Abstract

A two-dimensional (2D) compact finite-difference time-domain (FDTD) mode solver is developed based on wave equation formalism in combination with the matrix pencil method (MPM). The method is validated for calculation of both real guided and complex leaky modes of typical optical waveguides against the bench-mark finite-difference (FD) eigen mode solver. By taking advantage of the inherent parallel nature of the FDTD algorithm, the mode solver is implemented on graphics processing units (GPUs) using the compute unified device architecture (CUDA). It is demonstrated that the high-performance computing technique leads to significant acceleration of the FDTD mode solver with more than 30 times improvement in computational efficiency in comparison with the conventional FDTD mode solver running on CPU of a standard desktop computer. The computational efficiency of the accelerated FDTD method is in the same order of magnitude of the standard finite-difference eigen mode solver and yet require much less memory (e.g., less than 10%). Therefore, the new method may serve as an efficient, accurate and robust tool for mode calculation of optical waveguides even when the conventional eigen value mode solvers are no longer applicable due to memory limitation.

© 2010 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984).
    [CrossRef]
  2. M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
    [CrossRef]
  3. W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photon. Technol Lett. 3, 524–526 (1991).
  4. A. Fallahkhair, K. Li, and T. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
    [CrossRef]
  5. C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
    [CrossRef]
  6. B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984).
    [CrossRef]
  7. J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
    [CrossRef]
  8. M. D. Feit and J. A. Fleck., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19(7), 1154–1164 (1980).
    [CrossRef] [PubMed]
  9. C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).
  10. Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18(4), 618–623 (2000).
    [CrossRef]
  11. A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
    [CrossRef]
  12. S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
    [CrossRef]
  13. G. Zhou and X. Li, “Wave equation-based semivectorial compact 2-D-FDTD method for optical waveguide modal analysis,” J. Lightwave Technol. 22(2), 677–683 (2004).
    [CrossRef]
  14. T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995).
    [CrossRef]
  15. Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
    [CrossRef]
  16. S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
    [CrossRef]
  17. A. Taflove and S. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House Norwood, MA, 1995).
  18. S. Ryoo, C. Rodrigues, S. Baghsorkhi, S. Stone, D. Kirk, and W. Hwu, “Optimization principles and application performance evaluation of a multithreaded GPU using CUDA,” in (ACM, 2008), 73–82.
  19. S. Krakiwsky, L. Turner, and M. Okoniewski, “Acceleration of finite-difference time-domain (FDTD) using graphics processor units (GPU),” in Microwave Symposium Digest, IEEE MTT-S International, 2004), 1033- 1036.
  20. A. Balevic, L. Rockstroh, A. Tausendfreund, S. Patzelt, G. Goch, and S. Simon, “Accelerating simulations of light scattering based on finite-difference time-domain method with general purpose gpus,” in 2008), 327–334.
  21. M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. 3(9), 307–309 (1993).
    [CrossRef]
  22. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [CrossRef]
  23. J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
  24. J. Heaton, M. Bourke, S. Jones, B. Smith, K. Hilton, G. Smith, J. Birbeck, G. Berry, S. Dewar, and D. Wight, “Optimization of deep-etched, single-mode GaAs/AlGaAs optical waveguides using controlled leakage into the substrate,” J. Lightwave Technol. 17(2), 267–281 (1999).
    [CrossRef]
  25. W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
    [CrossRef]
  26. W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
    [CrossRef]

2008 (1)

2004 (1)

2000 (1)

1999 (1)

1997 (1)

A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
[CrossRef]

1996 (1)

S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
[CrossRef]

1995 (2)

T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995).
[CrossRef]

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

1994 (3)

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

1993 (3)

M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. 3(9), 307–309 (1993).
[CrossRef]

C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
[CrossRef]

1992 (1)

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
[CrossRef]

1991 (1)

J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
[CrossRef]

1988 (1)

M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

1984 (2)

E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984).
[CrossRef]

B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984).
[CrossRef]

1980 (1)

1969 (1)

J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

Berry, G.

Birbeck, J.

Bourke, M.

Bridges, W.

E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984).
[CrossRef]

Cendes, Z.

J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
[CrossRef]

Chaudhuri, S.

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).

Chew, W.

W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

Davies, J.

B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984).
[CrossRef]

Dewar, S.

Fallahkhair, A.

Feit, M. D.

Fleck, J. A.

Gedney, S.

S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
[CrossRef]

Goell, J.

J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).

Heaton, J.

Hilton, K.

Huang, W.

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
[CrossRef]

Jin, H.

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
[CrossRef]

Jones, S.

Juntunen, J.

A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
[CrossRef]

Kingsland, D.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

Koshiba, M.

Lee, J.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
[CrossRef]

Lee, R.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

Li, K.

Li, X.

Murphy, T.

Okoniewski, M.

M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. 3(9), 307–309 (1993).
[CrossRef]

Pereira, O.

T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995).
[CrossRef]

Rahman, B.

B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984).
[CrossRef]

Räisänen, A.

A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
[CrossRef]

Sacks, Z.

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

Sarkar, T.

T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995).
[CrossRef]

Schweig, E.

E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984).
[CrossRef]

Smith, B.

Smith, G.

Stern, M.

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

Sun, D.

J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
[CrossRef]

Tsuji, Y.

Vahldieck, R.

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
[CrossRef]

Weedon, W.

W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

Wight, D.

Xiao, S.

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
[CrossRef]

Xu, C.

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
[CrossRef]

Zhao, A.

A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
[CrossRef]

Zhou, G.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

J. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).

IEE Proc. Optoelectron. (1)

C. Xu, W. Huang, M. Stern, and S. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141(5), 281–286 (1994).
[CrossRef]

IEE Proc., Optoelectron. (1)

M. Stern, “Semivectorial polarised finite difference method for opticalwaveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

T. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag. 37(1), 48–55 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vectorbeam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (2)

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett. 2(5), 165–167 (1992).
[CrossRef]

M. Okoniewski, “Vector wave equation 2-D-FDTD method for guided wave problems,” IEEE Microw. Guid. Wave Lett. 3(9), 307–309 (1993).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

Z. Sacks, D. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbingboundary condition,” IEEE Trans. Antenn. Propag. 43(12), 1460–1463 (1995).
[CrossRef]

S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for thetruncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (3)

B. Rahman and J. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microw. Theory Tech. 32(1), 20–28 (1984).
[CrossRef]

J. Lee, D. Sun, and Z. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech. 39(8), 1262–1271 (1991).
[CrossRef]

E. Schweig and W. Bridges, “Computer analysis of dielectric waveguides: A finite-difference method,” IEEE Trans. Microw. Theory Tech. 32(5), 531–541 (1984).
[CrossRef]

J. Comput. Phys. (1)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (4)

Journalism (1)

C. Xu, W. Huang, and S. Chaudhuri, ““Efficient and accurate vector mode calculations by beam propagationmethod,” Lightwave Technology,” Journalism 11, 1209–1215 (1993).

Microw. Opt. Technol. Lett. (2)

A. Zhao, J. Juntunen, and A. Räisänen, “Analysis of hybrid modes in channel multilayer optical waveguides with the compact 2-D FDTD method,” Microw. Opt. Technol. Lett. 15(6), 398–403 (1997).
[CrossRef]

W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[CrossRef]

Other (5)

W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, “A scalar finite-difference time-domain approach to guided-wave optics,” IEEE Photon. Technol Lett. 3, 524–526 (1991).

A. Taflove and S. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House Norwood, MA, 1995).

S. Ryoo, C. Rodrigues, S. Baghsorkhi, S. Stone, D. Kirk, and W. Hwu, “Optimization principles and application performance evaluation of a multithreaded GPU using CUDA,” in (ACM, 2008), 73–82.

S. Krakiwsky, L. Turner, and M. Okoniewski, “Acceleration of finite-difference time-domain (FDTD) using graphics processor units (GPU),” in Microwave Symposium Digest, IEEE MTT-S International, 2004), 1033- 1036.

A. Balevic, L. Rockstroh, A. Tausendfreund, S. Patzelt, G. Goch, and S. Simon, “Accelerating simulations of light scattering based on finite-difference time-domain method with general purpose gpus,” in 2008), 327–334.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the square channel waveguide structure, n 1 = 1.52 , n 2 = 1.46 , w 1 = 2 μ m , PML Layers = 20.

Fig. 3
Fig. 3

Guided modal analysis by Goell’s model and solver in this paper.

Fig. 2
Fig. 2

Fundamental mode profile of the square channel waveguide.

Fig. 4
Fig. 4

Schematic diagram of a deep-etched GaAs-alGaAs optical waveguide structure.

Fig. 5
Fig. 5

TE20 mode profile of the ridge waveguide.

Fig. 6
Fig. 6

Percentage error of extracted wavelength of fundamental mode versus different FDTD time steps.

Fig. 7
Fig. 7

Percentage error of extracted wavelength of fundamental mode versus MPM sample numbers.

Fig. 8
Fig. 8

Memory consumptions of the scalar mode calculation by the accelerated FDTD and the conventional FD mode solvers.

Fig. 9
Fig. 9

Memory consumptions of the scalar, semi-vector and full-vector mode calculations by the accelerated FDTD solvers.

Fig. 10
Fig. 10

Time consumption of the scalar mode calculation by the standard FDTD, accelerated FDTD and the conventional FD mode solvers.

Fig. 11
Fig. 11

Time consumption of the scalar, semi-vector and full-vector mode calculations by accelerated FDTD mode solvers on GPUs.

Tables (1)

Tables Icon

Table 1 Leaky modal analysis by FD mode solver and solver in this paper

Equations (22)

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

Ψ t ( x , y , t ) = m = 0 A m exp ( j ω m t ) exp ( ξ m t ) Φ m ( x , y )
S ( t ) = m = 0 M R m exp ( s m t )
ω m = Im ( s m ) ξ m = Re ( s m )
E s r c ( x , y , t ) = A exp ( ( x x 0 ) 2 + ( y y 0 ) 2 R m ) exp ( ( t t 0 ) 2 T m ) sin ( ω 0 t )
Φ m ( x , y ) = Δ t n = 0 N t 1 Ψ t ( x , y , t ) exp ( j ω m n Δ t ) exp ( ξ m n Δ t )
t 2 E t + ( n 2 n eff 2 ) k 2 E t = t ( 1 n 2 t n 2 E t )
A x x E x + A x y E y = k 2 n eff 2 E x A y x E x + A y y E y = k 2 n eff 2 E y
A x x E x = x [ 1 n 2 x ( n 2 E x ) ] + 2 E x y 2 + n 2 k 2 E x A y y E y = y [ 1 n 2 y ( n 2 E y ) ] + 2 E y x 2 + n 2 k 2 E y A x y E y = x [ 1 n 2 y ( n 2 E y ) ] 2 E y x y A y x E x = y [ 1 n 2 x ( n 2 E x ) ] 2 E x y x
n 2 c 2 2 t 2 E x = x [ 1 n 2 x ( n 2 E x ) ] + 2 E x y 2 + x [ 1 n 2 y ( n 2 E y ) ] 2 E y x y β 2 E x n 2 c 2 2 t 2 E y = y [ 1 n 2 y ( n 2 E y ) ] + 2 E y x 2 + y [ 1 n 2 x ( n 2 E x ) ] 2 E x y x β 2 E y
n 2 c 2 2 t 2 E x = 1 s x x [ 1 n 2 1 s x x ( n 2 E x ) ] + 1 s y y [ 1 s y y E x ] β 2 E x + 1 s x x [ 1 n 2 1 s y y ( n 2 E y ) ] 1 s y y [ 1 s x x E y ] n 2 c 2 2 t 2 E y = 1 s y y [ 1 n 2 1 s y y ( n 2 E y ) ] + 1 s x x [ 1 s x x E y ] β 2 E y + 1 s y y [ 1 n 2 1 s x x ( n 2 E x ) ] 1 s x x [ 1 s y y E x ]
j ω D x _ x = 1 s x x ( n 2 E x ) ; j ω D x _ y = 1 s y y E x j ω D y _ x = 1 s x x E y ; j ω D y _ y = 1 s y y ( n 2 E y )
j ω D x _ x x = 1 s x x ( 1 n 2 j ω D x _ x ) ; j ω D x _ y y = 1 s y y ( j ω D x _ y ) j ω D y _ y x = 1 s x x ( 1 n 2 j ω D y _ y ) ; j ω D y _ x y = 1 s y y ( j ω D y _ x )
j ω D y _ y y = 1 s y y ( 1 n 2 j ω D y _ y ) ; j ω D y _ x x = 1 s x x ( j ω D y _ x ) j ω D x _ x y = 1 s y y ( 1 n 2 j ω D x _ x ) ; j ω D x _ y x = 1 s x x ( j ω D x _ y )
n 2 c 2 2 t 2 E x = j ω D x _ x x + j ω D x _ y y β 2 E x + j ω D y _ y x j ω D y _ x y n 2 c 2 2 t 2 E y = j ω D y _ y y + j ω D y _ x x β 2 E y + j ω D x _ x y j ω D x _ y x
n 2 c 2 2 t 2 E x = t D x _ x x + t D x _ y y β 2 E x + t D y _ y x t D y _ x y n 2 c 2 2 t 2 E y = t D y _ y y + t D y _ x x β 2 E y + t D x _ x y t D x _ y x t D x _ x + σ x ε 0 n 2 D x _ x = x ( n 2 E x ) ; t D x _ y + σ y ε 0 n 2 D x _ y = y E x t D y _ x + σ x ε 0 n 2 D y _ x = x E y ; t D y _ y + σ y ε 0 n 2 D y _ y = y ( n 2 E y ) t D x _ x x + σ x ε 0 n 2 D x _ x x = x ( 1 n 2 t D x _ x ) ; t D x _ y y + σ y ε 0 n 2 D x _ y y = y ( t D x _ y ) t D y _ y x + σ x ε 0 n 2 D y _ y x = x ( 1 n 2 t D y _ y ) ; t D y _ x y + σ y ε 0 n 2 D y _ x y = y ( t D y _ x ) t D y _ y y + σ y ε 0 n 2 D y _ y y = x ( 1 n 2 t D y _ y ) ; t D y _ x x + σ x ε 0 n 2 D y _ x x = x ( t D y _ x ) t D x _ x y + σ y ε 0 n 2 D x _ x y = y ( 1 n 2 t D x _ x ) ; t D x _ y x + σ x ε 0 n 2 D x _ y x = x ( t D x _ y )
σ x ε 0 n 2 D x = σ x ε 0 n 2 D x n + 1 + D x n 2 ​   σ y ε 0 n 2 D y = σ y ε 0 n 2 D y n + 1 + D y n 2
E x n + 1 = E x n 1 + 2 E x n + c 2 Δ t n 2 ( D x _ x x n D x _ x x n 1 ) + c 2 Δ t n 2 ( D x _ y y n D x _ y y n 1 ) β 2 c 2 Δ t 2 n 2 E x n             + c 2 Δ t n 2 ( D y _ y x n D y _ y x n 1 ) c 2 Δ t n 2 ( D y _ x y n D y _ x y n 1 ) E y n + 1 = E y n 1 + 2 E y n + c 2 Δ t n 2 ( D y _ y y n D y _ y y n 1 ) + c 2 Δ t n 2 ( D y _ x x n D y _ x x n 1 ) β 2 c 2 Δ t 2 n 2 E y n             + c 2 Δ t n 2 ( D x _ x y n D x _ x y n 1 ) c 2 Δ t n 2 ( D x _ y x n D x _ y x n 1 )
D x _ x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ x ( x n 2 E x ) D x _ y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ y ( y E x ) D y _ x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D y _ x n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ x ( x E y ) D y _ y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D y _ y n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ y ( y n 2 E y ) D x _ x x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x 1 n 2 D x _ x n x 1 n 2 D x _ x n 1 ] D x _ y y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y D x _ y n y D x _ y n 1 ] D y _ y x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D y _ y x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x 1 n 2 D y _ y n x 1 n 2 D y _ y n 1 ] D y _ x y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D y _ x y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y D y _ x n y D y _ x n 1 ] D y _ y y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D y _ y y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y 1 n 2 D y _ y n y 1 n 2 D y _ y n 1 ] D y _ x x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D y _ x x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x D y _ x n x D y _ x n 1 ] D x _ x y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ x y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y 1 n 2 D x _ x n y 1 n 2 D x _ x n 1 ] D x _ y x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ y x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x D x _ y n x D x _ y n 1 ]
E x n + 1 = E x n 1 + 2 E x n + c 2 Δ t n 2 ( D x _ x x n D x _ x x n 1 ) + c 2 Δ t n 2 ( D x _ y y n D x _ y y n 1 ) β 2 c 2 Δ t 2 n 2 E x n E y n + 1 = E y n 1 + 2 E y n + c 2 Δ t n 2 ( D y _ y y n D y _ y y n 1 ) + c 2 Δ t n 2 ( D y _ x x n D y _ x x n 1 ) β 2 c 2 Δ t 2 n 2 E y n
D x _ x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ x ( x n 2 E x ) D x _ y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ y ( y E x ) D y _ x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D y _ x n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ x ( x E y ) D y _ y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D y _ y n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ y ( y n 2 E y ) D x _ x x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x 1 n 2 D x _ x n x 1 n 2 D x _ x n 1 ] D x _ y y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y D x _ y n y D x _ y n 1 ] D y _ y y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D y _ y y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y 1 n 2 D y _ y n y 1 n 2 D y _ y n 1 ] D y _ x x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D y _ x x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x D y _ x n x D y _ x n 1 ]
Ψ n + 1 = Ψ n 1 + 2 Ψ n + c 2 Δ t n 2 ( D x _ x x n D x _ x x n 1 ) + c 2 Δ t n 2 ( D x _ y y n D x _ y y n 1 ) β 2 c 2 Δ t 2 n 2 Ψ n
D x _ x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ x ( x E x ) D x _ y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y n + 2 ε 0 n 2 Δ t 2 ε 0 n 2 + Δ t σ y ( y E x ) D x _ x x n + 1 = 2 ε 0 n 2 Δ t σ x 2 ε 0 n 2 + Δ t σ x D x _ x x n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ x [ x D x _ x n x D x _ x n 1 ] D x _ y y n + 1 = 2 ε 0 n 2 Δ t σ y 2 ε 0 n 2 + Δ t σ y D x _ y y n + 2 ε 0 n 2 2 ε 0 n 2 + Δ t σ y [ y D x _ y n y D x _ y n 1 ]

Metrics