Abstract

Spectral properties of one-dimensional tungsten gratings with various depths and widths of grooves were investigated by means of finite-difference time-domain simulation with a multi-Lorentz model. Shallow gratings showed a strong absorption peak due to surface plasmon polaritons when the oscillation of the electric field was parallel to the grating vector. On the other hand, deep gratings with wide grooves showed a different absorption attributed to the microcavity effect when the oscillation of the electric field was perpendicular to the grating vector. With narrowed grooves, another absorption with d-dependence occurred, which was probably due to vertically oscillating surface plasmons to the grooves.

© 2005 Optical Society of America

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    [CrossRef]
  27. Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
    [CrossRef]
  28. Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
  30. A. Taflove, S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
  31. J.-J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
    [CrossRef]
  32. D. W. Lynch, W. R. Hunter, “Tunsgsten,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1998), pp. 357–367.
  33. M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
    [CrossRef]

2004

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

H. Sai, H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85, 3399–3401 (2004).
[CrossRef]

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

2003

F. Kusunoki, J. Takahara, T. Kobayasi, “Qualitative change of resonant peaks in thermal emission from periodic array of microcavities,” Electron. Lett. 39, 23–24 (2003).
[CrossRef]

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

H. Sai, Y. Kanamori, H. Yugami, “High-temperature resistive surface grating for spectral control of thermal radiation,” Appl. Phys. Lett. 82, 1685–1687 (2003).
[CrossRef]

S. Y. Lin, J. G. Fleming, I. El-Kady, “Three-dimensional photonic-crystal emission through thermal excitation,” Opt. Lett. 28, 1909–1911 (2003).
[CrossRef] [PubMed]

2002

Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
[CrossRef]

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

2001

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, K. Hane, “Spectral control of thermal emission by periodic microstructures in the near-infrared region,” J. Opt. Soc. Am. A 18, 1471–1476 (2001).
[CrossRef]

2000

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

S. Hava, M. Auslender, “Design and analysis of low-reflection grating microstructures for a solar energy absorber,” Sol. Energy Mater. Sol. Cells 61, 143–151 (2000).
[CrossRef]

1999

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

1998

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

J.-J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
[CrossRef]

1988

N. F. Hartman, T. K. Gaylord, “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
[CrossRef] [PubMed]

J. N. Zemel, “Black body radiation and transport processes from microstructures,” Comments Condens. Matter Phys. 14, 1–20 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10795–10802 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” Proc. SPIE 883, 8–11 (1988).
[CrossRef]

1986

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986).
[CrossRef]

1982

S. J. Wilson, M. C. Hutely, “The optical properties of moth eye antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

1973

P. B. Clapham, M. C. Hutley, “Reduction of lens reflection by moth eye principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Akiyama, Y.

Auslender, M.

S. Hava, M. Auslender, “Design and analysis of low-reflection grating microstructures for a solar energy absorber,” Sol. Energy Mater. Sol. Cells 61, 143–151 (2000).
[CrossRef]

Blasi, B.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

Bläsi, B.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Boerner, V.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

Carminati, R.

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

Chen, Y.

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

Chou, S. Y.

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Clapham, P. B.

P. B. Clapham, M. C. Hutley, “Reduction of lens reflection by moth eye principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Dearholt, D. W.

D. W. Dearholt, W. R. McSpadden, Electromagnetic Wave Propagation (McGraw-Hill, 1973).

Diso, D.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Döll, W.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Dreibholtz, J.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

El-Kady, I.

Esashi, E.

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

Fleming, J. G.

Fukumoto, S.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

Gao, H.

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Gaylord, T. K.

Ge, H.

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Gebhart, B.

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10795–10802 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986).
[CrossRef]

Ghbara, T.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

Ghmari, F.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

Glaubitt, W.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Gombert, A.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Greffet, J.-J.

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

J.-J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Hane, K.

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
[CrossRef]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, K. Hane, “Spectral control of thermal emission by periodic microstructures in the near-infrared region,” J. Opt. Soc. Am. A 18, 1471–1476 (2001).
[CrossRef]

Hartman, N. F.

Hava, S.

S. Hava, M. Auslender, “Design and analysis of low-reflection grating microstructures for a solar energy absorber,” Sol. Energy Mater. Sol. Cells 61, 143–151 (2000).
[CrossRef]

Heinzel, A.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Herminghaus, S.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Hesketh, P. J.

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10795–10802 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986).
[CrossRef]

Hiroshima, T.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

Hunter, W. R.

D. W. Lynch, W. R. Hunter, “Tunsgsten,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1998), pp. 357–367.

Hutely, M. C.

S. J. Wilson, M. C. Hutely, “The optical properties of moth eye antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Hutley, M. C.

P. B. Clapham, M. C. Hutley, “Reduction of lens reflection by moth eye principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Ishimori, M.

Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
[CrossRef]

Joulain, K.

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

Joulian, K.

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

Kanamori, Y.

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

H. Sai, Y. Kanamori, H. Yugami, “High-temperature resistive surface grating for spectral control of thermal radiation,” Appl. Phys. Lett. 82, 1685–1687 (2003).
[CrossRef]

Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
[CrossRef]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, K. Hane, “Spectral control of thermal emission by periodic microstructures in the near-infrared region,” J. Opt. Soc. Am. A 18, 1471–1476 (2001).
[CrossRef]

Kashiwa, T.

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

Knoll, W.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Kobayashi, K.

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

Kobayashi, T.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

Kobayasi, T.

F. Kusunoki, J. Takahara, T. Kobayasi, “Qualitative change of resonant peaks in thermal emission from periodic array of microcavities,” Electron. Lett. 39, 23–24 (2003).
[CrossRef]

Kohama, T.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

Kreiter, M.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Kusunoki, F.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

F. Kusunoki, J. Takahara, T. Kobayasi, “Qualitative change of resonant peaks in thermal emission from periodic array of microcavities,” Electron. Lett. 39, 23–24 (2003).
[CrossRef]

Laroche, M.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

Licciulli, A.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Lin, S. Y.

Lomascolo, M.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Luther, J.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

Lynch, D. W.

D. W. Lynch, W. R. Hunter, “Tunsgsten,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1998), pp. 357–367.

Maffezzoli, A.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Mainguy, S.

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

Marquier, F.

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

Maruyama, S.

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

Mazzer, M.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

McSpadden, W. R.

D. W. Dearholt, W. R. McSpadden, Electromagnetic Wave Propagation (McGraw-Hill, 1973).

Mitter-Neher, S.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Moharam, M. G.

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” Proc. SPIE 883, 8–11 (1988).
[CrossRef]

Mulet, J.-P.

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

Mullet, J. P.

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

Nieto-Vesperinas, M.

Oster, J.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Preist, T. W.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Rose, K.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Sai, H.

H. Sai, H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85, 3399–3401 (2004).
[CrossRef]

H. Sai, Y. Kanamori, H. Yugami, “High-temperature resistive surface grating for spectral control of thermal radiation,” Appl. Phys. Lett. 82, 1685–1687 (2003).
[CrossRef]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, K. Hane, “Spectral control of thermal emission by periodic microstructures in the near-infrared region,” J. Opt. Soc. Am. A 18, 1471–1476 (2001).
[CrossRef]

Sambles, J. R.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Sambles, R.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

Sobnack, M. B.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Sporn, D.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Takahara, J.

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

F. Kusunoki, J. Takahara, T. Kobayasi, “Qualitative change of resonant peaks in thermal emission from periodic array of microcavities,” Electron. Lett. 39, 23–24 (2003).
[CrossRef]

Tan, W. C.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Torsello, G.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Tundo, S.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Wanstall, N. P.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Waymouth, J. F.

J. F. Waymouth, “Optical light source device,” U.S. patent 5,079,473 (January 7, 1992).

Wilson, S. J.

S. J. Wilson, M. C. Hutely, “The optical properties of moth eye antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Wittwer, V.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Wu, W.

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Yu, Z.

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Yugami, H.

H. Sai, H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85, 3399–3401 (2004).
[CrossRef]

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

H. Sai, Y. Kanamori, H. Yugami, “High-temperature resistive surface grating for spectral control of thermal radiation,” Appl. Phys. Lett. 82, 1685–1687 (2003).
[CrossRef]

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, K. Hane, “Spectral control of thermal emission by periodic microstructures in the near-infrared region,” J. Opt. Soc. Am. A 18, 1471–1476 (2001).
[CrossRef]

Zemel, J. N.

J. N. Zemel, “Black body radiation and transport processes from microstructures,” Comments Condens. Matter Phys. 14, 1–20 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10795–10802 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. Maruyama, T. Kashiwa, H. Yugami, E. Esashi, “Thermal radiation from two-dimensionally confined modes in microcavities,” Appl. Phys. Lett. 79, 1393–1395 (2001).
[CrossRef]

H. Sai, Y. Kanamori, H. Yugami, “High-temperature resistive surface grating for spectral control of thermal radiation,” Appl. Phys. Lett. 82, 1685–1687 (2003).
[CrossRef]

H. Sai, H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85, 3399–3401 (2004).
[CrossRef]

Comments Condens. Matter Phys.

J. N. Zemel, “Black body radiation and transport processes from microstructures,” Comments Condens. Matter Phys. 14, 1–20 (1988).

Electron. Lett.

F. Kusunoki, J. Takahara, T. Kobayasi, “Qualitative change of resonant peaks in thermal emission from periodic array of microcavities,” Electron. Lett. 39, 23–24 (2003).
[CrossRef]

IEEE Photonics Technol. Lett.

Y. Kanamori, M. Ishimori, K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photonics Technol. Lett. 14, 1064–1066 (2002).
[CrossRef]

J. Appl. Phys.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys. 96, 2656–2664 (2004).
[CrossRef]

J. Mod. Opt.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

Z. Yu, H. Gao, W. Wu, H. Ge, S. Y. Chou, “Fabrication of large area subwavelength antireflection structures using trilayer resist nanoimprint lithography and liftoff,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Jpn. J. Appl. Phys., Part 1

Y. Kanamori, K. Kobayashi, H. Yugami, K. Hane, “Subwavelength antireflection gratings for GaSb in visible and near-infrared wavelengths,” Jpn. J. Appl. Phys., Part 1 42, 4020–4023 (2003).
[CrossRef]

F. Kusunoki, T. Kohama, T. Hiroshima, S. Fukumoto, J. Takahara, T. Kobayashi, “Narrow-band thermal radiation with low directivity by resonant modes inside tungsten microcavity,” Jpn. J. Appl. Phys., Part 1 43, 5253–5258 (2004).
[CrossRef]

Nature (London)

J.-J. Greffet, R. Carminati, K. Joulian, J.-P. Mulet, S. Mainguy, Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986).
[CrossRef]

P. B. Clapham, M. C. Hutley, “Reduction of lens reflection by moth eye principle,” Nature (London) 244, 281–282 (1973).
[CrossRef]

Opt. Acta

S. J. Wilson, M. C. Hutely, “The optical properties of moth eye antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Opt. Commun.

M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mitter-Neher, W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168, 117–122 (1999).
[CrossRef]

F. Marquier, K. Joulain, J. P. Mullet, R. Carminati, J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. 237, 379–388 (2004).
[CrossRef]

Opt. Lett.

Phys. Rev. B

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10795–10802 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B 69, 155412 (2004).
[CrossRef]

Phys. Rev. Lett.

M. B. Sobnack, W. C. Tan, N. P. Wanstall, T. W. Preist, J. R. Sambles, “Stationary surface plasmons on a zero-order metal grating,” Phys. Rev. Lett. 80, 5667–5670 (1998).
[CrossRef]

Proc. SPIE

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” Proc. SPIE 883, 8–11 (1988).
[CrossRef]

Semicond. Sci. Technol.

A. Licciulli, D. Diso, G. Torsello, S. Tundo, A. Maffezzoli, M. Lomascolo, M. Mazzer, “The challenge of high-performance selective emitters for thermophotovoltaic applications,” Semicond. Sci. Technol. 18, S174–S183 (2003).
[CrossRef]

Sol. Energy

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholtz, B. Bläsi, A. Heinzel, D. Sporn, W. Döll, V. Wittwer, “Antireflective transparent covers for solar devices,” Sol. Energy 68, 357–360 (2000).
[CrossRef]

Sol. Energy Mater. Sol. Cells

S. Hava, M. Auslender, “Design and analysis of low-reflection grating microstructures for a solar energy absorber,” Sol. Energy Mater. Sol. Cells 61, 143–151 (2000).
[CrossRef]

Other

A. Taflove, S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

D. W. Lynch, W. R. Hunter, “Tunsgsten,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1998), pp. 357–367.

J. F. Waymouth, “Optical light source device,” U.S. patent 5,079,473 (January 7, 1992).

D. W. Dearholt, W. R. McSpadden, Electromagnetic Wave Propagation (McGraw-Hill, 1973).

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Figures (14)

Fig. 1
Fig. 1

Schematic of the geometry of a 1D grating.

Fig. 2
Fig. 2

Schematic diagram showing the FDTD calculation model in this study.

Fig. 3
Fig. 3

Spectral relative permittivity and reflectivity of W approximated by multi-Lorentz dispersion.

Fig. 4
Fig. 4

Comparison of the FDTD and RCWA approaches on spectral absorptivity of flat W and rectangular W gratings with Λ x = Λ y = Λ = 1.0 μ m , a x = a y = a = 0.8 μ m , and d = 0.8 μ m for θ = ϕ = 0 ° and ψ = 45 ° . FDTD calculations were performed under the conditions of cell size of 10 nm × 10 nm × 10 nm and time step 1.73 fs with a Gaussian pulse. In the RCWA calculation, diffraction orders up to ( ± 10 , ± 10 ) were considered.

Fig. 5
Fig. 5

Spectral absorptivity of a 1D shallow rectangular W grating with Λ = 1.0 μ m , a = 0.8 μ m , and d = 0.2 μ m ( a Λ = 0.8 , d a = 0.25 ) at normal incidence for three different ψ calculated by the FDTD method.

Fig. 6
Fig. 6

Spectral absorptivity of a 1D shallow rectangular W grating with a Λ = 0.8 and d a = 0.25 at ψ = 0 ° for three different Λ calculated by the FDTD method. The other calculation parameters are the same as those for Fig. 4. The downward arrows indicate the first-order SPP wavelength.

Fig. 7
Fig. 7

Spectral absorptivity of a 1D shallow rectangular W grating with Λ = 1.0 μ m and a = 0.8 μ m ( a Λ = 0.8 ) at ψ = 0 ° for four different d calculated by the FDTD method. The other calculation parameters are the same as those for Fig. 4.

Fig. 8
Fig. 8

Spectral absorptivity of a 1D deep rectangular W grating with Λ = 1.0 μ m , a = 0.8 μ m , and d = 0.8 μ m ( a Λ = 0.8 , d a = 1.0 ) at normal incidence for three different ψ calculated by the FDTD method.

Fig. 9
Fig. 9

Time series of E-field distribution scattered by the 1D W grating with Λ = 1.0 μ m and a = d = 0.8 μ m at ψ = 0 ° (upper) and 90° (lower). The cell size and the time step Δ t are set to 20 nm × 20 nm × 20 nm and 3.47 fs , respectively.

Fig. 10
Fig. 10

Spectral absorptivity of a 1D deep rectangular W grating with a = d = 0.8 μ m ( d a = 1.0 ) at normal incidence and ψ = 90 ° for three different Λ calculated by the FDTD method. The other calculation parameters are the same as those for Fig. 4.

Fig. 11
Fig. 11

Spectral absorptivity of a 1D deep rectangular W grating with Λ = 1.0 μ m , a = 0.2 μ m , and d = 0.4 μ m ( a Λ = 0.2 , d a = 2.0 ) at normal incidence for three different ψ calculated by the FDTD method.

Fig. 12
Fig. 12

Spectral absorptivity of a 1D deep rectangular W grating at ψ = 0 ° for several different d. The other calculation parameters are the same as those for Fig. 10.

Fig. 13
Fig. 13

Transient electric field at the four points on the W grating with Λ = 1.0 μ m , a = 0.2 μ m , and d = 0.4 μ m ( a Λ = 0.2 , d a = 2.0 ) excited with a Gaussian pulsed wave impinging at ψ = 0 ° . The cell size and the time step Δ t are set to 20 nm × 20 nm × 20 nm and 3.47 fs , respectively.

Fig. 14
Fig. 14

Fourier transformation results of Fig. 13.

Tables (1)

Tables Icon

Table 1 List of the Lorentz Parameters Used in This Study

Equations (23)

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

k s p x = k 0 ( ε m 1 + ε m ) 1 2 .
k s p = k 0 x ± m K x ,
K = K = 2 π Λ ,
λ spp = Λ m [ ( ε m 1 + ε m ) 1 2 sin θ cos ϕ ] ,
k s p = ( k 0 x ± m K x ) + ( k 0 y ± n K y ) ,
λ l m = 2 π k = 2 ( l L x ) 2 + ( m L y ) 2 ,
λ l m n = 2 ( l L x ) 2 + ( m L y ) 2 + ( n 2 L z ) 2 ,
ε r ( ω ) = ε + p = 1 P Δ ε p ω p 2 ω p 2 + 2 i δ p ω ω 2 ,
× E = μ 0 H t ,
× H = D t = ε 0 ε E t + ε 0 p = 1 P P p t ,
u ( t = n Δ t , r = ( x , y , z ) ) = u r n .
P ̆ p ( ω ) = Δ ε p ω p 2 ω p 2 + 2 i δ p ω ω 2 E ̆ ( ω ) .
( ω p 2 + 2 i δ p ω ω 2 ) P ̆ p ( ω ) = Δ ε p ω p 2 E ̆ ( ω ) .
ω p 2 P p + 2 δ p P p t + 2 P p t 2 = Δ ε p ω p 2 E .
2 P p t 2 r n + 1 2 2 P p t 2 r n .
P p r n + 1 = CP 1 p P p r n CP 2 p P p r n 1 + CP 3 p ( E r n + 1 + E r n ) ,
CP 1 p = ω p 2 Δ t 2 4 δ p Δ t 4 ω p 2 Δ t 2 + 4 δ p Δ t + 2 ,
CP 2 p = 2 ω p 2 Δ t 2 + 4 δ p Δ t + 2 ,
CP 3 p = Δ ε p ω p 2 Δ t 2 ω p 2 Δ t 2 + 4 δ p Δ t + 2 .
× H r n + 1 2 = ε 0 ε E t r n + 1 2 + ε 0 p = 1 P P p t r n + 1 2 = ε 0 ε Δ t ( E r n + 1 E r n ) + ε 0 Δ t p = 1 P ( P p r n + 1 P p r n ) .
E r n + 1 = CEa E r n + CEb Δ t ε 0 × H r n + 1 2 CEb p = 1 P [ ( CP 1 p 1 ) P p r n + CP 2 p P p r n 1 ] ,
CEa = ( ε p = 1 P CP 3 p ) ( ε + p = 1 P CP 3 p ) ,
CEb = 1 ( ε + p = 1 P CP 3 p ) .

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