Abstract

We develop a coherent-scattering model for the reflection of light from a monolayer of large particles and low surface coverage. The model takes into account multiple scattering between particles of the monolayer and with the substrate, and it can be used around the critical angle in an internal reflection configuration. We compare the results of the model with our own reflectivity data taken with latex particles adsorbed on a glass–water interface and with a simpler effective-medium model.

© 2006 Optical Society of America

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  1. R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
    [CrossRef]
  2. C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
    [CrossRef]
  3. E. M. Ortiz, F. González, J. M. Saiz, and F. Moreno, "Experimental measurements of the statistics of the scattered intensity from particles on surfaces," Opt. Express 10, 190-195 (2002), http://opticsexpress.org
    [PubMed]
  4. R. G. Barrera and A. García-Valenzuela, "Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach," J. Opt. Soc. Am. A 20, 296-311 (2003).
    [CrossRef]
  5. E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
    [CrossRef]
  6. E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
    [CrossRef]
  7. D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, 2004).
  8. M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
    [CrossRef]
  9. M. Peña-Gomar, Ma. L. González-González, A. García-Valenzuela, J. Antó-Roca, and E. Pérez, "Monitoring particle adsorption by laser reflectometry near the critical angle," Appl. Opt. 43, 5963-5970 (2004).
    [CrossRef] [PubMed]
  10. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (J. Wiley 1983).
  11. A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
    [CrossRef]
  12. P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
    [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

2004 (1)

2003 (2)

A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
[CrossRef]

R. G. Barrera and A. García-Valenzuela, "Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach," J. Opt. Soc. Am. A 20, 296-311 (2003).
[CrossRef]

2002 (1)

2000 (1)

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

1999 (1)

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

1998 (1)

M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
[CrossRef]

1996 (2)

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
[CrossRef]

1991 (1)

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Antó-Roca, J.

Barerra, R. G.

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Barrera, R. G.

R. G. Barrera and A. García-Valenzuela, "Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach," J. Opt. Soc. Am. A 20, 296-311 (2003).
[CrossRef]

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Bedeaux, D.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, 2004).

Beitia, C.

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Böhmer, M. R.

M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (J. Wiley 1983).

Borensztein, Y.

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Castillo-Mussot, M.

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Cuisinier, F. J. G.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Decher, G.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Garcia-Valenzuela, A.

A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
[CrossRef]

García-Valenzuela, A.

González, F.

González-González, Ma. L.

Haarmans, M. T.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (J. Wiley 1983).

Koper, G. J. M.

M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
[CrossRef]

E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
[CrossRef]

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

Lavalle, Ph.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Lazzari, R.

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Mann, E. K.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

Mochán, W. L.

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Monsivais, G.

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Moreno, F.

Ngankam, P. A.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Nieto, J.

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Ortiz, E. M.

Peña-Gomar, M.

Peña-Gomar, M. C.

A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
[CrossRef]

Pérez, E.

Sagis, L. M.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

Sagis, L. M. C.

E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
[CrossRef]

Saiz, J. M.

Schaaf, P.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Szyk, L.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

van der Zeeuw, E. A.

M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
[CrossRef]

E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
[CrossRef]

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

Villaseñor, P.

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

Villatoro, J.

A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
[CrossRef]

Vlieger, J.

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, 2004).

Voegel, J. C.

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

Appl. Opt. (1)

J. Am. Chem. Soc. (1)

P. A. Ngankam, Ph. Lavalle, J. C. Voegel, L. Szyk, G. Decher, P. Schaaf, and F. J. G. Cuisinier, "Influence of polyelectrolyte multilayer films on calcium phosphate nucleation," J. Am. Chem. Soc. 122, 8998-9005 (2000).
[CrossRef]

J. Chem. Phys. (1)

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, "The suitability of angles scanning reflectometry for colloidal particle sizing," J. Chem. Phys. 105, 1646-1653 (1996).
[CrossRef]

J. Colloid Interlace Sci. (1)

M. R. Böhmer, E. A. van der Zeeuw, and G. J. M. Koper, "Kinetics of particle adsorption in stagnation point flow studied by optical reflectometry," J. Colloid Interlace Sci. 197, 242-250 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Macromolecules (1)

E. A. van der Zeeuw, L. M. C. Sagis, and G. J. M. Koper, "Direct observation of swelling of non-cross-linked latex particles by scanning angle reflectometry," Macromolecules 29, 801-803 (1996).
[CrossRef]

Opt. Eng. (1)

A. Garcia-Valenzuela, M. C. Peña-Gomar, and J. Villatoro, "Sensitivity analysis of angle-sensitive-detectors based on a film resonator," Opt. Eng. 42, 1084-1092 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (2)

R. G. Barrera, M. Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, "Optical properties of two-dimensional disordered systems on a substrate," Phys. Rev. B 43, 13819-13826 (1991).
[CrossRef]

C. Beitia, Y. Borensztein, R. Lazzari, J. Nieto, and R. G. Barerra, "Substrate-induced multipolar resonates in supported free-electron metal spheres," Phys. Rev. B 60, 6018-6022 (1999).
[CrossRef]

Other (3)

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, 2004).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (J. Wiley 1983).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

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Figures (4)

Fig. 1
Fig. 1

(a) Geometry considered in the coherent reflection and transmission of a plane wave from a thin slab of a dilute random system of spherical particles; (b) geometry of the internal reflection configuration for reflectivity measurements from an adsorbed fractional monolayer of particles.

Fig. 2
Fig. 2

Schematic of the experimental setup.

Fig. 3
Fig. 3

Experimental data of the reflectance of a TM polarized He–Ne laser beam from a clean glass–water interface (open circles) and with an adsorbed monolayer of latex particles (full circles). The radii of the particles are (a) 137, (b) 256, (c) 344 nm. The refractive index of the glass, water, and particles is 1.515, 1.331, and 1.59, respectively. Theoretical curves with the coherent-scattering model (full curve) and with the effective-medium model (dotted curve) are also plotted. The values of the surface coverage used to adjust the scattering-theory model to the experimental data are (a) 0.08, (b) 0.03, and (c) 0.016. Microscopic images from the base of the prism after the experiments are also shown.

Fig. 4
Fig. 4

Plots of the change in reflectance, ΔR = (R modelRs )∕Rs , of a glass–water interface (refractive indices 1.515 and 1.331, respectively) due to the adsorption of (a) latex and (b) TiO2 (rutile) particles of radii 500 and 200 nm, respectively. A wavelength of 632.8 nm was assumed. The surface coverage fraction was assumed to be (a) 5% and (b) 2%. The full curve is with the coherent-scattering model and the dashed curve is with the effective-medium model.

Equations (13)

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

E s     + = E 0 α cos θ i S ( 0 ) ,
E s     = E 0 α cos θ i sin ( k z     i d ) d k z     i S n ( π 2 θ i ) ,
t s = 1 α cos θ i S ( 0 ) ,
r s = α cos θ i S n ( π 2 θ i ) ,
α 3 2 f x 3 k d = 2 Θ x 2 ,
t s E i = E i α S ( 0 ) cos θ i t s E i ,
r s E i = α S n ( π 2 θ i ) cos θ i t s E i .
t s = cos θ i cos θ i + α S ( 0 ) ,
r s = α S n ( π 2 θ i ) cos θ i + α S ( 0 ) .
r C S M = r 12 ( θ i ) + r s ( θ m ) 1 + r 12 ( θ i ) r s ( θ m ) ,
r s ( θ m ) = α S n ( π 2 θ m ) cos θ m + α S ( 0 ) exp ( 2 i k z     m a ) ,
R ( θ i ) = ω 0 k 1 2 π 0 π / 2 | r ( θ ) | 2 exp [ ( ω 0 k 1 ) 2 2 ( θ θ i ) 2 ]  d θ ,
r E M M = r 12 + r 23 exp ( 2 i k z 2 d ) 1 + r 12 r 23 exp ( 2 i k z 2 d ) .

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