Abstract

An integral equation method employing complex images Green's functions is developed for analyzing different devices fabricated in 2-D dielectric photonic crystals. The integral equation is written in terms of the unknown equivalent current sources flowing on the surfaces of the periodic 2-D cylinders. The method of moments is then employed to solve for the unknown current distributions. The required Green's function of the problem is represented in terms of a finite summation of complex images instead of the conventional slowly converging infinite series. It is shown that when the field-point is far from the periodic sources, it is just sufficient to consider the contribution of the propagating poles in the structure. This will result in a summation of plane waves that has an even smaller size compared with the conventional complex images Green's function. This will enable us to analyze the dielectric periodic structures efficiently and accurately. The method is applied to a number of waveguide structures and its results are compared with the existing literature.

© 2010 IEEE

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
  3. A. Scherer, T. Doll, E. Yablonovitch, H. O. Everitt, J. A. Higgins, "Guest Editorial: Electromagnetic crystal structures, design, synthesis, and applications," J. Lightw. Technol. 17, 1928-1930 (1999).
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  5. S. Shi, C. Chen, D. W. Prather, "Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers," J. Opt. Soc. Amer. A. 21, 1769-1775 (2004).
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  7. H. Yu, D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Tech. 44, 2688-2695 (1996).
  8. C. Mias, J. P. Webb, R. L. Ferrari, "Finite element modeling of electromagnetic waves in doubly and triply periodic structures," IEE Proc. Optoelectron. 146, 111-118 (1999).
  9. P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E. 75, 026703-1-026703-14 (2007).
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  13. R. Lampe, P. Klock, P. Mayes, "Integral transforms useful for the accelerated summation of periodic free-space Green's functions," IEEE Trans. Microw. Theory Tech. MTT-33, 734-736 (1985).
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  15. H. Alaeian, R. Faraji-Dana, "A fast and accurate analysis of 2-D periodic devices using complex images Green's functions," IEEE/OSA J. Lightw. Technol. 27, 2216-2223 (2009).
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  18. Y. Hua, T. K. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antennas Propag. 37, 229-234 (1989).
  19. Y. Mengtao, T. K. Sarkar, M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microw. Theory Tech. 54, 1025-1032 (2006).
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  21. R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  22. A. Mekis, S. Fan, J. D. Joannopoulos, "Bound state in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).
  23. J. D. Joannopolous, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals, Molding the Flow of Light (Princeton Univ. Press, 2008).
  24. S. Boscolo, M. Midrio, C. G. Someda, "Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides," IEEE J. Quantum Electron. 38, 47-52 (2002).

2009 (1)

H. Alaeian, R. Faraji-Dana, "A fast and accurate analysis of 2-D periodic devices using complex images Green's functions," IEEE/OSA J. Lightw. Technol. 27, 2216-2223 (2009).

2007 (1)

P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E. 75, 026703-1-026703-14 (2007).

2006 (1)

Y. Mengtao, T. K. Sarkar, M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microw. Theory Tech. 54, 1025-1032 (2006).

2004 (2)

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, H. Y. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Expr. 12, 1551-1561 (2004).

S. Shi, C. Chen, D. W. Prather, "Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers," J. Opt. Soc. Amer. A. 21, 1769-1775 (2004).

2002 (1)

S. Boscolo, M. Midrio, C. G. Someda, "Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides," IEEE J. Quantum Electron. 38, 47-52 (2002).

2000 (1)

M. Qiu, S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B. 61, 12872-12876 (2000).

1999 (2)

A. Scherer, T. Doll, E. Yablonovitch, H. O. Everitt, J. A. Higgins, "Guest Editorial: Electromagnetic crystal structures, design, synthesis, and applications," J. Lightw. Technol. 17, 1928-1930 (1999).

C. Mias, J. P. Webb, R. L. Ferrari, "Finite element modeling of electromagnetic waves in doubly and triply periodic structures," IEE Proc. Optoelectron. 146, 111-118 (1999).

1998 (2)

A. Mekis, S. Fan, J. D. Joannopoulos, "Bound state in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).

M. G. Khazinsky, A. R. McGurn, "Green's function method for waveguide and single impurity methods in 2-D photonic crystals: H-polarization," Phys. Lett. 237, 175-182 (1998).

1996 (2)

H. Yu, D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Tech. 44, 2688-2695 (1996).

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

1994 (1)

W. M. Robertson, S. A. Boothroyd, L. Chan, "Photonic band structure calculations using a two dimensional electromagnetic simulator," J. Mod. Opt. 41, 285-293 (1994).

1991 (1)

Y. L. Chow, J. J. Yang, D. G. Fang, G. E. Howard, "A closed-form spatial Green's function for the thick microstrip substrate," IEEE Trans. Microw. Theory Tech. 39, 588-592 (1991).

1989 (1)

Y. Hua, T. K. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antennas Propag. 37, 229-234 (1989).

1987 (2)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).

1985 (1)

R. Lampe, P. Klock, P. Mayes, "Integral transforms useful for the accelerated summation of periodic free-space Green's functions," IEEE Trans. Microw. Theory Tech. MTT-33, 734-736 (1985).

IEE Proc. Optoelectron. (1)

C. Mias, J. P. Webb, R. L. Ferrari, "Finite element modeling of electromagnetic waves in doubly and triply periodic structures," IEE Proc. Optoelectron. 146, 111-118 (1999).

IEEE J. Quantum Electron. (1)

S. Boscolo, M. Midrio, C. G. Someda, "Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides," IEEE J. Quantum Electron. 38, 47-52 (2002).

IEEE Trans. Antennas Propag. (1)

Y. Hua, T. K. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antennas Propag. 37, 229-234 (1989).

IEEE Trans. Microw. Theory Tech. (4)

Y. Mengtao, T. K. Sarkar, M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microw. Theory Tech. 54, 1025-1032 (2006).

Y. L. Chow, J. J. Yang, D. G. Fang, G. E. Howard, "A closed-form spatial Green's function for the thick microstrip substrate," IEEE Trans. Microw. Theory Tech. 39, 588-592 (1991).

H. Yu, D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microw. Theory Tech. 44, 2688-2695 (1996).

R. Lampe, P. Klock, P. Mayes, "Integral transforms useful for the accelerated summation of periodic free-space Green's functions," IEEE Trans. Microw. Theory Tech. MTT-33, 734-736 (1985).

IEEE/OSA J. Lightw. Technol. (1)

H. Alaeian, R. Faraji-Dana, "A fast and accurate analysis of 2-D periodic devices using complex images Green's functions," IEEE/OSA J. Lightw. Technol. 27, 2216-2223 (2009).

J. Lightw. Technol. (1)

A. Scherer, T. Doll, E. Yablonovitch, H. O. Everitt, J. A. Higgins, "Guest Editorial: Electromagnetic crystal structures, design, synthesis, and applications," J. Lightw. Technol. 17, 1928-1930 (1999).

J. Mod. Opt. (1)

W. M. Robertson, S. A. Boothroyd, L. Chan, "Photonic band structure calculations using a two dimensional electromagnetic simulator," J. Mod. Opt. 41, 285-293 (1994).

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

S. Shi, C. Chen, D. W. Prather, "Plane wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers," J. Opt. Soc. Amer. A. 21, 1769-1775 (2004).

Opt. Expr. (1)

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, H. Y. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Expr. 12, 1551-1561 (2004).

Phys. Lett. (1)

M. G. Khazinsky, A. R. McGurn, "Green's function method for waveguide and single impurity methods in 2-D photonic crystals: H-polarization," Phys. Lett. 237, 175-182 (1998).

Phys. Rev. B (1)

A. Mekis, S. Fan, J. D. Joannopoulos, "Bound state in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).

Phys. Rev. B. (1)

M. Qiu, S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B. 61, 12872-12876 (2000).

Phys. Rev. E. (1)

P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E. 75, 026703-1-026703-14 (2007).

Phys. Rev. Lett. (3)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).

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

Other (5)

R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).

W. F. Richards, J. R. Zinecker, D. R. Wilton, S. Singh, Y. T. Lo, S. M. Wright, "Acceleration of periodic Green's function in free space," URSI Symp. Proc. (1983) pp. 23-26.

J. D. Joannopolous, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals, Molding the Flow of Light (Princeton Univ. Press, 2008).

J. R. Mosig, F. E. Gardiol, Advances in Electronics and Electron Physics (Academic, 1982) pp. 139-239.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

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