Abstract

We present a theory for the multiple scattering of light by obstacles situated over a rough surface. This problem is important for applications in biological and chemical sensors. To keep the formulation of this theory simple, we study scalar waves. This theory requires knowledge of the scattering operator (t-matrix) for each of the obstacles as well as the reflection operator for the rough surface. The scattering operator gives the field scattered by the obstacle due to an exciting field incident on the scatterer. The reflection operator gives the field reflected by the rough surface due to an exciting field incident on the rough surface. We apply this general theory for the special case of point scatterers and a slightly rough surface with homogeneous Dirichlet and Neumann boundary conditions. We show examples that demonstrate the utility of this theory.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102-1106 (1997).
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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]
  26. J. G. Watson and J. B. Keller, “Reflection, scattering, and absorption of acoustic waves by rough surfaces,” J. Acoust. Soc. Am. 74, 1887-1894 (1983).
    [CrossRef]
  27. A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface Green's function based on the first-order modified perturbation and smoothed diagram methods,” Waves Random Media 10, 17-31 (2000).
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2009 (2)

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

L.-X. Guo, J. Li, and H. Zeng, “Bistatic scattering from a three-dimensional object above a two-dimensional randomly rough surface modeled with the parallel FDTD approach,” J. Opt. Soc. Am. A 26, 2383-2392 (2009).
[CrossRef]

2008 (1)

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

2007 (1)

C. D. Chin, C. Linder, and S. K. Sia, “Lab-on-a-chip devices for global health: past studies and future opportunities,” Lab Chip 7, 41 (2007).
[CrossRef]

2005 (2)

N. L. Rosi and C. A. Mirkin, “Nanostructures in biodiagnostics,” Chem. Rev. (Washington, D.C.) 105, 1547-1562 (2005).
[CrossRef]

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

2003 (1)

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528-539 (2003).
[CrossRef] [PubMed]

2002 (1)

J. T. Johnson, “A numerical study of scattering from an object above a rough surface,” IEEE Trans. Antennas Propag. 50, 1361-1367 (2002).
[CrossRef]

2001 (2)

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

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

2000 (2)

A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface Green's function based on the first-order modified perturbation and smoothed diagram methods,” Waves Random Media 10, 17-31 (2000).
[CrossRef]

Y.-Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10, 273-280 (2000).
[CrossRef]

1999 (4)

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antennas Propag. 47, 902-913 (1999).
[CrossRef]

S. Weiss, “Fluorescence spectroscopy of single biomolecules,” Science 283, 1676-1683 (1999).
[CrossRef] [PubMed]

X. Fang, X. Liu, S. Schuster, and W. Tan, “Designing a novel molecular beacon for surface-immobilized DNA hybridization studies,” J. Am. Chem. Soc. 121, 2921 (1999).
[CrossRef]

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447-466 (1998).
[CrossRef]

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

1996 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

1993 (1)

1992 (1)

T. Sano, C. L. Smith, and C. R. Cantor, “Immuno-PCR: very sensitive antigen detection by means of specific antibody-DNA conjugates,” Science 258, 120 (1992).
[CrossRef] [PubMed]

1991 (2)

1983 (1)

J. G. Watson and J. B. Keller, “Reflection, scattering, and absorption of acoustic waves by rough surfaces,” J. Acoust. Soc. Am. 74, 1887-1894 (1983).
[CrossRef]

1981 (1)

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

1952 (1)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
[CrossRef]

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Baber, P. W.

Bergman, J. G.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Bickel, W. S.

Brolo, A. G.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

Cantor, C. R.

T. Sano, C. L. Smith, and C. R. Cantor, “Immuno-PCR: very sensitive antigen detection by means of specific antibody-DNA conjugates,” Science 258, 120 (1992).
[CrossRef] [PubMed]

Chemla, D. S.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Chin, C. D.

C. D. Chin, C. Linder, and S. K. Sia, “Lab-on-a-chip devices for global health: past studies and future opportunities,” Lab Chip 7, 41 (2007).
[CrossRef]

Chiu, T.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antennas Propag. 47, 902-913 (1999).
[CrossRef]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447-466 (1998).
[CrossRef]

Doll, J. C.

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

Economou, N. P.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Emory, S. R.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

Fang, X.

X. Fang, X. Liu, S. Schuster, and W. Tan, “Designing a novel molecular beacon for surface-immobilized DNA hybridization studies,” J. Am. Chem. Soc. 121, 2921 (1999).
[CrossRef]

Ferri, C. G. L.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Fu, C. -C.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Ghosh, S.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Gopinathan, A.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Gordon, R.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

Grimes, A.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Guo, L. -X.

Hawryluk, A. M.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Homola, J.

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528-539 (2003).
[CrossRef] [PubMed]

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Iafelice, V. J.

Ishimaru, A.

A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface Green's function based on the first-order modified perturbation and smoothed diagram methods,” Waves Random Media 10, 17-31 (2000).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

Jin, Y. -Q.

Y.-Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10, 273-280 (2000).
[CrossRef]

Johnson, J. T.

J. T. Johnson, “A numerical study of scattering from an object above a rough surface,” IEEE Trans. Antennas Propag. 50, 1361-1367 (2002).
[CrossRef]

Kavanagh, K. L.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

Keller, J. B.

J. G. Watson and J. B. Keller, “Reflection, scattering, and absorption of acoustic waves by rough surfaces,” J. Acoust. Soc. Am. 74, 1887-1894 (1983).
[CrossRef]

Khine, M.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Kim, J.

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

Kong, J. A.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

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

Kuga, Y.

A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface Green's function based on the first-order modified perturbation and smoothed diagram methods,” Waves Random Media 10, 17-31 (2000).
[CrossRef]

Lagendijk, A.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447-466 (1998).
[CrossRef]

Lax, M.

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621-629 (1952).
[CrossRef]

Lee, L. P.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

Li, G.

Y.-Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10, 273-280 (2000).
[CrossRef]

Li, J.

Liao, P. F.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Lindell, I. V.

Linder, C.

C. D. Chin, C. Linder, and S. K. Sia, “Lab-on-a-chip devices for global health: past studies and future opportunities,” Lab Chip 7, 41 (2007).
[CrossRef]

Liu, G. L.

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

Liu, X.

X. Fang, X. Liu, S. Schuster, and W. Tan, “Designing a novel molecular beacon for surface-immobilized DNA hybridization studies,” J. Am. Chem. Soc. 121, 2921 (1999).
[CrossRef]

Long, M.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Lu, Y.

G. L. Liu, Y. Lu, J. Kim, J. C. Doll, and L. P. Lee, “Magnetic nanocrescents as controllable surface enhanced Raman scattering nanoprobes for biomolecular imaging,” Adv. Mater. 17, 2683-2688 (2005).
[CrossRef]

Lumme, K. A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Melngailis, J.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Mirkin, C. A.

N. L. Rosi and C. A. Mirkin, “Nanostructures in biodiagnostics,” Chem. Rev. (Washington, D.C.) 105, 1547-1562 (2005).
[CrossRef]

Muinonen, K. O.

Nie, S.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

Rich, B. D.

C.-C. Fu, A. Grimes, M. Long, C. G. L. Ferri, B. D. Rich, S. Ghosh, S. Ghosh, L. P. Lee, A. Gopinathan, and M. Khine, “Tunable nanowrinkles on shape memory polymer sheets,” Adv. Mater. 21, 4472-4476 (2009).
[CrossRef]

Rockway, J. D.

A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface Green's function based on the first-order modified perturbation and smoothed diagram methods,” Waves Random Media 10, 17-31 (2000).
[CrossRef]

Rosi, N. L.

N. L. Rosi and C. A. Mirkin, “Nanostructures in biodiagnostics,” Chem. Rev. (Washington, D.C.) 105, 1547-1562 (2005).
[CrossRef]

Sano, T.

T. Sano, C. L. Smith, and C. R. Cantor, “Immuno-PCR: very sensitive antigen detection by means of specific antibody-DNA conjugates,” Science 258, 120 (1992).
[CrossRef] [PubMed]

Sarabandi, K.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antennas Propag. 47, 902-913 (1999).
[CrossRef]

Schuster, S.

X. Fang, X. Liu, S. Schuster, and W. Tan, “Designing a novel molecular beacon for surface-immobilized DNA hybridization studies,” J. Am. Chem. Soc. 121, 2921 (1999).
[CrossRef]

Sia, S. K.

C. D. Chin, C. Linder, and S. K. Sia, “Lab-on-a-chip devices for global health: past studies and future opportunities,” Lab Chip 7, 41 (2007).
[CrossRef]

Sihvola, A. H.

Sinton, D.

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

Smith, C. L.

T. Sano, C. L. Smith, and C. R. Cantor, “Immuno-PCR: very sensitive antigen detection by means of specific antibody-DNA conjugates,” Science 258, 120 (1992).
[CrossRef] [PubMed]

Tan, W.

X. Fang, X. Liu, S. Schuster, and W. Tan, “Designing a novel molecular beacon for surface-immobilized DNA hybridization studies,” J. Am. Chem. Soc. 121, 2921 (1999).
[CrossRef]

Tsang, L.

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

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Turner, M. G.

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447-466 (1998).
[CrossRef]

Videen, G.

Watson, J. G.

J. G. Watson and J. B. Keller, “Reflection, scattering, and absorption of acoustic waves by rough surfaces,” J. Acoust. Soc. Am. 74, 1887-1894 (1983).
[CrossRef]

Weiss, S.

S. Weiss, “Fluorescence spectroscopy of single biomolecules,” Science 283, 1676-1683 (1999).
[CrossRef] [PubMed]

Wokaun, A.

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, “Surface-enhanced Raman scattering from microlithographic silver particle surfaces,” Chem. Phys. Lett. 82, 355-359 (1981).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wolfe, W. L.

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Zeng, H.

Acc. Chem. Res. (1)

R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, “A new generation of sensors based on extraordinary optical transmission,” Acc. Chem. Res. 41, 1049-1057 (2008).
[CrossRef] [PubMed]

Adv. Mater. (2)

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Figures (7)

Fig. 1
Fig. 1

Method for fabricating low-cost and nano-structured metallic substrates reported in [12]. A shape memory polymer is coated with a thin film of metal. Upon heating, the polymer retracts, but the metal does not lead to a buckling of the metal surface. The final image on the right shows a scanning electron micrograph of one such nano-structured metallic substrate fabricated using this method.

Fig. 2
Fig. 2

A sketch of the physical problem. A wave is incident on several obstacles situated over a rough surface. The rough surface is given by the function z = f ( x , y ) . Light scatters from the obstacles and the rough surface.

Fig. 3
Fig. 3

A diagram showing the interactions between the point obstacle and the slightly rough surface given by Eq. (5.3). In the first diagram, the incident field and the incident field reflected by the rough surface excite the point obstacle. Next, that exciting field is scattered by the point obstacle and reflected by the rough surface to excite the point obstacle again. This series continues to include infinitely many interactions between the point obstacle and the slightly rough surface.

Fig. 4
Fig. 4

A plot of the rough surface and point obstacle shown on the y = 0 plane. The point obstacle is located at position r 1 = ( 11.7 , 0.0 , 0.1 ) in units of wavelengths.

Fig. 5
Fig. 5

Contour plots of the image I ( x , y ) defined in Eq. (5.5) corresponding to a single point obstacle shown in Fig. 4 for the Dirichlet (top) and Neumann (bottom) cases.

Fig. 6
Fig. 6

A plot of the rough surface and two point obstacles shown on the y = 0 plane. The point obstacles are located at positions r 1 = ( 11.7 , 0.0 , 0.1 ) and r 2 = ( 9.7 , 0.0 , 0.1 ) in units of wavelengths.

Fig. 7
Fig. 7

Contour plots of the image I ( x , y ) defined in Eq. (5.5) corresponding to two point obstacles shown in Fig. 6 for the Dirichlet (top) and Neumann (bottom) cases.

Equations (67)

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2 u + k 2 u = k 2 m = 1 M V m u ,     in   z > f ( x , y ) ,
u = 0     on   z = f ( x , y ) ,
ν u = 0     on   z = f ( x , y ) ,
ν = ( x f , y f , 1 ) ( x f ) 2 + ( y f ) 2 + 1
2 u + k 2 u = 0 ,
S m u E ( r ) = χ m t m ( r , r ) u E ( r ) d r .
u = u i + m = 1 M S m ϕ m + R ψ .
ϕ m = u i + n = 1 n m M S n ϕ n + R ψ     in   χ m ,     m = 1 , , M ,
ψ = u i + m = 1 M S m ϕ m     on   z = f ( x , y ) .
S m u i ( r ) = σ m G 0 ( r ; r m ) u i ( r m ) ,
G 0 ( r ; r m ) = e i k | r r m | e | r r m | / a 4 π [ 1 + ( k a ) 2 ] | r r m |
ϕ m = u i ( r m ) + n = 1 n m M σ n G 0 ( r m ; r n ) ϕ n + R ψ ( r m ) ,     m = 1 , , M .
ψ ( r ) = u i ( r ) + m = 1 M σ m G 0 ( r ; r m ) ϕ m     on   z = f ( x , y ) .
u i ( x , y , z ) = A ( ξ , η ) e i ξ x + i η y i κ z d ξ d η ,
κ = κ ( ξ , η ) = { k 2 ξ 2 η 2 , ξ 2 + η 2 k 2 i ξ 2 + η 2 k 2 , ξ 2 + η 2 > k 2 } .
R u i ( x , y , z ) = R A ( ξ , η ) e i ξ x + i η y + i κ z d ξ d η ,
ψ ( x , y , 0 ) = u i ( x , y , 0 ) + m = 1 M σ m G 0 ( x , y , 0 ; r m ) ϕ m .
Ψ ( ξ , η ) = A ( ξ , η ) + m = 1 M σ m g ̂ 0 ( ξ , η ; r m ) ϕ m ,
Ψ ( ξ , η ) = 1 ( 2 π ) 2 ψ ( x , y , 0 ) e i ξ x i η y d x d y ,
g ̂ 0 ( ξ , η ; r m ) = i 8 π 2 κ e i ξ x m i η y m + i κ | z m | .
G 0 ( r ; r ) = 1 ( 2 π ) 2 i 2 κ exp [ i ξ ( x x ) + i η ( y y ) + i κ | z z | ] d ξ d η .
R G 0 ( r ; r n ) = R g ̂ 0 ( ξ , η ; r n ) e i ξ x + i η y + i κ z d ξ d η ,
R u i ( r ) = R A ( ξ , η ) e i ξ x + i η y + i κ z d ξ d η .
R ψ ( r m ) = R u i ( r m ) + n = 1 M σ n R G 0 ( r m ; r n ) ϕ n .
n = 1 M A m n ϕ n = u i ( r m ) + R u i ( r m ) ,     m = 1 , , M ,
A m n = { 1 σ m R G 0 ( r m ; r m ) , m = n σ n [ G 0 ( r m ; r n ) + R G 0 ( r m ; r n ) ] , m n . }
u s ( r ) = R u i ( r ) + m = 1 M σ m [ G 0 ( r ; r m ) + R G 0 ( r ; r m ) ] ϕ m .
[ 1 σ 1 R G 0 ( r 1 ; r 1 ) ] ϕ 1 = u i ( r 1 ) + R u i ( r 1 ) .
ϕ 1 = u i ( r 1 ) + R u i ( r 1 ) 1 σ 1 R G 0 ( r 1 ; r 1 ) .
ϕ 1 = n = 0 [ σ 1 R G 0 ( r 1 ; r 1 ) ] n [ u i ( r 1 ) + R u i ( r 1 ) ] .
u s ( r ) = R u i ( r ) + σ 1 [ G 0 ( r ; r 1 ) + R G 0 ( r ; r 1 ) ] ϕ 1 .
I ( x , y ) = | u s ( x , y , z 0 ) | 2 | R u i ( x , y , z 0 ) | 2
u s F 1 ( s ̂ ) e i k R R ,     k R ,
F 1 ( s ̂ ) = i 2 π k s z R A ( k s x , k s y ) + σ 1 ϕ 1 [ e i k s ̂ r 1 4 π i 2 π k s z R g ̂ 0 ( k s x , k s y ; r 1 ) ] .
[ A 11 A 12 A 21 A 22 ] [ ϕ 1 ϕ 2 ] = [ b 1 b 2 ] ,
[ b 1 b 2 ] = [ u i ( r 1 ) + R u i ( r 1 ) u i ( r 2 ) + R u i ( r 2 ) ] .
ϕ 1 = 1 det ( A ) { [ 1 σ 2 R G 0 ( r 2 ; r 2 ) ] b 1 σ 2 [ G 0 ( r 1 ; r 2 ) + R G 0 ( r 1 ; r 2 ) ] b 2 } ,
ϕ 2 = 1 det ( A ) { [ 1 σ 1 R G 0 ( r 1 ; r 1 ) ] b 2 σ 1 [ G 0 ( r 2 ; r 1 ) + R G 0 ( r 2 ; r 1 ) ] b 1 } ,
det ( A ) = 1 σ 1 R G 0 ( r 1 ; r 1 ) σ 2 R G 0 ( r 2 ; r 2 ) σ 1 σ 2 [ G 0 ( r 1 ; r 1 ) G 0 ( r 2 ; r 2 ) + G 0 ( r 1 ; r 1 ) R G 0 ( r 2 ; r 2 ) + R G 0 ( r 1 ; r 1 ) G 0 ( r 2 ; r 2 ) ] .
u s ( r ) = R u i ( r ) + σ 1 [ G 0 ( r ; r 1 ) + R G 0 ( r ; r 1 ) ] ϕ 1 + σ 2 [ G 0 ( r ; r 2 ) + R G 0 ( r ; r 2 ) ] ϕ 2 .
u s F 2 ( s ̂ ) e i k R R ,     k R ,
F 2 ( s ̂ ) = i 2 π k s z R A ( k s x , k s y ) + σ 1 ϕ 1 [ e i k s ̂ r 1 4 π i 2 π k s z R g ̂ 0 ( k s x , k s y ; r 1 ) ] + σ 2 ϕ 2 [ e i k s ̂ r 2 4 π i 2 π k s z R g ̂ 0 ( k s x , k s y ; r 2 ) ] .
2 u + k 2 u = 0     in   z > ϵ f ( x , y ) ,
u = 0     on   z = ϵ f ( x , y ) .
[ 1 + ϵ f z + ϵ 2 1 2 f 2 z 2 + O ( ϵ 3 ) ] ( u i + R u i ) = 0     on   z = 0.
R u i ( x , y , z ) = B ( ξ , η ) e i ξ x + i η y + i κ z d ξ d η ,
B ( ξ , η ) = A ( ξ , η ) + ϵ i F ( ξ ξ , η η ) κ ( ξ , η ) [ A ( ξ , η ) B ( ξ , η ) ] d ξ d η + ϵ 2 1 2 F ( ξ ξ , η η ) F ( ξ ξ , η η ) κ 2 ( ξ , η ) [ A ( ξ , η ) + B ( ξ , η ) ] d ξ d η d ξ d η .
B ( ξ , η ) = B 0 ( ξ , η ) + ϵ B 1 ( ξ , η ) + ϵ 2 B 2 ( ξ , η ) + O ( ϵ 3 ) .
B 0 ( ξ , η ) = A ( ξ , η ) .
B 1 ( ξ , η ) = i F ( ξ ξ , η η ) κ ( ξ , η ) [ A ( ξ , η ) B 0 ( ξ , η ) ] d ξ d η
B 1 ( ξ , η ) = i 2 F ( ξ ξ , η η ) κ ( ξ , η ) A ( ξ , η ) d ξ d η = i 2 F ( κ A ) .
B 2 ( ξ , η ) = 1 2 F ( ξ ξ , η η ) F ( ξ ξ , η η ) κ 2 ( ξ , η ) [ A ( ξ , η ) + B 0 ( ξ , η ) ] d ξ d η d ξ d η i F ( ξ ξ , η η ) κ ( ξ , η ) B 1 ( ξ , η ) d ξ d η .
B 2 ( ξ , η ) = 2 F ( ξ ξ , η η ) κ ( ξ , η ) F ( ξ ξ , η η ) κ ( ξ , η ) A ( ξ , η ) d ξ d η d ξ d η = 2 F [ κ F ( κ A ) ] .
B ( ξ , η ) = R D A = A + ϵ i 2 F κ A + ϵ 2 2 F [ κ F ( κ A ) ] + O ( ϵ 3 ) .
2 u + k 2 u = 0     in   z > ϵ f ( x , y ) ,
ν u = 0     on     z = ϵ f ( x , y ) .
[ z + ϵ ( f z 2 f x x f y y ) + ϵ 2 ( 1 2 f 2 z 3 f f x x z f f y y z ) + O ( ϵ 3 ) ] ( u i + R u i ) = 0     on   z = 0.
i κ B ( ξ , η ) = i κ A ( ξ , η ) + ϵ F ( ξ ξ , η η ) [ κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) ] [ A ( ξ , η ) + B ( ξ , η ) ] d ξ d η ϵ 2 i F ( ξ ξ , η η ) F ( ξ ξ , η η ) [ 1 2 κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) ] κ ( ξ , η ) [ A ( ξ , η ) B ( ξ , η ) ] d ξ d η d ξ d η .
B 0 ( ξ , η ) = A ( ξ , η ) .
i κ B 1 ( ξ , η ) = F ( ξ ξ , η η ) [ κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) ] [ A ( ξ , η ) + B 0 ( ξ , η ) ] d ξ d η .
κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) = k 2 ξ ξ η η ,
B 1 ( ξ , η ) = i 2 F ( ξ ξ , η η ) α ( ξ , η ; ξ , η ) A ( ξ , η ) d ξ d η = i 2 F ( α A ) ,
α ( ξ , η ; ξ , η ) = k 2 ξ ξ η η κ ( ξ , η ) .
i κ B 2 ( ξ , η ) = i F ( ξ ξ , η η ) F ( ξ ξ , η η ) [ 1 2 κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) ] κ ( ξ , η ) [ A ( ξ , η ) B 0 ( ξ , η ) ] d ξ d η d ξ d η + F ( ξ ξ , η η ) [ κ 2 ( ξ , η ) ( ξ ξ ξ 2 ) ( η η η 2 ) ] B 1 ( ξ , η ) d ξ d η .
B 2 ( ξ , η ) = i F ( ξ ξ , η η ) α ( ξ , η ; ξ , η ) B 1 ( ξ , η ) d ξ d η .
B 2 ( ξ , η ) = 2 F ( ξ ξ , η η ) α ( ξ , η ; ξ , η ) F ( ξ ξ , η η ) α ( ξ , η ; ξ , η ) A ( ξ , η ) d ξ d η d ξ d η = 2 F [ α F ( α A ) ] .
B ( ξ , η ) = R N A = A + ϵ i 2 F ( α A ) ϵ 2 2 F [ α F ( α A ) ] + O ( ϵ 3 ) .

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