Abstract

An efficient forward scattering model is constructed for penetrable 2D submicron particles on rough substrates. The scattering and the particle-surface interaction are modeled using discrete sources with complex images. The substrate micro-roughness is described by a heuristic surface transfer function. The forward model is applied in the numerical estimation of the profile of a platinum (Pt) submicron wire on rough silicon (Si) substrate, based on experimental Bidirectional Reflectance Distribution Function (BRDF) data.

© 2012 OSA

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  1. F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
    [CrossRef]
  2. E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
    [CrossRef]
  3. E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
    [CrossRef]
  4. P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
    [CrossRef]
  5. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
    [CrossRef]
  6. T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE 4779, 60–71 (2002).
  7. J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
    [CrossRef]
  8. P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
    [CrossRef] [PubMed]
  9. J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
    [CrossRef]
  10. B. W. Bell and W. S. Bickel, “Single fiber light scattering matrix: an experimental determination,” Appl. Opt.20(22), 3874–3879 (1981).
    [CrossRef] [PubMed]
  11. M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
    [CrossRef]
  12. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. (Springer, 1998).
  13. J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE, 1995).
  14. E. D. Palik, Handbook of Optical Constants of Solids. (Academic Press, 1985).
  15. I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984).
    [CrossRef]
  16. J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt.27(8), 1527–1533 (1988).
    [CrossRef] [PubMed]
  17. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications. (Wiley, 2007).
  18. M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
    [CrossRef]
  19. S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express19(10), 9820–9835 (2011).
    [CrossRef] [PubMed]

2011

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
[CrossRef]

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express19(10), 9820–9835 (2011).
[CrossRef] [PubMed]

2010

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

2007

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
[CrossRef] [PubMed]

E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
[CrossRef]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

2005

E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
[CrossRef]

2001

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

1999

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

1988

1984

I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984).
[CrossRef]

1981

Alanen, E.

I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984).
[CrossRef]

Albella, P.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
[CrossRef] [PubMed]

Bell, B. W.

Bickel, W. S.

Coriand, L.

de la Peña, J. L.

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Duparré, A.

Eremin, Yu.

E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
[CrossRef]

Eremina, E.

E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
[CrossRef]

González, F.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
[CrossRef] [PubMed]

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Hansen, P.-E.

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
[CrossRef]

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

Harvey, J. E.

Karamehmedovic, M.

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
[CrossRef]

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

Lavrinenko, A. V.

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

Lindell, I. V.

I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984).
[CrossRef]

Moran, E. C.

Moreno, F.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
[CrossRef] [PubMed]

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

Moreno, F. F.

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Penalver, D. H.

Saiz, J. M.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express15(11), 6857–6867 (2007).
[CrossRef] [PubMed]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Schröder, S.

Sørensen, M.-P.

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

Tünnermann, A.

Valle, P. J.

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Videen, G.

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

Wriedt, T.

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
[CrossRef]

Zmek, W. P.

Appl. Opt.

IEEE Trans. Antenn. Propag.

I. V. Lindell and E. Alanen, “Exact Image Theory for the Sommerfeld Half-Space Problem, Part I: Vertical Magnetic Dipole,” IEEE Trans. Antenn. Propag.32(2), 126–133 (1984).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, “Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy,” J. Eur. Opt. Soc. Rapid Publ.5, 10021 (2010).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “A fast inversion method for highly conductive submicron wires on a substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11039 (2011).
[CrossRef]

M. Karamehmedović, P.-E. Hansen, and T. Wriedt, “An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate,” J. Eur. Opt. Soc. Rapid Publ.6, 11021 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

F. González, G. Videen, P. J. Valle, J. M. Saiz, J. L. de la Peña, and F. Moreno, “Light scattering computational methods for particles on substrates,” J. Quant. Spectrosc. Radiat. Transf.70(4-6), 383–393 (2001).
[CrossRef]

P. Albella, F. Moreno, J. M. Saiz, and F. González, “2D double interaction method for modeling small particles contaminating microstructures located on substrates,” J. Quant. Spectrosc. Radiat. Transf.106(1-3), 4–10 (2007).
[CrossRef]

Opt. Commun.

J. M. Saiz, J. L. de la Peña, F. González, and F. Moreno, “Detection and recognition of local defects in 1D structures,” Opt. Commun.196(1-6), 33–39 (2001).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Analysis of the light scattering properties of a gold nanorod on a plane surface via discrete sources method,” Opt. Commun.273(1), 278–285 (2007).
[CrossRef]

E. Eremina, Yu. Eremin, and T. Wriedt, “Discrete sources method for simulation of resonance spectra of nonspherical nanoparticles on a plane surface,” Opt. Commun.246(4-6), 405–413 (2005).
[CrossRef]

Opt. Express

Part. Part. Syst. Charact.

J. L. de la Peña, J. M. Saiz, P. J. Valle, F. González, and F. F. Moreno, “Tracking Scattering Minima to Size Metallic Particles on Flat Substrates,” Part. Part. Syst. Charact.16(3), 113–118 (1999).
[CrossRef]

Other

T. A. Germer, G. W. Mulholland, J. H. Kim, and S. H. Ehrman, “Measurement of the 100 nm NIST SRM® 1963 by laser surface light scattering,” Advanced Characterization Techniques for Optical, Semiconductor, and Data Storage Components, Angela Duparré and Bhanwar Singh, Eds., Proc. SPIE 4779, 60–71 (2002).

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications. (Wiley, 2007).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. (Springer, 1998).

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE, 1995).

E. D. Palik, Handbook of Optical Constants of Solids. (Academic Press, 1985).

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Figures (4)

Fig. 1
Fig. 1

Pt submicron wire on Si substrate: a)-b) SEM. c) AFM. d) Cross-section schematic.

Fig. 2
Fig. 2

The forward scattering model. a) Interior sources with complex images. b) Exterior sources.

Fig. 3
Fig. 3

The heuristic surface roughness model. a) Fitting the bare-substrate BRDF.
b) The effect on the forward model output.

Fig. 4
Fig. 4

Objective function: a) with, and b) without the roughness model.
c)-d) Top view of a) and b), respectively.
Best cross-section match: e) reconstructed |(E)inc + (E)sca|, f) reconstructed total near field amplitude.

Equations (11)

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

BRDF( θ s )= d P s ( θ s )/d Ω s P i cos θ s P s / θ s P i cos θ s P s cos θ s .
| E inc ( r ) | 2 = | z ^ E inc ( r ) | 2 exp[ ( | r |sinθ ) 2 / σ inc 2 ],
E sca ( r ) z ^ ν=1 N C ν Φ 1/2,Si ( r, r ν ), r + 2 \σ.
Φ 1/2,Si ( r, r ) ( 4j ) -1 [ H 0 (2) ( k 0 | r r | ) H 0 (2) ( k 0 | r r ˜ | ) ],r + 2 ,
E tot ( r ) z ^ ν=1 M D ν Φ Pt ( r, r ν ), rσ.
E + sca E - tot =( E + ref + E + inc ), σ ^ ×( H + sca H - tot )= σ ^ ×( H + ref + H + inc )at t μ .
RMSE:= n 1 ν ( log 10 I num ( θ ν ) log 10 I meas ( θ ν ) ) 2 , θ ν [ 20°,20° ].
I sca,0 ( θ ) ν=1 3 A ν exp[ ( | r |sinθ ) 2 / σ ν 2 ] ,θ[ 20°,20° ],
log 10 I sca,0 ( θ ) α 0 + ν=1 11 [ α ν cos( νγθ )+ β ν sin( νγθ ) ] ,
H( ξ ) ( I sca,0 ) ( | E inc | 2 ) 1 σ inc ν=1 3 A ν σ ν exp[ ξ 2 ( σ ν 2 σ inc 2 )/4 | r | 2 ] .
f 1 ε 1 1+2 ε 1 +(1f) ε 2 ε 1 ε 2 +2 ε 1 =0.

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