Abstract

A higher-order finite-difference time-domain (HO-FDTD) numerical method is proposed for the time-domain analysis of planar optical waveguide devices. The anisotropic perfectly matched layer (APML) absorbing boundary condition for the HO-FDTD scheme is implemented and the numerical dispersion of this scheme is studied. The numerical simulations for the parallel-slab directional coupler are presented and the computing results using this scheme are in highly accordance with analytical solutions. Compared with conventional FDTD method, this scheme can save considerable computational resource without sacrificing solution accuracy and especially could be applied in the accurate analysis of optical devices.

© 2006 Optical Society of America

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  1. K. S. Yee, "Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1996).
  2. S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989).
    [CrossRef]
  3. W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
    [CrossRef]
  4. K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003).
    [CrossRef]
  5. Q. H. Liu, "The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations," IEEE Antennas Propag. Soc. Int. Symp. 1,122-125 (1997).
  6. M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996).
    [CrossRef]
  7. J. Fang, "Time Domain Finite Difference Computation for Maxwell’s Equations," PhD thesis, (Univ. California, Berkeley, 1989).
  8. C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).
  9. K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004).
    [CrossRef]
  10. M. L. Ghrist, "High order finite difference methods for wave equations," M. S. dissertation, (University of Colorado, 1997).
  11. E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
    [CrossRef]
  12. J. P. Berenger, "A perfectly match layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994).
    [CrossRef]
  13. S. D. Gedney. "An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices," IEEE Trans. Antennas and Propag. 44, 1630-1639 (1996).
    [CrossRef]
  14. A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001).
    [CrossRef]
  15. K. Okamoto, Fundamentals of optical waveguides, (Academic Press, 2000).

2004

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004).
[CrossRef]

2003

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003).
[CrossRef]

2001

A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001).
[CrossRef]

1999

E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
[CrossRef]

1997

Q. H. Liu, "The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations," IEEE Antennas Propag. Soc. Int. Symp. 1,122-125 (1997).

1996

M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1996).

S. D. Gedney. "An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices," IEEE Trans. Antennas and Propag. 44, 1630-1639 (1996).
[CrossRef]

1995

C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).

1994

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

1991

W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
[CrossRef]

1989

S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Berenger, J. P.

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

Broschat, S. L.

C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).

Chaudhuri, S. K.

S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Chu, S.

S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Gedney, S. D.

S. D. Gedney. "An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices," IEEE Trans. Antennas and Propag. 44, 1630-1639 (1996).
[CrossRef]

Goss, A.

W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
[CrossRef]

Harvey, J. F.

E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
[CrossRef]

Huang, W.

W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
[CrossRef]

Katehi, L. P. B.

M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996).
[CrossRef]

Krumpholz, M.

M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996).
[CrossRef]

Liu, Q. H.

Q. H. Liu, "The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations," IEEE Antennas Propag. Soc. Int. Symp. 1,122-125 (1997).

Manry, C. W.

C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).

Petropoulos, P. G.

A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001).
[CrossRef]

Robertson, R. L.

E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
[CrossRef]

Schneider, J. B.

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004).
[CrossRef]

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003).
[CrossRef]

C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).

Shlager, K. L.

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004).
[CrossRef]

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003).
[CrossRef]

Tentzeris, E. M.

E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
[CrossRef]

Xu, C. L.

W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1996).

Yefet, A.

A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001).
[CrossRef]

Appl. Comput. Electromagn. Soc. J.

C. W. Manry, S. L. Broschat, and J. B. Schneider, "Higher-order FDTD methods for large problems," Appl. Comput. Electromagn. Soc. J. 10, 17-29 (1995).

IEEE Antennas Propag. Soc. Int. Symp.

Q. H. Liu, "The pseudospectral time-domain (PSTD) method: a new algorithm for solution of Maxwell’s equations," IEEE Antennas Propag. Soc. Int. Symp. 1,122-125 (1997).

IEEE Photon. Technol. Lett.

W. Huang, C. L. Xu and A. Goss, "A scalar finite-difference time-domain approach to guided-wave optics," IEEE Photon. Technol. Lett. 3, 524-526 (1991).
[CrossRef]

IEEE Trans. Antennas and Propag.

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of several low dispersion finite-difference time-domain algorithms," IEEE Trans. Antennas and Propag. 51, 642-653 (2003).
[CrossRef]

K. L. Shlager and J. B. Schneider, "Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes," IEEE Trans. Antennas and Propag. 52,1095-1104 (2004).
[CrossRef]

S. D. Gedney. "An anisotropic perfectly matched layer-absorbing medium for truncation of FDTD lattices," IEEE Trans. Antennas and Propag. 44, 1630-1639 (1996).
[CrossRef]

IEEE Trans. Antennas Propag.

K. S. Yee, "Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1996).

IEEE Trans. Microwave Theory Tech.

M. Krumpholz and L. P. B. Katehi. "MRTD New time-domain schemes based on multiresolution analysis." IEEE Trans. Microwave Theory Tech. 44, 555-571 (1996).
[CrossRef]

E. M. Tentzeris, R. L. Robertson, and J. F. Harvey, "Stability and dispersion analysis of Battle -Lemarie-Based MRTD schemes," IEEE Trans. Microwave Theory Tech. 47, 1004-1012 (1999).
[CrossRef]

J. Comput. Phys.

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

A. Yefet and P. G. Petropoulos, "A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations," J. Comput. Phys. 168, 286-315 (2001).
[CrossRef]

J. Lightwave Technol.

S. Chu and S. K. Chaudhuri, "A finite-difference time-domain method for the design and analysis of guided-wave optical structures," IEEE, J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Other

J. Fang, "Time Domain Finite Difference Computation for Maxwell’s Equations," PhD thesis, (Univ. California, Berkeley, 1989).

M. L. Ghrist, "High order finite difference methods for wave equations," M. S. dissertation, (University of Colorado, 1997).

K. Okamoto, Fundamentals of optical waveguides, (Academic Press, 2000).

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Figures (5)

Fig. 1.
Fig. 1.

Image theory for two-dimensional TE wave

Fig.2.
Fig.2.

The numerical dispersion error versus Δs/λ (a) Δt=0.1Δs/c (b) Δt=0.5Δs/c

Fig.3.
Fig.3.

The numerical dispersion error versus ϕ for Δs=λ/10 (a)Δt=0.1Δs/c (b)Δt=0.5Δs/c

Fig.4.
Fig.4.

Sketch of parallel-slab directional coupler

Fig. 5.
Fig. 5.

Field distribution for the parallel-slab directional coupler

Tables (2)

Tables Icon

Table 1. The coefficient weights a(l) for some of HO-FDTD schemes

Tables Icon

Table 2. Computation results of couple length

Equations (22)

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

E z y = μ H x t
E z x = μ H y t
H y x H x y = ε E z t + σ E z
f ( x ) 1 Δ x l = M M 1 a ( l ) f ( x + ( l + 1 2 ) Δ x )
a ( l ) = ( 1 ) l 2 ( l + 1 2 ) 2 [ ( 2 M 1 ) ! ! ] 2 ( 2 M 2 2 l ) ! ! ( 2 M + 2 l ) ! ! l = M , M + 1 , . . . , M 1
H x n + 1 2 ( i , j + 1 2 ) = H x n 1 2 ( i , j + 1 2 ) Δ t μ Δ y l = M M 1 a ( l ) E z n ( i , j + l + 1 )
H y n + 1 2 ( i + 1 2 , j ) = H y n 1 2 ( i + 1 2 , j ) + Δ t μ Δ x l = M M 1 a ( l ) E z n ( i + l + 1 , j )
E z n + 1 ( i , j ) = 2 ε σ Δ t 2 ε + σ Δ t E z n ( i , j ) + 2 Δ t 2 ε + σ Δ t ( l = M M 1 a ( l ) ( H y n + 1 2 ( i + l + 1 2 , j ) Δ x H x n + 1 2 ( i , j + l + 1 2 ) Δ y ) )
Δ t ( v max l = 0 M 1 a ( l ) ( 1 Δ x ) 2 + ( 1 Δ y ) 2 ) 1
E z y = j ω μ s y s x H x
E z x = j ω μ s x s y H y
H y x H x y = j ω ε s x s y E z
s η = 1 + σ η j ω ε 0 ( η = x , y )
B x n + 1 2 ( i , j + 1 2 ) = 2 ε 0 σ y Δ t 2 ε 0 + σ y Δ t B x n 1 2 ( i , j + 1 2 ) 2 ε 0 Δ t 2 ε 0 + σ y Δ t l = M M 1 a ( l ) ( E z n ( i , j + l + 1 ) Δ y )
H x n + 1 2 ( i , j + 1 2 ) = H x n 1 2 ( i , j + 1 2 ) + 1 μ ( 1 + σ x Δ t 2 ε 0 ) B x n + 1 2 ( i , j + 1 2 ) 1 μ ( 1 σ x Δ t 2 ε 0 ) B x n 1 2 ( i , j + 1 2 )
B y n + 1 2 ( i + 1 2 , j ) = 2 ε 0 σ x Δ t 2 ε 0 + σ x Δ t B y n 1 2 ( i + 1 2 , j ) + 2 ε 0 Δ t 2 ε 0 + σ x Δ t l = M M 1 a ( l ) ( E z n ( i + l + 1 , j ) Δ x )
H y n + 1 2 ( i + 1 2 , j ) = H y n 1 2 ( i + 1 2 , j ) + 1 μ ( 1 + σ y Δ t 2 ε 0 ) B y n + 1 2 ( i + 1 2 , j ) 1 μ ( 1 σ y Δ t 2 ε 0 ) B y n 1 2 ( i + 1 2 , j )
D z n + 1 ( i , j ) = 2 ε 0 σ x Δ t 2 ε 0 + σ x Δ t D z n ( i , j ) + 2 ε 0 Δ t 2 ε 0 + σ x Δ t l = M M 1 a ( l ) ( H y n + 1 2 ( i + l + 1 2 , j ) Δ x H x n + 1 2 ( i , j + l + 1 2 ) Δ y )
E z n + 1 ( i , j ) = 2 ε 0 σ y Δ t 2 ε 0 + σ y Δ t E z n ( i , j ) + 2 ε 0 ε ( 2 ε 0 + σ y Δ t ) ( D z n + 1 ( i , j ) D z n ( i , j ) )
[ 1 c Δ t sin ( ω Δ t 2 ) ] 2 = [ 1 Δ x ( l = 0 M 1 a ( l ) sin ( k x ( l + 1 2 ) Δ x ) ) ] 2 + [ 1 Δ y ( l = 0 M 1 a ( l ) sin ( k y ( l + 1 2 ) Δ y ) ) ] 2
[ Δ s c Δ t sin ( ω Δ t 2 ) ] 2 = ( l = 0 M 1 a ( l ) sin ( k cos ϕ ( l + 1 2 ) Δ s ) ) 2 + ( l = 0 M 1 a ( l ) sin ( k sin ϕ ( l + 1 2 ) Δ s ) ) 2
v e = ( 1 v p c ) × 100 %

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