Abstract

The Backside Absorbing Layer Microscopy (BALM) is a recently introduced surface imaging technique in reflected light with an unprecedented combination of sensitivity and lateral resolution, hence very promising for the development of imaging sensors. This requires to turn BALM images into quantative analyte measurements. The usual way to analyze reflectivity is to compare the optical signal and a numerical model with many adjustable parameters. Here we demonstrate a universal relationship between the sample reflectivity and the physical thickness of the sample, ruled by three measurable quantities. Mapping the physical sample thickness becomes possible whatever the instrument setting and the sample refractive index. Application to kinetic measurements is discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  6. U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (1)

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

2018 (2)

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

2017 (2)

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

2015 (2)

2014 (1)

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

2006 (1)

D. Ausserré and M.-P. Valignat, “Wide-Field Optical Imaging of Surface Nanostructures,” Nano Lett. 6(7), 1384–1388 (2006).
[Crossref]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

1985 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

1954 (1)

L. G. Parratt, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

Abou Khachfe, R.

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Amra, C.

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Ausserre, D.

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Ausserré, D.

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

D. Ausserré and M.-P. Valignat, “Wide-Field Optical Imaging of Surface Nanostructures,” Nano Lett. 6(7), 1384–1388 (2006).
[Crossref]

Avci, O.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Aygun, U.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, “Ellipsometry and polarized light”, ISBN North-Holland 0 7204 0694 3, (1977).

Bai, J.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, “Ellipsometry and polarized light”, ISBN North-Holland 0 7204 0694 3, (1977).

Bing, D.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Brotons, G.

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Campidelli, S.

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Combellas, C.

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

Cornut, R.

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Derycke, V.

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Du, R.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Geim, A. K.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Jaouen, K.

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Kang, S.

H. H. Nguyen, J. Park, S. Kang, and M. Kim, “Surface Plasmon Resonance: A Versatile Technique for Biosensor Applications,” Sensors 15(5), 10481–10510 (2015)..
[Crossref]

Kanoufi, F.

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

Kim, M.

H. H. Nguyen, J. Park, S. Kang, and M. Kim, “Surface Plasmon Resonance: A Versatile Technique for Biosensor Applications,” Sensors 15(5), 10481–10510 (2015)..
[Crossref]

Kravets, V. G.

Lemarchand, F.

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Lemineur, J.-F.

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

Leng, J.

Liu, L.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Marshall, O. P.

Monteiller, J.

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Nair, R. R.

Nguyen, H. H.

H. H. Nguyen, J. Park, S. Kang, and M. Kim, “Surface Plasmon Resonance: A Versatile Technique for Biosensor Applications,” Sensors 15(5), 10481–10510 (2015)..
[Crossref]

Noël, J.-M.

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Nygren, H.

Ozkumur, A. Y.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Park, J.

H. H. Nguyen, J. Park, S. Kang, and M. Kim, “Surface Plasmon Resonance: A Versatile Technique for Biosensor Applications,” Sensors 15(5), 10481–10510 (2015)..
[Crossref]

Parratt, L. G.

L. G. Parratt, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

Rigorenko, A. N.

Roussille, L.

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Sandström, T.

Seymour, E.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Stenberg, M.

Thackray, B.

Ünlü, M. S.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Urey, H.

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Valignat, M.-P.

D. Ausserré and M.-P. Valignat, “Wide-Field Optical Imaging of Surface Nanostructures,” Nano Lett. 6(7), 1384–1388 (2006).
[Crossref]

Vonna, L.

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Wang, Y.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Wu, G.

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Zerrad, M.

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Zhukov, A.

ACS Sens. (1)

U. Aygun, O. Avci, E. Seymour, H. Urey, M. S. Ünlü, and A. Y. Ozkumur, “Label -Free and High-Throughput Detection of Biomolecular Interactions Using a Flatbed Scanner Biosensor,” ACS Sens. 2(10), 1424–1429 (2017)..
[Crossref]

Angew. Chem., Int. Ed. (1)

J.-F. Lemineur, J.-M. Noël, D. Ausserré, C. Combellas, and F. Kanoufi, “Combining electrodeposition and optical microscopy for probing size-dependent single nanoparticle electrochemistry,” Angew. Chem., Int. Ed. 57(37), 11998–12002 (2018).
[Crossref]

Appl. Opt. (1)

J. Nanomed. Nanotechnol. (1)

D. Ausserre, C. Amra, R. Abou Khachfe, L. Roussille, G. Brotons, L. Vonna, F. Lemarchand, and M. Zerrad, “Anti-Reflecting Absorbing Layers for Electrochemical and Biophotonic Applications,” J. Nanomed. Nanotechnol. 5(4), 1000214 (2014).
[Crossref]

Nano Lett. (1)

D. Ausserré and M.-P. Valignat, “Wide-Field Optical Imaging of Surface Nanostructures,” Nano Lett. 6(7), 1384–1388 (2006).
[Crossref]

Nanoscale (1)

K. Jaouen, R. Cornut, D. Ausserré, S. Campidelli, and V. Derycke, “Ideal optical contrast for 2D materials observation using bi-layer antireflection absorbing substrates,” Nanoscale 11(13), 6129–6135 (2019)..
[Crossref]

Opt. Commun. (1)

D. Bing, Y. Wang, J. Bai, R. Du, G. Wu, and L. Liu, “Optical contrast for identifying the thickness of two-dimensional materials,” Opt. Commun. 406, 128–138 (2018).
[Crossref]

Opt. Express (1)

Phys. Rev. (1)

L. G. Parratt, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Sci. Adv. (1)

S. Campidelli, R. Abou Khachfe, K. Jaouen, J. Monteiller, C. Amra, M. Zerrad, R. Cornut, V. Derycke, and D. Ausserré, “Backside absorbing layer microscopy: Watching graphene chemistry,” Sci. Adv. 3(5), e1601724 (2017).
[Crossref]

Science (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004).
[Crossref]

Sensors (1)

H. H. Nguyen, J. Park, S. Kang, and M. Kim, “Surface Plasmon Resonance: A Versatile Technique for Biosensor Applications,” Sensors 15(5), 10481–10510 (2015)..
[Crossref]

Other (2)

http://www.watchlive.fr/ taken on 2019, june 20th.

R. M. A. Azzam and N. M. Bashara, “Ellipsometry and polarized light”, ISBN North-Holland 0 7204 0694 3, (1977).

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Figures (4)

Fig. 1.
Fig. 1. a) Sketch of the BALM setup and layer numbering; b) Hypothetical reflectivity curves as a function of sample thickness. The minimum reflectivity ${\textrm{R}_{\textrm{min}}}$ may correspond to positive (red curve) or negative (blue curve) values of ${\textrm{e}_{\textrm{min}}}$. They can be simultaneously obtained when probing a sample with two different wavelengths. Examples may be found in further Fig. 3(e); c) When ${\textrm{e}_{\textrm{min}}}$ positive, to a given value of $\textrm{R}$ correspond two values ${e_1}$ and ${e_2}$ of the sample thickness e. They can be separated with the help of a second wavelength.
Fig. 2.
Fig. 2. a): Reflectivity as a function of the sample thickness ${e_2}$ for 24 configurations : [${e_1}$ = 3, 4 and 5 nm (gold)] x [λ=450, 480, 550 nm and addition of the three wavelengths (“white”)] x [$\theta = 0$ or 30 deg. aperture angle], with one graph per thickness; b): same with the 24 superimposed results; c): transformed reflectivity $f({R/{R_{min}},{R_0}/{R_{min}}} )$, noted $f(R )$, as a function of the reduced sample thickness $e/{e_{min}}$, with different ${e_{min}},\; {R_{min}}$ and ${R_0}$ for each parameter configuration; d): same curve as c) with each thickness isolated and the corresponding $e/{e_{min}}$ range compared to each other using dotted lines; Figure insert : color code for each (λ, aperture) setting. The refractive indices ${n_0}$, ${n_2},$ ${n_3}$, weakly dependent of λ, are respectively fixed to 1.52, 1.5 and 1.34, and ${n_1}\; $ (λ) was taken from [14].
Fig. 3.
Fig. 3. a),c) : GASn3W, ${\theta} = 0$ ; b),d): GAGOn3W and illumination cone $\theta = 0 - 30 \deg.$ a),b): ${e_{3min}}$ (probed layer) as a function of ${e_2}$ for a number of ${n_3}$ refractive indices ranging from 1.4 to 1.7; c),d): ${R_{min}}$ as a function of ${e_2}$ for the same ${n_3}$ values; e),f) GAGOn3W with $\theta = 0$ and ${e_1}$ = 3nm (gold). ${e_{3min}}$ and ${R_{min}}$ as a function of ${e_2}$ (GO) for four different wavelengths; inserts: color codes for the sample refractive index in 2a-d and for the wavelength in 2e-f. The GO refractive index as a function of λ was taken from [15].
Fig. 4.
Fig. 4. Schematic sequence of a kinetic experiment. The green arrows show two measurement and analysis sequences with two different illuminations. c) postulated kinetics ${e_3}({t/\tau } )$; a) Simulated experimental reflectivity $R({t/\tau } )$ for a normal incidence; e) Simulated reflectivity $R({t/\tau } )$ for a 30 deg. aperture; b) $f(R )$ as a function of $\textrm{e}/{\textrm{e}_{\textrm{min}}}$ resulting from the transform of a); e) same with the transform of e); d) measured kinetics $({t/\tau } )({{e_3}} )$ obtained from a) and b) or from e) and f); fixed values $e1$=3 nm (gold) and $e2$ = 2 nm (GO).

Equations (7)

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r 012 l ( l + 1 ) = r 012 l + r l ( l + 1 ) e 2 j β l 1 + r 012 l r l ( l + 1 ) e 2 j β l
R = R 0 [ 1 + A 1 β + A 2 β R + A 3 β 2 + A 4 β R β + A 5 β R 2 ] ,
R R 0 R 0 = U e + V e 2 ,
R R m i n R 0 R m i n = ( e e m i n e m i n ) 2
| e e m i n e m i n | = f ( R R m i n , R 0 R m i n ) ,
R θ , λ = R 0 θ , λ + R 0 U λ θ , λ e + R 0 V λ 2 θ , λ e 2
R θ , λ R m i n e f f R 0 θ , λ R m i n e f f = ( e e m i n e f f e m i n e f f ) 2