Abstract

We extend the continuity relations of field derivatives across an abrupt interface to arbitrary orders for transverse electric and magnetic waves in slab structures. Higher-order finite-difference formulation is then obtained by combining the systematically-obtained interface conditions with Taylor series expansion. Generalized Douglas scheme is also adopted to further enhance the convergence of truncation errors by two orders. We apply the derived finite-difference formulation, up to nine-points in this paper, to solve the guided modes in simple a slab waveguide and multiple quantum well waveguides. The results shows the truncation error is much higher, up to tenth order, as expected. Using those higher-order schemes, accurate results are obtained with much fewer sampled points, and hence with tremendously less computation time and memory.

© 2010 Optical Society of America

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  1. M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc. J. 135, 56-63 (1988).
  2. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
    [CrossRef]
  3. M. S. Stern, "Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 138, 185-190 (1991).
  4. P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992).
    [CrossRef]
  5. W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
    [CrossRef]
  6. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).
  7. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
    [CrossRef]
  8. Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, and H.-C. Chang, "Finite-difference modeling of dielectric waveguides with corners and slanted facets," J. Lightwave Technol. 27, 2077-2086 (2009).
    [CrossRef]
  9. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency domain analysis of 2-D photonic crytals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008).
    [CrossRef]
  10. G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998).
    [CrossRef]
  11. J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
    [CrossRef]
  12. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000).
    [CrossRef]
  13. H. Zhang, Q. Guo, and W. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightwave Technol. 25, 556-561 (2007).
    [CrossRef]
  14. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
    [CrossRef]
  15. B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
    [CrossRef]
  16. G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation II: vertical-cavity surface-emitting lasers," J. Lightwave Technol. 16, 142-151 (1998).
    [CrossRef]
  17. S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
    [CrossRef]
  18. S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
    [CrossRef]
  19. R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks Cole, 2004).
  20. G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992).
    [CrossRef] [PubMed]
  21. J. Yamauchi, Propagating Beam Analysis of Optical Waveguides, (Exeter, UK: Research Studies Press, 2003).

2009 (1)

2008 (2)

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency domain analysis of 2-D photonic crytals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008).
[CrossRef]

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

2007 (1)

2006 (1)

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

2002 (1)

2000 (1)

1999 (1)

J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
[CrossRef]

1998 (2)

1994 (1)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

1992 (4)

P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992).
[CrossRef]

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992).
[CrossRef] [PubMed]

1991 (1)

M. S. Stern, "Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 138, 185-190 (1991).

1989 (1)

S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
[CrossRef]

1988 (1)

M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc. J. 135, 56-63 (1988).

1985 (1)

S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
[CrossRef]

Benson, T. M.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Chang, H.-C.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Chiang, Y.-C.

Chiou, Y.-P.

Cho, Y.

S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
[CrossRef]

S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
[CrossRef]

Chu, S.-T.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Du, C.-H.

Guo, Q.

Hadley, G. R.

Hu, B.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Huang, W.

H. Zhang, Q. Guo, and W. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightwave Technol. 25, 556-561 (2007).
[CrossRef]

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Huang, W. P.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

Lai, C.-H.

Li, B.-J.

P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992).
[CrossRef]

Liu, P.-L.

P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992).
[CrossRef]

Nakano, H.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
[CrossRef]

Ohke, S.

S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
[CrossRef]

S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
[CrossRef]

Sewell, P.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Shibayama, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
[CrossRef]

Stern, M. S.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

M. S. Stern, "Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 138, 185-190 (1991).

M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc. J. 135, 56-63 (1988).

Takahashi, T.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

Umeda, T.

S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
[CrossRef]

S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
[CrossRef]

Vassallo, C.

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

Vukovic, A.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Wykes, J. G.

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Xu, C.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Xu, C. L.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

Yamauchi, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
[CrossRef]

Zhang, H.

IEEE J. Quantum Electron. (1)

P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992).
[CrossRef]

Inst. Elect. Eng. Proc. J. (1)

M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc. J. 135, 56-63 (1988).

J. Lightwave Technol. (8)

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
[CrossRef]

Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, and H.-C. Chang, "Finite-difference modeling of dielectric waveguides with corners and slanted facets," J. Lightwave Technol. 27, 2077-2086 (2009).
[CrossRef]

Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency domain analysis of 2-D photonic crytals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008).
[CrossRef]

G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998).
[CrossRef]

G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation II: vertical-cavity surface-emitting lasers," J. Lightwave Technol. 16, 142-151 (1998).
[CrossRef]

Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000).
[CrossRef]

H. Zhang, Q. Guo, and W. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightwave Technol. 25, 556-561 (2007).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008).
[CrossRef]

Opt. Commun. (2)

S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985).
[CrossRef]

S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999).
[CrossRef]

Photon. Tech. Lett. (1)

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006).
[CrossRef]

Proc. Inst. Elect. Eng. J. (3)

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994).
[CrossRef]

M. S. Stern, "Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 138, 185-190 (1991).

Other (2)

J. Yamauchi, Propagating Beam Analysis of Optical Waveguides, (Exeter, UK: Research Studies Press, 2003).

R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks Cole, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

Continuity of fields on interface and sample points.

Fig. 2.
Fig. 2.

Relative error of TE mode propagation constant with respect to grid size in a simple slab waveguide.

Fig. 3.
Fig. 3.

Relative error of TM mode propagation constant with respect to grid size in a simple slab waveguide.

Fig. 4.
Fig. 4.

Relative error of TE mode propagation constant with respect to grid size in a MQW waveguide.

Fig. 5.
Fig. 5.

Relative error of TM mode propagation constant with respect to grid size in a MQW waveguide.

Fig. 6.
Fig. 6.

Relative error of TE mode propagation constant with respect to grid size in a MQW waveguide with larger index difference.

Fig. 7.
Fig. 7.

Relative error of TM mode propagation constant with respect to grid size in a MQW waveguide with larger index difference.

Equations (62)

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

× E = B
× H = D
· D = 0
· B = 0 .
2 E + k 0 2 n 2 E = 0
2 H + k 0 2 n 2 H = 0 ,
E = E ¯ exp ( jβz )
H = H ¯ exp ( jβz )
2 E ¯ x 2 = ω 2 με E ¯ = β 2 E ¯
2 H ¯ x 2 = ω 2 με H ¯ = β 2 H ¯ .
E yR = E yL
H zR = H zL
E y x = jωμ H z .
E yR x = μ R μ L E yL x .
2 E y x 2 + ω 2 με E y = β 2 E y
E y = 1 β 2 [ 2 x 2 + ω 2 με ] E y .
[ 2 x 2 + ω 2 μ L ε L ] E yR = [ 2 x 2 + ω 2 μ R ε R ] E yL
2 E yR x 2 = [ 2 x 2 + ω 2 μ L ε L ] E yL ω 2 μ R ε R E yR = [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] E yL .
2 n x 2 n E yR = [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] n E yL .
2 n + 1 x 2 n + 1 E yR = μ R μ L x [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] n E yL .
2 k x 2k E yR = [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] k E yL .
2 k x 2 k { 1 β 2 [ 2 x 2 + ω R 2 μ R ε R ] E yR } = 1 β 2 { 2 k + 2 E yR x 2 k + 2 + ω 2 μ R ε R 2 k E yR x 2 k }
= [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] k { 1 β 2 [ 2 x 2 + ω 2 μ L ε L ] E yL } .
2 k + 2 E yR x 2 k + 2 = [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] k { [ 2 x 2 + ω 2 μ L ε L ] E yL }
ω 2 μ R ε R { [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] k E yL }
= [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] k + 1 E yL .
[ ψ R ψ R ψ R ψ R ( 3 ) ψ R ( 4 ) ψ R ( 2 N 1 ) ψ R ( 2 N ) ] = [ 1 0 0 0 0 0 0 a 0 0 0 0 b 0 1 0 0 0 0 ab 0 a 0 0 b 2 0 2 b 0 1 0 0 C 0 N 1 a b N 1 0 C 1 N 1 a b N 2 0 0 C 0 N b N 0 C 1 N b N 1 0 C 2 N b N 2 C N N ] [ ψ L ψ L ψ L ψ L ( 3 ) ψ L ( 4 ) ψ R ( 2 N 1 ) ψ L ( 2 N ) ]
Ψ R = M R : L Ψ L ,
H yR = H yL
E zR = E zL ,
H y x = jωε E z .
H yR x = ε R ε L H yL x
2 n x 2 n H yR = [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] n H yL .
2 n + 1 x 2 n + 1 H yR = ε R ε L x [ 2 x 2 + ( ω 2 μ L ε L ω 2 μ R ε R ) ] n H yL .
[ ψ L ψ L ψ L ψ L ( 2 N ) ] = [ 1 p p 2 2 ! p 2 N ( 2 N ) ! 0 1 p p 2 N 1 ( 2 N 1 ) ! 0 0 1 p 2 N 2 ( 2 N 2 ) ! 0 0 0 1 ] [ ψ i ψ i ψ i ψ i ( 2 N ) ] + HOT
Ψ L = M L : i Ψ i + HOT ,
[ ψ i + 1 ψ i + 1 ψ i + 1 ψ i + 1 ( 2 N ) ] = [ 1 q q 2 2 ! q 2 N ( 2 N ) ! 0 1 q q 2 N 1 ( 2 N 1 ) ! 0 0 1 q 2 N 2 ( 2 N 2 ) ! 0 0 0 1 ] [ ψ R ψ R ψ R ψ R ( 2 N ) ] + HOT
Ψ i + 1 = M i + 1 : R Ψ R + HOT .
Ψ i + 1 = M i + 1 : R Ψ R + HOT = M i + 1 : R M R : L Ψ L + HOT = M i + 1 : R M R : L M L : i Ψ i + HOT
= M i + 1 : i Ψ i + HOT .
Ψ i + 1 = M i + 1 : RT M RT : LT M LT : R ( T 1 ) M L ( t + 1 ) : Rt M Rt : Lt M R 1 : L 1 M L 1 : i Ψ i + HOT
= M i + 1 : i Ψ i + HOT
Ψ i + j = M i + j : i + j 1 Ψ i + j 1 + HOT = = M i + j : i + j 1 M i + 1 : i Ψ i + HOT
= M i + j : i Ψ i + HOT ,
ψ i + j = [ u j , 0 u j , 1 u j , 2 u j , 2 N ] Ψ i + O ( h 2 N + 1 ) .
[ ψ i N ψ i ψ i + N ] = [ u N , 0 u N , 1 u N , 2 N u N , 0 u N , 1 u N , 2 N ] [ ψ i ψ i ( j ) ψ i ( 2 N ) ] + O ( h 2 N + 1 )
F i = U i Ψ i + O ( h 2 N + 1 ) .
F i = U i D i ,
D i = B i F i ,
D x 2 ψ + ω 2 μ ε ψ = β 2 ψ .
F i = U i Ψ i + G 1 ψ i ( 2 N + 1 ) + G 2 ψ i ( 2 N + 2 ) + O ( h 2 N + 3 ) ,
D i = B i F i = Ψ i + B i ( G 1 ψ i ( 2 N + 1 ) ) + G 2 ψ i ( 2 N + 2 ) + O ( h 2 N + 1 ) ,
D x 2 ψ i = j = N j = N b 3 , j ψ i + j ψ i + v 1 ψ i ( 2 N + 1 ) + v 2 ψ i ( 2 N + 2 )
= ( 1 + v 1 2 N 1 x 2 N 1 + v 2 2 N x 2 N ) ψ i ,
D x 2 ψ i ( 1 + v 1 D x 2 N 1 + v 2 D x 2 N ) ψ i ,
ψ i = D x 2 1 + v 1 D x 2 N 1 + v 2 D x 2 N ψ i .
D x 2 ψ + ω μ ε ( 1 + v 1 D x 2 N 1 + v 2 D x 2 N ) ψ = β 2 ( 1 + v 1 D x 2 N 1 + v 2 D x 2 N ) ψ
Relative error ε r = β calculated β exact k 0 ( n core n clad ) ,
Relative error = β calculated β exact k 0 ( n well n clad ) .
n avg 2 = i w i n i 2 i w i
1 n avg 2 = i w i n i −2 i w i
D x 2 Ψ + ω 2 μ ε Ψ β ¯ 2 Ψ + 2 Ψ z 2 = j 2 β ¯ Ψ z .

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