Abstract

Light scattering and transmission by rough surfaces are of considerable interest in a variety of applications including remote sensing and characterization of surfaces. In this work, the finite-difference time-domain technique is applied to calculate the scattered and transmitted electromagnetic fields of an infinite periodic rough surface. The elements of the Mueller matrix for scattered light are calculated by an integral of the near fields over a significant number of periods of the surface. The normalized Mueller matrix elements of the scattered light and the spatial distribution of the transmitted flux for a monolayer of micrometer-sized dielectric spheres on a silicon substrate are presented. The numerical results show that the nonzero Mueller matrix elements for scattering from a surface consisting of a monolayer of dielectric spheres on a silicon substrate have specific maxima at some scattering angles. These maxima may be used in the characterization of features of the surface. For light transmitted through the monolayer of spheres, our results show that the transmitted energy focuses around the ray passing through centers of the spheres. At other locations, the transmitted flux is very small. Therefore, micrometer-sized dielectric spheres might be placed on a semiconductor surface to burn nanometer-sized holes in a layer using laser pulses. The method may also be useful in the assembly of periodic microstructures on surfaces.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  37. S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
    [CrossRef]
  38. D. E. Merewether, R. Fisher, and F. W. Smith, "On implementing a numeric Huygen's source in a finite difference program to illustrate scattering bodies," IEEE Trans. Nucl. Sci. NS-27, 1829-1833 (1980).
    [CrossRef]
  39. K. Umashanker and A. Taflove, "A novel method to analyze electromagnetic scattering of complex objects," IEEE Trans. Electromagn. Compat. EMC-24, 397-405 (1982).
    [CrossRef]
  40. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  41. P. Yang and K. N. Liou, "Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space," J. Opt. Soc. Am. A 13, 2072-2085 (1996).
    [CrossRef]
  42. W. Sun, N. G. Loeb, and Q. Fu, "Finite-difference time domain solution of light scattering and absorption by particles in an absorbing medium," Appl. Opt. 41, 5728-5743 (2002).
    [CrossRef] [PubMed]

2006

J. Mullins, "The stuff of beams," New Sci. 190, 44-47 (2006).

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
[CrossRef]

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

2002

W. Sun, N. G. Loeb, and Q. Fu, "Finite-difference time domain solution of light scattering and absorption by particles in an absorbing medium," Appl. Opt. 41, 5728-5743 (2002).
[CrossRef] [PubMed]

D. L. Schuler, J.-S. Lee, D. Kasilingam, and G. Nesti, "Surface roughness and slope measurements using polarimetric SAR data," IEEE Trans. Geosci. Remote Sens. 40, 687-698 (2002).
[CrossRef]

H. Lin and J. Zhu, "Characterization of nanocrystalline silicon films," Proc. SPIE 4700, 354-356 (2002).

A. K. Fung, Z. Li, and K. S. Chen, "An improved IEM model for bistatic scattering from rough surfaces," J. Electromagn. Waves Appl. 16, 689-702 (2002).
[CrossRef]

2001

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, 103-137 (2001).
[CrossRef]

1999

F. Mattia, "Backscattering properties of multi-scale rough surfaces," J. Electromagn. Waves Appl. 13, 493-527 (1999).
[CrossRef]

1998

K. S. Chen, T. D. Wu, and A. K. Fung, "A study of backscattering from multiscale rough surface," J. Electromagn. Waves Appl. 12, 961-979 (1998).
[CrossRef]

S. Gomez, K. Hale, J. Burrows, and B. Griffiths, "Measurements of surface defects on optical components," Meas. Sci. Technol. 9, 607-616 (1998).
[CrossRef]

1997

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

1996

P. Yang and K. N. Liou, "Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space," J. Opt. Soc. Am. A 13, 2072-2085 (1996).
[CrossRef]

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, "A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surfaces," IEEE Trans. Antennas Propag. 44, 504-513 (1996).
[CrossRef]

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

1995

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

F. D. Hastings, J. B. Schneider, and S. L. Broschat, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).

K. Demarest, R. Plumb, and Z. Huang, "FDTD modeling of scatterers in stratified media," IEEE Trans. Antennas Propag. 43, 1164-1168 (1995).
[CrossRef]

1994

A. K. Fung, M. R. Shah, and S. Tjuatja, "Numerical simulation of scattering from three-dimensional randomly rough surfaces," IEEE Trans. Geosci. Remote Sens. 32, 986-994 (1994).
[CrossRef]

R. J. Luebbers and C. Penney, "Scattering from apertures in infinite ground planes using FDTD," IEEE Trans. Antennas Propag. 42, 731-736 (1994).
[CrossRef]

1993

1992

A. K. Fung, Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. Geosci. Remote Sens. 30, 356-369 (1992).
[CrossRef]

1991

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, "Electromagnetic scattering of waves by rough surfaces: a finite-difference time-domain approach," Microwave Opt. Technol. Lett. 4, 355-359 (1991).
[CrossRef]

1986

A. K. Fung and G. W. Pan, "An integral equation method for rough surface scattering," in Proceedings of the International Symposium on Multiple Scattering of Waves in Random Media and Random Surfaces (1986), pp. 701-714.

1982

K. Umashanker and A. Taflove, "A novel method to analyze electromagnetic scattering of complex objects," IEEE Trans. Electromagn. Compat. EMC-24, 397-405 (1982).
[CrossRef]

1980

D. E. Merewether, R. Fisher, and F. W. Smith, "On implementing a numeric Huygen's source in a finite difference program to illustrate scattering bodies," IEEE Trans. Nucl. Sci. NS-27, 1829-1833 (1980).
[CrossRef]

1966

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

1954

H. Davies, "The reflection of electromagnetical waves from rough surfaces," Proc. Inst. Electr. Eng. 101, 209-214 (1954).

1953

C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 66-570 (1953).
[CrossRef]

1951

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[CrossRef]

1941

Ao, C. O.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Baron, J. E.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, "A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surfaces," IEEE Trans. Antennas Propag. 44, 504-513 (1996).
[CrossRef]

Beckmann, P.

P. Beckmann and A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Bohren, C. F.

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

Broschat, S. L.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).

Budiarto, H.

H. Budiarto and J. Takada, The Electromagnetic Wave Scattering from Building Surfaces for the Mobile Propagation Modeling, ITE Technical Report 25 (ITE, 2001), pp. 7-11.

Burrows, J.

S. Gomez, K. Hale, J. Burrows, and B. Griffiths, "Measurements of surface defects on optical components," Meas. Sci. Technol. 9, 607-616 (1998).
[CrossRef]

Chan, C. H.

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, "Electromagnetic scattering of waves by rough surfaces: a finite-difference time-domain approach," Microwave Opt. Technol. Lett. 4, 355-359 (1991).
[CrossRef]

Chen, K. S.

A. K. Fung, Z. Li, and K. S. Chen, "An improved IEM model for bistatic scattering from rough surfaces," J. Electromagn. Waves Appl. 16, 689-702 (2002).
[CrossRef]

K. S. Chen, T. D. Wu, and A. K. Fung, "A study of backscattering from multiscale rough surface," J. Electromagn. Waves Appl. 12, 961-979 (1998).
[CrossRef]

A. K. Fung, Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. Geosci. Remote Sens. 30, 356-369 (1992).
[CrossRef]

Coppo, P.

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

Davies, H.

H. Davies, "The reflection of electromagnetical waves from rough surfaces," Proc. Inst. Electr. Eng. 101, 209-214 (1954).

Demarest, K.

K. Demarest, R. Plumb, and Z. Huang, "FDTD modeling of scatterers in stratified media," IEEE Trans. Antennas Propag. 43, 1164-1168 (1995).
[CrossRef]

Ding, K. H.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Eckart, C.

C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 66-570 (1953).
[CrossRef]

Fano, U.

Ferrari, R. L.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers (Cambridge U. Press, 1990).

Fisher, R.

D. E. Merewether, R. Fisher, and F. W. Smith, "On implementing a numeric Huygen's source in a finite difference program to illustrate scattering bodies," IEEE Trans. Nucl. Sci. NS-27, 1829-1833 (1980).
[CrossRef]

Fu, Q.

Fung, A. K.

A. K. Fung, Z. Li, and K. S. Chen, "An improved IEM model for bistatic scattering from rough surfaces," J. Electromagn. Waves Appl. 16, 689-702 (2002).
[CrossRef]

K. S. Chen, T. D. Wu, and A. K. Fung, "A study of backscattering from multiscale rough surface," J. Electromagn. Waves Appl. 12, 961-979 (1998).
[CrossRef]

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

A. K. Fung, M. R. Shah, and S. Tjuatja, "Numerical simulation of scattering from three-dimensional randomly rough surfaces," IEEE Trans. Geosci. Remote Sens. 32, 986-994 (1994).
[CrossRef]

A. K. Fung, Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. Geosci. Remote Sens. 30, 356-369 (1992).
[CrossRef]

A. K. Fung and G. W. Pan, "An integral equation method for rough surface scattering," in Proceedings of the International Symposium on Multiple Scattering of Waves in Random Media and Random Surfaces (1986), pp. 701-714.

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

Gedney, S. D.

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

Gomez, S.

S. Gomez, K. Hale, J. Burrows, and B. Griffiths, "Measurements of surface defects on optical components," Meas. Sci. Technol. 9, 607-616 (1998).
[CrossRef]

Gonzalez, F.

Griffiths, B.

S. Gomez, K. Hale, J. Burrows, and B. Griffiths, "Measurements of surface defects on optical components," Meas. Sci. Technol. 9, 607-616 (1998).
[CrossRef]

Grzegorczyk, T. M.

Gurrola, E. M.

P. B. Wong, G. L. Tyler, J. E. Baron, E. M. Gurrola, and R. A. Simpson, "A three-wave FDTD approach to surface scattering with applications to remote sensing of geophysical surfaces," IEEE Trans. Antennas Propag. 44, 504-513 (1996).
[CrossRef]

Hale, K.

S. Gomez, K. Hale, J. Burrows, and B. Griffiths, "Measurements of surface defects on optical components," Meas. Sci. Technol. 9, 607-616 (1998).
[CrossRef]

Hastings, F. D.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).

Hsieh, C. Y.

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

Huang, Z.

K. Demarest, R. Plumb, and Z. Huang, "FDTD modeling of scatterers in stratified media," IEEE Trans. Antennas Propag. 43, 1164-1168 (1995).
[CrossRef]

Huffman, D. R.

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

Jin, J. M.

J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

Jordan, D. L.

Kasilingam, D.

D. L. Schuler, J.-S. Lee, D. Kasilingam, and G. Nesti, "Surface roughness and slope measurements using polarimetric SAR data," IEEE Trans. Geosci. Remote Sens. 40, 687-698 (2002).
[CrossRef]

Kemp, B. A.

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

Kong, J. A.

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
[CrossRef]

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, "Electromagnetic scattering of waves by rough surfaces: a finite-difference time-domain approach," Microwave Opt. Technol. Lett. 4, 355-359 (1991).
[CrossRef]

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Lee, J.-F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

Lee, J.-S.

D. L. Schuler, J.-S. Lee, D. Kasilingam, and G. Nesti, "Surface roughness and slope measurements using polarimetric SAR data," IEEE Trans. Geosci. Remote Sens. 40, 687-698 (2002).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

Li, Z.

A. K. Fung, Z. Li, and K. S. Chen, "An improved IEM model for bistatic scattering from rough surfaces," J. Electromagn. Waves Appl. 16, 689-702 (2002).
[CrossRef]

A. K. Fung, Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. Geosci. Remote Sens. 30, 356-369 (1992).
[CrossRef]

Lin, H.

H. Lin and J. Zhu, "Characterization of nanocrystalline silicon films," Proc. SPIE 4700, 354-356 (2002).

Liou, K. N.

Loeb, N. G.

Lou, S. H.

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, "Electromagnetic scattering of waves by rough surfaces: a finite-difference time-domain approach," Microwave Opt. Technol. Lett. 4, 355-359 (1991).
[CrossRef]

Luebbers, R. J.

R. J. Luebbers and C. Penney, "Scattering from apertures in infinite ground planes using FDTD," IEEE Trans. Antennas Propag. 42, 731-736 (1994).
[CrossRef]

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Mattia, F.

F. Mattia, "Backscattering properties of multi-scale rough surfaces," J. Electromagn. Waves Appl. 13, 493-527 (1999).
[CrossRef]

Merewether, D. E.

D. E. Merewether, R. Fisher, and F. W. Smith, "On implementing a numeric Huygen's source in a finite difference program to illustrate scattering bodies," IEEE Trans. Nucl. Sci. NS-27, 1829-1833 (1980).
[CrossRef]

Moreno, F.

Mullins, J.

J. Mullins, "The stuff of beams," New Sci. 190, 44-47 (2006).

Nesti, G.

D. L. Schuler, J.-S. Lee, D. Kasilingam, and G. Nesti, "Surface roughness and slope measurements using polarimetric SAR data," IEEE Trans. Geosci. Remote Sens. 40, 687-698 (2002).
[CrossRef]

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

Pan, G. W.

A. K. Fung and G. W. Pan, "An integral equation method for rough surface scattering," in Proceedings of the International Symposium on Multiple Scattering of Waves in Random Media and Random Surfaces (1986), pp. 701-714.

Penney, C.

R. J. Luebbers and C. Penney, "Scattering from apertures in infinite ground planes using FDTD," IEEE Trans. Antennas Propag. 42, 731-736 (1994).
[CrossRef]

Plumb, R.

K. Demarest, R. Plumb, and Z. Huang, "FDTD modeling of scatterers in stratified media," IEEE Trans. Antennas Propag. 43, 1164-1168 (1995).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, The Theory of Sound (MacMillan, 1896).

Rice, S. O.

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[CrossRef]

S. O. Rice, Reflection of EM from Slightly Rough Surfaces (Interscience, 1963).

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
[CrossRef]

Saillard, M.

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, 103-137 (2001).
[CrossRef]

Saiz, J. M.

Schneider, J. B.

F. D. Hastings, J. B. Schneider, and S. L. Broschat, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).

Schuler, D. L.

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K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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K. Umashanker and A. Taflove, "A novel method to analyze electromagnetic scattering of complex objects," IEEE Trans. Electromagn. Compat. EMC-24, 397-405 (1982).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

A. K. Fung, M. R. Shah, and S. Tjuatja, "Numerical simulation of scattering from three-dimensional randomly rough surfaces," IEEE Trans. Geosci. Remote Sens. 32, 986-994 (1994).
[CrossRef]

C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, "A further study of the IEM surface scattering model," IEEE Trans. Geosci. Remote Sens. 35, 901-909 (1997).
[CrossRef]

A. K. Fung, Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. Geosci. Remote Sens. 30, 356-369 (1992).
[CrossRef]

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[CrossRef]

IEEE Trans. Nucl. Sci.

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[CrossRef]

S. O. Rice, Reflection of EM from Slightly Rough Surfaces (Interscience, 1963).

H. Lin and J. Zhu, "Characterization of nanocrystalline silicon films," Proc. SPIE 4700, 354-356 (2002).

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Figures (8)

Fig. 1
Fig. 1

Surface consisting of a monolayer of dielectric spheres on a silicon substrate and the coordinate systems used in this study. Normal radiation incidence is treated, i.e., along the z direction.

Fig. 2
Fig. 2

Real and virtual surfaces associated with a unit periodicity. Used for implementing the FDTD algorithm for calculation of the scattered and transmitted fields near a periodic rough surface. We use periodic boundary conditions and UPML absorbing boundary conditions to truncate the computational domain.

Fig. 3
Fig. 3

Coordinate systems for the derivation of scattering matrix elements. The unit vectors α, β, and r are orthogonal. r is in the scattering direction, and β is perpendicular to the scattering plane.

Fig. 4
Fig. 4

Geometry of one period in the surface of a monolayer of dielectric spheres on a silicon substrate.

Fig. 5
Fig. 5

Normalized light-scattering phase function S 11 / ( k 2 s ) computed with the FDTD scheme for a flat silicon substrate of refractive index m = 4.90 + 3.84 i at a normal incidence of λ 0 = 308   nm . Calculations are performed using a 308   nm × 308   nm substrate period and integrations over 2001 × 2001 periods. Since normal incidence is used, this curve should be a delta function in the backscattering direction (180° of phase angle). The curve therefore shows the magnitude and character of the computational noise.

Fig. 6
Fig. 6

Normalized light-scattering phase function S 11 / ( k 2 s ) computed with the FDTD scheme for a monolayer of spheres with a radius of 510   nm and a refractive index of 1.64 on the silicon substrate of Fig. 5. Peaks with high signal-to-noise ratios emerge.

Fig. 7
Fig. 7

Normalized Muller matrix elements computed with the FDTD scheme for a monolayer of dielectric spheres with a radius of 510   nm and a refractive index of 1.64 on a silicon substrate under normal incidence: (a) S 12 ( S 21 ) / S 11 , (b) S 33 ( S 44 ) / S 11 , and (c) S 34 ( S 43 ) / S 11 . Also shown are the corresponding normalized Muller matrix elements of the noise, i.e., of computations for the flat silicon substrate, in dashed curves. These polarization quantities give more information about the surface than the total scattering.

Fig. 8
Fig. 8

Normalized transmitted light flux at a depth of λ 0 / 30 inside the silicon substrate for a period of the rough surface, as shown in Fig. 4. The unit for both x and y axes in this figure is the FDTD cubic cell size, i.e., Δ s = λ 0 / 60 . Micrometer-sized dielectric spheres might be placed on a semiconductor surface to burn nanometer-sized holes in a layer using laser pulses.

Equations (22)

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H x n + 1 / 2 ( i b , j + 1 2 , k + 1 2 ) H x n + 1 / 2 ( i a , j + 1 2 , k + 1 2 ) ,
H x n + 1 / 2 ( i , j b + 1 2 , k + 1 2 ) H x n + 1 / 2 ( i , j a + 1 2 , k + 1 2 ) ,
E x n + 1 ( i a + 1 2 , j , k ) E x n + 1 ( i b + 1 2 , j , k ) ,
E x n + 1 ( i + 1 2 , j a , k ) E x n + 1 ( i + 1 2 , j b , k ) ,
[ I s Q s U s V s ] = 1 k 2 R 2 [ S 11 S 21 S 31 S 41 S 12 S 22 S 32 S 42 S 13 S 23 S 33 S 43 S 14 S 24 S 34 S 44 ] [ I 0 Q 0 U 0 V 0 ] ,
J = n × H s ,
M = n × E s ,
E θ i k   exp ( i k R ) 4 π R ( L ϕ + η 0 N θ ) ,
E ϕ + i k   exp ( i k R ) 4 π R ( L θ η 0 N ϕ ) ,
L = s ( x M x + y M y + z M z ) exp ( i k r r ) d s ,
N = s ( x J x + y J y + z J z ) exp ( i k r r ) d s ,
L = δ s 0 ( x M x + y M y + z M z ) exp ( i k r r ) d s ,
N = δ s 0 ( x J x + y J y + z J z ) exp ( i k r r ) d s ,
δ = m = n = exp [ i k r ( m p x , m p y , 0 ) ] ,
E ( R ) = α E α ( R ) + β E β ( R ) ,
[ E α ( R ) E β ( R ) ] = exp ( i k R ) i k R [ s 2 s 4 s 3 s 1 ] ( E 0 , α E 0 , β ) ,
( E 0 , α E 0 , β ) = [ β x β y β y β x ] ( E 0 , y E 0 , x ) .
[ s 2 s 4 s 3 s 1 ] = [ F α , y F β , y F α , x F β , x ] [ β x β y β y β x ] ,
( F α , x F β , x ) = k 2 4 π ( L ϕ + η 0 N θ L θ η 0 N ϕ ) ;
( F α , y F β , y ) = k 2 4 π ( L ϕ + η 0 N θ L θ η 0 N ϕ ) .
[ I s / s Q s / s U s / s V s / s ] = S 11 / ( k 2 s ) R 2 [ 1 S 21 / S 11 S 31 / S 11 S 41 / S 11 S 12 / S 11 S 22 / S 11 S 32 / S 11 S 42 / S 11 S 13 / S 11 S 23 / S 11 S 33 / S 11 S 43 / S 11 S 14 / S 11 S 24 / S 11 S 34 / S 11 S 44 / S 11 ] × [ I 0 Q 0 U 0 V 0 ] ,
0 2 π π / 2 π [ S 11 / ( k 2 s ) ] sin   θ d θ d ϕ = Q s c a .

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