Abstract

In this study, we develop a numerical algorithm to calculate the interaction of an arbitrary electro magnetic beam with an arbitrary dielectric surface as one of the tools necessary to design and build a detector network based on surface-enhanced Raman scattering (SERS). By using the scattered-field finite-difference time-domain (FDTD) method with incident source terms in the FDTD equations, this development enables an arbitrary incident beam to be implemented onto an arbitrary dielectric surface or particle. Most importantly, in this study a scattered-field uniaxial perfectly matched layer (SF-UPML) absorbing boundary condition (ABC) is developed to truncate the computational domain of the scattered-field FDTD grid. The novel SF-UPML for the scattered-field FDTD algorithm should have a numerical accuracy similar to that of the conventional uniaxial perfectly matched layer for the source-free FDTD equations. Using the new ABC, the scattered-field FDTD method can accurately calculate electromagnetic wave scattering by an arbitrary dielectric surface or particles illuminated by an arbitrary incident beam.

© 2009 Optical Society of America

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2009

2008

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Surface inspection by monitoring spectral shifts of localized plasmon resonances,” Opt. Express 16, 12872-12879 (2008).
[CrossRef] [PubMed]

D. W. Mackowski, “Exact solution for the scattering and absorption properties of sphere clusters on a plane surface,” J. Quant. Spectrosc. Radiat. Transfer 109, 770-788 (2008).
[CrossRef]

W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, “Side-face effect of a dielectric strip on its optical properties,” IEEE Trans. Geosci. Remote Sens. 46, 2337-2344 (2008).
[CrossRef]

2007

N. J. Cassidy, “A review of practical numerical modelling methods for the advanced interpretation of ground-penetrating radar in near-surface environments,” Near Surface Geophys. 5, 5-21 (2007).

P. G. Venkata, M. M. Aslan, M. P. Menguc, and G. Videen, “Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates,” J. Heat Transfer 129, 60-70(2007).
[CrossRef]

A. Angell and C. Rappaport, “Computational modeling analysis of radar scattering by clothing covered arrays of metallic body-worn explosive devices,”Prog. Electromagn. Res. pier-76, 285-298 (2007).
[CrossRef]

W. Sun, G. Videen, B. Lin, and Y. Hu, “Modeling light scattered from and transmitted through dielectric periodic structures on a substrate,” Appl. Opt. 46, 1150-1156 (2007).
[CrossRef] [PubMed]

M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express 15, 17902-17911 (2007).
[CrossRef] [PubMed]

2005

G. Videen, M. M. Aslan, and M. P. Mengüç, “Characterization of metallic nano-particles via surface wave scattering. A. Theoretical framework and formulation,” J. Quant. Spectrosc. Radiat. Transfer 93, 195-206 (2005).
[CrossRef]

2004

2003

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, “Light scattering by Gaussian particles: A solution with finite-difference time domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083-1090 (2003).
[CrossRef]

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]

H. Lin and J. Zhu, “Characterization of nanocrystalline silicon films,” Proc. SPIE 4700, 354-356 (2002).
[CrossRef]

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]

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, R103-R137 (2001).
[CrossRef]

2000

1999

1998

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376-384 (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]

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

1997

1996

1995

G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1-7 (1995).
[CrossRef]

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 Propagat. 43, 1460-1463 (1995).
[CrossRef]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200(1994).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499 (1994).
[CrossRef]

1993

1992

Errata, J. Opt. Soc. Am. A 9, 844-845 (1992).
[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]

1991

1987

1986

S. B. Singham and G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658-2667 (1986).
[CrossRef]

1980

R. Holland, R. L. Simpson, and K. S. Kunz, “Finite-difference analysis of EMP coupling to lossy dielectric structures,” IEEE Trans. Electromagn. Compat. emc-22, 203-209 (1980).
[CrossRef]

1977

R. Holland, “Threde: a free-field EMP coupling and scattering code,” IEEE Trans. Nucl. Sci. 24, 2416-2421 (1977).
[CrossRef]

1973

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

1966

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

1954

H. Davies, “The reflection of electromagnetical waves from rough surfaces,” Proc. IEEE 101, 209-214 (1954).

1953

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566-570 (1953).
[CrossRef]

1951

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

1941

Albella, P.

Angell, A.

A. Angell and C. Rappaport, “Computational modeling analysis of radar scattering by clothing covered arrays of metallic body-worn explosive devices,”Prog. Electromagn. Res. pier-76, 285-298 (2007).
[CrossRef]

Ao, C. O.

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

Aslan, M. M.

P. G. Venkata, M. M. Aslan, M. P. Menguc, and G. Videen, “Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates,” J. Heat Transfer 129, 60-70(2007).
[CrossRef]

G. Videen, M. M. Aslan, and M. P. Mengüç, “Characterization of metallic nano-particles via surface wave scattering. A. Theoretical framework and formulation,” J. Quant. Spectrosc. Radiat. Transfer 93, 195-206 (2005).
[CrossRef]

Beckmann, P.

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

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200(1994).
[CrossRef]

Bickel, W. S.

Borghese, F.

Borghi, R.

Brock, R. S.

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]

Cassidy, N. J.

N. J. Cassidy, “A review of practical numerical modelling methods for the advanced interpretation of ground-penetrating radar in near-surface environments,” Near Surface Geophys. 5, 5-21 (2007).

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 multi-scale 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]

Chen, Z.

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. IEEE 101, 209-214 (1954).

Denti, P.

Ding, K. H.

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

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376-384 (1998).
[CrossRef]

A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, 2000).

Draine, B. T.

Durnin, J.

Eckart, C.

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566-570 (1953).
[CrossRef]

Eremin, Y.

A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, 2000).

Fano, U.

Feng, Q.

W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, “Side-face effect of a dielectric strip on its optical properties,” IEEE Trans. Geosci. Remote Sens. 46, 2337-2344 (2008).
[CrossRef]

Ferrari, R. L.

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

Flatau, P. J.

Frezza, F.

Fu, Q.

Fu, Y.

W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, “Side-face effect of a dielectric strip on its optical properties,” IEEE Trans. Geosci. Remote Sens. 46, 2337-2344 (2008).
[CrossRef]

Fucile, E.

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 multi-scale 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, 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, Microwave Scattering and Emission Models and their Applications (Artech House, 1994).

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.
[PubMed]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. 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]

González, F.

Gori, 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]

Hagness, S. C.

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

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]

Hirleman, E. D.

Hoekstra, A. G.

Holland, R.

R. Holland, R. L. Simpson, and K. S. Kunz, “Finite-difference analysis of EMP coupling to lossy dielectric structures,” IEEE Trans. Electromagn. Compat. emc-22, 203-209 (1980).
[CrossRef]

R. Holland, “Threde: a free-field EMP coupling and scattering code,” IEEE Trans. Nucl. Sci. 24, 2416-2421 (1977).
[CrossRef]

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]

Hu, Y.

W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, “Side-face effect of a dielectric strip on its optical properties,” IEEE Trans. Geosci. Remote Sens. 46, 2337-2344 (2008).
[CrossRef]

W. Sun, G. Videen, B. Lin, and Y. Hu, “Modeling light scattered from and transmitted through dielectric periodic structures on a substrate,” Appl. Opt. 46, 1150-1156 (2007).
[CrossRef] [PubMed]

Iafelice, V. J.

Jin, J. M.

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

Johnson, B. R.

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]

Kattawar, G. W.

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 Propagat. 43, 1460-1463 (1995).
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L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
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R. Holland, R. L. Simpson, and K. S. Kunz, “Finite-difference analysis of EMP coupling to lossy dielectric structures,” IEEE Trans. Electromagn. Compat. emc-22, 203-209 (1980).
[CrossRef]

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

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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 Propagat. 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 Propagat. 43, 1460-1463 (1995).
[CrossRef]

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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]

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W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, “Side-face effect of a dielectric strip on its optical properties,” IEEE Trans. Geosci. Remote Sens. 46, 2337-2344 (2008).
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H. Lin and J. Zhu, “Characterization of nanocrystalline silicon films,” Proc. SPIE 4700, 354-356 (2002).
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D. W. Mackowski, “Exact solution for the scattering and absorption properties of sphere clusters on a plane surface,” J. Quant. Spectrosc. Radiat. Transfer 109, 770-788 (2008).
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F. Mattia, “Backscattering properties of multi-scale rough surfaces,” J. Electromagn. Waves Appl. 13, 493-527 (1999).
[CrossRef]

Menguc, M. P.

P. G. Venkata, M. M. Aslan, M. P. Menguc, and G. Videen, “Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates,” J. Heat Transfer 129, 60-70(2007).
[CrossRef]

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G. Videen, M. M. Aslan, and M. P. Mengüç, “Characterization of metallic nano-particles via surface wave scattering. A. Theoretical framework and formulation,” J. Quant. Spectrosc. Radiat. Transfer 93, 195-206 (2005).
[CrossRef]

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Muinonen, K.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, “Light scattering by Gaussian particles: A solution with finite-difference time domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083-1090 (2003).
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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]

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[CrossRef]

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Nousiainen, T.

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W. Sun, N. G. Loeb, G. Videen, and Q. Fu, “Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm,” Appl. Opt. 43, 1957-1964 (2004).
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J. Heat Transfer

P. G. Venkata, M. M. Aslan, M. P. Menguc, and G. Videen, “Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates,” J. Heat Transfer 129, 60-70(2007).
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[CrossRef]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, “Light scattering by Gaussian particles: A solution with finite-difference time domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083-1090 (2003).
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A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, 2000).

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

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

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

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Figures (3)

Fig. 1
Fig. 1

Illustration of the FDTD computational domain enclosed by the SF-UPML and the beam incident on a dielectric surface. The interface between the material and free space passes through the center of the computational domain. The central axis of the incident beam passes through the center of the computational domain within the x y plane.

Fig. 2
Fig. 2

Total electric field intensity ( | E | 2 ) for an electromagnetic beam incident on a smooth surface at 45 ° . The refractive index of the material space is n 2 = 3 . The incident beam is a continuous wave described by Eqs. (5, 6) with a half-width δ of 12 Δ s , where Δ s = λ / 30 . The fields shown here are those on the x y plane passing through the center of the computational domain. The central axis of the incident beam and the polarization direction of the incident electric field are also in this x y plane. The upper panel and lower panel are from the 300th and 900th FDTD time step, respectively. The incident electric field intensity at the central axis of the beam is normalized to 1.

Fig. 3
Fig. 3

Same as in Fig. 2, but with a spherical particle with a size parameter of 2 π a λ = π and a refractive index of n particle = 3 resting on the smooth surface.

Tables (1)

Tables Icon

Table 1 Comparison of the Reflectivity From a Planar Substrate Calculated Using the FDTD Method and the Fresnel Equation

Equations (64)

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

H = H s + H i ,
E = E s + E i .
μ 0 H t = × E ,
ε E t + σ E = × H ,
μ 0 H i t = × E i ,
ε 0 E i t = × H i .
μ 0 H s t = × E s ,
ε E s t + σ E s = × H s σ E i ( ε ε 0 ) E i t .
H x s , n + 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) = H x s , n 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) + Δ t μ 0 Δ s × [ E y s , n ( i , j + 1 / 2 , k + 1 ) E y s , n ( i , j + 1 / 2 , k ) + E z s , n ( i , j , k + 1 / 2 ) E z s , n ( i , j + 1 , k + 1 / 2 ) ] ,
E x s , n + 1 ( i + 1 / 2 , j , k ) = ( 2 ε σ Δ t 2 ε + σ Δ t ) E x s , n ( i + 1 / 2 , j , k ) + ( 2 Δ t / Δ s 2 ε + σ Δ t ) [ H y s , n + 1 / 2 ( i + 1 / 2 , j , k 1 / 2 ) H y s , n + 1 / 2 ( i + 1 / 2 , j , k + 1 / 2 ) + H z s , n + 1 / 2 ( i + 1 / 2 , j + 1 / 2 , k ) H z s , n + 1 / 2 ( i + 1 / 2 , j 1 / 2 , k ) ] ( σ Δ t 2 ε + σ Δ t ) [ E x i , n + 1 ( i + 1 / 2 , j , k ) + E x i , n ( i + 1 / 2 , j , k ) ] ( 2 ε 2 ε 0 2 ε + σ Δ t ) [ E x i , n + 1 ( i + 1 / 2 , j , k ) E x i , n ( i + 1 / 2 , j , k ) ] ,
E i = A ( r ) exp ( i k · r ) ,
A ( r ) = A 0 exp [ ( ρ δ ) ] ,
E i ( r , t ) = R [ A ( r ) exp ( i ω t i k · r ) ] ,
× H ( x , y , z ) = ( i ω ε + σ ) s ¯ ¯ E ( x , y , z ) ,
× E ( x , y , z ) = i ω μ 0 s ¯ ¯ H ( x , y , z ) ,
s ¯ ¯ = [ s x 1 0 0 0 s x 0 0 0 s x ] [ s y 0 0 0 s y 1 0 0 0 s y ] [ s z 0 0 0 s z 0 0 0 s z 1 ] = [ s y s z s x 1 0 0 0 s x s z s y 1 0 0 0 s x s y s z 1 ] ,
κ x ( x ) = 1 + ( x / d ) m ( κ x , max 1 ) ,
σ x ( x ) = ( x / d ) m σ x , max ,
R ( θ ) = exp [ 2 cos θ ε 0 c 0 d σ ( x ) d x ] = exp [ 2 σ x , max d cos θ ε 0 c ( m + 1 ) ] .
σ x , max = ( m + 1 ) ε 0 c ln [ R ( 0 ) ] 2 d .
× [ H s ( x , y , z ) + H i ( x , y , z ) ] = ( i ω ε + σ ) s ¯ ¯ [ E s ( x , y , z ) + E i ( x , y , z ) ] ,
× [ E s ( x , y , z ) + E i ( x , y , z ) ] = i ω μ 0 s ¯ ¯ [ H s ( x , y , z ) + H i ( x , y , z ) ] .
× H i ( x , y , z ) = i ω ε 0 s ¯ ¯ E i ( x , y , z ) ,
× E i ( x , y , z ) = i ω μ 0 s ¯ ¯ H i ( x , y , z ) ,
× H s ( x , y , z ) = ( i ω ε + σ ) s ¯ ¯ E s ( x , y , z ) + [ i ω ( ε ε 0 ) + σ ] s ¯ ¯ E i ( x , y , z ) ,
× E s ( x , y , z ) = i ω μ 0 s ¯ ¯ H s ( x , y , z ) .
P x s ( x , y , z ) = ( s y s z s x ) E x s ( x , y , z ) ,
P y s ( x , y , z ) = ( s x s z s y ) E y s ( x , y , z ) ,
P z s ( x , y , z ) = ( s x s y s z ) E z s ( x , y , z ) ,
Q x s ( x , y , z ) = ( 1 s y ) P x s ( x , y , z ) ,
Q y s ( x , y , z ) = ( 1 s z ) P y s ( x , y , z ) ,
Q z s ( x , y , z ) = ( 1 s x ) P z s ( x , y , z ) ,
P x i ( x , y , z ) = ( s y s z s x ) E x i ( x , y , z ) ,
P y i ( x , y , z ) = ( s x s z s y ) E y i ( x , y , z ) ,
P z i ( x , y , z ) = ( s x s y s z ) E z i ( x , y , z ) ,
Q x i ( x , y , z ) = ( 1 s y ) P x i ( x , y , z ) ,
Q y i ( x , y , z ) = ( 1 s z ) P y i ( x , y , z ) ,
Q z i ( x , y , z ) = ( 1 s x ) P z i ( x , y , z ) .
[ H y s ( x , y , z ) z H z s ( x , y , z ) y H z s ( x , y , z ) x H x s ( x , y , z ) z H x s ( x , y , z ) y H y s ( x , y , z ) x ] = ( i ω ε + σ ) [ P x s ( x , y , z ) P y s ( x , y , z ) P z s ( x , y , z ) ] + [ i ω ( ε ε 0 ) + σ ] [ P x i ( x , y , z ) P y i ( x , y , z ) P z i ( x , y , z ) ] .
[ H y s ( x , y , z ) z H z s ( x , y , z ) y H z s ( x , y , z ) x H x s ( x , y , z ) z H x s ( x , y , z ) y H y s ( x , y , z ) x ] = ε [ P x s ( x , y , z ) t P y s ( x , y , z ) t P z s ( x , y , z ) t ] + σ [ P x s ( x , y , z ) P y s ( x , y , z ) P z s ( x , y , z ) ] + ( ε ε 0 ) [ P x i ( x , y , z ) t P y i ( x , y , z ) t P z i ( x , y , z ) t ] + σ [ P x i ( x , y , z ) P y i ( x , y , z ) P z i ( x , y , z ) ] .
P x s , n + 1 ( i + 1 / 2 , j , k ) = ( 2 ε σ Δ t 2 ε + σ Δ t ) P x s , n ( i + 1 / 2 , j , k ) + ( 2 Δ t / Δ s 2 ε + σ Δ t ) × [ H y s , n + 1 / 2 ( i + 1 / 2 , j , k 1 / 2 ) H y s , n + 1 / 2 ( i + 1 / 2 , j , k + 1 / 2 ) + H z s , n + 1 / 2 ( i + 1 / 2 , j + 1 / 2 , k ) H z s , n + 1 / 2 ( i + 1 / 2 , j 1 / 2 , k ) ] ( σ Δ t 2 ε + σ Δ t ) [ P x i , n + 1 ( i + 1 / 2 , j , k ) + P x i , n ( i + 1 / 2 , j , k ) ] ( 2 ε 2 ε 0 2 ε + σ Δ t ) [ P x i , n + 1 ( i + 1 / 2 , j , k ) P x i , n ( i + 1 / 2 , j , k ) ] .
Q x s , n + 1 ( i + 1 / 2 , j , k ) = ( 2 ε 0 κ y σ y Δ t 2 ε 0 κ y + σ y Δ t ) Q x s , n ( i + 1 / 2 , j , k ) + ( 2 ε 0 2 ε 0 κ y + σ y Δ t ) × [ P x s , n + 1 ( i + 1 / 2 , j , k ) P x s , n ( i + 1 / 2 , j , k ) ] .
E x s , n + 1 ( i + 1 / 2 , j , k ) = ( 2 ε 0 κ z σ z Δ t 2 ε 0 κ z + σ z Δ t ) E x s , n ( i + 1 / 2 , j , k ) + ( 1 2 ε 0 κ z + σ z Δ t ) × [ ( 2 ε 0 κ x + σ x Δ t ) Q x s , n + 1 ( i + 1 / 2 , j , k ) ( 2 ε 0 κ x σ x Δ t ) Q x s , n ( i + 1 / 2 , j , k ) ] .
Q x i ( x , y , z ) = ( 1 s y ) P x i ( x , y , z ) = s z s x E x i ( x , y , z ) ,
( κ x + σ x i ω ε 0 ) Q x i ( x , y , z ) = ( κ z + σ z i ω ε 0 ) E x i ( x , y , z ) .
( i ω ε 0 κ x + σ x ) Q x i ( x , y , z ) = ( i ω ε 0 κ z + σ z ) E x i ( x , y , z ) .
ε 0 κ x Q x i t + σ x Q x i = ε 0 κ z E x i t + σ z E x i .
Q x i , n + 1 ( i + 1 / 2 , j , k ) = ( 2 ε 0 κ x σ x Δ t 2 ε 0 κ x + σ x Δ t ) Q x i , n ( i + 1 / 2 , j , k ) + ( σ z Δ t 2 ε 0 κ x + σ x Δ t ) [ E x i , n + 1 ( i + 1 / 2 , j , k ) + E x i , n ( i + 1 / 2 , j , k ) ] + ( 2 κ z ε 0 2 ε 0 κ x + σ x Δ t ) [ E x i , n + 1 ( i + 1 / 2 , j , k ) E x i , n ( i + 1 / 2 , j , k ) ] .
P x i ( x , y , z ) = s y Q x i ( x , y , z ) = ( κ y + σ y i ω ε 0 ) Q x i ( x , y , z ) ,
ε 0 P x i t = ε 0 κ y Q x i t + σ y Q x i .
P x i , n + 1 ( i + 1 / 2 , j , k ) = P x i , n ( i + 1 / 2 , j , k ) + ( σ y Δ t 2 ε 0 ) [ Q x i , n + 1 ( i + 1 / 2 , j , k ) + Q x i , n ( i + 1 / 2 , j , k ) ] + κ y [ Q x i , n + 1 ( i + 1 / 2 , j , k ) Q x i , n ( i + 1 / 2 , j , k ) ] .
B x s ( x , y , z ) = μ 0 ( s z s x ) H x s ( x , y , z ) ,
B y s ( x , y , z ) = μ 0 ( s x s y ) H y s ( x , y , z ) ,
B z s ( x , y , z ) = μ 0 ( s y s z ) H z s ( x , y , z ) .
[ E y s ( x , y , z ) z E z s ( x , y , z ) y E z s ( x , y , z ) x E x s ( x , y , z ) z E x s ( x , y , z ) y E y s ( x , y , z ) x ] = i ω [ s y 0 0 0 s z 0 0 0 s x ] [ B x s ( x , y , z ) B y s ( x , y , z ) B z s ( x , y , z ) ] .
( i ω κ x + σ x ε 0 ) B x s ( x , y , z ) = ( i ω κ z + σ z ε 0 ) μ 0 H x s ( x , y , z ) ,
( i ω κ y + σ y ε 0 ) B y s ( x , y , z ) = ( i ω κ x + σ x ε 0 ) μ 0 H y s ( x , y , z ) ,
( i ω κ z + σ z ε 0 ) B z s ( x , y , z ) = ( i ω κ y + σ y ε 0 ) μ 0 H z s ( x , y , z ) .
[ E y s ( x , y , z , t ) z E z s ( x , y , z , t ) y E z s ( x , y , z , t ) x E x s ( x , y , z , t ) z E x s ( x , y , z , t ) y E y s ( x , y , z , t ) x ] = t [ κ y 0 0 0 κ z 0 0 0 κ x ] [ B x s ( x , y , z , t ) B y s ( x , y , z , t ) B z s ( x , y , z , t ) ] + 1 ε 0 [ σ y 0 0 0 σ z 0 0 0 σ x ] [ B x s ( x , y , z , t ) B y s ( x , y , z , t ) B z s ( x , y , z , t ) ] ,
κ x B x s ( x , y , z , t ) t + σ x ε 0 B x s ( x , y , z , t ) = μ 0 κ z H x s ( x , y , z , t ) t + μ 0 σ z ε 0 H x s ( x , y , z , t )
κ y B y s ( x , y , z , t ) t + σ y ε 0 B y s ( x , y , z , t ) = μ 0 κ x H y s ( x , y , z , t ) t + μ 0 σ x ε 0 H y s ( x , y , z , t )
κ z B z s ( x , y , z , t ) t + σ z ε 0 B z s ( x , y , z , t ) = μ 0 κ y H z s ( x , y , z , t ) t + μ 0 σ y ε 0 H z s ( x , y , z , t )
B x s , n + 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) = ( 2 ε 0 κ y σ y Δ t 2 ε 0 κ y + σ y Δ t ) B x s , n 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) + ( 2 ε 0 Δ t / Δ s 2 ε 0 κ y + σ y Δ t ) × [ E y s , n ( i , j + 1 / 2 , k + 1 ) E y s , n ( i , j + 1 / 2 , k ) + E z s , n ( i , j , k + 1 / 2 ) E z s , n ( i , j + 1 , k + 1 / 2 ) ] ,
H x s , n + 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) = ( 2 ε 0 κ z σ z Δ t 2 ε 0 κ z + σ z Δ t ) H x s , n 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) + ( 1 / μ 0 2 ε 0 κ z + σ z Δ t ) × [ ( 2 ε 0 κ x + σ x Δ t ) B x s , n + 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) ( 2 ε 0 κ x σ x Δ t ) B x s , n 1 / 2 ( i , j + 1 / 2 , k + 1 / 2 ) ] .

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