Abstract

We propose a model to calculate scattering from inhomogeneous three-dimensional, rough surfaces on top of a stratified medium. The roughness is made up of an ensemble of deposits with various shapes and permittivities whose heights remain small with respect to the wavelength of the incident light. This geometry is encountered in the remote sensing of soil surfaces, or in optics wherever there are contaminated planar components. Starting from a volume-integral equation involving the Green’s tensor of the stratified medium, we derive a height-perturbative expansion up to second order. Our formulation, which depends explicitly on the profiles of each deposit and on the Fresnel coefficients of the layered substrate, accounts for double-scattering events and permits an evaluation of depolarization in the plane of incidence. Comparisons with rigorous calculations in the simplified case of two-dimensional geometries are presented. It is shown that the second-order scattering term can be much more important for heterogeneous surfaces than for their homogeneous counterparts.

© 2004 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  34. T. Sondergaard, S. Boshelvonyi, “Vectorial model for multiple scattering by surface nanoparticles via surface polariton–polariton interactions,” Phys. Rev. B 67, 165405 (2003).
    [Crossref]
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2003 (1)

T. Sondergaard, S. Boshelvonyi, “Vectorial model for multiple scattering by surface nanoparticles via surface polariton–polariton interactions,” Phys. Rev. B 67, 165405 (2003).
[Crossref]

2001 (6)

A. Fuks, “Wave diffraction by a rough boundary of an arbitrary plane-layered medium,” IEEE Trans. Antennas Propag. 49, 630–639 (2001).
[Crossref]

P. Johansson, “Light scattering from disordered overlayers of metallic nanoparticles,” Phys. Rev. B 64, 165405 (2001).
[Crossref]

A. Soubret, G. Berginc, C. Bourrely, “A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional, randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

P. Dinesen, J. Hesthaven, “Fast and accurate modeling of waveguide grating couplers, three-dimensional vectorial case,” J. Opt. Soc. Am. A 18, 2876–2885 (2001).
[Crossref]

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[Crossref]

S. Linden, J. Kuhl, H. Giessen, “Controlling the interaction between light and gold nanoparticles with selective suppression of extinction,” Phys. Rev. B 86, 4688–4691 (2001).

2000 (1)

“Computational wave issues in remote sensing, imaging and target identification, propagation, and inverse scattering,” special issue, IEEE Trans. Geosci. Remote Sens. 38, (2000).

1999 (1)

1998 (2)

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
[Crossref]

1997 (3)

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[Crossref]

N. Zhuk, “Scattering of em waves from a slightly rough surface of a generally anisotropic plane-layered half space,” IEEE Trans. Antennas Propag. 45, 1774–1782 (1997).
[Crossref]

L. J. Lévesque, B. E. Paton, “Detection of defects in multiple layer structures by using surface plasmon resonance,” Atmos. Ocean. 36, 7199–7203 (1997).

1996 (2)

S. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[Crossref]

K. Sarabandi, O. Yisok, F. Ulaby, “A numerical simulation of scattering from one-dimensional, inhomogeneous, dielectric, random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[Crossref]

1995 (5)

A. Sentenac, J.-J. Greffet, “Scattering by 2D particles deposited on a dielectric planar waveguide, a near-field and far-field study,” Waves Random Media 5, 145–155 (1995).
[Crossref]

S. Dietrich, A. Haase, “Scattering of X-rays and neutrons at interfaces,” Phys. Rep. 260, (1995).

I. Ohlidal, K. Navratil, “Scattering of light from multilayer systems with rough boundary,” Prog. Opt. 34, 251–334 (1995).

R. Carminati, J.-J. Greffet, “Influence of dielectric contrast and topography on the near-field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[Crossref]

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces.” J. Opt. Soc. Am. 12, 1547–1557 (1995).
[Crossref]

1994 (2)

A. G. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media 4, 337–367 (1994).
[Crossref]

F. Pincemin, A. Sentenac, J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Acoust. Soc. Am. 11, 1117–1127 (1994).

1993 (3)

1991 (1)

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[Crossref]

1990 (1)

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. 7, 1185–1201 (1990).
[Crossref]

1988 (1)

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[Crossref]

1985 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces.” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

1941 (1)

Berginc, G.

A. Soubret, G. Berginc, C. Bourrely, “A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional, randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

Boshelvonyi, S.

T. Sondergaard, S. Boshelvonyi, “Vectorial model for multiple scattering by surface nanoparticles via surface polariton–polariton interactions,” Phys. Rev. B 67, 165405 (2003).
[Crossref]

Bourrely, C.

A. Soubret, G. Berginc, C. Bourrely, “A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional, randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

Bruno, O. P.

Carminati, R.

Chaumet, P.

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

Chen, M. F.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[Crossref]

de Fornel, F.

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

Dietrich, S.

S. Dietrich, A. Haase, “Scattering of X-rays and neutrons at interfaces,” Phys. Rep. 260, (1995).

Dinesen, P.

Ding, K. H.

L. Tsang, J. A. Kong, K. H. Ding, Scattering of electromagnetic waves, first volume of three in Wiley Series in Remote Sensing (Wiley-Interscience, New York, 2001).

Dufour, J. P.

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

Fano, U.

Friberg, A. T.

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. 7, 1185–1201 (1990).
[Crossref]

Fuks, A.

A. Fuks, “Wave diffraction by a rough boundary of an arbitrary plane-layered medium,” IEEE Trans. Antennas Propag. 49, 630–639 (2001).
[Crossref]

Fung, A. K.

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[Crossref]

Germer, T.

Giessen, H.

S. Linden, J. Kuhl, H. Giessen, “Controlling the interaction between light and gold nanoparticles with selective suppression of extinction,” Phys. Rev. B 86, 4688–4691 (2001).

Giovannini, H.

Greffet, J. J.

F. Pincemin, A. Sentenac, J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Acoust. Soc. Am. 11, 1117–1127 (1994).

Greffet, J.-J.

A. Sentenac, J.-J. Greffet, “Scattering by 2D particles deposited on a dielectric planar waveguide, a near-field and far-field study,” Waves Random Media 5, 145–155 (1995).
[Crossref]

R. Carminati, J.-J. Greffet, “Influence of dielectric contrast and topography on the near-field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[Crossref]

Haase, A.

S. Dietrich, A. Haase, “Scattering of X-rays and neutrons at interfaces,” Phys. Rep. 260, (1995).

Hesthaven, J.

Ishimaru, A.

Johansson, P.

P. Johansson, “Light scattering from disordered overlayers of metallic nanoparticles,” Phys. Rev. B 64, 165405 (2001).
[Crossref]

Kong, J. A.

T. Tsang, J. A. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

L. Tsang, J. A. Kong, K. H. Ding, Scattering of electromagnetic waves, first volume of three in Wiley Series in Remote Sensing (Wiley-Interscience, New York, 2001).

Kuhl, J.

S. Linden, J. Kuhl, H. Giessen, “Controlling the interaction between light and gold nanoparticles with selective suppression of extinction,” Phys. Rev. B 86, 4688–4691 (2001).

Lekner, J.

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

Lévesque, L. J.

L. J. Lévesque, B. E. Paton, “Detection of defects in multiple layer structures by using surface plasmon resonance,” Atmos. Ocean. 36, 7199–7203 (1997).

Li, L.

Linden, S.

S. Linden, J. Kuhl, H. Giessen, “Controlling the interaction between light and gold nanoparticles with selective suppression of extinction,” Phys. Rev. B 86, 4688–4691 (2001).

Maradudin, A. A.

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces.” J. Opt. Soc. Am. 12, 1547–1557 (1995).
[Crossref]

Mendez, E. R.

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces.” J. Opt. Soc. Am. 12, 1547–1557 (1995).
[Crossref]

Milder, D. M.

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[Crossref]

Mulholland, G.

Navratil, K.

I. Ohlidal, K. Navratil, “Scattering of light from multilayer systems with rough boundary,” Prog. Opt. 34, 251–334 (1995).

Nieto-Vesperinas, M.

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. 7, 1185–1201 (1990).
[Crossref]

Ohlidal, I.

I. Ohlidal, K. Navratil, “Scattering of light from multilayer systems with rough boundary,” Prog. Opt. 34, 251–334 (1995).

Paton, B. E.

L. J. Lévesque, B. E. Paton, “Detection of defects in multiple layer structures by using surface plasmon resonance,” Atmos. Ocean. 36, 7199–7203 (1997).

Pincemin, F.

F. Pincemin, A. Sentenac, J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Acoust. Soc. Am. 11, 1117–1127 (1994).

Rahmani, A.

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

Rayleigh,

Rayleigh, The Theory of Sound, 3rd ed. (MacMillan, London, 1896).

Reitich, F.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces.” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

Saillard, M.

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[Crossref]

H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
[Crossref]

Sanchez, J. A.

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces.” J. Opt. Soc. Am. 12, 1547–1557 (1995).
[Crossref]

Sarabandi, K.

K. Sarabandi, O. Yisok, F. Ulaby, “A numerical simulation of scattering from one-dimensional, inhomogeneous, dielectric, random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[Crossref]

Sentenac, A.

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[Crossref]

H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
[Crossref]

A. Sentenac, J.-J. Greffet, “Scattering by 2D particles deposited on a dielectric planar waveguide, a near-field and far-field study,” Waves Random Media 5, 145–155 (1995).
[Crossref]

F. Pincemin, A. Sentenac, J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Acoust. Soc. Am. 11, 1117–1127 (1994).

Shin, R.

T. Tsang, J. A. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Smith, S.

S. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[Crossref]

Sondergaard, T.

T. Sondergaard, S. Boshelvonyi, “Vectorial model for multiple scattering by surface nanoparticles via surface polariton–polariton interactions,” Phys. Rev. B 67, 165405 (2003).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. 7, 1185–1201 (1990).
[Crossref]

Soubret, A.

A. Soubret, G. Berginc, C. Bourrely, “A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional, randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

Sung, L.

Tsang, L.

L. Tsang, J. A. Kong, K. H. Ding, Scattering of electromagnetic waves, first volume of three in Wiley Series in Remote Sensing (Wiley-Interscience, New York, 2001).

Tsang, T.

T. Tsang, J. A. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Ulaby, F.

K. Sarabandi, O. Yisok, F. Ulaby, “A numerical simulation of scattering from one-dimensional, inhomogeneous, dielectric, random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[Crossref]

Voronovich, A. G.

A. G. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media 4, 337–367 (1994).
[Crossref]

A. G. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena (Springer, New York, 1994).

Winebrenner, D. P.

Yisok, O.

K. Sarabandi, O. Yisok, F. Ulaby, “A numerical simulation of scattering from one-dimensional, inhomogeneous, dielectric, random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[Crossref]

Zhuk, N.

N. Zhuk, “Scattering of em waves from a slightly rough surface of a generally anisotropic plane-layered half space,” IEEE Trans. Antennas Propag. 45, 1774–1782 (1997).
[Crossref]

Atmos. Ocean. (1)

L. J. Lévesque, B. E. Paton, “Detection of defects in multiple layer structures by using surface plasmon resonance,” Atmos. Ocean. 36, 7199–7203 (1997).

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces.” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

IEEE Trans. Antennas Propag. (2)

N. Zhuk, “Scattering of em waves from a slightly rough surface of a generally anisotropic plane-layered half space,” IEEE Trans. Antennas Propag. 45, 1774–1782 (1997).
[Crossref]

A. Fuks, “Wave diffraction by a rough boundary of an arbitrary plane-layered medium,” IEEE Trans. Antennas Propag. 49, 630–639 (2001).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

“Computational wave issues in remote sensing, imaging and target identification, propagation, and inverse scattering,” special issue, IEEE Trans. Geosci. Remote Sens. 38, (2000).

K. Sarabandi, O. Yisok, F. Ulaby, “A numerical simulation of scattering from one-dimensional, inhomogeneous, dielectric, random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[Crossref]

J. Acoust. Soc. Am. (2)

D. M. Milder, “An improved formalism for wave scattering from rough surfaces,” J. Acoust. Soc. Am. 89, 529–541 (1991).
[Crossref]

F. Pincemin, A. Sentenac, J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Acoust. Soc. Am. 11, 1117–1127 (1994).

J. Opt. Soc. Am. (3)

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. 7, 1185–1201 (1990).
[Crossref]

J. A. Sanchez, A. A. Maradudin, E. R. Mendez, “Limits of validity of three perturbation theories of the specular scattering of light from one-dimensional, randomly rough, dielectric surfaces.” J. Opt. Soc. Am. 12, 1547–1557 (1995).
[Crossref]

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31, 213–222 (1941).
[Crossref]

J. Opt. Soc. Am. A (8)

Opt. Lett. (1)

Phys. Rep. (1)

S. Dietrich, A. Haase, “Scattering of X-rays and neutrons at interfaces,” Phys. Rep. 260, (1995).

Phys. Rev. B (5)

P. Johansson, “Light scattering from disordered overlayers of metallic nanoparticles,” Phys. Rev. B 64, 165405 (2001).
[Crossref]

T. Sondergaard, S. Boshelvonyi, “Vectorial model for multiple scattering by surface nanoparticles via surface polariton–polariton interactions,” Phys. Rev. B 67, 165405 (2003).
[Crossref]

P. Chaumet, A. Rahmani, F. de Fornel, J. P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310 (1998).
[Crossref]

A. Soubret, G. Berginc, C. Bourrely, “A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional, randomly rough surfaces,” Phys. Rev. B 63, 245411 (2001).
[Crossref]

S. Linden, J. Kuhl, H. Giessen, “Controlling the interaction between light and gold nanoparticles with selective suppression of extinction,” Phys. Rev. B 86, 4688–4691 (2001).

Prog. Opt. (1)

I. Ohlidal, K. Navratil, “Scattering of light from multilayer systems with rough boundary,” Prog. Opt. 34, 251–334 (1995).

Radio Sci. (2)

S. Smith, “The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces,” Radio Sci. 31, 1377–1385 (1996).
[Crossref]

M. F. Chen, A. K. Fung, “A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models,” Radio Sci. 23, 163–170 (1988).
[Crossref]

Waves Random Media (3)

M. Saillard, A. Sentenac, “Rigorous solutions for electromagnetic scattering from rough surfaces,” Waves Random Media 11, R103–R137 (2001).
[Crossref]

A. G. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media 4, 337–367 (1994).
[Crossref]

A. Sentenac, J.-J. Greffet, “Scattering by 2D particles deposited on a dielectric planar waveguide, a near-field and far-field study,” Waves Random Media 5, 145–155 (1995).
[Crossref]

Other (6)

X-Ray and Neutron Reflectivity: Principles and Applications, J. Daillant, A. Gibaud, eds. (Springer, New York, 1999).

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

L. Tsang, J. A. Kong, K. H. Ding, Scattering of electromagnetic waves, first volume of three in Wiley Series in Remote Sensing (Wiley-Interscience, New York, 2001).

A. G. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena (Springer, New York, 1994).

T. Tsang, J. A. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Rayleigh, The Theory of Sound, 3rd ed. (MacMillan, London, 1896).

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Figures (8)

Fig. 1
Fig. 1

Homogeneous rough deposit on a multilayer substrate.

Fig. 2
Fig. 2

Heterogeneous rough deposit on a multilayer substrate.

Fig. 3
Fig. 3

Scattering cross section at 20° incidence for the geometry of Fig. 2 with εh1=εh2=εml(z)=2.25 (glass on glass).

Fig. 4
Fig. 4

Same as Fig. 3 with εh1=εh2=εml(z)=-3+0.8i (metal on metal).

Fig. 5
Fig. 5

Same as Fig. 3 with εh1=εh2=-3+0.8i, εml(z)=2.25 (metal on glass).

Fig. 6
Fig. 6

Same as Fig. 3 with εh1=εh2=2.25, εml(z)=-3+0.8i (glass on metal).

Fig. 7
Fig. 7

Same as Fig. 3 with εh1=4, εh2=2.25, εml(z)=-3+0.8i (silicon+glass on metal).

Fig. 8
Fig. 8

Same as Fig. 3 with εh1=-3+0.8i, εh2=2.25 and a bilayer εml(z)=4 for 0>z>-λ/4 and εml(z)=-3+0.8i for -<z<-λ/4 (metal+glass on a bilayer of silicon–metal).

Equations (75)

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ε(R)=1ifz>h(r)εhif0zh(r)εml(z)ifz<0,
εref(R)=εref(z)=1ifz>0εml(z)ifz<0.
××E(R)-K2εref(z)E(R)=J(R)+V(R)E(R),
R×R×G(R, R)-K2εref(z)G(R, R)
=Iδ(R-R),
E(R)=Eref(r, z)+dRG(R, R)V(R)E(R),
Eref(R)=exp(iK0-R)pinc+exp(iK0+R)R(k0)pinc,
G(R, R)=-K-2zˆzˆδ(R-R)+G˜(r-r, z, z)
G˜(r, z, z)=i8π2R2dkqexp(ikr){exp[iq(z+z)]R(k)+exp(iq|z-z|)Psign(z-z)(k)},
Eext(R)=Eref(R)+dRG˜(R, R)V(R)Eint(R),
Eint(R)=AEref(R)+dRAG˜(R, R)V(R)Eint(R),
A=I+(1/εh-1)zˆzˆ.
Eext(r, z)=Eref(r, z)+Vdr0h(r)dzG˜×(r-r, z, z)AEref(r, z)+V2drdr0h(r)dz0h(r)dzG˜×(r-r, z, z)AG˜(r-r, z, z)×Eint(r, z).
E0(R)=Eref(R),
E1(r, z)=Vdrh(r)×G˜(r-r, z, 0+)AEref(r, 0+),zD.
hˆ(k)=14π2exp(-ikr)h(r)dr,
E1(r, z)=dkhˆ(k-k0)×exp(ikr+iqz)B1(k, k0)pinc(k0),
B1(k, k0)=-V2iqC+(k)AC-(k0).
C±(k)=R(k)+P±(k).
I=Vdr0h(r)dzG˜(r-r, z, z)AEref(r, z)2,
J=V2drdr0h(r)dz0h(r)dz×G˜(r-r, z, z)AG˜(r-r, z, z)×AEref(r, z)2,
E2(r, z)=dkdξhˆ(k-ξ)hˆ(ξ-k0)exp(ikr+iqz)×B2(k, k0, ξ)pinc(k0),
B2(k, k0, ξ)=-V4q {q0C+(k)A[R(k0)-P-(k0)]+q[R(k)-P+(k)]AC-(k0)}-V28qK2C+(k)A[(k-k0)zˆ+zˆ(k-k0)]AC-(k0)-V28qqξC+(k)A[C+(ξ)+C-(ξ)]AC-(k0).
E(r, z)-Einc(r, z)=dkexp(ikr+iqz)×S(k, k0)pinc(k0),
S0(k, k0)=R0(k0)δ(k-k0),
S1(k, k0)=hˆ(k-k0)B1(k, k0),
S2(k, k0)=dξhˆ(k-ξ)hˆ(ξ-k0)×B2(k, k0, ξ).
h(r)=i=1Nhi(r),
ε(R)
=εhiif rDi,0<z<hi(r)(i=1,,N)εml(z)ifz<01elsewhere.
S0(k, k0)=R0(k0)δ(k-k0),
S1(k, k0)=i=1NS1(k, k0, εhi, hi).
S2(k, k0)=i=1NS2(k, k0, εhi, hi)+ijdξh^i×(k-ξ)h^j(ξ-k0)B0ij(k, k0, ξ),
B0ij(k, k0, ξ)=-ViVj8qqξC+(k)Ai[C+(ξ)+C-(ξ)]AjC-(k0).
EBornext(R)=Eref(R)+dRG˜(R, R)V(R)AEref(R).
EBornext(R)=Eref(R)+dRG˜(R, R)V(R)Eref(R),
I=V2drh2(r)dG˜dz (r-r, z, 0+)AEref(r, 0+)+G˜(r-r, z, 0+)AdErefdz (r, 0+).
G˜(r-r, z, z)=G˜s(r-r, z, z)+G˜as(r-r, z, z),
G˜s(r-r, z, z)=12 [G˜(r-r, z, z)+G˜(r-r, z, z)],
G˜as(r-r, z, z)=12 [G˜(r-r, z, z)-G˜(r-r, z, z)].
G˜as(r-r, z, z)
=-sign(z-z) i8π2K2R2dξexp[iξ(r-r)]
×exp(iqξ|z-z|)(ξz˜+zˆξ),
12q [C-(k)-C+(k)]=12q [P-(k)-P+(k)]=1K2 (kzˆ+zˆk).
Js=drdrh(r)h(r)G˜(r-r, z, 0+)×AG˜s(r-r, 0+, 0+)AEref(r, 0+).
Ψ(r, z)dr0h(r)dzG˜as(r-r, z, z)×AEref(r, z)1.
Ψ(r, z)=14π2K2drdξexp[iξ(r-r)]exp(ik0r)i(ξzˆ+zˆξ)ϕ(r, z)AC-(k0)Einc=(rzˆ+zˆr)[ϕ(r, z)×exp(ik0r)]AC-(k0)Einc,
V2K2dr0h(r)dzG˜(r-r, z, z)A(rzˆ+zˆr)
×[(z-hr/2)exp(ik0r)]AC-(k0)Einc}2
=-V22K2drh(r)G˜(r-r, z, 0+)A[h(r)zˆ
+zˆh(r)]exp(ik0r)AC-(k0)}Einc.
-V4dkqexp(ikr+iqz)h^2(k-k0){q0C+(k)A[R(k0)-P-(k0)]+q[R(k)-P+(k)]AC-(k0)}-V28K2dkqexp(ikr+iqz)h^2(k-k0)C+(k)×A[(k-k0)zˆ+zˆ(k-k0)]AC-a(k0)-V28dkqdξqξ×exp(ikr+iqz)hˆ(k-ξ)hˆ(ξ-k0)C+(k)A[C+(ξ)+C-(ξ)]AC-(k0)pinc.
p1±(k0)=k0zˆ±q0k^0K,p2±(k0)=zˆ×k^0.
P±(k)=p1±(k)p1±(k)+p2±(k)p2±(k),
R(k)r1(k)p1+(k)p1-(k)+r2(k)p2+(k)p2-(k).
r1(k)=εq-qq+εq,r2(k)=q-qq+q,
rj(k)=rj(01)(k)+rj(12)(k)exp(2iq1L)1+rj(01)(k)rj(12)(k)exp(2iq1L),
Bn(k, k0)=i,j=12(Bn)ji(k, k0)pj(k)pi(k0),
(B1)11(k, k0)=(εh-1)2iq[1-r1(k)]×[1-r1(k0)]qq0kˆk^0-[1+r1(k)]×[1+r1(k0)] kk0εh,
(B1)21(k, k0)=-(B1)12(-k0, -k)=-(εh-1)q0K2iq ×[1-r1(k0)][1+r2(k)]zˆ[kˆ,k^0],
(B1)22(k, k0)=-(εh-1)K22iq [1+r2(k0)]×[1+r2(k)]kˆk^0.
(B2)ji(k, k0, ξ)=εh-14 αji+(εh-1)24 βji.
α11=-(εh-1)2εhq [kq0(1+r1)(1-r10)(kˆk^0k-k0)+k0q(1-r1)(1+r10)(k-k0kˆk^0)]-1εhq [(kk0+εhqq0kˆk^0)(q-q0)(r1-r10)+(kk0-εhqq0kˆk^0)(q+q0)(r1r10-1)],
β11=(1-r1)(1-r10)K2q0qξzˆ[kˆ,ξˆ]zˆ[k^0, ξˆ](1+r2)-q0qξ(1-r1)ξˆkˆξˆk^0-1εh2q (1+r1)×kk0ξ2qξ (1+r1)(1+r10)+q0kξεhξˆk^0r1×(1-r10)+k0ξεh (1-r1)(1+r10)r1kˆξˆ;
α12=Kzˆ[kˆ, k^0](q+q0)(r1r20+1)+(q-q0)×(r1+r20)+k22qεh (1+r1)(1+r20),
β12=zˆ[ξˆ, k^0](1+r20)kξqεh r1(1+r1)-(1-r1)×[qξ(1+r1)kˆξˆ-K(1+r2)zˆ(ξˆ, kˆ)ξˆk^0];
α22=K2qkˆk^0[q0(1+r2)(1-r20)+q(1-r2)(1+r20)],
β22=-(1+r20)(1+r2) K2qK2qξ (ξˆkˆ)(ξˆk^0)(1+r2)+qξ(1-r1)zˆ[kˆ,ξˆ]zˆ[ξˆ, k^0].
(B2)21(k, k0, ξ)=-(B2)12(-k0, -k, -ξ),
(Bcij)11(k, k0, ξ)
=(εhi-1)(εhj-1)4 β11(εh2, εh, εh)(εhiεhj, εhj, εhi),
(Bcij)12(k, k0, ξ)
=(εhi-1)(εhj-1)4 β12εhεhj,
(Bcij)22(k, k0, ξ)
=(εhi-1)(εhj-1)4 β22.

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