Abstract

In this paper, we compare three different models that have been used to interpret reflectivity measurements of supported monolayers of nanoparticles. Two of them: (i) isotropic Maxwell Garnett and (ii) anisotropic two-dimensional-dipolar model are based on an effective-medium approach, while the third one (iii) coherent-scattering model, lies within the framework of multiple-scattering theory. First, we briefly review, on physical grounds, the foundations of each model and write down the corresponding formulas for the calculation of the reflectivity. In the two-dimensional-dipolar model, the dilute limit of the pair-correlation function (also called hole-correlation function) is always used in the calculation of the effective optical response. Then we use these formulas to plot and analyze graphs of the reflectivity of a monolayer of gold nanoparticles on a glass substrate, as a function of several relevant parameters, for two different commonly used experimental configurations. Finally, we discuss the importance of our results and how they can be used to infer the limits of validity of each model.

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

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References

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2017 (3)

X. Yang, H. Yu, X. Guo, Q. Ding, T. Pullerits, R. Wang, G. Zhang, W. Liang, and M. Sun, “Plasmon-exciton coupling of monolayer MoS2-Ag nanoparticles hybrids for surface catalytic reaction,” Mater. Today Energy 5, 72–78 (2017).
[Crossref]

Y. Bellouard, E. Block, J. Squier, and J. Gobet, “Plasmon-less surface enhanced Raman spectra induced by self-organized networks of silica nanoparticles produced by femtosecond lasers,” Opt. Express 25, 9587–9594 (2017).
[Crossref] [PubMed]

O. Vázquez-Estrada and A. García-Valenzuela, “Reflectivity and transmissivity of a surface covered by a disordered monolayer of large and tenuous particles: theory versus experiment,” Appl. Optics 56(25), 7158–7166 (2017).
[Crossref]

2016 (5)

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
[Crossref]

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

K. Kubiak, Z. Adamczyk, J. Maciejewska, and M. Oćwieja, “Gold nanoparticle monolayers of controlled coverage and structure,” J. Phys. Chem. C 120, 11807–11819 (2016).
[Crossref]

O. Vázquez-Estrada, G. Morales-Luna, A. Reyes-Coronado, A. Calles-Martinez, and A. García-Valenzuela, “Sensitivity of optical reflectance to the deposition of plasmonic nanoparticles and limits of detection,” J. Nanophotonics 10, 026019 (2016).
[Crossref]

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

2015 (2)

J. P. López-Neira, J. M. Galicia-Hernández, A. Reyes-Coronado, E. Pérez, and F. Castillo-Rivera, “Surface enhanced Raman scattering of amino acids assisted by gold nanoparticles and Gd3+ ions,” J. Phys. Chem. A 119, 4127–4135 (2015).
[Crossref]

O. Vázquez-Estrada, G. Morales-Luna, A. Calles-Martinez, A. Reyes-Coronado, and A. García-Valenzuela, “Optical reflectivity as an inspection tool for metallic nanoparticles deposited randomly on a flat substrate,” Proc. SPIE 9556, 1–9 (2015).

2014 (5)

A. L. Thorneywork, R. Roth, D. G. A. L. Aarts, and R. P. A. Dullens, “Communication: Radial distribution functions in a two-dimensional binary colloidal hard sphere system,” J. Chem. Phys. 140, 161106 (2014).
[Crossref] [PubMed]

Y. Battie, A. E. Naciri, W. Chamorro, and D. Horwat, “Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers,” J. Phys. Chem. C 118, 4899–4905 (2014).
[Crossref]

O. Vázquez-Estrada and A. García-Valenzuela, “Optical reflectivity of a disordered monolayer of highly scattering particles: Coherent scattering model versus experiment,” J. Opt. Soc. Am. A 31, 745–754 (2014).
[Crossref]

J. Caoa, T. Suna, and K. T. V. Grattana, “Gold nanorod-based localized surface plasmon resonance biosensors: A review,” Sensor Actuat. B-Chem 195, 332–351 (2014).
[Crossref]

G. Hajisalem, Q. Min, R. Gelfand, and R. Gordon, “Effect of surface roughness on self-assembled monolayer plasmonic ruler in nonlocal regime,” Opt. Express 22, 9604–9610 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (1)

M. A. Garcia, “Surface plasmons in metallic nanoparticles: fundamentals and applications,” J. Phys. D Appl. Phys. 44, 283001 (2011).
[Crossref]

2009 (1)

2007 (3)

X. D. Hoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress,” Biosens. Bioelectron. 23, 151–160 (2007).
[Crossref] [PubMed]

C. Noguez, “Surface plasmons on metal nanoparticles: The influence of shape and physical environment,” J. Phys. Chem. C 111, 3806–3819 (2007).
[Crossref]

P. N. Njoki, I-I. S. Lim, D. Mott, H-Y. Park, B. Khan, S. Mishra, R. Sujakumar, J. Luo, and C-J Zhong, “Size correlation of optical and spectroscopic properties for gold nanoparticles,” J. Phys. Chem. C 111, 14664–14669 (2007).
[Crossref]

2006 (1)

2005 (1)

C. Noguez, “Optical properties of isolated and supported metal nanoparticles,” Opt. Mater. 27(7), 1204–1211 (2005).
[Crossref]

2003 (1)

2000 (2)

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
[Crossref]

T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor gold colloid monolayers deposited upon glass substrates,” Opt. Lett. 25, 372–374 (2000).
[Crossref]

1998 (1)

1991 (1)

R. G. Barrera, M. del Castillo-Mussot, and G. Monsivais, “Optical properties of two-dimensional disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[Crossref]

1982 (1)

R. G. Barrera and P. A. Mello, “Statistical interpretation of the local field inside dielectrics,” Am. J. Phys. 50, 165–169 (1982).
[Crossref]

1972 (1)

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

1945 (1)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

1904 (1)

J. C. M. Garnett, “Colours in metal glasses and meta films,” Philos. Trans. R. Soc. Lond.,  203, 368 (1904).
[Crossref]

Aarts, D. G. A. L.

A. L. Thorneywork, R. Roth, D. G. A. L. Aarts, and R. P. A. Dullens, “Communication: Radial distribution functions in a two-dimensional binary colloidal hard sphere system,” J. Chem. Phys. 140, 161106 (2014).
[Crossref] [PubMed]

Adamczyk, Z.

K. Kubiak, Z. Adamczyk, J. Maciejewska, and M. Oćwieja, “Gold nanoparticle monolayers of controlled coverage and structure,” J. Phys. Chem. C 120, 11807–11819 (2016).
[Crossref]

Ahn, H.

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
[Crossref]

Aussenegg, F. R.

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
[Crossref]

Barrera, R. G.

Battie, Y.

Y. Battie, A. E. Naciri, W. Chamorro, and D. Horwat, “Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers,” J. Phys. Chem. C 118, 4899–4905 (2014).
[Crossref]

Bellouard, Y.

Block, E.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, (Wiley-Interscience, 1983).

Boriskina, S. V.

Bossard-Giannesini, L.

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

Calles-Martinez, A.

O. Vázquez-Estrada, G. Morales-Luna, A. Reyes-Coronado, A. Calles-Martinez, and A. García-Valenzuela, “Sensitivity of optical reflectance to the deposition of plasmonic nanoparticles and limits of detection,” J. Nanophotonics 10, 026019 (2016).
[Crossref]

O. Vázquez-Estrada, G. Morales-Luna, A. Calles-Martinez, A. Reyes-Coronado, and A. García-Valenzuela, “Optical reflectivity as an inspection tool for metallic nanoparticles deposited randomly on a flat substrate,” Proc. SPIE 9556, 1–9 (2015).

Caoa, J.

J. Caoa, T. Suna, and K. T. V. Grattana, “Gold nanorod-based localized surface plasmon resonance biosensors: A review,” Sensor Actuat. B-Chem 195, 332–351 (2014).
[Crossref]

Castillo, J. J. F.

Castillo-Rivera, F.

J. P. López-Neira, J. M. Galicia-Hernández, A. Reyes-Coronado, E. Pérez, and F. Castillo-Rivera, “Surface enhanced Raman scattering of amino acids assisted by gold nanoparticles and Gd3+ ions,” J. Phys. Chem. A 119, 4127–4135 (2015).
[Crossref]

Chamorro, W.

Y. Battie, A. E. Naciri, W. Chamorro, and D. Horwat, “Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers,” J. Phys. Chem. C 118, 4899–4905 (2014).
[Crossref]

Chan, V.Z-H

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
[Crossref]

Chang, Y-M.

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
[Crossref]

Chen, H-Y.

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
[Crossref]

Chen, W-L.

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
[Crossref]

Chen, Y.

Christy, R. W.

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

Cruguel, H.

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

Dai, J.

Dal Negro, L.

del Castillo-Mussot, M.

R. G. Barrera, M. del Castillo-Mussot, and G. Monsivais, “Optical properties of two-dimensional disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[Crossref]

Dick, V. P.

Ding, Q.

X. Yang, H. Yu, X. Guo, Q. Ding, T. Pullerits, R. Wang, G. Zhang, W. Liang, and M. Sun, “Plasmon-exciton coupling of monolayer MoS2-Ag nanoparticles hybrids for surface catalytic reaction,” Mater. Today Energy 5, 72–78 (2017).
[Crossref]

Ditlbacher, H.

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
[Crossref]

Dullens, R. P. A.

A. L. Thorneywork, R. Roth, D. G. A. L. Aarts, and R. P. A. Dullens, “Communication: Radial distribution functions in a two-dimensional binary colloidal hard sphere system,” J. Chem. Phys. 140, 161106 (2014).
[Crossref] [PubMed]

Feldmann, J.

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
[Crossref]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Galicia-Hernández, J. M.

J. P. López-Neira, J. M. Galicia-Hernández, A. Reyes-Coronado, E. Pérez, and F. Castillo-Rivera, “Surface enhanced Raman scattering of amino acids assisted by gold nanoparticles and Gd3+ ions,” J. Phys. Chem. A 119, 4127–4135 (2015).
[Crossref]

Garcia, M. A.

M. A. Garcia, “Surface plasmons in metallic nanoparticles: fundamentals and applications,” J. Phys. D Appl. Phys. 44, 283001 (2011).
[Crossref]

García-Valenzuela, A.

O. Vázquez-Estrada and A. García-Valenzuela, “Reflectivity and transmissivity of a surface covered by a disordered monolayer of large and tenuous particles: theory versus experiment,” Appl. Optics 56(25), 7158–7166 (2017).
[Crossref]

O. Vázquez-Estrada, G. Morales-Luna, A. Reyes-Coronado, A. Calles-Martinez, and A. García-Valenzuela, “Sensitivity of optical reflectance to the deposition of plasmonic nanoparticles and limits of detection,” J. Nanophotonics 10, 026019 (2016).
[Crossref]

O. Vázquez-Estrada, G. Morales-Luna, A. Calles-Martinez, A. Reyes-Coronado, and A. García-Valenzuela, “Optical reflectivity as an inspection tool for metallic nanoparticles deposited randomly on a flat substrate,” Proc. SPIE 9556, 1–9 (2015).

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X. Yang, H. Yu, X. Guo, Q. Ding, T. Pullerits, R. Wang, G. Zhang, W. Liang, and M. Sun, “Plasmon-exciton coupling of monolayer MoS2-Ag nanoparticles hybrids for surface catalytic reaction,” Mater. Today Energy 5, 72–78 (2017).
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P. N. Njoki, I-I. S. Lim, D. Mott, H-Y. Park, B. Khan, S. Mishra, R. Sujakumar, J. Luo, and C-J Zhong, “Size correlation of optical and spectroscopic properties for gold nanoparticles,” J. Phys. Chem. C 111, 14664–14669 (2007).
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ACS Photonics (1)

S. Gwo, C-Y. Wang, H-Y. Chen, M-H. Lin, L. Sun, X. Li, W-L. Chen, Y-M. Chang, and H. Ahn, “Plasmonic metasurfaces for nonlinear optics and quantitative SERS,” ACS Photonics 3, 1371–1384 (2016).
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Am. J. Phys. (1)

R. G. Barrera and P. A. Mello, “Statistical interpretation of the local field inside dielectrics,” Am. J. Phys. 50, 165–169 (1982).
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App. Phys. Lett. (3)

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

L. Bossard-Giannesini, H. Cruguel, E. Lacaze, and O. Pluchery, “Plasmonic properties of gold nanoparticles on silicon substrates: Understanding Fano-like spectra observed in reflection,” App. Phys. Lett. 109, 111901 (2016).
[Crossref]

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V.Z-H Chan, J. P. Spatz, and M. Möller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” App. Phys. Lett. 77, 2949 (2000).
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Appl. Opt. (1)

Appl. Optics (1)

O. Vázquez-Estrada and A. García-Valenzuela, “Reflectivity and transmissivity of a surface covered by a disordered monolayer of large and tenuous particles: theory versus experiment,” Appl. Optics 56(25), 7158–7166 (2017).
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Biosens. Bioelectron. (1)

X. D. Hoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress,” Biosens. Bioelectron. 23, 151–160 (2007).
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J. Chem. Phys. (1)

A. L. Thorneywork, R. Roth, D. G. A. L. Aarts, and R. P. A. Dullens, “Communication: Radial distribution functions in a two-dimensional binary colloidal hard sphere system,” J. Chem. Phys. 140, 161106 (2014).
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J. Nanophotonics (1)

O. Vázquez-Estrada, G. Morales-Luna, A. Reyes-Coronado, A. Calles-Martinez, and A. García-Valenzuela, “Sensitivity of optical reflectance to the deposition of plasmonic nanoparticles and limits of detection,” J. Nanophotonics 10, 026019 (2016).
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J. Opt. Soc. Am. A (4)

J. Phys. Chem. A (1)

J. P. López-Neira, J. M. Galicia-Hernández, A. Reyes-Coronado, E. Pérez, and F. Castillo-Rivera, “Surface enhanced Raman scattering of amino acids assisted by gold nanoparticles and Gd3+ ions,” J. Phys. Chem. A 119, 4127–4135 (2015).
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J. Phys. Chem. C (4)

K. Kubiak, Z. Adamczyk, J. Maciejewska, and M. Oćwieja, “Gold nanoparticle monolayers of controlled coverage and structure,” J. Phys. Chem. C 120, 11807–11819 (2016).
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C. Noguez, “Surface plasmons on metal nanoparticles: The influence of shape and physical environment,” J. Phys. Chem. C 111, 3806–3819 (2007).
[Crossref]

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J. Phys. D Appl. Phys. (1)

M. A. Garcia, “Surface plasmons in metallic nanoparticles: fundamentals and applications,” J. Phys. D Appl. Phys. 44, 283001 (2011).
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Mater. Today Energy (1)

X. Yang, H. Yu, X. Guo, Q. Ding, T. Pullerits, R. Wang, G. Zhang, W. Liang, and M. Sun, “Plasmon-exciton coupling of monolayer MoS2-Ag nanoparticles hybrids for surface catalytic reaction,” Mater. Today Energy 5, 72–78 (2017).
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Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. (1)

C. Noguez, “Optical properties of isolated and supported metal nanoparticles,” Opt. Mater. 27(7), 1204–1211 (2005).
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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

The so-called reflectivity of a surface corresponds to the collimated-to-collimated reflectance and in general is well approximated by the coherent reflectance of the surface. The difference is that the reflectivity includes all light, the coherent and diffuse components, reaching the detector in a collimated-to-collimated reflectance measurement, whereas the coherent reflectance is calculated only considering the coherent wave and thus, it does not consider that some diffuse light may also reach the detector. However, in practice, the amount of diffuse light captured by the photodetector in a collimated-to-collimated reflectance measurement is negligible.

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http://www.schott.com/d/advanced_optics/ac85c64c-60a0-4113-a9df-23ee1be20428/1.1/schott-optical-glass-collection-datasheets-english-17012017.pdf

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Figures (10)

Fig. 1
Fig. 1 Schematics of the supported spherical particles (yellow circles) forming the monolayer and their images induced in the substrate (gray-dashed circles).
Fig. 2
Fig. 2 Schematics of the three-layer system illuminated from the substrate side, internal incidence configuration, with wavevector k 1 at an angle of incidence θi respect to the normal to the monolayer direction a ^ z.
Fig. 3
Fig. 3 Schematic illustration of a plane wave incident on a random slab of particles of thickness d. The centers of the particles are restricted to lie within the slab. The dashed line indicates the plane that divides the slab in two equal halves. On the average, half of the particles are in the upper slab and half on the lower slab. The origin of coordinates is placed on the plane at the middle of the slab and the z axis points towards the transmission hemisphere.
Fig. 4
Fig. 4 Reflectivity as a function of the angle of incidence θi for fixed wavelength of 540 nm and surface coverage of 5%, for p (TM) polarization, and two different radius of the nanoparticles forming the monolayer and for both configurations: internal configuration for a radius of (a) 10 nm and (c) 30 nm, and external configuration for a radius of (b) 10 nm and (d) 30 nm. Inset in (b) applies to all the plots in this figure.
Fig. 5
Fig. 5 Reflectivity spectra for normal incidence and surface coverage of 5%, for two different radius of the nanoparticles forming the monolayer and for both configurations: internal configuration for a radius of (a) 10 nm and (c) 30 nm, and external configuration for a radius of (b) 10 nm and (d) 30 nm. Inset in (b) applies to all the plots in this figure.
Fig. 6
Fig. 6 Reflectivity spectra for an angle of incidence θi = 45° and surface coverage of 5%, for p (TM) polarization, and two different radii of the nanoparticles forming the monolayer and for both configurations: internal-reflectance configuration for a radius of (a) 10 nm and (c) 30 nm, and external-reflectance configuration for a radius of (b) 10 nm and (d) 30 nm. Inset in (a) applies to all the plots in this figure.
Fig. 7
Fig. 7 Reflectivity versus the radius of the particles forming the monolayer, at a fixed wavelength of 540 nm and surface coverage of 5%, for p (TM) polarization, and two different angles of incidence and for both configurations: internal-reflectance configuration for an angle of (a) θi = 0 and (c) θi = 45°, and external-reflectance configuration for an angle of (b) θi = 0 and (d) θi = 45°. Inset in (d) applies to all the plots in this figure.
Fig. 8
Fig. 8 Reflectivity versus the surface coverage, at a fixed wavelength of 540 nm and radius of the particles forming the monolayer of 10 nm for p (TM) polarization, and two different angles of incidence and for both configurations: internal-reflectance configuration for an angle of (a) θi = 0 and (c) θi = 45°, and external-reflectance configuration for an angle of (b) θi = 0 and (d) θi = 45°. Inset in (b) applies to all the plots in this figure.
Fig. 9
Fig. 9 Reflectivity versus the surface coverage, at a fixed wavelength of 540 nm and radius of the particles forming the monolayer of 30 nm for p (TM) polarization, and two different angles of incidence and for both configurations: internal configuration for an angle of (a) θi = 0 and (c) θi = 45°, and external configuration for an angle of (b) θi = 0 and (d) θi = 45°. Inset in (b) applies to all the plots in this figure.
Fig. 10
Fig. 10 Reflectivity versus the angle of incidence for a monolayer formed with gold (dielectric function taken from [38]) nanoparticles of 10 nm in radius, embedded in air and supported onto a silver substrate (dielectric function taken from [38]), illuminated with a fixed wavelength of 540 nm and surface coverage of 10%, for p polarization, using the 2D-DM considering images (continuous curve) and without them (long-dashed curve).

Equations (41)

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ϵ MG e f f ϵ m = ϵ p ( 1 + 2 f ) + 2 ϵ m ( 1 f ) ϵ p ( 1 f ) + ϵ m ( 2 + f ) ,
P = ϵ 0 χ e f f E ,
( E , ϵ e f f E ) ( E e x t , ϵ m E e x t ) ,
P = ϵ m χ e f f E
P = ( ϵ e f f ϵ m ) E e x t ,
P = ( 1 ϵ m ϵ e f f ) ϵ m E e x t .
p i = ϵ m α p o l ( ω ) [ E e x t + j i N t i j p j + j N t i j I M p j ] ,
p x = ϵ m α p o l [ E x e x t + j N u i j x x p x ] ,
p z = ϵ m α p o l [ E z e x t + j N u i j z z p z ] ,
ϵ x x e f f ϵ m = 1 + 2 Θ α ˜ p o l 1 A α ˜ p o l 8 Θ α ˜ p o l 1 2 ( G A G I ) ,
ϵ m ϵ z z e f f = 1 2 Θ α ˜ p o l 1 A α ˜ p o l 4 + Θ α ˜ p o l ( G + A G I ) ,
G = 0 ρ ( 2 ) ( 2 a x ) x 2 d x ,
G I = 0 ρ ( 2 ) ( 2 a x ) x ( x 2 2 ) ( x 2 + 1 ) 5 / 2 d x ,
ρ ( 2 ) ( r ) = { 0 r < 2 a 1 r 2 a ,
r 123 = r 12 + r 23 exp [ 2 i k 2 z d ] 1 + r 12 r 23 exp [ 2 i k 2 z d ] ,
r i j s = k i z k j z k i z + k j z and r i j p = ϵ j k i z ϵ i k j z ϵ j k i z + ϵ i k j z ,
k i z = k 0 ϵ ˜ i ϵ ˜ 1 sin 2 θ 1 , i = 1 , 2 , 3 ,
ϵ x x e f f ϵ 3 = 1 + 2 Θ α ˜ p o l 1 A α ˜ p o l 8 Θ α ˜ p o l 1 2 ( 1 A 2 4 ) ,
ϵ 3 ϵ z z e f f = 1 2 Θ α ˜ p o l 1 A α ˜ p o l 4 + Θ α ˜ p o l ( 1 A 2 4 ) ϵ 3 ϵ 1 ,
r 12 s = k 1 z k 2 z s k 1 z + k 2 z s and r 12 p = ϵ x x e f f k 1 z ϵ 1 k 2 z p ϵ x x e f f k 1 z + ϵ 1 k 2 z p ,
r 23 s = k 2 z s k 3 z k 2 z s + k 3 z and r 23 p = ϵ 3 k 2 z p ϵ x x e f f k 3 z ϵ 3 k 2 z p + ϵ x x e f f k 3 z ,
E s c a t t = E 0 α S ( 0 ) ,
E s c a t r = E 0 α sin ( k z i d ) k z i d S n ( π 2 θ i ) ,
α = k m d cos θ i 3 f 2 x m 3 ,
t c o h = 1 α S ( 0 ) ,
r c o h = α S n ( π 2 θ i ) .
α = 2 Θ x m 2 cos θ i ,
E e x c = E e x c + exp [ i k i r ] e ^ i + E e x c exp [ i k r r ] e ^ r ,
E e x c + exp [ i k i r ] a ^ i = [ E 0 1 2 α S ( 0 ) E e x c + 1 2 α S n ( π 2 θ i ) E e x c ] exp [ i k i r ] e ^ i
E e x c exp [ i k r r ] a ^ r = [ 1 2 α S n ( π 2 θ i ) E e x c + 1 2 α S ( 0 ) E e x c ] exp [ i k r r ] e ^ r ,
E e x c + = 1 + 1 2 α S n ( 0 ) 1 + α S ( 0 ) + 1 4 α 2 [ S 2 ( 0 ) S n 2 ( π 2 θ i ) ] E 0
E e x c = 1 2 α S n ( π 2 θ i ) 1 + α S ( 0 ) + 1 4 α 2 [ S 2 ( 0 ) S n 2 ( π 2 θ i ) ] E 0 .
E c o h t = E i + E s c a t t ( z > d 2 )
= [ E 0 E e x c + α cos θ i S ( 0 ) E e x c α cos θ i S n ( π 2 θ i ) ] exp [ i k i r ] e ^ i ,
E c o h r = E s c a t r ( z < d 2 )
= [ E e x c + α cos θ i S n ( π 2 θ i ) E e x c α cos θ i S ( 0 ) ] exp [ i k r r ] e ^ r ,
r coh = α S n ( π 2 θ i ) 1 + α S ( 0 ) + 1 4 α 2 [ S 2 ( 0 ) S n 2 ( π 2 θ i ) ]
t coh = 1 1 4 α 2 [ S 2 ( 0 ) S n 2 ( π 2 θ i ) ] 1 + α S ( 0 ) + 1 4 α 2 [ S 2 ( 0 ) S n 2 ( π 2 θ i ) ] .
r ( θ i ) = r coh ( θ i ) + r 13 ( θ i ) t coh 2 ( θ i ) exp [ i β 1 ] 1 r 13 ( θ i ) r coh ( θ i ) exp [ i β 1 ] ,
θ t = arcsin ( n 1 n 2 sin θ i ) .
r ( θ i ) = r 13 ( θ i ) + r coh ( θ t ) exp [ i β 2 ] 1 + r 13 ( θ i ) r coh ( θ t ) exp [ i β 2 ] ,

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