Abstract

We present a modified formulation of the Finite-Difference Time-Domain (FDTD) technique that facilitates the accurate modeling of curved plasmonic interfaces. These interfaces appear in structures of interest for the design of optical metamaterials, such as arrays of plasmonic nanorods. Our approach uses the standard rectangular FDTD mesh and tensor effective permittivities for the interface cells, implicitly enforcing field boundary conditions, and is readily applicable to thin curved dispersive layers. We demonstrate the accuracy and effectiveness of our approach with the periodic analysis of a silver nanorod array and the computation of scattering parameters from a thin dispersive ring in a waveguide.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23, 573–583 (2006).
    [Crossref]
  2. N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
    [Crossref]
  3. A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1667 (2006).
    [Crossref] [PubMed]
  4. N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
    [Crossref]
  5. A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
    [Crossref]
  6. T.-B. Wang, X.-W. Wen, C.-P. Yin, and H.-Z. Wang, “The transmission characteristics of surface plasmon polaritons in ring resonator,” Opt. Express 17, 24096 (2009).
    [Crossref]
  7. Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15, 1823–1831 (2007).
    [Crossref] [PubMed]
  8. A. Cangellaris and D. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
    [Crossref]
  9. S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett. 7, 273–275 (1997).
    [Crossref]
  10. Y. Zhao and Y. Hao, “Finite-Difference Time-Domain Study of Guided Modes in Nano-Plasmonic Waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
    [Crossref]
  11. A. Mohammadi, T. Jalali, and M. Agio, “Dispersive contour-path algorithm for the two-dimensional finite-difference time-domain method,” Opt. Express 16, 7397–7406 (2008).
    [Crossref] [PubMed]
  12. A. Deinega and I. Valuev, “Subpixel smoothing for conductive and dispersive media in the finite-difference time-domain method,” Opt. Lett. 32, 3429–3431 (2007).
    [Crossref] [PubMed]
  13. J. Liu, M. Brio, and J. V. Moloney, “Subpixel smoothing finite-difference time-domain method for material interface between dielectric and dispersive media,” Opt. Lett. 37, 4802–4804 (2012).
    [Crossref] [PubMed]
  14. Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
    [Crossref]
  15. N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
    [Crossref]
  16. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006).
    [Crossref] [PubMed]
  17. J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
    [Crossref]
  18. J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
    [Crossref]
  19. J. Maloney and G. Smith, “The use of surface impedance concepts in the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 40, 38–48 (1992).
    [Crossref]
  20. K.-P. Hwang and A. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wirel. Compon. Lett. 11, 158–160 (2001).
    [Crossref]
  21. J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces,” Microw. Opt. Technol. Lett. 23, 245–249 (1999).
    [Crossref]

2016 (1)

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

2012 (1)

2011 (1)

N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
[Crossref]

2009 (1)

2008 (3)

A. Mohammadi, T. Jalali, and M. Agio, “Dispersive contour-path algorithm for the two-dimensional finite-difference time-domain method,” Opt. Express 16, 7397–7406 (2008).
[Crossref] [PubMed]

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
[Crossref]

2007 (5)

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Y. Zhao and Y. Hao, “Finite-Difference Time-Domain Study of Guided Modes in Nano-Plasmonic Waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[Crossref]

Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
[Crossref]

Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15, 1823–1831 (2007).
[Crossref] [PubMed]

A. Deinega and I. Valuev, “Subpixel smoothing for conductive and dispersive media in the finite-difference time-domain method,” Opt. Lett. 32, 3429–3431 (2007).
[Crossref] [PubMed]

2006 (3)

2003 (1)

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

2001 (1)

K.-P. Hwang and A. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wirel. Compon. Lett. 11, 158–160 (2001).
[Crossref]

1999 (1)

J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces,” Microw. Opt. Technol. Lett. 23, 245–249 (1999).
[Crossref]

1997 (1)

S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett. 7, 273–275 (1997).
[Crossref]

1992 (1)

J. Maloney and G. Smith, “The use of surface impedance concepts in the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 40, 38–48 (1992).
[Crossref]

1991 (1)

A. Cangellaris and D. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[Crossref]

Agio, M.

Akyurtlu, A.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Alú, A.

N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
[Crossref]

A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23, 573–583 (2006).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1667 (2006).
[Crossref] [PubMed]

Azad, A. K.

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

Bermel, P.

Brio, M.

Burr, G. W.

Cangellaris, A.

K.-P. Hwang and A. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wirel. Compon. Lett. 11, 158–160 (2001).
[Crossref]

A. Cangellaris and D. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[Crossref]

Deinega, A.

Deuflhard, P.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

Dey, S.

S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett. 7, 273–275 (1997).
[Crossref]

Dong, Q.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Eleftheriades, G. V.

Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
[Crossref]

Engheta, N.

N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
[Crossref]

A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23, 573–583 (2006).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1667 (2006).
[Crossref] [PubMed]

Farjadpour, A.

Goodhue, W. D.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Hao, Y.

Y. Zhao and Y. Hao, “Finite-Difference Time-Domain Study of Guided Modes in Nano-Plasmonic Waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[Crossref]

Hwang, K.-P.

K.-P. Hwang and A. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wirel. Compon. Lett. 11, 158–160 (2001).
[Crossref]

Ibanescu, M.

Jalali, T.

Joannopoulos, J. D.

Johnson, S. G.

Lanteri, S.

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

Lee, J.-Y.

J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces,” Microw. Opt. Technol. Lett. 23, 245–249 (1999).
[Crossref]

Li, J.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Liu, J.

Liu, Y.

Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
[Crossref]

Maloney, J.

J. Maloney and G. Smith, “The use of surface impedance concepts in the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 40, 38–48 (1992).
[Crossref]

Marx, K. A.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Mittra, R.

S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett. 7, 273–275 (1997).
[Crossref]

Mohammadi, A.

Moloney, J. V.

Moreau, A.

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

Myung, N.-H.

J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces,” Microw. Opt. Technol. Lett. 23, 245–249 (1999).
[Crossref]

Nadobny, J.

J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
[Crossref]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

O’Hara, J. F.

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

Qiang, Z.

Rodriguez, A.

Roundy, D.

Salandrino, A.

N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
[Crossref]

A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14, 1557–1667 (2006).
[Crossref] [PubMed]

Sarris, C. D.

Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
[Crossref]

Scheid, C.

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

Schmitt, N.

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

Smirnova, E.

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

Smith, G.

J. Maloney and G. Smith, “The use of surface impedance concepts in the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 40, 38–48 (1992).
[Crossref]

Soref, R. A.

Sullivan, D.

J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
[Crossref]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

Taylor, A. J.

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

Valuev, I.

Viquerat, J.

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

Wang, H.-Z.

Wang, T.-B.

Wen, X.-W.

Wlodarczyk, W.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

Wongkasem, N.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

Wright, D.

A. Cangellaris and D. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[Crossref]

Wust, P.

J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
[Crossref]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

Yin, C.-P.

Zhao, Y.

Y. Zhao and Y. Hao, “Finite-Difference Time-Domain Study of Guided Modes in Nano-Plasmonic Waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[Crossref]

Zhou, W.

Appl. Phys. Lett. (1)

A. K. Azad, A. J. Taylor, E. Smirnova, and J. F. O’Hara, “Characterization and analysis of terahertz metamaterials based on rectangular split-ring resonators,” Appl. Phys. Lett. 92, 011119 (2008).
[Crossref]

IEEE Microw. Guided Wave Lett. (1)

S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett. 7, 273–275 (1997).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

K.-P. Hwang and A. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microw. Wirel. Compon. Lett. 11, 158–160 (2001).
[Crossref]

IEEE Trans. Antennas Propag. (6)

Y. Zhao and Y. Hao, “Finite-Difference Time-Domain Study of Guided Modes in Nano-Plasmonic Waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[Crossref]

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of Chiral Negative Refractive Index Metamaterials for the Terahertz Frequency Regime,” IEEE Trans. Antennas Propag. 55, 3052–3062 (2007).
[Crossref]

A. Cangellaris and D. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[Crossref]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag. 51, 1760–1770 (2003).
[Crossref]

J. Nadobny, D. Sullivan, and P. Wust, “A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform,” IEEE Trans. Antennas Propag. 56, 1027–1040 (2008).
[Crossref]

J. Maloney and G. Smith, “The use of surface impedance concepts in the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 40, 38–48 (1992).
[Crossref]

J. Comput. Phys. (1)

N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, and J. Viquerat, “A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects,” J. Comput. Phys. 316, 396–415 (2016).
[Crossref]

J. Light. Technol. (1)

Y. Liu, C. D. Sarris, and G. V. Eleftheriades, “Triangular-Mesh-Based FDTD Analysis of Two-Dimensional Plasmonic Structures Supporting Backward Waves at Optical Frequencies,” J. Light. Technol. 25, 938–945 (2007).
[Crossref]

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

A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23, 573–583 (2006).
[Crossref]

Microw. Opt. Technol. Lett. (1)

J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces,” Microw. Opt. Technol. Lett. 23, 245–249 (1999).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

N. Engheta, A. Salandrino, and A. Alú, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett. 95, 095504 (2011).
[Crossref]

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Figures (14)

Fig. 1
Fig. 1 (a) Yee cell of 2-D TE mode (Ex, Ey, Hz). (b) Update of flux density in y-direction from a line integral of magnetic field around a loop centered at (iΔx, (j+1/2)Δy).
Fig. 2
Fig. 2 (a) Sloped interface across two Yee cells centered at (iΔx, (j + 1/2)Δy). (b) Sloped thin sheet across two Yee cells centered at (iΔx, (j + 1/2)Δy).
Fig. 3
Fig. 3 Calculation of the tensor effective permittivity through surface integration at the grid point (iΔx, (j + 1/2)Δy) (a) Faces for the sloped interface. (b) Faces for the thin sheet.
Fig. 4
Fig. 4 Contour path integration for the update of magnetic field, E(1) and E(2) are both needed.
Fig. 5
Fig. 5 1-D array of periodic silver nonarods and its correspondent simulation structure realized by Bloch’s boundary conditions.
Fig. 6
Fig. 6 Spectrum comparison between staircased FDTD and tensor ADE-FDTD method. The cell size of the former varies from 2.5 nm to 0.625 nm; the cell size for the latter is 2.5 nm.
Fig. 7
Fig. 7 Magnetic field magnitude and electric field vector of the five modes of the nanorods array when ky α = π, using the tensor ADE-FDTD method with Δ = 2.5 nm. (a) f = 864 THz (b) f=990 THz. (c) f=1036 THz. (d) f=1069 THz. (e) f=1114 THz.
Fig. 8
Fig. 8 Magnetic field magnitude and electric field vector of the five modes of the nanorods array when ky α = π, using the staircased FDTD method with Δ = 2.5 nm. (a) f = 828 THz (b) f=969 THz. (c) f=1053 THz. (d) f=1081 THz. (e) f=1115 THz.
Fig. 9
Fig. 9 Convergence comparison between staircased FDTD and tensor ADE-FDTD method (log-log scale). RMS error of the resonant frequency of the first two modes is compared for the two methods; the cell sizes of both methods vary from 5 nm to 0.625 nm.
Fig. 10
Fig. 10 Dispersion diagram of the nanorods array, results in [10] are also compared.
Fig. 11
Fig. 11 (a) Sloped thin dielectric sheet array loaded cavity, 50 cm × 2.5 cm in size. The thin sheets has a width of w and inclination angle of θ. An actual cell for the tensor method is marked as a square. (b) Spectrum resolved by tensor FDTD (Δ = 5 mm) and FDTD with very fine mesh (Δ = 0.2 mm), with sheet width w = 1 mm and inclination angle θ = 45°.
Fig. 12
Fig. 12 (a) Error (compared with converged FDTD with very fine mesh) of tensor FDTD varies by inclination angle from θ = 0° to θ = 45°, with a fixed sheet width w = 1 mm. (b) Error (compared with converged FDTD with very fine mesh) of tensor FDTD varies by width from w = 0.5 mm to w = 3 mm, with a fixed inclination angle θ = 45°.
Fig. 13
Fig. 13 A thin ring of Lorentz media excited by plane wave in a parallel plate waveguide. Actual cells for the tensor ADE-FDTD method are marked as squares.
Fig. 14
Fig. 14 S-parameters comparison between the tensor ADE-FDTD method and standard FDTD with very fine mesh. The cell size of the former is 0.677 nm; the cell size for the latter varies from 0.037 nm to 0.0111 nm.

Tables (3)

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Table 1 Comparison of frequencies (THz) of the first five modes, computed by staircased FDTD, tensor ADE-FDTD and EPs (effective permitivities) [10] for k y α = π .

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Table 2 Comparison of execution time and RMS error in the resonant frequencies of the first two modes, computed by the staircased FDTD and the proposed method using various cell sizes.

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Table 3 Comparison of execution time for the staircased FDTD using various cell sizes and the proposed method using 0.667 nm cells.

Equations (26)

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D y ( i , j + 1 2 ) n + 1 = D y ( i , j + 1 2 ) n Δ t Δ x Δ l ( H z ( i + 1 2 , j + 1 2 ) n + 1 2 Δ l H z ( i 1 2 , j + 1 2 ) n + 1 2 Δ l )
n ^ × ( E ( 1 ) E ( 2 ) ) = 0
n ^ ( E ( 1 ) ϵ 1 E ( 2 ) ϵ 2 ( ω ) ) = 0
E ( 2 ) = [ A ] E ( 1 ) = [ { ε 1 / ε 2 ( ω ) } n x 2 + ( 1 n x 2 ) { ε 1 / ε 2 ( ω ) 1 } n x n y { ε 1 / ε 2 ( ω ) 1 } n y n x { ε 1 / ε 2 ( ω ) } n y 2 + ( 1 n y 2 ) ] E ( 1 )
E ( 2 ) = [ A ] E ( 1 ) = [ A ] E ( 3 )
S D d S = S ε E d S
D y ( i , j + 1 / 2 ) n + 1 Δ x Δ l = ϵ 1 E y ( 1 ) ( Δ x g ) Δ l + ϵ 2 ( ω ) E y ( 2 ) g Δ l
D y ( i , j + 1 / 2 ) n + 1 = ϵ 1 ( 1 g * ) E y ( 1 ) + ϵ 2 ( ω ) g * E y ( 2 )
D y ( i , j + 1 / 2 ) n + 1 = ϵ 1 ( 1 g * ) E y ( 1 ) + ϵ 2 ( ω ) g * ( A 21 E x ( 1 ) + A 22 E y ( 1 ) ) = { ε 1 ε 2 ( ω ) } n y n x g * E x ( 1 ) + ( ε 1 { ε 1 ε 2 ( ω ) } ( 1 n y 2 ) g * ) E y ( 1 )
D = [ ε ] E ( 1 ) = [ ε 1 { ε 1 ε 2 ( ω ) } ( 1 n x 2 ) f * { ε 1 ε 2 ( ω ) } n x n y f * { ε 1 ε 2 ( ω ) } n y n x g * ε 1 { ε 1 ε 2 ( ω ) } ( 1 n y 2 ) g * ] E ( 1 )
E ( 1 ) = [ ε ] 1 D
D x ( i , j + 1 / 2 ) = 0.25 ( D x ( i 1 / 2 , j + 1 ) + D x ( i + 1 / 2 , j + 1 ) + D x ( i 1 / 2 , j ) + D x ( i + 1 / 2 , j ) )
ε 2 ( ω ) = ε 0 ( 1 + ω p 2 ω t 2 + 2 j ω δ ω 2 )
P x = ε 2 ( ω ) E x ( 1 ) , P y = ε 2 ( ω ) E y ( 1 )
D = ε ¯ α E ( 1 ) + ε ¯ β [ P x P y ]
P x n + 1 = 1 c 1 ( c 2 P x n c 3 P x n 1 + e 1 E x ( 1 ) n + 1 + e 2 E x ( 1 ) n + e 3 E x ( 1 ) n 1 ) P y n + 1 = 1 c 1 ( c 2 P y n c 3 P y n 1 + e 1 E y ( 1 ) n + 1 + e 2 E y ( 1 ) n + e 3 E y ( 1 ) n 1 )
c 1 = ω t 2 4 + 1 Δ t 2 + δ Δ t e 1 = ε 0 ( ω t 2 + ω p 2 4 + 1 Δ t 2 + δ Δ t ) c 2 = ω t 2 2 2 Δ t 2 e 2 = ε 0 ( ω t 2 + ω p 2 2 2 Δ t 2 ) c 3 = ω t 2 4 + 1 Δ t 2 δ Δ t e 3 = ε 0 ( ω t 2 + ω p 2 4 + 1 Δ t 2 δ Δ t )
[ E x ( 1 ) n + 1 E y ( 1 ) n + 1 ] = ( ε ¯ α + e 1 c 1 ε ¯ β ) 1 ( [ D x n + 1 ( i + 1 2 , j ) D y n + 1 ( i + 1 2 , j ) ] ε ¯ β c 1 [ ( c 2 P x n c 3 P x n 1 + e 2 E x ( 1 ) n + e 3 E x ( 1 ) n 1 ) ( c 2 P y n c 3 P y n 1 + e 2 E y ( 1 ) n + e 3 E y ( 1 ) n 1 ) ] )
Q x = E x ( 1 ) / ε 2 ( ω ) , Q y = E y ( 1 ) / ε 2 ( ω )
E ( 2 ) = [ ( 1 n y 2 ) n y n z n y n z ( 1 n z 2 ) ] E ( 1 ) + [ ε 1 n y 2 ε 1 n y n z ε 1 n y n z ε 1 n z 2 ] [ Q x Q y ] = r ¯ α E + r ¯ β [ Q x Q y ]
Q x n + 1 = 1 e 1 ( e 2 Q x n e 3 Q x n 1 + c 1 E x ( 1 ) n + 1 + c 2 E x ( 1 ) n + c 3 E x ( 1 ) n 1 ) Q y n + 1 = 1 e 1 ( e 2 Q y n e 3 Q y n 1 + c 1 E y ( 1 ) n + 1 + c 2 E y ( 1 ) n + c 3 E y ( 1 ) n 1 )
[ E x ( 2 ) n + 1 E y ( 2 ) n + 1 ] = ( r ¯ α + c 1 e 1 r ¯ β ) [ E x ( 1 ) n + 1 E y ( 1 ) n + 1 ] + r ¯ β e 1 [ ( e 2 Q x n e 3 Q x n 1 + c 2 E x ( 1 ) n + c 3 E x ( 1 ) n 1 ) ( e 2 Q y n e 3 Q y n 1 + c 2 E y ( 1 ) n + c 3 E y ( 1 ) n 1 ) ]
y = j Δ y y = ( j + 1 ) Δ y E y ( y ) d y = l y E y ( 2 ) ( i , j + 1 2 ) + ( Δ y l y ) E y ( 1 ) ( i , j + 1 2 )
H z ( i + 1 2 , j + 1 2 ) n + 1 2 = H z ( i + 1 2 , j + 1 2 ) n 1 2 + Δ t Δ x Δ y ( l y E y ( 2 ) n ( i , j + 1 2 ) + ( Δ y l y ) E y ( 1 ) n ( i , j + 1 2 ) Δ y E y ( 1 ) n ( i + 1 , j + 1 2 ) + Δ x E x ( 1 ) n ( i + 1 / 2 , j + 1 ) l x E x ( 2 ) n ( i + 1 2 , j ) ( Δ x l x ) E x ( 1 ) n ( i + 1 2 , j ) )
E x ( 0 ) = E x ( α ) e k y α , H z ( 0 ) = H z ( α ) e k y α
R M S E = i = 1 2 ( f i n u m f i r e f ) 2 i = 1 2 ( f i r e f ) 2

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