Abstract

The scattering of electromagnetic waves from rough surfaces has been actively studied for more than a century now because of its involvement in vast application areas. In the past two decades, great advances have been made by incorporating multiple scattering effects into analytical approaches. However, no model can yet be applied to surfaces with arbitrary roughness. It is also very difficult to study the cross-polarization, shadowing, or multiple scattering effects. In order to study more fundamentally the interaction of polarized light with more general rough surfaces of general media, we have developed a rigorous numerical simulator to calculate the resulting speckle fields. The full Maxwell equations were solved using surface integral equations combined with a boundary element method. The rough surface was discretized by higher order quadrilateral edge elements. The effective tangential electric and magnetic fields in each element in terms of 10 edges were first solved. The scattered electric and magnetic fields everywhere in space were then calculated correspondingly. One of the great advantages of such a simulator is that both the near and far fields can be calculated directly. Preliminary results of different kinds of metallic structures are presented, by which the advantages of the method are demonstrated.

© 2014 Optical Society of America

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References

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    [CrossRef]
  2. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1963).
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    [CrossRef]
  7. T. M. Elfouhaily and C.-A. Gu’erin, Wave Random Media 14, R1 (2004).
  8. V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).
  9. J. A. DeSanto, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 211–235.
  10. T. A. Leskova and A. A. Maradudin, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 371–408.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).
  23. F. Shen and A. Wang, Appl. Opt. 45, 1102 (2006).
    [CrossRef]
  24. J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
    [CrossRef]
  25. http://www.fastmultipole.org/ .

2013

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

2010

I. Simonsen, Eur. Phys. J. Special Top. 181, 1 (2010).

2009

2007

J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
[CrossRef]

2006

2004

T. M. Elfouhaily and C.-A. Gu’erin, Wave Random Media 14, R1 (2004).

V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).

2001

K. F. Warnick and W. C. Chew, Wave Random Media 11, R1 (2001).

2000

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

1998

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

B. M. Kolundzija, IEEE Trans. Antennas Propag. 46, 1009 (1998).

1997

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

1992

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

1982

S. M. Rao, D. Wilton, and A. M. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

1913

L. I. Mandel’shtam, Ann. Phys. 346, 609 (1913).
[CrossRef]

Beckmann, P.

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

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Buchau, A.

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, Wave Random Media 11, R1 (2001).

DeSanto, J. A.

J. A. DeSanto, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 211–235.

Dhadwal, G.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Gu’erin, Wave Random Media 14, R1 (2004).

Ennos, A. E.

A. E. Ennos, Speckle interferometry, in Laser Speckle, and Related Phenomena, A. A. Maradudin, ed. (Springer-Verlag, 1984), pp. 203–253.

Glisson, A. M.

S. M. Rao, D. Wilton, and A. M. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts and Company, 2007).

Gu’erin, C.-A.

T. M. Elfouhaily and C.-A. Gu’erin, Wave Random Media 14, R1 (2004).

Haas, M.

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

Harvey, J. E.

J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
[CrossRef]

Hoschek, R.

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

Huber, C. J.

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Kalia, S.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Kern, A. M.

Kolundzija, B. M.

B. M. Kolundzija, IEEE Trans. Antennas Propag. 46, 1009 (1998).

Krywonos, A.

J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
[CrossRef]

Lee, T. K.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Leskova, T. A.

T. A. Leskova and A. A. Maradudin, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 371–408.

Lifante, G.

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).

Lui, H.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Mandel’shtam, L. I.

L. I. Mandel’shtam, Ann. Phys. 346, 609 (1913).
[CrossRef]

Maradudin, A. A.

V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).

T. A. Leskova and A. A. Maradudin, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 371–408.

Martin, O. J. F.

McLean, D. I.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Méndez, E. R.

V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).

Onate, E.

E. Onate, Structural Analysis with the Finite Element Method, Linear Statics (Springer-Verlag, 2009), Vol. 1.

Rao, S. M.

S. M. Rao, D. Wilton, and A. M. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Richter, K. R.

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

Rieger, W.

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

Rucker, W. M.

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

Schlemmer, E.

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

Shchegrov, V.

V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).

Shen, F.

Simonsen, I.

I. Simonsen, Eur. Phys. J. Special Top. 181, 1 (2010).

Spizzichino, A.

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

Steffan, J.

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

Tchvialeva, L.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
[CrossRef]

Wang, A.

Warnick, K. F.

K. F. Warnick and W. C. Chew, Wave Random Media 11, R1 (2001).

Wilton, D.

S. M. Rao, D. Wilton, and A. M. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Zeng, H.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

Ann. Phys.

L. I. Mandel’shtam, Ann. Phys. 346, 609 (1913).
[CrossRef]

Appl. Opt.

Eur. Phys. J. Special Top.

I. Simonsen, Eur. Phys. J. Special Top. 181, 1 (2010).

IEEE Trans. Antennas Propag.

S. M. Rao, D. Wilton, and A. M. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

B. M. Kolundzija, IEEE Trans. Antennas Propag. 46, 1009 (1998).

IEEE Trans. Magn.

C. J. Huber, W. Rieger, M. Haas, and W. M. Rucker, IEEE Trans. Magn. 34, 2441 (1998).
[CrossRef]

C. J. Huber, A. Buchau, W. Rieger, and W. M. Rucker, IEEE Trans. Magn. 36, 844 (2000).
[CrossRef]

C. J. Huber, W. M. Rucker, R. Hoschek, and K. R. Richter, IEEE Trans. Magn. 33, 1386 (1997).
[CrossRef]

E. Schlemmer, J. Steffan, W. M. Rucker, and K. R. Richter, IEEE Trans. Magn. 28, 1755 (1992).
[CrossRef]

J. Biomed. Opt.

L. Tchvialeva, G. Dhadwal, H. Lui, S. Kalia, H. Zeng, D. I. McLean, and T. K. Lee, J. Biomed. Opt. 18, 061211 (2013).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

J. E. Harvey, A. Krywonos, and C. L. Vernold, Opt. Eng. 46, 078002 (2007).
[CrossRef]

Prog. Opt.

V. Shchegrov, A. A. Maradudin, and E. R. Méndez, Prog. Opt. 46, 117 (2004).

Wave Random Media

T. M. Elfouhaily and C.-A. Gu’erin, Wave Random Media 14, R1 (2004).

K. F. Warnick and W. C. Chew, Wave Random Media 11, R1 (2001).

Other

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

E. Onate, Structural Analysis with the Finite Element Method, Linear Statics (Springer-Verlag, 2009), Vol. 1.

A. E. Ennos, Speckle interferometry, in Laser Speckle, and Related Phenomena, A. A. Maradudin, ed. (Springer-Verlag, 1984), pp. 203–253.

J. A. DeSanto, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 211–235.

T. A. Leskova and A. A. Maradudin, in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer-Verlag, 2007), pp. 371–408.

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

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts and Company, 2007).

A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer-Verlag, 2007).

http://www.fastmultipole.org/ .

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).

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Figures (5)

Fig. 1.
Fig. 1.

Unit sphere discretized with 168 quadrilateral elements, each having eight nodes and 10 vector edges.

Fig. 2.
Fig. 2.

(a) Calculated Ex-field distribution in the xy-plane at z=0 of a silver nanosphere using the SIE method developed in this work. The sphere is 250 nm in diameter and its dielectric constant is 18.0+i0.5124 for silver at λ=600nm, the wavelength of the incident light. The sphere is illuminated by a linearly polarized plane wave with an E-field along the x-direction propagating along the z-direction. (b) Comparison of the p-polarized field intensities at x=0 from the SIE method using 1008 (red dashed line) and 192 (green solid line) independent edges and the Mie-theory (black solid line) [20].

Fig. 3.
Fig. 3.

(a) Numerically generated rough surface with a Gaussian distribution function. The correlation length in the x-direction is lx=1μm, in the y-direction is ly=2μm, and the RMS=0.5μm. The size of the surface under simulation is 100μm×100μm. (b) The meshed surface has 256 quadrilateral elements in total. The surface was assumed to be penetrable metal (Ag) with a dielectric permittivity of 18.0+i0.5124 at λ=600nm, which is a plane wave with normal incidence with P-polarized light.

Fig. 4.
Fig. 4.

(a) Speckle field of Ex component at z=500μm above the surface as shown in Fig. 3(a). (b) Averaged angular differential reflectance spectral from the surface (S1) shown in (a) with RMS=0.5μm at normal incidence and from a surface with RMS=0.1μm (S2) at an oblique incident angle of θi=30°. The red curve represents a Beckmann–Kirchhoff model with a correlation length of 2.0 μm and RMS=0.59μm.

Fig. 5.
Fig. 5.

(a) Surface profile of a PTB surface standard sample which was taken as Ag for simulation at λ=600nm. The unit of the color bar is micrometers. (b) Profiles along the x-direction and the y-direction. (c) Speckle field distribution of the Ex-component at 100 μm above the rough surface under an illumination of p-polarized light at normal incidence. (d) Speckle distribution of the cross-polarized light with Ey component. In the calculation, the same parameters for the surface and incident light were used.

Equations (4)

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

c(rf)E(rf)=EincΓ(n×Et)×G(rf,rs)dΓΓ[n·×HtjωεG(rf,rs)jωμ(n×Ht)G(rf,rs)]dΓ.
c(rf)H(rf)=HincΓ(n×Ht)×G(rf,rs)dΓΓ[n·×EtjωμG(rf,rs)+jωε(n×Et)G(rf,rs)]dΓ.
Et=i=110Ni(ξ,η)SEi,
Ht=i=110Ni(ξ,η)SHi.

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