Abstract

We contribute to the study of the optical properties of high-permittivity nanostructures deposited on surfaces. We present what we believe is a new computational technique derived from the coupled-dipole approximation (CDA), which can accommodate high-permittivity scatterers. The discretized CDA equations are reformulated by use of the sampling theory to overcome different sources of inaccuracy that arise for high-permittivity scatterers. We first give the nonretarded filtered surface Green’s tensor used in the new scheme. We then assess the accuracy of the technique by comparing it with the standard CDA approach and show that it can accurately handle scatterers with a large permittivity.

© 2001 Optical Society of America

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  1. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  2. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  3. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  4. O. J. F. Martin, “3D Simulations of the experimental signal measured in near-field optical microscopy,” J. Microsc. (Oxford) 194, 235–239 (1999).
    [CrossRef]
  5. H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
    [CrossRef]
  6. M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
    [CrossRef]
  7. M. Paulus, O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
    [CrossRef]
  8. N. B. Piller, O. J. F. Martin, “Extension of the generalized multipole technique to anisotropic media,” Opt. Commun. 150, 1–6 (1998).
    [CrossRef]
  9. N. B. Piller, O. J. F. Martin, “Increasing the performances of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
    [CrossRef]
  10. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  11. J. R. Mosig, “Integral equation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 3, pp. 133–213.
  12. P. Gay-Balmaz, J. R. Mosig, “3D planar radiating structures in stratified media,” Int. J. Microwave Millimeter Wave CAE 3, 330–343 (1997).
    [CrossRef]
  13. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Institute of Electrical and Electronics Engineers, New York, 1994).
    [CrossRef]
  14. P. Gay-Balmaz, O. J. F. Martin, “Validity domain and limitation of non-retarded Green’s tensor for electromagnetic scattering at surfaces,” Opt. Commun. 184, 37–47 (2000).
    [CrossRef]
  15. H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Comments Pure Appl. Math. 3, 355–391 (1950).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  17. See, for example, the International Technology Roadmap for Semiconductors, http://public.itrs.net .
  18. O. J. F. Martin, C. Girard, A. Dereux, “Dielectric versus topographic contrast in near-field microscopy,” J. Opt. Soc. Am. A 13, 1801–1808 (1996).
    [CrossRef]
  19. The Computer Physics Communications Program Library, http://www.cpc.cs.qub.ac.uk .

2001 (1)

2000 (2)

P. Gay-Balmaz, O. J. F. Martin, “Validity domain and limitation of non-retarded Green’s tensor for electromagnetic scattering at surfaces,” Opt. Commun. 184, 37–47 (2000).
[CrossRef]

M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

1999 (1)

O. J. F. Martin, “3D Simulations of the experimental signal measured in near-field optical microscopy,” J. Microsc. (Oxford) 194, 235–239 (1999).
[CrossRef]

1998 (3)

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

N. B. Piller, O. J. F. Martin, “Extension of the generalized multipole technique to anisotropic media,” Opt. Commun. 150, 1–6 (1998).
[CrossRef]

N. B. Piller, O. J. F. Martin, “Increasing the performances of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

1997 (1)

P. Gay-Balmaz, J. R. Mosig, “3D planar radiating structures in stratified media,” Int. J. Microwave Millimeter Wave CAE 3, 330–343 (1997).
[CrossRef]

1996 (1)

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1988 (1)

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1950 (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Comments Pure Appl. Math. 3, 355–391 (1950).
[CrossRef]

Biebuyck, H.

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

Dereux, A.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Institute of Electrical and Electronics Engineers, New York, 1994).
[CrossRef]

Gay-Balmaz, P.

M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

P. Gay-Balmaz, O. J. F. Martin, “Validity domain and limitation of non-retarded Green’s tensor for electromagnetic scattering at surfaces,” Opt. Commun. 184, 37–47 (2000).
[CrossRef]

P. Gay-Balmaz, J. R. Mosig, “3D planar radiating structures in stratified media,” Int. J. Microwave Millimeter Wave CAE 3, 330–343 (1997).
[CrossRef]

Girard, C.

Goedecke, G. H.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

Levine, H.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Comments Pure Appl. Math. 3, 355–391 (1950).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Institute of Electrical and Electronics Engineers, New York, 1994).
[CrossRef]

Martin, O. J. F.

M. Paulus, O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
[CrossRef]

P. Gay-Balmaz, O. J. F. Martin, “Validity domain and limitation of non-retarded Green’s tensor for electromagnetic scattering at surfaces,” Opt. Commun. 184, 37–47 (2000).
[CrossRef]

M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

O. J. F. Martin, “3D Simulations of the experimental signal measured in near-field optical microscopy,” J. Microsc. (Oxford) 194, 235–239 (1999).
[CrossRef]

N. B. Piller, O. J. F. Martin, “Increasing the performances of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

N. B. Piller, O. J. F. Martin, “Extension of the generalized multipole technique to anisotropic media,” Opt. Commun. 150, 1–6 (1998).
[CrossRef]

O. J. F. Martin, C. Girard, A. Dereux, “Dielectric versus topographic contrast in near-field microscopy,” J. Opt. Soc. Am. A 13, 1801–1808 (1996).
[CrossRef]

Michel, B.

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

Mosig, J. R.

P. Gay-Balmaz, J. R. Mosig, “3D planar radiating structures in stratified media,” Int. J. Microwave Millimeter Wave CAE 3, 330–343 (1997).
[CrossRef]

J. R. Mosig, “Integral equation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 3, pp. 133–213.

O’Brien, S. G.

Paulus, M.

M. Paulus, O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Piller, N. B.

N. B. Piller, O. J. F. Martin, “Increasing the performances of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

N. B. Piller, O. J. F. Martin, “Extension of the generalized multipole technique to anisotropic media,” Opt. Commun. 150, 1–6 (1998).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Schmid, H.

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

Schwinger, J.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Comments Pure Appl. Math. 3, 355–391 (1950).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Schmid, H. Biebuyck, B. Michel, O. J. F. Martin, “Light-coupling masks for lensless, sub-wavelength optical lithography,” Appl. Phys. Lett. 72, 2379–2381 (1998).
[CrossRef]

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Comments Pure Appl. Math. (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Comments Pure Appl. Math. 3, 355–391 (1950).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. B. Piller, O. J. F. Martin, “Increasing the performances of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

Int. J. Microwave Millimeter Wave CAE (1)

P. Gay-Balmaz, J. R. Mosig, “3D planar radiating structures in stratified media,” Int. J. Microwave Millimeter Wave CAE 3, 330–343 (1997).
[CrossRef]

J. Microsc. (Oxford) (1)

O. J. F. Martin, “3D Simulations of the experimental signal measured in near-field optical microscopy,” J. Microsc. (Oxford) 194, 235–239 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

N. B. Piller, O. J. F. Martin, “Extension of the generalized multipole technique to anisotropic media,” Opt. Commun. 150, 1–6 (1998).
[CrossRef]

P. Gay-Balmaz, O. J. F. Martin, “Validity domain and limitation of non-retarded Green’s tensor for electromagnetic scattering at surfaces,” Opt. Commun. 184, 37–47 (2000).
[CrossRef]

Phys. Rev. B (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Phys. Rev. E (1)

M. Paulus, P. Gay-Balmaz, O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Other (5)

J. R. Mosig, “Integral equation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 3, pp. 133–213.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Institute of Electrical and Electronics Engineers, New York, 1994).
[CrossRef]

The Computer Physics Communications Program Library, http://www.cpc.cs.qub.ac.uk .

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

See, for example, the International Technology Roadmap for Semiconductors, http://public.itrs.net .

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Figures (9)

Fig. 1
Fig. 1

Typical scattering problem considered in this study: The scatterer ε(r) is deposited on, or embedded inside, a surface of permittivity ε1 and illuminated with an incident field E 0( r ). The upper medium has a permittivity ε2.

Fig. 2
Fig. 2

Geometry used for the calculations: A rectangular scatterer with dimensions d x = 110 nm, d y = 55 nm, d z = 55 nm is deposited on a surface. The scatterer and the surface have the same permittivity (see text). This system is illuminated with a plane wave propagating at θ = 23°. The incident electric field E 0 is linearly polarized, with a 30° angle from the p-polarization axis.

Fig. 3
Fig. 3

Amplitude of the E z field component at the center of the structure depicted in Fig. 2 (the field is calculated along the AB line shown in Fig. 2). The vertical dotted–dashed lines represent the scatterer boundary. Nonfiltered Green’s tensor.

Fig. 4
Fig. 4

Comparison between nonfiltered (continuous curve) and filtered (dashed curve) solutions for the two continuous field components E y and E z . (a) Same region as in Fig. 3. (b) Blowup of the region surrounding the right-hand interface.

Fig. 5
Fig. 5

Intensity E x Ex* of the x electric field component along the center of a scatterer with dimensions d x = 300 nm, d y = 300 nm, d z = 60 nm and permittivity ε = 10. The substrate has ε1 = 10, and the illumination wavelength in λ = 600 nm. (a) Nonfiltered Green’s tensor, (b) filtered Green’s tensor.

Fig. 6
Fig. 6

Field continuity at the left-hand interface of the scatterer depicted in Fig. 5 for three different permittivities ε = ε1 = 2, 6, or 10; λ = 600 nm. The intensity of the x electric field component is shown for the nonfiltered (continuous curve) and the filtered (dashed curve) Green’s tensor.

Fig. 7
Fig. 7

Simulation of defects on a high-permittivity substrate (ε1 = 10). (a) Trench (d x = 600 nm, d y = 100 nm, d z = 50 nm) is etched in the substrate. (b) Mesoscopic protrusion (d x = 600 nm, d y = 100 nm, d z = 50 nm) with similar permittivity is deposited on the surface.

Fig. 8
Fig. 8

Relative field intensity at a constant height x = 100 nm above the substrate, when a trench is etched in the dielectric [geometry of Fig. 7(a)]. Two incident polarizations are investigated: (a) p polarization and (b) s polarization. The wavelength is λ = 633 nm.

Fig. 9
Fig. 9

Relative field intensity at a constant height x = 100 nm above the substrate, when a protrusion is deposited on the surface [geometry of Fig. 7(b)]. Two incident polarizations are investigated: (a) p polarization and (b) s polarization. The wavelength is λ = 633 nm.

Equations (34)

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

Er=E0r+VdrGBr, r·k02ΔεrEr,
Δεr=εr-εB.
εB=ε1 for z<0  =ε2 for z0.
Ei=Ei0+j=1,jiNGi,jB·k02ΔεjEjVj+Mi·k02ΔεiEi-L·ΔεiεBEi,  i=1,  , N,
Mi=limδV0Vi-δVdrGBri, r.
*Gr, r=*G2r, rε1-ε2ε1+ε2 *G2r, r,
*Gr, r=*G1r, r±ε1-ε2ε1+ε2 *G1r, r,
*Gr, r=2ε1ε1+ε2 *G1r, r,
*Gr, r=2ε2ε1+ε2 *G2r, r.
g˜k=4π 0dRgRj0kRR2,
gR=12π20dkg˜kj0kRk2.
GBr, r=1+ikBR-1kB2R21+3-3ikBR-kB2R2kB2R4RRexpikBR4πR.
GFr, r=1+kB2gFr, r.
g˜k=1k2-kB2.
gFR=gR-12π2kFdkg˜kj0kRk2.
gFR=expjkBR4πR-sinkBR4π2RCikF+kBR-CikF-kBR-coskBR4π2R×π-SikF+kBR-SikF-kBR.
GFr, r=gF+gFkB2R1+-gFkB2R3+gFkB2R2RR+13kref2 hrr-rk.
GqsBr, r=-1kB2R21+3kB2R4RR14πR.
g˜k=1k2.
gqsF=14πR-14π2Rπ-2 SikFR.
GqsFr, r=gqsFkB2R1+-gqsFkB2R3+gqsFkB2R2RR+13kref2 hrr-rk.
hrr=sinkF|r|-kF|r|coskF|r|2π|r|3,
gF=gB-α4π2R,
gF=ikBgB-gBR+α4π2R2-α4π2R,
gF=-kB2gB-2ikBgBR+2gBR2-α2π2R3+α2π2R2-α4π2R,
α=sinkBRCi+-Ci-+coskBRπ-Si+-Si-,
α=kB sinkBR-π+Si-+Si++kB coskBR×Ci+-Ci--2 sinkFRR,
α=kB2 sinkBRCi--Ci++kB2 coskBRπ-Si-+Si++2 sinkFRR2-2kF coskFRR,
gqsF=gqsB-αqs4π2R,
gqsF=-gqsBR+αqs4π2R2-αqs4π2R,
gqsF=2gqsBR2-αqs2π2R3+αqs2π2R2-αqs4π2R,
αqs=π-2 SikFR,
αqs=-2 sinkFRR,
αqs=2 sinkFRR2-2kF coskFRR.

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