Abstract

In this work, we present a numerical method that remedies the instabilities of the conventional FDTD approach for solving Maxwell’s equations in a space-time dependent magneto-electric medium with direct application to the simulation of the recently proposed spacetime cloak. We utilize a dual grid FDTD method overlapped in the time domain to provide a stable approach for the simulation of a magneto-electric medium with time and space varying permittivity, permeability and coupling coefficient. The developed method can be applied to explore other new physical possibilities offered by spacetime cloaking, metamaterials, and transformation optics.

© 2014 Optical Society of America

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  1. J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1779 (2006).
    [CrossRef] [PubMed]
  3. M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
    [CrossRef]
  4. P. Kinsler, M. W. McCall, “Cloaks, editors, and bubbles: applications of spacetime transformation theory,” Ann. Phys. 526, 51–62 (2014).
    [CrossRef]
  5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  6. A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
    [CrossRef]
  7. A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005), 3rd edition.
  8. F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag. 56, 2150–2166 (2008).
    [CrossRef]
  9. J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
    [CrossRef]
  10. A. Akyurtlu, D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag. 52, 416–425 (2004).
    [CrossRef]
  11. A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
    [CrossRef]

2014 (1)

P. Kinsler, M. W. McCall, “Cloaks, editors, and bubbles: applications of spacetime transformation theory,” Ann. Phys. 526, 51–62 (2014).
[CrossRef]

2011 (1)

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

2010 (1)

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

2008 (1)

F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag. 56, 2150–2166 (2008).
[CrossRef]

2006 (3)

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1779 (2006).
[CrossRef] [PubMed]

2004 (1)

A. Akyurtlu, D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag. 52, 416–425 (2004).
[CrossRef]

1975 (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

1966 (1)

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

Akyurtlu, A.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

A. Akyurtlu, D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag. 52, 416–425 (2004).
[CrossRef]

Boardman, A.

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Bray, M. G.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Brio, M.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Brodwin, M. E.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

Favaro, A.

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Hagness, S.

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005), 3rd edition.

Hoyer, W.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Kern, D.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Kinsler, P.

P. Kinsler, M. W. McCall, “Cloaks, editors, and bubbles: applications of spacetime transformation theory,” Ann. Phys. 526, 51–62 (2014).
[CrossRef]

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Koch, S. W.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1779 (2006).
[CrossRef] [PubMed]

Liu, J.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

McCall, M. W.

P. Kinsler, M. W. McCall, “Cloaks, editors, and bubbles: applications of spacetime transformation theory,” Ann. Phys. 526, 51–62 (2014).
[CrossRef]

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Moloney, J. V.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Semichaevsky, A.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005), 3rd edition.

Teixeira, F. L.

F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag. 56, 2150–2166 (2008).
[CrossRef]

Werner, D. H.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

A. Akyurtlu, D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag. 52, 416–425 (2004).
[CrossRef]

Yee, K. S.

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

Zakharian, A.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Zeng, Y.

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

Ann. Phys. (1)

P. Kinsler, M. W. McCall, “Cloaks, editors, and bubbles: applications of spacetime transformation theory,” Ann. Phys. 526, 51–62 (2014).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

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

F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag. 56, 2150–2166 (2008).
[CrossRef]

A. Akyurtlu, D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag. 52, 416–425 (2004).
[CrossRef]

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

J. Comput. Phys. (1)

J. Liu, M. Brio, Y. Zeng, A. Zakharian, W. Hoyer, S. W. Koch, J. V. Moloney, “Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model,” J. Comput. Phys. 229, 5921–5932 (2010).
[CrossRef]

J. Opt. (1)

M. W. McCall, A. Favaro, P. Kinsler, A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Science (2)

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1779 (2006).
[CrossRef] [PubMed]

Other (1)

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005), 3rd edition.

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Figures (5)

Fig. 1
Fig. 1

(a) A spacetime cloak in the (t, x) domain. (b) Free space after the transformation to the (τ, ξ) domain.

Fig. 2
Fig. 2

Computation of Ez using (a) the time extrapolation under a conventional FDTD method and (b) the overlapping Yee algorithm without extrapolation.

Fig. 3
Fig. 3

Contour plots of the electric field intensity for spacetime cloak simulations. Left column: the overlapping Yee FDTD results with grid sizes (a) N = 1600, (c) N = 2000, and (e) N = 2400, respectively. Right column: the corresponding results using a conventional time extrapolation based FDTD method.

Fig. 4
Fig. 4

Evolution of the Electric field as the wave propagates through the spacetime cloak.

Fig. 5
Fig. 5

Distribution of the eigenvalues in the complex plane for (a) the overlapping Yee FDTD method and (b) the conventional FDTD method.

Equations (29)

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

˜ M ˜ = ˜ ( 0 D x D y D z D x 0 H z H y D y H z 0 H x D z H y H x 0 ) = 0 ,
˜ M ˜ = ˜ ( 0 D ξ D η D ζ D ξ 0 H ζ H η D η H ζ 0 H ξ D ζ H η H ξ 0 ) = 0 ,
Λ ˜ = ( τ , ξ , η , ζ ) ( t , x , y , z ) = ( τ t τ x 0 0 ξ t ξ x 0 0 0 0 1 0 0 0 0 1 ) = ( Λ 0 0 I 2 ) .
τ = t ,
ξ = σ ( x x S ) σ δ ( t t S ) / ( t 0 t S ) + x S ,
t S = { t p , if ( t , x ) lies in region I or II , t q , if ( t , x ) lies in region III or IV ,
x S = { x 0 + ( t t 0 ) c + σ , if ( t , x ) lies in region I or IV , x 0 + ( t t 0 ) c σ , if ( t , x ) lies in region II or III .
E z = α D z + β B y ,
H y = β D z + γ B y ,
{ α = 1 / a 12 , β = a 22 / a 12 , γ = ( a 12 a 21 a 11 a 22 ) / a 12 ,
( a 11 a 12 a 21 a 22 ) = Λ 1 ( 0 ε 1 / μ 0 ) Λ .
D z t = H y x ,
B y t = E z x ,
D i n + 1 2 = D i n 1 2 + Δ t Δ x ( H i + 1 2 n H i 1 2 n ) ,
B i + 1 2 n + 1 = B i + 1 2 n + Δ t Δ x ( E i + 1 n + 1 2 E i n + 1 2 ) ,
E i n + 1 2 = α i n + 1 2 D i n + 1 2 + β i n + 1 2 B i n + 1 2 ,
H i + 1 2 n + 1 = β i + 1 2 n + 1 D i + 1 2 n + 1 + γ i + 1 2 n + 1 B i + 1 2 n + 1 .
B i n + 1 2 3 B i n B i n 1 2
3 2 ( B i + 1 2 n + B i 1 2 n ) 1 2 ( B i + 1 2 n 1 + B i 1 2 n 1 ) 2
3 B i + 1 2 n + 3 B i 1 2 n B i + 1 2 n 1 B i 1 2 n 1 4 ,
D i + 1 2 n + 1 3 D i + 1 n + 1 2 + 3 D i n + 1 2 D i + 1 n 1 2 D i n 1 2 4 .
D i n + 1 2 = D i n 1 2 + Δ t Δ x [ H i + 1 2 n H i 1 2 n ] ,
B i n + 1 2 = B i n 1 2 + Δ t Δ x [ E i + 1 2 n E i 1 2 n ] .
E i n + 1 2 = α i n + 1 2 D i n + 1 2 + β i n + 1 2 B i n + 1 2 ,
H i n + 1 2 = β i n + 1 2 D i n + 1 2 + γ i n + 1 2 B i n + 1 2 .
B i + 1 2 n + 1 = B i + 1 2 n + Δ t Δ x [ E i + 1 n + 1 2 E i n + 1 2 ] ,
D i + 1 2 n + 1 = D i + 1 2 n + Δ t Δ x [ H i + 1 n + 1 2 H i n + 1 2 ] .
E i + 1 2 n + 1 = α i + 1 2 n + 1 D i + 1 2 n + 1 + β i + 1 2 n + 1 B i + 1 2 n + 1 ,
H i + 1 2 n + 1 = β i + 1 2 n + 1 D i + 1 2 n + 1 + γ i + 1 2 n + 1 B i + 1 2 n + 1 .

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