Abstract

Based on the Padé approximation and multistep method, we propose an implicit high-order unconditionally stable complex envelope algorithm to solve the time-dependent Maxwell’s equations. Unconditional numerical stability can be achieved simultaneously with a high-order accuracy in time. As we adopt the complex envelope Maxwell’s equations, numerical dispersion and dissipation are very small even at comparatively large time steps. To verify the capability of our algorithm, we compare the results of the proposed method with the exact solutions.

© 2008 Optical Society of America

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References

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  1. K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).
    [CrossRef]
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics--The Finite-Difference Time-Domain Method (Artech House, 2005).
  3. J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
    [CrossRef]
  4. J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
    [CrossRef]
  5. H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
    [CrossRef]
  6. G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge U. Press, 1996).
    [CrossRef]
  7. G. Ronald Hadley, Opt. Lett. 17, 1743 (1992).
    [CrossRef]
  8. S. D. Conte and C. E. Boor, Elementary Numerical Analysis (McGraw-Hill, 1972).

2003

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

2002

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
[CrossRef]

2001

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
[CrossRef]

1992

1966

K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).
[CrossRef]

Baker, G. A.

G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge U. Press, 1996).
[CrossRef]

Boor, C. E.

S. D. Conte and C. E. Boor, Elementary Numerical Analysis (McGraw-Hill, 1972).

Conte, S. D.

S. D. Conte and C. E. Boor, Elementary Numerical Analysis (McGraw-Hill, 1972).

De Raedt, H.

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
[CrossRef]

Figge, M. T.

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
[CrossRef]

Graves-Morris, P.

G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge U. Press, 1996).
[CrossRef]

Hagness, S. C.

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

Kole, J. S.

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
[CrossRef]

Michielsen, K.

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

Ronald Hadley, G.

Taflove, A.

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

Yee, K. S.

K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).
[CrossRef]

Opt. Lett.

Phys. Rev. E

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 64, 066705 (2001).
[CrossRef]

J. S. Kole, M. T. Figge, and H. De Raedt, Phys. Rev. E 65, 066705 (2002).
[CrossRef]

H. De Raedt, K. Michielsen, J. S. Kole, and M. T. Figge, Phys. Rev. E 67, 056706 (2003).
[CrossRef]

Other

G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge U. Press, 1996).
[CrossRef]

S. D. Conte and C. E. Boor, Elementary Numerical Analysis (McGraw-Hill, 1972).

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

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Figures (3)

Fig. 1
Fig. 1

Exact and numerical solutions of electric amplitude when the wave propagates 200 time units in vacuum, where the mesh size and time step are Δ x = 0.0025 and Δ t = 8 , respectively.

Fig. 2
Fig. 2

Numerical dispersion and dissipation error in different orders in time, where mesh size and pulse width are Δ x = 0.0025 and t p = 25 , respectively. (a) Kole et al.’s method [3], (b) our scheme.

Fig. 3
Fig. 3

Transmission curves of the quarter-wave reflector obtained by TMM and our scheme, where the mesh size and time step for our scheme are Δ x = 0.01 and Δ t = 8 , respectively. Inset, quarter-wave reflector.

Tables (1)

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Table 1 Time Step Refinement Analysis for a 1D System

Equations (16)

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H t + j ω H = 1 μ × E ,
E t + j ω E = 1 ϵ × H ,
t ( H ( t ) E ( t ) ) = ( j ω 1 μ × 1 ϵ × j ω ) ( H ( t ) E ( t ) ) M ( H ( t ) E ( t ) ) ,
t Ψ ( t ) = M Ψ ( t ) .
Ψ ( t + τ ) = exp ( τ M ) Ψ ( t ) .
exp 1 1 ( τ M ) = I + 1 2 τ M I 1 2 τ M ,
exp 2 2 ( τ M ) = [ I 3 + 3 i 12 τ M ] [ I 3 3 i 12 τ M ] [ I + 3 + 3 i 12 τ M ] [ I + 3 3 i 12 τ M ] ,
exp n n ( τ M ) = ( I a n τ M ) ( I a 2 τ M ) ( I a 1 τ M ) ( I + a n τ M ) ( I + a 2 τ M ) ( I + a 1 τ M ) ,
Ψ ( t + τ ) = ( I a n τ M ) ( I a 2 τ M ) ( I a 1 τ M ) ( I + a n τ M ) ( I + a 2 τ M ) ( I + a 1 τ M ) Ψ ( t ) .
Ψ ( t + i n τ ) = ( I a i τ M ) ( I + a i τ M ) Ψ ( t + i 1 n τ ) .
H y ( i , t ) t = 1 μ i [ 1 Δ x ( E z ( i + 1 , t ) E z ( i 1 , t ) ) ] j ω H y ( i , t ) ,
E z ( j , t ) t = 1 ϵ j [ 1 Δ x ( H y ( j + 1 , t ) H y ( j 1 , t ) ) ] j ω E z ( j , t ) ,
Ψ ( i , t ) = { H y ( i , t ) i odd E z ( i , t ) i even } .
M = i = 1 n 1 [ α i e i e i + 1 T α i + 1 e i + 1 e i T + β e i e i T ] ,
α i = ( 1 Δ x ) ( 1 η i ) , β = j ω , and η i = { μ i i odd ϵ i i even } .
Ψ ( t + τ ) = exp ( τ M x + τ M y ) Ψ ( t ) .

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