger and S. Fray, “Scattering from reflective diffraction gratings: the challenges of measurement and simulation,” Adv. Opt. Technol. 6, 379–386 (2017).

Appl. Opt. (5)

Appl. Phys. A (1)

U. D. Zeitner, M. Oliva, F. Fuchs, D. Michaelis, T. Benkenstein, T. Harzendorf, and E.-B. Kley, “High performance diffraction gratings made by e-beam lithography,” Appl. Phys. A 109, 789–796 (2012).
[Crossref]

Appl. Phys. B (1)

M. Heusinger, T. Flügel-Paul, and U.-D. Zeitner, “Large-scale segmentation errors in optical gratings and their unique effect onto optical scattering spectra,” Appl. Phys. B 122, 222 (2016).
[Crossref]

CEAS Space J. (1)

B. Guldimann, A. Deep, and R. Vink, “Overview on grating developments at esa,” CEAS Space J. 7, 433–451 (2015).
[Crossref]

J. Micro/Nanolithography, MEMS MOEMS (2)

C. A. Mack, “Analytical expression for impact of linewidth roughness on critical dimension uniformity,” J. Micro/Nanolithography, MEMS MOEMS 13, 020501 (2014).
[Crossref]

T. Verduin, P. Kruit, and C. W. Hagen, “Determination of line edge roughness in low-dose top-down scanning electron microscopy images,” J. Micro/Nanolithography, MEMS MOEMS 13, 033009 (2014).
[Crossref]

J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. (1)

V. Constantoudis, G. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. ii. scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. & Technol. B: Microelectron. Nanometer Struct. Process. Meas. Phenom. 21, 1019–1026 (2003).
[Crossref]

JOSA A (6)

M. Moharam, E. B. Grann, D. A. Pommet, and T. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” JOSA A 12, 1068–1076 (1995).
[Crossref]

H. Rigneault, F. Lemarchand, and A. Sentenac, “Dipole radiation into grating structures,” JOSA A 17, 1048–1058 (2000).
[Crossref] [PubMed]

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” JOSA A 27, 1083–1090 (2010).
[Crossref] [PubMed]

D. A. Pommet, M. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” JOSA A 11, 1827–1834 (1994).
[Crossref]

E. N. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” JOSA A 19, 702–715 (2002).
[Crossref] [PubMed]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” JOSA A 28, 1121–1138 (2011).
[Crossref] [PubMed]

Measurement (1)

H. Gross, S. Heidenreich, and M. Bär, “Impact of different stochastic line edge roughness patterns on measurements in scatterometry-a simulation study,” Measurement 98, 339–346 (2017).
[Crossref]

Microelectron. Eng. (2)

T. Schuster, S. Rafler, V. F. Paz, K. Frenner, and W. Osten, “Fieldstitching with kirchhoff-boundaries as a model based description for line edge roughness (ler) in scatterometry,” Microelectron. Eng. 86, 1029–1032 (2009).
[Crossref]

B. Schnabel and E.-B. Kley, “On the influence of the e-beam writer address grid on the optical quality of high-frequency gratings,” Microelectron. Eng. 57, 327–333 (2001).
[Crossref]

Opt. acta (1)

M. Sharpe and D. Irish, “Straylight in diffraction grating monochromators,” Opt. acta 25, 861–893 (1978).
[Crossref]

Opt. Eng. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Surf. Topogr. Metrol. Prop. (1)

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

Waves Random Media (1)

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[Crossref]

Other (13)

Y. Zhao, G.-C. Wang, and T.-M. Lu, Characterization of Amorphous and Crystalline Rough Surface – Principles and Applications (Elsevier, 2000).

T. A. Germer, “Modeling the effect of line profile variation on optical critical dimension metrology,” in Advanced Lithography, (SPIE, 2007), pp. 65180Z.
[Crossref]

M. Heusinger, M. Banasch, T. Flügel-Paul, and U. D. Zeitner, “Investigation and optimization of rowland ghosts in high efficiency spectrometer gratings fabricated by e-beam lithography,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics IX, 9759 (SPIE, 2016), p. 97590A.

C. A. Palmer and E. G. Loewen, Diffraction Grating Handbook (Newport Corporation, 2005).

D. Lobb and I. Bhatti, “Applications of immersed diffraction gratings in earth observation from space,” in ICSO 2010, 10565 (SPIE, 2017), p. 105651M.

S. Kraft, U. Del Bello, B. Harnisch, M. Bouvet, M. Drusch, and J.-L. Bézy, “Fluorescence imaging spectrometer concepts for the earth explorer mission candidate flex,” in ICSO 2012, 10564 (SPIE, 2017), p. 105641W.

G. Kopp, P. Pilewskie, C. Belting, Z. Castleman, G. Drake, J. Espejo, K. Heuerman, B. Lamprecht, P. Smith, and B. Vermeer, “Radiometric absolute accuracy improvements for imaging spectrometry with hysics,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, (IEEE, 2013), pp. 3518–3521.

B. Harnisch, A. Deep, R. Vink, and C. Coatantiec, “Grating scattering brdf and imaging performances: A test survey performed in the frame of the flex mission,” in ICSO 2012, 10564 (SPIE, 2017), p. 105642P.

J. Bischoff and K. Hehl, “Scatterometry modeling for gratings with roughness and irregularities,” in Metrology, Inspection, and Process Control for Microlithography XXX, 9778 (SPIE, 2016), p. 977804.

E. Gogolides, V. Constantoudis, and G. Kokkoris, “Towards an integrated line edge roughness understanding: metrology, characterization, and plasma etching transfer,” in Advanced Etch Technology for Nanopatterning II, 8685 (SPIE, 2013), p. 868505.

A. R. Pawloski, A. Acheta, S. Bell, B. La Fontaine, T. Wallow, and H. J. Levinson, “The transfer of photoresist ler through etch,” in Advances in Resist Technology and Processing XXIII, 6153 (SPIE, 2006), p. 615318.

T. H. Naylor, J. L. Balintfy, D. S. Burdick, and K. Chu, “Computer simulation techniques,” Tech. rep., Wiley (1966).

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering, 1995).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Power spectral density for different values of the roughness parameters: standard deviation σ (a) correlation length ξ (b) and roughness exponent α (c) and typical rough line edges according to each set of parameters.
Fig. 2
Fig. 2 (a) Illustration of the underlying principle of the applied one- and two-dimensional model that was used for simulating the scattered light distribution of LER-disturbed gratings. Left: 2D-model applies full LER-characteristics within an area Px × Py and calculates stray light orders (SO) in the full half space (not illustrated). Right: 1D-model deduced from the profile of the 2D-grating-structure calculates SOs in dispersion plane. (b) 1D- and 2D-simulation results (average of 32 single simualtions) for a monolithic grating in fused silica (ni = 1.457) with p = 667 nm, d = 1640 nm, b = 430 nm, θi = 22° and λ = 633 nm and a roughness characterized by σ = 3 nm, ξ = 50 nm, α = 0.5. The 99%-CI (colored shading and error bars) represents the error. A factor S harmonizes the different simulation results (blue solid curve, cf. Eq. (7)).
Fig. 3
Fig. 3 Comparison of the numerically calculated scattering spectra within the 1D- and 2D-approach (red line and black dots, respectively) for a contemporary spectrometer grating (p = 667 nm, b = 430 nm, d = 1640 nm, λ = 720 nm, θi = 22°, P ¯ = TE, ni = 1.457, nt = 1, σ = 4 nm, ξ = 500 nm, α = 0.5). (a) Complete 2D-simulation for a domain Px × Py = 12p × 5 µm with the 0th and −1st DO at (mx, my) = (0, 0) and (mx, my) = (−12,0), respectively. (b) Comparison with 1D-simulation (N = 100) for straylight along the dispersion direction. (c) Comparison with 1D-simulation (N = 100) for conical scattering parallel to the dispersion direction with θy ≈ 30°.
Fig. 4
Fig. 4 Comparison of the numerically calculated scattering spectra (mean of 32 simulations) within the 1D- and 2D-approach (red line and black dots, respectively) for shallow and deep monolithic gratings (with different periods). The gray circles belong to a single simulation. The LER of the grating lines was set to be σ = 3 nm, ξ = 50 nm and α = 0.5 and the illuminating wave is characterized by λ = 633 nm, P ¯ = TE.
Fig. 5
Fig. 5 Influence of the roughness parameters onto the stray light spectra of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1). The thick line represents the TEA-model whereas the thin line is the 1D-RCWA result.
Fig. 6
Fig. 6 Above: Influence of the grating thickness d onto the straylight spectrum of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1) for different refractive indices ni and nt. The thick line represents the TEA-model whereas the thin line corresponds to the 1D-rigorous simulation. Bottom: Interference term as a function of the grating depth with the colored marker indicating the depths of the corresponding stray light simulation. The blue marker corresponds to dc.
Fig. 7
Fig. 7 Influence of the grating period p (a), the duty cycle FF (b) and the angle of incidence θi (c) onto the stray light spectrum of a thin monolithic low frequency grating (parameter set as given in Sec. 4.1). The thick line represents the TEA-model whereas the thin line corresponds to 1D-RCWA Data.
Fig. 8
Fig. 8 Calculated phase shift (a) and Fresnel coefficient (and relative amplitude, respectively) (b) of every single diffraction order of a fused silica grating with p = 60 µm, FF = 0.645 and different grating depths compared to the phase function and Fresnel coefficient used for the TEA-model.

Equations (11)

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P S D ( f ) = P 0 ( 1 + ( 2 π f ξ ) 2 ) H + 0.5 ,
A R S 1 D ( θ x ( m ) ) = d P scat d θ x P i 2 η m | θ x ( m 1 ) θ x ( m + 1 ) | ,
A R S 2 D ( θ x , θ y ) = d P scat d Ω P i η m x m y Δ Ω m x m y ,
s scat ( k t , ) = [ F left Δ X ˜ left ( k i , , k i , ) + F right Δ X ˜ right ( k i , , k i , ) ] s i ( k i , ) ,
A R S 2 D = d P scat d Ω P i = k 0 n t k t , z 2 ( θ x ) k i , z ( θ i ) [ | F LER , left | 2 + | F LER , right | 2 ] P S D ( k t , y k i , y ) p ,
A R S 1 D = d P scat d θ x P i = 2 π k t , z 2 ( θ x ) k i , z ( θ i ) [ | F 1 D , left | 2 + | F 1 D , right | 2 ] σ 2 p .
A R S 2 D [ n t λ P S D ( k t , y k i , y ) σ 2 ] A R S 1 D = S A R S 1 D .
g ( x , y ) = l = 0 L 1 ( x l p p ) t e i φ g + l = 0 L 1 ( x l p Δ p l ( y ) p + Δ p l ( y ) ) [ t e i φ b + t e i φ g ] ,
A R S LER = 4 n t 3 cos 2 θ x p λ 2 n i cos θ i | t ( θ i ) | 2 [ 1 cos ( φ b ( θ x ) φ g ( θ x ) ) ] P S D LER ( θ y ) .
A R S SR = 1 λ 4 n t 3 n i cos 2 θ x cos θ i | r ( θ i ) | | r ( θ x ) | P S D SR .
A R S LER ( d λ ) = 1 λ 4 n t 3 n i cos 2 θ x cos θ i | r ( θ i ) | | ρ ( θ x ) | [ d 2 P S D LER p ] ,

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