Abstract

In this work, we propose a numerical solver combining the spectral element - boundary integral (SEBI) method with the periodic layered medium dyadic Green’s function. The periodic layered medium dyadic Green’s function is formulated under matrix representation. The surface integral equations (SIEs) are then implemented as the radiation boundary condition to truncate the top and bottom computation domain. After describing the interior computation domain with the vector wave equations, and treating the lateral boundaries with Bloch periodic boundary conditions, the whole computation domains are discretized with mixed-order Gauss- Lobatto-Legendre basis functions in the SEBI method. This method avoids the discretization of the top and bottom layered media, so it can be much more efficient than conventional methods. Numerical results validate the proposed solver with fast convergence throughout the whole computation domain and good performance for typical multiscale nano-optical applications.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function

Yongpin P. Chen, Wei E. I. Sha, Wallace C. H. Choy, Lijun Jiang, and Weng Cho Chew
Opt. Express 20(18) 20210-20221 (2012)

Boundary integral spectral element method analyses of extreme ultraviolet multilayer defects

Jun Niu, Ma Luo, Yuan Fang, and Qing Huo Liu
J. Opt. Soc. Am. A 31(10) 2203-2209 (2014)

References

  • View by:
  • |
  • |
  • |

  1. B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” J. Vac. Sci. Technol. 25, 1743–1761 (2007).
    [Crossref]
  2. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [Crossref] [PubMed]
  3. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
    [Crossref]
  4. H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
    [Crossref]
  5. I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
    [Crossref]
  6. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405 (2001).
    [Crossref]
  7. J. Niu, M. Luo, Y. Fang, and Q. H. Liu, “Boundary integral spectral element method analyses of extreme ultraviolet multilayer defects,” J. Opt. Soc. Am. A 31, 2203–2209 (2014).
    [Crossref]
  8. J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
    [Crossref]
  9. M. Luo, Y. Lin, and Q. H. Liu, “Spectral methods and domain decomposition for nanophotonic applications,” Proc. IEEE 101, 473–483 (2013).
    [Crossref]
  10. J. Niu, M. Luo, J. Zhu, and Q. H. Liu, “Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method,” Opt. Express 23, 4539–4551 (2015).
    [Crossref] [PubMed]
  11. J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33, 615–621 (2016).
    [Crossref]
  12. J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
    [Crossref]
  13. Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
    [Crossref]
  14. Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
    [Crossref]
  15. Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
    [Crossref]
  16. B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
    [Crossref]
  17. J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
    [Crossref]
  18. W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
    [Crossref]
  19. K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
    [Crossref]
  20. J. Jin, The Finite Element Method in Electromagnetics (Wiley-IEEE Press, 2014), 3rd ed.
  21. A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
    [Crossref]

2017 (1)

Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
[Crossref]

2016 (2)

Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
[Crossref]

J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33, 615–621 (2016).
[Crossref]

2015 (2)

J. Niu, M. Luo, J. Zhu, and Q. H. Liu, “Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method,” Opt. Express 23, 4539–4551 (2015).
[Crossref] [PubMed]

K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
[Crossref]

2014 (1)

2013 (2)

I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
[Crossref]

M. Luo, Y. Lin, and Q. H. Liu, “Spectral methods and domain decomposition for nanophotonic applications,” Proc. IEEE 101, 473–483 (2013).
[Crossref]

2012 (2)

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[Crossref]

2011 (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

2009 (1)

J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
[Crossref]

2007 (2)

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” J. Vac. Sci. Technol. 25, 1743–1761 (2007).
[Crossref]

2006 (2)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
[Crossref]

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
[Crossref]

2004 (1)

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

2001 (1)

2000 (1)

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

1992 (1)

J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
[Crossref]

Ahmed, I.

I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
[Crossref]

Alvarez, J.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Angulo, L. D.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Aygun, K.

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

Burger, S.

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

Chen, J.

J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
[Crossref]

Chen, K.

K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
[Crossref]

Chen, Y. P.

Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
[Crossref]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[Crossref]

Chew, W. C.

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[Crossref]

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
[Crossref]

Collins, J. D.

J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
[Crossref]

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Ergin, A. A.

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

Fang, Y.

Garcia, S. G.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Geuzaine, C.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

Guenneau, S.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

Huang, W. F.

Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
[Crossref]

Jiang, L.

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[Crossref]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley-IEEE Press, 2014), 3rd ed.

Jin, J. M.

J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
[Crossref]

Kamgaing, T.

K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
[Crossref]

Khoo, E. H.

I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
[Crossref]

Koshiba, M.

Kumar, A.

B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” J. Vac. Sci. Technol. 25, 1743–1761 (2007).
[Crossref]

Lee, J. H.

J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
[Crossref]

Lee, J.-H.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
[Crossref]

Li, E.

I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
[Crossref]

Lin, H.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Lin, Y.

M. Luo, Y. Lin, and Q. H. Liu, “Spectral methods and domain decomposition for nanophotonic applications,” Proc. IEEE 101, 473–483 (2013).
[Crossref]

Liu, Q. H.

Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
[Crossref]

Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
[Crossref]

J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33, 615–621 (2016).
[Crossref]

J. Niu, M. Luo, J. Zhu, and Q. H. Liu, “Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method,” Opt. Express 23, 4539–4551 (2015).
[Crossref] [PubMed]

J. Niu, M. Luo, Y. Fang, and Q. H. Liu, “Boundary integral spectral element method analyses of extreme ultraviolet multilayer defects,” J. Opt. Soc. Am. A 31, 2203–2209 (2014).
[Crossref]

M. Luo, Y. Lin, and Q. H. Liu, “Spectral methods and domain decomposition for nanophotonic applications,” Proc. IEEE 101, 473–483 (2013).
[Crossref]

J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
[Crossref]

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
[Crossref]

Luo, M.

Martin, R. G.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Michielssen, E.

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

Nicolet, A.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

Niu, J.

Pantoja, M. F.

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

Pomplun, J.

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

Ren, Y.

Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
[Crossref]

Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
[Crossref]

Saitoh, K.

Saville, M. A.

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
[Crossref]

Schmidt, F.

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

Shanker, B.

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

Song, J.

K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
[Crossref]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Volakis, J. L.

J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
[Crossref]

Wu, B.

B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” J. Vac. Sci. Technol. 25, 1743–1761 (2007).
[Crossref]

Xiao, T.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
[Crossref]

Xiong, J. L.

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
[Crossref]

Zhu, J.

Zolla, F.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

Zschiedrich, L.

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (1)

Y. Ren, W. F. Huang, J. Niu, and Q. H. Liu, “A hybrid solver based on domain decomposition method for the composite scattering in layered medium,” IEEE Antennas Wireless Propag. Lett. 16, 420–423 (2017).
[Crossref]

IEEE Microw. Compon. Lett. (2)

H. Lin, M. F. Pantoja, L. D. Angulo, J. Alvarez, R. G. Martin, and S. G. Garcia, “Fdtd modeling of graphene devices using complex conjugate dispersion material model,” IEEE Microw. Compon. Lett. 22, 612–614 (2012).
[Crossref]

I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the lod-fdtd method,” IEEE Microw. Compon. Lett. 23, 306–308 (2013).
[Crossref]

IEEE Trans. Antennas Propag. (6)

J. H. Lee, J. Chen, and Q. H. Liu, “A 3D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag. 57, 2666–2674 (2009).
[Crossref]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propag. 60, 4766–4776 (2012).
[Crossref]

Y. Ren, Q. H. Liu, and Y. P. Chen, “A hybrid fem/mom method for 3-d electromagnetic scattering in layered medium,” IEEE Trans. Antennas Propag. 64, 3487–3495 (2016).
[Crossref]

B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag. 48, 1064–1074 (2000).
[Crossref]

J. D. Collins, J. M. Jin, and J. L. Volakis, “Eliminating interior resonances in finite element-boundary integral methods for scattering,” IEEE Trans. Antennas Propag. 40, 1583–1585 (1992).
[Crossref]

K. Chen, J. Song, and T. Kamgaing, “Accurate and efficient computation of layered medium doubly periodic Green’s function in matrix-friendly formulation,” IEEE Trans. Antennas Propag. 63, 809–813 (2015).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Techn. 54, 437–444 (2006).
[Crossref]

IIEEE Trans. Antennas Propag. Lett. (1)

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IIEEE Trans. Antennas Propag. Lett. 5, 490–494 (2006).
[Crossref]

J. Lightwave Technol. (1)

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

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

J. Vac. Sci. Technol. (1)

B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” J. Vac. Sci. Technol. 25, 1743–1761 (2007).
[Crossref]

Journal of Computational and Applied Mathematics (1)

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” Journal of Computational and Applied Mathematics 168, 321–329 (2004). Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002).
[Crossref]

Opt. Express (1)

physica status solidi (b) (1)

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” physica status solidi (b) 244, 3419–3434 (2007).
[Crossref]

Proc. IEEE (1)

M. Luo, Y. Lin, and Q. H. Liu, “Spectral methods and domain decomposition for nanophotonic applications,” Proc. IEEE 101, 473–483 (2013).
[Crossref]

Science (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Other (1)

J. Jin, The Finite Element Method in Electromagnetics (Wiley-IEEE Press, 2014), 3rd ed.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Computational framework of the SEBI-PLMGF method.
Fig. 2
Fig. 2 Validation of the PLMGF: (a) electric field comparison; (b) magnetic field comparison.
Fig. 3
Fig. 3 Geometrical structure of a typical multi-layer nano structure.
Fig. 4
Fig. 4 Field pattern and validation of the electric field results. (a) Electric field distribution observed at z = 672.5 nm. (b) Validation of the electric fields observed along median lines of the x– and y– dimensions of the principal cell.
Fig. 5
Fig. 5 Typical structures impact the convergence performance. (a) The computation domain only contains scatterers with no field singularity corner. (b) The computation domain includes geometries for moderate field singularity. The layered medium structures below the computation domain are identical to that in Fig. 3.
Fig. 6
Fig. 6 p-Refinement study for three typical scenarios.
Fig. 7
Fig. 7 EUV lithography model.
Fig. 8
Fig. 8 Electric field distribution observed one wavelength above the EUV lithography pattern.

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

V ( × Φ i ) T μ r 1 ( × E ) + k 0 2 Φ i T r E d V + j k 0 S Φ i T ( n ^ × H ˜ ) d S = V Φ i T S e d V
n ^ × [ E ( r ) E inc ] = n ^ × [ E ( r , J ) + 𝒦 E ( r , M ) ]
n ^ × [ H ( r ) H inc ] = n ^ × [ H ( r , M ) + 𝒦 H ( r , J ) ]
E ( t , b ) ( r , X ) = j ω S ( t , b ) d r G ¯ P , e ( t , b ) ( r , r ) μ ( r ) X ( r )
𝒦 E ( t , b ) ( r , X ) = 1 ( r ) S ( t , b ) d r × G ¯ P , m ( t , b ) ( r , r ) ( r ) X ( r )
H ( t , b ) ( r , X ) = j ω S ( t , b ) d r G ¯ P , m ( t , b ) ( r , r ) ( r ) X ( r )
𝒦 H ( t , b ) ( r , X ) = μ 1 ( r ) S ( t , b ) d r × G ¯ P , e ( t , b ) ( r , r ) μ ( r ) X ( r )
V ( × Φ i ) T μ r 1 ( × E ) d V + k 0 2 V Φ i T r E d V j k 0 S ( n ^ × Φ i ) T [ η 0 H ( r , M ) + 𝒦 ( r , J ˜ ) ] d S = j k 0 S ( n ^ × Φ i ) T H ˜ inc d S + V Φ i T S e d V
j k 0 S ( n ^ × Φ i ) T E inc d S = j k 0 S ( n ^ × Φ i ) T [ E ( r ) + 1 η 0 E ( r , J ˜ ) + 𝒦 E ( r , M ) ] d S .
G ¯ P ( r , r ) = p = + q = + G ¯ p q ( r , r p q ) e j k x i p L x e j k y i q L y = 1 L x L y p = + q = + G ¯ ˜ p q ( k x p , k y q ; z , z ) e j k x p ( x x ) e j k y q ( y y )
G ¯ P , e ( r , r ) = p q ( I ¯ + k m 2 ) g ( r , r p q ) e j k x i p L x e j k y i q L y
g ( r , r ) = e j k m | r r | 4 π | r r | .
< n ^ × Φ i ( r ) , E , n ^ × Φ j ( r ) > = < n ^ × Φ i ( r ) , G ¯ P , e ( r , r ) , n ^ × Φ j ( r ) > = < n ^ × Φ i , p q g ( r , r p q ) e j k x i p L x e j k y i q L y , n ^ × Φ j > 1 k m 2 < ( n ^ × Φ i ) , p q g ( r , r p q ) e j k x i p L x e j k y i q L y , ( n ^ × Φ j ) > .
p q g ( r , r p q ) e j k x i p L x e j k y i q L y = 1 L x L y p q j 2 k m z e j k m z | z z | e j k x p ( x x ) e j k y q ( y y ) ,
× G ¯ P , e = p q × G ¯ e ( r , r p q ) e j k x i p L x e j k y i q L y = p q [ 0 z y z 0 x y x 0 ] g ( r , r p q ) e j k x i p L x e j k y i q L y . = p q [ 0 ( z z ) y y z z 0 ( x x ) ( y y ) x x 0 ] 1 R p q ( j k m 1 R p q ) e j k m R p q 4 π R p q e j k x i p L x e j k y i q L y .
× G ¯ P , e = 1 L x L y p q [ 0 z y z 0 x y x 0 ] j 2 k m z e j k m z | z z | e j k x p ( x x ) e j k y q ( y y ) , ( z z ) .
< n ^ × Φ i ( r ) , 𝒦 H , n ^ × Φ j ( r ) > = < n ^ × Φ i ( r ) , 𝒦 ˜ H , n ^ × Φ j ( r ) > + 1 2 < n ^ × Φ i ( r ) , Φ j ( r ) > = 1 2 < n ^ × Φ i ( r ) , Φ j ( r ) >
< n ^ × Φ i ( r ) , G ¯ P , e , n ^ × Φ j ( r ) > = < n ^ × Φ i ( r ) , G ¯ P , e TE ( r , r ) , n ^ × Φ j ( r ) > + 1 k n m 2 < n ^ × Φ i ( r ) , G ¯ P , e TM ( r , r ) , n ^ × Φ j ( r ) >
k n m 2 = ω 2 n μ m ,
G ¯ P , e TE ( r , r ) = p q ( × z ^ ) ( × z ^ ) g TE ( r , r p q ) e j k x i p L x e j k y i q L y ,
G ¯ P , e TM ( r , r ) = p q ( × × z ^ ) ( × × z ^ ) g TM ( r , r p q ) e j k x i p L x e j k y i q L y .
G ¯ P , e TE ( r , r ) = [ y y y x 0 x y x x 0 0 0 0 ] . p q g TE ( r , r p q ) e j k x i p L x e j k y i q L y ,
G ¯ P , e TM ( r , r ) = [ x x z z x y z z x z k ρ 2 y x z z y y z z y z k ρ 2 x z k ρ 2 y z k ρ 2 k ρ 4 ] p q g TM ( r , r p q ) e j k x i p L x e j k y i q L y .
g α ( r , r p q ) = 1 { g ˜ α ( k x p , k y q , z , z ) } = 1 { j 2 F α ( k ρ ; z , z ) k m z k ρ 2 } .
G ¯ P , e TE ( r , r ) = [ y y y x 0 x y x x 0 0 0 0 ] 1 L x L y p q g ˜ TE ( k x p , k y q ; z , z ) e j k x p ( x x ) e j k y q ( y y ) ,
G ¯ P , e TM ( r , r ) = [ x x z z x y z z x z k ρ 2 y x z z y y z z y z k ρ 2 x z k ρ 2 y z k ρ 2 k ρ 4 ] 1 L x L y p q g ˜ TM ( k x p , k y q ; z , z ) e j k x p ( x x ) e j k y q ( y y ) .
< n ^ × Φ i ( r ) , × G ¯ P , e , n ^ × Φ j ( r ) > = < n ^ × Φ i ( r ) , × G ¯ P , e TE ( r , r ) , n ^ × Φ j ( r ) > + 1 k n m 2 < n ^ × Φ i ( r ) , × G ¯ P , e TM ( r , r ) , n ^ × Φ j ( r ) >
× G ¯ P , e TE ( r , r ) = p q ( × × z ^ ) ( × z ^ ) g TE ( r , r p q ) e j k x i p L x e j k y i q L y
× G ¯ P , e TM ( r , r ) = p q k n 2 ( × z ^ ) ( × × z ^ ) g TM ( r , r p q ) e j k x i p L x e j k y i q L y .
× G ¯ P , e TE ( r , r ) = [ x z y x z x 0 y z y y z x 0 k ρ 2 y k ρ 2 x 0 ] 1 L x L y p q g ˜ TE ( k x p , k y q ; z , z ) e j k x p ( x x ) e j k y q ( y y )
× G ¯ P , e TM ( r , r ) = [ y x z y y z y k ρ 2 x x z x y z x k ρ 2 0 0 0 ] 1 L x L y p q k n 2 g ˜ TM ( k x p , k y q ; z , z ) e j k x p ( x x ) e j k y q ( y y ) .
[ K II K IS 0 K SI K SS + L H K H 0 K E L E ] [ E I E S H ˜ S ] = [ S e H ˜ inc E inc ]
E ( r + a ) = E ( r ) e j k a
H ( r + a ) = H ( r ) e j k a
A u = b .
u = [ u in , u x 1 , u x 2 , u y 1 , u y 2 , u x y 1 , u x y 2 , u x y 3 , u x y 4 ] T
u x 2 = u x 1 e j k x L x
u y 2 = u y 1 e j k y L y
u x y 2 = u x y 1 e j k x L x
u x y 3 = u x y 1 e j k y L y
u x y 4 = u x y 1 e j k x L x e j k y L y .
P = [ I 0 0 0 0 I 0 0 0 I e j k x L x 0 0 0 0 I 0 0 0 I e j k y L y 0 0 0 0 I 0 0 0 I e j k x L x 0 0 0 I e j k y L y 0 0 0 I e j k x L x e j k y L y ] ,
P AP u ˜ = P b
u ˜ = [ u in , u x 1 , u y 1 , u x y 1 ] T

Metrics