Abstract

Recently a multiple-scattering model for the reflectivity of a disordered monolayer of scattering particles on a flat surface was put forth [J. Opt. Soc. Am. 29, 1161 (2012)]. The approximate theoretical model provides relatively simple formulas for the reflectivity, but it was developed for a monodisperse monolayer. Here we extend the model to the case of a polydisperse monolayer and derive the appropriate formulas to calculate the optical transmissivity of the monolayer supported by a flat interface. A second objective of this paper is to test the approximate theoretical model against experimental data with highly scattering particles. We prepared monolayers of three different surface coverage fractions of polydisperse alumina and titanium dioxide particles deposited randomly on a glass slide. We measured their optical reflectivity and transmissivity versus the angle of incidence using a laser with a wavelength of 670 nm. Using the nominal values for the particles’ most probable radius and refractive index, we fitted the theoretical model to the experimental curves and found that it reproduces very well the experimental curves. Interestingly, a dip in the reflectivity curves at large angles of incidence is present for the alumina monolayers but not in the titanium dioxide monolayers. The dip corresponds to a maximum in the scattering efficiency by the alumina monolayers. The theoretical model reproduces very well this behavior.

© 2014 Optical Society of America

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References

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2012 (1)

2011 (1)

2009 (2)

2008 (2)

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16, 2288–2301 (2008).
[CrossRef]

2007 (2)

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

D.-S. Wang and C.-W. Lin, “Density-dependent optical response of gold nanoparticle monolayers on silicon substrates,” Opt. Lett. 32, 2128–2130 (2007).
[CrossRef]

2006 (1)

2004 (1)

2003 (1)

2000 (1)

1997 (1)

1996 (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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

1988 (1)

1982 (1)

J. J. H. Wang, “A unified and consistent view on the singularities of the electric dyadic Green’s function in the source region,” IEEE Trans. Antennas Propag. 30, 463–468 (1982).
[CrossRef]

1979 (1)

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824–4838 (1979).
[CrossRef]

1978 (1)

1971 (1)

Anto-Roca, J.

Babonneau, D.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Bagchi, A.

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824–4838 (1979).
[CrossRef]

Barrera, R. G.

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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Bohren, C. F.

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

Bosi, G.

Camelio, S.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Castillo, J. J. F.

Correia, R. R. B.

Dick, V. P.

García-Valenzuela, A.

Girardeau, T.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

González-González, M. L.

Gutiérrez-Reyes, E.

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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Horowitz, F.

Huffman, D. R.

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

Ivanov, A. P.

Kinbara, A.

Kobayashi, T.

Kong, J. A.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, 2001).

Koper, G. J. 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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Lin, C.-W.

Loiko, V. A.

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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Menegotto, T.

Mishchenko, M. I.

Okamoto, T.

Peña-Gomar, M. C.

Pereira, M. B.

Perez, E.

Pérez, E.

Pilloni, L.

Protopapa, M. L.

Rajagopal, A. K.

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824–4838 (1979).
[CrossRef]

Re, M.

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Rizzo, A.

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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

Simonot, L.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Sudoh, A.

Takahashi, H.

Toudert, J.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Truong, V. V.

Tsang, L.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, 2001).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Dover, 1981).

van der Zeeuw, E. A.

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

Wang, D.-S.

Wang, J. J. H.

J. J. H. Wang, “A unified and consistent view on the singularities of the electric dyadic Green’s function in the source region,” IEEE Trans. Antennas Propag. 30, 463–468 (1982).
[CrossRef]

Yamaguchi, I.

Yamaguchi, T.

Yoshida, S.

Appl. Opt. (6)

IEEE Trans. Antennas Propag. (1)

J. J. H. Wang, “A unified and consistent view on the singularities of the electric dyadic Green’s function in the source region,” IEEE Trans. Antennas Propag. 30, 463–468 (1982).
[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 angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Nanotechnology (1)

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (2)

A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824–4838 (1979).
[CrossRef]

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Other (3)

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, 2001).

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

H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Dover, 1981).

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Figures (8)

Fig. 1.
Fig. 1.

Coherent and diffuse components for a light beam reflected and transmitted by a polydisperse monolayer of particles.

Fig. 2.
Fig. 2.

Illustration of the monolayer supported by a flat interface. The particles are sitting on the interface and embedded in medium 1, with refractive index n1.

Fig. 3.
Fig. 3.

Schematic illustration of multiple reflections of a monochromatic beam incident on a monolayer of particles immersed in air and sitting on a flat substrate.

Fig. 4.
Fig. 4.

(a) Schematics of the experimental setup for measuring the coherent reflectance and (b) the coherent transmittance.

Fig. 5.
Fig. 5.

Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of alumina with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ=3.8%; (b) Θ=5.7%; and (c) Θ=10.6%.

Fig. 6.
Fig. 6.

Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of TiO2 with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ=1.3%; (b) Θ=4.9%; and (c) Θ=10.7%.

Fig. 7.
Fig. 7.

(a) Transmittance spectrum at normal incidence for a monolayer of particles of alumina with surface coverage factor of Θ5.7% and (b) transmittance spectrum at normal incidence for a monolayer of particles of TiO2 with surface coverage factor of Θ1.3%.

Fig. 8.
Fig. 8.

In the upper figure is shown a micrograph of a monolayer of alumina particles for the sample with a surface coverage fraction of Θ=3.8%. In the lower figure is shown the reconstructed image with the software, for this same sample, of the monolayer of particles without the substrate.

Tables (2)

Tables Icon

Table 1. Alumina Particles

Tables Icon

Table 2. TiO2 Particles

Equations (22)

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Es(r)=d3rd3rG(r,r)·Ta(rrn,rrn)·Enexc(r),
E(r)=Ei(r)+n=1Nd3rd3rG(r,r)·Ta(rrn,rrn)·Enexc(r).
E(r)=Ei(r)+0daρ(a)d3rnpn(rn)d3rG(r,r)·d3rTa(rrn,rrn)·Enexcn(r,rn),
Ejexc(r,rj)=Ei(r)+0daρ(a)d3rnd3rG(r,r)·d3rp(rn|rj)Ta(rrn,rrn)·Enexc(r,rn),
Epexc(r,rp)=E1exp(iki·r)e^i+E2exp(ikr·r)e^r,
Ejind(r,rj)η2E10daρ(a)[Sa(ki,ki)exp(ki·r)+Sa(kr,ki)exp(2ikzia)exp(ikr·r)]η2E20daρ(a)[Sa(ki,kr)exp(2ikzia)exp(ki·r)+Sa(kr,kr)exp(ikr·r)],
η=2πkm2cosθi,
Sa(q,p)=km4πi(Iq^q^)·Ta(q,p)·e^p.
E1e^i=Eie^iη2E10daρ(a)Sa(ki,ki)η2E20daρ(a)Sa(ki,kr)exp(2ikzia),
E2e^r=η2E10daρ(a)Sa(kr,ki)exp(2ikzia)η2E20daρ(a)Sa(kr,kr),
E1=Ei1+12βF1+βF+14(βF2βCβB),
E2=Ei12βC1+βF+14(βF2βCβB),
βF=η0daρ(a)S(0),βB=η0daρ(a)Sj,a(π2θi)exp(2ikzia),βC=η0daρ(a)Sj,a(π2θi)exp(2ikzia).
rcoh=βC1+βF+14(βF2βCβB),
tcoh=114(βF2βCβB)1+βF+14(βF2βCβB).
n(a)=12πalnσexp[ln2(a/a)2ln2σ],
ρST=Θπa2exp(2ln2σ).
r(θi)=r1+r2+r3++r1=rcoh(θi)+r12(θi)tcoh2(θi)exp(β1)+rcoh(θi)tcoh2(θi)r122(θi)exp(2β1)++tcoh2(θi)r21(θi)t12(θi)t21(θi)exp(β2),
r(θi)=rcoh(θi)+r12(θi)tcoh2(θi)1r12(θi)rcoh(θi)+tcoh2(θi)r21(θi)t12(θi)t21(θi).
t(θi)=t1+t2+=tcoh(θi)t12(θi)t21(θi)exp(β)+rcoh(θi)tcoh(θi)r12(θi)t12(θi)t21(θi)exp(β)exp(β1)+
t(θi)=[tcoh(θi)t12(θi)1r12(θi)rcoh(θi)]t21(θi).
Rcoh(θi)=|r(θi)|2andTcoh(θi)=|t(θi)|2.

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