Abstract

Using a multiple-scattering formalism, we derive closed-form expressions for the coherent reflection and transmission coefficients of monochromatic electromagnetic plane waves incident upon a two-dimensional array of randomly located spherical particles. The calculation is performed within the quasi-crystalline approximation, and the statistical correlation among the particles is assumed to be given simply by a correlation hole. In the resulting model, the size of the spheres and the angle of incidence are both unrestricted. The final formulas are relatively simple, making the model suitable for a straightforward interpretation of optical-sensing measurements.

© 2012 Optical Society of America

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2011

2009

2008

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]

C. Kuemin, T. Kraus, H. Wolf, and N. D. Spencer, “Matrix effects on the surface plasmon resonance of dry supported gold nanocrystals,” Opt. Lett. 33, 806–808 (2008).
[CrossRef]

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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

2007

2006

M. C. Peña-Gomar, J. J. F. Castillo, A. García-Valenzuela, R. G. Barrera, and E. Pérez, “Coherent optical reflectance from a monolayer of large particles adsorbed on a glass surface,” Appl. Opt. 45, 626–632 (2006).
[CrossRef]

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

2005

2004

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

A. Ponyavina, S. Kachan, and N. Sil’vanovich, “Statistical theory of multiple scattering of waves applied to three-dimensional layered photonic crystals,” J. Opt. Soc. Am. 21, 1866–1875 (2004).
[CrossRef]

2003

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

A. García-Valenzuela and R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective field approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 627–647 (2003).
[CrossRef]

2000

1998

1997

1996

G. Bosi, “Optical response of a thin film of spherical particles upon a dielectric substrate: retarded multipolar treatment including multiple reflections,” J. Opt. Soc. Am. B 13, 1691–1696(1996).
[CrossRef]

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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]

1994

R. G. Barrera and C. I. Mendoza, “Three-particle correlations in the optical properties of granular composites,” Solar Energy Mater. Solar Cells 32, 463–476 (1994).
[CrossRef]

G. Bosi, “Optical response of a thin film of spherical particles on a dielectric substrate: retarded multipolar treatment,” J. Opt. Soc. Am. B 11, 1073–1083 (1994).
[CrossRef]

1992

1991

R. G. Barrera, M. del 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]

1988

1982

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

A Bagchi, R. G. Barrera, and R. Fuchs, “Local-field effect in optical reflectance from adsorbed overlayers,” Phys. Rev. B 25, 7086–7096 (1982).
[CrossRef]

1980

1979

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

1958

J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Bagchi, A

A Bagchi, R. G. Barrera, and R. Fuchs, “Local-field effect in optical reflectance from adsorbed overlayers,” Phys. Rev. B 25, 7086–7096 (1982).
[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.

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]

M. C. Peña-Gomar, J. J. F. Castillo, A. García-Valenzuela, R. G. Barrera, and E. Pérez, “Coherent optical reflectance from a monolayer of large particles adsorbed on a glass surface,” Appl. Opt. 45, 626–632 (2006).
[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,” Opt J. Soc. Am. A 20, 296–311 (2003).
[CrossRef]

A. García-Valenzuela and R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective field approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 627–647 (2003).
[CrossRef]

R. G. Barrera and C. I. Mendoza, “Three-particle correlations in the optical properties of granular composites,” Solar Energy Mater. Solar Cells 32, 463–476 (1994).
[CrossRef]

R. G. Barrera, M. del 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]

A Bagchi, R. G. Barrera, and R. Fuchs, “Local-field effect in optical reflectance from adsorbed overlayers,” Phys. Rev. B 25, 7086–7096 (1982).
[CrossRef]

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]

Bedeaux, D.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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.

Calvo-Perez, O.

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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Castillo, J. J. F.

Chen, Y.

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE, 1995), Chap. 7.

Correia, R. R. B.

del Castillo-Mussot, M.

R. G. Barrera, M. del 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]

Dick, V. P.

Durant, S.

Emel’yanov, V. I.

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

Fedotov, V. A.

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

Fuchs, R.

A Bagchi, R. G. Barrera, and R. Fuchs, “Local-field effect in optical reflectance from adsorbed overlayers,” Phys. Rev. B 25, 7086–7096 (1982).
[CrossRef]

García-Valenzuela, 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]

M. C. Peña-Gomar, J. J. F. Castillo, A. García-Valenzuela, R. G. Barrera, and E. Pérez, “Coherent optical reflectance from a monolayer of large particles adsorbed on a glass surface,” Appl. Opt. 45, 626–632 (2006).
[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,” Opt J. Soc. Am. A 20, 296–311 (2003).
[CrossRef]

A. García-Valenzuela and R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective field approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 627–647 (2003).
[CrossRef]

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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Greffet, J. J.

Haarmans, M. T.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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]

Hong, K. M.

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.

Jin, P.

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

Kachan, S.

A. Ponyavina, S. Kachan, and N. Sil’vanovich, “Statistical theory of multiple scattering of waves applied to three-dimensional layered photonic crystals,” J. Opt. Soc. Am. 21, 1866–1875 (2004).
[CrossRef]

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, 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]

Kraus, T.

Kuemin, C.

Lin, C.-W.

Loiko, V. A.

MacDonald, K. F.

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

Mann,

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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]

Mendoza, C. I.

R. G. Barrera and C. I. Mendoza, “Three-particle correlations in the optical properties of granular composites,” Solar Energy Mater. Solar Cells 32, 463–476 (1994).
[CrossRef]

Menegotto, T.

Mishchenko, M. I.

Miskevich, A. A.

Mochán, W. L.

R. G. Barrera, M. del 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]

Molochko, V. I.

Monsivais, G.

R. G. Barrera, M. del 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]

Okamoto, T.

Peña-Gomar, M. C.

Percus, J. K.

J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Pereira, M. B.

Pérez, E.

Pilloni, L.

Ponyavina, A.

A. Ponyavina, S. Kachan, and N. Sil’vanovich, “Statistical theory of multiple scattering of waves applied to three-dimensional layered photonic crystals,” J. Opt. Soc. Am. 21, 1866–1875 (2004).
[CrossRef]

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, 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]

Sil’vanovich, N.

A. Ponyavina, S. Kachan, and N. Sil’vanovich, “Statistical theory of multiple scattering of waves applied to three-dimensional layered photonic crystals,” J. Opt. Soc. Am. 21, 1866–1875 (2004).
[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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Spencer, N. D.

Sudoh, A.

Takahashi, H.

Tazawa, M.

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

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 nanocluster 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 der Zeeuw, E. A.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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]

Villaseñor, P.

R. G. Barrera, M. del 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]

Vukadinovic, N.

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’s function in the source region,” IEEE Trans. Antennas Propag. 30, 463–468 (1982).
[CrossRef]

Wolf, H.

Xu, G.

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

Yamaguchi, I.

Yamaguchi, T.

Yevick, G. J.

J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110, 1–13 (1958).
[CrossRef]

Zheludev, N. I.

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. Xu, Y. Chen, M. Tazawa, and P. Jin, “Influence of dielectric properties of a substrate upon plasmon resonance spectrum supported Ag nanoparticles,” Appl. Phys. Lett. 88, 043114 (2006).
[CrossRef]

IEEE Trans. Antennas Propag.

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

J. Chem. Phys.

E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, 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. Pure Appl. Opt.

V. A. Fedotov, V. I. Emel’yanov, K. F. MacDonald, and N. I. Zheludev, “Optical properties of closely packed nanoparticle films: spheroids and nanoshells,” J. Opt. Pure Appl. Opt. 6, 155–160 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

A. García-Valenzuela and R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective field approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 627–647 (2003).
[CrossRef]

Nanotechnology

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 nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Opt J. Soc. Am. A

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

Opt. Express

Opt. Lett.

Phys. Rev.

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[CrossRef]

Phys. Rev. B

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]

R. G. Barrera, M. del 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]

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]

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[CrossRef]

Solar Energy Mater. Solar Cells

R. G. Barrera and C. I. Mendoza, “Three-particle correlations in the optical properties of granular composites,” Solar Energy Mater. Solar Cells 32, 463–476 (1994).
[CrossRef]

Other

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

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE, 1995), Chap. 7.

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

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Figures (12)

Fig. 1.
Fig. 1.

Sketch of the (a) side view and (b) top view of a free-standing monolayer of particles. The incident wave comes from z<0 at an angle θi with respect to the monolayer’s normal (the z-axis). The incident wave vector is assumed to be in the yz plane.

Fig. 2.
Fig. 2.

(a) Illustration of the averaging procedure keeping the jth particle fixed at rj and moving the nth particle across the plane of the monolayer, except where |rjrn|<2a. (b) Different sections of the monolayer plane in which we divide the integral over d3rn around rj.

Fig. 3.
Fig. 3.

Illustration of the monolayer supported by a flat interface. (a) The particles are sitting on the interface and embedded in medium 1 and (b) the particles are below the interface in medium 2. Note that, in (b) the angle of incidence to the monolayer plane is θ2 and is related to θ1 by Snell’s law between medium 1 and medium 2. In both cases the plane of the monolayer is indicated by the dashed line parallel to the interface and separated by one particle’s radius (a).

Fig. 4.
Fig. 4.

Plot of the coherent (a) transmittance and (b) reflectance of a free-standing monolayer of particles of size parameter xm=10 immersed on air (nm=1.0) as a function of the particle refractive index (assumed real). The angle of incidence is θi=45°, surface-coverage-fraction is Θ=0.3, and the polarization of light is assumed TE. The solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM), and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 5.
Fig. 5.

Plot of the coherent (a) transmittance and (b) reflectance of a free-standing monolayer of particles of size parameter xm=10 and a refractive index with its real part, Re(np), equal to 1.3, immersed on air (nm=1.0) as a function of the imaginary part of the particle refractive index. The angle of incidence is θi=45°, surface-coverage-fraction is Θ=0.3, and the polarization of light is assumed TE. The solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM) and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 6.
Fig. 6.

Plot of the coherent reflectance and transmittance of a free-standing monolayer of particles in air (nm=1.0) as a function of the particle radius at two different angles of incidence (a) θi=30° and (b) θi=60° at a fixed surface-coverage-fraction Θ=0.3. The refractive index of the particles is np=2+4i, the wavelength of radiation is assumed to be λ=500nm and the polarization of light is TE. The solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM) and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 7.
Fig. 7.

Calculated spectra of the coherent reflectance of a free-standing monolayer of gold particles in air. (a) For and angle of incidence of θi=0°, (b) For an angle of incidence of θi=60° and TE polarization, (c) For an angle of incidence of θi=0° and TM polarization. The particle radius is a=50nm and the surface-coverage fraction is Θ=0.25. The solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM) and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 8.
Fig. 8.

Calculated spectra of the coherent reflectance of a monolayer of gold particles sitting on top of an air-glass interface (nm=1.0 and ns=1.5) (a) for an angle of incidence of θi=0°, (b) for an angle of incidence of θi=60° and TE polarization, (c) for an angle of incidence of θi=60° and TM polarization. The particle radius is a=50nm and the surface-coverage fraction is Θ=0.25. The reflectance of the substrate alone (without the monolayer) is the dotted line (brown on line), the solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM) and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 9.
Fig. 9.

Calculated spectra of the coherent reflectance of a monolayer of silver particles sitting on top of silver half-space (nm=1.0 and ns(ω) is that of silver). The particle radius is a=50nm, the angle of incidence is θi=30°, the surface-coverage-fraction Θ=0.25 and the polarization is (a) TE and (b) TM. The reflectance of the substrate alone (without the monolayer) is the dotted line (brown on line), the solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM), and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 10.
Fig. 10.

Plot of the coherent reflectance of a monolayer of particles adsorbed below a glass–water interface (from the water’s side) (n1=1.5, n2=1.33, nm=1.33) as a function of the angle of incidence. The particle refractive index is np=1.59 (e.g, polystyrene) and the particle radius is (a) a=100nm, (b) a=200nm, and (c) a=300nm. The wavelength in vacuum of the light is λ=635nm. The polarization of light is TM. The reflectance of the substrate alone (without the monolayer) is the dotted line (brown on line), the solid line (black on line) is for the multiple-scattering model (MSM), the dashed line (red on line) is for the heuristic model (HM), and the short dash (navy on line) is for the single-scattering model (SS).

Fig. 11.
Fig. 11.

(a) First half and (b) second half of the path of integration in Eq. (B1) in the complex u— plane. The open circle indicates schematically where u=km and the function kzs=[km2u2]1/2 has a branch point.

Fig. 12.
Fig. 12.

Closed contours used to evaluate the principal value of the integrals on (a) Γ1 and (b) Γ2 in Eq. (B2). The open circle indicates schematically where u=km and the function kzs=[km2u2]1/2 has a branch point.

Equations (83)

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Enind(r;ω)=iωμ0G(r,r;ω)·Jind(n)(r;ω)d3r,
(××kM2)G(r,r;ω)=1δ(rr),
Jind(n)(r)=d3rσ(rrn,rrn)·Enexc(r),
T(r,r)=U(r)[δ(rr)1+VSd3rG(r,r)·T(r,r)],
E(r)=Ei(r)+n=1Nd3rd3rG(r,r)·T(rrn,rrn)·Enexc(r),
Ejexc(r)=Ei(r)+njNEnind(r),
E(r)=d3r1d3r2d3rnp(R)E(r,R),
E(r)=Ei(r)+n=1NG(r,r)·T(rrn,rrn)·Enexc(r,R)d3rd3r.
T(rrn,rrn)·Enexc(r,R)=d3rnp(rn)jnNd3rjp(R|rn)T(rrn,rrn)·Enexc(r,R),
T(rrn,rrn)·Enexc(r,R)=d3rnp(rn)T(rrn,rrn)·Enexc(r,R)n,
Enexc(r,R)n=jnNd3rjp(R|rn)·Enexc(r,R)
E(r)=Ei(r)+ρd3rG(r,r)d3rd3rnp(rn)T(rrn,rrn)·Enexcn(r,rn),
Ejexc(r,R)j=Ei(r)+d3rG(r,r)·d3rnjNT(rrn,rrn)·Enexc(r,R)njj,
T(rrn,rrn)·Enexc(r,R)njj=d3rnp(rn|rj)T(rrn,rrn)·Enexc(r,R)nj,
Ejexc(r,R)j=Ei(r)+d3rG(r,r)·d3rρd3rng(rn,rj)T(rrn,rrn)·Enexc(r,R)nj,
Enexc(r⃗,R)jn=mn,jNd3rmp(R|r⃗n,r⃗j)E⃗nexc(r⃗,R),
Enexc(r,R)njEnexc(r,R)nEnexcn(r,rn),
Ejexc(r,rj)=Ei(r)+d3rG(r,r)·d3rρd3rng(rn,rj)T(rrn,rrn)·Enexc(r,rn),
Epexc(r,rp)=exp[ik||i·rp]F(rrp),
F(rrp)=E1exp[ik1·(rrp)]e^1+E2exp[ik2·(rrp)]e^2,
Epexc(r,rp)=E1exp[i(k||ik1)·rp]exp(ik1·r)e^1+E2exp[i(k||ik2)·rp]exp(ik2·r)e^2.
g(rjrn)={0if|rjrn|<2a1if|rjrn|>2a.
T(r,r)=1(2π)6d3pd3pexp(ip·r)T(p,p)exp(ip·r),
Bd3rn(·)0ΔzLyj2aLLdxndyndzn(·)andTd3rn(·)0Δzyj+2aLLLdxndyndzn(·),
G±(r,r)=a^ya^yδ(rr)+i2dkxsdkzs(2π)21kys(Ik^±sk^±s)·exp[ik±s·(rr)],
Ejexc(r,rj)=Ei(r)+iρ2Eexcd3rdkxsdkzs(2π)21kys(Ik^±sk^±s)·exp[ik±s·(rr)]d3rd3p(2π)3d3p(2π)3×d3rng(|rjrn|)exp(ip·[rrn])T(p,p)·exp(ip·[rrn])exp(ikD·rn)exp(ikexc·r)e^exc,
Ejexc(r,rj)=Ei(r)+iρ2Eexcdkxsdkzs(2π)21kys(Ik^±sk^±s)·exp[ik±s·r]×d3rng(|rjrn|)exp(ik±s·rn)T(k±s,kexc)·exp(ik||i·rn)e^exc.
Ejind(r,rj)=EBind(r,rj)+ETind(r,rj),
EBind(r,rj)=iρs2Eexcdkzs2π1kys(Ik^+sk^+s)·T(k+s,kexc)·e^excexp[ik+s·r]Lyj2adynexp[i(kyikys)yn],
ETind(r,rj)=iρs2Eexcdkzs2π1kys(Ik^sk^s)·T(ks,kexc)·e^excexp[iks·r]yj+2aLdynexp[i(kyi+kys)yn],
EBind(r,rj)=iρs2Eexcdkzs2π(i4π/km)S(k+s,kexc)kysexp[i(kyikys)(yj2a)]exp[i(kyikys)L]i(kyikys)exp[ik+s·r],
ETind(r,rj)=iρs2Eexcdkzs2π(i4π/km)S(ks,kexc)kysexp[i(kyi+kys)L]exp[i(kyi+kys)(yj+2a)]i(kyi+kys)exp[iks·r],
S(q,p)kmi4π(Iq^q^)·T(q,p)·e^p.
EBind(r,rj)=12αEexc[S(ki,kexc)exp(iki·r)+S(kr,kexc)exp(ikr·r)]ρsηBC3(r)ρsηBC3(r),
ETind(r,rj)=ρsηTC3(r)ρsηTC3(r),
Ejind(r,rj)=α2Eexc[S(ki,kexc)exp(ki·r)+S(kr,kexc)exp(ikr·r)]ρsηevanes(r),
Ejind(r,rj)=α2E1[S(ki,k1)exp(ki·r)+S(kr,k1)exp(ikr·r)]α2E2[S(ki,k2)exp(ki·r)+S(kr,k2)exp(ikr·r)]ρsηevanes(1)(r)ρsηevanes(2)(r),
Epexc(r,rp)=E1exp(iki·r)ei+E2exp(ikr·r)er,
E1e^i=Eie^iα2E1S(ki,ki)α2E2S(ki,kr),
E2e^r=α2E1S(kr,ki)α2E2S(kr,kr).
E1=Ei(1+α2e^r·S(kr,kr))1+αe^i·S(ki,ki)+α24[e^i·S(ki,ki)]2α24[e^r·S(kr,ki)]2,
E2=Eiα2e^r·S(kr,ki)1+αe^i·S(ki,ki)+α24[e^i·S(ki,ki)]2α24[e^i·S(kr,ki)]2.
α=2Θxm2cosθi,
G±(r,r)=a^za^zδ(rr)+i2dkxsdkys(2π)21kzs(Ik^±sk^±s)·exp[ik±s·(rr)],
Et(r)=[EieiαE1S(ki,ki)αE2S(ki,kr)]exp(iki·r),
Er(r)=α[E1S(kr,ki)+E2S(kr,kr)]exp(ikr·r).
S(ki,ki)=km4πi(Ik^ik^i)·T(ki,ki)·e^i=S(0)e^i,
S(kr,kr)=km4πi(Ik^rk^r)·T(kr,kr)·e^r=S(0)e^r,
S(kr,ki)=km4πi(Ik^rk^r)·T(kr,ki)·e^i=Sj(π2θi)e^r,
S(ki,kr)=km4πi(Ik^ik^i)·T(ki,kr)·e^r=Sj(π2θi)e^i.
rcoh=αSj(π2θi)1+αS(0)+α24(S2(0)Sj2(π2θi)),
tcoh=(1α24(S2(0)Sj2(π2θi))1+αS(0)+α24(S2(0)Sj2(π2θi))),
tcoh=11+αS(0),
rcoh=αSj(π2θi)1+αS(0).
r=rcoh(θ1)+r12(θ1)tcoh2(θ1)exp(β1)+rcoh(θ1)r122(θ1)tcoh2(θ1)exp(2β1)+rcoh2(θ1)r123(θ1)tcoh2(θ1)exp(3β1)+,
r(θ1)=rcoh(θ1)+r12(θ1)tcoh2(θ1)exp(β1)1r12(θ1)rcoh(θ1)exp(β1),
r=r12(θ1)+t12(θ1)t21(θ2)rcoh(θ2)exp(β2)[1+rcoh(θ2)r21(θ2)exp(β2)+(rcoh(θ2)r21(θ2))2exp(2β2)+]
r(θ1)=r12(θ1)+rcoh(θ2)exp(β2)1+r12(θ1)rcoh(θ2)exp(β2),
α24(S2(0)Sj2(π2θi))4Θ2xm4S(0)Sj(0),
rcohTEiβχ1iβχ,tcohTE11iβχ,
rcohTMiβχcos(π2θi)1iβχ14β2χ2sin2(π2θi),tcohTM1+14β2χ2sin2(π2θi)1iβχ14β2χ2sin2(π2θi),
Lssd3rn(·)0Δzyj2ayj+2aLxj2adxndyndzn(·),Rssd3rn(·)0Δzyj2ayj+2axj+2aLdxndyndzn(·).
G(r,r)=a^xa^xδ(rr)+i2dkysdkzs(2π)21kxs(Ik^±sk^±s)·exp[ik±s·(rr)],
ELssind(r)=iρs2Eexcdkysdkzs(2π)21kxs(Ik^+sk^+s)·T(k+s,kexc)·e^exc×(exp[ikxs(xj2a)]exp(ikxsL)ikxs)(2isin[(kyikys)2a]i(kyikys))exp[i(kyikys)yj]exp[ik+s·r],
EBind(r,rj)=iρs2Eexcdkzsdkxs(2π)21kys(Ik^+sk^+s)·T(k+s,kexc)·e^excexp[ik+s·r]×[2isin(kxsL)ikxs][exp[i(kyikys)(yj2a)]exp[i(kyikys)L]i(kyikys)].
dkysdkzs02π0dθρdρ,dkxsdkzs02π0dθρdρ,
ELssind(r)=iρs2Eexc02π0dθρdρ(2π)21km2ρ2(Ik^+sk^+s)·T(k+s,kexc)·e^excexp[ik+s·r]×(exp[ikm2ρ2(xj2a)]exp[ikm2ρ2L]ikm2ρ2)(2isin[(kyiρsinθ)2a]i(kyiρsinθ))exp[i(kyiρsinθ)yj],
EBind(r,rj)=iρs2Eexc02π0dθρdρ(2π)21km2ρ2(Ik^+sk^+s)·T(k+s,kexc)·e^excexp[ik+s·r](2isin(ρsinθL)iρsinθ)(exp[i(kyikm2ρ2)(yj2a)]exp[i(kyikm2ρ2)L]i(kyikm2ρ2)).
EBind(r,rj)=iρs4πEexcΓdukm2u2(i4πkm)S(k+s,kexc)exp[ik+s·r]exp[i(kyiu)(yj2a)]exp[i(kyiu)L]i(kyiu),
EBind(r,rj)=ρskmEexcΓ1f(u;Re(kzs)<0)du+ρskmEexcΓ2f(u;Re(kzs)>0)du,
f(u)=exp[ik+s·r]km2u2S(k+s,kexc)exp[i(kyiu)(yj2a)]exp[i(kyiu)L]i(kyiu).
EBind(r,rj)=ρskmEexc[PΓ1f(u;Re(kzs)<0)du+PΓ2f(u;Re(kzs)>0)du],
Im(kzs)±ui2+ui44ur2ui22=±ui2+ui214ur2/ui22±ui2+ui2(12ur2/ui2)2±ur,
EC3(r,rj)=ρskmEexcexp[ikyi(yj2a)]C3duS(k+s,kexc)km2u2exp[iu(yyj+2a)]i(kyiu)exp[ikm2u2z],
EC1+EC2iπρskmEexc[(kyiu)fsing(u;Re(kzs)<0)]u=kyi+EC3=0,
EC1+EC2+iπρskmEexc[(kyiu)fsing(u;Re(kzs)>0)]u=kyi+EC3=0.
EBind(r,rj)=πρskzikmEexc(S(ki,kexc)exp[iki·r]+S(kr;kexc)exp[ikr·r])ρsηLC3(r,rj)ρsηLC3(r,rj),
ETind(r,rj)=iρs2Eexcdkzs2πS(ks,kexc)kysexp[i(kyi+kys)L]exp[i(kyi+kys)(yj+2a)]i(kyi+kys)exp[iks·r].
ETind(r,rj)=EC1+EC2+EC1+EC2=EC3EC3ρsηTC3ρsηTC3,
ηTC3(r,rj)=1kmEexcexp[ikyi(yj+2a)]0dεS(ks,kexc)km2(km+iε)2exp[i(km+iε)(yyj2a)](kyi+km+iε)exp(ikm2(km+iε)2z)
E(r)=Ei(r)+iρ2d3rdkxsdkys(2π)21kzs(Ik^±sk^±s)·exp[ik±s·(rr)]d3rd3p(2π)3d3p(2π)3×Vd3rnexp(ip·[rrn])T(p,p)·exp(ip·[rrn])[E1exp(iki·r)e^i+E2exp(ikr·r)e^r],
E(r)=Ei(r)+iρ2dkxsdkys(2π)21kzs(Ik^±sk^±s)·exp[ik±s·r]Vd3rn[E1exp[i(kik±s)·rn]T(k±s,ki)·exp(iki·rn)e^i+E2exp[i(krk±s)·rn]T(k±s,kr)·e^rexp(ikr·rn)].
E(r)=Ei(r)αE1S(k±s,ki)exp[ik±s·r]αE2S(k±s,kr)exp[ik±s·r],

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