Abstract

The fourth-order finite-difference time-domain (FDTD) method using a symplectic integrator propagator can calculate the propagation of the electromagnetic waves with very low dispersion error in the region of a constant or smoothly varying index profile. An additional technique is required for the problem with the discontinuous dielectric interfaces. We derived the third-order effective permittivities at dielectric interfaces for the fourth-order FDTD method in the case of 2D TE polarization. As the required accuracy level is increased, the memory resources used by the fourth-order FDTD method with the effective permittivities are reduced severalfold or more compared with the standard FDTD method. The accurate performance of the proposed method is demonstrated through numerical examples.

© 2007 Optical Society of America

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2005 (3)

2003 (2)

N. Feng, G. Zhou, and W. P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

W. Cai and S. Deng, "An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case," J. Comput. Phys. 190, 159-183 (2003).
[CrossRef]

2002 (2)

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

T. Hirono, Y. Yoshikuni, and Y. Shibata, "Third-order effective permittivities for the 4th-order FDTD method in the 2D TM polarization case," in Physics and Simulation of Optoelectronic Devices X, P. Blood, M. Osinski, and Y. Arakawa, eds., Proc. SPIE 4646, 630-640 (2002).
[CrossRef]

2001 (4)

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]

A. Ditkowski, K. Dridi, and J. S. Hesthaven, "Convergent Cartesian grid methods for Maxwell's equations in complex geometries," J. Comput. Phys. 170, 39-80 (2001).
[CrossRef]

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propag. 49, 749-756 (2001).
[CrossRef]

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

2000 (4)

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 12, 813-815 (2000).
[CrossRef]

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

1999 (1)

1998 (2)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. Yamauchi, H. Kanbara, and H. Nakano, "Analysis of optical waveguides with high-reflection coatings using the FD-TD method," IEEE Photon. Technol. Lett. 10, 111-113 (1998).
[CrossRef]

1997 (3)

C. Zhang and R. J. LeVeque, "The immersed interface method for acoustic wave equations with discontinuous coefficients," Wave Motion 25, 237-263 (1997).
[CrossRef]

J. L. Young, D. Gaitonde, and J. J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach," IEEE Trans. Antennas Propag. 45, 1573-1580 (1997).
[CrossRef]

M. F. Hadi and M. Piket-May, "A modified FDTD (2, 4) scheme for modeling electrically large structures with high-phase accuracy," IEEE Trans. Antennas Propag. 45, 254-264 (1997).
[CrossRef]

1996 (2)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

1994 (1)

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

1991 (3)

S. T. Chu, W. P. Huang, and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits," Comput. Phys. Commun. 68, 451-484 (1991).
[CrossRef]

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).
[CrossRef]

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

1989 (1)

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

1988 (1)

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Ahmad, R. U.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

Balanis, C. A.

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

Balestra, C. L.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Beaulieu, L.

T. Deveze, L. Beaulieu, and W. Tabbara, "A fourth-order scheme for the FDTD algorithm applied to Maxwell's equations," in IEEE Antennas Propagation Society International Symposium (IEEE, 1992), pp. 346-349.

Berenger, J.-P.

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

Birtcher, C. R.

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

Byers, A.

A. Byers, I. Rumsey, Z. Popovic, and M. Piket-May, "Surface-wave guiding using periodic structures," in IEEE Antennas Propagation Society International Symposium (IEEE, 2000), Vol. 1, pp. 342-345.

Cai, W.

W. Cai and S. Deng, "An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case," J. Comput. Phys. 190, 159-183 (2003).
[CrossRef]

Camarda, G. S.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

Celuch-Marcysiak, M.

M. Celuch-Marcysiak and W. K. Gwarek, "Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits," in Proceedings of the 24th European Microwave Conference (1994), pp. 1530-1535.

Chaudhuri, S. K.

S. T. Chu, W. P. Huang, and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits," Comput. Phys. Commun. 68, 451-484 (1991).
[CrossRef]

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).
[CrossRef]

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

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

Chew, W. C.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

S. T. Chu, W. P. Huang, and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits," Comput. Phys. Commun. 68, 451-484 (1991).
[CrossRef]

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).
[CrossRef]

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

Chuang, S-L.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Deng, S.

W. Cai and S. Deng, "An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case," J. Comput. Phys. 190, 159-183 (2003).
[CrossRef]

Deveze, T.

T. Deveze, L. Beaulieu, and W. Tabbara, "A fourth-order scheme for the FDTD algorithm applied to Maxwell's equations," in IEEE Antennas Propagation Society International Symposium (IEEE, 1992), pp. 346-349.

Ditkowski, A.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, "Convergent Cartesian grid methods for Maxwell's equations in complex geometries," J. Comput. Phys. 170, 39-80 (2001).
[CrossRef]

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propag. 49, 749-756 (2001).
[CrossRef]

Dong, P.

Dridi, K.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, "Convergent Cartesian grid methods for Maxwell's equations in complex geometries," J. Comput. Phys. 170, 39-80 (2001).
[CrossRef]

Dridi, K. H.

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propag. 49, 749-756 (2001).
[CrossRef]

Espinola, R. L.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

Fan, S.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "High-density integrated optics," J. Lightwave Technol. 17, 1682-1692 (1999).
[CrossRef]

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Fang, J.

J. Fang, "Time domain finite difference computation for Maxwell's equations," Ph.D. dissertation (University of California at Berkeley, 1989).

Feng, N.

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Gaitonde, D.

J. L. Young, D. Gaitonde, and J. J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach," IEEE Trans. Antennas Propag. 45, 1573-1580 (1997).
[CrossRef]

Georgakopoulos, S. V.

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

Goss, A.

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).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Guo, W.-H.

W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Gwarek, W. K.

M. Celuch-Marcysiak and W. K. Gwarek, "Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits," in Proceedings of the 24th European Microwave Conference (1994), pp. 1530-1535.

Hadi, M. F.

M. F. Hadi and M. Piket-May, "A modified FDTD (2, 4) scheme for modeling electrically large structures with high-phase accuracy," IEEE Trans. Antennas Propag. 45, 254-264 (1997).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haus, A.

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

Haus, H. A.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "High-density integrated optics," J. Lightwave Technol. 17, 1682-1692 (1999).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

He, S.

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Herrick, R. W.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Hesthaven, J. S.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, "Convergent Cartesian grid methods for Maxwell's equations in complex geometries," J. Comput. Phys. 170, 39-80 (2001).
[CrossRef]

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propag. 49, 749-756 (2001).
[CrossRef]

Hirono, T.

T. Hirono, Y. Yoshikuni, and Y. Shibata, "Third-order effective permittivities for the 4th-order FDTD method in the 2D TM polarization case," in Physics and Simulation of Optoelectronic Devices X, P. Blood, M. Osinski, and Y. Arakawa, eds., Proc. SPIE 4646, 630-640 (2002).
[CrossRef]

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

T. Hirono, W. W. Lui, and S. Seki, "Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space," in IEEE MTT-S International Microwave Symposium Digest (IEEE, 1999), pp. 1293-1296.

Huang, W. P.

N. Feng, G. Zhou, and W. P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

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).
[CrossRef]

S. T. Chu, W. P. Huang, and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits," Comput. Phys. Commun. 68, 451-484 (1991).
[CrossRef]

Huang, Y.-Z.

W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Joannopoulos, J. D.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "High-density integrated optics," J. Lightwave Technol. 17, 1682-1692 (1999).
[CrossRef]

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Johnson, S. G.

Kanbara, H.

J. Yamauchi, H. Kanbara, and H. Nakano, "Analysis of optical waveguides with high-reflection coatings using the FD-TD method," IEEE Photon. Technol. Lett. 10, 111-113 (1998).
[CrossRef]

Kattawar, G. W.

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Kirk, A. G.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Lee, S. M.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

LeVeque, R. J.

C. Zhang and R. J. LeVeque, "The immersed interface method for acoustic wave equations with discontinuous coefficients," Wave Motion 25, 237-263 (1997).
[CrossRef]

Li, C.

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Lu, Y.

Lui, W.

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

Lui, W. W.

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

T. Hirono, W. W. Lui, and S. Seki, "Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space," in IEEE MTT-S International Microwave Symposium Digest (IEEE, 1999), pp. 1293-1296.

Luo, H.

Manolatou, C.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974), pp. 7-17.

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Moghaddam, M.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Nakano, H.

J. Yamauchi, H. Kanbara, and H. Nakano, "Analysis of optical waveguides with high-reflection coatings using the FD-TD method," IEEE Photon. Technol. Lett. 10, 111-113 (1998).
[CrossRef]

Nasir, M. A.

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

Osgood, R. M.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

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]

Piket-May, M.

M. F. Hadi and M. Piket-May, "A modified FDTD (2, 4) scheme for modeling electrically large structures with high-phase accuracy," IEEE Trans. Antennas Propag. 45, 254-264 (1997).
[CrossRef]

A. Byers, I. Rumsey, Z. Popovic, and M. Piket-May, "Surface-wave guiding using periodic structures," in IEEE Antennas Propagation Society International Symposium (IEEE, 2000), Vol. 1, pp. 342-345.

Pizzuto, F.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

Popovic, Z.

A. Byers, I. Rumsey, Z. Popovic, and M. Piket-May, "Surface-wave guiding using periodic structures," in IEEE Antennas Propagation Society International Symposium (IEEE, 2000), Vol. 1, pp. 342-345.

Qiu, M.

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Rao, H.

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

Renaut, R. A.

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

Rumsey, I.

A. Byers, I. Rumsey, Z. Popovic, and M. Piket-May, "Surface-wave guiding using periodic structures," in IEEE Antennas Propagation Society International Symposium (IEEE, 2000), Vol. 1, pp. 342-345.

Scki, S.

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

Seki, S.

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

T. Hirono, W. W. Lui, and S. Seki, "Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space," in IEEE MTT-S International Microwave Symposium Digest (IEEE, 1999), pp. 1293-1296.

Shang, J. J. S.

J. L. Young, D. Gaitonde, and J. J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach," IEEE Trans. Antennas Propag. 45, 1573-1580 (1997).
[CrossRef]

Shibata, Y.

T. Hirono, Y. Yoshikuni, and Y. Shibata, "Third-order effective permittivities for the 4th-order FDTD method in the 2D TM polarization case," in Physics and Simulation of Optoelectronic Devices X, P. Blood, M. Osinski, and Y. Arakawa, eds., Proc. SPIE 4646, 630-640 (2002).
[CrossRef]

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Tabbara, W.

T. Deveze, L. Beaulieu, and W. Tabbara, "A fourth-order scheme for the FDTD algorithm applied to Maxwell's equations," in IEEE Antennas Propagation Society International Symposium (IEEE, 1992), pp. 346-349.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Turkel, E.

E. Turkel and A. Yefet, "Fourth-order method for Maxwell equations on a staggered mesh," in IEEE Antennas Propagation Society International Symposium (IEEE, 1997), pp. 2156-2159.

Vassallo, C.

Villeneuve, P. R.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "High-density integrated optics," J. Lightwave Technol. 17, 1682-1692 (1999).
[CrossRef]

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Wang, Q.-M.

W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Yamauchi, J.

J. Yamauchi, H. Kanbara, and H. Nakano, "Analysis of optical waveguides with high-reflection coatings using the FD-TD method," IEEE Photon. Technol. Lett. 10, 111-113 (1998).
[CrossRef]

Yang, P.

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

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]

E. Turkel and A. Yefet, "Fourth-order method for Maxwell equations on a staggered mesh," in IEEE Antennas Propagation Society International Symposium (IEEE, 1997), pp. 2156-2159.

Yoshikuni, Y.

T. Hirono, Y. Yoshikuni, and Y. Shibata, "Third-order effective permittivities for the 4th-order FDTD method in the 2D TM polarization case," in Physics and Simulation of Optoelectronic Devices X, P. Blood, M. Osinski, and Y. Arakawa, eds., Proc. SPIE 4646, 630-640 (2002).
[CrossRef]

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

Young, J. L.

J. L. Young, D. Gaitonde, and J. J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach," IEEE Trans. Antennas Propag. 45, 1573-1580 (1997).
[CrossRef]

Zhai, P.-W.

Zhang, C.

C. Zhang and R. J. LeVeque, "The immersed interface method for acoustic wave equations with discontinuous coefficients," Wave Motion 25, 237-263 (1997).
[CrossRef]

Zhou, C.

Zhou, G.

Zingg, D. W.

D. W. Zingg, "High-order finite-difference methods in computational electromagnetics," in IEEE Antennas Propagation Society International Symposium (IEEE, 1997), pp. 110-113.

Appl. Opt. (2)

Comput. Phys. Commun. (1)

S. T. Chu, W. P. Huang, and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits," Comput. Phys. Commun. 68, 451-484 (1991).
[CrossRef]

IEEE Antennas Propag. Magazine (1)

S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, "Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration, part I: Theory," IEEE Antennas Propag. Magazine 44, 134-142 (2002).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

T. Hirono, Y. Shibata, W. W. Lui, S. Scki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme," IEEE Microw. Guid. Wave Lett. 10, 359-361 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

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).
[CrossRef]

J. Yamauchi, H. Kanbara, and H. Nakano, "Analysis of optical waveguides with high-reflection coatings using the FD-TD method," IEEE Photon. Technol. Lett. 10, 111-113 (1998).
[CrossRef]

W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 12, 813-815 (2000).
[CrossRef]

R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Osgood, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Pade approximation," IEEE Photon. Technol. Lett. 14, 65-67 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

J. L. Young, D. Gaitonde, and J. J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach," IEEE Trans. Antennas Propag. 45, 1573-1580 (1997).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

M. F. Hadi and M. Piket-May, "A modified FDTD (2, 4) scheme for modeling electrically large structures with high-phase accuracy," IEEE Trans. Antennas Propag. 45, 254-264 (1997).
[CrossRef]

K. H. Dridi, J. S. Hesthaven, and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propag. 49, 749-756 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, "A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Trans. Microwave Theory Tech. 49, 1640-1648 (2001).
[CrossRef]

J. Comput. Phys. (4)

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

A. Ditkowski, K. Dridi, and J. S. Hesthaven, "Convergent Cartesian grid methods for Maxwell's equations in complex geometries," J. Comput. Phys. 170, 39-80 (2001).
[CrossRef]

W. Cai and S. Deng, "An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case," J. Comput. Phys. 190, 159-183 (2003).
[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. (5)

S. M. Lee, W. C. Chew, M. Moghaddam, M. A. Nasir, S-L. Chuang, R. W. Herrick, and C. L. Balestra. "Modeling of rough-surface effects in an optical turning mirror using the finite-difference time-domain method," J. Lightwave Technol. 9, 1471-1480 (1991).
[CrossRef]

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

J. C. Chen, A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightwave Technol. 14, 2575-2580 (1996).
[CrossRef]

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, "High-density integrated optics," J. Lightwave Technol. 17, 1682-1692 (1999).
[CrossRef]

N. Feng, G. Zhou, and W. P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Rev. B (1)

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. Hirono, Y. Yoshikuni, and Y. Shibata, "Third-order effective permittivities for the 4th-order FDTD method in the 2D TM polarization case," in Physics and Simulation of Optoelectronic Devices X, P. Blood, M. Osinski, and Y. Arakawa, eds., Proc. SPIE 4646, 630-640 (2002).
[CrossRef]

Wave Motion (1)

C. Zhang and R. J. LeVeque, "The immersed interface method for acoustic wave equations with discontinuous coefficients," Wave Motion 25, 237-263 (1997).
[CrossRef]

Other (9)

M. Celuch-Marcysiak and W. K. Gwarek, "Higher-order modelling of media interfaces for enhanced FDTD analysis of microwave circuits," in Proceedings of the 24th European Microwave Conference (1994), pp. 1530-1535.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974), pp. 7-17.

A. Byers, I. Rumsey, Z. Popovic, and M. Piket-May, "Surface-wave guiding using periodic structures," in IEEE Antennas Propagation Society International Symposium (IEEE, 2000), Vol. 1, pp. 342-345.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

J. Fang, "Time domain finite difference computation for Maxwell's equations," Ph.D. dissertation (University of California at Berkeley, 1989).

T. Deveze, L. Beaulieu, and W. Tabbara, "A fourth-order scheme for the FDTD algorithm applied to Maxwell's equations," in IEEE Antennas Propagation Society International Symposium (IEEE, 1992), pp. 346-349.

D. W. Zingg, "High-order finite-difference methods in computational electromagnetics," in IEEE Antennas Propagation Society International Symposium (IEEE, 1997), pp. 110-113.

E. Turkel and A. Yefet, "Fourth-order method for Maxwell equations on a staggered mesh," in IEEE Antennas Propagation Society International Symposium (IEEE, 1997), pp. 2156-2159.

T. Hirono, W. W. Lui, and S. Seki, "Successful applications of PML-ABC to the symplectic FDTD scheme with 4th-order accuracy in time and space," in IEEE MTT-S International Microwave Symposium Digest (IEEE, 1999), pp. 1293-1296.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the analyzed xy plane for the third-order effective permittivities in the 2D TE z polarization case.

Fig. 2
Fig. 2

Offset ratio range for methods EP-I and EP-II in the numerical examples.

Fig. 3
Fig. 3

Stability reduction S r of the effective permittivities for (a) EP-I and (b) EP-II.

Fig. 4
Fig. 4

Proposed technique for effective permittivity assignment at corners for method EP-I. In this example, ε 3 is assumed to be larger than ε 2 . The superscript ( i j ) means the effective permittivity for the interface between the regions with permittivity ε i and ε j . Subscript 2nd means the second-order effective permittivity.

Fig. 5
Fig. 5

Phase velocity errors of the TM modes of a 2D step-profile waveguide. The offset ratio of one core-cladding interface is 0. EP-I is used for the interface. The offset ratio and the applied effective permittivity method at the other interface are listed in the legend. The second-order effective permittivities were used for the standard FDTD method.

Fig. 6
Fig. 6

H z component of the TM5 mode at a waveguide section after 200   μm propagation by the (4, 4) scheme with EP-I. The grid resolution was ten cells per wavelength in the core region. The offset ratio of the core-cladding interface at x = 0.75   μm was 0.0 and that of the other interface at x = 0.75   μm was 0.5 .

Fig. 7
Fig. 7

Simulated reflectivity error at a waveguide facet as a function of the cell number per wavelength in the core region. Here, d w is the offset ratio at the core-cladding interfaces, and d f is the offset ratio at the facet. The second-order effective permittivities were used for the standard FDTD method.

Fig. 8
Fig. 8

Simulated reflectivity at a coated facet as a function of the cell number per wavelength in the core region. The second-order effective permittivities were used for the standard FDTD method. (a) Thicknesses were 0.118   μm for the first coating and 0 .189   μm for the second. (b) Thicknesses were 0 .1133   μm for the first coating and 0.1911 μm for the second.

Fig. 9
Fig. 9

Simulated cross talk at a waveguide crossing as a function of the cell number per wavelength in the core region. The second-order effective permittivities were used for the standard FDTD method. (a) Refractive indexes were 3.6 in the core and 2.0 in the cladding. The wavelength in vacuum was 1 .2   μm . (b) The refractive indexes were 4.0 in the core and 1.0 in the cladding. The wavelength in vacuum was 1 .55   μm .

Equations (85)

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ε E x t = H z y ,
ε E y t = H z x ,
μ   H z t = ( E y x E x y ) ,
( D 4 , x f ) i = 27 ( f i + 1 / 2 f i 1 / 2 ) ( f i + 3 / 2 f i 3 / 2 ) 24 Δ x
E x ( n Δ t , i Δ x , j Δ y ) = exp ( j n Δ t ω ) E x s ( i Δ x , j Δ y ) ,
E y ( n Δ t , i Δ x , j Δ y ) = exp ( j n Δ t ω ) E y s ( i Δ x , j Δ y ) ,
H z ( n Δ t , i Δ x , j Δ y ) = exp ( j n Δ t ω ) H z s ( i Δ x , j Δ y ) ,
j ε ω E x s = D 4 , y H z s + O ( Δ x 4 ) ,
j ε ω E y s = D 4 , x H z s + O ( Δ x 4 ) ,
j μ ω H z s = ( D 4 , x E y s D 4 , y E x s ) + O ( Δ x 4 ) .
E x s ( i Δ x , j Δ y ) = exp ( j k y j Δ y ) E x ( i Δ x ) ,
E y s ( i Δ x , j Δ y ) = exp ( j k y j Δ y ) E y ( i Δ x ) ,
H z s ( i Δ x , j Δ y ) = exp ( j k y j Δ y ) H z ( i Δ x ) .
   E ( i Δ x ) = E y ( i Δ x ) / C ,
H ( i Δ x ) = H z ( i Δ x ) ,
C = 1 K y ( k y ) 2 ε μ ω 2 ,
K y ( k y ) = j   27 ( e j k y Δ y / 2 e j k y Δ y / 2 ) ( e 3 j k y Δ y / 2 e 3 j k y Δ y / 2 ) 24 Δ y .
j ε C ω E ( i Δ x ) = D 4 , x H ( i Δ x ) + O ( Δ x 4 ) ,
j μ C ω H ( i Δ x ) = D 4 , x E ( i Δ x ) + O ( Δ x 4 ) .
E ( i Δ x ) = E 0 exp ( j k x i Δ x ) ,
H ( i Δ x ) = H 0 exp ( j k x i Δ x ) ,
( u 2 ) ( u 26 ) 2 + ( 24 K x Δ x ) 2 = 0 ,
u = e j k x Δ x + e j k x Δ x ,
K x = ε μ C ω .
j ε 1 C 1 ω E 1 ( i Δ x ) = D 4 , x H 1 ( i Δ x ) + O ( Δ x 4 ) ,
j μ C 1 ω H 1 ( i Δ x ) = D 4 , x E 1 ( i Δ x ) + O ( Δ x 4 ) ,
C 1 = 1 K y ( k y ) 2 ε 1 μω 2 .
E 1 ( i ) = E 1 , F ( i d ) + R E 1 , B ( i d ) + A 1 , p p , a E 1 , p p , a ( i d ) + A 1 , p p , b E 1 , p p , b ( i d ) ,
H 1 ( i ) = H 1 , F ( i d ) + R H 1 , B ( i d ) + A 1 , p p , a H 1 , p p , a ( i d ) + A 1 , p p , b H 1 , p p , b ( i d ) ,
j ε 2 C 2 ω E 2 ( i Δ x ) = D 4 , x H 2 ( i Δ x ) + O ( Δ x 4 ) ,
j μ C 2 ω H 2 ( i Δ x ) = D 4 , x E 2 ( i Δ x ) + O ( Δ x 4 ) ,
C 2 = 1 K y ( k y ) 2 ε 2 μω 2 .
E 2 ( i ) = T E 2 , F ( i d ) + A 2 , p n , a E 2 , p n , a ( i d ) + A 2 , p n , b E 2 , p n , b ( i d ) ,
H 2 ( i ) = T H 2 , F ( i d ) + A 2 , p n , a H 2 , p n , a ( i d ) + A 2 , p n , b H 2 , p n , b ( i d ) ,
  ( 1 ε p , 3 / 2 1 ε 1 ) { 27 ( H 1 ( 1 ) H 1 ( 2 ) ) + H 1 ( 3 ) } H c ε p , 3 / 2 + H 1 ( 0 ) ε 1 = O ( Δ x 4 ) ,
( ε o , 1 ε 1 ) ( 1 C 1 2 ) { 27 ( E 1 ( 1 2 ) E 1 ( 3 2 ) ) E 1 ( 1 2 ) + E 1 ( 5 2 ) } + ε o , 1 C 1 ( C 2 E 2 ( 1 2 ) C 1 E 1 ( 1 2 ) ) = O ( Δ x 4 ) ,
( 1 ε p , 1 / 2 1 ε 1 ) ( 27 H 1 ( 1 ) + H 1 ( 2 ) ) H 2 ( 1 ) ε p , 1 / 2 + H 1 ( 1 ) ε 1 + 27 ( H c ε p , 1 / 2 H 1 ( 0 ) ε 1 ) = O ( Δ x 4 ) ,
( 1 ε p , 1 / 2 1 ε 2 ) ( 27 H 2 ( 1 ) H 2 ( 2 ) ) + H 1 ( 1 ) ε p , 1 / 2 H 2 ( 1 ) ε 2 + 27 ( H c ε p , 1 / 2 + H 2 ( 0 ) ε 2 ) = O ( Δ x 4 ) ,
( ε o , 1 ε 2 ) ( 1 C 2 2 ) { 27 ( E 2 ( 3 2 ) E 2 ( 1 2 ) ) E 2 ( 5 2 ) + E 2 ( 1 2 ) } + ε o , 1 C 2 ( C 2 E 2 ( 1 2 ) C 1 E 1 ( 1 2 ) ) = O ( Δ x 4 ) ,
( 1 ε p , 3 / 2 1 ε 2 ) { 27 ( H 2 ( 2 ) H 2 ( 1 ) ) H 2 ( 3 ) } + H c ε p , 3 / 2 H 2 ( 0 ) ε 2 = O ( Δ x 4 ) ,
  H c = j 24 μ ω Δ x ε o , 0 ε o , 0 ε 1 ( 1 C 1 2 ) { 27 ( C 2 E 2 ( 1 2 ) C 1 E 1 ( 1 2 ) ) C 2 E 2 ( 3 2 ) + C 1 E 1 ( 3 2 ) } .
R = cos θ i ε 1 cos θ t ε 2 cos θ i ε 1 + cos θ t ε 2 + R 1 ( K 0 Δ x ) + R 2 ( K 0 Δ x ) 2 + R 3 ( K 0 Δ x ) 3 + O ( Δ x 4 ) ,
T = 2 cos θ i ε 1 cos θ i ε 1 + cos θ t ε 2 + T 1 ( K 0 Δ x ) + T 2 ( K 0 Δ x ) 2 + T 3 ( K 0 Δ x ) 3 + O ( Δ x 4 ) ,
K 0 = ε 0 μ ω .
ε p , 3 / 2 + ε p , 1 / 2 + ε p , 1 / 2 + ε p , 3 / 2 4
= ( 1 + d / 2 ) ε 1 + ( 1 d / 2 ) ε 2 2 ,
ε o , 1 1 + ε o , 0 1 + ε o , 1 1 3
      = ( 1 + 2 d / 3 ) ε 1 1 + ( 1 2 d / 3 ) ε 2 1 2 .
12 ( ε p , 1 / 2 ε p , 1 / 2 ) + 36 ( ε p , 3 / 2 ε p , 3 / 2 ) ( 47 12 d 2 ) × ( ε 2 ε 1 ) = 0 ,
8 ε 1 ε 2 { ε o , 1 1 ( a 1 ε p , 1 / 2 + a 2 ε p , 1 / 2 + a 3 ) + ε o , 1 1 ( a 4 ε p , 1 / 2 + a 5 ε p , 1 / 2 + a 6 ) } + 8 a 7 ε p , 1 / 2 + 8 a 8 ε p , 1 / 2 + a 9 = 0 ,
ε p , 1 / 2 = ε 1 + 1 24 ( ε 2 ε 1 ) ( 1 12 d + 12 d 2 ) ,
ε p , 1 / 2 = ε 2 + 1 24 ( ε 1 ε 2 ) ( 1 + 12 d + 12 d 2 ) ,
ε o , 1 = L M ε 1 ,
ε o , 0 = 1 ( 3 2 + d ) / ε 1 + ( 1 2 d ) / ε 2 1 / ε o , 1 ,
L = 24 ( 6 + 42 ) ε 2 { 6 ( ε 1 ε 2 ) d 2 + 42 ( ε 2 ε 1 ) d ε 2 ( 6 + 42 ) ε 1 } ,
M = 72 ( ε 2 ε 1 ) 2 d 3 + { ( 864 138 42 ) ε 2 + 6 42 ε 1 } ( ε 2 ε 1 ) d 2 + { ( 1038 144 42 ) ε 2 30 ε 1 } ( ε 2 ε 1 ) d + ( 144 25 42 ) ε 2 2 144 ε 1 ε 2 + 42 ε 1 2 .
ε p , 1 / 2 = ε 1 + 1 24 ( ε 2 ε 1 ) ( 25 36 d + 12 d 2 ) ,
ε p , 3 / 2 = ε 1 + 1 24 ( ε 2 ε 1 ) ( 23 + 12 d 12 d 2 ) ,
ε o , 1 = N O ε 1 ,
ε o , 0 = 1 ( 3 2 + d ) / ε 1 + ( 1 2 d ) / ε 2 1 / ε o , 1 ,
N = 24 ( 6 + 42 ) ε 2 { 6 ( ε 1 ε 2 ) d 2 + ( 12 + 42 ) ( ε 2 ε 1 ) d ( 7 + 42 ) ε 2 ( 168 + 26 42 ) ε 1 } ,
O = 72 ( ε 2 ε 1 ) 2 d 3 + { ( 1008 138 42 ) ε 2 + ( 864 + 150 42 ) ε 1 } ( ε 2 ε 1 ) d 2 + { ( 762 + 144 42 ) ε 2 + 42 ε 1 } ( ε 2 ε 1 ) d ( 42 + 31 42 ) ε 2 2 ( 2100 + 300 42 ) ε 1 ε 2 + ( 126 + 19 42 ) ε 1 2 .
R e s = ε eff , min / ε 1 ( ε eff , min < ε 1 ) = 1 ( ε eff , min ε 1 ) .
S r = 1 R e s .
k x = K x + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
k x = K x + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
k x Δ x = ln ( 13 + 2 42 ) j + 1 2 7 K x Δ x + j   9 3 112 14 ( K x Δ x ) 2 45 1568 7 ( K x Δ x ) 3 + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
k x Δ x = ln ( 13 + 2 42 ) j 1 2 7 K x Δ x + j   9 3 112 14 ( K x Δ x ) 2 + 45 1568 7 ( K x Δ x ) 3 + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
k x Δ x = ln ( 13 + 2 42 ) j + 1 2 7 K x Δ x j   9 3 112 14 ( K x Δ x ) 2 45 1568 7 ( K x Δ x ) 3 + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
k x Δ x = ln ( 13 + 2 42 ) j 1 2 7 K x Δ x j   9 3 112 14 ( K x Δ x ) 2 + 45 1568 7 ( K x Δ x ) 3 + O ( Δ x 4 ) ,
H 0 E 0 = ε μ .
a 1 = 24 ( 20 + 3 42 ) ,
a 2 = 24 ,
a 3 = 6 ( ε 1 ε 2 ) d 2 + 3 42 ( ε 1 ε 2 ) d + ( 25 + 6 42 ) ε 1 ( 4 3 42 ) ε 2 ,
a 4 = 24 ,
a 5 = 24 ( 20 + 3 42 ) ,
a 6 = 6 ( ε 1 ε 2 ) d 2 3 42 ( ε 1 ε 2 ) d + ( 4 3 42 ) ε 1 ( 25 + 6 42 ) ε 2 ,
a 7 = ( 156 + 24 42 ) ( ε 1 ε 2 ) d ( 228 + 39 42 ) ε 1 ( 228 + 33 42 ) ε 2 ,
a 8 = ( 156 + 24 42 ) ( ε 1 ε 2 ) d + ( 228 + 33 42 ) ε 1 + ( 228 + 39 42 ) ε 2 ,
a 9 = { ( 696 + 108 42 ) ( ε 1 ε 2 ) d 3 ( 1002 + 162 42 ) × ( ε 1 + ε 2 ) d 2 + ( 1682 + 285 42 ) ( ε 1 ε 2 ) d + ( 2159 + 363 42 ) ( ε 1 + ε 2 ) } ( ε 1 ε 2 ) .

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