Abstract

The Green’s functions are physical responses due to a single point source in a periodic lattice. The single point source can also correspond to an impurity or a defect. In this paper, the Green’s functions, including the scatterers, for periodic structures such as in photonic crystals and metamaterials are calculated. The Green’s functions are in terms of the multiband solutions of the periodic structures. The Green’s functions are broadband solutions so that the frequency or wavelength dependences of the physical responses can be calculated readily. They are obtained by integrating the periodic Green’s function including the scatterers in the Brillouin zone. Low wavenumber extraction methods are used to accelerate the convergence rate of the multiband expansions. The low wavenumber component represents the reactive near field. The band solutions of the periodic structure are obtained from a surface integral equation solution, which is effectively converted to a linear eigenvalue problem, giving multiple band solutions simultaneously. Numerical results are illustrated for the band modal fields, the periodic Green’s functions, and the single point source Green’s functions for two-dimensional (2D) perfect-electric-conductor (PEC) scatterers in a 2D lattice.

© 2017 Optical Society of America

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References

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  13. F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
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  16. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. 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]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  35. L. Tsang and S. Huang, “Broadband Green’s function with low wave number extraction for arbitrary shaped waveguide with applications to modeling of vias in finite power/ground plane,” Prog. Electromagn. Res. 152, 105–125 (2015).
    [Crossref]
  36. S. Huang and L. Tsang, “Fast electromagnetic analysis of emissions from printed circuit board using broadband Green’s function method,” IEEE Trans. Electromagn. Compat. 58, 1642–1652 (2016).
    [Crossref]
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  38. T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
    [Crossref]
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  40. M. G. Silveirinha and C. A. Fernandes, “Efficient calculation of the band structure of artificial materials with cylindrical metallic inclusions,” IEEE Trans. Microwave Theory Tech. 51, 1460–1466 (2003).
    [Crossref]
  41. M. G. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D metallic crystals,” IEEE Trans. Microwave Theory Tech. 52, 889–902 (2004).
    [Crossref]
  42. M. G. Silverinha and C. A. Fernandes, “Computation of the electromagnetic modes in two-dimensional photonic crystals: a technique to improve the convergence rate of the plane wave method,” Electromagnetics 26, 175–187 (2006).
    [Crossref]

2017 (1)

T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
[Crossref]

2016 (2)

S. Huang and L. Tsang, “Fast electromagnetic analysis of emissions from printed circuit board using broadband Green’s function method,” IEEE Trans. Electromagn. Compat. 58, 1642–1652 (2016).
[Crossref]

L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24, 945–965 (2016).
[Crossref]

2015 (3)

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

L. Tsang and S. Huang, “Broadband Green’s function with low wave number extraction for arbitrary shaped waveguide with applications to modeling of vias in finite power/ground plane,” Prog. Electromagn. Res. 152, 105–125 (2015).
[Crossref]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

2013 (1)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

2008 (1)

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

2007 (3)

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

2006 (1)

M. G. Silverinha and C. A. Fernandes, “Computation of the electromagnetic modes in two-dimensional photonic crystals: a technique to improve the convergence rate of the plane wave method,” Electromagnetics 26, 175–187 (2006).
[Crossref]

2005 (1)

F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
[Crossref]

2004 (2)

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D metallic crystals,” IEEE Trans. Microwave Theory Tech. 52, 889–902 (2004).
[Crossref]

2003 (2)

M. G. Silveirinha and C. A. Fernandes, “Efficient calculation of the band structure of artificial materials with cylindrical metallic inclusions,” IEEE Trans. Microwave Theory Tech. 51, 1460–1466 (2003).
[Crossref]

M. Silveirinha and C. Fernandes, “Effective permittivity of metallic crystals: a periodic Green’s function formulation,” Electromagnetics 23, 647–663 (2003).
[Crossref]

2002 (1)

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green’s functions of a two-dimensional photonic crystal excited by a line source—the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380–1391 (2002).
[Crossref]

2001 (2)

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

1999 (2)

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

1998 (1)

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

1997 (1)

S. G. Johnson, “Erratum: accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 55, 15942 (1997).

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]

1995 (1)

1994 (1)

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[Crossref]

1993 (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[Crossref]

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 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, 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, 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. 65, 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]

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, 392–400 (1947).
[Crossref]

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[Crossref]

Alvarez-Melcon, A.

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[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]

Caloz, C.

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green’s functions of a two-dimensional photonic crystal excited by a line source—the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380–1391 (2002).
[Crossref]

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

Capolino, F.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
[Crossref]

Chan, C. T.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

W. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8, 203–208 (2001).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref]

Chong, Y.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Cowan, M. L.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

Curcio, D.

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

Ding, K.-H.

T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
[Crossref]

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]

Felsen, L. B.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

Fernandes, C.

M. Silveirinha and C. Fernandes, “Effective permittivity of metallic crystals: a periodic Green’s function formulation,” Electromagnetics 23, 647–663 (2003).
[Crossref]

Fernandes, C. A.

M. G. Silverinha and C. A. Fernandes, “Computation of the electromagnetic modes in two-dimensional photonic crystals: a technique to improve the convergence rate of the plane wave method,” Electromagnetics 26, 175–187 (2006).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D metallic crystals,” IEEE Trans. Microwave Theory Tech. 52, 889–902 (2004).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “Efficient calculation of the band structure of artificial materials with cylindrical metallic inclusions,” IEEE Trans. Microwave Theory Tech. 51, 1460–1466 (2003).
[Crossref]

M. G. M. V. Silveirinha and C. A. Fernandes, “Radiation from a short vertical dipole in a disk-type PBG material,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2003), Vol. 3, pp. 990–993.

Gao, F.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Gao, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Gardiol, F. E.

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green’s functions of a two-dimensional photonic crystal excited by a line source—the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380–1391 (2002).
[Crossref]

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[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, 13962–13972 (1992).
[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. 65, 3152–3155 (1990).
[Crossref]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Huang, S.

S. Huang and L. Tsang, “Fast electromagnetic analysis of emissions from printed circuit board using broadband Green’s function method,” IEEE Trans. Electromagn. Compat. 58, 1642–1652 (2016).
[Crossref]

L. Tsang and S. Huang, “Broadband Green’s function with low wave number extraction for arbitrary shaped waveguide with applications to modeling of vias in finite power/ground plane,” Prog. Electromagn. Res. 152, 105–125 (2015).
[Crossref]

Inguva, R.

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

Jackson, D. R.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
[Crossref]

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref]

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. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[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.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref]

S. G. Johnson, “Erratum: accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 55, 15942 (1997).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[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]

Korringa, J.

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

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[Crossref]

Leung, K. M.

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 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]

Liao, T.-H.

T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
[Crossref]

Lin, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[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]

Liu, Z.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[Crossref]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 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.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Page, J. H.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[Crossref]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 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, 7767–7771 (1993).
[Crossref]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[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]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[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]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Sheng, P.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

W. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8, 203–208 (2001).
[Crossref]

Shi, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Silveirinha, M.

M. Silveirinha and C. Fernandes, “Effective permittivity of metallic crystals: a periodic Green’s function formulation,” Electromagnetics 23, 647–663 (2003).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D metallic crystals,” IEEE Trans. Microwave Theory Tech. 52, 889–902 (2004).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “Efficient calculation of the band structure of artificial materials with cylindrical metallic inclusions,” IEEE Trans. Microwave Theory Tech. 51, 1460–1466 (2003).
[Crossref]

Silveirinha, M. G. M. V.

M. G. M. V. Silveirinha and C. A. Fernandes, “Radiation from a short vertical dipole in a disk-type PBG material,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2003), Vol. 3, pp. 990–993.

Silverinha, M. G.

M. G. Silverinha and C. A. Fernandes, “Computation of the electromagnetic modes in two-dimensional photonic crystals: a technique to improve the convergence rate of the plane wave method,” Electromagnetics 26, 175–187 (2006).
[Crossref]

Skrivervik, A. K.

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green’s functions of a two-dimensional photonic crystal excited by a line source—the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380–1391 (2002).
[Crossref]

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[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. 65, 3152–3155 (1990).
[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, 13962–13972 (1992).
[Crossref]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Suzuki, T.

Tan, S.

L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24, 945–965 (2016).
[Crossref]

S. Tan and L. Tsang, “Scattering from finite periodic arrays using broadband Green’s function of periodic scatterers with low wavenumber extraction (BBGFL),” in IEEE Progress in Electromagnetics Research Symposium (PIERS), St. Petersburg, Russia, May 22–25, 2017.

S. Tan, “Multiple volume scattering in random media and periodic structures with applications in microwave remote sensing and wave functional materials,” Ph.D. dissertation (University of Michigan, 2016).

Tretyakov, S.

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

Tsang, L.

T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
[Crossref]

S. Huang and L. Tsang, “Fast electromagnetic analysis of emissions from printed circuit board using broadband Green’s function method,” IEEE Trans. Electromagn. Compat. 58, 1642–1652 (2016).
[Crossref]

L. Tsang and S. Tan, “Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green’s function with low wavenumber extraction (BBGFL),” Opt. Express 24, 945–965 (2016).
[Crossref]

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

L. Tsang and S. Huang, “Broadband Green’s function with low wave number extraction for arbitrary shaped waveguide with applications to modeling of vias in finite power/ground plane,” Prog. Electromagn. Res. 152, 105–125 (2015).
[Crossref]

S. Tan and L. Tsang, “Scattering from finite periodic arrays using broadband Green’s function of periodic scatterers with low wavenumber extraction (BBGFL),” in IEEE Progress in Electromagnetics Research Symposium (PIERS), St. Petersburg, Russia, May 22–25, 2017.

Tse, W.-K.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Villeneuve, P. R.

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]

Wang, Z.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Wilton, D. R.

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Yang, S.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

Yang, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Yu, P. K. I.

Zhang, B.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Zhang, W.

Appl. Phys. Lett. (1)

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]

Electromagnetics (2)

M. Silveirinha and C. Fernandes, “Effective permittivity of metallic crystals: a periodic Green’s function formulation,” Electromagnetics 23, 647–663 (2003).
[Crossref]

M. G. Silverinha and C. A. Fernandes, “Computation of the electromagnetic modes in two-dimensional photonic crystals: a technique to improve the convergence rate of the plane wave method,” Electromagnetics 26, 175–187 (2006).
[Crossref]

IEEE Trans. Antennas Propag. (2)

F. Capolino, D. R. Jackson, and D. R. Wilton, “Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source,” IEEE Trans. Antennas Propag. 53, 91–99 (2005).
[Crossref]

F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag. 55, 1644–1655 (2007).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

S. Huang and L. Tsang, “Fast electromagnetic analysis of emissions from printed circuit board using broadband Green’s function method,” IEEE Trans. Electromagn. Compat. 58, 1642–1652 (2016).
[Crossref]

IEEE Trans. Microwave Theory Tech. (4)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “Efficient calculation of the band structure of artificial materials with cylindrical metallic inclusions,” IEEE Trans. Microwave Theory Tech. 51, 1460–1466 (2003).
[Crossref]

M. G. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D metallic crystals,” IEEE Trans. Microwave Theory Tech. 52, 889–902 (2004).
[Crossref]

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green’s functions of a two-dimensional photonic crystal excited by a line source—the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380–1391 (2002).
[Crossref]

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

Nat. Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

Opt. Express (3)

Phys. Rev. (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]

Phys. Rev. B (10)

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

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]

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

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

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[Crossref]

S. G. Johnson, “Erratum: accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 55, 15942 (1997).

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

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[Crossref]

Phys. Rev. Lett. (6)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 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]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
[Crossref]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett. 93, 024301 (2004).
[Crossref]

Physica (1)

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

Proc. SPIE (1)

C. Caloz, D. Curcio, A. Alvarez-Melcon, A. K. Skrivervik, and F. E. Gardiol, “Slot antenna on a photonic crystal substrate Green’s function study,” Proc. SPIE 3795, 176–187 (1999).
[Crossref]

Prog. Electromagn. Res. (3)

L. Tsang, “Broadband calculations of band diagrams in periodic structures using the broadband Green’s function with low wavenumber extraction (BBGFL),” Prog. Electromagn. Res. 153, 57–68 (2015).
[Crossref]

L. Tsang and S. Huang, “Broadband Green’s function with low wave number extraction for arbitrary shaped waveguide with applications to modeling of vias in finite power/ground plane,” Prog. Electromagn. Res. 152, 105–125 (2015).
[Crossref]

T.-H. Liao, K.-H. Ding, and L. Tsang, “High order extractions of broadband Green’s function with low wavenumber extractions for arbitrary shaped waveguide,” Prog. Electromagn. Res. 158, 7–20 (2017).
[Crossref]

Other (5)

S. Tan and L. Tsang, “Scattering from finite periodic arrays using broadband Green’s function of periodic scatterers with low wavenumber extraction (BBGFL),” in IEEE Progress in Electromagnetics Research Symposium (PIERS), St. Petersburg, Russia, May 22–25, 2017.

S. Tan, “Multiple volume scattering in random media and periodic structures with applications in microwave remote sensing and wave functional materials,” Ph.D. dissertation (University of Michigan, 2016).

M. G. M. V. Silveirinha and C. A. Fernandes, “Radiation from a short vertical dipole in a disk-type PBG material,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2003), Vol. 3, pp. 990–993.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

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Figures (12)

Fig. 1.
Fig. 1.

Illustration of periodic scatterers in 2D periodic lattice in the xy plane. Spq denotes the surface of the pqth scatterer. ρ¯ and ρ¯ represent the locations of arbitrary source and field points, respectively.

Fig. 2.
Fig. 2.

(a) Geometry of the cylinder (red circle) and the source point (black cross) inside the unit cell. Blue circles denote 16×16 uniformly distributed grid points where we probe the fields. The black plus sign denotes a special field point to be examined more closely. (b) Magnitude of surface currents on the cylinder.

Fig. 3.
Fig. 3.

Field distribution of gPS(kL,k¯i;ρ¯,ρ¯) over the lattice: (a) magnitude, (b) real part, and (c) imaginary part.

Fig. 4.
Fig. 4.

Modal field distribution (magnitude) for the lowest three modes: (a) fn=0.216, (b) fn=0.368, and (c) fn=0.413.

Fig. 5.
Fig. 5.

Spatial variation of gP,BS(k,kL,k¯i;ρ¯,ρ¯) at fn=0.2 with fnL=0.001 and k¯i=0.1b¯1+0.05b¯2: (a) magnitude, (b) real part, and (c) imaginary part.

Fig. 6.
Fig. 6.

Magnitude of gPS(k,k¯i;ρ¯,ρ¯) at (a) fn=0.1, (b) fn= 0.2, and (c) fn= 0.4. The numbers of modes included in gBS are 12, 49, and 116, respectively.

Fig. 7.
Fig. 7.

gPS(k,k¯i;ρ¯,ρ¯) as a function of the normalized frequency fn: (a) ϵb=8.9ϵ0 and (b) ϵb=8.9(1+0.11i)ϵ0.

Fig. 8.
Fig. 8.

Magnitude of the integrand as a function of k¯i: (a) gPS(k,k¯i;ρ¯,ρ¯), (b) gP0(k,k¯i;ρ¯,ρ¯), and (c) gPR(k,k¯i;ρ¯,ρ¯). Here, ϵb=8.9ϵ0.

Fig. 9.
Fig. 9.

Magnitude of gPS(k,k¯i;ρ¯,ρ¯) as a function of k¯i at fn=0.26: (a) ϵb=8.9ϵ0 and (b) ϵb=8.9(1+0.11i)ϵ0.

Fig. 10.
Fig. 10.

Spatial variations (magnitudes) of gS(k;ρ¯,ρ¯) in a lossless background of ϵb=8.9ϵ0: (a) fn=0.1, (b) fn=0.2, and (c) fn=0.4.

Fig. 11.
Fig. 11.

gS(k;ρ¯,ρ¯) as a function of the normalized frequency: (a) ϵb=8.9ϵ0 and (b) ϵb=8.9(1+0.11i)ϵ0.

Fig. 12.
Fig. 12.

Spatial variations (magnitudes) of gS(k;ρ¯,ρ¯) in a lossy background of ϵb=8.9(1+0.11i)ϵ0: (a) fn=0.1, (b) fn=0.2, and (c) fn=0.4.

Tables (1)

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Table 1. CPU Time Decomposition

Equations (34)

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δ(k¯i;ρ¯ρ¯)=p=q=δ(ρ¯(ρ¯+R¯pq))exp(ik¯i·R¯pq).
gP0(k,k¯i;ρ¯,ρ¯)=1Ω0αexp(ik¯iα·(ρ¯ρ¯))|k¯iα|2k2,
gP0(k,k¯i;ρ¯,ρ¯)=gP0(kL,k¯i;ρ¯,ρ¯)+k2kL2Ω0αexp(ik¯iα·(ρ¯ρ¯))(|k¯iα|2k2)(|k¯iα|2kL2),
Ω00dρ¯ψβS*(k¯i;ρ¯)ψβS(k¯i;ρ¯)=δββ.
gPS(k,k¯i;ρ¯,ρ¯)=βψβS(k¯i;ρ¯)ψβS*(k¯i;ρ¯)[kβS(k¯i)]2k2.
gP,BS(k,kL,k¯i;ρ¯,ρ¯)=(k2kL2)βψβS(k¯i;ρ¯)ψβS*(k¯i;ρ¯)([kβS(k¯i)]2k2)([kβS(k¯i)]2kL2),
gPS(k,k¯i;ρ¯,ρ¯)=gPS(kL,k¯i;ρ¯,ρ¯)+gP,BS(k,kL,k¯i;ρ¯,ρ¯).
gS(k;ρ¯,ρ¯)=01dβ101dβ2gPS(k,k¯i(β1,β2);ρ¯,ρ¯).
δ(ρ¯ρ¯)=01dβ101dβ2δ(k¯i(β1,β2);ρ¯ρ¯).
gS(k;ρ¯,ρ¯)=01dβ101dβ2gPS(k,k¯i(β1,β2);ρ¯,ρ¯)=01dβ101dβ2[gPS(kL,k¯i(β1,β2);ρ¯,ρ¯)+gP,BS(k,kL,k¯i(β1,β2);ρ¯,ρ¯)].
gS(k;ρ¯,ρ¯)=(Δβ)2m=1Nbn=1Nb[gPS(kL,k¯i(βm,βn);ρ¯,ρ¯)+gP,BS(k,kL,k¯i(βm,βn);ρ¯,ρ¯)],
S00dρ¯[gP0(k,k¯i;ρ¯,ρ¯)J(ρ¯)]=0,
S00dρ¯gP0(kL,k¯i;ρ¯,ρ¯)J(ρ¯)+αRα(ρ¯)bα=0,
bα=S00dρ¯Rα*(ρ¯)λλαJ(ρ¯),
Rα(ρ¯)=λαψ˜α0(k¯i;ρ¯),
L¯¯q¯+R¯b¯=0,
Lmn=1ΔtnS00(n)dρ¯gP0(kL,k¯i;ρ¯,ρ¯).
b¯=(λI¯¯D¯¯)1R¯¯q¯,
q¯=L¯¯1R¯¯b¯,
A¯¯b¯=λb¯,
S00dρ¯[gP0(kβS(k¯i),k¯i;ρ¯,ρ¯)Jβ(ρ¯)]={ψβS(k¯i;ρ¯)forρ¯outside scatterer,0forρ¯inside scatterer,
ψβS(k¯i;ρ¯)=S00dρ¯gP0(kL,k¯i;ρ¯,ρ¯)Jβ(ρ¯)+αRα(ρ¯)bαβ,
ψβ(k¯i;ρ¯)=[2ψβ(k¯i;ρ¯)]/[kβS(k¯i)]2,
2ψ˜α0(k¯i;ρ¯)=|k¯iα|2ψ˜α0(k¯i;ρ¯),
2gPS(kL,k¯i;ρ¯,ρ¯)=δ(k¯i;ρ¯ρ¯)kL2gPS(kL,k¯i;ρ¯,ρ¯).
ψβS(k¯i;ρ¯)=kL2[kβS]2S00dρ¯gP0(kL,k¯i;ρ¯,ρ¯)Jβ(ρ¯)+1[kβS]2α|k¯iα|2ψ˜α0(k¯i;ρ¯)(|k¯iα|2kL2)bαβ.
ψβS(k¯i;ρ¯)1[kβS]2αψ˜α0(k¯i;ρ¯)bαβ.
ψ˜βS(k¯i;ρ¯)=ψβS(k¯i;ρ¯)Ω00dρ¯|ψβS(k¯i;ρ¯)|2.
Ω00dρ¯ψβS*(k¯i;ρ¯)ψβS(k¯i;ρ¯)=1[kβSkβS]2αbαβ*bαβ.
Ω00dρ¯ψβS*(ρ¯,k¯i)ψβS(ρ¯,k¯i)=1[kβS]4δββ.
ψ˜βS(k¯i;ρ¯)=[kβS]2ψβS(k¯i;ρ¯)=αψ˜α0(k¯i;ρ¯)bαβ.
gPS(k,k¯i;ρ¯,ρ¯)gP0(k,k¯i;ρ¯,ρ¯)=S00dρ¯[gPS(k,k¯i;ρ¯,ρ¯)n^·gP0(k,k¯i;ρ¯,ρ¯)gP0(k,k¯i;ρ¯,ρ¯)n^·gPS(k,k¯i;ρ¯,ρ¯)],
S00dρ¯[gP0(k,k¯i;ρ¯,ρ¯)J(k¯i;ρ¯i)]=gP0(k,k¯i;ρ¯,ρ¯).
ψ(k¯i;ρ¯+R¯pq)=ψ(k¯i;ρ¯)exp(ik¯i·R¯pq).

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