Abstract

We derive generalized full-vectorial continuity relations of field derivatives across an abrupt curved interface. Using the Helmholtz wave equation, we can extend the interface conditions by two orders. Repeating the process, we obtain interface conditions of even and odd orders from the zeroth- and first-order interface conditions, respectively, which can be extended to arbitrary orders. The interface conditions combined with Taylor series expansion are applied in higher order full-vectorial finite-difference analysis of several waveguide structures. From effective index convergence analysis of optical fiber modes, the 6-, 15-, and 28-point schemes give second-, fourth-, and sixth-order convergence, respectively. The higher order formulation is also applied to guided mode analysis of photonic crystal fibers and terahertz pipe waveguides, where improved accuracy is obtained when using higher order scheme. Our proposed method allows coarser discretization, which can greatly reduce the computation time and memory. The ultimate accuracy can also be higher due to smaller accumulated roundoff error.

© 2011 IEEE

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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, S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Inst. Elect. Eng. Proc. J. 141, 281-286 (1994).
  3. Y.-C. Chiang, Y.-P. Chiou, H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightw. Technol. 20, 1609-1618 (2002).
  4. T. Ando, H. Nakayama, S. Numata, J. Yamauchi, H. Nakano, "Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface," J. Lightw. Technol. 20, 1627-1634 (2002).
  5. C.-P. Yu, H.-C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers," Opt. Exp. 12, 6165-6177 (2004).
  6. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Inst. Elect. Eng. Proc. J. 139, 137-142 (1992).
  7. J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides," IEEE Photon. Technol. Lett. 9, 961-963 (1997).
  8. Y.-P. Chiou, Y.-C. Chiang, H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).
  9. Y.-P. Chiou, C.-H. Du, "Arbitrary-order interface conditions for slab structures and their applications in waveguide analysis," Opt. Exp. 18, 4088-4102 (2010).
  10. J. G. Wykes, P. Sewell, A. Vukovic, T. M. Benson, "Subsampling of fine features in finite-difference frequency-domain simulations," Microw. Opt. Technol. Lett. 44, 95-101 (2005).
  11. Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, H.-C. Chang, "Finite-difference modeling of dielectric waveguides with corners and slanted facets," J. Lightw. Technol. 27, 2077-2086 (2009).
  12. Y.-C. Chiang, Y.-P. Chiou, H.-C. Chang, "Finite-difference frequency domain analysis of 2-D photonic crytals with curved dielectric interfaces," J. Lightw. Technol. 26, 971-976 (2008).
  13. Y.-C. Lu, L. Yang, W.-P. Huang, S.-S. Jian, "Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry," J. Lightw. Technol. 26, 1868-1876 (2008).
  14. H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microw. Theory Tech. 52, 1166-1174 (2004).
  15. R. B. Lehoucq, D. C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (SIAM, 1998).
  16. T. Davis, "Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method," ACM Trans. Math. Softw. 30, 196-199 (2004) http://www.cise.ufl.edu/research/sparse/umfpack/.
  17. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, "Multipole method for microstructured optical fibers—I: Formulation," J. Opt. Soc. Amer. B 19, 2322-2330 (2002).
  18. K. Saitoh, M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).
  19. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, C.-K. Sun, "Low-index terahertz pipe waveguides," Opt. Lett. 34, 3457-3459 (2009).
  20. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, H.-C. Chang, "Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding," Opt. Exp. 18, 309-322 (2010).
  21. G. Gallot, S. P. Jamison, R. W. McGowan, D. Grischkowsky, "Terahertz waveguides," J. Opt. Soc. Amer. B 17, 851-862 (2000).
  22. K. Wang, D. M. Mittleman, "Metal wires for terahertz wave guiding," Nature 432, 376-379 (2004).
  23. R. Mendis, D. Grischkowsky, "Plastic ribbon THz waveguides," J. Appl. Phys. 88, 4449-4451 (2000).
  24. D. Chen, H. Chen, "A novel low-loss terahertz waveguide: Polymer tube," Opt. Exp. 18, 3762-3767 (2010).

2010 (3)

Y.-P. Chiou, C.-H. Du, "Arbitrary-order interface conditions for slab structures and their applications in waveguide analysis," Opt. Exp. 18, 4088-4102 (2010).

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, H.-C. Chang, "Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding," Opt. Exp. 18, 309-322 (2010).

D. Chen, H. Chen, "A novel low-loss terahertz waveguide: Polymer tube," Opt. Exp. 18, 3762-3767 (2010).

2009 (2)

C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, C.-K. Sun, "Low-index terahertz pipe waveguides," Opt. Lett. 34, 3457-3459 (2009).

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

2008 (2)

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

Y.-C. Lu, L. Yang, W.-P. Huang, S.-S. Jian, "Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry," J. Lightw. Technol. 26, 1868-1876 (2008).

2005 (1)

J. G. Wykes, P. Sewell, A. Vukovic, T. M. Benson, "Subsampling of fine features in finite-difference frequency-domain simulations," Microw. Opt. Technol. Lett. 44, 95-101 (2005).

2004 (4)

H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microw. Theory Tech. 52, 1166-1174 (2004).

T. Davis, "Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method," ACM Trans. Math. Softw. 30, 196-199 (2004) http://www.cise.ufl.edu/research/sparse/umfpack/.

C.-P. Yu, H.-C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers," Opt. Exp. 12, 6165-6177 (2004).

K. Wang, D. M. Mittleman, "Metal wires for terahertz wave guiding," Nature 432, 376-379 (2004).

2002 (4)

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

T. Ando, H. Nakayama, S. Numata, J. Yamauchi, H. Nakano, "Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface," J. Lightw. Technol. 20, 1627-1634 (2002).

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, "Multipole method for microstructured optical fibers—I: Formulation," J. Opt. Soc. Amer. B 19, 2322-2330 (2002).

K. Saitoh, M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).

2000 (3)

G. Gallot, S. P. Jamison, R. W. McGowan, D. Grischkowsky, "Terahertz waveguides," J. Opt. Soc. Amer. B 17, 851-862 (2000).

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

R. Mendis, D. Grischkowsky, "Plastic ribbon THz waveguides," J. Appl. Phys. 88, 4449-4451 (2000).

1997 (1)

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides," IEEE Photon. Technol. Lett. 9, 961-963 (1997).

1994 (1)

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

1992 (1)

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

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).

ACM Trans. Math. Softw. (1)

T. Davis, "Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method," ACM Trans. Math. Softw. 30, 196-199 (2004) http://www.cise.ufl.edu/research/sparse/umfpack/.

IEEE J. Quantum Electron. (1)

K. Saitoh, M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002).

IEEE Photon. Technol. Lett. (1)

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama, H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides," IEEE Photon. Technol. Lett. 9, 961-963 (1997).

IEEE Trans. Microw. Theory Tech. (1)

H. A. Jamid, "Enhanced PML performance using higher order approximation," IEEE Trans. Microw. Theory Tech. 52, 1166-1174 (2004).

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

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

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

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

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. Lightw. Technol. (1)

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

J. Appl. Phys. (1)

R. Mendis, D. Grischkowsky, "Plastic ribbon THz waveguides," J. Appl. Phys. 88, 4449-4451 (2000).

J. Lightw. Technol. (1)

T. Ando, H. Nakayama, S. Numata, J. Yamauchi, H. Nakano, "Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface," J. Lightw. Technol. 20, 1627-1634 (2002).

J. Lightw. Technol. (4)

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

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

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

Y.-C. Lu, L. Yang, W.-P. Huang, S.-S. Jian, "Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry," J. Lightw. Technol. 26, 1868-1876 (2008).

J. Opt. Soc. Amer. B (1)

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, "Multipole method for microstructured optical fibers—I: Formulation," J. Opt. Soc. Amer. B 19, 2322-2330 (2002).

J. Opt. Soc. Amer. B (1)

G. Gallot, S. P. Jamison, R. W. McGowan, D. Grischkowsky, "Terahertz waveguides," J. Opt. Soc. Amer. B 17, 851-862 (2000).

Microw. Opt. Technol. Lett. (1)

J. G. Wykes, P. Sewell, A. Vukovic, T. M. Benson, "Subsampling of fine features in finite-difference frequency-domain simulations," Microw. Opt. Technol. Lett. 44, 95-101 (2005).

Nature (1)

K. Wang, D. M. Mittleman, "Metal wires for terahertz wave guiding," Nature 432, 376-379 (2004).

Opt. Exp. (4)

C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, H.-C. Chang, "Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding," Opt. Exp. 18, 309-322 (2010).

D. Chen, H. Chen, "A novel low-loss terahertz waveguide: Polymer tube," Opt. Exp. 18, 3762-3767 (2010).

Y.-P. Chiou, C.-H. Du, "Arbitrary-order interface conditions for slab structures and their applications in waveguide analysis," Opt. Exp. 18, 4088-4102 (2010).

C.-P. Yu, H.-C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers," Opt. Exp. 12, 6165-6177 (2004).

Opt. Lett. (1)

Other (1)

R. B. Lehoucq, D. C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (SIAM, 1998).

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