Abstract

This paper describes a novel Boundary Source Method (BSM) applied to the vector calculation of electromagnetic fields from a surface defined by the interface between homogenous, isotropic media. In this way, the reflected and transmitted fields are represented as an expansion of the electric fields generated by a basis of orthogonal electric and magnetic dipole sources that are tangential to, and evenly distributed over the surface of interest. The dipole moments required to generate these fields are then calculated according to the extinction theorem of Ewald and Oseen applied at control points situated at either side of the boundary. It is shown that the sources are essentially vector-equivalent Huygens’ wavelets applied at discrete points at the boundary and special attention is given to their placement and the corresponding placement of control points according to the Nyquist sampling criteria. The central result of this paper is that the extinction theorem should be applied at control points situated at a distance d = 3s (where s is the separation of the sources) and consequently we refer to the method as 3sBSM. The method is applied to reflection at a plane dielectric surface and a spherical dielectric sphere and good agreement is demonstrated in comparison with the Fresnel equations and Mie series expansion respectively (even at resonance). We conclude that 3sBSM provides an accurate solution to electromagnetic scattering from a bandlimited surface and efficiently avoids the singular surface integrals and special basis functions proposed by others.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (1)

2014 (2)

2013 (1)

2012 (1)

2011 (1)

A. Bouzidi and T. Aguili, “Numerical optimization of the method of auxiliary sources by using level set technique,” Prog. Electromagn. Res. B 33, 203–219 (2011).
[Crossref]

2010 (1)

J. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46(1), 69–78 (2010).
[Crossref]

2008 (1)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

2002 (2)

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19(1), 101–110 (2002).
[Crossref]

D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44(3), 48–64 (2002).
[Crossref]

2001 (1)

1998 (1)

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation,” Waves Random Media 8(1), 53–66 (1998).
[Crossref]

1989 (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applicat. 3(1), 1–15 (1989).
[Crossref]

1981 (1)

1979 (1)

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366(1725), 155–171 (1979).
[Crossref]

1972 (1)

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6(3), 217–220 (1972).
[Crossref]

1948 (1)

W. Franz, “Zur Formulierung des Huygensschen Prinzips,” Z. Naturforsch. 3a, 500–506 (1948).

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

1923 (1)

F. Kottler, “Electromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Berlin, Ger.) 376(15), 457–508 (1923).
[Crossref]

Aguili, T.

A. Bouzidi and T. Aguili, “Numerical optimization of the method of auxiliary sources by using level set technique,” Prog. Electromagn. Res. B 33, 203–219 (2011).
[Crossref]

Anastassiu, H. T.

D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44(3), 48–64 (2002).
[Crossref]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle (Oxford University, 1939).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1993).

Bouzidi, A.

A. Bouzidi and T. Aguili, “Numerical optimization of the method of auxiliary sources by using level set technique,” Prog. Electromagn. Res. B 33, 203–219 (2011).
[Crossref]

Chamot, S.

Chew, W. C.

W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves (Morgan & Claypool, 2009).

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle (Oxford University, 1939).

Coupland, J.

Coupland, J. M.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

Cullen, A. L.

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366(1725), 155–171 (1979).
[Crossref]

Delacrétaz, Y.

Depeursinge, C.

Erni, D.

Ettemeyer, A.

Franz, W.

W. Franz, “Zur Formulierung des Huygensschen Prinzips,” Z. Naturforsch. 3a, 500–506 (1948).

Gao, F.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

Gaylord, T. K.

Gibson, W. C.

W. C. Gibson, The method of moments in electromagnetics (Taylor & Francis Group, 2008).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics, (Roberts, 2005).

Hafner, C.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Harrington, R. F.

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applicat. 3(1), 1–15 (1989).
[Crossref]

R. F. Harrington, Field computation by moment methods (Wiley, 1993).

Hu, B.

W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves (Morgan & Claypool, 2009).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1993).

Humphries, S.

S. Humphries, Field Solutions on Computers (CRC Press, 1997).

Kaklamani, D. I.

D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44(3), 48–64 (2002).
[Crossref]

Kauranen, M.

J. Mäkitalo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89(16), 165429 (2014).
[Crossref]

Kottler, F.

F. Kottler, “Electromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Berlin, Ger.) 376(15), 457–508 (1923).
[Crossref]

Leach, R.

Leach, R. K.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

Liu, M.

M. Liu, N. Senin, and R. Leach, “Defect detection for structured surfaces via light scattering and machine learning,” presented at the 14th International Symposium on Measurement Technology and Intelligent Instruments (ISMTII), Niigata, Japan, 1-4 September 2019.

Lobera, J.

J. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46(1), 69–78 (2010).
[Crossref]

Macdonald, J.

P. Mouroulis and J. Macdonald, Geometrical optics and optical design (Oxford University, 1997).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE Publication, 1998).

Mäkitalo, J.

J. Mäkitalo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89(16), 165429 (2014).
[Crossref]

Mandal, R.

Mansfield, D.

Marathay, A. S.

McCalmont, J. F.

Moharam, M. G.

Moreno, E.

Mouroulis, P.

P. Mouroulis and J. Macdonald, Geometrical optics and optical design (Oxford University, 1997).

Nikolaev, N.

Orfanidis, S. J.

S. J. Orfanidis, Electromagnetic waves and antennas, https://www.ece.rutgers.edu/∼orfanidi/ewa/

Palodhi, K.

Pattanayak, D. N.

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6(3), 217–220 (1972).
[Crossref]

Petzing, J.

N. Nikolaev, J. Petzing, and J. Coupland, “Focus variation microscope: linear theory and surface tilt sensitivity,” Appl. Opt. 55(13), 3555–3565 (2016).
[Crossref]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

Senin, N.

M. Liu, N. Senin, and R. Leach, “Defect detection for structured surfaces via light scattering and machine learning,” presented at the 14th International Symposium on Measurement Technology and Intelligent Instruments (ISMTII), Niigata, Japan, 1-4 September 2019.

Sertel, K.

J. L. Volakis and K. Sertel, Integral Equation Methods for Electromagnetics (SciTech Publishing, 2012).

Seydoux, O.

Sheppard, C. J. R.

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation,” Waves Random Media 8(1), 53–66 (1998).
[Crossref]

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Suuriniemi, S.

J. Mäkitalo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89(16), 165429 (2014).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tong, M. S.

W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves (Morgan & Claypool, 2009).

Vahldieck, R.

Van Bladel, J. G.

J. G. Van Bladel, Singular electromagnetic fields and sources (Wiley, 1996).

Volakis, J. L.

J. L. Volakis and K. Sertel, Integral Equation Methods for Electromagnetics (SciTech Publishing, 2012).

Wolf, E.

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6(3), 217–220 (1972).
[Crossref]

Yu, P. K.

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366(1725), 155–171 (1979).
[Crossref]

Ann. Phys. (Berlin, Ger.) (1)

F. Kottler, “Electromagnetische Theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Berlin, Ger.) 376(15), 457–508 (1923).
[Crossref]

Appl. Opt. (3)

IEEE Antennas Propag. Mag. (1)

D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44(3), 48–64 (2002).
[Crossref]

J. Electromagn. Waves Applicat. (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applicat. 3(1), 1–15 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008).
[Crossref]

Opt. Commun. (1)

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6(3), 217–220 (1972).
[Crossref]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Phys. Rev. B (1)

J. Mäkitalo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89(16), 165429 (2014).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366(1725), 155–171 (1979).
[Crossref]

Prog. Electromagn. Res. B (1)

A. Bouzidi and T. Aguili, “Numerical optimization of the method of auxiliary sources by using level set technique,” Prog. Electromagn. Res. B 33, 203–219 (2011).
[Crossref]

Strain (1)

J. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46(1), 69–78 (2010).
[Crossref]

Waves Random Media (1)

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation,” Waves Random Media 8(1), 53–66 (1998).
[Crossref]

Z. Naturforsch. (1)

W. Franz, “Zur Formulierung des Huygensschen Prinzips,” Z. Naturforsch. 3a, 500–506 (1948).

Other (14)

W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic and elastic waves (Morgan & Claypool, 2009).

R. F. Harrington, Field computation by moment methods (Wiley, 1993).

M. Liu, N. Senin, and R. Leach, “Defect detection for structured surfaces via light scattering and machine learning,” presented at the 14th International Symposium on Measurement Technology and Intelligent Instruments (ISMTII), Niigata, Japan, 1-4 September 2019.

S. Humphries, Field Solutions on Computers (CRC Press, 1997).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

J. L. Volakis and K. Sertel, Integral Equation Methods for Electromagnetics (SciTech Publishing, 2012).

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle (Oxford University, 1939).

J. W. Goodman, Introduction to Fourier optics, (Roberts, 2005).

J. G. Van Bladel, Singular electromagnetic fields and sources (Wiley, 1996).

S. J. Orfanidis, Electromagnetic waves and antennas, https://www.ece.rutgers.edu/∼orfanidi/ewa/

W. C. Gibson, The method of moments in electromagnetics (Taylor & Francis Group, 2008).

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1993).

P. Mouroulis and J. Macdonald, Geometrical optics and optical design (Oxford University, 1997).

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE Publication, 1998).

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Figures (7)

Fig. 1.
Fig. 1. Scattering of Electromagnetic field, ${{\textbf E}_r}({\textbf r} )$, from a closed homogenous medium.
Fig. 2.
Fig. 2. Huygens’ wavelet generated by electric and magnetic dipoles ${\textbf p} = ({0,{p_0},0} )$ and ${\textbf m} = \left( {0,0,\sqrt {{\mu_A}/{\varepsilon_A}} \; {p_0}} \right).$ Vectors in the direction of the electric field (red), magnetic field (blue), Poynting vector (black) and the absolute value of the Poynting vector [A.U.] in xy and xz planes.
Fig. 3.
Fig. 3. a) Source and control point positions (general surface) b) Regularly spaced vector Huygens’ sources and control points (planar surface).
Fig. 4.
Fig. 4. Spectrum of the Ex component of vector Huygens’ wavelet (a) and (b) its profile along kx and ky axes at distance d = s = λ /100; (c) and (d) similar plots at distance d = 3s =  /100. [Note ${k_B} = 1/{\lambda _B}$]
Fig. 5.
Fig. 5. Scattering from plane interface Lx = 20${\lambda _{B}}$, Ly = 20${\lambda _{B}}$, ${n_B} = 1$, ${n_A} = 1.5$, discretization s = ${\lambda _{B}}$/5.
Fig. 6.
Fig. 6. Scattering from a sphere with radius r=λB, ${n_B} = 1$, ${n_A} = 1.5$, at discretization s = λB/10; (a) extinction of the sum of the scattered field and incident field inside of the sphere and the extinction of the transmitted field outside the sphere, (b) comparison of the scattered fields with Mie series. Fields are plotted in xz plane, outside the sphere at radius a = 1.4λB as a function of the angle θ between radius vector of the point and the x-axis (such that in forward scatter $ \theta = \pi /2$).
Fig. 7.
Fig. 7. Scattering cross section for a dielectric sphere with refractive index nA = 2, nB = 1, radius r = 1 [A.U.] as a function of the incident wavelength λB. At the resonance wavelengths denoted with A,B,C,D comparison between scattered Ex components calculated with Mie series and 3sBSM is presented.

Equations (21)

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

E A ( r ) = × S [ ( n ^ × E A ) G A ( r , r ) ] d S ( r ) 1 i ω ε A × × S [ ( n ^ × H A ) G A ( r , r ) ] d S ( r )
E B ( r ) = E r ( r ) + × S [ ( n ^ × E B ) G B ( r , r ) ] d S ( r ) + 1 i ω ε B × × S [ ( n ^ × H B ) G B ( r , r ) ] d S ( r )
n ^ × E A = n ^ × E B n ^ × E
n ^ × H A = n ^ × H B n ^ × H .
E A ( r ) = i ω i = 1 N × [ m i G A ( r i r ) ] 1 ε A i = 1 N × × [ p i G A ( r i r ) ] ,
E B ( r ) = E r ( r ) i ω i = 1 N × [ m i G B ( r i r ) ] + 1 ε B i = 1 N × × [ p i G B ( r i r ) ] ,
i ω i = 1 N × [ m i G A ( r i r ) ] 1 ε A i = 1 N × × [ p i G A ( r i r ) ] = 0 ,
E r ( r ) i ω i = 1 N × [ m i G B ( r i r ) ] + 1 ε B i = 1 N × × [ p i G B ( r i r ) ] = 0 ,
m o = μ A ε A p o .
E H ( r ) = p o [ i ω μ A ε A × [ k G A ( r ) ] 1 ε A × × [ j G A ( r ) ] ]
c i A = s i 3 s n ^ i c i B = s i + 3 s n ^ i
C = W S
S = ( [ p 1 u , p 2 u , p i u ] , [ p 1 v , p 2 v , p i v ] , [ m 1 u , m 2 u , m i u ] , [ m 1 v , m 2 v , m i v ] ) T
C = ( [ A 1 u , A 2 u , A i u ] , [ A 1 v , A 2 v , A i v ] , [ B 1 u , B 2 u , B i u ] , [ B 1 v , B 2 v , B i v ] ) T ,
W = ( pA i j u u pA i j u v pA i j v u pA i j v v mA i j u u mA i j u v mA i j v u mA i j v v pB i j u u pB i j u v pB i j v u pB i j v v mB i j u u mB i j u v mB i j v u mB i j v v ) .
p A i j u u = 1 ε B [ × × [ u ^ j G B ( c i A s j ) ] ] . u ^ i , p A i j u v = 1 ε B [ × × [ v ^ j G B ( c i A s j ) ] ] . u ^ i , m A i j u u = i ω [ × [ u ^ j G B ( c i A s j ) ] ] . u ^ i , m A i j u v = i ω [ × [ v ^ j G B ( c i A s j ) ] ] . u ^ i ,
p B i j v u = 1 ε A [ × × [ u ^ j G A ( c i B s j ) ] ] . v ^ i , p B i j v v = 1 ε A [ × × [ v ^ j G A ( c i B s j ) ] ] . v ^ i , m B i j v u = i ω [ × [ u ^ j G A ( c i B s j ) ] ] . v ^ i , m B i j v v = i ω [ × [ v ^ j G A ( c i B s j ) ] ] . v ^ i .
S = ( [ p 1 u , p 2 u , p i u ] , [ p 1 v , p 2 v , p i v ] ) T ,
C = ( [ A 1 u , A 2 u , A i u ] , [ A 1 v , A 2 v , A i v ] ) T ,
W = ( pA i j u u pA i j u v pA i j v u pA i j v v ) .
C s c a t = 2 π k 2 n = 1 ( 2 n + 1 ) ( | a n | 2 + | b n | 2 )

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