Abstract

The broadband Green’s function with low wavenumber extraction (BBGFL) is applied to the calculations of band diagrams of two-dimensional (2D) periodic structures with dielectric scatterers. Periodic Green’s functions of both the background and the scatterers are used to formulate the dual surface integral equations by approaching the surface of the scatterer from outside and inside the scatterer. The BBGFL are applied to both periodic Green’s functions. By subtracting a low wavenumber component of the periodic Green’s functions, the broadband part of the Green’s functions converge with a small number of Bloch waves. The method of Moment (MoM) is applied to convert the surface integral equations to a matrix eigenvalue problem. Using the BBGFL, a linear eigenvalue problem is obtained with all the eigenmodes computed simultaneously giving the multiband results at a point in the Brillouin zone Numerical results are illustrated for the honeycomb structure. The results of the band diagrams are in good agreement with the planewave method and the Korringa Kohn Rostoker (KKR) method. By using the lowest band around the Γ point, the low frequency dispersion relations are calculated which also give the effective propagation constants and the effective permittivity in the low frequency limit.

© 2016 Optical Society of America

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References

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  1. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 16, 3152–3155 (1990).
    [Crossref]
  2. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [Crossref] [PubMed]
  3. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44 (16), 8565–8571 (1991).
    [Crossref]
  4. R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
    [Crossref]
  5. M. Kafesaki and CM Soukoulis, “Historical perspective and review of fundamental principles in modelling three-dimensional periodic structures with emphasis on volumetric EBGs,” in Metamaterials, ed by N Engheta and RW Ziolkowski, eds. (John Wiley and Sons, 2006), Chap. 8.
    [Crossref]
  6. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University, 2011).
  7. H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
    [Crossref]
  8. J Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13(6), 392–400 (1947).
    [Crossref]
  9. W Kohn and N. Rostoker, “Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium,” Phys Rev. 94, 1111–1120 (1954).
    [Crossref]
  10. K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
    [Crossref]
  11. Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
    [Crossref]
  12. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
    [Crossref]
  13. R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electr. 31, 843–855 (1999).
    [Crossref]
  14. B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
    [Crossref]
  15. J.-M. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays, (Wiley, 2009)
  16. M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
    [Crossref]
  17. L. Tsang and S. Huang, “Full Wave Modeling and Simulations of The Waveguide Behavior of Printed Circuit Boards Using A Broadband Green’s Function Technique,” Provisional U.S. Patent No. 62/152.702 (April242015).
  18. S. Huang, “Broadband Green’s Function and Applications to Fast Electromagnetic Analysis of High-Speed Interconnects,” Ph.D. dissertation, Dept. Elect. Eng., Univ.Washington, Seattle, WA (2015).
  19. S. Huang and L. Tsang, “Broadband Green’s Function and Applications to Fast Electromagnetic Modeling of High Speed Interconnects,” IEEE International Symposium on Antennas and Propagation, Vancouver, BC, Canada (2015).
  20. L. Tsang and S. Huang, “Broadband Green’s Function with Low Wavenumber Extraction for Arbitrary Shaped Waveguide with Applications to Modeling of Vias in Finite Power/Ground Plane,” Prog. Electromag. Res. 152, 105–125 (2015).
    [Crossref]
  21. L. Tsang, “Broadband Calculations of Band Diagrams in Periodic Structures Using the Broadband Green’s Function with Low Wavenumber Extraction (BBGFL),” Prog. Electromag. Res. 153, 57–68 (2015).
    [Crossref]
  22. P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).
  23. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations (Wiley Interscience, 2001).
  24. L. Tsang and J. A. Kong, “Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formulism,” J. Appl. Phys. 51(7), 3465–3485 (1980).
    [Crossref]
  25. L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16(3), 303–320 (1981).
    [Crossref]
  26. L. Tsang, J. A. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).
  27. A. H. Sihvola, Electromagnetic mixing formulas and applications, (IET, 1999).
    [Crossref]
  28. A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
    [Crossref]
  29. J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
    [Crossref]

2015 (2)

L. Tsang and S. Huang, “Broadband Green’s Function with Low Wavenumber Extraction for Arbitrary Shaped Waveguide with Applications to Modeling of Vias in Finite Power/Ground Plane,” Prog. Electromag. Res. 152, 105–125 (2015).
[Crossref]

L. Tsang, “Broadband Calculations of Band Diagrams in Periodic Structures Using the Broadband Green’s Function with Low Wavenumber Extraction (BBGFL),” Prog. Electromag. Res. 153, 57–68 (2015).
[Crossref]

2009 (1)

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[Crossref]

2003 (1)

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[Crossref]

2002 (1)

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

2000 (1)

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

1999 (1)

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electr. 31, 843–855 (1999).
[Crossref]

1998 (1)

J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
[Crossref]

1996 (1)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

1993 (1)

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

1992 (2)

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44 (16), 8565–8571 (1991).
[Crossref]

1990 (2)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 16, 3152–3155 (1990).
[Crossref]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[Crossref] [PubMed]

1981 (1)

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16(3), 303–320 (1981).
[Crossref]

1980 (1)

L. Tsang and J. A. Kong, “Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formulism,” J. Appl. Phys. 51(7), 3465–3485 (1980).
[Crossref]

1954 (1)

W Kohn and N. Rostoker, “Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium,” Phys Rev. 94, 1111–1120 (1954).
[Crossref]

1947 (1)

J Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13(6), 392–400 (1947).
[Crossref]

Ao, C. O.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations (Wiley Interscience, 2001).

Arcioni, P.

P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).

Beckett, D. H.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

Bozzi, M.

P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).

Bressan, M.

P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).

Brommer, K. D.

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[Crossref]

Chan, C. T.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 16, 3152–3155 (1990).
[Crossref]

Conciauro, G.

P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).

Cox, S. J.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

Ding, K. H.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations (Wiley Interscience, 2001).

Fan, S.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

Generowicz, J. M.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

Goertzen, A. L.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

Haus, J. W.

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Hiett, B. P.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 16, 3152–3155 (1990).
[Crossref]

Holden, J. A.

J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
[Crossref]

Huang, S.

L. Tsang and S. Huang, “Broadband Green’s Function with Low Wavenumber Extraction for Arbitrary Shaped Waveguide with Applications to Modeling of Vias in Finite Power/Ground Plane,” Prog. Electromag. Res. 152, 105–125 (2015).
[Crossref]

L. Tsang and S. Huang, “Full Wave Modeling and Simulations of The Waveguide Behavior of Printed Circuit Boards Using A Broadband Green’s Function Technique,” Provisional U.S. Patent No. 62/152.702 (April242015).

S. Huang, “Broadband Green’s Function and Applications to Fast Electromagnetic Analysis of High-Speed Interconnects,” Ph.D. dissertation, Dept. Elect. Eng., Univ.Washington, Seattle, WA (2015).

S. Huang and L. Tsang, “Broadband Green’s Function and Applications to Fast Electromagnetic Modeling of High Speed Interconnects,” IEEE International Symposium on Antennas and Propagation, Vancouver, BC, Canada (2015).

Inguva, R.

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Ishimaru, A.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[Crossref]

Jandhyala, V.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[Crossref]

Jin, J.-M.

J.-M. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays, (Wiley, 2009)

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University, 2011).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University, 2011).

Kafesaki, M.

M. Kafesaki and CM Soukoulis, “Historical perspective and review of fundamental principles in modelling three-dimensional periodic structures with emphasis on volumetric EBGs,” in Metamaterials, ed by N Engheta and RW Ziolkowski, eds. (John Wiley and Sons, 2006), Chap. 8.
[Crossref]

Kohn, W

W Kohn and N. Rostoker, “Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium,” Phys Rev. 94, 1111–1120 (1954).
[Crossref]

Kong, J. A.

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16(3), 303–320 (1981).
[Crossref]

L. Tsang and J. A. Kong, “Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formulism,” J. Appl. Phys. 51(7), 3465–3485 (1980).
[Crossref]

L. Tsang, J. A. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations (Wiley Interscience, 2001).

Korringa, J

J Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica 13(6), 392–400 (1947).
[Crossref]

Kuga, Y.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[Crossref]

Lee, S. W.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[Crossref]

Leung, K. M.

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[Crossref] [PubMed]

Li, Z.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[Crossref]

Liu, Q. H.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[Crossref]

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[Crossref] [PubMed]

Liu, Z.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

Luo, M.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[Crossref]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44 (16), 8565–8571 (1991).
[Crossref]

Mead, R. D.

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University, 2011).

Molinari, M.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,”, IEE Proc-Sci Meas. Technol. 149, 293–296 (2002).
[Crossref]

Page, J. H.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

Pendry, J. B.

J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
[Crossref]

Perregrini, L.

P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “The BI-RME method: An historical overview,” in 2014 International Conference on Numerical Electromagnetic Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), 1–4 (2014).

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44 (16), 8565–8571 (1991).
[Crossref]

Qiu, Y.

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48(11), 7767–7771 (1993).
[Crossref]

Rappe, A. M.

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic bandgap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).
[Crossref]

Riley, D. J.

J.-M. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays, (Wiley, 2009)

Robbins, J. D.

J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
[Crossref]

Rostoker, N.

W Kohn and N. Rostoker, “Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium,” Phys Rev. 94, 1111–1120 (1954).
[Crossref]

Sheng, P.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, “Elastic wave scattering by periodic structures of spherical objects: Theory and experiment,” Phys. Rev. B 62, 2446–2457 (2000).
[Crossref]

Shin, R.

L. Tsang, J. A. Kong, and R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Sihvola, A. H.

A. H. Sihvola, Electromagnetic mixing formulas and applications, (IET, 1999).
[Crossref]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 16, 3152–3155 (1990).
[Crossref]

Soukoulis, CM

M. Kafesaki and CM Soukoulis, “Historical perspective and review of fundamental principles in modelling three-dimensional periodic structures with emphasis on volumetric EBGs,” in Metamaterials, ed by N Engheta and RW Ziolkowski, eds. (John Wiley and Sons, 2006), Chap. 8.
[Crossref]

Sozuer, H. S.

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45(24), 13962–13972 (1992).
[Crossref]

Stewart, J. W.

J. B. Pendry, J. A. Holden, J. D. Robbins, and J. W. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Mat. 10, 4785–4809 (1998).
[Crossref]

Tanaka, M.

R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches,” Opt. Quantum Electr. 31, 843–855 (1999).
[Crossref]

Thomas, K. S.

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S. Huang and L. Tsang, “Broadband Green’s Function and Applications to Fast Electromagnetic Modeling of High Speed Interconnects,” IEEE International Symposium on Antennas and Propagation, Vancouver, BC, Canada (2015).

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[Crossref]

L. Tsang, “Broadband Calculations of Band Diagrams in Periodic Structures Using the Broadband Green’s Function with Low Wavenumber Extraction (BBGFL),” Prog. Electromag. Res. 153, 57–68 (2015).
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J.-M. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays, (Wiley, 2009)

L. Tsang and S. Huang, “Full Wave Modeling and Simulations of The Waveguide Behavior of Printed Circuit Boards Using A Broadband Green’s Function Technique,” Provisional U.S. Patent No. 62/152.702 (April242015).

S. Huang, “Broadband Green’s Function and Applications to Fast Electromagnetic Analysis of High-Speed Interconnects,” Ph.D. dissertation, Dept. Elect. Eng., Univ.Washington, Seattle, WA (2015).

S. Huang and L. Tsang, “Broadband Green’s Function and Applications to Fast Electromagnetic Modeling of High Speed Interconnects,” IEEE International Symposium on Antennas and Propagation, Vancouver, BC, Canada (2015).

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Figures (8)

Fig. 1
Fig. 1 Geometry of the 2D scattering problem in 2D lattice. 2D dielectric scatterers form a periodic array with primitive lattice vectors ā1 and ā2, and lattice constant a. The primitive cell area Ω0 = |ā1 × ā2|. The (0, 0)-th scatterer has boundary S00 and cross section A00. The scatterer and background has dielectric constant εrp and εrb, respectively.
Fig. 2
Fig. 2 Band diagram of the hexagonal structure with background dielectric constant of 8.9 and air voids of radius b = 0.2a. The results of BBGFL are shown by the solid curves for TMz polarization and by the dashed curves for the TEz polarization. The circles and crosses are the results of planewave method for the TMz and TEz polarizations, respectively.
Fig. 3
Fig. 3 Modal surface currents distribution near Γ point at k ¯ i = 0.05 b ¯ 1 corresponding to the first few modes of the hexagonal structure with background dielectric constant of 8.9 and air voids of radius b = 0.2a. (a) TMz, surface electric currents J z ψ / n; (b) TMz, surface magnetic current Mtψ; (c) TEz, surface magnetic currents M z ψ / n; (d) TEz, surface electric current Jtψ. The corresponding normalized mode frequencies are 1: 0.0208, 2: 0.3736, 3: 0.3864 for TMz wave, and 1: 0.02224, 2: 0.3847, 3: 0.404 for TEz wave, respectively.
Fig. 4
Fig. 4 Band diagram of the hexagonal structure with background dielectric constant of 12.25 and air voids of radius b = 0.48a. The results of BBGFL are shown by the solid curves for TMz polarization and by the dashed curves for the TEz polarization. The circles and crosses are the results of planewave method for the TMz and TEz polarizations, respectively. The triangles and squares are the results of the KKR method for the TMz and TEz polarizations, respectively.
Fig. 5
Fig. 5 Modal surface currents distribution near Γ point at k ¯ i = 0.05 b ¯ 1 corresponding to the first few modes of the hexagonal structure with background dielectric constant of 12.25 and air voids of radius b = 0.48a. (a) TMz, surface electric currents J z ψ / n; (b) TMz, surface magnetic current Mtψ; (c) TEz, surface magnetic currents M z ψ / n; (d) TEz, surface electric current Jtψ. The corresponding normalized mode frequencies are 1: 0.03437, 2: 0.4425, 3: 0.6154 for TMz wave, and 1: 0.04145, 2: 0.7669, 3: 0.7710 for TEz wave, respectively.
Fig. 6
Fig. 6 Dispersion relationship of the hexagonal structure with background dielectric constant of 8.9 and air voids of radius b = 0.2a.
Fig. 7
Fig. 7 Dispersion relationship of the hexagonal structure with background dielectric constant of 12.25 and air voids of radius b = 0.48a.
Fig. 8
Fig. 8 Dispersion relationship of the hexagonal structure with background dielectric constant of 8.9 and PEC cylinders of radius b = 0.2a.

Tables (1)

Tables Icon

Table 1 The convergence of the lowest mode with respect to the number of Bloch waves used in BBGFL using different low wavenumber kL. The results are tabulated for TMz polarization with k ¯ i = 0.05 b ¯ 1, where 0.020798 is the first band solution.

Equations (92)

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p , q S p q d l [ ψ ( ρ ¯ ) n ^ g ( ρ ¯ , ρ ¯ ) g ( ρ ¯ , ρ ¯ ) n ^ ψ ( ρ ¯ ) ] = 0
S p q d l [ ψ 1 ( ρ ¯ ) n ^ g 1 ( ρ ¯ , ρ ¯ ) g 1 ( ρ ¯ , ρ ¯ ) n ^ ψ 1 ( ρ ¯ ) ] = 0
S 00 d l [ ψ ( ρ ¯ ) n ^ g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) n ^ ψ ( ρ ¯ ) ] = 0
S 00 d l [ ψ 1 ( ρ ¯ ) n ^ g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) n ^ ψ 1 ( ρ ¯ ) ] = 0
g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) = 1 Ω 0 m n exp ( i k ¯ i m n ( ρ ¯ ρ ¯ ) ) | k ¯ i m n | 2 k 2
g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) = 1 Ω 0 m n exp ( i k ¯ i m n ( ρ ¯ ρ ¯ ) ) | k ¯ i m n | 2 k 1 2
S 00 d l [ ψ ( ρ ¯ ) n ^ g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) n ^ ψ ( ρ ¯ ) ] = 0 , ρ ¯ S 00
S 00 d l [ ψ ( ρ ¯ ) n ^ g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) 1 s n ^ ψ ( ρ ¯ ) ] = 0 , ρ ¯ S 00 +
S ¯ ¯ p ¯ L ¯ ¯ q ¯ = 0
S ¯ ¯ ( 1 ) p ¯ 1 s L ¯ ¯ ( 1 ) q ¯ = 0
S m n = 1 Δ t n S 00 ( n ) d l n ^ g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) , ρ ¯ ρ ¯ m
L m n = 1 Δ t n S 00 ( n ) d l g P ( k , k ¯ i ; ρ ¯ m , ρ ¯ )
S m n ( 1 ) = 1 Δ t n S 00 ( n ) d l n ^ g 1 P ( k 1 , k ¯ i ; ρ ¯ , ρ ¯ ) , , ρ ¯ ρ ¯ m +
L m n ( 1 ) = 1 Δ t n S 00 ( n ) d l g 1 P ( k 1 , k ¯ i ; ρ ¯ m , ρ ¯ )
p n = Δ t n ψ ( ρ ¯ n )
q n = Δ t n [ n ^ ψ ( ρ ¯ ) ] ρ ¯ = ρ ¯ n
g P ( k , k ¯ i , ρ ¯ , ρ ¯ ) = g P ( k L , k ¯ i , ρ ¯ , ρ ¯ ) + g B ( k , k L , k ¯ i , ρ ¯ , ρ ¯ )
g B ( k , k L , k ¯ i , ρ ¯ , ρ ¯ ) = k 2 k L 2 Ω 0 α exp ( i k ¯ i α ( ρ ¯ ρ ¯ ) ) ( | k ¯ i α | 2 k 2 ) ( | k ¯ i α | 2 k L 2 )
g 1 P ( k 1 , k ¯ i , ρ ¯ , ρ ¯ ) = g 1 P ( k 1 L , k ¯ i , ρ ¯ , ρ ¯ ) + g 1 B ( k 1 , k 1 L , k ¯ i , ρ ¯ , ρ ¯ )
g 1 B ( k 1 , k 1 L , k ¯ i , ρ ¯ , ρ ¯ ) = k 1 2 k 1 L 2 Ω 0 α exp ( i k ¯ i α ( ρ ¯ ρ ¯ ) ) ( | k ¯ i α | 2 k 1 2 ) ( | k ¯ i α | 2 k 1 L 2 )
g B ( k , k L , k ¯ i , ρ ¯ , ρ ¯ ) = 1 Ω 0 α 1 1 k 2 k L 2 1 | k ¯ i α | 2 k L 2 exp ( i k ¯ i α ( ρ ¯ ρ ¯ ) ) ( | k ¯ i α | 2 k L 2 ) 2
α R α ( k L , ρ ¯ ) W α ( k , k L ) R α * ( k L , ρ ¯ )
R α ( k L , ρ ¯ ) = 1 Ω 0 exp ( i k ¯ i α ρ ¯ ) | k ¯ i α | 2 k L 2
W α ( k , k L ) = 1 λ ( k , k L ) D α ( k L )
λ ( k , k L ) = 1 k 2 k L 2
D α ( k L ) = 1 | k ¯ i α | 2 k L 2
n ^ g B ( k , k L , k ¯ i , ρ ¯ , ρ ¯ ) = α R α ( k L , ρ ¯ ) W α ( k , k L ) Q α * ( k L , ρ ¯ )
Q α ( k L , ρ ¯ ) = n ^ R α ( k L , ρ ¯ ) = [ n ^ ( i k ¯ i α ) ] R α ( k L , ρ ¯ )
S ¯ ¯ ( k ) = S ¯ ¯ ( k L ) + R ¯ ¯ ( k L ) W ¯ ¯ ( k L , k ) Q ¯ ¯ ( k L )
L ¯ ¯ ( k ) = L ¯ ¯ ( k L ) + R ¯ ¯ ( k L ) W ¯ ¯ ( k L , k ) R ¯ ¯ ( k L )
W ¯ ¯ ( k L , k ) = ( λ ( k , k L ) I ¯ ¯ D ¯ ¯ ( k L ) ) 1
D α α = D α ( k L )
S ¯ ¯ ( k L ) p ¯ L ¯ ¯ ( k L ) q ¯ + R ¯ ¯ ( k L ) W ¯ ¯ ( k L , k ) Q ¯ ¯ ( k L ) p ¯ R ¯ ¯ ( k L ) W ¯ ¯ ( k L , k ) R ¯ ¯ ( k L ) q ¯ = 0
S ¯ ¯ ( 1 ) ( k 1 L ) p ¯ 1 s L ¯ ¯ ( 1 ) ( k 1 L ) q ¯ + R ¯ ¯ ( 1 ) ( k 1 L ) W ¯ ¯ ( 1 ) ( k 1 L , k 1 ) Q ¯ ¯ ( 1 ) ( k 1 L ) p ¯ 1 s R ¯ ¯ ( 1 ) ( k 1 L ) W ¯ ¯ ( 1 ) ( k 1 L , k 1 ) R ¯ ¯ ( 1 ) ( k 1 L ) q ¯ = 0
k 1 L = k L ε r
λ ( k 1 , k 1 L ) = 1 ε r λ ( k , k L )
W ¯ ¯ ( 1 ) ( k 1 L , k 1 ) = ( 1 ε r λ ( k , k L ) I ¯ ¯ D ¯ ¯ ( 1 ) ( k L ) ) 1
b ¯ = W R ¯ ¯ q ¯
c ¯ = W Q ¯ ¯ p ¯
b ¯ ( 1 ) = W ¯ ¯ ( 1 ) R ¯ ¯ ( 1 ) q ¯
c ¯ ( 1 ) = W ¯ ¯ ( 1 ) Q ¯ ¯ ( 1 ) p ¯
W ¯ ¯ 1 b ¯ = ( λ I ¯ ¯ D ¯ ¯ ) b ¯ = R ¯ ¯ q ¯
W ¯ ¯ 1 c ¯ = ( λ I ¯ ¯ D ¯ ¯ ) c ¯ = Q ¯ ¯ p ¯
λ b ¯ = D ¯ ¯ b ¯ + R ¯ ¯ q ¯
λ c ¯ = D ¯ ¯ c ¯ + Q ¯ ¯ p ¯
λ b ¯ ( 1 ) = ε r D ¯ ¯ ( 1 ) b ¯ ( 1 ) + ε r R ¯ ¯ ( 1 ) q ¯
λ c ¯ ( 1 ) = ε r D ¯ ¯ ( 1 ) c ¯ ( 1 ) + ε r Q ¯ ¯ ( 1 ) p ¯
S ¯ ¯ p ¯ L ¯ ¯ q ¯ + R ¯ ¯ c ¯ R ¯ ¯ b ¯ = 0
S ¯ ¯ ( 1 ) p ¯ 1 s L ¯ ¯ ( 1 ) q ¯ + R ¯ ¯ ( 1 ) c ¯ ( 1 ) 1 s R ¯ ¯ ( 1 ) b ¯ ( 1 ) = 0
[ p ¯ q ¯ ] = A ¯ ¯ [ b ¯ c ¯ b ¯ ( 1 ) c ¯ ( 1 ) ]
A ¯ ¯ [ S ¯ ¯ L ¯ ¯ S ¯ ¯ ( 1 ) 1 s L ¯ ¯ ( 1 ) ] 1 [ R ¯ ¯ R ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ 1 s R ¯ ¯ ( 1 ) R ¯ ¯ ( 1 ) ]
P ¯ ¯ x ¯ = λ x ¯
P ¯ ¯ = [ D ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ D ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ ε r D ¯ ¯ ( 1 ) O ¯ ¯ O ¯ ¯ O ¯ ¯ O ¯ ¯ ε r D ¯ ¯ ( 1 ) ] + [ O ¯ ¯ R ¯ ¯ Q ¯ ¯ O ¯ ¯ O ¯ ¯ ε r R ¯ ¯ ( 1 ) ε r Q ¯ ¯ ( 1 ) O ¯ ¯ ] A ¯ ¯
x ¯ = [ b ¯ T c ¯ T b ¯ ( 1 ) T c ¯ ( 1 ) T ] T
a ¯ 1 = a 2 ( 3 x ^ + y ^ ) a ¯ 2 = a 2 ( 3 x ^ + y ^ )
b ¯ 1 = 2 π a ( 1 3 x ^ + y ^ ) b ¯ 2 = 2 π a ( 1 3 x ^ + y ^ )
Γ : k ¯ i = 0 b ¯ 1 + 0 b ¯ 2 = 0 M : k ¯ i = 1 2 b ¯ 1 + 0 b ¯ 2 = π a ( 1 2 x ^ + y ^ ) K : k ¯ i = 1 3 b ¯ 1 + 1 3 b ¯ 2 = 4 π 3 a y ^
f N ( k ) = k a 2 π ε 0 ε b
g P ( k , k ¯ i ; ρ ¯ ) = g ( k , ρ ¯ ) + g R ( k , k ¯ i ; ρ ¯ )
g ( k , ρ ¯ ) = i 4 H 0 ( 1 ) ( k ρ ) = i 4 J 0 ( k ρ ) = 1 4 N 0 ( k ρ )
g R ( k , k ¯ i ; ρ ¯ ) = n = E n J n ( k ρ ) exp ( i n ϕ )
g P ( k , k ¯ i ; ρ ¯ ) = 1 4 N 0 ( k ρ ) + n = D n J n ( k ρ ) exp ( i n ϕ )
D n = E n + i 4 δ n 0
g P = 1 Ω 0 α 1 | k ¯ i α | 2 k 2 n i n exp ( i n ϕ i α ) J n ( | k ¯ i α | ρ ) exp ( i n ϕ )
D n = 1 J n ( k ρ ) [ i n 1 Ω 0 α exp ( i n ϕ i α ) J n ( | k ¯ i α | ρ ) | k ¯ i α | 2 k 2 + 1 4 N 0 ( k ρ ) δ n 0 ]
D n = D n
n ^ g P ( k , k ¯ i ; ρ ¯ , ρ ¯ ) = n ^ g P a ( k ; ρ ¯ , ρ ¯ ) + n ^ g R ( k , k ¯ i ; ρ ¯ , ρ ¯ )
n ^ g ( k ; ρ ¯ , ρ ¯ ) = i 4 k H 1 ( 1 ) ( k | ρ ¯ ρ ¯ | ) ( n ^ ρ ¯ ρ ¯ | ρ ¯ ρ ¯ | )
n ^ g R ( k , k ¯ i ; ρ ¯ , ρ ¯ ) = n ^ n E n J n ( k | ρ ¯ ρ ¯ | ) exp ( i n ϕ ρ ¯ ρ ¯ )
J n ( k | ρ ¯ ρ ¯ | ) exp ( i n ϕ ρ ¯ ρ ¯ ) = m = J m ( k ρ ) J m n ( k ρ ) exp ( i m ϕ i ( m n ) ϕ )
n ^ g R ( k , k ¯ i , ρ ¯ ρ ¯ ) = n E n m = J m ( k ρ ) exp ( i m ϕ ) n ^ ( ρ ^ k J m n ( k ρ ) + ϕ ^ 1 ρ J m n ( k ρ ) [ i ( m n ) ] ) exp ( i ( m n ) ϕ )
S m n = S m n ( P ) + S m n ( R )
L m n = L m n ( P ) + L m n ( R )
S m n ( P ) = 1 Δ t n S 00 ( n ) d l n ^ g P ( k , ρ ¯ ρ ¯ ) , ρ ¯ ρ ¯ m ¯ = { 1 2 Δ t n n = m [ n ^ g P ( k , k ¯ i , ρ ¯ ρ ¯ ) ] ρ ¯ = ρ ¯ m , ρ ¯ = ρ ¯ n n m
S m n ( R ) = 1 Δ t n S 00 ( n ) d l n ^ g R ( k , k ¯ , ρ ¯ m ρ ¯ ) = [ n ^ g R ( k , k ¯ i , ρ ¯ ρ ¯ ) ] ρ ¯ = ρ ¯ m , ρ ¯ = ρ ¯ n
L m n ( P ) = 1 Δ t n S 00 ( n ) d l g P ( k , ρ ¯ m ρ ¯ ) = { i 4 [ 1 + i 2 π l n ( γ k 4 e Δ t n ) ] n = m [ g P ( k , ρ ¯ ρ ¯ ) ] ρ ¯ = ρ ¯ m , ρ ¯ = ρ ¯ n n m
L m n ( R ) = 1 Δ t n S 00 ( n ) d l g R ( k , k ¯ , ρ ¯ m ρ ¯ ) = [ g R ( k , k ¯ i , ρ ¯ ρ ¯ ) ] ρ ¯ = ρ ¯ m , ρ ¯ = ρ ¯ n
S m m ( 1 ) ( P ) = 1 2 Δ t , ρ ¯ ρ ¯ m +
E ¯ e x = E ¯ E ¯ P
D ¯ = ε e f f E ¯
D ¯ = ε E ¯ + P ¯
P ¯ = n 0 α E ¯ e x
E ¯ P = χ P ¯ ε
P ¯ = n 0 α ε ε n 0 α χ E ¯
ε e f f = ε + n 0 α ε ε n 0 α χ = ε 1 + n 0 α ( 1 χ ) / ε 1 n 0 α χ / ε
α = A 0 y 2 ε
χ = 1 2
y = ε p ε ε p + ε
ε e f f = ε 1 + n 0 A 0 y 1 n 0 A 0 y = ε 1 + f v y 1 f v y
α = A 0 ( ε p ε )
χ = 0
ε e f f = ε + n 0 A ( ε p ε ) = ( 1 f v ) ε + f v ε p

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